Spatial Dynamics and Pattern Formation in Biological Populations [1 ed.] 0367555506, 9780367555504

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Spatial Dynamics and Pattern Formation in Biological Populations

Spatial Dynamics and Pattern Formation in Biological Populations

Ranjit Kumar Upadhyay

Professor, Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, 826004, Jharkhand, India

Satteluri R. K. Iyengar

Professor of Mathematics (Retired), Indian Institute of Technology, New Delhi, India

MATLAB® is a trademark of Te MathWorks, Inc. and is used with permission. Te MathWorks does not warrant the accuracy of the text or exercises in this book. Tis book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by Te MathWorks of a particular pedagogical approach or particular use of the MATLAB® software

First edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2021 Ranjit Kumar Upadhyay and Satteluri R. K. Iyengar CRC Press is an imprint of Taylor & Francis Group, LLC Te right of Ranjit Kumar Upadhyay and Satteluri R. K. Iyengar to be identifed as authors of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. Reasonable eforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. Te authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microflming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identifcation and explanation without intent to infringe. ISBN: 978-0-367-55550-4 (hbk) ISBN: 978-0-367-55551-1 (pbk) ISBN: 978-1-003-09395-4 (ebk) Typeset in Palatino by codeMantra

Contents Foreword .........................................................................................................................................ix Preface..............................................................................................................................................xi About Authors ............................................................................................................................. xiii 1 Introduction to Diffusive Processes ...................................................................................1 1.1 Introduction ...................................................................................................................1 1.2 Diffusion, Convection, Advection and Dispersion Processes ................................2 1.3 Some Basic Laws of Diffusion .....................................................................................7 1.3.1 Fick’s Laws of Diffusion..................................................................................7 1.3.2 Darcy’s Law ......................................................................................................9 1.4 Diffusion Equation........................................................................................................9 1.4.1 Linear Diffusion Equation in One Dimension .......................................... 10 1.4.1.1 Time-Dependent/Concentration-Dependent Diffusion Coeffcient Problems ...................................................................... 13 1.4.2 Linear Diffusion Equation in Two and Three Dimensions ..................... 14 1.4.2.1 Two-Dimensional Diffusion on a Disk ....................................... 15 1.4.2.2 Linear Diffusion Equation in Three Dimensions...................... 15 1.4.2.3 Reaction–Diffusion Equations in Diffusion Processes ............. 16 1.4.3 Diffusion in a Heterogeneous Environment ............................................. 17 1.5 Stochastic Reaction–Diffusion (SRD) Systems........................................................ 18 1.6 Hopf Bifurcation Analysis .........................................................................................22 1.7 Multiple-Scale Analysis/Weakly Nonlinear Analysis .......................................... 25 1.7.1 Linear Stability Analysis of the Amplitude Equation .............................. 31 1.8 Overview of the Book.................................................................................................34 References ............................................................................................................................... 36 2 Reaction–Diffusion Modeling........................................................................................... 41 2.1 Introduction ................................................................................................................. 41 2.2 Reaction–Diffusion Equations ..................................................................................43 2.2.1 Derivation of Reaction-Diffusion Equation ...............................................43 2.3 Hyperbolic Reaction–Diffusion Equations..............................................................44 2.4 Single-Species Reaction–Diffusion Models............................................................. 46 2.4.1 Model 1: Linear Model of Kierstead and Slobodkin................................. 47 2.4.1.1 KISS Model in Two Dimensions .................................................. 49 2.4.2 Model 2: Nonlinear Fisher Equation........................................................... 50 2.4.2.1 Spatial Steady-State Solution ........................................................ 51 2.4.2.2 Some Analytical Solutions ............................................................ 53 2.4.3 Model 3: Nagumo Equation ......................................................................... 55 2.4.3.1 Numerical Solutions ...................................................................... 58 2.5 Two-Species Reaction–Diffusion Models ................................................................ 60 2.5.1 Turing Instabilities of Two-Species Reaction–Diffusion Systems .......... 60 2.5.1.1 Predator–Prey Reaction–Diffusion Systems...............................64

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2.6

Applications in Biochemistry: Belousov–Zhabotinsky Reaction–Diffusion Systems ......................................................................................................................... 72 2.6.1 Model 1: Oregonator Model ......................................................................... 73 2.6.2 Model 2: Brusselator Model.......................................................................... 75 2.6.3 Model 3: Schnakenberg Model .................................................................... 79 2.6.4 Model 4: Lengyel–Epstein Model ................................................................80 2.6.5 Model 5: Sel’kov Model................................................................................. 82 2.6.6 Model 6: Gray–Scott Model ..........................................................................85 2.7 Multispecies Reaction–Diffusion Models................................................................ 88 2.7.1 Model 1: Hastings–Powell Model................................................................ 88 2.7.2 Model 2: Modifed Upadhyay–Rai Model .................................................. 91 2.7.3 Model 3: Modifed Leslie–Gower-Type Three-Species Model ................ 96 References ............................................................................................................................. 100 3 Modeling Virus Dynamics in Time and Space ............................................................ 111 3.1 Introduction ............................................................................................................... 111 3.1.1 Next-Generation Operator Method........................................................... 117 3.2 Susceptible-Infected (SI) Models ............................................................................ 118 3.2.1 Models with Nonlinear Incidence Rate .................................................... 119 3.2.2 Models with Self and Cross-Diffusion ..................................................... 123 3.2.3 Infuenza Epidemic Models........................................................................ 126 3.2.3.1 A Simple Spatial SI Epidemic Model......................................... 127 3.2.3.2 Turing Instability.......................................................................... 133 3.2.3.3 Two-time Scale Infuenza Models.............................................. 138 3.3 Susceptible-Infected-Susceptible (SIS) Models ..................................................... 142 3.4 Susceptible-Infected-Removed (SIR) Models ........................................................ 153 3.4.1 SIR Models with Vital Dynamics .............................................................. 158 3.4.2 SIR Models with Treatment Rate............................................................... 161 3.5 Susceptible-Infected-Removed-Susceptible (SIRS) Models ................................ 170 3.6 Susceptible-Exposed-Infected-Recovered (SEIR) Models ................................... 179 3.6.1 Infuenza Model Revisited.......................................................................... 194 Exercise 3............................................................................................................................... 201 References ............................................................................................................................. 202 4 Modeling the Epidemic Spread and Outbreak of Ebola Virus................................. 215 4.1 Introduction ............................................................................................................... 215 4.1.1 Source and Symptoms................................................................................. 215 4.1.2 Transmission and Control of Epidemics .................................................. 216 4.2 Formulation of Ebola Epidemic Models ................................................................ 217 4.3 Model 1: Ebola Epidemic SEIR Model.................................................................... 226 4.3.1 Spatial SEIR Ebola Epidemic Model ......................................................... 232 4.4 Model 2: Ebola Epidemic SEIRHD Model ............................................................. 238 4.4.1 Sensitivity Indices of 0 .............................................................................. 241 4.5 Model 3: Ebola Epidemic SEIORD Model and Its Extension.............................. 243 4.6 Model 4: Ebola Epidemic SEIRD Model with Time Delay.................................. 248 4.6.1 Existence of Endemic Equilibrium and Stability Analysis ................... 252 4.7 Model 5: General Ebola Transmission Model for Population in a Community................................................................................................................254

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Exercise 4............................................................................................................................... 261 References ............................................................................................................................. 261 5 Modeling the Transmission Dynamics of Zika Virus................................................ 267 5.1 Introduction ............................................................................................................... 267 5.1.1 Symptoms and Clinical Features............................................................... 269 5.2 Formulation of Zika Epidemic Model.................................................................... 270 5.3 Model 1: Zika Virus SIR Transmission Model...................................................... 273 5.3.1 Optimal Control Analysis .......................................................................... 280 5.4 Model 2: Zika Virus SEIR Transmission Model ................................................... 283 5.4.1 Bifurcation Analysis .................................................................................... 286 5.4.2 Optimal Control Analysis .......................................................................... 288 5.5 Model 3: Zika Virus SEIR Horizontal and Vertical Transmission Model......... 290 5.6 Model 4: Zika Virus with Vertical Transmission ................................................. 299 5.7 Model 5: Zika Virus SIR Transmission Model with Human and Vector Mobility.......................................................................................................... 303 5.7.1 Existence of Travelling Wave Solutions ....................................................306 5.8 Model 6: Zika Virus Transmission with Criss-Cross Interactions Model ........308 5.9 Model 7: Zika Virus SEIR Transmission Model.................................................... 314 5.9.1 Model with Diffusion.................................................................................. 320 Exercise 5............................................................................................................................... 325 References ............................................................................................................................. 325 6 Brain Dynamics: Neural Systems in Space and Time................................................. 331 6.1 Introduction ............................................................................................................... 331 6.2 Properties of Neurons ..............................................................................................334 6.2.1 Electrophysiological Properties of Neurons ............................................ 335 6.2.2 Ionic Conductance ....................................................................................... 335 6.2.3 Generation of Action Potential, Its Activity, and Signal Propagation ... 337 6.2.3.1 Synapse and Its Functional Mechanism ................................... 338 6.2.4 Ionic Currents, Neuronal Activity and Neuronal Responses ............... 339 6.3 Hodgkin–Huxley (HH) Model................................................................................344 6.3.1 Simulation Results ....................................................................................... 349 6.4 FitzHugh-Nagumo (FHN) Model...........................................................................354 6.4.1 Linear Stability Analysis and Hopf Bifurcation...................................... 356 6.4.2 Amplitude Equation .................................................................................... 357 6.4.2.1 Linear Stability Analysis of the Amplitude Equation ............ 361 6.4.3 Secondary Bifurcation of the Turing Pattern........................................... 365 6.4.3.1 Dynamics of 1D Diffusion in FHN Model ............................... 369 6.5 Morris–Lecar (M–L) Model...................................................................................... 370 6.5.1 Stability and Bifurcation Analysis ............................................................ 372 6.5.1.1 Bifurcation Analysis .................................................................... 373 6.5.2 Spatial Morris–Lecar Model ...................................................................... 374 6.5.3 Multiple-Scale Analysis (Amplitude Equations)..................................... 377 6.5.3.1 Amplitude Stability...................................................................... 383 6.5.4 Spiking and Bursting in Single M-L Neuron Model ..............................384 6.6 Hindmarsh–Rose (H-R) Model ............................................................................... 388 6.6.1 Formulation of a Modifed H-R System.................................................... 392

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6.6.2 6.6.3 6.6.4

Bifurcation Analysis .................................................................................... 394 Modifed Reaction–Diffusion H-R System............................................... 395 Construction of Traveling Front Solution................................................. 398 6.6.4.1 Numerical Results ........................................................................ 403 References .............................................................................................................................405 Solutions to Odd-Numbered Problems................................................................................. 413 Index ............................................................................................................................................. 431

Foreword Mathematical modelling is playing an increasingly important role in ecology, epidemiology, biology, and medicine. As I write this foreword, mathematical models are being used to inform governments across the world on the impact of different strategies for dealing with the COVID-19 pandemic. This book is therefore very timely as it provides an introduction to the deterministic (and some stochastic) modelling of spatiotemporal phenomena in ecology, epidemiology, and neural systems, surveying classical models in the feld with up-to-date applications. It begins with a detailed description of spatial dynamics, showing how movement phenomena from many different areas can be modelled within the same general mathematical framework. The methods of analysis of the resultant partial differential equations are presented. This is built upon by adding kinetics, leading to systems of reaction– diffusion equations. The classical models in this area are motivated and their analyses presented. The next three chapters discuss modelling virus dynamics in space and time, with applications to the recent Ebola and Zika virus pandemics. The fnal chapter presents an introduction to the biology of the brain, together with the classical models for neural dynamics. This book is written in a very accessible way, carefully and clearly explaining all steps. It describes in detail the scientifc problem, development of the appropriate mathematical models, subsequent analysis (including techniques such as linear stability analysis, weakly nonlinear analysis, control theory, and numerical simulation) and resultant insights gained into the scientifc problem. It is an ideal introduction to modelling spatiotemporal dynamics for anyone wishing to enter the feld of mathematical biology. Philip K. Maini, FRS, FMedSci, FNA Oxford, UK June 2020

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Preface During the past few years, the world has experienced the emergence of major devastating epidemic outbreaks and the spread of viruses like Ebola and Zika. As this manuscript was getting ready to be sent to the publishers, Coronavirus has emerged and has been threatening the whole world. Understanding the modeling of virus dynamics of infectious diseases holds the key for designing control strategies from the public health perspective. Therefore, we felt that collecting all the available literature at one place may be useful for students and research workers in these areas. Emphasis is made on mathematical modeling using reaction–diffusion systems in biological populations with applications to ecology, epidemiology, and neural systems. The temporal and spatial dynamics can be essentially different, and while its results may be misleading in some situations, temporal analysis provides an important direction for understanding the spatial/spatiotemporal dynamics. The analysis of patterns enables us to study the dynamics of macroscopic and microscopic behavior of underlying systems. The travelling wave-type patterns can be especially observed in dispersive systems. Chapter 1 introduces the basic concepts of spatial dynamics and spatial/diffusive processes of biological populations, Hopf bifurcation analysis, multiple-scale analysis, and neural systems that play a key role in understanding the functions of the brain and its dynamics. In Chapter 2, reaction–diffusion modeling is presented to describe the diffusive dispersal of the population, developmental processes, etc. Three single-species models, a two-species model, six models in applications in biochemistry, and three models of multispecies reaction–diffusion are analyzed. In Chapter 3, transmission dynamics of infectious diseases are studied. Five different types of models are analyzed: (i) susceptible–infected (SI), (infuenza epidemic); (ii) susceptible–infected–susceptible (SIS); (iii)  susceptible–infected–removed (SIR); (iv) susceptible–infected–removed–susceptible (SIRS); and (v)  susceptible–exposed–infected–recovered (SEIR). Chapter 4 deals with fve models of epidemic spread and outbreak dynamics of the Ebola virus. Chapter 5 deals with seven models of transmission and outbreak dynamics of the Zika virus. The frst four models discuss the temporal dynamics and the remaining three models discuss the temporal and spatial dynamics. In Chapter 6, four biophysical neuron models are examined for the evolution of the functional mechanism of the brain. Using conductance-based mathematical models, patterns of spiking activity and qualitative behavior of temporal activity such as periodic fring, bursting, chattering, mixed-mode oscillations, and chaotic fring are studied. Most of the models discussed in the book are solved using the software MATLAB or MATHEMATICA. We are extremely grateful to Prof. Dr. Philip K. Maini, FRS, FMedSci, FNA, Wolfson Centre for Mathematical Biology, Mathematical Institute, Oxford University, Oxford, for writing the foreword for the book. We thank all the Professors who reviewed the book and provided constructive suggestions, which gave us the proper direction and impetus for writing the book. We express our gratitude to all the following authors and their co-authors for giving approval and providing an opportunity to include their works in the book: Dr. Abid, W.; Dr. Agusto, F. B.; Dr. Al-Darabsah, I.; Dr. Allen, L. J. S.; Dr. Ambrosio, B.; Dr. Aziz Alaoui, Dr. Bonyah, E.; Dr. Brauer, F.; Dr. Cai, Y.; Dr. Camacho, A.; Dr. Capone, F.; xi

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Dr. Carrero, G.; Dr. Charles, W. M.; Dr. Chinviriyasit, S.; Dr. Cui, R.; Dr. Deng, K.; Dr. Djiomba Njankou; Dr. Do, T. S.; Dr. Dubey, B.; Dr. Fitzgibbon, W.; Dr. Funk, S.; Dr. Han, W.; Dr. Harko, T.; Dr. Hille, B.; Dr. Imran, M.; Dr. Izhikevich, E. M.; Dr. Jin, Z.; Dr. Kao, Y.; Dr.  Kim, K. Ik.; Dr. King, J. R.; Dr. Kucharski, A. J.; Dr. Kumari, N.; Dr. Legrand, J.; Dr.  Li, T.; Dr. Li, X.; Dr. Liu, P.; Dr. Liu, Q. X.; Dr. Ma, M.; Dr. Madzvamuse, A.; Dr.  Mao, X.; Dr. Mariana Ruiz Villarreal.; Dr. Mark Kot.; Dr. Mazin, W.; Dr. Mondal, A.; Dr. Morris, C.; Dr. Pankaj Seth.; Dr. Parshad, R.D.; Dr. Purves, D.; Dr. Röst, G.; Dr. Roy, P.; Dr.  Ruan, S.; Dr.  Samsuzzoha, M.; Dr. Schwartz, Ira. B.; Dr. Schwiening, C. J.; Dr. Shen, J.; Dr. Shi, J.; Dr.  Sun, G. Q.; Dr. Tang, B.; Dr. Tang, Q. L.; Dr. Tsaneva-Atanasova, K.; Dr.  Wang,  W.; Dr.  Wang, X.; Dr. Wu, C.; Dr. Yang, F.; Dr. Yang, J.; Dr. Yi, F.; Dr. Zhang, J.; Dr. Zheng, Q.; Dr. Zhou, J. Our grateful thanks to the following publishers who have given copyright permission to reproduce fgures from their reputed International journals and books: American Institute of Physics, American Physical Society, Cambridge University Press, Elsevier, Hindawi Publishing Corporation, IOP Publishing, John Wiley and Sons, MIT Press, Oxford Publishing House, Society for Industrial and Applied Mathematics, Springer Nature, Springer Open Journal, the Physiological Society, and World Scientifc Publishing Company. We profusely thank Taylor and Francis for accepting to publish the book and bringing it in a nice form. We shall be extremely happy to receive suggestions and comments to improve the quality of the book. MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

About Authors

Dr. Ranjit Kumar Upadhyay is Professor in the Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, India. He is an acknowledged researcher and has contributed to various areas of applied mathematics, mathematical modeling, and nonlinear dynamics. His research areas are interdisciplinary in nature such as dynamical systems theory; chaotic dynamics of real-world situations; population dynamics for marine and terrestrial ecosystems; spatial dynamics in epidemic (infuenza, Ebola, Zika and Corona viruses), e-epidemic and neural models; and reaction–diffusion modeling. He has published 160 research papers in different International journals of repute, and a number of these publications are with international collaborators. He has supervised nine students for their Ph.D. Currently, six students are working under his guidance for their Ph.D. degree. He has handled a number of sponsored R&D projects. He is the co-author of the book titled Introduction to Mathematical Modeling and Chaotic Dynamics published by Taylor & Francis group (CRC Press, USA). He is on the editorial board for many reputed journals. He is an associate editor of the Food Webs journal by Elsevier and Advances in Difference Equations and Differential Equations and Dynamical Systems journals by Springer. He is a guest editor of a special issue entitled Nonlinear Models in Biosignaling, Biosensor and Neural Systems – Modeling, Simulations, and Applications, being brought out by the Differential Equations and Dynamical Systems, a Springer journal. He was a visiting research fellow under the Indo-Hungarian educational exchange program in Eötvös University, Budapest, Hungary. He was a visiting scientist and delivered invited lectures in many renowned institutions such as University of Cambridge; University of Leicester, UK; and University of Le Havre Normandie, France. He is a member of the International Society of Computational Ecology, Hong Kong. Dr. Satteluri R. K. Iyengar was Professor and former Head of the Department of Mathematics, Indian Institute of Technology (IIT), New Delhi, India. He worked as a professor for more than 22 years. His areas of research work are numerical analysis and mathematical modeling. He is the co-author of the books titled Introduction to Mathematical Modeling and Chaotic Dynamics published by Taylor  & Francis group (CRC Press, USA) and Advanced Engineering Mathematics, published by Narosa Publishing House. He is a coauthor of a number of books on Numerical Analysis like “Numerical Methods for Scientifc and Engineering Computation” etc. He has many research publications in international journals of repute. He was a post-doctoral fellow at Oxford University Computing Laboratory, Oxford, United Kingdom, and the University of Saskatchewan, Canada. He was awarded the Distinguished Service Award by the Indian Institute of Technology, New Delhi, India, during its golden jubilee year in 2011. He was also awarded the Distinguished Indian Award in 2007 by the Pentagram Research Center (P) Limited, Hyderabad, India, for his contributions. xiii

1 Introduction to Diffusive Processes

1.1 Introduction Diffusive/spatial processes like evolution of new species, dynamics of invading species, maintenance of biodiversity, and movements of animals and plants play central roles in ecology. A key factor in how ecological communities are shaped is the spatial components of ecological interactions. Modern tools give us information about rates of dispersal among subpopulations. The study of the data of ecological species reveals that species spread as traveling waves and also provide information about the interplay between ecological and evolutionary dynamics in the spread of the population. The spatial dynamics of pathogens and immune system cells within individual organisms is important for understanding infectious disease outbreaks [89]. A common feature of earlier ecological population models is that the interactions were based on the mass action law, an approach that has its conceptual foundation in modeling chemical reactions [79]. When the reactants are well mixed and have to collide in order to react, the mass action law states that “the collision rate (hence the reaction rate) is proportional to the product of the concentrations of the reacting molecules”. The law assumes that the population of interacting species is large enough to guarantee conditions of well mixing. Ecological situations in which well-mixed condition does not hold can lead to incorrect predictions, and a spatial model with local interactions is more appropriate in such cases. The following are two examples of spatial movements: Andrewartha and Birch [2] observed that some insect populations become frequently extinct but persisted globally due to recolonization from local populations. The second example is Huffaker’s [43] laboratory experiment with two mites, one of which feeds on oranges (Eotetranychus sexmaculatus) and the other feeds on the predatory mite (Typhlodromus occidentalis) that attacks E. sexmaculatus. Huffaker set up an array of oranges and rubber balls with different spatial complexity that controlled dispersal. He demonstrated that a complex spatially heterogeneous array promoted species coexistence and later confrmed that spatial subdivision is important for the persistence of populations. In modeling the population dynamics, space can be included either implicitly or explicitly. The Levins model [55] describes the dynamics of a population in a spatially subdivided habitat. Population may go to extinction in patches and may subsequently be recolonized from other occupied patches. Space is implicit in the sense that recolonization is equally likely from all occupied patches regardless of their locations. The Father of the studies of ecological diffusion, John Gorden Skellam [96], employed models that include space explicitly to describe the invasion of species. These models are similar to Fisher’s model [51] for the spread of novel allele. Fisher and the famous troika of researchers – Kolmogorov, Petrovskii, and Piscunov – studied the diffusion models in population dynamics as early 1

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Spatial Dynamics and Pattern Formation in Biological Populations

as 1936. These models included space, but they did not allow for spatial correlations, since local populations are effectively infnite [79]. A key aspect of the spatial processes is the degree to which the subunits of a system or network are connected to one another, and this degree governs the population persistence, patterns of biodiversity, and ecosystem function [7]. Complex ecosystems exhibit patterns that are bound to each other and are observed over different spatial and time scales [36]. Ecosystem theory is full of macroecological patterns for which causal relationships are still debated [34,68]. Species–area and diversity–stability relationships of ecological patterns are some of the important areas of study.

1.2 Diffusion, Convection, Advection and Dispersion Processes Diffusion: The word diffusion means “to spread out”. If a substance is “spreading out”, it is moving from an area of high concentration to an area of low concentration. Diffusion is a phenomenon by which a particle group as a whole spreads according to the irregular motion of each particle. Diffusion results in mixing or mass transport, without requiring bulk fow. When the microscopic irregular motion of each particle gives rise to a regularity of motion of the total particle group, then the phenomenon of diffusion arises. A consideration of the long-term statistical trend of the irregular motion of a single particle also leads to the concept of diffusion (concept of randomness). In terms of randomness, diffusion can be defned to be a basically irreversible phenomenon by which matter, particle groups, population, etc., spread out within a given space according to individual random motion. It was shown that even the motion of humans can be very well approximated with random walks if considered on a relevant spatiotemporal scale [5]. It is important to know how the mean square displacement x 2 depends on time. A study describes this dependence by the power law x 2 ~ t˜ . We have (i) Brownian motion for ˜ = 1/2, (ii) subdiffusion for ˜ < 1/2, and (iii) superdiffusion for ˜ > 1/2. Individual random walks are of standard Brownian type which eventually result in diffusion. Diffusion can be introduced in the following two ways: (i) an approach starting with Fick’s laws of diffusion and their mathematical consequences and (ii) an approach that considers the random walk of the diffusing particles. In the frst case, diffusion is the movement of a substance from a region of high concentration to a region of low concentration without bulk motion. According to Fick’s law, “the diffusion fux is proportional to the negative gradient of concentrations”. Generalizations of Fick’s laws were developed in various frameworks of thermodynamics and nonequilibrium thermodynamics. Diffusion is a mixture of molecules due to Brownian motion, and it depends on the magnitude of the concentration gradient. It was observed that diffusion and dispersion always take place together and that dispersion has something to do with the concentration gradient. The molecule transport by dispersion is of magnitudes bigger compared to diffusion. Since diffusion is a phenomenon of random motion, it causes the system to decay toward uniform conditions. For example, diffusing molecules will move randomly between areas of high and low concentrations but since there are more molecules in the high concentration region, more molecules will leave the high concentration region than the low concentration one. Therefore, there will be a net movement of molecules from high concentration to low concentration. Initially, a concentration gradient (a smooth decrease in concentration from high to low) will form between the two regions.

Introduction to Diffusive Processes

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As  time progresses, the gradient will grow increasingly shallow until the concentrations are equalized. It is possible that nonreversible processes such as diffusion can be generated by microscopic deterministic chaos. Diffusion increases entropy, decreases Gibbs free energy, and therefore is thermodynamically favorable. Diffusion operates under the second law of thermodynamics. The second law of thermodynamics gives Gibbs equation which determines the direction of an irreversible process, relating entropy to a change in internal energy, volume, and partial masses. Diffusion is important in systems experiencing an applied force. In a conducting material, the net motion of electrons in an electrical feld quickly reaches a terminal velocity (resulting in a steady current described by Ohm’s law) because of the thermal (diffusive) motions of atoms. Einstein’s relation, D = kT × mobility (discovered by Nernst 1884), relates the diffusion coeffcient to the mobility of particles [22]. Pais in his book titled, Subtle is the Lord: The Science and the Life of Albert Einstein [82], gave an interesting review of Einstein’s work on diffusion. Einstein employed a random walk model for his analysis. Einstein’s work on diffusion can be applied to dairy technology (colloidal properties of micelle suspension in milk) and construction industry (link with granular matter) [17,82]. In cell biology, diffusion is the main form of transport within cells and across cell membranes. Convection: Convection is the collective movement of groups or aggregates of molecules within fuids, through advection or through diffusion or as a combination of both of them. Convection of mass cannot take place in solids, since neither bulk current fows nor signifcant diffusion can take place in solids. Diffusion of heat can take place in solids, but that is called heat conduction. Convection can be demonstrated by placing a heat source at the side of a glass full of a liquid and observing the changes in temperature in the glass caused by the warmer fuid moving into cooler areas. Convective heat transfer is one of the major types of heat transfer, and convection is also a major mode of mass transfer in fuids. Convective heat transfer and mass transfer both take place by diffusion due to the random Brownian motion of individual particles in the fuid. In advection, matter or heat is transported by the larger-scale motion of currents in the fuid. In the context of heat transfer and mass transfer, the term convection is used to refer to the sum of advective and diffusive transfers. Advection: Advection is a transport mechanism of a fuid due to the fuid’s bulk motion. A simple example of advection is the transport of pollutants or silt in a river by bulk water fow downstream. Another example of advection is energy or enthalpy. The fuid’s motion is described mathematically as a vector feld, and the transported material is described by a scalar feld showing its distribution over space. Since advection requires currents in a fuid, it cannot happen in rigid solids. It does not include transport of substances by molecular diffusion. In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean such as heat, humidity (moisture), or salinity. Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle. Dispersion/dispersal: Biological dispersal refers to those processes by which a species maintains or expands the distribution of its population. Dispersal is necessary in populations because members of the species compete for the same limited resources within an ecosystem. Dispersal relieves pressure on resources in an ecosystem. Dispersal mechanisms depend on the competition for these resources. Dispersal may involve replacement of a parent generation by a new generation with only minor changes in the geographic area occupied. Dispersal enables the species population to occupy much of the available habitat and maximize its resources in its favor. Some organisms (plants and especially sedentary animals) have evolved adaptations for dispersal by taking advantage of

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Spatial Dynamics and Pattern Formation in Biological Populations

various forms of kinetic energy occurring naturally in the environment like water fow and wind. Often,  dispersal may be purely random, density-dependent, or random plus density-dependent. Dispersal over long distances can be approximated mathematically by deterministic partial differential equations or integro-differential equations. Dispersal over very short distances often results in large spatial correlations. In case of interacting particle systems, local dispersal can result in spatial correlations. The global dispersal results in a Poisson distribution which allows one to study spatially homogeneous models that are at the onset of exhibiting spatial correlation, such as when offspring are dispersed over intermediate distances [79]. Difference between diffusion and dispersion: Dispersive mass transfer in fuid dynamics is the spreading of mass from highly concentrated areas to less concentrated areas. Dispersive mass fux is analogous to diffusion. It can be described using Fick’s frst law, J = −E ( dc/dx ), where c is the mass concentration of the species being dispersed, E is the dispersion coeffcient, and x is the position in the direction of the concentration gradient. The following physical model of dispersion is explained well in the book Analysis of Transport Phenomena by William M. Deen [18]. Consider a convective fow superposed over the diffusion. If the fow velocity is uniform everywhere (plug fow), then molecules at different places in the fow will move with the same convective velocity, and it is only the diffusion rate that will differentiate between them. Now, consider a fow in which gradients exist, for example, a fully developed pipe fow. The fuid at the center of the pipe is moving much faster than the fuid close to the walls. If there is diffusion in the radial direction, then molecules can hop over from one streamline to the next and thereby will be transported over different distances due to the difference in velocities. This is caused indirectly by diffusion in the radial direction. The amount of dispersion reduces with increasing diffusion coeffcient. This is because the molecules will just be hopping from one streamline to another constantly and will not get suffcient time to be transported far from each other. Depending on the situation, a diffusive process is classifed as self-diffusion, crossdiffusion, mutual-diffusion, etc. A detailed review of the interpretations of these forms of diffusion was given by Vanag and Epstein [111]. Wolfenbarger [113] introduced the following defnition: When transportation takes place due to energy from within the organism, the dispersion is termed as active, and when it takes place due to energy from outside the body, the dispersion is termed as passive. The smaller the organism, the more it is subject to the effect of environmental turbulence. Thus, the diffusion of small animals should be considered as partly passive and partly active. For instance, bacteria and pollen in the air and phytoplankton in the water diffuse almost passively, while many insects in fight undergo varying proportions of passive diffusion and active diffusion, according to the degree of movement of the environmental fuid. Multicomponent diffusion: In gas mixtures or concentrated solutions where more than one chemical species is present, diffusion coeffcient is not a constant or composition-independent. In these cases, diffusion depends on intermolecular dependencies. The diffusion equation is to take into account relations between the mass fux of one chemical species to the concentration gradients of all chemical species present. The mathematical equations are formulated from the Maxwell–Stefan description of diffusion [16,69]. This formulation is often applied to describe gas mixtures, such as syngas in a reactor or the mix of oxygen, nitrogen, and water in a fuel cell cathode. In Maxwell–Stefan diffusion, the choices of dependent variables are not the species concentrations, but rather the species mole or mass fractions. The diffusive mass fux of each species is, in turn, expressed in terms of the gradients of the mole or mass fractions, using multicomponent diffusion coeffcients ese are symmetric, so that an n-component system requires n(n − 1)/2 independent coeffcients to

Introduction to Diffusive Processes

5

parametrize the rate of diffusion of its components. Simplifcations to the Maxwell–Stefan equations are derived in order to employ the equivalent Fick’s law of diffusivity. Self- and cross-diffusion: During the last two decades, there has been considerable interest in the study of the stability behavior of a system of interacting populations by taking into account the effect of self- as well as cross-diffusion [63]. Cross-diffusion can change the stability of a constant positive steady state of a self-diffusive system. That is, cross-diffusion can lead to the occurrence and disappearance of the Turing instability of the system. This implies that cross-diffusion is not always helpful to create the Turing instability for the predator–prey system. Biologically, self-diffusion means the movement of individuals from a higher to a lower concentration region. Cross-diffusion implies the population fuxes of one species due to the presence of the other species. The value of the cross-diffusion coeffcient may be positive, negative, or zero. Positive cross-diffusion coeffcient denotes the movement of the species in the direction of lower concentration of another species, and negative cross-diffusion coeffcient denotes that one species tends to diffuse in the direction of higher concentration of another species. For example, Enrique Peacock-Lopez [83] considered the case when negative cross-diffusion (which represents the lack of predator mobility) is compensated by their ability to camoufage and attract their prey. The dynamics of interacting populations with self- and cross-diffusion was investigated by several researchers [13,19,20,44,49,104,105,111]. By exploring the idea of cross-diffusion, Chattopadhyay and Tapaswi [13] observed two-dimensional (2D) spatial patterns in the same system without environmental heterogeneity. It was observed that constant coeffcient cross-diffusions are necessary to maintain spatial pattern in 2D Lotka–Volterra competitive systems [13]. In developmental biology, experimental fndings demonstrated that cross-diffusion can be quite signifcant in generating spatial structures [111]. In molecular biology, cross-diffusion processes appear in multicomponent systems containing at least two solute components [70,112]. Multicomponent systems containing nanoparticles, surfactants, polymers, and other macromolecules in solution play an important role in industrial applications and biological functions [70]. The effects of crossdiffusion on reaction–diffusion-type models for pattern formations have been studied by many authors [8,9,32,33,44,52,57,90,91,107,114,118,119]. Introducing linear cross-diffusion for a two-component reaction–diffusion system with activator-depleted reaction kinetics, many authors derived cross-diffusion-driven instability conditions [35,65,88,92] and showed that they are a generalization of the classical diffusion-driven instability conditions in the absence of cross-diffusion. Cross-diffusion induced diffusion-driven instability on stationary domains and volumes, occurs when a uniform steady state which is linearly stable in the absence of cross-diffusion and regular diffusion, becomes unstable when cross-diffusion and regular diffusion are present [65]. Why diffusion in ecology? No life is possible if spatial and temporal processes do not interact. Existence and infuence of the spatial processes (active and passive movement of species, passive movement of plant species, etc.) on the temporal evolution of species densities necessitate mathematical biologists to model the real movements of animals as random and study the same. Currently, these movements are represented mathematically by Fick’s law of diffusivity (dispersal, migration). Diffusion in ecology is a phenomenon by which the biological population spreads according to the irregular motion of each individual of the population. The conceptual applicability of diffusion terms to describe redistribution of species in space due to random motion of the individuals for any value of population density was shown by Okubo [80]. Since the dispersal rates in the prey–predator model are assumed to be the same for both the prey and predator, the patterns cannot appear due to the Turing instability [93] but due to some other mechanisms [85]. Depending on the

6

Spatial Dynamics and Pattern Formation in Biological Populations

details of the species distribution, there can be two different patterns – regular or chaotic corresponding to two different regimes of the system dynamics [86]. Reaction–diffusion models in ecology were studied by many authors [1,21,54,77,78,95]. Skellam [96,97,98] suggested that the process of biological diffusion cannot be purely random. Animals often concentrate together to form groups. In such cases, an effect that opposes diffusion occurs due to behavioral patterns and interaction between individuals. One of the important features that distinguish the movement of animals from the random motion of the inorganic material is this delicate balance between “spreading” and “concentrating”. Dispersion in the physical environment of species cannot be ignored while considering organism dispersal. Horizontal density variation of plankton in the form of plankton patchiness has been of interest to biological oceanographers [11,101]. While a tendency for the formation of patches exists, the effect of oceanic diffusion is to supply a mechanism to intra- and interspecies relationships which causes instability of the ecosystem [56,93]. When movement is diffusive, the traveling wave moves from the stronger population to the weaker. However, by incorporating behaviorally induced directed movement toward the stronger population, the weaker one can slow the traveling wave down, even reversing its direction. Hence, movement responses can switch the predictions of traditional mechanistic models [87]. Why diffusion in neural systems? The electrical activities in the neurons are governed via movement of ionic currents through neuron membranes. Izhikevich [45] has discussed the mechanism of how ions diffuse down the concentration gradient through the membrane and produce outward current. The positive and negative charges accumulate on the opposite sides of the membrane surface creating an electric potential gradient across the membrane called as the transmembrane potential or membrane voltage. This potential slows the diffusion of K+ ions which are attracted to the negatively charged interior and repelled from the positively charged exterior of the membrane. Most of these membrane currents involve one of the four ionic species: sodium (Na+), potassium (K+), calcium (Ca2+), and chloride (Cl−). The concentrations of these ions are different on the inside and the outside of a cell, which creates electrochemical gradients – the major driving forces of neural activity. The extracellular medium has high concentration of Na+, Cl−, and Ca2+ ions, and the intracellular medium has high concentration of K+ ions. According to Fick’s frst law, the ions will move from a region of high concentration to a region of low concentration gradient. In cell membrane, Na+ and Ca2+ ions move from outside the cell membrane to inside the cell membrane, whereas the K+ ions move from inside to outside the cell membrane. The movement of these ions is described mathematically by the diffusion processes. For example, in models of nerve conduction, only the membrane potential spatially interacts; the recovery and other variables interact only through the membrane potential [24,109]. Reaction–diffusion equations describe a variety of phenomena in neuroscience. The Hodgkin–Huxley equations describe the propagation of nerve pulses and the potassium and calcium ion concentrations in cortical structures. Reaction–diffusion systems have been shown to be able to propagate structured wave patterns without attenuation to form stable patterns [39,67,73] and to select the maximum stimulus using only local interactions [60,71,72,94,108]. A reaction–diffusion system requires the presence of a pair of antagonistic neurotransmitters. Note that a neurotransmitter may interact with more than one other neurotransmitter. The same system may also model diffusion neurotransmission of a single neurotransmitter without the antagonistic inhibitor. This can be treated as a special case of a reaction–diffusion (RD) system [60]. The solutions of RD equations display a wide range of behaviors including the formation of traveling waves in nerve conduction, for example, traveling wave solutions of the full Hodgkin–Huxley equations

7

Introduction to Diffusive Processes

[6]. Some of the processes that can be modeled by the RD equations are the following: (i) propagation of an action potential along an electric cable (HH equations), (ii) concentration changes of ionic species under the infuence of local reactions, and diffusion of ions down their concentration gradients without expense of energy in an aqueous medium (Kolmogorov–Petrovsky–Piskunov equations), and (iii) electrodiffusion (diffusion of charged particles under the infuence of an electric feld), which is a nonlinear transport process whose essence is diffusion of ions combined with their migration in an electric feld (Nernst–Planck equations). However, electrodiffusion is the principal means of migration only in the presence of large number of ions. Electrodiffusion has not yet been reconciled with action potential propagation, and alternative ways (based on Maxwell’s equations instead of RD systems) are appearing in theoretical models of information handling in the brain [6].

1.3 Some Basic Laws of Diffusion 1.3.1 Fick’s Laws of Diffusion Fick’s frst law: The law states that “the net movement of diffusing substance per unit area of section (the fux) is proportional to the concentration gradient (how steeply the concentration changes in space) and is toward lower concentration”. Consider a collection of particles performing a random walk in one dimension with length scale ˜x and time scale ˜t. Let N(x, t) be the number of particles at position x at time t. At a given time step, half of the particles would move to left and the other half would move to right. Since half of the particles at a point x move to right and half of the particles at point x + °x move to left, the net movement to the right is − [ N ( x + ˙x , t) − N ( x , t)] / 2 . The fux J is this net movement of particles across some area element of area a, normal to the random walk during a time interval ˜t. Hence, we may write the rate of transfer per unit area of section F as ( ˝x)2 ˙ N ( x + ˝x , t) − N ( x , t) ˘ ˙ N ( x + ˝x , t) − N ( x , t) ˘ F = −ˇ = − .  2 a˝t 2 ˝t ˇˆ a( ˝x)2 ˆ  The concentration per unit volume C and the diffusion coeffcient D are respectively, defned as C( x , t) =

( °x)2 N ( x , t) . , and D = 2°t a°x

Hence, F=−

D [C(x + ˙x, t) − C(x, t)]. ˙x

In the limit, we obtain the equation F = −DCx, which is Fick’s frst law of diffusion. Here, x is the space coordinate measured normal to the section. The modern mathematical form of the law is written as N i = − Di˛ci , where N i is the molar fux, Di is the diffusion coeffcient, and ci is the concentration of the species i. Crank [15] in his book The Mathematics of Diffusion gave an excellent account of methods for the solution of diffusion equations.

8

Spatial Dynamics and Pattern Formation in Biological Populations

Fick’s second law: Consider the case when there is diffusion at the front and rear surfaces of an incremental planar volume. Fick’s second law states that “the rate of accumulation (or depletion) of concentration within the volume is proportional to the local curvature of the concentration gradient”. The second law of diffusion is a linear equation with the dependent variable being the concentration of the chemical species under consideration. Diffusion of each chemical species occurs independently. The mass transport systems described by Fick’s second law are easy to simulate numerically. Accumulation is positive when the curvature is positive (that is, when the concentration gradient is more negative on the front end of the planar volume and less negative on the rear end so that more fux is driven into the volume at the front end than is driven out of the volume at the rear end) [102]. The accumulation (˜ C/˜ t) is proportional to the diffusivity D and the second derivative (or curvature) of the concentration ˜ 2 C/˜ x 2 . Fick’s second law is given by

(

)

Ct = DCxx .

(1.1)

The dimensions of accumulation, diffusion, and curvature of the concentration are cm −3 s −1 , cm 2 s −1 , and cm −5, respectively. When C is at a steady state, the solution for the concentration is a linear change of concentration along x. In two and three dimensions, we have the following equations: ˝ ˜ 2C ˜ 2C ˇ ˜C = Dˆ 2 + ,  ˜t ˜ y 2 ˘ ˙ ˜x

˝ ˜ 2C ˜ 2C ˜ 2C ˇ ˜C = Dˆ 2 + + . ˜t ˜ y 2 ˜ z 2 ˘ ˙ ˜x

(1.2)

If diffusion is dependent on concentration, then we have the equations:

˜C ˜ ˛ ˜C ˆ ˜ ˛ ˜C ˆ = D ,  ˙D ˘+ ˜ t ˜ x ˝ ˜ x ˇ ˜ y ˙˝ ˜ y ˘ˇ

˜C ˜ ˛ ˜C ˆ ˜ ˛ ˜C ˆ ˜ ˛ ˜C ˆ D + = ˙D ˘+ ˙D ˘. ˜ t ˜ x ˝ ˜ x ˇ ˜ y ˙˝ ˜ y ˘ˇ ˜ z ˝ ˜ z ˇ

(1.3)

In the cylindrical and spherical polar coordinates, we have the following equations in three dimensions: Diffusion in cylinder:

Diffusion in sphere:

˜C 1  ˜ ˝ ˜C ˇ ˜ ˝ D ˜C ˇ ˜ ˝ ˜C ˇ  =  ˆ rD + ˆ + ˆ rD  ˜t r ˜ r ˙ ˜ r ˘ ˜° ˙ r ˜° ˘ ˜ z ˙ ˜z ˘

˜C 1  ˜ ˙ 2 ˜C ˘ 1 ˜ ˙ ˜C˘ D ˜ 2C  Dsin° = 2  ˇr D + +   ˇ  ˜t r ˜ r ˆ ˜ r  sin° ˜° ˆ ˜°  sin 2° ˜˛ 2  or as

(1.4)

(1.5)

˜C = div(D grad C). ˜t

By taking suitable correspondence between the parameters, variables, and equations, it can be observed that the phenomena of heat fow and diffusion are essentially the same. Dimensional analysis of Fick’s second law reveals that in diffusive processes, there is a fundamental relation between the elapsed time and the square of the length over which diffusion takes place. Understanding this relation helps in accurate numerical simulation of diffusion.

9

Introduction to Diffusive Processes

1.3.2 Darcy’s Law In 1856, Henry Darcy published a monograph which contained a law which was later called Darcy’s law. He was determining the “Law of Flow of Water through Sand”. An excellent review of the law was given by Hubbert [42]. We present here the law as discussed by Hubbert. The law states that “the total discharge Q is proportional to the crosssectional area A, head difference ( h1 − h2 ) and inversely proportional to the length ˜l”. That is, Q = KA ( h1 − h2 )/˙l. In the limit, Q/A is called the specifc discharge and the equation becomes q = −Kdh/dl (the specifc discharge is directly proportional to the derivative of the head in the direction of the fow). Here, h is the height of water in the nanometer. For fow in the three-dimensional space of water-flled sand, the law is written as q = −K grad h. This is a kinetic equation expressing the coupling between the fow feld of the vector q and the scalar feld h. In order to make it a dynamical equation, energies and forces are to be taken into consideration. The height of water is a function of the fuid density p, gravity g, and fuid pressure ˜ at a given point of elevation z. The relationship is given by h = z + p/( ˜ g), or hg = gz + ( p/˜ ).

(1.6)

Each term of the equation represents a potential energy per unit mass of water at a given point. Here, hg = ° is the total energy per unit mass and also the potential ˜ of the water, gz is the gravitational energy, and p/˜ is the pressure energy. Since hg = °, we get q = −K grad h = −

K grad ˛. g

(1.7)

The force per unit mass acting upon the water at a given point is given by E = − g grad h = −grad ˛.

(1.8)

Let ( K/g ) = ˜ . Darcy’s law becomes [42] q = −˜ g grad h = −˜ grad ˝ = ˜ E.

(1.9)

1.4 Diffusion Equation Diffusion equation can be derived from Fick’s law. The constant of proportionality is the diffusion coeffcient, which depends on the diffusing species and the material through which diffusion occurs. Fick’s law may not hold for a diffusive system (for example, diffusion may depend on concentration in addition to concentration gradient). An analogous statement of Fick’s law for heat instead of concentration is Fourier’s law. Diffusion can also be described using discrete quantities. A common model of discrete diffusion is the random walk. A random walk model is connected to the diffusion equation by considering an infnite number of random walkers starting from a nonuniform confguration, where the evolution of the concentration is described by the diffusion equation.

10

Spatial Dynamics and Pattern Formation in Biological Populations

1.4.1 Linear Diffusion Equation in One Dimension The conservation law states that “the rate of change of number of individuals in a given interval of space is equal to the growth rate of population in (x, x + dx) plus the rate of entry in x minus rate of departure at (x + dx)”. A simple example is the following: The heat distribution in a rod of length L or the spread of a contaminant in a stationary medium is governed by the linear constant coeffcient parabolic diffusion equation: ut = ˜ 2 uxx , 0 < x < L,

(1.10)

where ˜ = k/(c° ) is the thermal diffusivity of the material. This is the simplest diffusion equation. The domain may be semi-infnite, 0 < x < ° ,t > 0, or infnite −° < x < °, t > 0. Suitable initial and boundary conditions are provided to solve the problem completely. The diffusion equation (1.10) is also written as ut = Duxx , where D is the diffusion coeffcient. If density ρ or specifc heat c depends on the location x, then α is a function of x. In this case, we may write the equation in the self-adjoint form as ut = ˙˝ K ( x ) ux ˆˇ x . However, it is not always possible to write the equation in self-adjoint form. If the material properties depend on t also, then the diffusion equation may be written as ut = [ a( x , t)ux ]x . Consider the case of diffusion in a cylinder of infnite length with unit cross section. Let 2

( 4° 2 t)

− x2

u denote the concentration. Observe that u = Pt −1/2 e is a solution of the above equation (1.10). If M is the amount of substance deposited in the plane x = 0 at time t = 0, then the amount of substance diffusing remains constant [15]. The total amount of substance M diffusing in the cylinder is given by (Crank [15]): ˇ

M=

˜

ˇ

udx =

−ˇ

˜

Pt −1/2 e

(

− x 2 / 4˛ 2 t

)dx.

−ˇ

(

)

Under the change of variables, ˜ 2 = x 2 4° 2t , we get M = 2P°

˜

ˇ

−ˇ

2

e − ˛ d ˛ = 2P° (˙)1/2 ,

which shows that M is independent of time. The concentration is given by u = Pt −1/2 e

(

− x 2 / 4˜ 2 t

)=

M − x 2 /( 4˜ 2 t ) . 1/2 e 2˜ ( ˙t )

(

)

It is a function of the dimensionless parameter x/ 2˜ t . Therefore, concentration is inversely proportional to t and the distance of penetration of concentration is proportional to t. The solutions of the diffusion equation (1.10) under some given initial and Dirichlet boundary conditions are the following: a. Diffusion out of a plane sheet of thickness L: Initially, the concentration is uniformly distributed and the surfaces are kept at zero concentration for t > 0. Consider the initial and boundary conditions as u( x , 0) = K, 0 < x < L ; u(0, t) = 0 = u(L, t), t > 0. Separation of variables technique gives the Fourier series solution as u( x , t) =

4K ˙

˘

˜ 2n1+ 1 e n= 0

− ˇ 2 dn2 t

sin ( dn x ) ,

(1.11)

11

Introduction to Diffusive Processes

where dn = [(2n + 1)π/L ]. b. Diffusion in a semi-infinite medium x > 0: Let initially, concentration is zero throughout the medium and the boundary is kept at constant concentration K. Consider the initial and boundary conditions as u( x , 0) = 0, x > 0; u(0, t) = K , t > 0. Using the Laplace transforms, the solution is obtained as  x  u( x ,t) = K erfc  ,  2α t 

(1.12) ∞

2 2 e −t dt. where erfc(z) = 1 − erf(z) = π

∫ z

The solution of the diffusion equation in composite mediums in which the diffusion coefficients are different, for example, for x > 0, the diffusion coefficient is D1 , and for x < 0, the diffusion coefficient is D2 can be obtained in terms of the basic solutions erf  x 2 D1t  and erf  x 2 D2t  .     c. Consider the initial and boundary conditions as (generalization of (a)) u( x , 0) = f ( x), ∀x ∈[ 0, L ] , u(0, t) = u(L,t) = 0, ∀t > 0. The general solution is given by

(

)

 2 L n= 1  ∞

u( x ,t) =

(

)



L



∑ ∫ f (ξ )sin  nπL ξ  sin  nπL x  exp −α  0

2

2  nπ     t  dξ . L 

(1.13)

d. Temperature distribution in a finite, thin, insulated rod/bar: Initial condition: The rod was initially at constant temperature T1 : u( x , 0) = T1 , 0 < x ≤ L. Boundary conditions: Left end of the rod is insulated. The other end is maintained at constant temperature T0 : ux (0, t) = 0, u(L,t) = T0 , t > 0. Using the Laplace transforms, the solution is obtained as u( x ,t) = T1 + (T0 − T1 )

∞

∑(−1) erfc  L −2xα+√t2nL  + erfc  L +2xα+√t2nL   . n

(1.14)

n= 0

e. Temperature distribution in a semi-infinite, thin, insulated rod/bar: i. Initial condition: The rod is kept at zero temperature. u ( x , 0 ) = 0, 0 ≤ x < ∞. Boundary condition: Left end of the rod is maintained at an arbitrary timedependent temperature. u ( 0, t ) = f ( t ) , t > 0. Using the Laplace transforms, the solution is obtained as u( x ,t) =

x 2α π

t

∫ 0

f (τ ) − x 2 /  4α 2 ( t − τ )  e dτ . 3/2 (t − τ )

(1.15)

If f (t) is taken as the unit step temperature, f (t) = uτ (t) = 0, for t < τ , and = 1, for t ≥ τ . Then, u( x ,t) reduces to u( x , t) =

x 2α π

t

∫τ 0

1 3/2

e

(

− x 2 / 4α 2 τ

)dτ = erfc  x  = 1 − erf  x  .  2α t   2α t 

(1.16)

12

Spatial Dynamics and Pattern Formation in Biological Populations

ii. Initial condition: Initial temperature u ( x, 0 ) = f ( x ) , 0 < x < °. Boundary condition: Left end of the rod is maintained at zero temperature: u ( 0,t ) = 0,t > 0. The solution obtained by using the Fourier sine transform is given by u( x , t) =

2 ˙



ˆ  2 2 ˘ f (° )sin(˛° )d°  e − ˛ t sin(˛ x)d˛ . ˘  ˇ0 

˜˜ 0

(1.17)

f. Temperature distribution in an infnite, thin rod/bar: Initial condition: Initial temperature u( x , 0) = f ( x), −° < x < °. u( x , t) is fnite as t ˜ ±˛. The solution obtained by using the Fourier transform is given by [46] u( x , t) =

1 2ˇ

 

˜˜

f (° )cos [˛ (° − x)] e −

2

˛ 2t

d ° d˛ .

(1.18)

−−

g. Diffusion in a long circular cylinder [15]: In the case of a long solid circular cylinder, diffusion is in the radial direction. The governing differential equation becomes

˜u 1 ˜ ˛ ˜uˆ = ˙ rD ˘ˇ , ˜t r ˜r ˝ ˜r

(1.19)

where D is the diffusion coeffcient. When D is a constant, u = U ( r ) e −D˝ t is a solution where U ( r ) satisfes Bessel’s differential equation of order zero 2

d 2U 1 dU + + ˜ 2U = 0. dr 2 r dr The concentration u is obtained in terms of Bessel’s function of order zero. h. Radial diffusion in a sphere [15]: When the diffusion coeffcient D is a constant, radial diffusion in a sphere is governed by the differential equation: ˝ ˜ 2u 2 ˜uˇ ˜u = Dˆ 2 + . ˜t r ˜ r ˘ ˙ ˜r

(1.20)

Under the transformation U = ur, the equation reduces to the linear fow equation:

˜U ˜ 2U =D 2 . ˜t ˜r Therefore, in many problems, it may be possible to express the solution in terms of the solutions of the corresponding linear problems. Consider the initial value or Cauchy problem for the heat equation on the real line: ut = uxx + f ( x , t ) , −ˆ < x < ˆ , t > 0,

(1.21)

with u( x , 0) = g( x), where f and g are given smooth functions. Let uˆ (˜ , t) denote the Fourier transformation of u in the space variable defned by uˆ (° , t) =

1 2˝

˜e

− i° x

u( x , t)dx ,

13

Introduction to Diffusive Processes

where ˜ ° is a parameter. Applying the Fourier transform to the heat equation (1.21), we obtain duˆ = − ˜ 2 uˆ + fˆ (˜ , t), uˆ (˜ , 0) = gˆ (˜ ). dt The solution is given by t

˜

2

uˆ (° , t) = gˆ (° )e − ° t + e − °

2

(t − s)

fˆ (° , s)ds.

0

By inverse Fourier transform, the solution of heat equation (1.21) is obtained as u( x , t) =

1 2˝

˜e

i° x

uˆ (° , t)d° t

=

˜ K(x − y, t)g(y)dy + ˜˜ K(x − y, t − s) f (y, s)dyds,

(1.22)

0

where Green’s function K is the Gaussian or heat kernel given by K ( x , t) = ˛˝1/(4°t)1/2 ˙ˆ e − x

2

(4 t )

.

1.4.1.1 Time-Dependent/Concentration-Dependent Diffusion Coeffcient Problems In this case, the one-dimensional diffusion equation becomes

˜u ˜2u = D(t) 2 , or ˜t ˜x Under the transformation T =

1 ˜u ˜2u = . D(t) ˜ t ˜ x 2

(1.23)

t

˜ D(° ) d° , or dT = D(t)dt, this equation becomes 0

˜u ˜2u = . ˜ T ˜ x2 Therefore, the solution of the equation with constant D can be used to fnd u as a function of T. The solution can then be written in terms of t. When diffusion is dependent on concentration, the diffusion equation becomes

˜u ˜ ˛ ˜uˆ = ˙D ˘. ˜t ˜ x ˝ ˜ x ˇ

(1.24)

Under the Boltzmann variable transformation ˜ = x/(2 ˛ t), the diffusion equation reduces to an ordinary differential equation as −2˜

du d ˝ du ˇ = D . d˜ d˜ ˆ˙ d˜ ˘

14

Spatial Dynamics and Pattern Formation in Biological Populations

The above transformation is usually used when diffusion takes place in an infnite or semi-infnite media when the concentration is initially constant. Further, the initial and boundary conditions must be expressible in terms of ˜ alone. For example, the typical initial conditions in an infnite medium u = u1 , x < 0, t = 0; u = u2 , x > 0, t = 0; transform to u = u1 , ˜ = −˝ ; u = u2 , ˜ = +˝. 1.4.2 Linear Diffusion Equation in Two and Three Dimensions Diffusion from an instantaneous point source on an infnite plane is governed by the equation: ˝ ˜2u ˜2uˇ ˜u = Dˆ 2 + 2  , ˜t ˜y ˘ ˙ ˜x

(1.25)

where the diffusion coeffcient D is a constant. By substitution, we fnd that u = Pt −1exp ˆˇ − x 2 + y 2 (4Dt) ˘ is the solution of the diffusion equation. The total diffusing substance M is

(

M=

)



ˆ x2 + y 2  P P exp ˇ−  dx dy = 4 Dt t − t ˘ 

˜ ˜ −





˜ ˜ 0

2

0

ˆ r2  exp ˇ−  r dr d° = 4DP. ˘ 4Dt 

˝ r2 ˇ M exp ˙− ˘. 4°Dt ˆ 4Dt  The heat distribution in a rectangular plate is governed by the linear constant coeffcient parabolic diffusion equation: Thus, concentration is given by u =

ut = uxx + uyy , 0 < x < x0 , 0 < y < y 0 ,

(1.26)

(where the diffusion coeffcient is taken as 1) under suitable initial condition and boundary conditions. This is the simplest diffusion equation in two dimensions. Assume the conditions as the following: Initial condition: u( x , y , 0) = f ( x , y ). Boundary condition: u( x , y , t) = 0, on the boundary. The separable solution of (1.26) is given by 

u( x , y , t) =



˜˜ A

mn

m = 1 n= 1

where Amn =

4 x0 y 0

x0

˜ ˜ 0

y0

0

˝ n˛y ˇ − mnt ˝ m˛x ˇ sin ˆ sin ˆ , e  ˙ x0 ˘ ˙ y 0 ˘

(1.27)

˙ m 2 n2 ˘ ˙ n˝y ˘ ˙ m˝x ˘ f ( x , y )sin ˇ sin ˇ dy dx , and °mn = ˝ 2 ˇ 2 + 2  .   ˆ x0  ˆ y0  ˆ x0 y 0 

The time-dependent portion of the solution to the diffusion equation is given by Tm , n (t) = C(m, n)e − °mnt. The decay time ˜ at which the (m, n)th mode decays to (1/e) of its initial value is given by ˜ m , n = 1/°mn , where ˜mn = ° m + ˛ n , ˜ m is the eigenvalue corresponding to the x-dependent solution and ˜ n is the eigenvalue corresponding to the y-dependent solution.

15

Introduction to Diffusive Processes

1.4.2.1 Two-Dimensional Diffusion on a Disk Consider the case of diffusion current at a circular electrode. The problem is also called Weber’s disk problem [15]. When the radius of the circular electrode is a and u is the concentration, the diffusion equation becomes

˜2u 1 ˜u ˜2u + + = 0. ˜ r 2 r ˜ r ˜ z2

(1.28)

The boundary conditions become

˜u = 0, z = 0, r > a; ˜z

u = 0, z = 0, r ˛ a;

u = K , r ° 0, z = ˛; u = K , z ° 0, r = ˛ ; where K is the concentration in the bulk of the solution. The solution is obtained in terms of Bessel’s function of order zero as K−u=

2K ˝



˜ 0

sin(ma) J 0 (mr )e −mz dm m

ˇ  2K a˙2 2 2 2 = tan −1   , where R = r + z − a . 2 2 ˝ ˘ R + R + 4z a 

(1.29)

The concentration gradient at the disk surface is 2K ˙ °u˘ =− ˇˆ  ° z z= 0 



˜ sin ( ma) J (mr ) dm =  0

0

2K a2 − r 2

.

(1.30)

1.4.2.2 Linear Diffusion Equation in Three Dimensions Isotropic diffusion of Fickian type with constant diffusivity D is governed by the equation: ˝ ˜ 2S ˜ 2S ˜ 2Sˇ ˜S = Dˆ 2 + 2 + 2  , ˜t ˜y ˜z ˘ ˙ ˜x

(1.31)

where S is the concentration of pheromones. Suppose that M molecules of pheromone are released instantaneously at the origin. The solution of (1.31) in an infnite domain is given by (Carslaw and Jaeger [10]): S( x , y , z, t) =

(4˝Dt)

3/2

M , r 2 = x2 + y 2 + z2 . 2 exp −r (4Dt)

(

)

The ground is assumed to be a refecting plane at z = 0, implying the boundary condition:

(˜ S/˜ z ) = 0, at z = 0. If the point source is located at the ground, the solution of (1.31) on a

16

Spatial Dynamics and Pattern Formation in Biological Populations

semi-infinite domain (z > 0) with this boundary condition is twice the solution in infinite space. Thus, for a ground source, S( x , y , z,t) =

2M . (4πDt)3/2 exp − r 2 (4Dt)

(

)

The amount of release during an infinitesimal time dt is denoted by Qdt. Replacing M by Qdt in the above solution and integrating with respect to time, the solution is obtained as t

S ( r,t ) =

∫ 0

 r2  2Q exp  − dt′ = 4πDt′  4Dt′ 

Q ,   r 1/2   2 πDr 1 − ϕ    ( 4Dt )   

where ϕ is the error function. 1.4.2.3 Reaction–Diffusion Equations in Diffusion Processes An equation of the form

∂u − ∂t

3





∑ ∂∂x  d ∂∂xu  = f (t, x, u), j

j=1

(1.32)

j

where x = ( x1 , x2 , x3 ) , u = u( x , t) is the density function, d is a diffusion coefficient, is known as a reaction–diffusion equation. If the reaction term f depends on the density function u, this is known as facilitated diffusion. It occurs when the flux of an ionic species is amplified by a reaction that takes place in the diffusing medium. If the diffusion coefficient d depends on the density, d = d(u) > 0, for u ≥ 0, then we obtain a quasi-linear reaction– diffusion equation of the form:

∂u − ∂t

3





∑ ∂∂x  d(u) ∂∂xu  = f (t, x, u). j

j =1

(1.33)

j

If the diffusion process is time-dependent or non-stationary, then u = u( x , t), and we call (1.33), a non-stationary reaction–diffusion equation. If the reaction process is steady state, then we have an equation of the form: 3

−





∑ ∂∂x  d(u) ∂∂xu  = f (x, u). j =1

j

(1.34)

j

If the diffusion process involves r density functions ui = ui (t , x), i = 1, 2,…, r , and admits convection, then the system of coupled reaction–diffusion–convection equations can be written as m

∂ ui ∂  i ∂ ui  d ( t , x) + − jk ∂ t j , k = 1 ∂ x j  ∂ xk 

∑

m

∑

b ij (t, x)

j  = 1

i = 1, 2,…, r ,

∂ ui = f i t , x , u1 , u2 ,…, ur , ∂ xj

(

)

17

Introduction to Diffusive Processes

where d ijk are coeffcients of diffusion and b ij = b ij (t , x), i = 1,2, …., r, is a drift vector. A reaction–diffusion (RD) model for biochemical cell polarization was proposed by Mori et al. [75]. They found a wave-based phenomenon whereby a traveling wave is initiated at one end of a fnite, homogeneous 1D domain, moves across the domain, but stalls before arriving at the opposite end. They refer to this behavior as wave-pinning and observed that this phenomenon was obtained from a two-component RD system obeying the following assumptions: (i) Mass is conserved and limited (there is no production or removal, only exchange between one species and the other), (ii) one species is far more mobile than the other (due to binding to immobile structures, or embedding in a lipid membrane), and (iii) there is feedback (autocatalysis) from one form to further conversion to that form. Mori et al. [76] also analyzed a bistable reaction–diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of a form of wave-pinning (a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops inside the domain, forming a stationary front). Köhnke and Malchow [50] studied the emergence of stationary fronts in two-species competition–diffusion models with particular emphasis on the stability against environmental perturbations. Wave-pinning in the considered one-dimensional models is not stable against environmental noise. They have demonstrated the wave-pinning in competition–diffusion models in variable environments. 1.4.3 Diffusion in a Heterogeneous Environment There have been many efforts to fnd a correct diffusion equation for physical particles in a heterogeneous environment. In particular, Fickian and Fokker-Planck type diffusions are often claimed as the correct models. Choi and Kim [14] studied the biological diffusion models in spatially heterogeneous environment in terms of microscopic scale dynamics. They showed that the density of the total population of the two phenotypes converges to the solution of a Fokker-Planck type diffusion equation if turning frequencies are of higher order than the state transition frequencies. If it is the other way around, (if the state changes many times between each turning), the density converges to the solution of a Fickian diffusion equation. For example, if temperature is spatially heterogeneous, Brownian particles often aggregate and form a non-constant steady state. In this case, diffusivity is a function of x, D = D( x). The researchers considered the following three diffusion models which are all identical if the diffusivity is constant. Fickian equation: ut = ˝. ( D( x)˝u) , Wereide equation [48]: ut = ˝.

(

(

(1.35)

))

D( x) ˝ D( x)u ,

Chapman equation [11] : ut = ˝ ( D( x)u) .

(1.36) (1.37)

For the Fickian equation, any constant state is a steady state. For the Wereide and Chapman equations, steady states are non-constant if the diffusivity is not constant. The Chapman equation is a Fokker-Planck-type diffusion equation. These two equations are (i) satisfed by the probability density functions of a stochastic process when the Stratonovich and Itô integrals are considered, respectively, and (ii) considered as mathematical diffusion models in a heterogeneous environment [4,110]. All the three diffusion laws show different behaviors. Choi and Kim [14] have calculated and compared their steady states as follows: Consider the one-dimensional case with the Neumann boundary conditions.

18

Spatial Dynamics and Pattern Formation in Biological Populations

˝ ut = ˙ˇ a( x) ( b( x)u)˝ ˘ , for 0 < x < 1, t > 0,  ˆ

(1.38)

a(0) ( b(0)u(0))˝ = c0 , a(1) ( b(1)u(1))˝ = c1 . ˛ The steady state of the problem exists only if c0 = c1 and satisfes ˝ˆ a( x) ( b( x)u)˛ ˇ = 0. ˘ ˙ The steady state is given by 1 u( x) = b( x)

x

c1

c

˜ a(y) dy + b(x) ,

0 < x < 1.

0

The steady states corresponding to the three diffusion laws (1.35)–(1.37) are x

u( x) =

c1

˜ D(y) dy + c,

(1.39)

0

u( x) =

1 D( x)

x

˜ 0

u( x) =

c1 c dy + , D( y ) D( x)

(1.40)

c1 x + c . D( x)

(1.41)

If the zero fux boundary condition is given, c1 = 0, then the steady states are given by u( x) = c, u( x) = c/ D( x) , and u( x) = c/D( x), respectively. Therefore, the steady state of Fick’s law is constant, while the other steady states are functions of x, even if there is no fux across the boundary.

1.5 Stochastic Reaction–Diffusion (SRD) Systems In a pioneering paper, Freidlin [31] studied large deviations for small noise limit of stochastic reaction–diffusion equations. Flandoli [27] presented a new method of solution based on semigroup techniques to prove the global existence and uniqueness of solution for a stochastic reaction–diffusion equation with multiplicative noise and polynomial nonlinearity in a bounded domain. Following the works of Sowers [99], Peszat [84], Cerrai and Röckner [12] derived the large deviation estimates for the small noise limit for systems of stochastic reaction–diffusion equations with globally Lipschitz but unbounded diffusion coeffcients (assuming the reaction terms to be only locally Lipschitz with polynomial growth). Luo et al. [64] discussed the “Theory and application of stability for stochastic reaction diffusion systems”. Lyapunov direct method is an effective technique in the study of stability for ordinary differential equations and stochastic differential equations. However, this useful method was not popular in stochastic partial differential equations as the

19

Introduction to Diffusive Processes

corresponding Itô formula was not available. The authors [64] extended the Lyapunov direct method to the Itô stochastic reaction–diffusion systems. They formulated the corresponding Lyapunov stability theory and discussed stability in probability, asymptotic stability in probability, and exponential stability in mean square. They applied their theory to study the stability of the Hopfeld neural network. Their results generalized the results of Holden et al. [40] and Liao [61]. In the following, we briefy report the work of Luo et al. [64]. The authors [47,64] consider the stochastic reaction–diffusion equation:

(

)

dv(t , x) = ˜ x2 ( v(t , x)) + f ( t , x , v(t , x)) dt + g ( t , x , v(t , x)) dW (t , x)

(1.42)

( t, x ) ˛t+ × G, with the initial condition v(t0 , x) = ˜ (x), x ˛G, and the boundary condition (˜ v (t, x )/˜ N ) = 0, (t, x ) ˙t+ × ˜ G, where G = {x, x < l < +ˆ} ˇ  r ; and 0

0

f ˜˛˝  + × G ×  n ,  n ˙ˆ ,

g ˜ ˛˝  + × G ×  n ,  n × m ˆ˙

are both Boral measurable functions and  ° ( v ( t , x ))    2 x

r

˜ k =1

° ° xk

° v1 ˘ ˙ ˇ D1k ( t , x , v ( t , x )) ° x  ,…, k  ˆ

r

° ° xk

˜ k =1

T

° vn ˘  ˙ ˇ Dnk ( t , x , v ( t , x )) ° x  . k  ˆ

Dik ( x , t , v) ˜ 0 is smooth, W (t) = ( w1 , w2 ,…, wm ) is a m-dimensional Brownian motion defned on complete probability space ˛, , ( t )t ˝I ,  with natural fltration {t }t ˛ 0 , N is the outside unit normal vector of ˜G, and ˜ (x ) is a suitably smooth known function. The authors assume that g ( t , x , v(t , x)) satisfes the integral linear growth condition, and in addition, f and g satisfy the Lipschitz condition, that is, there exists a constant C > 0 such that T

(

)

(

)

g (t , x , v ) ˛ C 1 + v G , f ( t , x , v1 ) − f ( t, x , v2 ) G ˝ C v1 − v2 G , g ( t , x , v1 ) − g ( t, x , v2 ) G ˝ C v1 − v2 G , where

v ( x ,.) G 

˜v ( x,.) dx .

Without loss of generality, it was assumed that

G

f (t , x , 0) ˜ g (t , x , 0) ˜ 0, t ° t0 . Then, the system (1.42) has trivial solution v(t , x ) = 0. The authors [64] defned the following: Stochastically stable: If for all ˜ 1 °(0,1) and for all ˜ 2 > 0, there exists ˜ = ˜ ( ° 1 , ° 2 ,t0 ) > 0 such that  v ( t , x , t0 , v0 ) G < ˜ 2 ,t ˇ t0 ˇ 1 − ˜ 1 holds for v0 G ˛ ˜ (x ) G < ° , then the trivial solution of the system (1.42) is said to be stochastically stable or stable in probability. Otherwise, the trivial solution is stochastically unstable or unstable in probability. Stochastically asymptotically stable: If the trivial solution of system (1.42) is stochastically stable, and for all ˜ °(0,1), there exists ˜ 0 = ˜ (° ,t0 ) > 0, such that  lim v ( t , x , t0 , v0 ) G = 0 ˇ 1 − ˜ , holds for v0 G ˛ ˜ (x ) G < ° 0 , then the trivial solution of the

{

{

t

}

}

system (1.42) is said to be stochastically asymptotically stable.

20

Spatial Dynamics and Pattern Formation in Biological Populations

Stochastically globally asymptotically stable: If the trivial solution of system (1.42) is stochastically stable and for all ˜ > 0 whenever v0 G ˛ ˜ (x ) G < ° such that  lim v ( t , x , t0 , v0 ) G = 0 = 1, holds, then the trivial solution of the system (1.42) is said to be

{

}

tˆˇ

stochastically globally asymptotically stable. Mean square stable: The trivial solution of the system (1.42) is said to be mean square  }, we have stable if for any ˜ > 0, there exists ˜ = ˜ (° ) such that for all i0 ˜  = {1, 2,…,˝ 2 2 E v( t , x;0, ˜ , i0 G < ° , t ˜ 0, when ˜ satisfes E ˜ ˙ °.

{

{ }

}

G

Class K function If µ (.) ˆ C ˇ˘[ 0, r ] ,   is a strictly increasing function and µ(0) = 0, then the function µ is said to be a Class K function. Suppose that µ °K. If µ (.) ˝C ˙ˆ  + ,  + ˇ˘ and µ °K, lim µ(r ) = +˝, then µ °K. r˙+˝

As the authors in their works [58,115,117,120] have pointed out, a continuous function V (t , ˜ ) is said to be positive-defnite if V (t , 0) = 0, and for some µ °K, V (t , ˜ ) ˙ µ ˜ . Write C1,2  + ×  n ;  + for the family of all nonnegative functions V (t , ˜ ) on  + ×  n that are continuously twice differentiable in ˜ and once in t. If V ( t, ˜ ) ˝C1,2  + ×  n ;  + , then defne an operator V ( t, ˜ ) from  + ×  n to  with respect to (1.42) by

(

( )

)

(

V ( t , ˜ ) = Vt ( t, ˜ ) + V˜T ( t, ˜ ) f ( t , x , ˜ ) + where Vt (t , ˜ ) =

)

1 trace ˆˇ g T ( t , x , ˜ ) V˜˜ ( x , t ) g ( t , x , ˜ ) ˘ , 2

(1.43)

˙ ° 2 V (t , ˜ ) ˘ ˙ ° V (t , ˜ ) ° V (t , ˜ ) ° V (t , ˜ ) ˘ , V˜T (t , ˜ ) = ˇ ,…, , and V ( x , t ) = ˜˜  . ˇ °t °˜ n  ˆ °˜1 ˆ °˜ i °˜ j  n × n

˜

Applying the Itô formula to V ( t , v(t , x )) dx along system (1.42), one obtains that for all t ˜ t0 , G

˜

d V ( t , v(t , x)) dx G

= (1)

˜ ˘( LV (t, v(t, x)) + V (t, v)°ˇv(t, x)) dt +V (t, v)g(t, x, v(t, x))dW(t) dx. T v

T v

G

The existence of the function V (t , v ) ˜C1,2 and other conditions in the classical Lyapunov theorem on the stability of (1.42) are needed [66]. The following are some defnitions: Lyapunov-A and Lyapunov-B functions: V ˛ C1,2  + ×  n ;  + is called as a Lyapunov-A

(

)

˜

function for (1.42), if L V (t , v)dx ° 0, and is called as a Lyapunov-B function for (1.42), if G

˜

˜

L V (t , v) dx ° −b V (t , v) dx , in which b > 0. G

G

The fundamental quantities in reaction–diffusion models are individual entities such as atoms, molecules, bacteria, cells, or animals, which move and/or react in a stochastic manner. If the number of individuals is large, then accounting for each individual is ineffcient. In this case, often PDE models describing the average or mean behavior of the system are used. If the number of individuals is large in certain regions and small in others, then a stochastic model may be ineffcient in one region, and a PDE model is inaccurate in the other. When a small number of individuals are involved, stochastic effects can play an

Introduction to Diffusive Processes

21

important role in the survival and spatiotemporal distribution of individuals. For example, when chemical reactions occur in discrete steps at the molecular level, then the processes are invariantly stochastic, which have been demonstrated experimentally for single-cell gene expression events [56,81]. Bates et al. [3] discussed the random attractors for stochastic reaction–diffusion equations on unbounded domains. Ferm et al. [26] derived an adaptive hybrid method for simulation of stochastic reaction–diffusion equations. Kelkel and Surulescu [48] studied the stochastic reaction–diffusion system modeling the pattern formation on seashells. Within the same species of seashells, they exhibit a huge variety of beautiful and highly complex patterns. The importance of the work was to study whether the diversity is due to a single or more mechanisms. Following the work of Gierer-Meinhardt [35], the authors [48] proposed a model and proved the existence of a positive solution for the resulting system. The authors have also performed numerical simulations and compared them with the solutions obtained using a deterministic approach. Li and Kao [59] investigated the mean square stability of a class of SRD equations with the Markovian switching and impulsive perturbations by means of the Lyapunov function and stochastic analysis. Jifeng Hu et al. [41] discussed the stochastic analysis of RD processes. The authors derived an estimator for the appropriate compartment size for simulating the RD systems and introduced a measure of fuctuations in a discretized system. They have also given a computational algorithm for implementing the method. Stochastic RD simulations were successfully used in many biological applications. Some examples are the following: (i) models of signal transduction in E. Coli Chemotaxis [62], (ii) oscillation of Min proteins in cell division [25], (iii) MAPK pathway [106], (iv) intracellular calcium dynamics [29], (v) models of Hes1 gene regulatory network [103], and (vi) Actin dynamics in flopodia [23]. Fabian Spill et al. [100] derived “Hybrid approaches for multiple-species stochastic reaction-diffusion models”. They considered the case where the fundamental quantities in such models are individual entities that move and/or react in a stochastic manner. The authors developed a scheme that couples a stochastic reaction–diffusion system in one part of the domain with its mean-feld analog (a discretized PDE model) in the other part of the domain. The interface between the two domains occupies exactly one lattice site and is chosen such that the mean-feld description is still accurate there and the errors due to the fux between the domains are small. Each species comprises individuals that can migrate to neighboring lattice sites, or react locally with entities of the same or other species. The algorithm preserves mass at the interface between the stochastic and deterministic domains, and these domains need to be neither static nor connected. The scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The authors showed that the method is signifcantly faster to simulate on a computer than the pure stochastic model. The authors applied a hybrid algorithm to solve the stochastic Fisher-Kolmogorov equation:

˜n n ˜2n ˆ = °n˘ 1 −  + D 2 , ˇ ˜t ˜x ˙

(1.44)

where D is the diffusion coeffcient, ˜ is the growth rate, and ˜ is the carrying capacity. The algorithm was used to simulate traveling wave solutions as the model parameters are varied. The authors have also applied the hybrid algorithm for solving the spatial stochastic Lotka-Volterra system. They considered the interaction reaction such that each time a

22

Spatial Dynamics and Pattern Formation in Biological Populations

prey (N) is eaten by a predator (M), a single new predator is born. The mean feld limit and continuum limit correspond to the classical spatial Lotka-Volterra equations:

˜n ˜2n = an − bnm + DN 2 , ˜t ˜x

(1.45a)

˜m ˜2m , = − cm + bnm + DM ˜t ˜ x2

(1.45b)

where n = n( x , t) and m = m( x , t) are prey and predator densities related to N ( k ) and M( k ), respectively. In the stochastic model, each species can jump to neighboring lattice sites, the prey reproduce at rate a, predators die at rate c, and consume prey and produce at rate b. They considered the following four cases depending on whether each of the prey and predator evolves deterministically or stochastically: (i) deterministic prey and deterministic predator, (ii) deterministic predator and stochastic prey, (iii) stochastic predator and deterministic prey, and (iv) stochastic prey and stochastic predator. Many authors focused on constructing SRD models to solve specifc biological questions or to analyze SRD algorithms or to develop SRD software. There are a number of recent SRD approaches like (i) combining together in a multiscale framework [28,38] or (ii) combining stochastic models with mean-feld descriptions [30]. There are many other works on SRD models that we have not quoted or discussed.

1.6 Hopf Bifurcation Analysis A good presentation of the Hopf bifurcation is given in the book Theory and Application of Hopf Bifurcation authored by Hassard et al. [37]. Kuznetsov [53] in his book Elements of Applied Bifurcation Theory presented Hopf bifurcation in RD systems on an interval with the Dirichlet boundary conditions. Hopf bifurcation is an instability induced by the transformation of the stability of a focus. In fact, the space-independent Hopf bifurcation breaks the temporal symmetry of a system and gives rise to oscillations that are periodic in time and uniform in space. Yi et al. [116] discussed the Hopf bifurcation for a diffusive predator–prey system with Holling type II functional response and also derived an explicit algorithm for determining the properties of the Hopf bifurcation (direction of the Hopf bifurcation and stability of bifurcating periodic solutions) for a general reaction–diffusion system. In the following, we present some salient features of analysis from their work. The authors considered the following general RD system subject to the Neumann boundary conditions on the spatial domain ˜ = (0, ˛), ut = f1 ( ˜ , u, v ) + d1uxx , vt = f2 ( ˜ , u, v ) + d2 vxx , ux ( 0,t ) = vx ( 0,t ) = ux ( ˝,t ) = vx ( ˝,t ) = 0, u ( x , 0 ) = u0 ( x ) , v ( x, 0 ) = v0 ( x ) , x ˙( 0, ˜ ) ,t > 0, d1 , d2 , ° ˙ + , f1 , f2 are C r , r ˇ 5, and f k ( ° , 0, 0 ) = 0, k = 1, 2. Defne F ( ˜ ,U ) by ˝ F(˜ , U) = ˆ ˆ˙

f1 (˜ , u, v) − A(˜ )u − B(˜ )v ˇ  , where U = (u, v)T X. f2 (˜ , u, v) − C(˜ )u − D(˜ )v ˘

(1.46)

23

Introduction to Diffusive Processes

Then, system (1.46) can be rewritten into the following abstract form: dU = L(˜ )U + F(˜ , U). dt

(1.47)

When ˜ = ˜0, the above system reduces to dU = L ( ˜0 )U + F0 (U ), where F0 (U ) = F ( ˜0 ,U ) . dt

(1.48)

The authors [116] considered the real and complex-valued Sobolev spaces. Let A(˜ ) = f1u (˜ , 0, 0), B(˜ ) = f1v (˜ , 0, 0), C(˜ ) = f2 u (˜ , 0, 0), and D(˜ ) = f2 v (˜ , 0, 0). Defne the linear operator L(˜ ) with the domain DL ( ° ) = X  as ˙ °2 ˇ d1 2 + A(˜ ) °x L(˜ ) = ˇ ˇ C(˜ ) ˇ ˆ

˘  .  °2 d2 + D(˜ )  2 °x  B(˜ )

(1.49)

Assume that, for some ˜0 °, the following Condition 1 holds: Condition 1 There exists a neighborhood  of ˜0 such that, for ˜ ˝ , L ( ˜ ) has a pair of simple, complex conjugate eigenvalues ˜ (° ) ± i˛ (° ), continuously differentiable in ˜ with ˜ ( °0 ) = 0, ˛ ( °0 ) = ˛ 0 > 0, ˜ ˙ ( °0 ) ˆ 0; and all other eigenvalues of L(˜ ) have nonzero real parts for ˜ ° . Then, from Hassard et al. [37], it is known that the system (1.46) undergoes a Hopf bifurcation at (0, 0) when ˜ crosses through ˜0 . Defne a second-order matrix sequence Ln (˜ ) as ˆ ˆ d n2  ˘ A(˜ ) − ˘ 1 2  ˇ   ˘ Ln ( ˜ ) = ˘ ˘ C(˜ ) ˘ ˇ

    , n  0 . 2 ˆ d2 n   D(˜ ) − ˘ 2  ˇ    B(˜ )

(1.50)

The characteristic equation is given by

˜ 2 − ˜ Tn ( ° ) + Dn ( ° ) = 0, n ˘ 0 where Tn (˜ ) = A(˜ ) + D(˜ ) − Dn (˜ ) =

( d1 + d2 ) n2 , 2

n2 d1d2 n4 − [ d1D(˜ ) + d2 A(˜ )] 2 + A(˜ )D(˜ ) − B(˜ )C(˜ ) 4   2

(1.51)

d˜ ( °0 ) = ˘Aˆ ( °0 ) + Dˆ ( °0 ) . d°

Condition 1 has the following equivalent form: Tn ( ˜0 ) = 0, Dn ( ˜0 ) > 0, Aˆ ( ˜0 ) + Dˆ ( ˜0 ) ˘ 0, for some n .  0

(1.52) (1.53)

24

Spatial Dynamics and Pattern Formation in Biological Populations

d˜ ( °0 ) ˙ 0. Now, ˜ 0 = Dn ( °0 ) , and B ( ˜0 ) ,C ( ˜0 ) cannot be equal to zero simuld° taneously. Therefore, the transversality condition holds and the system (1.46) undergoes Hopf bifurcation. This establishes the existence of Hopf bifurcation. The authors used the central manifold theorem and normal form as defned in Hassard et al. [37] to obtain the conditions under which a family of periodic solutions bifurcates from the positive steadystate solution of (1.46), when the control parameter crosses through the critical value. The authors derived the formulas to decide the direction of the Hopf bifurcation, stability, and period of bifurcating periodic solutions arising through the Hopf bifurcation. The authors transformed the system (1.48) into the following system in (z, w) coordinates: Therefore,

dw = L ( °0 ) w + H ( z, z , w), dt

dz = i˜ 0 z + q* , F0 , dt

(1.54)

where H ( z, z , w ) = F0 − q* , F0 q − q * , F0 q , F0 = F0 ( zq + z q + w ) . The frst equation of (1.54) gives the equation of reaction–diffusion system (1.46) restricted on the center manifold at ( ˜0 , 0, 0 ) as defned by Hassard et al. [37]. The application of the theory gives the equation:

( )

˜

g kj k j dz 4 = i° 0 z + z z +˛ z , dt k ! j ! 2˘ k + j ˘ 3

(1.55)

where g 20 = q* , Qqq , g11 = q* , Qqq , g 02 = q* , Qq q , and g 21 = 2 q* ,Qw11q + q* ,Qw20 q + q* , Cqqq . The dynamics of (1.54) can be determined by the dynamics of (1.55). In addition, it can be observed from the work of Hassard et al. [37] that when ˜ approaches ˜0 suffciently closely, the Poincare normal form of (1.47) has the form z = (° (˛ ) + i˝ (˛ )) z + z

M

˜c (˛ )(zz ) , j

j

(1.56)

j=1

where z is a complex variable, M ˜ 1, and c j (˜ ) are complex-valued coeffcients with c1 ( ˜ 0 ) =

g 21 1 i  * i ˙ 2 2˘ = q , Qqq q* , Qqq − 2 q* , Qqq ˇˆ g 20 g11 − 2 g11 − g 02  + 3 2 2° 0 2° 0  −

1 * q , Qq q 3

2

1 * 1 * ° * ˝˛ + q , Qw 11q + 2 q , Qw 20 q + 2 q , Qqqq ,

µ2 = ° ˘˘(0) = −

(1.57)

1 Re C1 ( °0 ) , ˝ 2 = 2Re C1 ( °0 ) . ˛ ˘ ( °0 )

(

)

(

2

)

(1.58)

The direction of Hopf bifurcation and stability of the bifurcating periodic solutions of (1.46) at ( ˜0 , 0, 0 ) can be determined by the sign of Re ˝˙ c1 ( ˜0 ) ˆˇ . For ˜ = ˜ ( s), where s is suffciently small, there exists a family of T ( s) periodic, continuously differentiable solutions ( u(s), v(s))(x, t) of system (1.46) such that u(0) = 0 = v(0), and T2 = Tˇˇ(0) = −

(

)

Re c1 ( °0 ) ˜ ˇ ( °0 )  4  Im c1 ( °0 ) − . 2  ˜ 0  ˛ ˇ ( °0 ) 

(

)

(1.59)

25

Introduction to Diffusive Processes

The following Hopf bifurcation theorem for the general RD equations (1.46) summarizes the above results. Theorem 1.1 (Yi et al. [116]) Assume that Condition 1 holds. Then, the model system (1.46) undergoes i. a supercritical (or subcritical) Hopf bifurcation at (0,0) when ˜ = ˜0 , if µ2 > 0 (or < 0), i.e., ˘1/  ˜ ˇ ( °0 ) ]Re[ c1 ( °0 )  < 0 (or > 0). ii. In addition, if all the other eigenvalues of L ( ˜0 ) have negative real parts, then the bifurcating periodic solutions are stable (or unstable) when ˜ 2 < 0 (or > 0), i.e., Re ˝˙ c1 ( ˜0 ) ˆˇ < 0 (or > 0). iii. T2 determines the period of the bifurcating periodic solutions. The period increases if T2 > 0 and decreases if T2 < 0.

1.7 Multiple-Scale Analysis/Weakly Nonlinear Analysis To study the dynamics near a bifurcation point, multiple-scale analysis method can be applied near this point. The relevant patterns can be expressed in terms of three active resonant pairs of modes k j , −k j , such that k j = kc , for j = 1, 2, 3, where kc is the critical wave number (where the instability occurs). Following the analysis given in the work of Zhang et al. [121], Zheng and Shen [122] consider the following general reaction–diffusion problem:

(

)

˜u = f (u, v) + d1˝ 2 u + d12 ˝ 2 v = a11u + a12 v + d1˝ 2 u + d12 ˝ 2 v , ˜t

(1.60a)

˜v = g(u, v) + d21˝ 2 u + d2 ˝ 2 v = a21u + a22 v + d21˝ 2 u + d2 ˝ 2 v , ˜t

(1.60b)

where ˜ 2 is the two-dimensional Laplace operator and ( x , y ) ˜ R 2 . The initial and boundary conditions are u( x , y , 0) > 0, v( x , y , 0) > 0,( x , y ) °˛,

(1.61a)

˜ u( x , y , t) ˜ v( x , y , t) = = 0, t > 0, ( x , y ) ˙˜ ˆ, ˆ ˙ R 2 . ˜° ˜°

(1.61b)

Amplitude of the modes in the solution of (1.60) cannot be directly determined. One can obtain an approximation to the amplitude using the Taylor series expansion. Expand the right-hand sides in (1.60) about the equilibrium point ( u0 , v0 ), and truncate the expansion at third order. We can use multiple-scale analysis to derive the amplitude equations, when cr k = kc . When the controlled parameter d12 = d12 is larger than the critical value of the Turing point, the solution of the system (1.60) can be expanded as (Zheng and Shen [122]), N

c = c0 +

˜(Z e ) , i

i=1

iki ˆr

26

Spatial Dynamics and Pattern Formation in Biological Populations

with k = kc , Z j and the conjugate Z j are the amplitudes associated with the modes k j and cr −k j. Close to onset of d12 = d12 , one obtains

˜ Zi = si Zi + Fi ( Zi , Z j ,) , ˜t where si is the coeffcient of the linear term of the variable Zi . Applying the center manifold theory near the Turing bifurcation point, it can be concluded that amplitude Z j satisfes the equation:

˜ Zi = Fi Zi , Zi , Z j , Z j , . ˜t

(

)

From the standard multiple-scale analysis, the spatiotemporal evolution of the amplitudes up to the third order in the perturbations can be written as

°0

˛ Zi = µZi + ˛t

˜h

lm

lm

Zl Zm +

˜g

lmn

Zl Zm Zn .

(1.62)

lm

Due to spatial translational symmetry, the following equation holds:

°0

˛ Zi iki r = µZi e iki r + e ˛t

˜h

lm

Zl Zm e i( kl + km )r +

lm

˜g

lmn

Zl ZmZne i( kl + km + kn )r .

(1.63)

lm

Comparing (1.62) with (1.63), it can be concluded that the two equations hold only if ki = kl +  + km. Let the system (1.60) be rewritten as ˜c = Lc + N(c), ˜t ° a + d 2 ° u ˙ 11 1 L , where c = ˝ = ˝ ˇ ˝˛ a21 + d21 2 ˛ v ˆ N=

˛ f u2 + 2 f uv + f v 2 uu uv vv 1 ˙ 2 ˙ g uuu2 + 2g uv uv + g vv v 2 ˝

(1.64) a12 + d12 2 ˙ ˇ, a22 + d2 2 ˇˆ

ˆ ˛ f u3 + 3 f u2 v + 3 f uv 2 + f v 3 uuu uuv uvv vvv 1 ˘+ ˙ ˘ˇ 6 ˙˝ g uuuu3 + 3g uuv u2 v + 3g uvv uv 2 + g vvv v 3

ˆ ˘. ˇ˘

cr Here, L is the linear term and N is the nonlinear term. When d12 is close to d12 , expand d12 as

( )

cr (1) (2) (3) d12 − d12 = ˜ d12 + ˜ 2 d12 + ˜ 3 d12 + o ˜3 ,

where ε is a small parameter. Expand c and N in terms of ˜ as

27

Introduction to Diffusive Processes

ˆ u2  2 ˆ u  ˆ u1  c=˘ =˘  ˜ +… ˜ +˘  ˇ v  ˇ v1  ˇ v2  N=

ˆ f u2 + 2 f u ° + f ° 2 uu 1 uv 1 1 uv 1 1 ˘ 2 2 ˘ g uu u1 + 2g uv u1°1 + g uv° 12 ˇ +

ˆ f u u + f ( u ° + u ° ) + f ° °  uu 1 2 uv uv 1 2 2 1 1 2   ˜ 2 + ˘ ˘ g uu u1u2 + g uv ( u1° 2 + u2° 1 ) + g vv° 1° 2  ˇ

ˆ f u3 + 3 f u2 v + 3 f u v 2 + f v 3 uuu 1 uvv 1 1 uuv 1 1 vvv 1 1 ˘ 3 2 2 6 ˘ g uuu u1 + 3g uuv u1 v1 + 3g uvv u1 v1 + g vvv v13 ˇ

  

   ˜ 3.  

Linear operator L can be written as

( )

(1) (2) L = Lc + ˜ Md12 + ˜ 2 Md12 + o ˜2 ,

˝ a + d ˛2 11 1 where Lc = ˆ ˆ˙ a21 + d21˛ 2

cr 2 ˇ a12 + d12 ˛ , a22 + d2˛ 2 ˘

˝ M=ˆ 0 ˙ 0

(1.65) ˛2 ˇ .  0 ˘

Let T0 = t , T1 = ˜ t , T2 = ˜ 2t…. Here, Ti is a dependent variable. Then,

˜ ˜ ˜ ˜ = +° + °2 + ˜ t ˜ T0 ˜ T1 ˜ T2 The solutions of the system (1.64) have the following form: ˛ u ˆ c=˙ ˘= ˝ v ˇ

3

˛ xi ˆ ik r ˘ e i + c  c. yi ˇ

˜ ˙˝ i=1

This expression implies that the bases of the solutions have nothing to do with time and the amplitude Z is a variable that changes slowly. As a result, one can write

˜Z ˜Z ˜Z =° + °2 + ˜t ˜ T1 ˜ T2

(1.66)

Comparing different orders of ˜ in (1.65), one obtains ˜ u1 ˝ Lc ˛ ˆ = 0, ° v1 ˙ ° u2 ˙ ° u1 ˙ 1 ° fuuu12 + 2 fuv u1v1 + f vv v12 ˜ ° u1 ˙ (1) Lc ˝ ˇ= ˝ ˇ − d12 M ˝ ˇ− ˝ 2 2 ˛ v2 ˆ ˜ T1 ˛ v1 ˆ ˛ v1 ˆ 2 ˝˛ g uuu1 + 2 g uv u1v1 + g vv v1

(1.67) ˙ ° F ˙ ˇ = ˝ u ˇ (1.68) ˇˆ ˛ Fv ˆ

28

Spatial Dynamics and Pattern Formation in Biological Populations

˝ u3 ˇ ˝ u2 ˇ ˝ u1 ˇ ˜ ˝ u2 ˇ ˜ ˝ u1 ˇ (2) (1) Lc ˆ =  − d12 M ˆ  ˆ + ˆ  − d12 M ˆ ˙ v2 ˘ ˙ v1 ˘ ˙ v3 ˘ ˜ T1 ˙ v2 ˘ ˜ T2 ˙ v1 ˘ ˝ fuu u1u2 + fuv ( u1 v2 + u2 v1 ) + f vv v1 v2 −ˆ ˙ˆ guu u1u2 + guv ( u1 v2 + u2 v1 ) + g vv v1 v2

ˇ  ˘

˝ f u3 + 3 f u 2 v + 3 f u v 2 + f v 3 uuu 1 uuv 1 1 uvv 1 1 vvv 1 1 − ˆ 6 ˆ guuu u13 + 3guuv u12 v1 + 3guvv u1 v12 + g vvv v13 ˙

ˇ ˝ G ˇ u  =ˆ . G ˘ ˙ v ˘

(1.69)

Consider the case of the frst order in ˜ . Since Lc is the linear operator of the system close T to the onset, ( u1 , v1 ) is the linear combination of the eigenvectors that correspond to the zero eigenvalue. Since ° u ˙ ˝ v ˇ= ˛ ˆ we obtain that xi = By i , where B =

3

° xi ˙ ik r ˇ e i + c  c, yi ˆ

˜ ˝˛ i=1

a22 − d2 kcr2 . If y i = 1 is assumed, then xi = B, and d21kcr2 − a21

° u1 ˙ ° B ˙ ° ˝ ˇ =˝ ˇ˝ ˛ v1 ˆ ˛ 1 ˆ ˛

3

˜W e i

i=1

iki r

˙ + c  c ˇ , i = 1, 2, 3 ˆ

where Wi is the amplitude of the mode e iki ˜r . Now, consider the second-order terms in ˜ 2 . According to the Fredholm solvability condition, the vector function of the right hand of the above equation must be orthogonal with the zero eigenvectors of operator L+c . The zero eigenvectors of adjoint operator L+c ˜ 1 ˝ −iki ˘r a11 − d1kcr2 are ˛ e A , where = . From the orthogonality condition, one gets the ˆ d21kcr2 − a21 ° A ˙ equations: ˛ u2 ˆ ˘ = 0. ˝ v2 ˇ

(1 A ) e −ik .r Lc ˙ i

Putting the value of ( u1 , v1 ) in the above equation and equating the coeffcient of e ik1r, one obtains the equations (A + B)

˜ W1 (1) = − kcr2 d12 W1 + ( f2 + Af3 ) W2W3 . ˜ T1

( A + B)

˝W2 (1) = −kcr2 d12 W2 + ( f2 + Af3 ) W1W3 . ˝T1

(A + B)

˝W3 (1) = −kcr2 d12 W3 + ( f2 + Af3 ) W1W2 , ˝T1

(1.70)

29

Introduction to Diffusive Processes

where f2 = fuuB2 + 2 fuv B + f vv and f3 = g uuB2 + 2g uv B + g vv . ˛ u2 ˆ ˛ U 0 ˆ Write  ˙ ˘+ ˘ =˙ ˝ ° 2 ˇ ˝ V0 ˇ

˛ Uj ˙ ˙ Vj j=1 ˝

ˆ ik r ˘e j + ˘ˇ

3

˜

˛ U jj ˙ ˙ Vjj j =1 ˝

ˆ ˛ U 12 ˆ i( k − k )r 2ik r 1 2 ˘e j +˙ ˘e V ˘ˇ 12 ˝ ˇ

3

˜

˛ U 23 ˆ i( k − k )r ˛ U 31 ˆ i( k − k )r 2 3 3 1 +˙ +˙ + c.c. ˘e ˘e ˝ V23 ˇ ˝ V31 ˇ

(1.71)

Substituting (1.71) into (1.68), collecting the coeffcients and comparing, one gets the equations: −1

a12 ˆ ˛ − f2 ˘ ˙ a22 ˇ ˙˝ − f3

˛ U 0 ˆ ˛ a11 ˙ ˘ =˙ ˝ V0 ˇ ˝ a21

ˆ 2 2 2 ˘ W1 + W2 + W3 ˘ˇ

(

˛ Zu0 ˆ 2 2 2 =˙ ˘ W1 + W2 + W3 ˝ Zv0 ˇ

(

U j = BVj ,

˜ U 12 ˝ ˜ a11 − 3 kcr2 d1 ˛ ˆ =˛ 2 ° V12 ˙ ˛° a21 − 3 kcr d21

(1.72)

)

j = 1, 2, 3

(1.73)

˜ f2 −1 cr ˝ ˛ − a12 − 4 kcr2 d12 2 ˆ ˛ a22 − 4 kcr2 d2 ˆ˙ ˛ − f3 ˛° 2

˜ U 11 ˝ ˜ a11 − 4 kcr2 d1 ˛ ˆ =˛ 2 ° V11 ˙ ˛° a21 − 4 kcr d21

)

cr ˝ a12 − 3 kcr2 d12 ˆ 2 a22 − 3 kcr d2 ˆ˙

−1

˜ − f2 ˛ ˛° − f3

˝ ˆ ˜ Zu1 ˝ 2 ˆ W12 = ˛ ˆ W1 , ˆ ° Zv 1 ˙ ˆ˙

(1.74)

˝ ˜ Zu2 ˝ ˆ W1 W2 = ˛ ˆ W1 W2 . ˆ˙ ° Zv 2 ˙

(1.75)

By permuting the suffxes, one can get the coeffcients of other terms of (1.71). T For the third order of ˜ as in (1.69), the coeffcient of e ik1 .r . , denoted by Gu1 Gv1 , is given by

(

˛ G1 u ˙ ˙˝ Gv1

˛ ˛ W1 V1 ˆ ˆ ˙ B ˙˝ T + T ˘ˇ 2 1    ˘ = ˙˙ ˘ˇ W1 V1 ˙ + ˙˝ T2 T1  

((( f

ˆ ˘ ˛ 0  ˘ + k2 cr ˙ ˘ ˙˝ 0  ˘ ˘ˇ

(1) ˆ ˛ d12   ˛ BV1 ˆ + k 2 0  ˘˙ cr ˙ ˘ ˙˝ 0  0 ˘ˇ ˝ V1 ˇ

)

(2) ˆ d12   ˛ BW1 ˆ ˘˙ ˘ 0 ˘ˇ ˝ W1 ˇ

˛ ˙ ˙ ˙ −˙ ˙ ˙ ˙ ˙ ˙˝

(( f B + f ) ( Z + Z ) + ( f B + f ) ( Z + Z )) ( W + W )) W + f (W V + W V ) ((( g B + g ) ( Z + Z ) + ( g B + g ) ( Z + Z ))) W + (( g B + g ) ( Z + Z ) + ( g B + g ) ( Z + Z )) ( W + W )) W + f (W V + W V )

ˆ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ˇ

˛ ˙ −˙ ˙ ˝

(W (W

(1.76)

uu

)

B + f uv ) ( Zu0 + Zu1 ) + ( f uv B + f vv ) ( Zv0 + Zv1 ) W1 + uv

uu

uv

vv

u0

u1

uv

1

1

2 2

2

+ W2 + W3 2

+ W2 + W3

v0

vv

)( f )( g

2 2

uv

v0

uuu

2

v2

2

vv

v2

3

v0

2

2

2

2

1

1

v1

3

2

B3 + 3 f uuv B2 + 3 f uvv B + f vvv

2

1

) )

3 2 uuu B + 3g uuv B + 3g uvv B + g vvv

uu

2

2

uv

3

uu

3

ˆ ˘ ˘ W1 . ˘ ˇ

2

3

3

u0

2

u0

uv

3

u2

2

u2

30

Spatial Dynamics and Pattern Formation in Biological Populations

(

)

(

T

)

T

Similarly, Gu2 Gv2 and Gu3 Gv3 can be obtained by permuting the subscripts. Using the Fredholm solvability condition as in the second order of ˜ , we obtain ˛ Gj ˆ u ˘ = 0, j G u ˘ ˝ ˇ

(1 A ) ˙˙

j = 1, 2, 3,

which on simplifcation gives

) (

˙ ˝W1 ˝V1 ˘ 2 = −kcr2 d12(1)V1 + d12(2)W1 + h1 W2V3 + W3V2 − G1 W1 (A + B) ˇ + ˆ ˝T2 ˝T1 

(

(

)

2

+ G2 W2 + W3

2

(

)) W ,

(1.77)

1

where h1 = f2 + Af3 ,

(

)

ˆ ( fuuB + fuv ) + A ( g uuB + g uv ) ( Zu0 + Zu1 )  ˘ , G1 = − ˘  ˇ + ( fuv B + f vv ) + A ( g uv B + g vv ) ( Zv 0 + Zv 1 ) + ( f 4 + Ag 4 )

(

)

(

)

ˆ ( fuuB + fuv ) + A ( g uuB + g uv ) ( Zu0 + Zu2 )  ˘ , G2 = − ˘  ˇ + ( fuv B + f vv ) + A ( g uv B + g vv ) ( Zv 0 + Zv2 ) + ( f 4 + Ag 4 )

(

)

f 4 = fuuuB3 + 3 fuuv B2 + 3 fuvv B + f vvv , g 4 = g uuuB3 + 3g uuv B2 + 3 g uvv B + g vvv , Similarly, the other two equations can be obtained by permuting the subscripts. The amplitudes, Z j , j = 1, 2, 3, are taken as variables that change slowly with respect to time so that ˜ Z j ˜ T0 = 0. Hence,

(

)

˜ Zj ˜ Zj ˜ Zj =° + °2 + o °2 . ˜t ˜ T1 ˜ T2

( )

(1.78)

( )

Setting Z j = ˜ Wj + ˜ 2Vj + o ˜ 2 , and using the above results, one can get the amplitude equations corresponding to Z1 , Z2 , Z3 as follows:

(

)

(

)

(

)

˜0

° Z1 2 2 2 = µZ1 + hZ2 Z3 − g1 Z1 + g 2 Z2 + g 2 Z3 Z1 , °t

˜0

° Z2 2 2 2 = µZ2 + hZ1Z3 − g1 Z2 + g 2 Z1 + g 2 Z3 Z2 , °t

˜0

° Z3 2 2 2 = µZ3 + hZ2 Z1 − g1 Z3 + g 2 Z2 + g 2 Z1 Z3 , °t

(1.79)

31

Introduction to Diffusive Processes

where µ =

cr d12 − d12 A+B −h G G , ° 0 = − cr 2 , h = cr 12 , g1 = − cr 1 2 , g 2 = − cr 2 2 . cr d12 d12 kcr d12 kcr d12 kcr d12 kcr

1.7.1 Linear Stability Analysis of the Amplitude Equation The dynamics of amplitude equations can be investigated by using the linear stability analysis. Assume that the amplitude in (1.79) can be expressed as Zi = ˜i e i˛ i , where ˜i = Zi and ˜ i is the phase. Substituting into (1.79), separating the real and imaginary parts, and simplifying, one obtains the following equations [122]:

˜0

°˛ ˝ 2 ˝ 2 + ˝12 ˝32 + ˝32 ˝22 = −h 1 2 sin(˙ ), °t ˝1 ˝ 2 ˝ 3

(1.80a)

(

)

(1.80b)

(

)

(1.80c)

(

)

(1.80d)

˜0

° ˛1 2 2 2 = µ˛1 + h˛2 ˛3 cos(˙ ) − g1 ˛1 + g 2 ˛2 + g 2 ˛3 ˛1 , °t

˜0

° ˛2 2 2 2 = µ˛2 + h˛1 ˛3 cos(˙ ) − g1 ˛2 + g 2 ˛1 + g 2 ˛3 ˛2 , °t

˜0

° ˛3 2 2 2 = µ˛3 + h˛2 ˛1 cos(˙ ) − g1 ˛3 + g 2 ˛2 + g 2 ˛1 ˛3 , °t

where ˜ = ˜1 + ˜2 + ˜3 . From this system, it can be observed that the phase of amplitude lies only in the phase ˜ = 0 and ˜ = ˛, when the system lies in the stationary state. Since ˜i ≥ 0, it can be concluded that the solution of the equation (1.80a) is stable for ˜ = 0, when h > 0, and is stable for ˜ = ˛, when h < 0. If we consider only the stable solution of this equation, then the mode equation has the following form:

˜0

(

)

° ˛1 2 2 2 = µ˛1 + h ˛2 ˛3 − g 1 ˛1 + g 2 ˛2 + g 2 ˛3 ˛1 . °t

(1.81)

The dynamical system (1.80) possesses four kinds of stationary solutions. Pattern formations can be investigated by performing linear stability analysis. Considering a perturbation to ( ˜1 , ˜2 , ˜2 ) as (˜°1 , ˜°2 , ˜°2 ) , substituting it in (1.81), one can derive the linear perturbation equation. The matrix of mode equation is given by

(

ˇ µ − 3g ° 2 − g ° 2 + ° 2 1 1 2 2 3   h °3 − 2g 2 °2 °1   h °2 − 2g 2 °3 °1 ˘

)

h °3 − 2g 2 °2 °1

(

µ − 3g 1 °22 − g 2 °32 + °12 h °1 − 2g 2 °2 °3

h °2 − 2g 2 °3 °1

)

h °1 − 2g 2 ° 2 °3

(

µ − 3g 1 °32 − g 2 °22 + °12

)

     

(1.82)

i. The stationary state: ˜1 = ˜2 = ˜3 = 0. The stationary state corresponding to the linear perturbation equation is ( ˙ˆ˜i / ˙t ) = µˆ˜i. The stationary solution is stable, when µ < 0 = µ2; otherwise, it is unstable. ii. Stripe patterns: ˜1 = µ/g1 , µ > 0, ˜2 = ˜3 = 0. Substituting ( ˜1 , 0, 0 ) in the perturbation equation (1.82), we obtain

32

Spatial Dynamics and Pattern Formation in Biological Populations

˙ ˝°1 dˇ ˜ 0 ˇ ˝°2 dt ˇ ˆ ˝°3

˘ ˙ µ − 3g1 °12  ˇ 0  =ˇ  ˇ 0  ˇˆ

0

0

µ − g 2 °12

h °1

h °1

µ − g2 °

˘˙ ˝°1 ˇ  ˇ ˝°2 ˇ  ˆ ˝°3

2 1

˘  .  

(1.83)

For ˜1 = µ/g1 , the characteristic equation of the coeffcient matrix is given by

{(

) }

(˜ + 2 µ )  µ 1 − g * − ˜ 

2

− h2 ˛12  = 0, 

(

g * = g 2 g1 ,

)

which has eigenvalues ˜1 = −2 µ < 0, ˜2,3 = − µ g * − 1 ± h ˛1 . When g 2 > g1, that is, g * > 1, ˜3 < 0. For µ >

h2 g1

( g 2 − g 1 )2

= µ3 , ˜2 < 0. Therefore, all the perturbations to the

stripe patterns will disappear when the above two conditions are satisfed. iii Hexagon patterns: ˜1 = ˜2 = ˜3 =

h ± h2 + 4 ( g1 + 2g 2 ) µ 2 ( g1 + 2g 2 )

= ˜.

˜ satisfes the equation ( g1 + 2g 2 ) ˜ 2 − h ˜ − µ = 0. One of the values is positive ˜ + , and the other ˜ − is negative. Setting ˜ = ˜1 = ˜2 = ˜3 in the perturbation equation (1.82), we obtain

( )

( )

˝ ˛°1 d ˆ ˜ 0 ˆ ˛°2 dt ˆ ˙ ˛°3

ˇ ˝ p  ˆ  =ˆ q  ˆ q ˘ ˙

q p q

q ˇ ˝ ˛°1 ˆ q  ˆ ˛°2 p ˘ ˆ˙ ˛°3

ˇ  ,  ˘

(1.84)

where p = µ − ( 3g1 + 2g 2 ) ° 2 , q = h ° − 2g 2 ° 2 . The characteristic equation is given 3 by ( p − ˜ ) − 3q 2 ( p − ˜ ) + 2q 3 = 0, whose eigenvalues are ˜1 = ˜2 = p − q, ˜3 = 2q + p. Now,

˜3 = 2 q + p = 2 h ° − 3 ° 2 ( g1 + 2g 2 ) + µ = 2 h ° − 3 ( h ° + µ ) + µ = − ( h ° + µ ) < 0, for

° = °+.

˜1 = ˜2 = p − q = µ − 3g1 ˛ 2 − h ˛ = ( g1 + 2g 2 ) ˛ 2 − 2 h ˛ − 3g1 ˛ 2 = −2  ˛ 2 ( g1 − g 2 ) + h ˛  < 0, for

˛ = ˛+ ,

when g1 > g 2 . Therefore, the pattern is stable for ˜ = ˜ + , when g1 > g 2 . When g 2 > g1, we obtain the condition µ
0, g1 + g 2 > 0, and g1 + g 2

33

Introduction to Diffusive Processes

µ − g1 °12 = µ −

g1 h2

(   g 2 − g 1 )2

> 0.

We obtain ˝ ˛°1 d ˆ ˜ 0 ˆ ˛°2 dt ˆ ˙ ˛°3

ˇ ˝ p1  ˆ  = ˆ q1  ˆ q1 ˘ ˙

(

p2 q2

)

where p1 = µ − 3g1 °12 − g 2 °22 + °32 =

(

q2 = h ˜1 − 2g 2 ˜22 = −

ˇ  ,  ˘

(1.85)

( 3g1 + g2 ) g1h2 , g1 − g 2 µ− g1 + g 2 ( g1 + g2 )( g1 − g2 )2

)

p2 = µ − 3g1 °22 − g 2 °22 + °12 = −

q1 = h ˜2 − 2g 2 ˜2 ˜1 = −

q1 ˇ ˝ ˛°1 ˆ q2  ˆ ˛°2 p2 ˘ ˆ˙ ˛°3

q1

(

)

3g12 − g 22 h2 2g1 µ+ , g1 + g 2 ( g1 + g2 )( g1 − g2 )2

g1 + g 2 ˘ g1 h2   h, 2 µ − ( g1 − g2 )  ( g1 − g2 )2 

(

)

g 22 − g12 + 2g1 g 2 h2 2 g2 . µ+ g1 + g 2 ( g1 + g2 )( g1 − g2 )2

The characteristic equation is ˆˇ ˜ 2 − ( p1 + p2 + q2 ) ˜ + p1 ( p2 + q2 ) − 2q12 ˘ ( ˜ − p2 + q2 ) = 0. The relationships between the roots are given by

˜1 = p2 − q2 , ˜2 + ˜3 = p1 + p2 + q2 , ˜2 ˜3 = p1 ( p2 + q2 ) − 2 q12 . The eigenvalues are negative, if (i)˜1 < 0, (ii)˜2 + ˜3 < 0, and (iii)˜2 ˜3 > 0. From (1.86i), we get

˜1 =

2 ( g 2 − g1 ) ( 4g1 + 2g2 )( g2 − g1 ) h2 = 2 ( g2 − g1 ) ˘ µ − ( 2g1 + g2 ) h2 . µ− g1 + g 2 g1 + g 2  ( g1 + g2 )( g1 − g2 )2 ( g1 − g2 )2  

If g 2 > g1 , and µ
0, µ − 2  µ − g1 + g 2  ( g1 − g2 )   ( g1 − g2 )2  

(1.86)

34

Spatial Dynamics and Pattern Formation in Biological Populations

˙ g1 h2 ˘ ˙ ( 2g1 + g2 ) h2 ˘ > 0, g > g .  ˇ or ˇ µ − µ − 2 1 ( g1 − g2 )2  ˇˆ ( g1 − g2 )2  ˆˇ But the mixed solution gives g 2 − g1 > 0, and

˜2 = ˜3 =

µ − g1 ˜12 g1 h2 > 0,and g1 + g 2 > 0. > 0, that is, µ − g1 ˜12 = µ − g1 + g 2 ( g 2 − g 1 )2

Hence, we get the condition µ >

( 2g1 + g2 ) h2 , which is a contradiction. Hence, the mixed ( g 1 − g 2 )2

structure Turing pattern is always unstable. The following is the summary of the results: 1. The stationary solution is stable when µ < 0 = µ2; otherwise, it is unstable. g1 h2 2. Stripe pattern solution is stable for µ > = µ3 , g 2 > g1 . Otherwise, it is ( g 1 − g 2 )2 unstable. 3. Hexagon pattern solution exists when µ > 0. If g1 , g 2 can take negative values, then ˇ  h2 + the condition is µ >  −  = µ1 . The solution ˜ = ˜ , is stable when (i) + 2g 4 g ( ) 1 2   ˘

( 2g1 + g2 ) h2 = µ . For ˜ = ˜ −, the patterns are g1 > g 2 ; or when (ii) g 2 > g1, and µ < 4 ( g 2 − g 1 )2 always unstable. 4. The mixed-state solution exists when g 2 > g1, g1 + g 2 > 0, and µ > µ3 = g1 °12 . The solution is always unstable.

Mishra and Upadhyay [74] used the multiscale analysis in studying the strategies for the existence of spatial patterns in predator–prey communities generated by cross-diffusion.

1.8 Overview of the Book The book is divided into six chapters. Chapter 2 gives a detailed study of single-species, two-species, and multiple-species reaction–diffusion systems. Linear model of Kierstead and Slobodkin; nonlinear Fisher equation; and Nagumo equation are the three singlespecies RD models studied, and their analytical and numerical solutions are discussed. Six models in two-species reaction–diffusion systems in biochemistry (Belousov-Zhabotinsky reaction–diffusion systems) are presented, and their Turing instability is discussed. Ecological research on the spatial spread of traits and species was the starting point for the mathematical theory of reaction–diffusion waves, and Alan Turing’s study on morphogenesis led to the development of the theory of self-organized pattern formations. In this chapter, three ecological multispecies reaction–diffusion systems are presented and their Turing and non-Turing pattern formations are studied.

Introduction to Diffusive Processes

35

Understanding the modeling of virus dynamics of infectious diseases holds the key for designing control strategies from public health perspective. In Chapter 3, terminologies used in epidemic modeling, types of incidence rates, and next-generation operator method are discussed. Spatial and temporal dynamics of the following fve models are discussed: (i) Susceptible-Infected (SI), (ii) Susceptible-Infected-Susceptible (SIS), (iii) Susceptible-Infected-Removed (SIR), (iv) Susceptible-Infected-RemovedSusceptible (SIRS), and (v) Susceptible-Exposed-Infected-Recovered (SEIR) epidemic compartment models. Infuenza is a communicable acute respiratory disease and one of the major infectious disease threats to the human population. SI- and SEIR-type infuenza models are studied, and their spatiotemporal dynamics is discussed. Recently, the world has experienced the emergence of major devastating epidemic outbreaks and spreads of Ebola and Zika viruses. In chapters 4 and 5, the concepts introduced in Chapter 3 are extended to study the dynamic models for the epidemic outbreaks and spread of Ebola and Zika viruses. Five models for Ebola virus transmission and seven models for Zika virus transmissions are presented. Several aspects in the modeling of these viruses adopted by different researchers have been discussed. Equilibria, stability, bifurcation, sensitivity analysis and optimal control analysis for the temporal models, Turing instability, and existence of traveling wave solutions for the spatial models are discussed. Modeling strategies and design of effective optimal control problem and its analysis using Pontryagin’s maximum principle are also included. Chapter 6 gives an introduction to the modern theory of brain dynamics and in particular to the notion of neuron and its neuronal dynamics. Excitability properties of neurons and the process of generation of action potentials are described. Action potentials are the basic characteristics of the mechanism of signal propagation across the neurons. It is related to synapses and its functional mechanism. The synaptic strength can be increased or decreased, and it is important to many brain functions. Neuronal excitability, various spike patterns, and other diverse neuronal responses are described. The spiking activities of the excitable cell membrane are studied. The basic theories of biophysical models of the Hodgkin-Huxley prototype formalism are described from a mathematical point of view. No single model can include all the neurocomputational properties. Biophysical models such as spiking and bursting Morris–Lecar model, Hindmarsh–Rose neuron model, and an improved version of Hindmarsh–Rose model are presented. The Hodgkin and Huxley (HH) model, FitzHugh–Nagumo (FHN) model, Morris–Lecar model (M-L) (1981), and Hindmarsh–Rose model (H-R) were formulated to model various neuronal responses like spiking and bursting dynamics. Stability analysis and bifurcation analysis of these models are discussed, and various aspects like diffusive instabilities, multiple-scale analysis, and traveling wave solutions are studied. In the book, we have tried to include all references on the relevant topics in the chapters. It is possible that we might have missed some references. While discussing the research works of some authors, we took the liberty of deriving some conditions in alternate ways. Where ever, some minor errors were found in the works, we have communicated the same to the authors. We do not claim any originality for the same and express our gratitude to all the authors for providing an opportunity to include their works.

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Spatial Dynamics and Pattern Formation in Biological Populations

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84. Peszat, S. 1994. Large deviation estimates for stochastic evolution equations. Prob. Theory Relat. Fields. 98, 113–136. 85. Petrovskii, S. V., Malchow, H. 2001. Wave of Chaos: New mechanism of pattern formation in spatiotemporal population dynamics. Theor. Pop. Biol. 59, 157–174. 86. Petrovskii, S., Li, B. L., Malchow, H. 2004. Transition to spatiotemporal chaos can resolve the paradox of enrichment. Ecol. Compl. 1(1), 37–47. 87. Potts, J. R., Petrovskii, S. V. 2017. Fortune favours the brave: Movement responses shape demographic dynamics in strongly competing populations. J. Theor. Biol. 420, 190–199. 88. Prigogine, I., Lefever, R. 1968. Symmetry breaking instabilities in dissipative systems, II. J. Chem. Phys. 48, 1695–1700. 89. Reyes-Silveyra, J., Mikler, A. R. 2016. Modeling immune response and its effect on infectious disease outbreak dynamics. Theor. Biol. Med. Model. 13(1), 1–21. 90. Rossi, F., Vanag, V. K., Tiezzi, E., Epstein, I. R. 2010. Quaternary cross-diffusion in water-inoil microemulsions loaded with a component of the Belousov-Zhabotinsky reaction. J. Phys. Chem. B 114, 8140–8146. 91. Ruiz-Baier, R., Tian, C. 2013. Mathematical analysis and numerical simulation of pattern formation under cross-diffusion. Nonlinear Anal. Real World Appl. 14, 601–612. 92. Schnakenberg, J. 1979. Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81, 389–400. 93. Segel, L. A., Jackson, J. L. 1972. Dissipative structure: An explanation and an ecological example. J. Theor. Biol. 37, 545–559. 94. Segel, L.A. 1984. Modeling Dynamic Phenomena in Molecular and Cellular Biology. London: Cambridge University Press. 95. Sherratt, J. A., Smith, M. J. 2008. Periodic travelling waves in cyclic populations: Field studies and reaction-diffusion models. J. R. Soc. Interface 5(22), 483–505. 96. Skellam, J. G. 1951. Random dispersal in theoretical populations. Biometrika 38(1/2), 196–218. 97. Skellam, J. G. 1972. Some philosophical aspects of mathematical modelling in empirical science with special reference to ecology. Math. Models Ecol. 13(8), 13–28, Ed. Jeffers, London: Blackwell Scientifc Publication. 98. Skellam, J. G. 1973. The formulation and interpretation of mathematical models of diffusional process in population biology. In The Mathematical Theory of the Dynamic of Biological Populations, pp. 63–85. Eds. M. S. Bartlett, R. W. Hiorns. New York: Academic Press. 99. Sowers, R. 1992. Large deviations for a reaction–diffusion equation with non-Gaussian perturbation. Ann. Probab. 20, 504–537. 100. Spill, F., Guerrero, P., Alarcon, T., Maini, P. K., Byrne, H. 2015. Hybrid approaches for multiplespecies stochastic reaction–diffusion models. J. Comp. Phys. 299, 429–445. 101. Steele, J. H. 1976. Patchiness. In The Ecology of the Seas, pp. 98–115. Eds. D. H. Cushing, J. J. Walsh. Oxford, London, Edinburgh, Melbourne: Blackwell Scientifc Publications. 102. Steven, L., Jacques, Scott A. Prahl 1998. Oregon Graduate Institute, Diffusion theory, ECE532 Biomedical Optics. 103. Sturrock, M., Hellander, A., Matzavinos, A., Chaplain, M. A. 2013. Spatial stochastic modelling of the Hes1 gene regulatory network: Intrinsic noise can explain heterogeneity in embryonic stem cell differentiation. J. R. Soc. Interface 10(80), 20120988. 104. Sun, G. Q., Jin, Z., Liu, Q. 2008. Pattern formation induced by cross-diffusion in a predator– prey system. Chinese Phys. B 17(11), 3936–3941. 105. Sun, G. Q., Jin, Z., Liu, Q. X., Li, L. 2009. Spatial pattern in an epidemic system with crossdiffusion of the susceptible. J. Biol. Sys. 17(01), 141–152. 106. Takahashi, Y., Tanase-Nicola, S., ten Wolde, P. 2010. Spatiotemporal correlations can drastically change the response of a MAPK pathway. Proc. Natl. Acad. Sci. U. S. A. 107, 19820–19825. 107. Tian Lin, Z., Pedersen, M. 2010. Instability induced by cross-diffusion in reaction-diffusion systems. Nonlinear Anal. Real World Appl. 11, 1036–1045. 108. Turing, A. 1952. The chemical basis of morphogenesis. Phil. Trans. R. Soc. London. Ser. B, Biol. Sci. 237(641), 37–72.

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109. Tyson, J. J. 1989. Cyclic-AMP waves in Dictyostelium: Specifc models and general theories. In Cell to Cell Signalling, pp. 521–537. Ed. A. Goldbeter. New York: Academic Press. 110. Van Milligen, B.P., Bons, P.D., Carreras, B.A., Sanchez, R. 2005. On the applicability of Fick’s law to diffusion in inhomogeneous systems. Eur. J. Phys. 26(5), 913–926. 111. Vanag, V. K., Epstein, I. R. 2009. Cross-diffusion and pattern formation in reaction-diffusion systems. Phys. Chem. Chem. Phys. 11(6), 897–912. 112. Vergara, A., Capuano, F., Paduano, L., Sartorio, R. 2006. Lysozyme mutual diffusion in solutions crowded by poly (ethylene glycol). Macromolecules 39(13), 4500–4506. 113. Wolfenbarger, D.O. 1975. Factors Affecting Dispersal Distances of Small Organisms. Hicksville, NY: Exposition Press. 114. Xie, Z. 2012. Cross-diffusion induced Turing instability for a three species food chain model. J. Math. Anal. Appl. 388, 539–547. 115. Yang, R., Gao, H., Shi, P. 2008. Novel robust stability criteria for stochastic Hopfeld neural networks with time delays. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 39(2), 467–474. 116. Yi, F., Wei, J., Shi, J. 2009. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system. J. Diff. Eqs. 246(5), 1944–1977. 117. Yuan, C., Jiang, D., O’Regan, D., Agarwal, R.P. 2012. Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation. Comm. Nonlinear Sci. Num. Simul. 17(6), 2501–2516. 118. Zemskov, E. P., Vanag, V. K., Epstein, I. R. 2011. Amplitude equations for reaction-diffusion systems with cross diffusion. Phys. Rev. E 84, 036216. 119. Zhang, J. F., Li, W. T., Wang, Y. X. 2011. Turing patterns of a strongly coupled predator-prey system with diffusion effects. Nonlinear Anal. 74, 847–858. 120. Zhang, L., Boukas, E.K., Lam, J. 2008. Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities. IEEE Trans. Auto. Control 53(10), 2458–2464. 121. Zhang, X., Sun, G., Jin, Z. 2012. Spatial dynamics in a predator-prey model with Beddington– DeAngelis functional response. Phys. Rev. E 85, 1–14, 021924. 122. Zheng, Q., Shen, J. 2015. Pattern formation in the Fitzhugh–Nagumo model. Comput. Math. App. 70, 1082–1097.

2 Reaction–Diffusion Modeling

2.1 Introduction Reaction–diffusion modeling has gained importance in recent years. Reaction–diffusion equations are used for modeling many biological processes like birth–death processes, random movements in the populations, intracellular signaling, and metabolic processes. In reaction–diffusion models, we consider time and space as continuous. There are two fundamental approaches to the mathematical modeling of these processes [41]: a. Deterministic (mean-feld) models: Deterministic models with global dispersal are called mean-feld models. It leads to PDEs for concentrations of biochemical species or for densities of individuals. Complex local interactions are replaced by an “effective feld” generated by all other particles, and the law of mass action is used to describe the dynamics. b. Stochastic models: In these models, the individual events of reaction and diffusion are followed. When the number of individuals involved is small, stochastic effects can play an important role in the survival and spatiotemporal distribution of individuals. For example, chemical reactions occur in discrete steps at the molecular level, the processes are inherently stochastic, and the inherent “irreproducibility” in these dynamics has been demonstrated experimentally for singlecell gene expression events [110,157]. If we consider discrete time and continuous space (DTCS), then the models are governed by integro-difference equations. This case is useful for modeling populations with nonoverlapping generations, such as annual plant species or seasonal outbreak. The conceptual difference between the reaction–diffusion equation and the integro-differential equation is that the individuals take many small steps in a random-walk-like fashion between reproductive events like zooplankton in water column and one big step right after they are born like plant seeds, respectively. When the initial distribution is constant in space, both the PDEs and the integro-differential equations reduce to the Levins model [109,144], which includes space implicitly. Kot and collaborators [98] developed a modeling approach that enables us to fnd out the invasion speed of a biological invasion. This approach involves setting up an integro-differential equation which needs a dispersal kernel to be specifed [173]. Lewis et al. [112] have defned biological invasion as “the uncontrolled spread and proliferation of species to areas outside their native range” and compiled the work in his book The Mathematics behind Biological Invasions. For discrete time and discrete space, models are governed by coupled map lattices [89] and cellular automata [24]. For continuous time and discrete space (CTDS), models are governed by coupled-patch models and 41

42

Spatial Dynamics and Pattern Formation in Biological Populations

meta-population models. It can be used for modeling a number of spatially isolated populations, such as populations on islands. Meta-population (a population of populations, with links between them such as a collection of cities and towns connected by a transportation network) models are often systems of ODEs of high dimensions. A meta-population may be divided into patches, with each patch corresponding to a separate location. For example, spatial differences were created by making the prey birth rate in patch 2 greater than that in patch 1. Prey moves symmetrically from one patch to other. Meta-population is neutrally stable when birth rates of prey are equal. When signifcant difference in birth rates is created, oscillations in prey abundance in the two patches become increasingly less correlated. This is associated with per capita prey immigration into a patch becoming increasingly temporally density-dependent. The density dependence arises as the number of immigrants into a patch is weakly correlated with the number of residents in the patch. Meta-population modeling plays an important role in landscape ecology and conservation biology [73]. Cantrell and Cosner [19] in their book, Spatial Ecology via Reaction-Diffusion Equations, gave an excellent review of the works on Spatial Ecology via Reaction-Diffusion Equations. Skellam [187] in his remarkable paper, Random Dispersal in Theoretical Populations, laid the foundations for the study of spatial ecology. His contributions are the following: (i)  Described the movements of the individual members of some theoretical biological species as random walks and connected them with the diffusion equation describing the dispersal of the species. (ii) Introduced reaction–diffusion equations into theoretical ecology to describe the diffusive dispersal of the population with population dynamics. (iii) Modeled the population density of a species in a bounded habitat using the reaction– diffusion equations. (iv) Employed both the linear Malthusian and logistic population growth terms. One of the important conclusions of Skellam’s work is that “if an isolated terrestrial habitat is less than a certain critical size, the population cannot survive. If the habitat is slightly greater than the critical size, the surface which expresses the density at all points is roughly dome-shaped, and for very large habitats this surface has the form of a plateau”. After his work, tremendous advances were made in the mathematical studies of the reaction–diffusion theory and its application to the study of populations and communities of populations in bounded habitats. In their remarkable book, Cantrell and Cosner [19] dealt with the mathematical development of reaction–diffusion theory and its applications to ecology. Skellam [187] derived the following general reaction–diffusion equation: ut = d° 2 u + c1 ( x , y )u − c2 ( x , y )u2 , which was a formidable equation to solve in 1951. Extensive use of the reaction–diffusion theory in ecology has been made to study ecological population invasions and Turing pattern formations. The essence of reaction–diffusion models is the following: (i) The models are spatially explicit and can incorporate most of the essential parameters of the ecological system like growth rates, carrying capacities, and dispersal rates, and (ii) studies with these models can lead to global conclusions about extinction, persistence, and coexistence of populations. In the case of biological systems, diffusion is the central issue in molecular motors and intracellular transport [27]. Reaction–diffusion equations are at the core of morphogenesis, pattern formation, fractal colony growth of biological species, etc. Reaction–diffusion models have been proposed to describe developmental processes such as skin pigmentation

43

Reaction–Diffusion Modeling

patterning [11,94], hair follicle patterning [186], and skeletal development in limbs  [136]. Some synthetic multicellular systems have been programmed de novo to generate simplifed patterns using quorum sensing mechanisms [12,116], envisioning future applications in tissue engineering and developmental processes [95,188]. Tomography studies of microemulsions have revealed three-dimensional Turing patterns [10] and selection criteria for patterns [124].

2.2 Reaction–Diffusion Equations If a particle or an individual reacts or interacts according to some rate law F(˜ ) and at the same time undergoes diffusion, then the diffusion equation can be combined with the rate equation (˜°/˜ t ) = F(° ). The result is the reaction–diffusion (RD) equation:

˜t = D˜xx + F(˜ ),

(2.1)

which provides a theoretical framework for spatiotemporal dynamics in biological, chemical, ecological, epidemiological, physical, neural systems and material sciences. We refer to equation (2.1) as the standard reaction–diffusion equation or simply as the reaction–diffusion equation. This equation preserves positivity, if the rate law F satisfes F (˜1 , , ˜i−1 , , ˜n ) ˝ 0, where the densities satisfy the conditions ˜i ° 0,  i = 1, , n. To derive the RD equation (2.1), there are two fundamental approaches: (i) a phenomenological approach based on the law of conservation and (ii) a mesoscopic approach based on a description of the underlying random motion. The mesoscopic approach is based on the idea that one can introduce mean-feld equations for the particle density involving a detailed description of the movement of particles on the microscopic level. At the same time, random fuctuations around the mean behavior can be neglected due to larger number of individual particles [129]. 2.2.1 Derivation of Reaction-Diffusion Equation Let S be an arbitrary surface enclosing a time-independent volume V. The law of conservation for the particle density [such as atoms, molecules, cells, bacteria, and chemicals] states that the rate of change of the amount of particles in V is due to the fow of particles across the surface S plus the net production of particles in the volume V. The law is given by

° ˛ ( x ,  t) dV = − J ˆ dS + F(˛ , ,x t ) dV , °t

˜

˜

V

˜

S

(2.2)

V

where ˜ ( x ,  t) represents the density of particles, J the particle fux, and F the net rate of production of ˜ . The application of the divergence theorem,

˜ J ° dS = ˜ ˝ ° J  dV, to (2.2) gives S

˘ °˛



˜    ° t + ˙ ˆ J − F  dV = 0. V

V

(2.3)

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Spatial Dynamics and Pattern Formation in Biological Populations

Since the integration volume is arbitrary, we obtain the conservation equation, or continuity equation, for ˜ as

˜t = −˝.J + F(˜ ).

(2.4)

The continuity equation (2.4) needs to be closed via a constitutive equation for the fux J. If  the transport corresponds to classical diffusion, then the constitutive equation is given by the Fick’s frst law, J = − D˝˜ . Substituting in (2.4), we obtain

˜t = ˛ ˝ (D˛˜ ) + F(˜ ).

(2.5)

If D is constant, equation (2.5) reduces to (2.1) in one-dimensional case. In some models like insect populations, bacterial chemotaxis, or for the animal grouping due to social aggregation, the diffusion coeffcient can be an explicit function of the particle density ˜ or a function of other chemicals. Since F is also in general a nonlinear function of ˜ , one may call equation (2.5) as the general nonlinear reaction–diffusion equation for single species. For n species, in the one-dimensional case, the RD equation can be written as n

°˛i ° ˙ °˛i ˘ ° ˙ °˛ j ˘ + Fi (˛ ), i = 1, , .n = Dij ˇˆ Dii  + °t ° x °x ° x ˆˇ ° x  j ,  j i

˜

(2.6)

The diagonal elements Dii of D are called the “main-term” diffusion coeffcients, and the off-diagonal elements are called the “cross-term” diffusion coeffcients or cross-diffusion terms. The cross-diffusion term links the gradient of species j to the fux of the species i. If the cross-diffusion terms are positive, Dij > 0, then the fux of species i is directed toward decreasing values of the concentration of species j, whereas if Dij < 0, it implies that the fux is directed toward increasing values of the concentration of species j. If the concentration (population density) of the species i goes to zero, then the cross-diffusion terms Dij become zero since there can be no fux of species i, if ˜i = 0. The term cross-diffusion implies the population fuxes of one species due to the presence of other, whereas the selfdiffusion implies the movement of individuals from a higher to lower region of concentration. Gambino et al. [63] discussed the pattern formations due to cross-diffusion for the Lotka–Volterra model. The effects of cross-diffusion on RD-type models for pattern formation have been studied by many authors [62,63,179,211,219]. If the cross-diffusion terms are negligible, that is, the diffusion matrix → diagonal matrix and the positive diffusion coeffcients Di do not depend upon ˜ , then the general n-variable RD system becomes

˜°i ˜ 2 °i = Di + Fi (° ), i = 1, , n. ˜t ˜ x2

(2.7)

2.3 Hyperbolic Reaction–Diffusion Equations Mendez et al. [129] explained that the origin of the unphysical behavior of the solutions of the reaction–diffusion equation can be understood from three different points of views of mathematical, macroscopic or phenomenological, and mesoscopic. From the mathematical

45

Reaction–Diffusion Modeling

point of view, the origin of the infnitely fast-spreading local disturbances can be traced to the parabolic character of the equation. This issue can be addressed by adding a small term ˜ (° 2 ˛/° t 2 ), ˜ small, to the reaction–diffusion equation (2.1) to make it a hyperbolic (HRDE) equation:

˜t + °˜tt = D˜xx + F(˜ ).

(2.8)

This type of equation is encountered in many areas such as nonlinear waves, nucleation theory, and phase-feld models of phase transitions, where it is known as the damped nonlinear Klein–Gordon equation [66,177]. In the limit as ˜ ° 0, equation (2.8) becomes the RD equation. Zemskov and Horsthemke [217] studied the hyperbolic RD systems and derived the criteria for diffusion-driven instabilities, Turing instability, and wave instability. They found that the wave instability requires only two species in HRDEs, in contrast to standard parabolic RDEs where at least three species are required. The critical values for the Turing instability are independent of inertial time and coincide with the critical values for standard parabolic RD systems. The critical values for the wave instability depend on the inertial time, and it cannot occur for arbitrarily small inertial times. A minimum distance from the parabolic limit case is required. From a macroscopic or phenomenological point of view, the RD equation follows from the continuity equation:

˜t = − J x + F(˜ ),

(2.9)

and the use of Fick’s frst law as the constitutive equation. Fick’s frst law implies that the fux adjusts instantaneously to the gradient of the density. This is physically unrealistic, and it gives rise to the pathological feature of infnitely fast spreading of local disturbances in the diffusion equation. It was argued that the fux adjusts to the gradient with a small but nonzero relaxation time ˜ (see Joseph and Preziosi [87]). It was suggested that the Fick’s frst law should be replaced by the Cattaneo equation [22,23]

˜ Jt + J = − D°x .

(2.10)

The hyperbolic system (2.9) and (2.10) is a reaction–Cattaneo (RC) system. This RC system can also be obtained from extended irreversible thermodynamics [88]. Al-Ghoul [4], Al-Ghoul and Eu [5] have derived such systems from generalized hydrodynamic theory. Differentiating (2.9) with respect to t and (2.10) with respect to x and eliminating the mixed derivative, we obtain

˜°tt + ˘1 − ˜ Fˇ (° ) °t = D°xx + F(° ).

(2.11)

This is called the reaction–telegraph (RT) equation or hyperbolic reaction–diffusion equation for reaction–telegraph equations. Dunbar [35] derived this equation as the kinetic equation for a branching random evolution. Fedotov [45–49] studied the traveling wave front solution for the RT equation. To ensure the dissipative character of the RT equation, the damping coeffcient [ 1 − ˜ Fˆ(° )] must be positive, that is, F ˛(˜ ) < (1/° ) for all ˜ . In the study of propagation of an electric signal in a cable of transmission line and wave phenomena, we use the 1D hyperbolic telegraph equation:

˜tt + 2°˜t + ˛ 2˜ = ˜xx + F( x , t), a ˆ x ˆ b , t ˇ 0,

46

Spatial Dynamics and Pattern Formation in Biological Populations

with initial conditions ˜ ( x , 0) = f1 ( x), ˜t ( x , 0) = f2 ( x) and boundary conditions ˜ ( a, t) = g1 (t), ˜ (b , t) = g 2 (t), t ˛ 0, where ˜ ,  ° are known real constants. For ˜ > 0, ° = 0, it represents a damped wave equation, and for ˜ , ° > 0, it is called telegraph equation. Dehghan and Shorki [32] proposed a numerical method for solving the telegraph equation using collocation points and approximated the solution using thin plates spline radial basis functions. Mohebbi and Dehghan [137] solved the problem using a higher-order compact fnite difference approximation of fourth order in space and collocation method in time direction. Dehghan and Lakestani [31] used the Chebyshev cardinal functions, whereas Saadatmandi and Dehghan [180] used the Chebyshev-Tau method for fnding the approximate solution. Lakestani and Saray [101] used interpolating scaling function technique to solve the problem. Dehghan and Ghesmati [30] solved the problem using dual reciprocity boundary integral equation method. Numerical schemes have been developed for solving telegraph equation using techniques such as radial basis function method [44], quadratic B-spline collocation method [34], differential quadrature method [84], and modifed cubic B-spline basis functions [132]. For a particular case of the above problem ˜tt + 12˜t + 4˜ = ˜xx + F( x , t), 0 ˜ x ˜ 1, t ° 0, with ˜ ( x, 0 ) = sin x , ˜t ( x , 0) = 0 = ˜ (0, t), ˜ (1, t) = cos t sin 1, and F( x , t) = −12 sin t sin x + 4 cos t  sin x, the exact solution is given by ˜ ( x , t ) = cos t sin x [30] Dehghan and Shorki [32], and Saadatmandi and Dehghan [180] considered the problem ˜tt + ˜t + ˜ = ˜xx + F( x , t), 0 ˜ x ˜ 1, t ° 0, with ˜ ( x , 0) = ˜t ( x , 0) = 0, ˜ (t , 0) = ˜ (1, t) = 0 and F( x ,  t) = 2 − 2t + t 2 x − x 2 e −t + 2t 2 e −t and found the exact solution as ˜ ( x , t) = x − x 2 t 2 e − t . From the mesoscopic viewpoint, the pathology of reaction–diffusion equation can be traced to the lack of inertia of the Brownian particles; their direction of motion in successive time intervals is uncorrelated, which has two consequences [129]: (i) The particles move with infnite velocity, and (ii) the motion of the dispersing individuals is unpredictable even on the smallest time scales. It is therefore desirable to adopt a model for dispersion that leads to more predictable motion with fnite speed at smaller time scales and approaches diffusive motion on longer time scales. The natural choice is a persistent random walk which is also called correlated random walk. This approach shed light on the foundational problems of the RC system (2.9)–(2.10) and the reaction–telegraph equation (2.11).

(

)(

)

(

)

2.4 Single-Species Reaction–Diffusion Models The simplest single-species population model is a contact process (that is, there exists a nontrivial stationary distribution with positive density of occupied sites) which describes growth when the birth rate is density-dependent but the death rate is density-independent. If deaths also depend on the density, then the corresponding model is known as the annihilating branching process. Individuals die at a rate equal to the fraction of occupied neighbors, and the birth process is the same as in the contact process. Despite local dispersal, there is no spatial correlation build-up at equilibrium [143,144]. In the case, when only one morphogen is present and the domain is convex, Casten and Holland [20], and Ni [146] have shown that the only stable patterns are spatially constant. Therefore, all the nontrivial patterns (nonhomogeneous patterns for single-species case or scalar case) are unstable and therefore are diffcult to fnd numerically. In dynamical terms, the criterion for the scalar case states that if the associated energy of the system is bistable (has two minima) but has a unique global minimum, then spots will form corresponding to the existence of

47

Reaction–Diffusion Modeling

homoclinic solutions and a stripe pattern will form for heteroclinic solutions (a transition layer) [124]. Holmes et al. [79] studied a general nonlinear RD equation of the form: ut = f (u) + Duxx , for 0 < x < L,  t > 0. Some of the applications of RD models are the following: (i) study of critical patch size [91], (ii) spread of advantageous genes [57], and (iii) pattern formation [193]. A simple model that relates the rate of horizontal diffusion to the rate of algal growth and calculates a critical patch size below which the effects of diffusion override any growth effects was given by Skellam [187] and Kierstead and Slobodkin [91]. 2.4.1 Model 1: Linear Model of Kierstead and Slobodkin Kierstead and Slobodkin [91] studied a phytoplankton population which is assumed to be increasing logarithmically in a mass of water surrounded by water which is unsuitable for the survival of the population. In the following, we present the model and its analysis as given by Kierstead and Slobodkin. They have shown that there is a minimum critical size for the water mass below which no increase in concentration of phytoplankton can occur. The population growth of phytoplankton in diffusing water mass is governed by the equation: N t = (K − ˜ )N,

(2.12)

where K is the growth rate per unit population, N is the total number of organisms in the water mass, and ˜ N is the rate of loss due to diffusion. Population increases if K > ˜ . Consider a one-dimensional case of a long narrow body of water of length L (0 ˜ x ˜ L) bounded on two sides by natural boundary. Outside this region, the water is unsuitable. Taking into account a linear growth of population, the concentration C is governed by the equation: Ct = DCxx + KC,

(2.13)

under the conditions C( x , 0) = g( x),  C(0, t) = 0 = C(L, t). The variable transformation f = Ce − Kt reduces the problem to ft = Df xx , f ( x , 0) = g( x), f (0, t) = 0 = f (L, t).

(2.14)

The variable separable solution of this problem is given by 

f =





˜B sin ˝ˆ˙ nL˛x ˇ˘ exp − n ˛L Dt  , B n

2

2

2

n

 0.

(2.15)

n=1

L

Bn =

2 ˝ n˛x ˇ   g( x)sin ˆ dx, ˙ L ˘ L

˜ 0



C=

˝ n2 ˛ 2 D ˇ  ˝ n˛x ˇ t. Bn sin ˆ exp ˆ K −  ˙ L ˘ L2 ˘  ˙ n=1

˜

(2.16)

48

Spatial Dynamics and Pattern Formation in Biological Populations

If the initial distribution is symmetric with respect to x = L 2, then even numbered terms vanish. The values of N and ˜ depend only on odd numbered terms. The population will be able to maintain itself against diffusion if there is at least one term in the series for which K − n2 ˝ 2 D L2 ˙ 0. If B1 ˜ 0, a necessary condition is K ˜ ° 2 D L2 . Even if B1 = 0, there would be at least one value Bm which is positive. Therefore, from the values of K and D, the critical length Lc is given by Lc = ° D K . Population will increase if L > Lc and decreases if L < Lc . The total number of organisms N is given by

(

)

L

°

N = A C dx = 0

2AL ˝

ˇ

 

˜ 2nB + 1 exp ˘ K − (2n + L1) ˝ D  t . 2n+1

2

2

2

(2.17)

n= 0

The net number of organisms diffusing in the forward direction in unit time is n( x , t) = −AD(˜ C/˜ x), which can be computed using (2.16). The number lost from the system per unit time by diffusion through the boundaries is n(L, t) − n(0, t). The leakage ˜ is defned by

˜ = [ n(L, t) − n(0, t)] N .

(2.18)

Therefore, the growth of the population is governed by (2.12) where ˜ is defned by (2.18). After a suitable lapse of time, the frst terms in the series in the numerator and denominator dominate. As an approximation, we get

˜°

˛2D L2 , if t  2 . 2 L 8˛ D

(2.19)

The authors have also shown the generality of these results by considering diffusion through a cylinder of radius R and depth h assuming cylindrical symmetry, and the water mass outside the cylinder is unsuitable. The equation corresponding to (2.13) is D ˜ ˛ ˜C ˆ ˙r ˘ + KC. r ˜r ˝ ˜r ˇ

Ct =

(2.20)

The variable transformation f = Ce − Kt reduces the problem to 1 ˆ ˛ ft = D ˙ frr + fr ˘ , f (r , 0) = g(t),  f (R , t) = 0. ˝ r ˇ

(2.21)

Further variable transformation reduces the differential equation to Bessel’s differential equation of order zero. The solution is obtained as 

C=

˝

ˇ 

˜ B   J ˝ˆ˙ °Rr ˘ˇ  exp ˆ˙ K − °RD ˘ t  , n

0

n

2 n

2

n=1

where J 0 is Bessel’s function of order zero and ˜ n are the roots of Bessel’s function of order zero, J 0 ( ˜ n ) = 0. Applying the initial condition, we obtain Bn =

R

 ° nr   dr. 2 ( rg r ) J 0   R  ˘ ˆRJ ˇ 1 ( °n ) 0 2

˜

49

Reaction–Diffusion Modeling

(

)

If B1 ˜ 0, a necessary condition for the maintenance of the population is ˙ˆ K − ˜12 D R 2 ˇ˘  0, or K ˛ 2.40482 D R 2 ˝˙. The critical radius is Rc ˜ 2.4048 D K , and

(

)

˜°

2.40482 D , if R2

t

R2 . 25D

The population will increase if R > Rc and decreases if R < Rc. The above model is often referred to as KISS model. A generalization of the above model is the nonlinear model: 

ut = r °˛ f (u)˝˙ + Duxx , 0 ˇ x ˇ L, t > 0, f (0) > 0,

(2.22)

where u( x ,  t) is the population density, r is the growth rate, D is the diffusion coeffcient, and ˜ > 0 is a critical exponent parameter. When f (u) = u, ˜ determines whether the model is linear (˜ = 1) or nonlinear (˜ ° 2). The boundary conditions are taken as u(0, t) = u(L,  t) = 0, and the initial condition is taken as u( x , 0) = u0 ( x). The critical patch size corresponds to the borderline between species extinction and coexistence. If u = 0 is stable, we have total extinction of the species population. But, if u = 0 is unstable (nontrivial case), we have a state that corresponds to the persistence or survival of the species. Owolabi and Patidar [155,156] studied the model for f (u) = u, ˜ = 2 and u( x , 0) = ˙ˆ1 {2 cosh(˜ x)} ˇ˘ , the solutions of which have exponential decay e −˜ x as x ˜ °. As D decreases, the species population oscillates in phase, but a stable steady-state solution was obtained for D > 1. For ˜  2; as t ˜ °, blow-up phenomena occurs which corresponds to the total extinction of the species. Consider the linear case, ˜ = 1. Linearize f (u) about u = 0 , in (2.22) to obtain ut = rf °(0)u + Duxx , u(0, t) = u(L,  t) = 0, u( x , 0) = u0 ( x),  f (0) = 0, f °(0) > 0. The solution is given by u( x ,  t) =

˜

2   ˇ ˆk    ˇ ˆkx  ° k exp  rf ˝(0) − D    t  sin  , ˘ L    ˘ L  k =1  



where the coeffcients ˜ k are determined using the initial condition u( x , 0) = u0 ( x). The state u0 ( x) = 0 loses, its stability at the point Lc = ˝ D ˇˆ rf ˙ ( 0 ) ˘ , [173]. If L <   Lc , u( x ,  t) ˜ 0 as t ˜ °. The population is wiped out from its initial condition, and no nontrivial steady state develops [91]. 2.4.1.1 KISS Model in Two Dimensions Consider a single-species population u( x , y , t ) inhabiting in an Lx × Ly rectangular domain. The population’s growth is described by the equation:

(

)

ut = D uxx + uyy + ru, 0 < x < Lx , 0 < y < Ly .

(2.23)

50

Spatial Dynamics and Pattern Formation in Biological Populations

Assume that the environment outside the domain is hostile. The boundary condition is of Dirichlet type, u(r ,  t) = 0, for all r ∈Ω and all t ≥ 0. The solution obtained by the separable variable method is given by (see Crank [26]) ∞

u( x ,  y ,  t) =

∞

∑∑C

nm

n =1 m =1

 πmy   πnx  e ( r − µn − µm )t sin  sin   ,  Lx   Ly 

(2.24)

2

2  πm   πn  where µn = D   , µm = D   and Cnm are the coefficients of the two-dimensional  Lx   Ly  Fourier sine series given by

Cnm

2 =  Lx 

Lx

∫ 0

  πnx    2  dx u0 ( x)sin   Lx    Ly 

Ly

∫ 0

 πny    u0 ( y )sin   dy .  Ly   

All the exponents in (2.24) are negative, and all terms therefore decay with time. Lewis et  al. [112] showed that the effect of the domain geometry obtained in the simple KISS model remains effectively the same in the more realistic case of invasion in an open space. For an invasive species in an unbounded 2D space, equation (2.23) is defined in the infinite domain −∞ < x ,  y < ∞. Consider that the initial population distribution is described by the 2D Gaussian distribution as u( x ,  y , 0) =

 x2 y2  K exp  − − , 2 4σ y 2  4πσ xσ y  4σ x

(2.25)

where K is the total initial population size, and σ x , σ y are the characteristic dimensions of the initial invaded domain in the x and y directions, respectively. The maximum initial population density is K 4πσ xσ y . The solution of the 2D KISS model (2.23) with the initial condition (2.25) is given by

(

u( x ,  y ,  t) =

4π

)

  y2 x2  exp  − − + rt  . 2 2 4 σ y + Dt σ x2 + Dt + σ y2 + Dt  4 σ x + Dt 

(

K

) (

)

(

)

(

)

2.4.2 Model 2: Nonlinear Fisher Equation Fisher [57] proposed the nonlinear diffusion equation ut = ru(1 − u) + Duxx , a ≤ x ≤ b , t > 0,

(2.26)

as a model for the propagation of a mutant gene with advantageous selection intensity r. The same equation occurs in flame propagation, in the branching Brownian motion process, in neurophysiology, in autocatalytic chemical reaction, and in nuclear reactor theory [18]. Fisher’s equation also represents the evolution of populations due to two competing physical processes, diffusion and nonlinear local multiplication. The initial and boundary conditions are u( x , 0) = u0 ( x),  a ≤ x ≤ b ;  u(a,  t) = f (t), u(b ,  t) = g(t), t ∈[0, T ].

51

Reaction–Diffusion Modeling

Setting t * = rt ,  x * = ( r D )

1/2

x, and dropping the asterisk, we obtain ut = uxx + u(1 − u).

(2.27)

A traveling wave solution u( x ,  t) = u( x − ct = z), propagating with speed c, obeys the equation uzz + cuz + u(1 − u) = 0.

(2.28)

Phase plane analysis can be used to characterize the solution of this equation, and for most applications, u( z) is restricted to be positive and bounded. The boundary conditions for the traveling wave solution are usually taken as u( z → −∞) → 1, u( z → ∞) → 0. The speed of the waves is to be found as the solution of an eigenvalue problem. Kolmogorov et al. [93] showed that if the initial datum satisfies 0 ≤ u( x, 0) ≤ 1,  u( x, 0) = 1 for x < a, and u( x , 0) = 0 for x > b , then the unique solution of the nondimensionalized form approaches a traveling wave of speed c = 2. It was shown that (2.27) has infinite number of traveling wave solutions for which 0 ≤ u( x, 0) ≤ 1, and wave speeds c ≥ cmin = 2. The velocity of Fisher’s traveling waves may depend on the wave amplitude. It was shown that (2.27) also admits a one-parameter family of standing wave solutions. McKean [128] showed that any wave speed c > 2 is stable if the initial datum has the right behavior at the tails. Modified form of Fisher’s equation ut = ru(1 − u) + uxx ,

(2.29)

was introduced by Li et al. [114] for which the nonlinear reactive term is made arbitrarily large than the diffusion term with nonlocal initial and boundary conditions. The reaction coefficient is generally chosen as r ≥ 1. 2.4.2.1 Spatial Steady-State Solution Kot [97] obtained the solution of the steady-state system corresponding to (2.26) in 0 < x < L, with homogeneous boundary conditions. The problem is given by ru(1 − u) + Du′′ = 0, u(0) = 0,  u(L) = 0.

(2.30)

We can write the equation as a system of first-order equations as u′ = v , v′ = −

r u(1 − u). D

(2.31)

This system has two-phase plane equilibria, E0 (0, 0) and E1 (1, 0). Linearization of the system (2.31) about (0, 0) gives  u′   0  v′  =  −r/D   

1  u  . 0   v 

This linear system has purely imaginary eigenvalues ±i r D. The linearized system about (1, 0) has eigenvalues ± r D. Therefore, E0 (0, 0) is a center for the linearized system and

52

Spatial Dynamics and Pattern Formation in Biological Populations

E1 (1, 0) is a saddle point for both linearized and the original nonlinear systems. But, linearization is unrealistic for nonhyperbolic fixed points. However, we can consider the solution of (2.31) in an alternate way. Multiplying it with u′, we get Du′′u′ + ru(1 − u)u′ = 0.

(2.32)

Integrating, we get  u2 u3  D (u′)2 +   r  −  = c * , or 2 3  2

v2 r  u2 u3  −  = c , where v = u′. +   2 D 2 3

(2.33)

The level curves of this equation are orbits in the phase plane. The phase portrait is symmetric in v = u′. Thus, origin is a center. Each orbit in this phase portrait satisfies (2.31). However, we are interested in a few special orbits. Equation (2.33) can be rewritten as v2 r r +   F(u) =   F( µ ), 2 D D

(2.34)

 u2 u3  where F(u) =  −  , u = µ when v = u′ = 0 at x = L 2. Thus, 3  2 v=

2r   du =+ {F(µ) − F(u)} , 0 < x < L 2 , dx D =−

2r   {F(µ) − F(u)} , L 2  < x < L.  D

(2.35)

µ

L2 du = dx. F( µ ) − F(u) 0 0 L D 0 −du Similarly, for the second half of the orbit, we obtain   = dx. 2r µ F( µ ) − F(u) L2 2D µ du Both the integrals give L =   . r F( µ ) − F(u) 0 2D 1 µdz . Substituting z ≡ u µ, we obtain L = r 0 F ( µ ) − F ( µ z) 2D 1 µdz 2D 1 µdz   = lim   Thus, Lc = lim µ→0 r F ( µ ) − F ( µ z) µ → 0 r 0 0 1 µ 1 − z 2 − 1 − z 3  µ  2 3 

Integrating over the first half of the orbit, we get

D   2r

∫

∫

∫

∫

∫

∫

∫

∫

=2

D r

∫

1

0

1 1− z

2

dz = 2

(

)

(

)

D −1 1 D sin z|0 = π   . r r

It may be evaluated as an elliptic integral. It is an increasing function of µ for 0 ≤ µ < 1, concave up for 0 ≤ µ < 1, lim L( µ ) → ∞ and lim L( µ ) = Lc = π D r . This formula is referred µ ↑1

µ ↓0

to as a time-map since it maps orbits to the time space it takes to traverse those orbits [97].

53

Reaction–Diffusion Modeling

King and McCabe [92] examined the Cauchy problem for a nonlinear Fisher–Kolmogorov– Petrovsky–Piskunov (FKPP) reaction–diffusion equation with fast diffusion:

(

)

ut = ˝ ˙ u−n˝u + u(1 − u), x ˘ N , t > 0, 0 < n < max ( 1, 2 N ) .

(2.36)

The authors considered the radially symmetric case of (2.36) as ut = r 1− N

˜ ˛ N −1 −n ˜ u ˆ ˙r u ˘ + u(1 − u), r > 0, t > 0, ˜r ˝ ˜r ˇ

(2.37)

with the initial and boundary conditions u(r , 0) = I (r ), r N −1u−n where

˜

˛

0

˜u = 0 at   r = 0; u(r , t) ˛ 0 as r ˛ ˝ ˜r

r N −1I (r ) dr is fnite.

The authors derived the large time asymptotic solution as u(r , t) =

1n Q , Q = ˇˆ2 ( 2 − nN ) ˘ e t , as t  , r = O e nt 2 . (nr )2 n + Q

(

)

(2.38)

( )

The large time behavior of u( x , t) was also derived in the region r = O t 1/2 . A wavefront (defned for n < 0 only in the limit n ˜ 0+) moves whose speed and wavelength grow exponentially with t. This implies that the problem does not have permanent-form traveling wave solutions. The authors have estimated the speed and position of the wavefront for small values of the fast-diffusion parameter n with t = O ( 1 n) and found an accelerating wave. Finally, the authors discussed the behavior of the solutions of the general quasilinear reaction–diffusion equation: ut =

˜ ˛ ˜uˆ n ˙ D(u) ˘ˇ + f (u), x  , t > 0, ˜x ˝ ˜x

(2.39)

in which D(u) ~ um ,  f (u) ~ u p as u ˜ 0+, f (u) > 0, 0 < u < 1,  f (1) = 0, for initial data I ( x) satisfying 0 < I < 1, °x and I = 0 for x > 0. 2.4.2.2 Some Analytical Solutions Ablowitz and Zepetella [3] obtained the frst known explicit solution for the traveling wave equation by substituting u = u( z) = u( x − ct). Equation (2.27) reduces to u˜˜ + cu˜ + u(1 − u) = 0, c ˙ 2.

(2.40)

The authors observed that if the equation has Painlevé property (equations whose solutions as functions of complex variable have only poles as movable singularities), then the equation may be explicitly solvable. They looked for the Laurent series solution of the form: u( z) =

6 a−1 + + a0 + a1z +  z2 z

54

Spatial Dynamics and Pattern Formation in Biological Populations

(

)

They have shown that for c = 5 6 ≈ 2.04, the equation is of Painlevé type. They have derived a one-parameter family of solutions as

(

u( z) = 1 − re Z

(

with u( x , 0) = 1 − re x/

6

)

−2

6

)

−2

, r < 0,

(2.41)

, u(−∞) = 1, u(+∞) = 0. The wave starts at u = 1 at z = −∞ and

decreases monotonically to zero as z → +∞. For finite real z, all solutions blow up for r > 0. The solution can also be written as (Wang [208]) u=

(

)

 x − 5 √6 t b  2 1 *  , x* =  − tanh x + 1 +  , b a constant.   4 2 √6 2 

( )

(2.42)

Rinzel and Keller [174] gave a method for finding traveling wave solutions for a bistable medium. The right-hand side of Equation (2.27) is replaced by a piecewise linear approximation. The roots u = 0 and u = 1 are connected by a continuous function with its unique maxima located between the roots. Equation (2.27) is replaced by ut = uxx + u, for u ≤ ( 1 2 ) ,

(2.43)

ut = uxx + (1 − u), for u ≥ ( 1 2 ) .

(2.44)

The solution is continuous at u = 1 2. The authors found that the velocity of the traveling waves is not uniquely determined but lies in (2, ∞),  cmin = 2. For c = 5 √ 6, the solution was obtained as u(ξ ) =  2 e −2θξ − e −3θξ  2 , u ≤ 1 2 ,

(2.45)

u(ξ ) =  2 − eθξ  2 , u ≥ 1 2 , θ = 1 √ 6 .

(2.46)

Wang [208] introduced the transformation u = w 2 α . Observing that ( dw dt ) = aw ( 1 − w ), where a is an undermined parameter and w( x ,  t) = w( x − ct) ≡ w( z), the author obtained the exact solution of the generalized Fisher’s equation:

(

)

ut = uxx + u 1 − uα .

(2.47)

The exact solution is given by 1 1 u =  tanh( x * ) +  2 2  

2/α

, x* = −

α α+4  b  x− t + , 2α + 4  2 2 2α + 4 

(2.48)

where b is a constant. For α = 1, this solution reduces to the solution given in (2.42). Wazwaz and Gorguis [209] studied Fisher’s equation, the general Fisher’s equation, and a nonlinear diffusion equation of the Fisher type by the Adomian decomposition method for some particular cases of initial conditions. Solution was obtained in a series form,

55

Reaction–Diffusion Modeling

which was then used to construct closed-form solutions. The authors mention that the method reduces the amount of computational work. For the problem

(

ut = uxx + 6u(1 − u), u( x , 0) = 1 + e x

)

−2

,

(2.49)

the authors obtained the solution as

(

u( x , t) = 1 + e x− 5t

)

−2

.

(2.50)

The method constructs and uses the Adomian polynomials. For the general problem

(

ut = uxx + ˜ u(1 − u), u( x , 0) = 1 + e(

)

˜ 6  x

)

−2

,

(2.51)

the solution was obtained as ˛ {( u( x , t) = ˙ 1 + e ˝

)

 6  x − (5 6)t

−2

}ˆ . ˘ˇ

(2.52)

For ˜ = 6, solution (2.52) reduces to solution (2.50). For the generalized Fisher’s equation (2.47), the authors obtained the solution using the Adomian polynomials as 1 u( x , t) = ˆ˝ tanh( x * ) + ˙2

1ˇ 2 ˘

2/˜

, x* = −

˜ ˜+4  b  x− t + ,  2 2˜ + 4  2˜ + 4  2

(2.53)

where b is a constant which is the same as given by Wang [208]. Hilhorst and Kim [76] have shown that traveling waves of Fisher’s equation (2.26) (with r = 1) with wave speed c > 0 converge to the inviscid traveling wave with speed c > 0 as the diffusion vanishes. The authors have illustrated through a diagram the relation between the diffusive and inviscid traveling waves. 2.4.3 Model 3: Nagumo Equation The general Fisher’s nonlinear reaction–diffusion equation is given by ut = uxx + u(1 − u)(u − ˜ ), 0 < ˜ < 1.

(2.54)

This equation is also called the Nagumo reaction–diffusion equation [141]. For ˜ = −1, the Nagumo equation reduces to the Newell–Whitehead equation. The equation has three constant solutions u = 0, 1 and ˜ . Linear stability analysis shows that (i) when −1 ˛ ˜ < 0, u = 1 and ˜ are stable while u = 0 is unstable and (ii) when 0 < ˜ < 1, u = 1 and 0 are stable while u = ˜ is unstable. Similar equation arises in the study of transmission of nerve impulses, circuit theory, population genetics, etc. [8,58,90,141]. Kawahara and Tanaka [90] gave exact solution to the equation using nonclassical symmetry reduction approach. Nucci and Clarkson [150] gave exact solutions to the equation using the Jacobi elliptic functions. The conditions used for fnding the solution are ux (0, t) = 0 = ux (L, t), u( x , 0) = ˜   ˛[0,1].

(2.55)

56

Spatial Dynamics and Pattern Formation in Biological Populations

Nagumo et al. [141] modeled an active pulse transmission line simulating Nerve Axon. This line shapes the signal waveform during transmission. Hodgkin–Huxley (H-H) equations [77] describe the phenomena of excitation of a Nerve Axon and propagation of this excitation. The case where the excitation of a Nerve Axon is spatially uniform is called a “space clamp”. In the case of space clamp, FitzHugh [58] simplifed the H-H equations by proposing the Bonhoeffer–van der Pol (BVP) model ˛ 1 u3 ˆ J = ut − w − ˙ u − ˘ , c 3ˇ ˝ cwt + bw = a − u, 1 > b > 0,  c 2 > b , 1 > a > 1 −

(2.56) 2b . 3

(2.57)

Nagumo et al. [141] experimentally simulated this BVP model. For a suitable circuit model, the distributed BVP model simplifying H-H equations is given by (2.56) and (2.57), where J = h ˜ 2 u ˜ s2 . For the case b = 0, eliminating w, we get

(

)

(

)

chusst = utt − c 1 − u2 ut + u − a, c > 0, 2 > a > 1,  h > 0. Using the transformations x =

s 2a a2 − 1 , z = 2 ( a − u),  µ = c a 2 − 1 ,  ° = , the 4a 2 ˆ (ch) a −1

(

authors reduce (2.58) as

(

(2.58)

)

zxxt = ztt + µ 1 − z + ° z 2 zt + z,

)

(

µ > 0, (3/16) > ° > 0.

)

(2.59)

This equation was solved under the boundary conditions at t = 0: z = 0, (˜ z ˜ t ) = 0; on x = 0: z = F(t) is given. The asymptotic waveform (steady-state solution) of (2.59) is obtained by solving the ordinary differential equation

˜° ˙˙˙ − ° ˙˙ − µ(1 − ° + ˝° 2 )° ˙ − ° = 0.

(2.60)

˜ ° 0 is always a solution. Asymptotic solutions ˜ (° ) ˛ 0 when ˜ ° ±˝ for some ˜ were obtained. Linearization about ˜ = 0 gives the equation ˜° ˝˝˝ − ° ˝˝ − µ° ˝ − ° = 0. Wazwaz and Gorguis [209] derived the exact solution of the equation (2.54) under the −1 − 1/2 x initial condition u( x , 0) = ˛1 + e ( ) ˆ . When 0 < ˜ < 1, the equation models heterozygote ˝˙ ˇ˘ inferiority. Using the Adomian decomposition method, the solution of the equation (2.54) was obtained in a series form which was then used to construct closed-form solutions. The −1 traveling wave solution was obtained as u( x , t) = ˙ˆ1 + e − ˜ / 2 ˇ˘ , ˜ = x + ct ,  c = (1 − 2° )  2 , which is the same solution as obtained by Jone and Sleeman [86]. The solution satisfes u(−°) = 0, u(°) = 1, which implies that u(˜ ) is a wave front traveling from right to left with speed c. Kawahara and Tanaka [90] derived a series of exact solutions as u( x , t) =

ˆ 1 1 2x 1 − 2˜  + tanh ˘ ± + t , 2 2 4  ˇ 4

(2.61)

57

Reaction–Diffusion Modeling

u( x , t) =

u( x , t) =

ˆ ˜ ˜ 2˜ x ˜ 2 − 2˜  + tanh ˘ ± + t , 2 2 4 4 ˇ 

(

(2.62)

)

2 ˘  (1 + ˜ ) (1 − ˜ ) 2(1 − ˜ )x 1 − ˜ + tanh ± + t. 2 2 4 4  

(2.63)

Li and Guo [113] used the frst integral method due to Feng [50–53] to obtain new exact solutions of the equation (2.54). Assume that equation (2.54) has traveling wave solutions of the form u ( x , t ) = u (˜ ) , ˜ = x − ct, where c is the velocity of the traveling wave. Equation (2.54) reduces to u°° + cu° + u(1 − u)(u − ˜ ) = 0.

(2.64)

Let x = u,  y = u° . Then, (2.64) reduces to the frst-order system x° = y ,  y ° = −cy + x(1 − x)( x − ˜ ).

(2.65)

Using the frst integral method to solve (2.65), they obtained the solution of (2.64) as u(˜ ) = ˝1 − e( ˙ˆ

)

˜ / 2 + a ˇ

˘

−1

, a arbitrary constant.

(2.66)

One of the values of c is c = ( 1 − 2˜ ) ˆ 2. The exact solution of (2.54) was obtained as u( x , t) =

1 2˜ − 1 ˘ ˙ x + t + a  , 1 − coth ˇ  ˆ2 2 2  4

(2.67)

where a is an arbitrary constant. Some more exact solutions containing the coth function were also derived. Abdusalam [1] obtained an analytic solution using the Picard iteration method for the boundary value problem of the telegraph equation:

˜ utt + ( 1 − ˜ fu ) ut = uxx + f (u),  f (u) = u( a − u)(1 − u),  x ˇ[0, L],  t ˇ ˘ 0,  t * 

(2.68)

with the boundary conditions ux (0, t) = 0 = ux (L, t). This is also called the Nagumo telegraph equation. The author has also considered the following form of the Nagumo telegraph equation:

˜ utt + ( 1 − ˜ fu ) ut = uxx + f (u), f (u) = − u + H (u − a), 0 ˇ a ˇ 1,

(2.69)

where H(u) is the Heaviside function. Setting z = x + ct, the problem is reduced to

(˜ c (˜ c

) − 1) u˙˙+c(1 + ˜ )u˙ + u = 1,

2

− 1 u˙˙+c(1 − ˜ )u˙ + u = 0,    u ˘ a,

(2.70)

2

u ˘ a,

(2.71)

u( z) ˜ 0 as z ˜ °,  u(0) = a.

58

Spatial Dynamics and Pattern Formation in Biological Populations

The traveling wave solutions are given by u( z) = ae k1 z , 

z ° 0,

(2.72)

= 1 + (a − 1)e k2 z , z ˙ 0,

 − ˜ ± ˜ 2 − 4°   , ° = ˛ c 2 − 1,  ˜ = c ( 1 + ˛ ) , c > c = 2 a . where ( k1 , k2 ) =  min 2° 4a − 1 Van Gordor and Vajravelu [203] obtained approximate solutions using a variational technique for Nagumo reaction–diffusion equation (2.54) and Nagumo telegraph equation (2.68). In his book, Murray [139,140] had given an excellent exposition of reaction–diffusion systems, in particular multispecies waves, spatial pattern formations, etc. 2.4.3.1 Numerical Solutions Gazdag and Canosa [65] presented a numerical solution of the nonlinear diffusion equation (2.26) using a pseudospectral approach. Parekh and Puri [158] and Twizell et al. [195] have presented implicit and explicit fnite difference algorithms. Tang and Weber [190] proposed a Galerkin fnite element method. Mavoungou and Cherruault [125] solved (2.26) using the Adomian method. Mickens [130] had introduced a fnite difference scheme for Fisher’s equation. Qiu and Sloan [171] used a moving mesh method. Al-Khaled [6] proposed the sinc collocation method. Olmos and Shizgel [152] constructed the numerical solutions using a pseudospectral method. Mittal and Kumar [135] and El-Azab [38] used the wavelet Galerkin method. Mittal and Arora [131] considered the numerical solution using B-spline collocation of the following form of Fisher equation: ut = ru − r1u2 + Duxx , 0 < t < ˙ , − ˙ < x < ˙

(2.73)

with the initial condition u( x , 0) = −

r ˝sech 2 (−˜ x) − 2tanh( − ˜ x) − 2ˆˇ , ˜ = 4r1 ˙

r 24c

(2.74)

and boundary conditions lim u( x , t) = 0.5 and lim u( x , t) = 0. We fnd that lim u( x , t) = r r1 . x°−˝

x°˛

x°−˝

The authors take this value as 0.5. The exact solution of equation (2.73) with D = 1 is u( x ,  t) = −

r ˆ 5r sech 2 x* − 2tanh x* − 2 ˘ , x* = −˜ x + t. 12 4r1 ˇ

( )

( )

(2.75)

The solution predicts a wave front of increasing allele frequency that propagates through the population. Equation (2.73) states that the change in the density of labeled particles at a given time depends on the infection rate ru − r1u2 and the diffusion of the neighboring area. The frst term (ru) measures the infection rate, which is proportional to the product of the density of the infected population and uninfected population. The second term −r1u2 shows how fast the infected populations are diffusing.

(

)

(

)

59

Reaction–Diffusion Modeling

Mittal and Jain [133] obtained the numerical solution of Fisher’s reaction–diffusion equation: ut = ˜ uxx + ° u(1 − u), 0 < t < ˇ ,  a < x < b

(2.76)

with the initial and boundary conditions u( x , 0) = u0 ( x), a ° x ° b ; u( a, t) = g 0 (t), u(b, t) = g1 (t) using a modifed cubic B-spline collocation method. Tang et al. [189] studied numerically the solution of the two-dimensional problem: ut = ˜1uxx + ˜2 uyy + µ u(1 − u), in ˆ:  x < a, y < b, u( x , y , 0) = u0 ( x , y ), 0 ° u0 ° 1, n ° ˛u = 0, or u = 0, t > 0, ( x , y ) ˆ˜ ˇ: x = a, .y = b The authors solved the problem using (i) an explicit fnite difference method, (ii) a Galerkin method and (iii) a Petrov–Galerkin fnite element method. The authors concluded that any local initial disturbance evolves into a hump in a larger space at the frst stage and then forms as a circular quasi-traveling wavefront. The velocity V of the front increases with distance and approaches 2 which is the minimum planar wavefront velocity for the one-dimensional equation. This value of V was used to predict the velocity of the elliptic quasi-traveling wavefront of the two-dimensional problem. Roessler and Hüssner [176] discussed the numerical solution of Fisher’s problem in two dimensions:

(

)

ut = ˜ uxx + uyy + ° u(l − u), 0 < x < h, 0 < y < k ,  t > 0, u( x , y , 0) = u0 ( x , y ), u( x , y , t) = 0 on the boundary, and l is the carrying capacity. The authors have also given the numerical solution of an application of Fisher’s equation to a coral reef. Brazhnik and Tyson [16] derived explicit solutions and approximations to the solution of Fisher’s equation in two dimensions in rescaled variables: ut = uxx + uyy + u(1 − u).

(2.77)

First, they assumed that a stationary wave moves along the x-axis. The velocity Vp of this wave is different from the velocity c of the plane wave. Substituting ˜ = x − Vpt , the equation governing u(˜ , t) is obtained as u˝˝ + Vp u˝ + uyy + u(1 − u) = 0.

(2.78)

The wave satisfes the conditions u(±°, y) ˛ u± ( y ). Smoothness condition requires u° = 0, and u°° = 0 as ˜ ° ±˝. Next, the authors consider the general case and have shown that Fisher’s traveling waves in a two-dimensional bistable medium have different geometry which affects the propagation velocity of waves. They obtained fve bounded solutions and characterized them as plane waves, V-waves, Y-waves, separatrix, and space-oscillating

60

Spatial Dynamics and Pattern Formation in Biological Populations

fronts. The slowest wave is an oscillation front, but its velocity increases with the increase of the wavelength. The authors have mentioned many application areas where the above patterns appear.

2.5 Two-Species Reaction–Diffusion Models 2.5.1 Turing Instabilities of Two-Species Reaction–Diffusion Systems In this section, we study the solution of two-species systems with the Neumann boundary conditions, which are most relevant for exponential systems. Consider the two-species system:

∂u ∂2u ∂v ∂2v = d1 2 + F(u,  v), = d2 2 + G(u,  v), ∂t ∂x ∂t ∂x

(2.79)

with zero flux boundary conditions on the interval [0, L]

∂ u(0,  t) ∂ u(L, t) ∂ v(0,  t) ∂ v(L, t) = = 0, = = 0, t ≥ 0. ∂x ∂x ∂x ∂x

(2.80)

Turing [193] showed that a system of coupled RD systems can be used to describe patterns and forms in biological systems. The Turing theory shows that the interplay of chemical reaction and diffusion may cause the stable equilibrium of the local system to become unstable for the diffusive system and lead to spontaneous formulation of a spatially periodic stationary structure. This kind of instability is called the Turing instability or diffusion-driven instability. We derive the conditions under which the Turing instability sets in for the above system. Consider the case without diffusion ( d1 = d2 = 0 ). The homogeneous steady-state solution ( u ,  v ) satisfies the equations F ( u , v )  =  G ( u ,  v )  = 0. The Jacobian of the system is given by  b11 J=  b21

b12 b22

 ∂F ∂F ∂G ∂G , b12 = , b21 = , b22 = .  , where b11 = ∂ u ∂ v ∂ u ∂v 

The characteristic equation of J is λ 2 − Tλ + R = 0, where T = trace ( J ) = b11 + b22 < 0, and R = det( J ) = b11b22 − b12b21 > 0.

(2.81)

These conditions are necessary for the diffusively driven instability, that is, the conditions for asymptotic stability of the system. To discuss the stability of the uniform steady state ( u ,  v ), perturb the solution of equations (2.79) as u = u + U ,  v = v + V and linearize. We obtain the system U t = b11U + b12V + d1U xx , Vt = b21U + b22V + d2  Vxx .

(2.82)

Write the solutions of (2.82) in the form U = se λ t cos ( nπx L ) , V = we λ t cos ( nπx L ), where λ is the frequency and ( L nπ ) is the wavelength. The homogeneous system in s and w has solution if the determinant of the coefficient matrix is zero. We obtain

61

Reaction–Diffusion Modeling

λ 2 − pλ + q = 0, where p = b11 + b22 − ( d1 + d2 ) l 2 , and q = d1d2l 4 − ( d1b22 + d2b11 ) l 2 + ( b11b22 − b12b21 ) , where l = nπ L .

(2.83)

This characteristic equation is called the dispersion relation. It relates the growth rates of the spatial modes to the parameter values of the system. A spatial Hopf bifurcation, commonly known as wave bifurcation, corresponds to a pair of purely imaginary eigenvalues for some l ≠ 0, that is, when p = 0 and q > 0. But, for stability, T = b11 + b22 < 0, and therefore, p < 0 for all l. Hence, the uniform steady state of a 2D RD system (2.79) cannot undergo a wave bifurcation, that is, an oscillatory instability to a standing wave pattern [147]. A Turing bifurcation corresponds to a zero eigenvalue. We require q = 0, which leads to d1d2 K 2 − ( d1b22 + d2b11 ) K + ( b11b22 − b12b21 ) = 0, K = l 2 .

(2.84)

Since the roots of the equation are positive, we require the condition ( d2b11 + d1b22 ) > 0, since R = b11b22 − b12b21 > 0. This is a necessary but not a sufficient condition for the Turing bifurcation. The Turing bifurcation can occur only if (i) the diffusion coefficients are not equal and (ii) b11 and b22 do not have the same sign. In other words, the Turing instability can occur only in pure or cross activator–inhibitor system. For such systems, b11 > 0,  b22 < 0, which together with (2.81) implies that the Turing bifurcation can occur if b22 > b11 since T < 0 and b12b21 < 0 and b12b21 > b11b22 since R > 0. Defining θ RD = d2 d1 , we obtain θ RD > − ( b22 /b11 ) > 1. In other words, for the Turing instability to set in, the activator must diffuse slower than the inhibitor and this characteristic is known as the principle of “short-range activation and long-range inhibition”. It is also known as “Local autocatalysis with lateral inhibition” or “Local Auto-activation-Lateral Inhibition” (LALI) [145,153], or “Self-Enhancement and Lateral Inhibition” (SELI) [142] and has been applied to mechanisms other than reaction– diffusion systems. Further, a stable uniform steady state undergoes a Turing bifurcation when the quadratic (2.84) is a perfect square, that is, when the discriminant of (2.84) van2 ishes. That is, when ( d2b11 + d1b22 ) = 4d1d2 R or equivalently at the critical ratio of diffusion coefficients

θ RD, cr

 1 =  b11

(

R + −b12b21

)

2

  . 

(2.85)

The critical wave number is given by lT2 ,  RD = R d1d2 . Therefore, the uniform steady state ( u ,  v ) of the system (2.79) satisfying the stability conditions (2.81) will be driven to instability by diffusion if and only if (i) ( d2b11 + d1b22 ) > 0, and (ii) ( d2b11 + d1b22 ) > 4d1d2 R. 2

(2.86)

The band of unstable modes is given by l−2 < l 2 < l+2 . Madzvamuse et al. [121] performed the analysis and simulation of the cross-diffusiondriven instability for a two-component reaction–diffusion system. Simulation was done using the finite element method. We briefly report the important results of the authors. The evaluation equations for reaction–diffusion equations with cross-diffusion are obtained from the application of the law of mass conservation and the extended Fick’s first law. ut = Du   ∇ 2 u + Duv∇ 2 v + f1 (u,  v),

62

Spatial Dynamics and Pattern Formation in Biological Populations

vt = Dv∇ 2 v + Dvu∇ 2 u + f2 (u,  v), x ∈ Ω, t > 0,

(2.87)

n ⋅ ∇u = 0 = n ⋅ ∇v , u( x , 0) = u0 (x), v( x, 0) = v0 ( x), x ∈Ω, t ≥ 0. Du > 0,  Dv > 0 are the diffusion coefficients and Duv ,  Dvu are the cross-diffusion coefficients and n is the outward unit normal. The diffusion coefficients Du ,  Dv describe the flux of a solute due to its own concentration gradient. The cross-diffusion coefficients Duv ,  Dvu describe the flux of a solute due to the concentration gradient of the other solute. In the onedimensional case, the authors considered the activator-depleted substrate model (Brusselator model), where f1 = k1a1 − k2 u + k3 u2 v, f2 = k 4b − k3 u2 v (all constants are positive). The nondimensional 1D reaction–diffusion system with cross-diffusion was obtained as

(

)

(2.88)

vt = G(u,  v) + d∇ 2 v + du∇ 2 u, G(u, v) = γ g(u, v) = γ b − u2 v ,

(2.89)

ut = F(u,  v) +   ∇ 2 u + dv∇ 2 v , F(u, v) = γ f (u, v) = γ a − u + u2 v ,

(

)

where d is the ratio of the diffusion coefficients and du , dv are the ratios of cross-diffusion and diffusion coefficients, respectively. The authors performed the linear stability analysis of the one-dimensional system (2.88) and (2.89). The model has a unique positive uniform steady state (in the absence of diffusion and cross-diffusion): 



( us , vs ) =  a + b, (a +bb)2  . Linearization about ( us , vs ) gives the system (see also Equation (2.82)) U t = b11U + b12V +   ∇ 2U + dv∇ 2V ,

(2.90a)

Vt = b21U + b22V + d∇ 2V + du∇ 2U,

(2.90b)

where b11 = γ fu , b12 = γ f v , b21 = γ g u , b22 = γ g v , evaluated at ( us , vs ) and ∇ 2 = ∂ 2 ∂ x 2 . Following the above procedure, the necessary conditions for the diffusion-driven instability, that is, the conditions for asymptotic stability of the system are obtained as (i) T = b11 + b22 < 0 and (ii) R = b11b22 − b12b21 > 0 (see (2.81)). Corresponding to Equation (2.83), we obtain

λ 2 − pλ + q = 0, where p = b11 + b22 − (d + 1)l 2 , and q = ( d − dudv ) l 4 − ( b22 + db11 − dub12 − dvb21 ) l 2 + R , where l = nπ L.

(2.91)

Since p < 0, the Hopf bifurcation does not take place. A Turing bifurcation corresponds to a zero eigenvalue. We require q = 0, which leads to

( d − dudv ) K 2 − ( b22 + db11 − dub12 − dvb21 ) K + R = 0. K = l2 , R > 0.

(2.92)

Since the roots of the equation are positive, we get the necessary condition as (iii) ( b22 + db11 − dub12 − dvb21 ) > 0.

(2.93)

63

Reaction–Diffusion Modeling

For Equation (2.92) to be a perfect square, the following equation is to be satisfied

(b22 + db11 − dub12 − dvb21 )2 = 4 ( d − dudv ) R.

(2.94)

(iv) Since, the right-hand side is positive, we get d − dudv > 0.

(2.95)

This implies DuDv − Duv Dvu > 0. That is, (product of primary diffusion coefficients) > (product of cross-diffusion coefficients). The above are necessary conditions for instability. For cross-diffusion-driven instability, we require (v) ( db11 + b22 − dub12 − dvb21 ) > 4 ( d − dudv ) R. 2

(2.96)

Using the linear stability analysis, the authors have (a) identified the parameter spaces for cross-diffusion-driven instability, (b) shown that introduction of cross-diffusion enhances the mechanism of diffusion-driven instability, and (c) shown that negative cross-diffusion can also induce cross-diffusion-driven instability. For d = 1,  dv = 1, − 1 < du < 0, and the above conditions hold, then the reaction–diffusion system with negative cross-diffusion in the v component only is the only system that can induce cross-diffusion-driven instability. The authors have given a number of examples like salt–salt model, polymer–salt model, polyethylene–glycol model, micelle–salt model, and polymer–micelle model in which d − dudv > 0. Finite element simulations were done to confirm the theoretical predictions. Madzvamuse et al. [122] discussed the effects of cross-diffusion on the stability of reaction–diffusion models on evolving domains. The authors have derived the stability conditions. Finite element simulations were done to confirm the theoretical predictions. We now define the cross activator–inhibitor and pure activator–inhibitor-type systems. Definition 2.1 [139] The homogeneous system F ( u , v )  =  G ( u ,  v )  = 0 is said to be cross activator–inhibitor type  + +   − −  if the Jacobian J has the structure J =  , or  , and it is of pure activa − −    + +   + tor–inhibitor type if the Jacobian has the structure   J =   +

 − −  , or   −   −

+  + 

Remark 2.1 We call u (respectively v) as a self-activator if b11 > 0, ( b22 > 0 ) at the homogeneous steady state since it upregulates its own production during the initiation of an instability and a self-inhibitor if b11 < 0 ( b22 < 0 ), as it downregulates its own production during the initiation of an instability. Thus, the Turing instability necessitates the pairing of a self-activator with a self-inhibitor [96]. Marquez-Lago and Padilla [124] considered the solution of the reaction–diffusion system ut = D1   ∇ 2 u + f (u,  v),

(2.97)

vt = D2∇ 2 v + g(u,  v),

(2.98)

64

Spatial Dynamics and Pattern Formation in Biological Populations

in a two-dimensional space with zero flux boundary conditions. The authors derived an analytic selection criterion for predicting patterns depending on the nonlinearities in the reaction terms. They considered two types of pattern generating systems: (i) a short-range positive feedback coupled to a large range negative feedback (Gierer–Meinhardt model) and (ii) FitzHugh–Nagumo equation. In Gierer–Meinhardt model, the observed patterns are generally either spots or stripes. In FitzHugh–Nagumo model, the observed patterns are generally either spots or labyrinth-like patterns. The authors have also considered the Fokker–Planck associated equation of (2.97) and used the calculus of variations for its solution. Numerical simulations were done using MATLAB, in which discretization of the system was done using finite differences in space and backward Euler time stepping. Analytical evidence and numerical evidence support the fact that the solution of the associated Fokker–Planck equation of the reaction–diffusion system can be used to determine the type of patterns produced. They conjectured the following: “Assume that the stationary solution of the Fokker–Planck equation has exactly two global maxima of different (equal) height. Then, the system admits a spot-like (a stripe-like) solution”. 2.5.1.1 Predator–Prey Reaction–Diffusion Systems Consider the general predator–prey models described by nonlinear growth of prey, general predator responses including Holling types I–IV, predator mortality, and diffusion of prey and predator. A general nondimensional nonlinear RD system can be written as

∂u = f (u) − g(u)v + ∆u, ∂t

(2.99)

∂v = α g(u)v − mν + D∆v. ∂t

(2.100)

Here, f (u) is the growth rate of prey population. The second term on the right-hand side of (2.99) describes the predation where g(u) is the functional response of the predator population, α is the prey consumption efficiency, mν stands for predator mortality, and D is the ratio of the diffusivities of the predator and the prey. The prototype of f (u) can be logistic prey growth or bistability/nonlinearity in the case of the Allee effect [111,151]. Yi et al. [216] performed Hopf and steady-state bifurcation analysis for a diffusive predator–prey system with Holling type II predator functional response subject to the Neumann boundary conditions. The authors considered the Rosenzweig–MacArthur reaction– diffusion model which is used to describe many situations like the predator– prey interactions and spatiotemporal dynamics of an aquatic community of zooplankton and phytoplankton system. The nondimensional system considered was u  muv  ut = d1∆u + u  1 −  − , x ∈Ω,  t > 0,  k u+1 vt = d2 ∆u − θ v +

muv , x ∈Ω, t > 0, u+1

∂ v u = ∂ v v = 0, x ∈∂ Ω,  t > 0, u( x , 0) = u0 ( x) ≥ 0, v( x , 0) = v0 ( x) ≥ 0,  x ∈∂ Ω,  t > 0,

(2.101) (2.102) (2.103)

Reaction–Diffusion Modeling

65

where k is the rescaled carrying capacity, ˜ is the death rate of the predator, and m is the strength of interaction. The other parameters are all positive constants. When diffusion is absent ( d1 = 0 = d2 ), the system has a unique limit cycle. The authors considered the solution of the system (2.101) and (2.102) in the one-dimensional case on the domain ( 0,  l˛ ) under the Neumann boundary conditions, using the center manifold theory and normal form method. The authors performed detailed bifurcation analysis for the constant coexistence equilibrium solution. They followed the geometric approach and selected coordinate λ of the vertical nullcline, that is, solution of ˝− ˙ ˜ + mu (u + 1) ˆˇ = 0 as the main bifurcation parameter. Under certain conditions of other parameters, they have shown that there exist exactly 2n Hopf bifurcation points where spatially nonhomogeneous periodic orbits bifurcate. These periodic orbits correspond to the spatial eigenmode cos ( kx l ) where l˜ is the length in spatial direction. The Hopf bifurcation points always exist in pairs, and for each fxed eigenmode, there corresponds exactly one pair of the Hopf bifurcation points. For some different parameter ranges, both the Hopf and steady-state bifurcations occur along the curve of the constant coexistence steady-state solutions. The authors showed the existence of multiple spatially nonhomogeneous periodic orbits. Their numerical studies show the existence of loops of spatially nonhomogeneous periodic orbits and steady-state solutions. Zhang et al. [220] studied the existence of local Hopf bifurcation in two-species spatial predator–prey model with the Beddington–DeAngelis (BD)-type functional response and homogeneous Neumann boundary conditions. To study the bifurcation, the authors used the method of Bogdanov–Takens. The study provided valuable information about periodic behavior and global dynamics that can occur in the diffusive systems, but cannot occur in corresponding local ODE system. Guo et al. [70] considered the spatially homogeneous and inhomogeneous autocatalysis models of arbitrary order. For spatially homogeneous model with zero fux boundary conditions, the authors studied the Turing instability and also the existence and stability of the Hopf bifurcation surrounding the interior equilibrium using the center manifold theory and normal form method. Numerical simulations were done to verify the theoretical results. Holzer and Scheel [80,81] investigated a class of two-component coupled Fisher– KPP equations where one species decouples from the other species. They showed that the  evolution of positive, compactly supported perturbations of the unstable homogeneous steady state can give rise to a pair of diverging waves propagating with different speeds for different species. This phenomenon is called anomalous spreading [61]. Their theory suggests that anomalous spreading arises due to poles of the pointwise Green’s function of the linearized system around the unstable homogeneous steady state. Shi and Ruan [185] studied the spatial, temporal, and spatiotemporal dynamics of a reaction–diffusion predator–prey system with mutual interference and with homogeneous Neumann boundary conditions. They used the Crowley–Martin-type functional response and used the following model:

˜u u˘ muv ˙ − d1˛u = ru ˇ 1 −  − , ˆ ˜t K  (1 + au)(1 + bv)

(2.104)

˜v v˘ ˙ − d2 ˛u = sv ˇ 1 −  , x , t > 0, ˆ ˜t hu 

(2.105)

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Spatial Dynamics and Pattern Formation in Biological Populations

∂u ∂v = = 0,  n is the outward unit normal vector of the boundary ∂ Ω, ∂n ∂n u( x , 0) = u0 ( x) ≥ 0,  v( x, 0) = v0 ( x) ≥ 0, where p(u, v) = mu [(1 + au)(1 + bv)] is the Crowley–Martin functional response. The authors have shown that spatial inhomogeneous patterns occur due to diffusion-driven instability (Turing instability), temporal inhomogeneous patterns occur due to the Hopf bifurcation, and spatiotemporal patterns occur at the points where the Turing instability curve and Hopf bifurcation curves intersect. They had performed numerical simulations to verify the theoretical results. Upadhyay et al. [202] studied the dynamics of a spatial nonlinear predator–prey model under harvesting. They used a modified Leslie–Gower-type model with Holling type IV functional response and nonlinear harvesting of prey. The modified Leslie–Gower-type model with Holling type IV functional response considered was (in nondimensional form)

∂u uv hu = d1∆u + u(1 − u) − 2 − , ∂t u α + u+γ c + u

(2.106)

∂v βv   = d2 ∆v + δ v  1 − .  ∂t u 

(2.107)

(

)

with no-flux conditions at the boundary and positive initial conditions. Numerical simulations were used to study the formation of the Turing patterns. For the spatial model, they discussed the stability and Hopf bifurcation and determined the direction of the Hopf bifurcation curves. Zhang et al. [222] studied the Hopf and steady-state bifurcations in a ratio-dependent predator–prey interaction with modified Holling–Tanner formalism and derived explicit conditions for the existence of nonconstant steady states that emerge through the steady spatiotemporal model system:

∂u mv   = u  a − bu −  + ∆u,  ∂t u + αv

(2.108)

∂v m1v   = v c − + d∆v,  ∂t u + α 1 

(2.109)

for ( X ,  t ) ∈Ω × R +, and subject to a nonnegative initial condition u(X , 0) = u0 (X ) ≥ 0, ∂u ∂v v(X , 0) = v0 (X ) ≥ 0, X = ( x ,  y ) ∈ Ω ⊂ R 2 , and no-flux boundary condition = = 0, n is ∂n ∂n the outward unit normal vector of the boundary ∂ Ω of a square bounded domain Ω. From their studies, the authors concluded the following: (i) Mobility of individuals for both species (diffusion) within their habitat can promote species coexistence which otherwise goes to extinction in the absence of diffusion. (ii) Coupling diffusion into a temporal model can significantly enrich the population dynamics by inducing alternative nonconstant steady states (two stable and two unstable states were observed), in particular when diffusion interacts with different types of bifurcation like Hopf and homoclinic bifurcations. (iii) Turing domain is identical to the parametric domain where there exists only steady-state

67

Reaction–Diffusion Modeling

bifurcation. Turing patterns are stable nonconstant steady states bifurcating from stable constant steady state. (iv) In non-Turing domain, steady-state bifurcation and Hopf bifurcation act in concert to determine the emergent spatial patterns, i.e., nonconstant steady state emerges through steady-state bifurcation, but it may be unstable if the destabilizing effect of the Hopf bifurcation counteracts the stabilizing effect of diffusion leading to nonstationary spatial patterns. The following are some examples of predator–prey RD systems: Example 2.1 Derive the conditions for the asymptotic stability and Turing instability for the spatially homogeneous equilibrium solutions of the following 1D RD Holling–Tanner predator– prey model studied by Li et al. [115] with no-flux boundary conditions.

∂u muv ∂2u = u(1 − bu) − + d1 2 , x ∈(0, π), t > 0, ∂t u+ a ∂x

(2.110)

∂v v ∂2v  = sv  1 −  + d2 2 , x ∈(0, π),  t > 0,  ∂t u ∂x

(2.111)

u( x , 0) = u0 ( x),  v( x, 0) = v0 ( x), ux (0,  t) = ux (π ,  t) = 0, vx (0,  t) = vx (π ,  t) = 0, t > 0. (The authors had considered the case a = 1 and used β, x, y for b, u, v, respectively). Solution First, consider the corresponding local ODE system and asymptotic stability of the equilibrium points. The model system has two nontrivial equilibrium points, E0 ( 1 b , 0 ) and E1 u* ,  v * where

(

)

u* =  

(

(

)

)(

)

1 − bu* u* + a 1 − P + P 2 + 4 ab , P = ( ab + m − 1), v * = . 2b m

(2.112)

The boundary equilibrium point E0 ( 1 b , 0 ) is a saddle point with the positive x-axis as its stable manifold. The Jacobian matrix J at E1 u* ,  v * is

(

 a J =  11  s

)

a12   −s 

where a11 = 1 − 2bu* −

mav *

(u

*

+a

)

2

, a12   = −

(

mu* . u* + a

)

The characteristic equation is given by

λ 2 − λT ( s) + D(s) = 0, where T (s) = a11 − s,  D(s) = − s ( a11 + a12 ) . Now, a11 + a12 = 1 − 2bu* −

mav *

(u

*

+a

−

) ( 2

mu* . u* + a

)

68

Spatial Dynamics and Pattern Formation in Biological Populations

Using the value of v * from (2.112) and the expression u* (m − 1 + ab) = a − bu* 2 , we obtain  a + bu*2  a + bu*2 < 0, and D( s) = s  * > 0. * u +a  u + a  If T( s) < 0, then the signs of the coefficients of the characteristic equation are (+, +, +) and the roots are either negative or a complex pair with negative real parts. The equilibrium point E1 u* ,   v * is asymptotically stable. Now, T( s) < 0 gives s > a11. Now, a11 + a12 = −

(

)

a11 = 1 − 2bu* −

= 1 − 2bu* −

2

u* + a *

( (u

a 1 − bu*

) u ( 1 − ab − 2bu ) u m − = = (u + a) (u + a)  *

(

mav *

*

*

*

+a

)

)

P 2 + 4 ab  . 

*

We have the following cases: a. a11 > 0, that is, 0 < a11 < s : We require m2 > P 2 + 4ab. Simplifying, we obtain 2 m(1 − ab) > (1 + ab)2 , or m >

(1 + ab)2 , ab < 1. 2(1 − ab)

(2.113)

b. a11 ≤ 0. We obtain m2 ≤ P 2 + 4ab. Simplifying, we obtain (1 + ab)2 + 2 m( ab − 1) ≥ 0.

(2.114)

If ab ≥ 1, then the condition is always satisfied. Otherwise, we obtain (1 + ab)2 . 2(1 − ab)

m≤

(2.115)

Since s > 0, the above condition implies that s > a11 always. Therefore, in this case, the

(

)

equilibrium point E1 u* ,  v * is globally asymptotically stable. If (2.113) holds, then

(

*

*

)

E1 u ,  v is locally asymptotically stable. If (2.113) holds and s < a11, then the characteristic equation has two positive roots. Then, equilibrium point E1 u* ,  v * is unstable. Now, consider the spatially homogeneous equilibrium solution of (2.110) and (2.111). The linearized system of (2.110) and (2.111) at E1 u* ,  v * has the form

(

)

(

 ut   u   u   d1   = L( s)   : = J ( s)  v  +  0 v v       t 

)

0   ∆u  ,  d2   ∆v 

(2.116)

where L( s) is a linear operation with domain DL = X C, where 2 2 X = (u,  v) ∈ H [(0, π)] × H [(0,  π)] : u′(0) = u′(π) = v′(0) = v′(π) = 0 is a real-valued

{

}

Sobolev space. Following the eigenvalue analysis, we obtain the Jacobian J k of the linearized system at the equilibrium point E1 as  a − k 2d 11 1 Jk =   s 

a12 2

− s − k d2

  , for k = 0, 1, 2, …  

69

Reaction–Diffusion Modeling

The characteristic equation of J k is

µ 2 − µTk + Dk = 0, k = 0, 1, 2,…

(2.117)

where Tk = a11 − s − ( d1 + d2 )   k 2 , Dk = ( d1d2 )   k 4 + [ d1 s − d2 a11 ] k 2 − s ( a11 + a12 ) .  The authors (Li et al. [115]) had derived the conditions for asymptotic stability, Turing instability, Hopf bifurcation, etc. Here, we derive the conditions in a slightly different way.

(

)

(i) E1 u* ,  v * is an unstable equilibrium point if one or both the roots of (2.117) are positive or have positive real parts. One of the roots of (2.117) is positive when Dk < 0. The sign of Tk is immaterial. Then, the coefficients have the signs ( +, −, − ) or ( +, +, − ).

Dk < 0 gives a11 < s
0. The term on the right is positive when, (a) a11 − k 2 d1 > 0, and t − ( a11 − k 2 d1 ) > 0, that is,

(

)

(

)

0 < a11 − k 2 d1 < t or a11 − t < k 2 d1 < a11 ;

(

)

(

)

or (b) a11 − k 2 d1 < 0, and t − a11 − k 2 d1 < 0, that is,

(

)

t < a11 − k 2 d1 < 0, or a11 < k 2 d1 < a11 − t. Note that the above conditions hold for a finite number of eigenmodes k. Therefore, the equilibrium point E1 u* , ν * is Turing unstable if s belongs to the

(

)

interval given in (2.118). That is, for s belonging to this interval, the equilibrium

(

)

E1 u* , ν * is locally asymptotically stable with respect to the temporal model and it is unstable with respect to the spatial model (2.110) and (2.111). For the sake of completeness, we have performed numerical simulations. Consider the following set of parameter values: a = 1, b = 0.2, m = 1.5, d1 = 0.008, d2 = 1. For k = 1, we obtain the equilibrium point as E1 (1.0895, 1.0895), a11 = 0.1899, Condition (a) is satisfied, and the interval is 0.1899 < s < 0.30306. Now, for s = 0.01, 0.05, 0.1, and 0.15, the eigenvalues of the characteristic equation are (−1.0034, 0.17528), (−1.01739, 0.14928), (−1.03577, 0.11765), and (−.05516, 0.087050), respectively. Therefore, one root of the characteristic equation (2.117) is always positive for these values of s and the equilibrium point E1 u* , ν * is unstable for the spatial model (2.110) and (2.111). However, for all these values of s, the temporal model is also unstable. Now, if we choose s = 0.25 > a11, the characteristic roots are (−1.09712, 0.0290111), and in this case, the temporal model becomes stable and the spatial model remains unstable and the equilibrium point E1 u* , ν * is Turing unstable. Can be easily checked that (2.118) is satisfied (0.1899 < s < 0.30306) and the equilibrium point of the corresponding ODE system is asymptotically stable. For s = 0.25, and with the other parameter values taken as above, the

(

)

(

)

70

Spatial Dynamics and Pattern Formation in Biological Populations

PDE system is Turing unstable in the range 0.8163 < k2 < 22.6696. Since k ˜, the range is 1 ˜ k ˜ 4. For the above set of parameter values, and with the initial conditions taken as u( x , 0) = 1.0895 + 0.1cos x = ˜ ( x , 0), the Turing instability of the equilibrium solution of the spatial model system (2.110) and (2.111) is presented in Figure 2.1. For this case, the graph of Dk for k2 is plotted in Figure 2.2. (ii) Both the roots of (2.117) are positive or have positive real parts when Tk > 0, Dk > 0. Then, the coeffcients have the signs (+ , − , +). In this case, the condition to be satisfed is

(

k 2 d2 a11 − k 2 d1

(

t − a11 − k 2 d1

(

)

) < s < (a

11

)

− k 2 d1 − k 2 d2 .

)

(2.119)

Since s > 0, we require that a11 − k 2 d1 − k 2 d2 > 0. If the ratio on the lefthand side is negative, then the ratio is replaced by zero. The values of

FIGURE 2.1 The Turing instability of the equilibrium solution of the model (2.110) and (2.111).

FIGURE 2.2 Graph of the function Dk vs. k2.

71

Reaction–Diffusion Modeling

k = 1, d1 = 0.008, d2 = 1, a11 = 0.1899 used in case (i) do not satisfy the above condition.

(

)

iii. E1 u* , ν * is an asymptotically stable equilibrium point if both the roots of (2.117) are negative or have negative real parts. This is possible when Tk < 0, and Dk > 0. Then, the coefficients have the signs (+ , + , +). We obtain the condition as

(

)

2 2   k d2 a11 − k d1 , a11 − ( d1 + d2 ) k 2  , s > max  2   t − a11 − k d1

(

)

(2.120)

provided that at least one of the values is positive. For the set of values considered in case (i), we obtain the condition as s > 0.30306. Example 2.2 The Ivlev-type predator–prey models have important applications in ecology, host– parasitoid systems, animal coat pattern, etc. Discuss the Hopf bifurcation for the following spatial diffusive predator–prey model with Ivlev-type functional response, in nondimensional form (Ivlev [83], Wang and Wei [207])

∂u = u(1 − u) − ν 1 − e −γ u + ∆u, x ∈(0, lπ), ∂t

(2.121)

∂u = βν α − 1 − α e −γ u + d∆ν , x ∈(0,  lπ), ∂t

(2.122)

(

(

)

)

ux (0,  t) = ux (lπ , t) = 0,  vx (0, t) = vx (lπ ,  t) = 0, t > 0, u( x , 0) = u0 ( x), v( x , 0) = v0 ( x),  x ∈(0,  lπ). Solution

(

)

The model system has three equilibrium points E0 (0, 0), E1 (1, 0), and E2 u* , ν * , where u* =

(

)

∗ * m * u 1− u α ,v = , with m = ln , α > 1, γ > m. * γ α −1 1 − e −γ u

The conditions assure that the steady state is in the positive quadrant. The model has saddle points at E0 (0, 0), E1 (1, 0). The Jacobian of the system at E2 u* , ν * is given by

(

 a11 J=  a21

a12 a22

)

  , where a11 = γ − 2 m − m (α − 1) (γ − m)  γ , 

a12 = −1 α ,  a21 = αβ m (α − 1) (γ − m) m , a22 = 0. Following the eigenvalue analysis, the characteristic equation of J n , ( k = n l ) at the equilibrium point E2 is given by

µ 2 − µTn (γ ) + Dn (γ ) = 0, n = 0, 1, 2,

(2.123)

72

Spatial Dynamics and Pattern Formation in Biological Populations

where Tn (˜ ) = a11 − (1 + d)k 2 , Dn (˜ ) = d ,k 4 − da11 k 2 + s s = ° m (˛ − 1) (˜ − m) ˜ . The authors [207] have analyzed the Hopf bifurcation occurring at the point E2 u* , ˜ * by choosing ˜ as a bifurcation parameter. Letting T0 = a11 = 0, we get

(

)

˜ = ˜ 0H =

m [ 2 − m(° − 1)] m = m+ > m. 1 − m(° − 1) 1 − m(° − 1)

This implies that the denominator > 0, that is, 1 − m(˜ − 1) > 0, which is true. For ˜ ˝ m, ˜ 0H , we have Tn (˜ ) < 0,  Dn (˜ ) > 0. This implies that the roots of the characteristic

(

)

(

)

equation (2.123) are negative or have negative real parts and the steady state E2 u* , ˜ * is locally asymptotically stable. Setting Tn (˜ ) = 0, we obtain

˜ nH =

 1 − m(° − 1)]  m [ 2 − m(° − 1)] n , k = , or n < l   2 1 − m(° − 1) − k (1 + d) 1+ d l  

1/2

.

Let N denote the integer part of the right-hand side. Then, 0 ˜ n ˜ N . The values of m > 0, ˜   and  ° are fxed, and l is chosen appropriately. Also, ˜ 0H must be a Hopf bifurcation point for any l > 0, since T0 ˜ 0H = 0, Tj ˜ 0H < 0 for any j ˜ 1, and Dm ˜ 0H > 0 for

( )

( )

( )

any m ˜ 0. It corresponds to the Hopf bifurcation of spatially homogeneous periodic

˜ ( 1 − m(° − 1)) > , then the spatial model d 4m(° − 1) H (2.121) and (2.122) undergoes a Hopf bifurcation at ˜ = ˜ n , (1 ˛ n ˛ N), and the bifurcating periodic solutions starting from ˜ = ˜ nH , (1 ˛ n ˛ N ) are spatially nonhomogeneous. Garvie [64] presented two fnite difference algorithms to get the numerical solution. Using numerical simulation, Wang et al. [206] showed the evolution process of pattern formations of the model. 2

solution. The authors [207] have shown that if

2.6 Applications in Biochemistry: Belousov–Zhabotinsky Reaction–Diffusion Systems Belousov–Zhabotinsky (BZ) reaction is an important class of oscillating chemical reactions discovered by Boris Belousov in 1951. In this reaction, an organic molecule (malonic acid, CH2(COOH)2) is oxidized by bromate ions BrO −3 . The basic reactants are Ce2(SO4)3, NaBrO3, CH2(COOH)2, H2SO4 to which a color indicator is added. He discovered the frst reaction of this class with the Ce3+/Ce4+ couple as catalyst and citric acid as reductant. He observed that the color of the reaction solution oscillated between colorless and yellow and found that the frequency of oscillations increased with rise of temperature. Above a certain critical temperature, a complicated variation of concentration of the chemicals occurs. The mechanism of the BZ reaction was elucidated by Field, Körös, and Noyes (FKN mechanism) in 1972 [54] and reduced the mechanism to fve essential steps by Field and Noyes [58]. Zhabotinsky’s group [218] conducted detailed studies of the reaction and constructed the frst mathematical model which was able to display the oscillatory behavior. Patterns arising in reaction–diffusion processes can be observed in well-known oscillatory models such as the Brusselator [170] and the Oregonator model [55] of the

(

)

73

Reaction–Diffusion Modeling

Belousov–Zhabotinsky chemical reaction. One can also observe distinct geometric patterns through the Schnakenberg and Brandeisator models of the CIMA reaction [123]. In the following, we discuss the asymptotic stability and Turing instability of some models based on the above theories. 2.6.1 Model 1: Oregonator Model The Oregonator is a mathematical model which captures the essence of the FKN mechanism and was developed by Field and Noyes [56]. Some approximations were made to reduce the number of variables. Tyson and Fife [198] derived the dimensionless rate equations in the model as

˜

du = aw − uw + u − u2 , dt dv = u − v, dt

˜°

dw = −aw − uw + bv. dt

(2.124) (2.125) (2.126)

From the measured rate constants for the BZ reaction, the authors estimated the values of the parameters as ˜ °  ˜  1, a  1, b ˛ 1. Since the variable w evolves on the fastest time scale, it is eliminated by setting dw dt = 0. From (2.126), we obtain w = bv (u + a). Then, the three-variable Oregonator model that describes the Field–Körös–Noyes mechanics of BZ reaction reduces to the two-variable RD Oregonator model as given by Zhou [223]

˜u ˙ u − a˘ ˜2u + d 2 , x , t > 0, = ˇ u − u2 − bv  ˜t ˆ u + a ˜x

(2.127)

˜v ˜2v = u − v + 2 , x ˙ ˆ, t > 0, ˜t ˜x

(2.128)

˜ˆ u = ˜ˆ v = 0, x ˛˜ ˝, t > 0, u ( x, 0 ) = u0 ( x ) ˝ 0, ˆ/ 0, v ( x , 0 ) = v0 ( x ) ˝ 0, ˆ/ 0, x ˇ˘, where ˜ °  N is a bounded domain with smooth boundary ˜ Ω, ° is the outward unit normal vector of the boundary ˜Ω. The homogeneous Neumann boundary conditions indicate that the system is self-contained with zero fux across the boundary. The constants a,  b , d are positive (here, we have used different notations for the constants). The existence of oscillatory cycles and Hopf bifurcation from the trivial steady state (0, 0) were shown in [197,198]. Turing patterns, existence and nonexistence of positive nonconstant steady-state solutions were studied by Peng and Sun [167]. From the work of Zhou [223], we briefy present the analysis of asymptotic stability and Turing instability of the system (2.127) and (2.128). The nontrivial equilibrium point of the model system is given by u* = v * =

1˛ P + P 2 + 4 a(1 + b) ˙ , P = 1 − a − b. ˆ 2˝

(2.129)

74

Spatial Dynamics and Pattern Formation in Biological Populations

(

)

The Jacobian matrix J at E* u* , v * is given by ˙ a J = ˇ 11 ˆ 1

Now, a11 + a12 = 1 − 2u* −

( (

) )

b u* − a a12 ˘ 2abv * * = − = − , where a 1 2u − , . a 11 12 2  −1  u* + a a + u*

(

2abu*

(a + u )

* 2

−

( (u

)

) = − ˆ˘ u + 2abu ˘ + a) (a + u ) ˇ = a(1 + b) − ( u ) is used.

b u* − a

*

*

*

*

2

  2 = − p < 0,  

* where the expression ( a + b − 1)u* The Jacobian matrix of the linearized system corresponding to model system (2.127) and (2.128) is given by

˛ a11 − d   kn Jn = ˙ 1 ˝

ˆ a12 ˘ ,  = (0,  l). −1 − kn ˇ

The characteristic equation of J n is ˜ 2 − ˜Tn + Dn = 0, where Tn = a11 − 1 − (d + 1)kn , Dn = dkn2 − ( a11 − d ) kn + p ,  p > 0.

(

)

The equilibrium point E* u* , v * is locally asymptotically stable when the roots of the characteristic equation are negative or have negative real parts. We require the conditions Tn < 0 and Dn > 0. Then, the signs of the coeffcients are (+ , + , +). (i) A suffcient condition for Dn > 0 is a11 ˜ d. Tn < 0 gives a11 < 1 + (d + 1)kn. That is, a suffcient condition is a11 ˛ min [ d , 1 + (d + 1)kn ]. (ii) For a11 − d > 0, the roots of Dn = H ( kn ) are real and positive. The roots are given by s1 , s2 = ˙( a11 − d )  ˆˇ

( a11 − d )2 − 4pd ˘ 

(2d).

(2.130)

and Dn = d ( s − s1 )( s − s2 ). Now, Dn > 0, when kn < s1, or when kn > s2 and Tn < 0, when a11 < 1 + (d + 1)kn. Hence, asymptotic stability is obtained when d < a11, a11 < 1 + (d + 1)kn and (a) kn < s1 or (b) kn > s2 . Zhou [223] derived a similar stronger condition. If Dn = H ( kn ) = dkn2 − ( a11 − d ) kn + p < 0 , then the characteristic equation has one real positive root irrespective of the sign of Tn (the signs of the coeffcients are either +,  +, − or +, −, −) and diffusive instability sets in. From (2.130), Dn < 0 when s1 < kn < s2. Hence, diffusive instability occurs when d < a11, and kn lies in the interval, s1 < kn < s2. H ( kn ) is quadratic in kn, and the graph of y = H ( kn ) is a parabola opening upward. The minimum occurs at the vertex of the parabola,  i.e., for kn = km , where km = [ a11 − d ] (2 d). For the sake of completeness, we have performed numerical simulations. Consider the set of parameter values as a = 0.02, b = 1, d = 0. We obtain a11 = 0.44735, a12 = −0.80975, a11 + a12 = −0.36240, E* = (0.19025, 0.19025). The Oregonator model without diffusion is asymptotically stable since Tn (0) = −0.55265 < 0, and Dn (0) = 0.3624 > 0. For the diffusive model, (d = 0.05) < a11 , km = 3.97348, H ( kn ) < 0, for 1.05107 < kn < 6.89589. For all values of kn lying in this range, the system is unstable. For the above set of parameter values, and with the initial condition taken as u( x , 0) = 0.19025 + 0.02 cos x = v(x, 0), the plot of the Turing instability is given in Figure 2.3. The plot of Dn  vs. kn is plotted in Figure 2.4.

Reaction–Diffusion Modeling

75

FIGURE 2.3 The Turing instability of the equilibrium solution of model (2.127) and (2.128).

FIGURE 2.4 Graph of the function Dn vs. kn.

The Oregonator system also belongs to the class of pure activator–inhibitor systems. Murray [138–140] and Tyson [197] have studied the model thoroughly and gave estimates of the wave speed. Ye and Wang [213] provided a general method for proving the existence of traveling wavefront solutions for a kind of reaction–diffusion systems. By using Schauder’s fxed point theorem, Ma [120] proved some existence results for traveling wavefronts of delayed reaction–diffusion systems with the Belousov–Zhabotinsky reaction. Rodrigo and Mimura [175] proposed a general method for fnding the exact traveling and standing wave solutions of reaction–diffusion systems. These authors [175] applied this method to several well-known systems, including simplifcation of the Field–Noyes model for the Belousov–Zhabotinsky reaction. Zhang [221] established the explicit traveling wave solutions of fve kinds of nonlinear evolution equations which include the Belousov– Zhabotinsky system of reaction–diffusion equations. Turing patterns do not occur in the aqueous BZ system, because the condition of a fast-diffusing inhibitor cannot be realized, since all the small molecule species in this system have nearly equal diffusion coeffcients. 2.6.2 Model 2: Brusselator Model The formation of patterns in chemical reactions was studied by means of a number of RD type models. “Brusselator” appearing in the modeling of chemical morphogenetic processes (Pierre, [168]) is one of the best studied models among them. Prigogine and

76

Spatial Dynamics and Pattern Formation in Biological Populations

Lefever [169] had shown that the model displays sustained oscillatory behavior. It was subsequently named as Brusselator by Tyson [196]. The rate equations of the Brusselator model are du = k 1 a − ( k 2 b + k 4 ) u + k 3 u2 v , dt dv = k2bu − k3 u2 v. dt After appropriate scaling, the evolution of the concentration of two intermediary reactants u and v with diffusion rates d1 ,  d2 with d2 > d1 is described by

˜u ˜2u = a − (b + 1)u + u2 v + d1 2 , x ˙(0, l) ˜t ˜x

(2.131)

˜v ˜2v = bu − u2 v + d2 2 . ˜t ˜x

(2.132)

u( x , 0) = u0 ( x), v( x , 0) = v0 ( x), x °(0, l),

˜u ˜v = 0,  = 0, at x = 0, and x = l, ˜x ˜x where a, b , d1 ,  d2 are positive constants. The Brusselator model has a unique steady state ( a, b a ) and is a cross activator–inhibitor system if b > 1. At the steady state, the trace and determinant of the Jacobian are b − 1 − a 2 and a 2, respectively. Therefore, the steady state of the model cannot undergo a stationary bifurcation (for stationary bifurcation, determinant should be zero). For the Hopf bifurcation, we require trace = 0, which yields bH = 1 + a 2 . For b < bH , the steady state is stable, and for b > bH , it is unstable and the asymptotic state of the Brusselator is a stable limit cycle, an attracting nonconstant periodic solution [129,196]. The frequency of the limit cycle oscillations at the bifurcation point is given by ˜ H = a. Stationary bifurcation and Hopf bifurcation occur as one parameter is varied and are known as co-dimension-one bifurcations. They represent the generic ways in which a steady state of a two-variable system can become unstable. Sometimes, it is possible to make the stationary and Hopf instability threshold coalesce by varying two parameters. Such an instability where trace and determinant are zero is known as Takens–Bogdanov bifurcation or a double-zero bifurcation, since both eigenvalues are zero at such point [69]. This bifurcation is a co-dimension-two bifurcation, since it requires the fne tuning of two system parameters. Ma and Hu [119] studied the bifurcation and performed the stability analysis of steady states to a Brusselator model. Pena and Garcia [165] obtained a generalized amplitude equation of a Brusselator model and studied the stability of stripes and hexagons arising due to spatial perturbations. This system exhibits rich dynamics, including oscillations, spatiotemporal chaos, and Turing instabilities. Many researchers have proposed methods for the numerical solution and studied the stability of the Brusselator system:

(

)

˜u = a − (b + 1)u + u2 v + d˙ 2 u, ˜t

(2.133)

77

Reaction–Diffusion Modeling

∂v = bu − u2 v + d∇ 2 v, ∂t

(2.134)

with initial conditions ( u( x ,  y , 0),  v( x ,  y , 0)) = ( f ( x, y ),  g(x ,  y )) and Neumann boundary conditions on the boundary ∂ C. Twizell et al. [194] have given a second-order finite difference scheme with the Neumann boundary conditions and initial condition f ( x ,  y ) = 2.0 + 0.25y and  g( x ,  y ) = 1.0 + 0.8x. The values of the parameters were taken as a = 2, b = 1 and d = 0.002. Ang [7] had given the dual reciprocity boundary element method for the numerical solution of the Brusselator system with f ( x ,  y ) = 0.5 x 2 − x 3 3  and  g( x ,  y ) = 0.5y 2 − y 3 3 . The values of the parameters were taken as a = 1, b = 0.5, and d = 0.002. They found that the numerical method is stable and tested the convergence and accuracy of the numerical solutions. Mittal and Jiwari [134] solved the 2D RD Brusselator system using polynomial-based differential quadrature method (DQM). They considered the system (2.133) and (2.134) on the square C defined by the lines x = 0,  y = 0, x = 5,  y = 5, with initial conditions u( x ,  y , 0) = f ( x , y), v( x ,  y , 0) = g( x , y ) and Neumann boundary conditions on the boundary ∂ C. Jiwari and Yuan [85] studied the 2D RD Brusselator system using modified cubic B-spline differential quadrature method (MCB-DQM). The MCB-DQM reduces the Brusselator system into a system of nonlinear ODEs and this system was solved by a four-stage RK4 scheme. They studied the model considered by Mittal and Jiwari [134] with the exact solution u( x ,  y , t) = exp(− x − y − 0.5t);  v(x ,  y , t) = exp( x + y + 0.5t). The initial conditions were taken from the exact solution. We discuss the Turing instability of the above system in the following example.

(

)

(

)

Example 2.3 Derive the Turing instability conditions for the standard Brusselator RD system (2.131) and (2.132). Solution The uniform steady state ( a, b a ) of the system is stable if b <   bH = 1 + a 2. The Jacobian matrix of the linearized system corresponding to the model system (2.131) and (2.132) is given by  b − 1 − d  k 2 1 Jn =   −b

a2 − a − d2 k 2

2

 nπ . ,k=  l

The characteristic equation of J n is given by λ 2 − λTn + Dn = 0, where

(

)

Tn = b − 1 − a 2 − ( d1 + d2 ) k 2 , 2 2 2 Dn = d1d2 k 4 − d  2 (b − 1) − d1 a  k + a .

The Turing instability occurs when one or both the roots of the characteristic equation are positive. For Dn < 0, one of the roots of the characteristic equation is positive, irrespective of the sign of Tn . A necessary condition is that the roots of Dn are real and positive. The roots of Dn are given by 2

2 2 2 d  2 (b − 1) − d1 a  ±  d2 (b − 1) − d1 a  − 4d1d2 a k ,  k = . 2d1d2 2 1

2 2

78

Spatial Dynamics and Pattern Formation in Biological Populations

We obtain the conditions 2

2 2 2  d2 (b − 1) − d1 a 2  > 0, or  d2 d1  > a  ( b − 1)  ; and d2 ( b − 1) − d1 a  ≥ 4d1d2 a .

(2.135)

Equality gives repeated positive roots for Dn . In this case, Dn becomes a perfect square. Define  d2 d1  = θ RD. Since d2 > d1, the first condition can be taken as

2 2 2 2 θ RD > a  ( b − 1)  > 1. For k1 < k 1), that is, the activator must diffuse slower than 2

2 2 the inhibitor. The critical value of θ RD is obtained from d  2 (b − 1) − d1 a  = 4d1d2 a . The

(

)

2

critical ratio of the diffusion coefficients is given by θ RD, cr =  a 1 + b ( b − 1)  . If we con  sider b as the control parameter, then the threshold for the Turing instability is given by

(

bRD,  Thr = 1 + a d1 d2

) = (1 + aθ ) . The critical wave number is given by l 2

−1/2 2 RD

2 T ,  RD

= a 2 d1d2

. In order that Turing bifurcation occurs first, the Turing threshold must lie below the

(

−1/2 Hopf threshold of the model system, that is bT  2 θ RD (θ RD − 1)  , which implies that a becomes arbitrarily large as θ RD → 1. Infinite

a corresponds to an infinite Hopf frequency and an infinite wave number of the Turing instability [129]. For the sake of completeness, we have performed numerical simulations. For example, consider a set of parameter values as a = 2,  b = 4.6, d1 = 0.25,  d2 = 1. We obtain E* u* ,  v * = (2, 2.3). The Brusselator model without diffusion is locally asymptotically stable since Tk (0) = −0.4 < 0, and Dk (0) = 4 > 0. For diffusive model, D k c2 < 0, for 1.87735 < k c2 < 8.52265. For the above set of parameter values, and with the initial condition taken as u( x , 0) = 2 + 0.01cos x ,  v(x , 0) = 2.3 + 0.01cos x , the plot of the Turing instability is given in Figure 2.5. The plot of Dk vs. k 2 is plotted in Figure 2.6.

(

)

( )

Ecological systems (like marine systems or plankton systems) can display temporal oscillations with changes in parameter values due to seasonal variations. The effect of time-varying diffusivities on the Turing instabilities was first considered by Timm and

FIGURE 2.5 The Turing instability of the equilibrium solution of the model (2.131) and (2.132).

Reaction–Diffusion Modeling

79

FIGURE 2.6 Graph of the function Dk vs. k2.

Okubo [191]. Mendez et al. [129] derived conditions for the Turing instability of time- and space-varying diffusivities. Trinh and Ward [192] derived and studied the differential algebraic equation that characterizes the slow dynamics for spot patterns for the Brusselator RD model on the surface of a sphere. The authors presented the asymptotic and numerical solutions for the system governing the spot strengths. Localized spot patterns can undergo fast time instability, and the authors derived the conditions for the phenomena, which depend on the spatial confguration of the spots and the parameters in the system. 2.6.3 Model 3: Schnakenberg Model The Schnakenberg model is a modifcation of the Brusselator model. Liu et al. [117] derived the dimensionless reaction–diffusion equations as

˜u ˜2u = a − u + u2 v + d1 2 , x ˙(0, lˆ), t > 0, ˜t ˜x

(2.136)

˜v ˜2v = b − u2 v + d2 2 , x ˙(0, lˆ), t > 0, ˜t ˜x

(2.137)

u( x , 0) = u0 ( x) ° 0, v( x , 0) = v0 ( x) ° 0, x ˛(0, l˝), ux (0, t) = ux (l°,  t) = vx (0,  t) = vx (l° ,  t) = 0,  t > 0. where u( x ,  t) and v( x ,  t) are the concentrations of the chemical products at time t and location x ˜(0, l°), with Neumann boundary conditions so that the chemical reactions are in a closed environment, and a,  b ,  d1 ,  d2 are positive constants. Denote a + b = ˜ , and b − a = ˜ ; that is, a = (˜ − ° ) 2, and b = (˜ + ° ) 2 . Then, the model has a unique positive steady state E(u, v) =   ˜ , [˜ + ° ] ˘ˇ 2 ˜ 2  . The model is a cross activator–inhibitor system if b > a. The

(

)

determinant of the Jacobian at the equilibrium point is ˜ 2 which is always positive, and no stationary bifurcation can occur. The condition for a Hopf bifurcation is trace = 0, which 3 yields bH − a = ( a + bH ) . Following the eigenvalue analysis, the characteristic equation of J n , ( k = n l ) at the equilibrium point E is given by

µ 2 − µTn (° ) + Dn (° ) = 0, n = 0, 1, 2,

(2.138)

80

Spatial Dynamics and Pattern Formation in Biological Populations

where

(

)

Tn (˜ ) = p − ° 2 − ( d1 + d2 ) k 2 , Dn (˜ ) = d1d2   k 4 + d1° 2 − d2 p k 2 + ° 2 , p = ˜ ° . The authors [117] have analyzed the Hopf bifurcation occurring at the point E(u, v) by choosing ˜ as a bifurcation parameter. Letting Tn (˜ ) = 0, Dn (˜ ) > 0, Tj (˜ ) ˝ 0,  Dj (˜ ) ˝ 0 for 2 any j ˜ n , we obtain. ˜ = ˜ jH = ° 3 + ° ( d1 + d2 ) ( j l ) Then, Tj ˜ jH = 0 and Ti ˜ jH ˝ 0 for i ˜ j.

( )

The authors have also shown the following results: (i) Defne ln = n

( d1

( ) + d ) ( 1 − ˜ ) , n ˜ . 2

2

0

Then, for ln < l ° ln+1 , we have exactly (n + 1) possible Hopf bifurcation points˜ = ˜ , (0 ˛ j ˛ n). (ii) Di ˜ iH ˝ 0 for all i ˜ 0. (iii) At ˜ = ˜ jH , let the eigenvalues close to the pure imaginary ones be ˜ (° ) ± i˛ (° ). Then, ˜ ˆ ° jH = ˘Tjˆ ° jH 2  = ˘1 ˛  > 0. Therefore, the bifurcating periodic orbits from ˜ = ˜ jH are spatially nonhomogeneous. For the sake of completeness, we have performed numerical simulations. For example, consider the set of parameter values as a = 0.14,b = 0.16, d1 = 0.01, d2 = 1. For these values b − a = ˜ = 0.02 andb + a = ˜ = 0.3, p = 1/15. We obtain E* u* ,  v * = (0.3, 1.7778). The Schnakenberg model without diffusion is locally asymptotically stable since Tn (0) = −0.0233 < 0, and Dn (0) = 0.09 > 0. For the diffusive model, D kc2 < 0, for 1.94172 < kc2 < 4.63508. For the above set of parameter values, and with the initial conditions taken as u( x , 0) = 0.3 + 0.001sin ( 5 x 3 ), v( x , 0) = 1.7778 + 0.001cos ( 5 x 3 ), the Turing instability of the equilibrium solution of the spatial model system (2.136) and (2.137) is presented in Figure 2.7. For this case, the graph of Dk vs. k 2 is plotted in Figure 2.8. H j

( )

( )

(

( )

)

( )

2.6.4 Model 4: Lengyel–Epstein Model The chlorite–iodide–malonic acid (CIMA) reaction played a key role in the frst observation of stationary spatial chemical patterns in nonlinear chemical dynamics. A major variant of the CIMA reaction is the simpler chlorine dioxide–iodine–malonic acid (CDIMA) reaction. Lengyel, Rabai, and Epstein (LRE) [105,106] proposed an empirical rate law description of the CDIMA reaction, which is referred to as LRE model. Lengyel and Epstein [102,103] have simplifed the LRE model to refect the behavior only of the iodide and chlorite ions. De Kepper et al. [29] discovered the formation of stationary three-dimensional (but almost 2D) structures with characteristic wavelengths of 0.2 mm which is the frst experimental

FIGURE 2.7 The Turing instability of the equilibrium solution of the model (2.136) and (2.137).

81

Reaction–Diffusion Modeling

FIGURE 2.8 Graph of the function Dk vs. k2.

evidence to the Turing patterns. A detailed historical account of development of CIMA reaction model and experiments can be found in Epstein and Pojman [40]. Bansagi Jr. and Taylor [9] investigated the formation of helical Turing patterns in the cylindrical layers using the LE model of the CDIMA reaction. They found that helices were obtained from random initial conditions in cylinders where spots were observed in two dimensions. Assume that the reactor ˜ is a bounded domain in  n, with a smooth boundary ˜ °. Let u( x ,  t) and v( x ,  t) denote the chemical concentrations of the activator iodide (I − ) and the inhibitor chlorite ClO −2 , respectively, at a point x ˜° and at time t > 0. The Lengyel– Epstein reaction–diffusion model is given by [102–104,214,215]

(

)

˜u 4uv ˜2u + = a−u− , ˜t 1 + u2 ˜ x 2

(2.139)

 ˆ ˜v ˜2v  uv  + d = ° b ˘ u − .  ˜t ˜ x 2  1 + u2   ˇ

(2.140)

The parameters a and b are related to the feed concentrations, d is the ratio of the diffusion coeffcients, and ˜ > 0 is a rescaling parameter depending on the concentration of the starch enlarging the effective diffusion ratio to ˜ d. In laboratory conditions, the parameters were taken in the ranges 0 < a < 35, 0 < b < 8, d = 1.5, and ˜ = 8. Ni and Tang [147] studied the existence of the steady states of the RD system, subject to the initial conditions u( x , 0) = u0 ( x) > 0,  v( x , 0) = v0 ( x) > 0 and with no-fux boundary conditions. It was shown that the system possesses spatially homogeneous periodic solutions for some parameter ranges, and the interaction of the Hopf and Turing bifurcations could be the driving force of more complicated spatiotemporal phenomena [178]. Rovinsky and Menzinger [178] have also derived the parameter ranges of the Hopf and Turing instability as well as bifurcation directions. Lengyel–Epstein model has a unique steady state E ˜ ,1 + ˜ 2 , ˜ = a 5.

(

(

2

)

)

The model is an activator–inhibitor system under the condition 3a − 125 > 0 [140]. The characteristic equation of the Jacobian J at the equilibrium point is given by ˜ 2 − p˜ + q = 0, where p = trace J = ˇˆ 3a 2 − 5˜ ab − 125˘ a 2 + 25 and q = det J = [25˜ ab] a 2 + 25 . The equi-

(

)

(

librium point E (u , v ) is locally asymptotically stable if 0 < 3a *

*

2

( ) − 125 ) < 5˜ ab (when p < 0

and q > 0). The condition for the Hopf bifurcation of the steady state E is p = 0, which yields

(

)

bH = 3a 2 − 125 (5a˜ ). Consequently, for a > 125 3 , the unique steady state of the CDIMA

82

Spatial Dynamics and Pattern Formation in Biological Populations

reaction in a CSTR is stable for b > bH , and for b < bH , the reaction oscillates. The iodide ion is an activator for a > 125 3 , and the chlorite ion is the inhibitor for all values of a and b [129]. Ni and Tang [147] and Yi et al. [214] have derived the conditions for diffusion-driven instability for the equilibrium solution (u*, v*) and Turing bifurcation for the CIMA reaction. Yi et al. [214] considered the model system (2.139) and (2.140) with no-flux boundary conditions ux (0,  t) = ux (π ,  t) = 0, vx (0,  t) = vx (π ,  t) = 0. The linearized form of the system at E α ,1 + α 2 , α = a 5 , is

(

)

 ut   uxx   u   u  = J + D  ,  = L    v   v   vxx   vt    where J =  

( 3α

2

)

−5 s

2σ bα s 2

  1 ,D=  0  −σ bα s  −4α s

0  , s = 1 + α 2 , with domain σ d 

{(u,  v) ∈ H [(0,  π)] × H [(0,  π)] : u (0, t) = u (π, t) = 0, v (0, t) = v (π, t) = 0} , 2

2

x

x

x

x

where H 2 (0, π)  is the standard Sobolev space. It is well known that the operator u → − uxx with given no-flux conditions has eigenvalues and eigenfunctions as

µ0 = 0, ϕ 0 ( x) =

1  , µ k = k 2 , ϕ k ( x) = π

2  cos ( kx ) , k = 1, 2, 3, π

From the standard linear operator theory [21], if all the eigenvalues of the operator L have negative real parts, then the equilibrium solution (u*, v*) is asymptotically stable, and if some eigenvalues have positive real parts, then (u*, v*) is unstable. The authors [214] have derived the following results: Suppose that b > b0 =: 3α 2 − 5 (σα ). If the equilibrium solution (u*, v*) is locally asymptotically stable for the temporal model, then it is locally asymptotically stable for the spatial model when α 2 > 3, 0 < d < 3bα α 2 − 3 , and unstable when α 2 > 3, and d > 3bα α 2 − 3 . For the sake of completeness, we have performed numerical simulations. For example, consider the set of parameter values as a = 15, σ = 8, b = 1.2, d = 2. We obtain E* u* , v * = (3, 10). The Lengyel–Epstein model without diffusion is locally asymptotically stable since Tk (0) = −0.68 < 0, and Dk (0) = 14.4 > 0. For the diffusive model, D kc2 < 0, for 0.6634  < kc2 <   1.3565. For the above set of parameter values, and with the initial conditions taken as u( x , 0) = 3 + sin x, v( x , 0) = 10 + cos x, the Turing instability of the equilibrium solution of the spatial model system (2.139) and (2.140), is presented in Figure 2.9. The graph of Dk vs. k 2 is plotted in Figure 2.10.

(

(

( )

) (

)

)

( )

2.6.5 Model 5: Sel’kov Model Sel’kov [184] introduced an autocatalytic model for glycolysis. It is now used for the study of morphogenesis, population dynamics, autocatalytic oxidation reactions, etc. [82,140,182]. The reaction–diffusion equations take the following form [184]:

∂u ∂2u = λ 1 − uv p + d1 2 , ∂t ∂x

(

)

(2.141)

83

Reaction–Diffusion Modeling

FIGURE 2.9 The Turing instability of the equilibrium solution of the model (2.139) and (2.140).

FIGURE 2.10 Graph of the function Dk vs. k2.

∂v ∂2v = λ uv p − v + d2 2 , ∂x ∂t

(

)

(2.142)

(x ,  t) ∈(0,  kπ) × (0,  ∞), ux ( x ,  t) = vx ( x ,  t) = 0, at x = 0, kπ , t > 0, where u and v are the nonnegative concentrations of the two reactants or densities of the two species, d1 and d2 are the diffusion coefficients of u and v, respectively, and λ and p are fixed positive constants. In its simplified nondimensional form, the steady-state model is given by the following coupled elliptic system (Peng [166]) −θ∆u = λ 1 − uv p ,

(

)

(2.143)

(

)

(2.144)

−∆v = λ uv p − v ,

in a bounded domain Ω ⊂  n (n ≥ 1) with smooth boundary ∂ Ω. The homogeneous Neumann boundary conditions are ∂ν u = ∂ν v = 0 on ∂Ω. The above system was extensively studied both analytically and numerically by the following authors: in 1D space dimension by Eilbeck and Furter [37] and Lopez-Gomez et al. [118]; in 2D space dimensions by Cameron [17]; in n space dimensions (n = 1, 2, 3, …) by Eilbeck [36] and Davidson and Rynne [28]. Davidson and Rynne [28] had also obtained a priori upper bounds for positive classical solution of the system (2.143) and (2.144) in the two cases 0 < p < ∞ for n = 1, 2, and 0 < p < 3 for n = 3. Wang [205] had established refined a priori estimates of upper and lower bounds for the positive solution of the system. Combining these results with the theory of topological degree [148], the local and global bifurcations were studied by Rabinowitz [172],

84

Spatial Dynamics and Pattern Formation in Biological Populations

Crandall and Rabinowitz [25]. Eilbeck and Furter [37] performed numerical bifurcation computations to show that the 1D problem has nonconstant solutions for suitable ranges of the parameters. Peng [166] studied the existence of nonconstant positive solutions of the system and performed qualitative analysis of the steady states of the Sel’kov model. Han and Bao [72] performed detailed Hopf bifurcation analysis and also derived conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution of the system (2.141) and (2.142). In the following, we briefy present the analysis. The system (2.141) and (2.142) has a unique fxed point (1, 1). The linearized form of the system at E(1,1) is ˜ ut ˝ ˜ uxx ˝ ˜ u ˝ ˜ u ˝ = J˛ + D˛ ˛ ˆ = L˛ ˆ, ˆ ˆ ° v ˙ ° v ˙ ° vt ˙ ° vxx ˙ ˛ −˜ ˆ ˛ d1         0 ˆ − p˜ where D = ˙ ˘. ˘, J =˙ ˙˝ ˜ ˜ ( p − 1) ˘ˇ ˝ 0         d2 ˇ It is well known that the operator u ˜ −uxx, x ˜(0, k°) with homogeneous Neumann boundary conditions has eigenvalues and eigenfunctions as

µ0 = 0, ° 0 ( x) = 1; µn = n2 k 2 , ° n ( x) = cos ˝˙(nx) k ˆˇ , n = 1, 2, 3, ˙ − ˜ − d1µn ˘ −p˜ Introduce the operator, J n ( p) = ˇ . ˜ ˜ (p − 1) −  d2 µn  ˆˇ Following the eigenvalue analysis, the characteristic equation of J n ( p) at the equilibrium point E is given by

µ 2 − µTn ( p) + Dn ( p) = 0, n = 0, 1, 2,

(2.145)

where Tn ( p) = (p − 2)˜ + ( d1 + d2 ) µn , Dn ( p) = d1d2 µn2 − ˜ ˘ d1 (1 − p) + d2  µn + ˜ 2 . For the Hopf bifurcations to occur, there exist n ˜  {0} such that Tn ( p) = 0, Dn ( p) > 0;  Tj ( p) ˛ 0,  Dj ( p) ˛ 0 for j ˛ n; Let the unique pair of complex eigenvalues that exist near the imaginary axis be ˜ ( p) ± i° ( p). Then, the transversality condition ˜ °( p) ˛ 0 holds. When Tn ( p) = 0, we have p = ˘ˇ( d1 + d2 ) µn °  + 2. Inserting the value of p in Dn ( p) > 0, we obtain d12 ( µn ) − ° ( d2 − d1 ) µn − ° 2 < 0. 2

2 n2 ( d2 − d1 ) + ( d2 − d1 ) + 4d1 < ˜. k2 2d12 Therefore, all the possible bifurcating values of the parameter p can be labeled as 2

Then, 0 ˝

{ }

˛ = pnH

N n= 0

for some N ˜˜ ° {0} satisfying 2 = p0H < p1H < p2H <  < pNH < +˝ ,

85

Reaction–Diffusion Modeling

(p

H N

)

− 2 ˜k2

( d2 − d1 ) + ( d2 − d1 )2 + 4d12

˜k2. d1 + d2 2d12 1 ( d1 + d2 ) n2  ,  ˛ ( p) = D ( p) − ˜ 2 ( p). Recall that ˜ ( p) = (p − 2)° − n  2 k2  T˛ ° We fnd that ˜ ˛( p)|p = pnH = n ( p)|p = pnH =   > 0. 2 2 The model system (2.141) and (2.142) undergoes a Hopf bifurcation at p = pnH , at which the bifurcating periodic solution is spatially dependent for n ˜ 0. The authors have also proved that the spatially independent bifurcating periodic solution is supercritical and asymptotically stable at p = pnH .

such that 0 ˝


0, and v = 1 d ,˜ = . 2 F u 2 F+k

86

Spatial Dynamics and Pattern Formation in Biological Populations

The two additional steady states are

((

)

) )

(

(

) )

(

E2 (u, v) = 1 − d 2 , α 1 + d 2 and E3 (u, v) = (1 + d ) 2 , α 1 − d 2 , where α measures the rate at which U is supplied in terms of the rate at which V is removed. E2 (u, v) is called the blue state. For discussing the stability of the steady states, consider the Jacobian of the system  − F − v2 J= v2 

 −2uv . 2 uv − ( F + k)  

(2.148)

For the red state E1 (1, 0), trace  J = −2 F − k < 0,    J   = F( F + k ) > 0. Hence, the eigenvalues have negative real parts and the red state is always linearly stable. For the steady state E3 (u, v), the authors [126] have shown that trace  J > 0,     J   < 0. Therefore, E3 (u, v) is always unstable. For F > 1 4, the production of V is effectively suppressed and the system exhibits only a single steady state, the red state. For F < 1 4, the production of V may take off and the two steady states E2 (u, v) and E3 (u, v) may exist. In order that Turing structures may appear, the diffusion constant of the depleter must be significantly larger than the diffusion coefficient of the activator, which implies short-range activation and long-range depletion. For Turing bifurcation to appear, we require the passage of a real eigenvalue through zero. The red state is always stable. There is a possibility that the blue state E2 (u, v) may provide the Turing instability. Using the eigenvalue analysis, the Jacobian of the system (2.146) and (2.147) is obtained as  − F − v 2 − d1q 2 J=  v2 

−2uv 2 uv − ( F + k ) − d2 q 2

 .  

(2.149)

The characteristic equation is given by

( )

( )

λ 2 + T q 2 λ + D q 2 = 0,

(2.150)

( )

(2.151)

where T q 2 = ( d1 + d2 ) q 2 + v 2 − k, and

{ (

( )

}

)

(

)

D q 2 =   d1d2 q 4 + d2 v 2 + F − d1 ( F + k ) q 2 + ( F + k ) v 2 − F . 

(2.152)

( )

The system is stable without diffusion implies that Trace  J = k − v 2 < 0 . Hence, T q 2 > 0. For one of the eigenvalues to be positive, we require D q 2 < 0 (actually, irrespective of the sign of T q 2 , one of the eigenvalues is positive when D q 2 < 0). Therefore, the Turing instability occurs when D q 2 < 0. Now, y = D q 2 is a parabola which attains its minimum value for

( )

( )

( ) ( )

( ) (

)

qc2 =  d1 ( F + k ) − d2 v 2 + F  ( 2 d1d2 ) .

(2.153)

87

Reaction–Diffusion Modeling

This determines the value of the critical wave number, that is, the wave number of the modes that frst becomes unstable. For q = qc , we have

( )

(

)

D qc2 = ( F + k ) v 2 − F −

2 1 ˇ d ˜ ( F + k ) − v 2 + F  , ˜ = 1 . ˘ 4˜ d2

(

)

(2.154)

( )

D qc2 < 0, gives the condition

(

)

(

2

)

ˆ˜ ( F + k ) − v 2 + F ˘ > 4˜ (F + k ) v 2 − F . ˇ 

(2.155)

Together with the condition that q is real, (2.155) determines the Turing space. Alternately, we may write D q 2 < 0 as

( )

(

ˇ d1 ( F + k ) − d2 v 2 + F q 4 − c1q 2 + c2 < 0, where c1 = ˘ d1d2 

)  , and c  

2

=

(

(F + k ) v 2 − F d1d2

).

2 = ˛ c1  c12 − 4c2 ˙ 2 , c12 − 4c2 > 0. We have from (2.152) The roots are q1,2 ˝ ˆ

( )

(

)(

)

D q 2 = d1d2 q 2 − q12 q 2 − q22 .

(2.156)

( )

For q12 < q 2 < q22 , D q 2 < 0. Therefore, for all wave numbers lying in this range, the Turing instability appears. For the sake of completeness, we have performed numerical simulations. For example, consider the set of parameter values as F = 0.064, k = 0.062, d1 = 0.01, d2 = 0.005. We obtain E* u* , v * = (0.455983, 0.276326). The model without diffusion is locally asymptotically stable, since T(0) = −0.0143 < 0, and D(0) = 0.001557 > 0. For the diffusive model, D q 2 < 0, for 5.42903 < q 2 < 5.73535. For the above set of parameter values, the plot of the Turing instability is given in Figure 2.11. The plot of D q 2 vs. q 2 is plotted in Figure 2.12. The initial condition is taken as u0  ( x) = 0.455983 + 0.01sinx, and v0  ( x) = 0.276326 + 0.01cosx. Nishiura and Ueyama [149] presented a new geometrical criterion for the transition to spatiotemporal chaos (STC) arising in the GS model. The geometrical characterization

(

)

( )

FIGURE 2.11 The Turing instability results of the equilibrium solution of the model (2.146) and (2.147).

( )

88

Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 2.12 Graph of D (q2) vs. q2.

gives us a universal view point about the onset and termination of STC. The GS model is known to produce STC due to interplay between a stable state and a limit cycle. Berenstein and Decker [15] showed that the limit cycle alone can produce STC of the defect-mediated type with equal diffusivity constants for both the species and the GS model is able to produce defect-mediated turbulence. Berenstein [14] has shown that standing wave-like patterns are obtained in the model when the dynamics that correspond to defect-mediated turbulence for equal diffusivities interact with the Turing instabilities. The Turing instability can be caused by either differential or cross-diffusion.

2.7 Multispecies Reaction–Diffusion Models The complex dynamics of tritrophic food chain models when the species undergoes spatial movements is of importance. Reaction–diffusion systems have been used to represent temporal evolution and spatial interaction among the species. The two model systems are differing in an essential way that the top predators are specialist (Hastings–Powell model [75]) or generalist (Upadhyay-Rai model [201]). Constraints on different parameters under which Turing and non-Turing patterns occur may be derived analytically. 2.7.1 Model 1: Hastings–Powell Model Consider the three-species model proposed by Hastings and Powell [75] with prey– specialist intermediate predator–specialist top predator interaction. At any location (u, v) in space and time t, the prey population of density x is the favorite food of the intermediate predator of density y, which serves as favorite food for the specialist top predator of density z. This food chain is described by the following reaction–diffusion system [99,199,212]: wxy ˜x x˘ ˙ = rx ˇ 1 −  − + ° 1x, ˆ ˜t K  1 + d1x

(2.157)

w1 xy w2 yz ˜x = −a2 y + − + ° 2 ˆy, ˜t 1 + d1 x 1 + d2 y

(2.158)

w3 yz ˜z = −a3 z + + ° 3 ˆz, ˜t 1 + d2 y

(2.159)

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Reaction–Diffusion Modeling

with the initial conditions x(u,  v , 0) ≥ 0, y(u,  v , 0) ≥ 0, z(u,  v , 0) ≥ 0,

(2.160)

and no-flux boundary conditions

∂y ∂z ∂x =0= = , at u = 0, and Lu , ∂u ∂u ∂u

(2.161)

∂y ∂z ∂x =0= = , at v = 0, and Lv , ∂v ∂v ∂v

(2.162)

where 0 < u < Lu , 0 < v < Lv. All the parameters are positive and defined as follows: r and K are intrinsic growth rate and carrying capacity of the prey population. a2 is the rate at which intermediate predator will die in the absence of prey.  a3 is the death rate of the specialist top predator z in the absence of its prey y, and w3 is a measure of its assimilation efficiency. w ,  wi , i = 1, 2 are the maximum values attainable by the per capita functional and numerical response of the specialist predator y and the functional response of the top predator z, respectively. The parameters d1 , d2 are constants; δ i , i = 1, 2, 3 are diffusion coefficients, and ∆ is the Laplace operator in one or two dimensions. The systems are defined on a bounded habitat and are augmented with appropriate initial conditions and zero flux boundary conditions. Kuznetsov et al. [100] showed that the temporal part of the model (2.157)–(2.159) admits a sequence of pairs of Belyakov bifurcation, then fold- and period-doubling cyclic bifurcation curves associated with each pair of Belyakov points. Yang and Fu [212] studied the asymptotic behavior of solutions of the RD system in 1D and the global existence of the solutions of the cross-diffusive system. The nontrivial interior equilibrium point E x * ,  y * , z* of the spatial model (2.157)–(2.159) is given by

(

)

y * = a3 ( w3 − a3 d2 ) , w3 > a3 d2 .

(A

A ±  2 x =

)

* * * − 4A1 A3   * D1  w1x − a2 D  ,z = , where A1 = rd1 , A2 = r ( Kd1 − 1) , 2A1  w2 D*  2 2

*

(

)

(

)

(

)

A3 = K wy * − r , D* = 1 + d1 x * , D1* = 1 + d2 y * , with Kd1 > 1, and wy * > r. For studying the linear stability of the spatial model (2.157)–(2.159), it is perturbed as x = x * + a  exp ( λ k t + i ( kuu + k v v )) , y = y * + b   exp ( λ k t + i ( kuu + k v v )) , z = z* + c  exp ( λ k t + i ( kuu + k v v )) ,

(2.163)

where a, b, c are sufficiently small constants, ku and k v are the components of wave number k along u and v directions, respectively, and λ k is the wave length. The system is linearized about the nontrivial interior equilibrium point E x * ,  y * , z* . The Jacobian of the linearized version of the spatial model system (2.157)–(2.159) is given by J xyz − λ I 3 x  = 0. The characteristic equation is given by

(

λ 3 + s1λ 2 + s2 λ + s3 = 0,

)

(

)

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Spatial Dynamics and Pattern Formation in Biological Populations

where

( )

s1 = −trace J xyz = − ( a11 + a22 ) + (  δ 1 + δ 2 + δ 3 ) k 2 ,

{

}

s2 = (δ 1δ 2 + δ 2δ 3 + δ 3δ 1 ) k 4 − a11 (δ 2 + δ 3 ) + a22 (δ 3 + δ 1 ) k 2   + ( a11a22 − a12 a21 − a23 a32 ) ,

( )

s3 = det J xyz = a11a23 a32 + k 2 {( a11a22 − a12 a21 )δ 3 − a23 a32δ 1 } − k 4 ( a11δ 2 + a22δ 1 )δ 3 + k 6 (δ 1δ 2δ 3 ) ,   2rx *  wy *  2 w1 y * wx * * * + D1* , a a with a11 = r −  , = − ,   = 21 12 2  2 ,  a22 = w2 d2 y z *  * * D D  K  D  a23 =

−w2 y * w3 z * 2 2 2 , a = 32 2 , a13 = 0,  a31 = 0, a33 = 0,  k = k u + k v . D1* D1*

Now, Re(λ ) < 0 provided that s1 > 0, s2 > 0,  s3 > 0 and s1s2 − s3 > 0 (R-H criteria), which are therefore the conditions for the steady state to be stable. Diffusion-driven instability requires that the stable homogeneous steady state is driven unstable by the interaction of the dynamics and diffusion of the species. Two-species RD systems have been studied by many authors [79,138,183,199,200]. Diffusion-driven instability can occur for some finite ranges of wave numbers, producing stable spatial patterns (essentially independent of initial conditions) which are small disturbances away from the homogeneous stable steady state [210]. In the case of three-species interaction, the complexity of observable patterns increases and the complex wave number may also produce complex spatial structures [154]. If any of the four inequalities given by R-H criteria becomes negative, then spatial patterning will be observed. White and Gilligan [210] showed that a change in sign of s3 k 2

( )

( ) − s ]( k ) can produce spatiotemporal patterns. For the sake of completeness, we 2

(provided s2 k   > 0) produces fixed spatial patterning, whereas a change in the sign of [ s1s2 3 2 have performed numerical simulations of the temporal model corresponding to the system (2.157)–(2.159), using the parameter values as given in Upadhyay and Iyengar [199]: r = 1,  K = 1, w = 5, d1 = 0.5, a2 = 0.4, w1 = 5, w2 = 0.1,  w3 = 0.1, a3 = 0.01,  d2 = 2. The interior equilibrium point is (0.5, 0.125, 20). The values of the coefficients of the characteristic equation are s1 (0) = 0.08,  s2 (0) = 0.6848, s3 (0) = 0.00512, and [ s1s2 − s3 ](0) = 0.049664. Therefore, the temporal model system is stable. Now, for the spatial model (2.157)–(2.159), the values of the diffusion coefficients are chosen as δ 1 = 0.8, δ 2 = 0.009, δ 3 = 0.01. We obtain s3 k 2 < 0, for 9.51564 < k 2 < 25.8292. Hence, we obtain the Turing instability at these parameter values (Fig 2.13a). The model shows complex spatial patterns at t = 800, for the parameter values r = 1.75,  K = 35, w = 0.1, d1 = 0.1, a2 = 0.2,  w1 = 0.08, w2 = 0.145,  w3 = 0.05, a3 = 0.1, d2 = 0.05, δ 1 = 1, δ 2 = 1, δ 3 = 1 (Figure 2.13b), (Kumari [99]).

( )

Reaction–Diffusion Modeling

91

FIGURE 2.13a Graph of S3 (k2) vs. k2.

FIGURE 2.13b 2D complex spatial pattern of prey, specialist, and top speciality predators populations of model system (2.157)–(2.159) at time t = 800. (Reproduced with permission from Kumari, N. 2013. Pattern formation in spatially extended tritrophic food chain model systems: generalist versus specialist top predator. ISRN Biomath., 2013, 1–12, ID 198185, [99]; and Hindawi Publishing Corporation. Copyright 2013.)

2.7.2 Model 2: Modified Upadhyay–Rai Model Upadhyay and Rai [201] proposed a three-species food chain model with generalist top predator and specialist intermediate predator population. The model based on a modifed version of the Leslie–Gower scheme incorporates mutual interference in all the three populations and generalizes several other known models in the ecological literature. In this model, prey population of size x1 serves as the only food for the specialist intermediate predator population of size x2 . This predator population, in turn, serves as a favorite food for the generalist predator population of size x3. The equations for rate of change of population size for prey and specialist predator were written following the Volterra scheme; that is, predator population dies out exponentially in the absence of its lone prey. The interaction between this predator x2 and the generalist predator x3 was modeled by the modifed version of the Leslie–Gower scheme where the loss in a predator population is proportional to the reciprocal of per capita availability of its most favorite food. This interaction is represented by the following system of a simple prey–specialist

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Spatial Dynamics and Pattern Formation in Biological Populations

predator–generalist predator interaction with the inclusion of spatial spread (Parshad et al. [159]):  x1  ˜ x1 = d1ˆx1 + a1x1 − b1x12 − ° 0   x1 + D0  ˜t

m1

x2m2

= g1 ( x1 , x2 , x3 ) + d1ˆx1 ,

˜ x2  x1  = d2 ˆx2 − a2 x2 + ° 1   x1 + D1  ˜t

(2.164)

m1

x

m2 2

 x2  − °2   x2 + D2 

= g 2 ( x1 , x2 , x3 ) + d2 ˆx2 ,

m2

x3m2 (2.165)

1 ˜ x3 = d3 ˆx3 + cx3m3 − ° 3 x m3   ˜t ( x2 + D3 ) 3 = g 3 ( x1 , x2 , x3 ) + d3 ˆx3 ,  x1 ,  x2 ,  x3     2 ,

(2.166)

where a1 , a2 , b1 , c , ˜ 0 , ˜ 1 , ˜ 2 , ˜ 3 are positive constants, D0 , D1 , D2 , D3 ˜ 0 and , mi , i = 1, 2, 3 are mutual interference parameters that model the intraspecifc competition among predators when hunting for prey [13,42,43,59,60,74]. The initial conditions are x1 ( x , 0) = x10 , x2 ( x , 0) = x20 , x3 ( x , 0) = x30 , and the boundary conditions are x1 = x2 = x3 = 0 on the smooth boundary ˜Ω. a1 is the intrinsic growth rate of the prey population x1, a2 is the intrinsic death rate of the predator population x2 in the absence of the only food x1, c measures the rate of self-reproduction of generalist predator x3, and ˜ 0 , ˜ 1 , ˜ 2 , ˜ 3 are the maximum values which per capita growth rate can attain. b1 measures the strength of intraspecifc competition among the individuals of the prey species x1. D0 and D1 quantify the extent to which environment provides protection to the prey x1 and may be thought of as a refuge or a measure of the effectiveness of the prey in evading predator’s attack. D2 is the value of x2 at which per capita removal rate of x2 becomes ˜ 2 2. D3 represents the residual loss in x3 population due to severe scarcity of its favorite food x2 . For m1 = m2 = 1, the coeffcient ˜ 0 ( x1 + D0 ) of the third term on the right-hand side of (2.164) is obtained by considering the probable effect of the density of the prey’s population on predator’s attack rate. If this coeffcient is multiplied by x1 (the prey population at any instant of time), it gives the attack rate on the prey per predator. Denote p ( x1 ) = (˜ 0 x1 ) ( x1 + D0 ). When x1 ˝ ˙, p ( x1 ) ˝ ˜ 0, which is the maximum that it can reach. The third term (˜ 2 x2 x3 ) ( x2 + D2 ) on the right-hand side of (2.165) represents the per capita functional response of the invertebrate predator x3 and was frst introduced by Holling [78] in the ecological literature. The interaction terms appearing in the rate equation restore to some extent the symmetry which characterizes the Lotka–Volterra model. The generalist predator x3 in (2.166) is a sexually reproducing species. It is assumed that males and females are equal in number and every individual has got equal opportunity to meet an individual of opposite sex. The frst term of (2.166) represents growth rate of the sexually reproducing species in well-mixed conditions. ˜ 3 measures the limitation on the growth of the generalist predator x3 by its dependence on per capita availability of its favorite prey x2 . One aspect of the community interactions is the mutual interference which is generally a “stabilizing” process among the interacting subpopulations [42,59,60], which are represented by the parameters mi. The parameters d1 , d2 and d3 are diffusion coeffcients of the populations. Parshad et al. [159,161] have shown

Reaction–Diffusion Modeling

93

that the model exhibits fnite time blow-up of the Lp norm of x3 for all p in the parameter range c > w3 D3 , or even if c ˜ w3 D3 is possible, if x20 and x30 are large enough. The blowup can be understood as concentration phenomenon for bacteria chasing a food source or the sharp increase in the amplitude of a pulse along a nerve leading to neuron fring or the sharp increase in an insect population from an outbreak. The authors (Parshad et al. [159]) have shown that the blow-up results hold for the Neumann boundary conditions as well. To illustrate the blow-up, the system was numerically solved by the authors using MATLAB (R2010) via PDEPE solver over 200 × 200 mesh points on a domain of size Lx × Ly , with spatial resolution ˜x = ˜y = 1 and time step ˜t = 0.1. The initial condition used is a small perturbation about (2.1, 2.9, 1.9) and no-fux boundary condition. The blow-ups in one and two dimensions are given in Figures 2.14a and b for m1 = m2 = 1, m3 = 2, at times t = 41 and 29.4, respectively. Parshad et al. [161] have investigated the model system (2.164)–(2.166) (for m1 = m2 = 1, m3 = 2) analytically and numerically. The model exhibits very rich dynamics including limit cycle, diffusion-induced chaos, and Turing instability. The Turing patterns in 1D include stripe, spot, and mesh patterns, whereas 2D patterns include spot, labyrinth, as well as weaving patterns. They have also calculated the Turing space in terms of the model parameters in certain parameter regimes, reconstructed a low-dimensional chaotic attractor, and estimated the fractal dimension of the attractor. Haile and Xie [71] extended the work of Parshad et al. [161] by introducing the intraspecies competition into middle predator and cross-diffusion. They have shown that the classical Turing instability induced by self-diffusion does not occur but the positive equilibrium solution becomes unstable and the model generates spatial patterns only in the presence of crossdiffusion and the phenomenon is known as the Turing instability induced by crossdiffusion. The spot, stripe, and almost periodically fuctuating patterns are obtained due to diffusion and cross-diffusion in 2D. Parshad et al. [162] have shown that the spatial version of Upadhyay–Rai model [201] is the frst example of a three-species RD system that can blow up in fnite time, starting from near equilibrium (small) initial conditions, whereas the ODE system can also blow up in fnite time, for far from equilibrium

FIGURE 2.14 (a) Finite time blow-up for the model system (2.164)–(2.166) in (a) 1D at t = 41 and (b) 2D at t = 29.4. (Reproduced with permission from Parshad, R.D. et al. 2013. Finite time blowup in a realistic food-chain model. ISRN Biomath. 2013, 1–12, ID 424062, [159]; and Hindawi Publishing Corporation. Copyright 2013.)

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Spatial Dynamics and Pattern Formation in Biological Populations

(large) initial condition. However, numerical simulations show a rich variety of blow-up profles. The authors conjectured that a form of Turing–Hopf/Turing-induced blow-up occurs for the system as well. Example 2.4 Derive the necessary and suffcient conditions for the equilibrium point (u*, v*, r*) to produce diffusion-driven instability which lead to emergence of patterns in the following model system with suitable initial and Neumann boundary conditions (Parshad et al. [160]).

˜u uv = u(1 − u) − + d1 ˆu, ˜t u +°4

(2.167)

˜° ˛ uv vr + d2 ˇ° , = −˛ 5° + 6 − ˜t u + ˛ 7 v + ˛ 8r + ˛ 9

(2.168)

˜r ° r2 = ° 10 r 2 − 11 + d3 ˇr. ˜t ˛ + ° 12

(2.169)

Solution The equilibrium point is given by ˘ 1−˜ )+ 4 ( u = *

where

M=

(1 − ˜ 4 )2 − 4M 

(

)

*  , ° * = ˜ 11 − ˜ 10˜ 12 , r * = ° + ˜ 9 N , ˜ 10 (1 − ˜ 8 N )

2

˜ 11 − ˜ 10 (˜ 4 + ˜ 12 ) ˜ u* , N = −˜ 5 + * 6 . ˜ 10 u + ˜7

Linearize the model system about the homogeneous steady state as u = u* + uˆ (˜ , t), v = v * + vˆ (˜ , t), r = r * + rˆ(˜ , t), where uˆ (˜ , t)  u* , vˆ (˜ , t)  v * and rˆ(˜ , t)  r *. In the linearized system, substitute uˆ (˜ , t) ˙ ° 1 ˇ ˝ vˆ (˜ , t) ˇ = ˝ 2 ˇ rˆ(˜ , t) ˇ ˝˛ 3 ˆ

° ˝ ˝ ˝ ˝˛

˙ ˇ  t + ik˜ , ˇe ˇ ˆ

(2.170)

where i , i = 1, 2, 3 are the corresponding amplitudes, k is the wave number, λ is the growth rate of perturbation in the time t, and ˜ is the space coordinate. The characteristic equation is obtained as

(

˙ 1 ˇ J − ˜ I − k 2 D ˇ 2 ˇ 3 ˆ

)

˘   = 0,  

(2.171)

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Reaction–Diffusion Modeling

 d1  where D =  0  0 

 a11  =  a21  a  31

0 d2 0

a12 a22 a32

0 0 d3

 *    u* −1 + ν 2   α∗        ω 6ω 7ν * , J =  β∗2      0  

u* α ∗2

0

ν *r* γ ∗2

−

(

ν* ν* + ω9

(ω r ) 10

γ ∗2

* 2

0

ω 11

)

          

  ,  

a13 a23 a33

α ∗ = u* + ω 4 , β∗ = u* + ω 7 , γ ∗ = v * + ω 8  r * + ω 9 . For nontrivial solution of (2.171), we require that a11 − λ − k 2 d1

a12

a21

a22 − λ − k d2

a23

a31

a32

a33 − λ − k 2 d3

a13 2

= 0,

which gives a dispersion relation corresponding to (u*, v*, r*). To determine the stability domain associated with (u*, v*, r*), rewrite the dispersion relation as a cubic polynomial function as

( ( )) = λ

P λ k2

3

( )

( )

( )

+ µ 2 k 2 λ 2 + µ1 k 2 λ + µ 0 k 2 ,

(2.172)

( )

where µ2 k 2 = ( d1 + d2 + d3 ) k 2 − ( a11 + a22 + a33 ) ,

( )

µ1 k 2 = a11 a33 + a22 a11 + a33 a22 − a32 a23 − a12 a21 − k 2 ( d3 + d1 ) a22

)

+ ( d2 + d1 ) a33 + ( d3 + d2 ) a11 + k 4 ( d2 d3 + d2 d1 + d1d3 ) ,

( )

µ0 k 2 = a11 a32 a23 − a33 a22 a11 + a33 a12 a21 + k 2 ( d1 ( a33 a22 − a32 a23 ) + d2 a11 a33 + d3 ( a22 a11 − a12 a21 )) − k 4 ( d2 d1 a33 + d1 d3 a22 + d2 d3 a11 ) + k 6 d1 d2 d3 . According to the Routh–Hurwitz criterion for stability, the system is stable if Re(λ ) < 0. The necessary and sufficient conditions for stability are

( )

( )

( )

µ2 k 2 > 0, µ1 k 2 > 0, and [ µ2 µ1 − µ0 ] k 2 > 0.

(2.173)

If any of the above conditions are not satisfied, the system is unstable. We now require conditions such that the homogeneous steady state (u*, v*, r*) is stable to small perturbations in the absence of diffusion and unstable in the presence of diffusion for

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Spatial Dynamics and Pattern Formation in Biological Populations

certain k values (diffusion-driven instability). For stability in the absence of diffusion, we require

(

)

(

)

(

)

(

)

µ2 k 2 = 0 > 0,  µ1 k 2 = 0 > 0,  µ0 k 2 = 0 > 0, and [ µ2 µ1 − µ0 ] k 2 = 0 > 0.

( )

Irrespective of the value of k 2 , µ2 k 2 > 0, since a11 + a22 + a33 is always negative. Now,

( )

( )

µ0 k 2 and [ µ2 µ1 − µ0 ] k 2 are cubic functions of k 2 , which are of the form

( )

( )

G k 2 = H H + k 2 DD + k 2

2

( )

CC + k 2

( )

3

BB , with BB > 0 and H H > 0.

( )

To show that either µ0 k 2 or [ µ2 µ1 − µ0 ] k 2 is negative for some k, we need to fnd the

(

2

)

minimum value of k referred to as the minimum Turing point kT2 such that G k 2 = kT2 < 0.

( )

The minimum Turing point occurs when ˜ G ˜ k 2 = 0, which gives k 2 = kT2 =

−CC + CC2 − 3BB DD . 3BB

For k 2 to be real and positive, we require CC2 − 3BB DD > 0, and either DD < 0 or CC < 0.

(2.174)

( )

Therefore, G k 2 < 0, if at k 2 = kT2

( )

{

(

Gmin k 2 = ˘ 2CC3 − 9DDCC BB − 2 CC2 − 3DD BB 

)

3/2

+ 27H H BB2

} 27B  < 0. 2 B

Therefore, we get the condition

(

2CC3 − 9DDCC BB − 2 CC2 − 3DD BB

)

3/2

+ 27 H H BB2 < 0.

(2.175)

Hence, (2.174) and (2.175) are the necessary and suffcient conditions for (u*, v*, r*) to produce diffusion-driven instability, which lead to emergence of patterns. Also, to establish stability when k = 0, H H in each case has to be positive.

2.7.3 Model 3: Modified Leslie–Gower-Type Three-Species Model Abid et al. [2] considered a reaction–diffusion model with three species, prey (U), intermediate predator (V), and top predator (W). The prey species is the only food source of the intermediate predator V, and the intermediate predator V is the only prey of a top predator W. Local interactions between species U and V are modeled by the Lotka–Volterra type scheme (the predator population dies out exponentially in the absence of its prey), and the interaction between species W and its prey V has been modeled by the Leslie–Gower scheme [107,108] (the loss in predator population is proportional to the reciprocal of per capita availability of its most favorite food). The model studied here is mainly based on a modifed version of Leslie–Gower scheme. The diffusion term describes the ability to move in a domain of  2 . The model is given by (Abid et al. [2]) ˆ ˜U vV  = D1˛U + ˘ a0 − b0U − 0  U, ˇ U + d0  ˜T

(2.176)

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Reaction–Diffusion Modeling

 ∂V vW  vU − 2 V, = D2 ∆V +  − a1 + 1  U + d0 V + d2  ∂T

(2.177)

 ∂W vW  W, = D3 ∆W +  c3 − 3  V + d3  ∂T

(2.178)

∂U ∂ V ∂ W = = = 0, ∂n ∂n ∂n U(0, x , y ) = U 0 ( x , y ) ≥ 0, V (0, x , y ) = V0 ( x , y ) ≥ 0, W (0, x , y ) = W0 ( x , y) ≥ 0. U(T, x, y) is the density of prey species, V(T, x, y) the density of intermediate predator species, and W (T, x, y) the density of top predator species at time T. The position (x, y) is defined on a circular domain (or disk domain) with radius R, ∆ is the Laplacian operator, and n is the outward unit normal. The three species are assumed to diffuse at rates Di ,  i = 1, 2, 3. a0 , b0 , v0 , d0 , a1 , v1 , v2 , d2 , c3 , v3 and d3 are assumed to be positive and are defined as follows: a0 is the rate of growth of the prey U, b0 measures mortality due to competition between individuals of the species U , v0 is the maximum extent that the rate of reduction by individual U can reach, d0 measures protection that prey U and intermediate predator V benefit through the environment, a1 represents the mortality rate V in the absence of U, v1 is the maximum value that the rate of reduction by the individual U can reach, v2 is the maximum value that the rate of reduction by the individual V can reach, v3 is the maximum value that the rate of reduction by the individual W can reach, d2 is the value of V for which the rate of elimination by individual V becomes v2 /2, c3 describes the growth rate of W , assuming that there are same number of males and females, and d3 represents the residual loss caused by high scarcity of prey V of the species W . The homogeneous Neumann boundary condition signifies that the system is self-contained and there is no population flux across the boundary ∂ Ω. Applying the transformation, x = r cosθ , y = r sin θ , 0 < r < R , 0 ≤ θ < 2 π, r = x 2 + y 2 and θ = tan −1 ( y/x), the Laplacian operator in polar coordinates is given by ∆ rθ u =

∂2u 1 ∂u 1 ∂2u +   +   . ∂ r 2 r ∂ r r 2 ∂θ 2

Introduce the transformations of the variables as U=

a=

a0 a2 a0 3 t r′ u, V = 0 v , W = w , T = , r = , θ = θ ′, b0 b0 v0 b0 v0 v2 a0 a0

a v dvb ca2 v dvb b0 d0 , b = 1 , c = 1 , d = 2 02 0 , p = 3 0 , q = 3 , s = 3 02 0 , a0 v0b0 v2 v2 a0 a0 a0 a0

δ1 =

D1 D D , δ 2 = 2 , δ 3 = 3 . a0 a0 a0

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Spatial Dynamics and Pattern Formation in Biological Populations

Then, the spatiotemporal system (2.176)–(2.178) in polar coordinates becomes

∂u = δ 1∆ rθ u + f (u,  v ,  w), ∂t

(2.179)

∂v = δ 2 ∆ rθ v + g(u,  v ,  w), ∂t

(2.180)

∂w = δ 3 ∆ rθ w + h(u, v, w), for all ( r , θ ) ∈ Γ , t > 0, ∂t

(2.181)

∂u ∂v ∂w = = = 0, for all ( r, θ ) ∈∂ Γ ∂η ∂η ∂η u(0, r , θ ) = u0 (r , θ ) ≥ 0, v(0, r , θ ) = v0 (r, θ ) ≥ 0, w(0, r , θ ) = w0 (r , θ ) ≥ 0, qw  v  cu w     − f (u, v , w) =  1 − u −  u, g(u, v, w) =  −b +  v , h(u, v , w) =  p −  w,  u+ a v + d v + s u+ a where all the variables are now functions of t , r , θ . Without diffusion, the system can be written as dE = L( E), dt where T

E = [u,  v ,  w]T , and L( E) =  f (u, v , w), g(u, v , w), h(u, v , w) . The system has five equilibrium points E0 = (0, 0, 0), E1 = (1, 0, 0), E2 = (0, 0, sp q) , E3 = (1, 0,  sp q), and E* = u* , v * , w * , where

(

u* =

)

(

)

p   v* + s a ( bq + p ) , , v * = 1 − u* u* + a , and w * = qc − bq − p q

(

)(

)

qc > bq + p and qc − bq − p > a(bq + p). The authors did not consider the sixth equilibrium point E4 = ( uˆ , vˆ , 0 ), where uˆ =

ac ( c − b − ab ) ab , vˆ = , c > b + ab. c−b ( c − b )2

For studying the Turing instability, set  u − u*  W =  v − v* *   w−w

  λ t + ikr  ϕ (r , θ )e  

99

Reaction–Diffusion Modeling

where k is the wave number and ˜ (r , ° ) is an eigenfunction of the Laplacian operator on a disk domain with zero fux on the boundary, that is, ˛ r°˜ = − k 2˜ , ˜ r (R , ° ) = 0. Linearizing about u* , v * , w * , we obtain the equation:

(

)

dW = D˝W + LE E* W, dt

( )

˛ ˜1 ˙ D=˙ 0 ˙ 0 ˝ ˆ ˘ av * ˘ 1 − 2u* − ˘ u* + a ˘ ˘ acv * LE = ˘ 2 u* + a ˘ ˘ ˘ 0 ˘ ˘ ˇ

(

(

)

0 ˜2 0

ˆ ˘ ˘, ˘ ˇ

0 0 ˜3

u* − * u +a

2

0

cu* dw * −b− * u +a u* + a

)

(

( )

q w* *

2

u +a

−b−

(v

(2.182)

)

dw * *

+d

2

)

2

v* − * v +d

)

2qw * v* + d

)

(

−

(

     ˆ a11  ˘  = ˘ a21  ˘ a31  ˇ    

a12 a22 a32

a13 a23 a33

  .  

From the third expression p2 qw  2qw qw 2 ˆ , and a33 = p − h ( u, v , w ) = ˘ p − w, we find a32 = = −p.  2 = ˇ v + s q v +s ( v + s) These expressions are in simplifed form compared to the expressions given by the authors. Note that a12 < 0,  a21 > 0, a23 < 0,  a32 > 0, a33 < 0. Without diffusion, the characteristic equation of LE E˛ can be written as

( )

˝ ( ˜ ) = ˜ 3 + B1˜ 2 + B2 ˜ + B3 = 0. The authors have shown that B1 > 0,  B2 > 0, B3 > 0,  B1B2 −   B3 > 0, (Routh–Hurwitz criterion) under certain conditions and the equilibrium point E* = u* , v * , w * is locally asymptotically stable. For the Hopf bifurcation, the authors consider p as a bifurcation parameter and pcr as the critical value of the parameter. The necessary and suffcient conditions for the occurrence of the Hopf bifurcation are B1 > 0,  B2 > 0, B3 > 0,  B1B2 −   B3 = 0, and ° d˜ ˙ Re ˝ i ˇ ˘ 0,  i = 1, 2, 3 at p = pcr . The authors have shown the existence of a value p = pcr ˛ dp ˆ at which the Hopf bifurcation occurs. Using a suitable Lyapunov function, the equilibrium point E* = u* , v * , w * was also shown as globally asymptotically stable under certain conditions. For the system with diffusion (2.179)–(2.181), we get

(

(

)

( )

˜° = LE E* − Dk 2° .

)

100

Spatial Dynamics and Pattern Formation in Biological Populations

The characteristic equation is given by

( )

˜ I 3 − LE E* − k 2 D = 0, or ˜ − a11 + ° 1k 2

−a12

−a21

˜ − a22 + ° 2 k

−a31

−a32

−a13 2

−a23

˜ − a33 + ° 3 k

= 0.

(2.183)

2

Expanding, we get

( )

( )

( )

( )

H k 2 = ˜ 3 + ˆ1 k 2 ˜ 2 + ˆ 2 k 2 ˜ + ˆ 3 k 2 = 0,

( ) ( k ) = k (˜ ˜

(2.184)

where ˝1 k 2 = k 2 (˜ 1 + ˜ 2 + ˜ 3 ) + B1 , ˝2

2

4

1 2

+ ˜ 2˜ 3 + ˜ 3˜ 1 ) − k 2 (˜ 1 ( a22 + a33 ) + ˜ 2 ( a33 + a11 ) + ˜ 3 ( a11 + a22 )) + B2 ,

( )

˝ 3 k 2 = k 6˜ 1˜ 2˜ 3 − k 4 (˜ 1˜ 2 a33 + ˜ 2˜ 3 a11 + ˜ 3˜ 1a22 ) + k 2 (˜ 3 ( a11a22 − .a12 a21 ) + a11a33 ) + B3 The equilibrium point is stable if Re(˜ ) < 0. The Turing instability requires that the stable homogeneous equilibrium becomes unstable due to the interaction and diffusion of species. The authors have shown that the Turing instability occurs under certain conditions. Assuming λ as complex, they have studied the formation of spatiotemporal patterns. Numerical simulations were done to demonstrate the Turing instability. The authors have taken the diffusion parameters for simulations as ˜ 1 = 0.02, ˜ 2 = 0.01, ˜ 3 = 0.05, and ˜ 1 = 2.5, ˜ 2 = 1.25, ˜ 3 = 6. It was observed that two waves burst at center of the disk, then these spirals burst leading to an aperiodic spatial distribution of some domain and this aperiodicity spreads throughout the area and remains in time (for the frst set of diffusion coeffcients) and then spatiotemporal chaos was obtained.

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3 Modeling Virus Dynamics in Time and Space

3.1 Introduction Emerging and re-emerging infectious diseases pose big challenges to public health, agriculture, and wildlife management [25,28,135]. The recent outbreaks of infuenza, Ebola, Zika and SARS-CoV-2 viruses and their spread are challenging topics of study for the scientists specializing in mathematical epidemiology. In particular, one is interested in the  following: (i) The time period between one gets infected and passes the infection to the next person(s). (ii) How can the spread of disease be reduced/stopped?, and (iii) How to control the disease/infection? Human or animal invasions of new ecosystems, climate changes, increased international travel, and environmental degradation provide opportunities for the existing diseases to spread and new infectious diseases to emerge [181]. Studies by various authors showed that about 60% of emerging infections in humans originated from wildlife [129,259]. Therefore, it is important to study how the infections spread in wildlife communities and how they are transmitted to humans. Conservation biologists are also interested to know how parasites affect biodiversity and ecosystem dynamics because an infectious disease can be a factor regulating host population [187]. Epidemiological or eco-epidemiological modeling is an important tool for simulating and analyzing the long-term dynamics of complex epidemiological and ecological systems, which also includes spatial structure and the spread of diseases. Modeling can help in identifying trends, making forecasts, and optimizing various detection, prevention, therapy, and control programs [107,110,114]. Understanding the transmission dynamics of infectious diseases in communities, regions, and countries can help us in designing control programs for these diseases. The following are some terminologies used in epidemiology: Susceptible population (S): These are units of population that are free from infection at a particular point of time but have a potential threat of infection by the infective agent. Exposed population (E): These are units of population that are in contact with the disease agent but are yet to show any infective effects. Infected population (I): These are units of population that have been infected and who have the potential to transmit the infectious disease to the rest of the population on having adequate contacts with the susceptible class of the population. Quarantine population (Q 1): It refers to the separation and restricted movement of sick persons who have a contagious disease (like viral disease COVID -19), in order to prevent its transmission to others. Normally, quarantine is implemented in a hospital setting. In special cases, it may be done at home under suitable guidance. Isolation (Q 2): It refers to the restriction of movement or separation of persons who are susceptible or possibly exposed to a contagious disease (coming in contact with a sick 111

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person), before it is known whether they fall ill. Isolation usually takes place in the home and may be applied at the individual level or to a group or community of susceptible or exposed population. Recovered population (R): These are units of population who have ceased to be infectious and have acquired immunity, which may be permanent or temporary based on whether they remain in recovered class or move to susceptible class. Incubation period: It is the interval between the effective exposure of the susceptible host to an infectious agent and the appearance of signs and clinical symptoms of the disease in that host. During this period, viral genomes replicate and the host responds by producing interferon leading to the classical symptoms of infection (e.g., fever, aches, pains, and nausea). As an example, during the incubation period, Ebola-infected patients do not pass the virus to others. Endemic disease/infectious agent: It refers to the constant presence of a disease or infectious agent within a given geographic area or population group. For example, chickenpox, malaria, cholera, and typhus are endemic diseases. Epidemic disease: An epidemic is a sudden outbreak of a disease that spreads rapidly among many people in a community at the same time. Examples are plague, severe acute respiratory syndrome (SARS), and so on. Pandemic disease: A pandemic is a disease outbreak that spreads across countries or continents. In comparison to an epidemic, it is more lethal and adversely infuences more people. An outbreak is declared when an illness happens in unexpected high numbers. It may stay in one region or extend more widely. An outbreak can last for days, months, or years. Examples are 2009 swine fu, 1918 Spanish fu, and COVID-19. Epidemic models are used to describe rapid outbreaks that occur in less than one year, while endemic models are used for studying diseases over longer periods, during which there is a renewal of susceptible by births and recovery from temporary immunity [110]. Basic reproductive number/ratio 0: It is defned as the average number of secondary infections that occur when one infected person is introduced into a completely susceptible host population [67]. It is also called the basic reproduction ratio [65] or basic reproductive rate [12]. It is an important metric predicting whether a disease will spread or die out in a deterministic population and is used in communicable disease theory [7]. If 0 > 1, one infectious individual generally produces more than one infection leading to spread of an epidemic, whereas if 0 < 1, one infectious individual generates less than one infection on average [53], and epidemic may die out [65]. For simple spatial models with just one type of individual, the original defnition as the average number of infections generated by one infectious individual in an otherwise susceptible population can be used. For nonspatial homogeneous mixing models, the critical value is 0 = 1. That is, when 0 ˜ 1, the expected outbreak size is small; when 0 > 1, there is a signifcant probability of a large outbreak [223]. When the population includes individuals of different infection types, 0 is defned as the largest eigenvalue of the next-generation operator for those types [64,101]. This is appropriate for most nonspatial models, for which branching process approximation can be applied [19,59]. For a spatial model, when the numbers of infective individuals often grow only quadratically rather than exponentially, this generalized defnition is not applicable [64]. It was used in the study of demographics [73,149,246], epidemic theory [102], vector-borne diseases such as malaria [180,229], and human infectious diseases [67,105,138]. It is now widely used in the study of infectious diseases and in-host population dynamics. The estimation of 0 was found to be important in the studies of the following: (i) outbreak of acute respiratory syndrome (SARS) [50,165,173,224], (ii) bovine spongiform encephalitis [61,86,291], (iii) foot and mouth disease [85,183], (iv) novel strains of infuenza [192,251],

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(v)  West Nile virus [290], (vi) incidence and spread of dengue [177], (vii) malaria [96,250], and (viii) Ebola virus [9,52,142,266]. White et al. [287] proposed a novel extension of a network-based approach for estimating the reproductive number, which was originally proposed by Wallinga and Teunis [274]. The extension allows us to incorporate spatial and/or demographic information through a similarity matrix. Xue and Scoglio [298] studied the network level reproduction and extinction threshold for vector-borne diseases. Contact number (σ): The contact number σ is the average number of adequate contacts of a typical infected individual during the infectious period. An adequate contact is one that is suffcient for transmission, if the individual contacted by the susceptible is infected. Replacement number (R): The replacement number R is defned as the average number of secondary infections produced by a typical infected person during the entire period of infectiousness [106,110]. A relation between these numbers is taken as 0 ° ˜ ° R, with equality of the three quantities at the time of invasion. Note that 0 = ˜ for most models and ˜ > R after the invasion for all models [110]. Contact rate of infection (β): It is the number of individuals contacted by an infective individual per unit of time. It is also called the disease-transmitting coeffcient. Incidence: Let S(t) and I(t) denote the number of susceptible and infective individuals respectively at time t and N be the total population size. Then, s ( t ) = S ( t ) N and i ( t ) = I ( t ) N are the susceptible and infectious fractions respectively. If ˜ is the average number of adequate contacts of a person per unit time, then ˇˆ ˜ I ( t ) ˘ N = ˜ i ( t ) is the average number of contacts with infective individuals per unit time of susceptible ones, and ˛° ˜ IS N ˝˙ =   ˜ is N is the number of new cases per unit time due to the number of susceptible individuals, S = sN. This form of the horizontal incidence is called the standard incidence [108,110]. The horizontal incidence is the infection rate of susceptible individuals through their contacts with infective individuals. The standard incidence is a better formulation than the simple mass action law for animal populations such as mice in a mouse-room or animals in a herd [60] because transmission primarily occurs locally from nearby animals. The spread of an infectious agent from one individual to another through contact with bodily excretion fuids is also called the horizontal transmission. Colonization (horizontal transmission) is sometimes seen as the result of close interactions between disease-free host and infected individuals. Vertical incidence is the infection rate of newborns, the infection being passed on by their mothers, assuming that a fxed fraction of the newborns is infected vertically [33]. For many diseases such as AIDS, hepatitis B, and hepatitis C, newborns from the infected individuals can be infected as well. The term, vertical transmission, is restricted by some authors to the genetic transmission and extended by others to include also the transmission of infection from one generation to the next, as by milk or through the placenta. Assume that the infected and susceptible hosts mix with each other and move randomly within an area of fxed size. Then, bilinear or simple mass action incidence or densitydependent transmission is defned by ˜ S ( t ) I ( t ). The standard or frequency-dependent transmission is defned by ˜ S ( t ) I ( t ) N, where N = S ( t ) + I ( t ). The saturated incidence is defned as ˜ S ( t ) I ( t ) ˇˆ° + S ( t ) + I ( t ) ˘, [10] or as ˜ S ( t ) I ( t ) ˇˆ1 + ° 1S ( t ) + ° 2 I ( t ) ˘ [131,286,297]. These expressions are called transmission functions. Many authors have used different nonlinear incidences (functional responses) in their studies [115,169,170,174]. Hethcote and Levin [111] presented a survey of mechanisms including nonlinear incidences that can lead to periodicity in epidemic models. Enatsu et al. [83] studied the global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. They had considered the following assumptions on the nonlinear incidence rate. The incidence

{

}

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f ( S, I ) :  2+0 →  +0 is a continuously differentiable function on  2+0 satisfying the following conditions: a. f ( 0,  I ) = f ( S, 0 ) = 0, for S,  I ≥ 0, b. f ( S, I ) is a strictly monotone increasing function of S ≥ 0, for any fixed I > 0, and a monotone increasing function of I ≥ 0, for any fixed S ≥ 0, c. ϕ ( S,  I ) =  f ( S,  I )/I is a bounded and monotone decreasing function of I > 0, for any fixed S ≥ 0, and K ( S) ≡ lim ϕ ( S,  I ) is continuous on S ≥ 0 and a monotone increasI→ 0 + ing function of S ≥ 0. Some of the nonlinear incidences are the following: Yorke and London [303]: β S ( t ) I ( t ) ( 1 − CI ( t )) . Capasso and Serio [41]: S(t)I (t) (1 + AI(t)). May and Anderson [185]: β S ( t ) I ( t ) . Liu et al. [170], Ruan and Wang [232]: (i) β Sq ( t ) I p ( t ) , (ii) S p ( t ) I q ( t ) B + I q ( t ) , p > 0, q > 0. May and Anderson [186]: β S ( t ) I ( t ) ( S ( t ) + I ( t )) . Diekmann and Kretzschmar [66], Roberts [226]: β S ( t ) I ( t ) ( c + S ( t ) + I ( t )) . Alexander and Moghadas [4], Li and Wang [155], Cui and Li [56]:

(

)

(

)

β S ( t ) I ( t ) 1 + vI q ( t ) , v ≥ 0, 0 < q ≤ 1. Xiao and Ruan [295]: (i) kS ( t ) I ( t ) 1 + α I 2 ( t ) , (ii) kI 2 ( t )  S ( t ) 1 + α I 2 ( t ) . Hu et al. [121], Liu et al. [170]: (i) kI p ( t )  S ( t ) 1 + α I q ( t ) , (ii) β Sq ( t ) I p ( t ) 1 + α I s ( t ) .

(

) (

)

(

) (

)

Feline leukemia (FeLV) and feline immunodeficiency (FIV) viruses are transmitted both horizontally and vertically. The deadly septicemia, which manages to kill 80% of septicemia-infected birds, gets lodged in the ovary of surviving birds and is passed later to the birds’ eggs (vertical transmission), spreading horizontally within the hatcher and brooder. Hilker et al. [116] investigated the impact of a strong Allee effect on a diffusive SI model with logistic growth and standard incidence (also called frequency-dependent transmission) and no vertical transmission. Courchamp et  al. [54] constructed a deterministic/ temporal version of this model for studying the circulation of the FIV, a feline retrovirus homologous to human immunodeficiency virus (HIV), within populations of domestic cats. The introduced disease propagated in the form of a traveling infection wave with a constant asymptotic rate of spread in the model with logistic growth. Knowledge of the working of pathogens, statistical data analysis, and mathematical modeling play important roles in the study of infectious diseases. SIS and SIR-type mathematical models are the first building blocks in the study of these diseases. A simple idealized example is the swine-flu infection (H1N1-2009). This pandemic was unknown to the population at that time. The individuals in the geographic area where the virus originated are the susceptible population (S). The individuals that contract the virus are the infected population (I). Of these individuals, many fight off the virus and survive. This set of individuals form the recovered population (R). Normally, the recovered individuals do not transmit the virus. This example gives a simple compartmentalization of the population. Spatial heterogeneity: It is generally defined as the complexity and variability of the system variables in space. It is a challenge to study how spatial heterogeneity of the

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environment and movement of individuals have an impact on the persistence and extinction of a species/disease. To understand the impact of spatial heterogeneity of the environment and movement of individuals on the persistence and extinction of a disease, Allen et al. [5] proposed a frequency-dependent SIS reaction-diffusion model for a population in a continuous spatial habitat. When we consider spatially heterogeneous interventions, it is essential to represent the location of hosts and the pattern of transmission. The spatial heterogeneities of intervention add another layer of complexity to the system and provide a challenge for modeling [223]. Examples of spatially localized interventions include ring culling (as carried out in 2001 UK foot and mouth epidemic [135] and ring vaccination [264]), school closure [120], and local top-up vaccination campaigns. Every intervention is in some sense local and therefore spatially heterogeneous. Rock et al. [228] gave an excellent presentation of the dynamics of infectious diseases in their review article. The study of the transmission dynamics of diseases may help us to control or even prevent the spread. Sir Ronald Ross [229] studied the spread of malaria. His studies tell us that it is suffcient to control or eliminate the malaria-carrying mosquito. Epidemiologists constructed and studied the models for the spread of diseases. Their theoretical results suggest that under certain conditions, a disease can go to extinction. Kermack and McKendrick [138] formulated an SIR compartmental model to study the outbreak of the Great Plague of London during 1665–1666, and the outbreak of plague in Mumbai in 1906. They examined a series of models based on healthy, infected, and immune individuals in the scenario of a constant population (no births or deaths). Many models for infectious diseases were proposed based on the basic model given by Kermack and McKendrick [138,139]. The frst study of the dynamic modeling of infectious diseases was carried out by Anderson and May [12]. The dynamic behavior of the SIR epidemic models was investigated by many workers [1,79,188,249,252]. Chinviriyasit and Chinviriyasit [49] studied a spatial SIR reaction-diffusion model for the transmission of diseases such as whooping cough. Lotf et al. [176] studied the dynamics of a reaction-diffusion SIR epidemic model with a specifc nonlinear incidence rate. Liu and Xiao [166] modeled an SIS epidemic diffusion model with the population migrating between two cities. The SIRS model with simple mass action was frst used to describe the spread of the disease in the predator population [17]. Gan et al. [88] studied the existence of traveling waves in an SIRS epidemic model with bilinear incidence rate, spatial diffusion, and time delay. The SEIR epidemic model was extensively studied by many researchers [8,154,156,158,159,305] and by the authors whose references are given therein. Analysis and study of many mathematical models describing the infectious diseases are given by Hethcote [110]. A good presentation of mathematical epidemiology of infectious diseases and their analysis and interpretation is given in the book of Diekmann and Heesterbeek [64]. Some of the challenges for deterministic epidemic modeling were highlighted recently by Roberts et  al. [227]. Spatial models have been introduced into epidemiology to resolve vividly the spatial transmission dynamics of the epidemic. From a biological perspective, individual organisms are distributed in space and typically interact with the physical environment and other organisms in their spatial neighborhood [37]. The diffusion of individuals may be connected with other things, such as searching for food, escaping high infection risk, and so on. [276]. Spatial models can be used to estimate the formation of spatial patterns on a large scale and the transmission of diseases. Hosono and Ilyas [119] investigated the existence of traveling wave solutions for the infective-susceptible two-component epidemic model. Cruickshank et al. [55] reported the development of a highly effcient numerical method for determining the principal characteristics (velocity, leading edge width, and peak height) of spatial invasions or epidemics described by deterministic one-dimensional (1D) reaction-diffusion models whose

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dynamics include the Allee effect. Ferguson et al. [85] presented an analysis of the footand-mouth disease epidemic in Britain over the frst two months of the spread of the virus, and the net transmission potential of the pathogen and the increasing impact of control measures were estimated over the course of the epidemic. Grenfell et al. [94] demonstrated recurrent epidemic traveling waves in an exhaustive spatiotemporal data set for measles in England and Wales. Mulone et al. [197] considered an epidemic model where the diffusion of individuals is infuenced by intraspecifc competition pressure and the individuals are weakly affected by different classes. Wang and Wang [275], Wang et al. [276], and Xu and Ma [296] studied the complex dynamics of a spatial HBV (Hepatitis B virus) model. Zhang et al. [308] studied the transmission dynamics of the HBV epidemic in Xinjiang, China. Kim et al. [144] investigated a diffusive infuenza epidemic model and suggested that the best policy to prevent the occurrence of a pandemic is not only to exterminate the infected birds with avian infuenza but also to reduce the contact rate for susceptible humans with the individuals infected with mutant avian infuenza virus. There are a few studies on modeling the spatial spread of specifc diseases using partial differential equation models. Riley et al. [223] highlighted several currently open challenges of spatial epidemic models. A wide variety of methods such as cellular automata [69,87], networks [23,202], meta-populations [133,172], reaction-diffusion equations [44,222], and integro-differential equations [136], which are useful tools in studying geographic epidemic spreads, have been used for understanding spatially structured epidemics. Modeling the spatial spread of vector-borne diseases is a challenging task [14], but one possible approach is to consider a meta-population as a directed graph, or a network, with each vertex representing a subpopulation in a location and links placed between two locations if there is a possibility of transmission, such as movement or proximity [26]. Network models are more widely used in epidemiology to understand the spread of infectious diseases through connected populations [201,272]. For spatial models in fnite domains, stationary states and their stability have been investigated by Capasso [40]. For many types of spatial epidemiology models in infnite domains, one often determines the thresholds above which a traveling wave exists, fnds the minimum speed of propagation and the asymptotic speed of propagation (which is usually shown to be equal to the minimum speed), and determines the stability of the traveling wave to perturbations [148,191,194,221,253]. For stochastic spatial endemic models, there is also a threshold condition so that the disease dies out below the threshold and approaches an endemic stationary distribution above the threshold [78]. Some of the ways of formulating spatial epidemiology models are the following [194,222,269]: i. Diffusion-based epidemiology models are formulated from nonspatial models, by adding diffusion terms corresponding to the random movements of susceptible and infective population on each day [189,203]. ii. Distributed contact models are formulated by using integral equations with kernels describing daily contacts of infective individuals with their neighbors [137,195]. iii. Restricted-movement models are formulated in which each individual has a home position about which the individual moves in a biased random walk. In such models, Reluga et al. [222] assumed that the probability distribution of an individual’s position approaches a stationary Gaussian distribution for a large time. In the limit, when the homeward attraction is much larger than the transmission rate,

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their model approximates a distributed-contacts model. When the transmission rate is much larger than the homeward attraction and diffusion is much faster than homeward attraction, their model approximates a diffusive distributedinfectives model. When the transmission rate is much larger than the homeward attraction and diffusion effects are weak, their model approximates an advective distributed-infectives model [222]. 3.1.1 Next-Generation Operator Method The next-generation operator method is used to compute the reproduction number 0 and is defned as the spectral radius of the next-generation matrix [65]. The formation of the matrix involves determination of the two compartments, infected and noninfected for the model system. A detailed explanation of the formation of the next-generation matrix/ operator is given by Diekmann and Heesterbeek [64]. We briefy describe their method of formation of the next-generation matrix. Assume that there are n compartments of which m are infected. Defne the vector x = xi ,  i = 1, 2,  , n, where xi denotes the number or proportion of individuals in the ith compartment and X s = { x ˝ 0|xi = 0,  i = 1, 2, , m} is defned as the disease-free states of the model. Suppose that the given disease transmission model, with nonnegative initial conditions, can be written in terms of the following autonomous system: x˜ i = f ( xi ) = Fi ( x ) − Vi ( x ) , i = 1, 2,…, n

(3.1)

where Fi ( x ) is the rate of appearance of new infections in compartment i. Let Vi ( x ) = Vi− ( x ) − Vi+ ( x ) , where Vi+ ( x ) is the rate of transfer of individuals into the compartment i by all means and Vi− ( x ) is the rate of transfer of individuals out of the ith compartment. The difference Fi ( x ) − Vi ( x ) gives the rate of change of xi . Fi ( x ) includes infections that are newly emerging, but does not include terms that describe the transfer of infectious individuals from one infected compartment to another. It is assumed that these functions are at least twice continuously differentiable in each variable [270]. Assume that Fi ( x ) and Vi ( x ) satisfy the axioms outlined by Diekmann et al. [65] and van den Driessche and Watmough [270] as given below: A1. If x ˜ 0, then Fi ( x ) , Vi− ( x ) , Vi+ ( x ) ˛ 0, for i = 1, 2, , n. A2. If xi = 0, then Vi− = 0. In particular, if x ˜ X s , then Vi− = 0, for i = 1, 2, , m. A3. Fi = 0, if i > m. A4. If x ˜ X s , then Fi ( x ) = 0, and Vi+ ( x ) = 0, for i = 1, 2, , m. A5. When F ( x ) is set to zero, all eigenvalues of Df ( x0 ) have negative real parts. The next-generation matrix (operator) FV −1 is formed from the matrices of partial derivatives of Fi and Vi . F and V are the m × m matrices defned by ˙ ˝F ˘ ˙ ˝V ˘ F = ˇ i ( x )  , and V = ˇ i ( x )  , 1  i, j  m, ˆ ˝x j  ˆ ˝x j 

(3.2)

where x is the disease-free equilibrium (DFE). The entries of FV −1 give the rate at which infected individuals in x j produce new infections in xi   times the average length of time an

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individual spends in a single visit to compartment j. 0 is defined as the spectral radius (dominant eigenvalue in magnitude) of the matrix FV −1. Applications of this method are illustrated in the works of many authors [46,117,184,212,290]. The following theorem of van den Driessche and Watmough [270] gives a result for studying the local asymptotic stability of DFE. Theorem 3.1 Consider the disease transmission model given by (3.1) with f ( x ) satisfying axioms A1 to A5. If x is a DFE of the model, then x is locally asymptotically stable, if 0 = ρ FV −1 < 1, where ρ is the spectral radius and is unstable if 0 > 1. Now, we discuss the spatial and temporal dynamics of some epidemic compartment models.

(

)

3.2 Susceptible-Infected (SI) Models When epidemics are modeled as reaction-diffusion systems, it is assumed that the susceptible and infected individuals move randomly. Let S and I denote the densities of susceptible and infected individuals respectively and N = S + I. An SI model assumes that the infection passes through encounters between S and I. If spatial movements are included, then the system can be written as

∂N = −∇ ⋅ J + f ( N ) , ∂t where N ( x , t ) is the population density, J is the spatial flux of the population, and f represents the births and deaths of the population. Mulone and Straughan [196] used the reduction method for studying the stability of constant solutions of some ecological systems with diffusion, which include Cantrell–Cosner and May–Leonard systems. They have derived a new canonical energy (Lyapunov function) to study the system. For the models considered, they have shown that the regions of linear and nonlinear stabilities coincide with a known radius of attraction for the initial data. A model without diffusion is written as [196] dS dI = µ ( 1 − S) − β SI = f ( S,  I ) , = β SI − ( µ +  ) I = g ( S,  I ) , dt dt

(3.3)

S ( 0 ) = S0 ≥ 0, I ( 0 ) = I 0 ≥ 0, where µ , β , and ε are the recruitment rate of the population, the per capita death rate of the population, disease-transmitting coefficient, and enhanced death rate respectively. Hethcote [109,110] has shown that the model (3.3) is well posed in the triangle T = {( S,  I )|S ≥ 0,  I ≥ 0,  S + I ≤ 1} in the SI plane. The model has two equilibrium points, DFE point (1, 0) and endemic equilibrium (EE) point S* ,  I * , where S* = 1 σ , I * = µ (σ − 1)/β , and σ = β ( µ +  ) . The Jacobian matrix J is given by

(

)

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Modeling Virus Dynamics in Time and Space

° fS J=˝ ˝˛ gS

fI ˙ ˇ. g I ˇˆ

The equilibrium points are locally asymptotically stable when Trace ( J ) = f s + g I < 0, and det ( J ) = f s g I − f I g s > 0. For the equilibrium point S* , I * , these conditions are obtained as

(

)

f s + g I = − ˜µ ( µ +  ) < 0, and f s g I − f I g s = µ ( ˜ − µ −  ) > 0.

(3.4)

The frst condition is always satisfed. For the equilibrium point (1, 0), we obtain these conditions as f s + g I = ˜ − 2 µ −  < 0, and f s g I − f I g s = − µ ( ° − µ −  ) > 0. The threshold quantity is given by 0 = ˜ . If ˜ ° 1,then the solution paths starting in T approach the DFE (1, 0). If ˜ > 1, then all solution paths approach the EE point S* ,  I * . For ˜ < 1, the I co-ordinate of the EE point is negative and coincides with the DFE value zero at ˜ = 1. The EE is unstable for ˜ < 1 and locally asymptotically stable for ˜ > 1, while the DFE is locally stable for ˜ < 1 and unstable for ˜ > 1. Therefore, these two equilibrium points exchange stabilities (transcritical bifurcation) as the EE moves through DFE when ˜ = 1 and becomes a distinct, epidemiologically feasible, locally asymptotically stable equilibrium when ˜ > 1 [110]. A typical pseudo diffusion model analog involving a constant diffusion coeffcient is given by the following system:

(

)

˜S ˜I = f ( S,  I ) + DS ˆS, = g ( S, .I ) + DI ˆI ˜t ˜t

(3.5)

(

)

The model system (3.3) supports diffusive instability (or Turing effect) if S* ,  I * is locally asymptotically stable for (3.3) but unstable for the system (3.5) at least for some values of DS ,  DI . That is, the general SI system (3.3) can have diffusive instability only if ( µ +  ) < °  or ˛ > 1. Since fs = − ( µ + ° I ) < 0 and g I = ˘ ˜ S − ( µ +  )  = 0, the condition for diffusive instability for the RD model (3.5), which requires (DI f s + DS g I ) > 0 , fails (see equation 2.86). Hence, the RD model (3.5) never supports diffusive instability. 3.2.1 Models with Nonlinear Incidence Rate Consider a population, in which a pathogenic agent is active and comprises two subgroups: the healthy individuals who are susceptible (S) to infection and the already infected individual (I) who can transmit the disease to the healthy one. Only the susceptible individuals have the capability of reproducing. Sun et al. [253] investigated the pattern formation in the following spatial SI model with nonlinear incidence rate and obtained the conditions for transcritical Turing and Hopf bifurcations. In the absence of infective individuals, the susceptible grows according to a logistic model with carrying capacity K   K ˛ R + and intrinsic birth rate constant r   r ˛ R +

(

)

(

)

˜S S˘ ˙ = rS ˇ 1 −  − ° S p I q + DS 2S, ˆ ˜t K

(3.6a)

˜I = ° S p I q − dI + DI ˆ 2 I, ˜t

(3.6b)

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Spatial Dynamics and Pattern Formation in Biological Populations

where ° 2 = ˜ 2 /˜ x 2 + ˜ 2 /˜ y 2 , p + q = 1, and p ,  q > 0. d represents the death coeffcient of I. The nonspatial model has at most three equilibrium points – the disease-free states (0, 0), (K, 0), and the endemic stationary state S* ,  I * , where

(

)

1/ p 1/ p 1/ p  d˝˜ˇ  ˝˜ˇ ˝˜ˇ S* = K 1 − ˆ   , I * = ˆ  S* , with r > d ˆ  . ˙ d˘ ˙ d˘ r ˙ d ˘  

(

)

The elements of the Jacobian matrix J at S* ,  I * are given by 2S ˇ ˝ Sˇ ˝ fS = r ˆ 1 −  − ˜ p ˆ  ˙ ˙ I˘ K˘

p−1

p

˝ Sˇ ˝ Sˇ ,  f I = − ˜ q ˆ  ,  gS = ˜ p ˆ  ˙ I˘ ˙ I˘

p−1

p

˝ Sˇ ,  g I = ˜ q ˆ  − d. ˙ I˘

Using the eigenvalue analysis, the Jacobian of the system (3.6a) and (3.6b) is obtained as ˙ J (k) = ˇ ˇ ˆ

fS − DS k 2

fI

gS

g I − DI k 2

˘ .  

(3.7)

The characteristic equation is given by ˜ 2 − ˜T + ˝ = 0, where  T = trace ( J ) = fS + g I − k 2 ( DS + DI ) , ˛ = det ( J ) = fS g I − f I g s − k 2 ( fSDI + g I DS ) + k 4 ( DSDI ) . The system is stable without diffusion, implying that 2S ˘ ˙ ˙ S˘ fS + g I = r ˇ 1 −  − d + ˜ ˇ  ˆ I ˆ K

p−1

 ˙ S˘   q ˇˆ I  − p  < 0.  

Hence, T < 0. Irrespective of the sign of T, one of the eigenvalues is positive when ˜ < 0. Therefore, Turing instability occurs when ˜ < 0. Now, y = ˝ k 2 is a parabola which attains its minimum value for

( )

˛ fSDI + g I DS ˙ˆ = kc2 = ˝ 2DSDI

fS g I − f I g S . DSDI

This determines the critical wave number (the wave number of the modes that frst becomes unstable). For k 2 = kc2, we have

( )= f g

˛ k

( )

2 c

S

I

− fI gs

2 fSDI + g I DS ) ( − .

4 ( DSDI )

˛ kc2 < 0 gives the condition ( fSDI + g I DS ) > 4DSDI ( fS g I − f I g s ). 2

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Modeling Virus Dynamics in Time and Space

Together with the condition that k is real, this inequality determines the Turing space. Hence, the conditions for yielding Turing patterns are given by (i) fS + g I < 0, (ii) ( fS g I − f I g s ) > 0, (iii) ( fSDI + g I DS ) > 0,

(3.8)

(iv) ( fSDI + g I DS ) > 4DSDI ( fS g I − f I g s ). 2

Numerical simulations are performed using the parameter values K = 1, ˜ = 0.5, d = 0.6, r = 0.4, p = 0.4, q = 0.6, DS = 0.1, and DI = 0.5. We obtain S* ,  I * = (0.4909,  0.0311), kc2 = 0.802903, ˝ kc2 = −0.0275198 < 0, for 0.0610169 < k 2 < 1.54479. For these parameter values, conditions (i)–(iv) are satisfed. Therefore, Turing patterns are obtained. The plots of ˛ k , k 2 and ( ˛ k , k ) are given in Figures 3.1 and 3.2 respectively. Using MATLAB 8.1, we have solved the two-dimensional (2D) reaction-diffusion model (3.6) using a fnite-difference technique with zero fux boundary conditions. The initial population densities are taken as S ( x ,  y , 0 ) = 0.183128 + 0.05 × rand, I ( x ,  y , 0 ) = 0.0997283 + 0.05 × rand, which has been perturbed randomly. Susceptible and infected population at t = 750 (at iteration 15,000) for p = 0.3 is plotted in Figure 3.3. From the fgure, we observe that the regular spotted patterns prevail over the whole domain fnally, and the dynamics of the model system does not undergo any further changes.

( )

(

)

FIGURE 3.1 Plot of ˜ k vs. k2.

FIGURE 3.2 Plot of ˜ k vs. k as p varies [p = 0.4 (black), 0.35 (grey), and 0.3 (white)].

(

)

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 3.3 Susceptible and infected population at t = 750 (at iteration 15,000) for p = 0.3.

Considering the intrinsic birth rate constant r as the bifurcation parameter, the authors [253] have obtained the following critical values for the transcritical bifurcation parameter (rt ), the transition Hopf bifurcation parameter (rH ), and Turing bifurcation parameter (rT ): β rt = d    d

1/p

(

β ,  rH = d    d

( p −1) p   p−1 1/p 2 2 − β 2( ) rT =  dpβ 1/p  2 − β 2 d  + kT DI β  

(

)

p

1/p

2 − p  − dp ,

) − k D d( 2 T

S

p − 1) p

( dp + k D ) 2 T

I

 d( p −1) 

p

( dp + k D ) . 2 T

I

The authors have also considered the effect of the relative diffusion coefficient ( D = DS DI ) and found that an increase in D does not change the transcritical and Hopf bifurcations, but causes the Turing bifurcation line to shift toward higher p values and lose steepness. This leads to successive contractions of the Turing space and its extinction for D > 1. Also, for D > 1, the infective class moves faster than the susceptible class S, which violates the instability condition. Biologically, in the case of transcritical bifurcation, two branches of equilibria meet and exchange their stability properties. The Hopf bifurcation is space independent and breaks the temporal symmetry of a system, which gives rise to oscillations that are uniform in space and periodic in time (USPT). The Turing bifurcation breaks spatial symmetry, leading to the formation of patterns that are stationary in time and oscillatory in space (STOS). The transcritical bifurcation occurs when one real eigenvalue of the local model vanishes. One can find this kind of bifurcation by solving λ1λ2 = ∆  = 0 at k = 0. The Hopf bifurcation occurs when a pair of imaginary eigenvalues crosses the real axis from the negative to the positive side in the absence of diffusion. One can find this kind of bifurcation by solving Im ( λ ( k )) ≠ 0,   Re ( λ ( k )) = 0 at k = 0. At the Hopf bifurcation threshold, the temporal symmetry of the system is broken and gives rise to USPT oscillations with frequency ω H = Im ( λ ( k )) = ∆ 0 . The Turing bifurcation occurs when Im ( λ ( k )) = 0 =   Re ( λ ( k )) at k = kT ≠ 0, and the wave number kT satisfies kT2 = ∆ 0 ( DSDI ) . At the Turing threshold, the spatial symmetry of the system is broken and gives rise to STOS with wavelength λT = 2 π kT [253].

Modeling Virus Dynamics in Time and Space

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Wang et al. [285] studied the cross-diffusion induced patterns in the model system

˜S S˘ ˙ = rS ˇ 1 −  − ° S p I q + DS 2S + D 2 I ,  ˆ ˜t K

(3.9a)

˜I = ° S p I q − dI + DI ˆ 2 I.  ˜t

(3.9b)

Cross-diffusion can lead the infected population into patched distribution, which may prevent the spread of the epidemic. By considering the cross-diffusion in the susceptibles, the authors obtained typical Turing patterns, spotted and strip-like spatial patterns (Problem 3.1, Exercise 3). Sun [254] studied the pattern formation in the following epidemic model with nonlinear incidence rates

˜S = A − dS − ° S p I q + DSˆ 2S,  ˜t

(3.10a)

˜I = ° S p I q − ( d + µ ) I + DI  2 I, ˜t

(3.10b)

where A is the recruitment rate of the population, d is the natural death rate of the population, and μ is the disease-related death rate from the infected. The author obtained the conditions for the Hopf bifurcation and Turing bifurcation, and in particular, the exact Turing domain was found in the two parameters space (Problem 3.2, Exercise 3). Ducrot [74] studied the following model

˜S = A − ° S − ˛ SI + dˇ 2S, ˜t

(3.11a)

˜I = ° SI − (˛ + µ ) I +  2 I, ˜t

(3.11b)

where S ( x, 0 ) = S0 ( x ) ,  I ( x , 0 ) = I 0 ( x ) , and x ˝ N are assumed to be bounded, nonnegative, and uniformly continuous on  N . In the absence of the disease, that is, when I ( t ,  x ) ˛ 0, the spatiotemporal evolution of the population satisfes the simple reaction-diffusion equation (˜ t − dˆ ) P(t, x) = A − ° P(t, x) involving some constant external supply A > 0 and a natural death rate γ > 0. Here, d > 0 describes the spatial mobility of individuals. In this model, the contamination process is assumed to follow the usual mass-action incidence with a contact rate β > 0. It was also assumed that the disease induces additional mortality with a given rate µ > 0. Note that when A = γ = 0, the above system reduces to the well-known diffusive Kermack and McKendrick model [138–140]. 3.2.2 Models with Self and Cross-Diffusion Cross-diffusion was frst proposed by Kerner [141]. Sun et  al. [256] considered a closed system with (i) a birth process in which susceptibles (S) are assumed to grow logistically; (ii) infection, where the susceptible individuals become infected by standard mass action after which they are removed from the S class and entered in the I class; and (iii) death, due

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Spatial Dynamics and Pattern Formation in Biological Populations

to which infected and susceptible individuals are removed from the infected and susceptible classes respectively. The model system with both self and cross-diffusion is written as [256]

∂S S SI  = rS  1 −  − β + DS∇ 2S + Dc ∇ 2 I ,  ∂t K S+ I

(3.12a)

∂I SI =β − dI + DI ∇ 2 I ,  ∂t S+ I

(3.12b)

with nonzero initial conditions and zero-flux boundary conditions: S ( x , y , 0 ) > 0,  I ( x ,  y , 0 ) > 0, ( x ,  y ) ∈ Ω = [ 0,  R ] × [ 0,  R ] ,

∂S ∂ I = = 0, ( x ,  y ) ∈∂ Ω. ∂n ∂n The parameter Dc is the cross-diffusion coefficient and other parameters have the same meanings as defined in the previous model system. The nonspatial model has at most two equilibrium points: (i) E0 ( K, 0 ), which corresponds to the disease-free point, and (ii) E* S* ,  I * , which corresponds to an endemic stationary state, where S* = K ( r − β + d )/r , I * = S* ( β − d )/d , with d < β < r + d. The elements of the Jacobian matrix J are given by

(

)

βI2 β S2 βI2 β S2 2S   = − = = fS = r  1 −  − , f , g , g − d. S I I  K  (S + I )2 (S + I )2 (S + I )2 ( S + I )2 The equilibrium point (K, 0) is locally asymptotically stable if r + d > β ,  and r ( d −   β ) > 0. For the equilibrium point E* S* ,  I * , we obtain the conditions as

(

)

  S* − I *  2S*  fS + g I = r  1 − − d + β  * *  = β − ( r + d ) < 0, K   S + I   2S*   β S* 2 fS g I − f I g s =   r  1 − K   S* + I *  

(

)

2

 β dI *2 − d + * *   S +I

(

)

2

> 0.

Using the eigenvalue analysis, the Jacobian of the system (3.12a) and (3.12b) is obtained as  J(k ) =   

fS − DS k 2

fI

gS

g I − DI k 2

The characteristic equation in the spatial case is given by

( ) ( )

λ 2 + λ a k 2 + b k 2 = 0,

 .  

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Modeling Virus Dynamics in Time and Space

( )

where a k 2 = k 2 ( DS + DI ) − ( fS + g I ) ,

( )

b k 2 = ( DSDI ) k 4 − ( fSDI + g I DS − gSD ) k 2 + ( fS g I − f I g s ) , where fS ,  f I ,    gS , and g I are as given above. The model system (3.12a) and (3.12b) will be unstable if at least one root of the characteristic equation is positive. A suffcient condition is that b k 2 < 0. Now, b k 2 is a quadratic polynomial with respect to k 2 . Its minimum is obtained at the vertex of the parabola. We obtain

( )

( )

2 kmin =

fSDI + g I DS − Dc gS . 2DSDI

2 1 ˆ fSDI + g I DS − gSDc ) − 4DSDI ( fS g I − f I g s ) ˘ . ( ˇ  4DSDI 2 At the critical point k = kc , b k = 0 [206]. For fxed kinetics parameters, this defnes a critical cross-diffusion coeffcient Dc as the appropriate root of

( )

2 At k 2 = kmin , b k2 = −

( )

( fSDI + gI DS − gSDc )2 − 4DSDI ( fS gI − fI gs ) = 0. The critical wave number kc is then given by kc2 =

fSDI + g I DS − Dc gS = 2DSDI

fS g I − f I g s . DSDI

Note that the critical wave number of the cross-diffusion system is the same as the critical wave number without the cross-diffusion term. That is, cross-diffusion has no effect on the critical wave number but has an effect on the roots of the characteristic equation. The authors [256] numerically obtained the emergence of the Turing pattern in 1D spatial oscillatory and anti-phase dynamics of different spatial points, which may play an important role in the extinction of the epidemic. We have performed numerical simulations for the model (3.12a) and (3.12b), using the parameter values r = 0.3,  ˜ = 0.5,  K = 1000,  d = 0.25,  DS = 0.1, and Dc = 0.01. To illustrate the dispersion relation, the variation of the real part of the characteristic value of the model system as DI increases is plotted in Figure 3.4. From Figure 3.4, we observe that as we

FIGURE 3.4 Plot of the real part of the characteristic value as DI is increased. DI = (a) 3.5, (b) 2.0, (c) 0.6, and (d) 0.5.

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Spatial Dynamics and Pattern Formation in Biological Populations

increase the value of the diffusion coeffcient for the infected population, we obtain Turing instability. For DI = 2 and DI = 3.5, we obtain diffusion-induced instability but for DI = 0.5 and DI = 0.6, the system shows stable behavior both in temporal and spatial cases. Wang et al. [281] presented the Turing pattern selection for a model system (3.12a) and (3.12b) with zero-fux boundary conditions. The authors have given a general survey of Hopf and Turing bifurcations and have derived amplitude equations for the excited modes. They found that the model dynamics exhibits a diffusion-controlled formation growth not only to stripes and stripes-spots but also to holes, stripes-holes, and spots replication. Ducrot et al. [75] analyzed a spatially structured SI epidemic model with vertical transmission, a logistic effect, and a density-dependent incidence and studied the existence of traveling wave solutions connecting the endemic and the disease-free states. Using the analysis of the center-unstable manifold around the interior equilibrium, Ducrot et al. [76] proved the existence of an infnite number of traveling wave solutions. Shu and Weng [247] investigated the diffusive SI model with a strong Allee effect and established the existence of traveling wave solutions connecting a DFE to the EE for wave speeds c > c°. 3.2.3 Influenza Epidemic Models Infuenza is a communicable acute respiratory disease and one of the major infectious disease threats to the human population. Infuenza virus affects individuals of all ages, causes repeated infections throughout life, and is responsible for annual worldwide epidemics of varying severity, commonly referred to as seasonal infuenza. Seasonal infuenza epidemics offer unique opportunities to study the invasion and re-invasion waves of a pathogen in a partially immune population [47]. Due to a lack of suitable disease data, detailed patterns of the spread remain elusive. The underlying mechanisms dictating the spatial spread of seasonal infuenza are not well understood, in part due to a lack of spatially resolved disease data to quantify patterns of spread. There are three types of infuenza viruses that infect humans – A, B, and C – which are classifed based on their immunological and biological properties. Infuenza viruses are negative-strand RNA viruses with a segmented genome; infuenza A and B viruses contain eight RNA segments, and infuenza C contains seven RNA segments. Infuenza A subtypes currently circulating among humans are infuenza A (H1N1) and A (H3N2) [99,234]. Infuenza causes severe illness and death in high-risk populations [211]. Infuenza is a contagious disease, which broke out during 1918–1920 (Spanish fu), 1957–1958 (Asian fu), 1968–1969 (Hong Kong fu), 2002–2003 (SARS), and 2009–2010 (A/H1N1), which killed millions of people. Mathematical modeling of infuenza was proposed from different points of view by various authors [217,233,240,258,283]. Viboud et al. [273] used infuenza-related mortality data to analyze the between-state progression of inter-pandemic infuenza in the United States over the past 30 years. A simple epidemiological model, based on the gravity formulation, captures the observed increase of infuenza spatial synchrony with transmissibility; high transmission allows infuenza to spread rapidly beyond local spatial constraints. Eifert et al. [81] used a variant of the logistic equation (as intrinsic growth) to describe a oneparameter discrete dynamical model for the spread of avian infuenza. This model utilizes the Lindblad dissipation dynamics [92,164] for the biological rate equation. Vaccination and antiviral treatment are two important prevention and control measures for the spread of infuenza. Qiu and Feng [217] developed a mathematical model that includes both drug-sensitive and resistant strains to explore the impact of vaccination and antiviral treatment on the transmission dynamics of infuenza. Eggo et al. [80] presented a statistical analysis of the spatiotemporal spread of the 1918 infuenza pandemic and demonstrated the degree of spatial locality in the large-scale geographical spread of infuenza between cities

Modeling Virus Dynamics in Time and Space

127

in England, Wales, and the United States. Wang [279] considered the infuence of behavioral changes on the infuenza spread. Li et al. [153] formulated a stochastic SIRS epidemic model with nonlinear incidence rate and varying population size to investigate the effect of stochastic environmental variability on inter-pandemic transmission dynamics of infuenza A. Suffcient conditions for extinction and persistence of the disease were established. 3.2.3.1 A Simple Spatial SI Epidemic Model Upadhyay et al. [267] assumed the following in formulating an infuenza epidemiological model. i. The susceptible (S) and infectious (I) populations move randomly, described as Brownian random motion, which is referred to as the irregular and unceasing movement of individuals. Many authors have suggested that the motion of humans can also be approximated with random walks if considered on a relevant spatiotemporal scale. For example, human travel is responsible for the geographical spread of human infections. In the light of increasing international trade, intensive human mobility, and the imminent threat of an influenza A epidemic, the knowledge of dynamical and statistical properties of human travel is of fundamental importance [31]. ii. In the absence of infection, the susceptible population grows logistically and in the presence of infection, the population is divided into two disjoint classes, namely, susceptible population S and infected population I. The total population, N ( t ) = S ( t ) + I ( t ), is not a constant but varies according to some growth law. For simplicity, they assumed that the birth and death rates depend on the population sizes as b ( N ) = rN   and d ( N ) = rN 2 K , respectively and have the logistic form [30]. The total population size satisfies the logistic diferential equation dN N˘ ˙ = b ( N ) − d ( N ) = rN ˇ 1 −  , where K > 0 is the carrying capacity. ˆ dt K iii. It was assumed that all members of the susceptible population are equally susceptible and all members of the infected population are equally infectious. It was also assumed that the disease is transmitted by contact between infected and susceptible populations only and the disease is not genetically inherited. The infected population does not recover or become immune. iv. The infection rate, ˜ I/( S + I + c ), is a function of the number of infective individuals present at a given point of time and ˜ is the maximum that this function can reach. This signifes the fact that the number of contacts an individual carrying the virus can have with other individuals reaches some fnite maximum value due to the spatial or social distribution of the population and/or limitation of time [66]. Based on the above assumptions, the authors [267] considered a simple spatial SI epidemic model as follows:

˜S S+ I ° SI ˆ = rs ˘ 1 − + DS 2S,  − ˇ ˜t K S+ I +c

(3.13a)

˜I ° SI = − aI + DI ˘ 2 I , ( x ,  y ) Ω,  t > 0, ˜t S + I + c

(3.13b)

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Spatial Dynamics and Pattern Formation in Biological Populations

where r is the intrinsic growth rate, K is the carrying capacity, ˜ is the contact rate between infective and susceptible population, c is the half saturation constant (see the derivation of a Holling type II functional response in prey–predator models), and a is the diseaseinduced mortality. The nonnegative constants DS and DI are the diffusion coeffcients of S and I respectively, and ˜ 2 is the Laplacian operator. Biologically, the constant c lowers the infection rate due to spatial or social distribution and limitation of time. The basic reproduction number 0 is the number of new infections produced in the lifetime of an infected host when introduced to a wholly susceptible population of specifed density. This measure is useful because it helps to determine whether or not an infectious disease will become endemic in the susceptible population. For the model (3.13a) and (3.13b), the basic reproduction number is defned by 0 = ˜ a. The disease will successfully invade when 0 > 1, but will die out if 0 < 1. Large values of 0 may indicate the possibility of major epidemics [179,280]. The initial and the boundary conditions are taken as S ( p , 0 ) ˙ 0,  I ( p, 0 ) ˙ 0, where p = ( x ,  y ) ˇ˘ = [ 0,  L ] × [ 0, L ]. ˜S ˜I = = 0, n is the outward unit normal vector on the boundary ˜˛. ˜n ˜n The model system (3.13a) and (3.13b) has the following equilibrium points: i. The trivial equilibrium point E0 = (0, 0) always exists. For E0 , the eigenvalues are r and − a. There is an unstable manifold along the S-direction and a stable manifold along the I- direction. Therefore, the equilibrium point E0 is a saddle point. ii. The DFE point E1 = ( K , 0) exists on the boundary of the frst octant. For E1, the eigenvalues are −r and ( ˜ K − a ) ( K + c ). Therefore, the equilibrium point E1 is locally asymptotically stable provided ˜ K < a . Also, E1 is a saddle point, if ˜ K > a . iii. The nontrivial equilibrium point E* (S* , I * ) exists if and only if there is a positive solution to the following set of equations: S+ I ˜I ˜S ˇ g 1 ( S, I ) = r  1 − = 0, g 2 ( S, I ) = − a = 0. − ˘ K  S+ I + c S+ I + c Subtracting the two equations, we obtain N ˆ ˜N ˛ r˙1− ˘ − + a = 0, N = S + I, ˝ Kˇ N+c or rN 2 − ˙ˆ r ( K − c ) − ( ˜ − a ) K ˘ˇ N − ( r + a ) Kc = 0. Solving for N, we obtain N = ˆ p ± p 2 + 4rKc ( r + a ) ˘ ( 2r ) , ˇ  p = r ( K − c ) − ( ˜ − a ) K = ( r + a ) K − ( rc + ˜ K ) .

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Modeling Virus Dynamics in Time and Space

From the second equation, we get ˜ S = a ( N + c ) . This equation gives a solution for S. Now, I = N − S gives the solution for I. The expression under square root is always positive. Now, p > 0, for ( r + a ) K > ( rc + ˜ K ) ; p < 0, for ( r + a ) K < ( rc + ˜ K ) ; and p = 0, for ( r + a ) K = ( rc + ˜ K ) . In terms of 0 = ˜ a , we can write these conditions as aK0 < or > or = r ( K − c ) + aK . The nontrivial equilibrium point E* (S* , I * ) exists and is given by N = ˙ p + p 2 + 4rKc ( r + a ) ˇ ( 2r ) . ˆ ˘ The value of N is largest when p > 0. Alternately, the solution of g1 ( S, I ) = 0 and g 2 ( S, I ) = 0 is given by a −  S =  *

( 1 − 0 )2 B + ( 1 − 0 ){( K + c ) r + Ka ( 1 − 0 )}

−  =

I* =

=

( a − ˜ )2 B + ( a − ˜ ){cr + K ( a − ˜ + r )} 2(a − ˜ )˜r .

2R0 r ( 1 − 0 )

( a − ˜ )2 B − ( a − ˜ )2 K − {˜ ( c − K ) + a ( c + K )} r 2˜ r

( 1 − 0 )2 B − a ( 1 − 0 )2 K − r {c ( 1 + 0 ) + K ( 1 − 0 )} 20r

, 0 =

˜ , a

where B = ( a − ˜ ) K 2 + 2K ˇ˘ ˜ ( c − K ) + a ( c + K )  r + ( c + K ) r 2 2

2

= ˘ˇ aK ( 1 − 0 )  + 2aKr ˘ˇ c ( 1 + 0 ) + K ( 1 − 0 )  r + ( c + K ) r 2 . 2

2

Now, to investigate the local behavior of the model system (3.13a) and (3.13b) about each equilibrium point, the variational matrix A at the point ( S, I ) is computed as * ° a11 A=˝ * ˝ a21 ˛

(

* * ˇ 2S* + I *  ˜ I c + I * a11 = r1− − K  c + I * + S* ˘

(

* a21 =

(

) +S )

˜ I* c + I*

(c + I

*

* 2

* ˙ a12 ˇ, * ˇ a22 ˆ

) )

2

(

) )

r ˜ c + S* * = −S*  + , a12 K c + I * + S* 

* = , a22

(

(

) − a. +S )

˜ S* c + S*

(c + I

*

* * Note that a12 < 0 and a21 > 0. From g1 ( S, I ) = 0, we obtain

S+ Iˇ ˜I ˝ rˆ1− . = ˙ K ˘ S+ I + c

* 2

2

 ,  

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Spatial Dynamics and Pattern Formation in Biological Populations

* Then, a11 simplifes as

(

˜ I* c + I* ˇ S* + I *  rS* * − − a11 = r1− K  K ˘ c + I * + S*

(

(

˜ I* c + I* rS* ˜ I* − − S* + I * + c K c + I * + S*

=

(

From g 2 ( S, I ) = 0, we obtain * Then, a22 simplifes as

* a22 =

) )

2

) )

2

 ˜ I* = S*   c + I * + S* 

(

)

2

−

r  . K 

˜S = a. S+ I + c

(

)− S +S )

˜ S* c + S*

(c + I

*

* 2

˜ S* ˜ S* I * = − * + I* + c c + I * + S*

(

)

2

< 0.

The characteristic equation is given by ˜ 2 − trace ( A ) ˜ + det ( A ) = 0. * * * * Now, det ( A ) = a11 a22 − a12 a21

=−

=

(

( )

˜ S*

(c + I

*

2

I*

) (

* 2

+S

ˇ ˜ I*   c + I * + S* ˘

)

2

˜ S* I * ˇ r ˜c + * * c+I +S K c + I * + S* ˘

)

(

−

)

2

(

) ˇ r +

* * * r  ˜ S I c + I + K  c + I * + S* 2 

(

)

K ˘

(

) +S )

˜ c + S*

(c + I

*

* 2

   

  > 0,  

* * trace ( A ) = a11 + a22 =−

rS* < 0. K

(

)

The characteristic equation has negative roots. Therefore, the EE E* S* ,  I * is locally asymptotically stable for all values of the parameters. In Figure 3.5, the phase portrait of the model system is plotted for r = 2.19, K = 400, c = 10, ˜ = 5.1, and a = 0.86. The susceptible population S and the infected population I are plotted on the horizontal axis and the vertical axis respectively. The white-colored curve is the susceptible nullcline and the grey-colored curve is the infected nullcline. In this fgure, (i) the equilibrium point E0 = ( 0, 0 ) is a saddle point; (ii) the equilibrium point E1 = ( 400, 0 ) is also a saddle point; and (iii) the equilibrium point E* = ( 4.0441,  9.93836 ) is locally asymptotically stable. To study the effect of diffusion, consider perturbation of the system about the positive equilibrium point E* as S ( x ,  y ,  t ) = S* + s ( x ,  y ,  t ) , I ( x, y ,  t ) = I * + i ( x ,  y ,  t ) . The linearized system is obtained as

˜s = b11s + b12 i + Ds˝ 2 s, ˜t

(3.14a)

˜i = b21s + b22 i + DI ˝ 2 i, ˜t

(3.14b)

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Modeling Virus Dynamics in Time and Space

FIGURE 3.5 Phase portrait of model system (3.13) with r = 2.19, K = 400, β = 5.1, c = 10, and a = 0.86. (Reprinted with permission from Upadhyay, R. K. et al. 2014. Deciphering dynamics of epidemic spread: the case of infuenza virus. Int. J. Bif. Chaos 24(5), 1450064, [267], and World Scientifc Publishing Company. Copyright 2014.)

where b11 = −

b21 =

(

) +S )

˜I* c + I*

(c + I (

*

* 2

) +S )

˜ I* c + I*

(c + I

*

* 2

(

)  , ) 

 rS* ˜ S* c + S* ˇ 2S* + I *   , = − + b + r1− 12 K  ˘  K c + I * + S* 

(

, b22 = −a +

(

) +S )

˜ S* c + S*

(c + I

*

* 2

2

.

Write the solution of this system in the form

((

))

((

))

s ( x ,  y ,  t ) ~ u exp ( ˜ k t ) exp i k x x + k y y , i ( x , y ,  t ) ~ v exp ( ˜ k t ) exp i k x x + k y y . Substitute the above expressions for s and i into equations (3.14a) and (3.14b). The homogeneous equations in u and v have solutions if the determinant of the coeffcient matrix is zero. Ak − ˜ I = 0, Ak = A − k 2 D,

(3.15)

° b11 b12 ˙ where D = diag ( Ds ,  DI ) and A = ˝ ˇ. ˛ b21 b22 ˆ The characteristic equation is given by

˜ 2 − trace ( Ak ) ˜ + det ( Ak ) = 0,

(3.16)

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Spatial Dynamics and Pattern Formation in Biological Populations

where trace ( Ak ) = ( b11 + b22 ) − ( Ds + DI ) k 2 , k 2 = k x2 + k y2 ,

(

)(

)

det ( Ak ) = b11 − Ds k 2 b22 − DI k 2 − b12b21 = DSDI k 4 − ( DI b11 + DSb22 ) k 2 + b11b22 − b12b21 ,

) + r  1 − 2S + I    − a + β S ( c + S ) K  ( c + I + S )  (c + I + S ) β (S − I )  2S + I  = + r1−   − a, K  (c + I + S ) 

b11 + b22 = −

(

β I* c + I*

*

*

*

* 2

*

*

* 2

*

*

*

*

*

*

*

(

 βI* c + I* 2S* + I *    − and b11b22 − b12b21 = r  1 − K   c + I * + S* 

(

(

) )

r β c + S* + S*  + K c + I * + S* 

(

2

(

(

(

(

) )

2

(

) )

  β S* c + S*  2  * *   c + I + S

  β I* c + I*   * *    c + I + S

* *   2S* + I *   β S c + S = r1−  K   c + I * + S*  

(

) )

) )

2

2

  − a 

   

(

* *     βI c + I − a +  * *   c + I + S

(

) )

 rS*   . 2  a +   K   

By the Routh–Hurwitz criterion, the roots of equation (3.16) are negative or have negative real parts, if trace ( Ak ) < 0 and det ( Ak ) > 0. A sufficient condition for trace ( Ak ) < 0 is b11 + b22 < 0. Now, b11 + b22 < 0 when

(

(

)

r 2S* + I * β S* β I* + r < + + a. K c + I * + S* c + I * + S*

)

(

)

(3.17)

Write det ( Ak ) = DSDI [ k 4 − tk 2 + z], where t=

1 1 ( DI b11 + DSb22 ) , and z = [b11b22 − b12b21 ]. DSDI DSDI

Now, y = [ k 4 − tk 2 + z] is a parabola opening upwards (in the k 2 -y plane) with vertex at

{( ) }

V ( t 2 ) ,  − t 2 4 − z  , t 2 > 4 z. Minimum of y occurs at V. Denote k12 , k22 = t ± t 2 − 4 z  2.     Then, det ( Ak ) = DSDI  k 2 − k12   k 2 − k22  . For k22 < k 2 < k12 , det ( Ak ) < 0,  and the characteristic equation has one positive root. Hence, the equilibrium point E* is unstable. Now, for all wavelengths violating this condition, that is, for k 2 > k12 ,  or  k 2 0, and the equilibrium point E* is asymptotically stable in the presence of diffusion. We have tested the stability of the equilibrium point for the following set of parameter values r = 2.19,   K = 400,   β = 6,  a = 0.86,   c = 10,  DS = 0.065,  and DI = 2.5. We obtain S* = 2.86229,  I * = 7.1072, t = 4.34533,   z = 5.8652,  and  t 2 ( ≈ 18.882 ) < 4 z ( ≈ 23.46 ) and hence the system is stable. For β = 5.1,  kT2 = 2.49983, and with the other parameter values being the same, we obtain S* = 4.0441,   I * = 9.93836, trace ( Ak ) = −6.4342 < 0, t = 4.99965,  z = 4.95179,

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Modeling Virus Dynamics in Time and Space

FIGURE 3.6 Plot of H(k2) vs. (k2).

k22 = 1.36082,  k12 = 3.63883, and det ( Ak ) = −0.210817, and the system is unstable. The plot of H k 2 vs.( k 2 ) given in Figure 3.6 shows the possible occurrence of Turing instability as

( ) H ( k ) becomes negative in some range of k . 2

2

3.2.3.2 Turing Instability Turing instability occurs when Im ( ˜ ( k )) = 0 and Re ( ˜ ( k )) = 0 at k = kT   ° 0. In fact, the Turing instability sets in when at least one of the solutions of equation (3.16) crosses the imaginary axis. In other words, the spatially homogeneous steady state will become unstable due to heterogeneous perturbation when at least one solution of equation (3.16) is positive. For this reason, one or both the following two inequalities are to be violated: trace ( Ak ) < 0 and det ( Ak ) > 0. Instability always occurs when H k 2 = det ( Ak ) < 0, for some k.

( )

( ) 2

As discussed earlier, H k is minimum (H min) for some value kT2 of k 2 at the vertex of the parabola V, where kT2 = ( DSb22 + DI b11 ) ( 2 DSDI ). Then, the condition that H kT2 < 0 gives

( )

t 2 > 4 z, or ( DSb22 + DI b11 ) > 4DSDI ( b11b22 − b12b21 ) . As shown earlier, for all wavelengths satisfying the condition k22 < k 2 < k12 , det ( Ak ) < 0, the characteristic equation has one positive root and the equilibrium point E* is unstable. The threshold value of the parameter ˜ , which corresponds to the onset of diffusive instability when the values of other parameters are fxed, can be determined. The discussion of the general case in which all the parameters are varying is very diffcult. The authors take the values of the parameters as K = 400, c = 10, r = 2.19, a = 0.86, DS = 0.065, and DI = 2.5   and consider ˜ as the bifurcation parameter. Suppose that the onset of diffusive instability occurs at a bifurcation value ˜ = ˜ i. 2 The value of ˜ i is obtained from H kT2 = 0, that is, ( DSb22 + DI b11 ) − 4DSDI ( b11b22 − b12b21 ) = 0. For the above parameter values, solving the nonlinear equation in ˜ , two real bifurcation values are obtained as ˜1 = 2.50305 and ˜ 2 = 5.54171, (these values differ slightly from the values given by the authors). For these values, Re ( ˜ ) cuts the axis, which gives the critical value for Turing instability. If ˜ < ˜1 = 2.50305, or ˜ > ˜ 2 = 5.54171, Re ( ˜ ) becomes negative. Thus, under the infuence of diffusion, the original (non-diffusive) stable system becomes unstable for ˜1 < ˜ < ˜ 2 . It can be concluded that diffusion destabilizes the system in some situations. The plot of the relationship between Re ( ˜ ) and ˜ is given in Figure 3.7. In Figure 3.7a, the plot for the temporal model with r = 2.19, K = 400, a = 0.86, and c = 10 is given. Since Re ( ˜ ) < 0, the temporal model is stable. In Figure 3.7b, the plot for the spatial model system with r = 2.19,  K = 400,  a = 0.86,  c = 10,  DS = 0.065,  and DI = 2.5 is given. In the range ˜1 = 2.50305 < ˜ <   ˜ 2 = 5.54171, Re ( ˜ ) > 0. In this region, the solutions of the spatial model system are unstable. Hence, for some sets of parameter values, if ˜1 < ˜ < ˜ 2 , 2

( )

134

Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 3.7 Relationship between Re(λ) and β.

FIGURE 3.8 Variation of the dispersion relation of the model system (3.13).

the solutions of model system with diffusion are unstable. In this region, despite that the solutions are unstable, diffusion can further destabilize the symmetric solutions so that the system with diffusion added to it can have symmetry-breaking capabilities, that is, it forms the Turing patterns. For r = 2.19, K = 400, a = 0.86, c = 10, DS = 0.065, and DI = 2.5,  the dispersion relation corresponding to several values of the bifurcation parameter ˜ is plotted in Figure 3.8. The gray line (a) corresponds to ˜ = 3.5, which lies in the interval, 2.50305 < ˜ < 5.54171. In this case, Turing instability occurs. The black line (b) and the blue line (c) correspond to the bifurcation parameter values ˜ = 2.51 and ˜ = 6.5 respectively.In the case of line (c), which corresponds to ˜ = 6.5, Turing instability decays and hence only a stable, steady-state solution is obtained. Example 3.1 Discuss the existence of the Hopf bifurcation for the model system (3.13), [267]. Solution Following the approach of Zhang et al. [305], the authors [267] showed the existence of the Hopf bifurcation for the model system (3.13a) and (3.13b). Here, we present the same analysis in a slightly modifed way. The Hopf bifurcation occurs when the eigenvalues of the characteristic equation satisfy the conditions ˝˙ ˜ ( k ) ˆˇ ˘ 0 and Re ˝˙ ˜ ( k ) ˆˇ = 0, at k = 0. Substitute S = S − S* and I = I − I * in the model system (3.13a) and (3.13b). The positive

135

Modeling Virus Dynamics in Time and Space

constant steady-state solution E* of the model system is transformed into the zero equilibrium of the new model system. The system is transformed as (dropping the bar for the simplicity of notation)

(

ˇ S + S* + I + I * St = r S + S*  1 − ˘ K

(

)

It =

(

(

)(

˜ S + S* I + I *

)  − ˜ (S + S )( I + I ) + D  S,  ( c + S + S + I + I ) *

*

*

)

c + S + S* + I + I *

)

(

S

*

(3.18a)

2

)

(3.18b)

− a I + I * + DI ˇ 2 I .

Using the Taylor series expansions of the terms about (S, I) = (0, 0), the system reduces to the system  ˆ  I * + 2 S*  St =  r ˘ 1 − − a1  S − a2 I + DS  2 S + f ( S, I , h ) ,  K    ˇ

(3.19a)

It = b1S + ( b2 − a ) I + DI ˆ 2 I + g ( S, I , h ) ,

(3.19b)

where f ( S, I , h ) and g ( S, I , h ) contain the nonlinear terms, and

( c + I ) h,  a *

a1 =

S

*

2

(

)

c + S* ˙ rS* ˘ ˜ I * S* h, h = =ˇ + b2  , b1 = a1 ,  b2 = * I ˆ K  c + I * + S*

(

)

2

.

In the following, we use h as the control parameter (in fact, β is the control parameter representing disease transmission rate). For fnding the stability of the steady-state solution and the existence of the Hopf bifurcation, h plays an important role. Defne the variables U 1 ( t ) , U 2 ( t ) and U ( t ) as U 1 ( t ) = S ( t , ˝) , U , 2 ( t ) = I ( t ˝) , U ( t ) = (U 1 ( t ) , U 2 ( t )) . T

Then, the system (3.19a) and (3.19b) can be written as U t = L (U ) + G (U ) ,

(3.20)

where  ˆ * *   r 1 − I + 2 S − a1 0  ˘  K  , KM =  ˇ  ˙2  b1 

ˆ ˙2 L = KM + D˘ ˇ 0

−a2

(b2 − a )

  ˆ DS , D = ˘  ˇ 0 

and G (U ) = ( f , g ) . Note that a1 ,  a2 ,  b1 ,  b2 , and a are positive. T

 ˆ  I * + 2 S*  det ( K M ) =  r ˘ 1 −  − a1  ( b2 − a ) + a2 b1 , K ˇ    ˆ   rS*  I * + 2 S*  = r ˘ 1 − − a1  ( b2 − a ) +  + b2  a1  K K   ˇ     ˆ  rS*  I * + 2 S*   + a  a1 . = r ˘ 1 −  ( b2 − a ) +   K    ˇ  K 

0  , DI 

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Spatial Dynamics and Pattern Formation in Biological Populations

There exist parameter values such that det ( K M ) > 0. Strong suffcient conditions are K < I * + 2S* and b2 < a, or K > I * + 2S* and b2 > a. Linearizing the system (3.20) about the origin (0, 0), we obtain U t = L (U ) .

(3.21)

2 ˝ 0 ˇ where L = K M + D ˆ ˛ . ˛2 ˘ ˙ 0 The characteristic equation of system (3.21) is [ L − ˜ I ] y = 0,

˙ ˝2 Dˇ ˆ 0

or

0 ˘  y + K M ( y ) − ˜ y = 0, ˝2 

(3.22)

˙ ˙ ˝2 0 ˘˘ where y ˛dom ˇ D ˇ   \{0} . ˇˆ ˆ 0 ˝ 2   The stability of the trivial solution of the system (3.20) depends on the locations of roots of (3.22). When all the roots of (3.22) have negative real parts, the trivial solution of (3.20) is stable; otherwise, it is unstable. The eigenvalue problem, −˝ 2˜ = °˜ ,  x ˆˇ,  ˘˜v = 0, ,x ˆ˘ˇ has eigenvalues 0 = ˜0 < ˜1 <    < ˜ k <  , and the corresponding eigenfunctions are ˜ k = °k ( x ) , ˝ ° ˇ ˝ 0 ˇ k ˛ N 0 = {0, 1, 2,} . Let ˜ k1 = ˆ k  ,  ˜ k2 = ˆ  . Then, Bk = ˙ 0 ˘ ˙ °k ˘ basis of the phase space of the system (3.20) and ˙ ˙ ˝2 y ˛dom ˇ D ˇ ˇˆ ˆ 0

˜(

1 k

2 k

ˇ k=0

forms a

0 ˘˘   \{0} can be decomposed as ˝ 2  



y=

{(˜ ,  ˜ )}



) ˜(

y ,  ° k1 ° k1 + y,  ° k2 ° k2 =

k=0

k=0

ˇ ° k1 ,  ° k2     ˘

)

y,  ° k1 y,  ° k2

 .  

(3.23)

Denote the Jacobian matrix of the model (3.20) by J k = K M − ˜ k2 D . Then, the characteristic equation of the system (3.20) is given by

˜ 2 − trace ( J k ) ˜ + det ( J k ) = 0,

(3.24)

where trace ( J k ) = trace ( K M ) − ( DS + DI ) ˜ k2 ,

    ˇ I * + 2S*  det ( J k ) = det ( K M ) + DSDI ˜ k4 − ( b2 − a ) DS +  r  1 − − a1  DI  ˜ k2 .  K  

 ˘   It was shown earlier that there exist parameter values such that det ( K M ) > 0. Strong suffcient conditions are K < I * + 2S* and b2 < a, or K > I * + 2S* and b2 > a. When the frst set of conditions, K < I * + 2S* and b2 < a , are satisfed, it is found that the third term on the right-hand side of det ( J k ) is positive. Hence, det ( J k ) is also positive under the same two

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Modeling Virus Dynamics in Time and Space

conditions. When the second set of conditions K > I * + 2S* and b2 > a are satisfied, it is found that the third term on the right-hand side of det ( J k ) is positive when      I * + 2S*  D I * + 2S*   − a1  < 0, and S <  a1 − r  1 −  r  1 −  K  DI  K      

(b2 − a ) .

(3.25)

Hence, det ( J k ) is also positive under these four conditions. Now, det ( J k ) < 0 implies that the characteristic equation has a positive root and the positive constant steady-state solution E* is unstable. When trace ( J k ) < 0 and det ( J k ) > 0, the characteristic equation has two negative roots and the positive constant steady-state solution E* is stable. Now, suppose that the characteristic equation has pure imaginary roots. Let iω be a pure imaginary root. By substituting in equation (3.24) and separating the real and imaginary parts, we get det ( J k ) = ω 2 > 0, and trace ( J k ) =  ( K M ) − ( Ds + DI ) λ k2 = 0.

β is the bifurcation parameter representing disease transmission rate and it appears in the control parameter h. Now, trace ( J k ) = trace ( K M ) − ( Ds + DI ) λ k2

(

 I * + 2S* = − ( a + a1 ) + b2 + r  1 −  K

(

 I * + 2S* = −a + r  1 −  K

)  − ( D 

s

+ DI ) λ k2

)  − ( D 

(

) (

 c + S* c + I* 2 + D λ + − )  k s I  I * S*

(

)  h. 

)(

)

Note that the last term on the right-hand side simplifies as β S* − I * / c + I * + S* . Then, the only value of h at which the homogeneous Hopf bifurcation occurs is h = h0 . Near h0 , substituting λ = p1 + q1 i in equation (3.24) and separating the real and imaginary parts, we get p12 − q12 − p1  trace ( J k ) + det ( J k ) = 0,

(3.26a)

2p1q1 − q1  trace ( J k ) = 0.

(3.26b)

Since q1 ≠ 0, 2 p1 =  trace ( J k ) ,

(3.27)

4det ( J k ) − trace ( J k ) = 4q12 , or 4det ( J k ) = 4q12 + trace ( J k ) > 0.  2

2

By differentiating both sides of equation (3.27) with respect to h, we obtain

(

)

 c + S* + I * (S* − I * )   dp1  1 d  trace ( J k )  =  sgn  =  ≠ 0.     S* I *  dh  2 dh Therefore, the transversality condition holds. Hence, the model system (3.13a) and (3.13b) undergoes Hopf bifurcation [51].

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Spatial Dynamics and Pattern Formation in Biological Populations

3.2.3.3 Two-time Scale Infuenza Models Wang [279] formulated a model to simulate human mobility response to infuenza infection for two-time scales. For a long time scale, the natural birth and death of human population were included and for a short time scale, the natural birth and death of human population were ignored and concentrated on the seasonal transmission of infuenza. The models for the two cases were taken as follows:

Long time scale: dS ° mI = ˝ − µS − S, dt 1 + hI dI ˜ mI =  S − ( µ + ˛ ) I, dt 1 + hI dm ˜I ˘ ˙   = m ˇ b − am − . ˆ dt 1 + hI 

(3.28)

Short time scale: ˜ mI dS S,  = − dt 1 + hI dI ˜ mI S − ° I, = dt 1 + hI dm ˜I ˘ ˙   = m ˇ b − am − . ˆ dt 1 + hI 

(3.29)

The meanings of the variables and parameters are as follows: S, I, and m respectively represent the number of susceptible individuals, number of infected individuals, and the intensity of population mobility, which could be the fraction of time to stay in public places in unit time and is affected by infected risk and economic benefts of mobility.  ˜ is the constant recruitment rate of the susceptible population, µ is the per capita birth/death rate of the population, and ˜ is the sum of recovery and treatment rates. The infection force is ˜ ( t ) = pCI N , where C is the contact number of a susceptible individual with all individuals per unit time. CI N gives the number of infectious contacts from those contacts, and p = p0 ( 1 + hI ) (p0 and h are positive constants) is the valid transmission probability which is a decreasing function of the infective number I under CI N infectious contact. It is assumed that the contact number C is a bilinear function of population size and intensity of population mobility, C = k1mN , where k1 is the proportionality constant and m is the intensity of population mobility. Then, the infection force becomes

˜ (t ) =

p 0 k 1m   I ° mI = , ( 1 + hI ) ( 1 + hI )

where ˜ = p0 k1 is the transmission rate. ˜ = wp0 k1 (w is the proportional constant) measures the human mobility response to infection risks. From the author’s work [279], we briefy discuss the behavior of the equilibrium points of the model system (3.28). The model system has three equilibrium points: (i) equilibrium point E0 ( ˝ µ , 0, 0 ), where there is no

139

Modeling Virus Dynamics in Time and Space

(

)

population mobility, (ii) a DFE, E1 ( Λ µ , 0, b a ), and (iii) an EE E* S* , I * ,  m* , where I * satisfies the equation pI *2 + qI * + r = 0, with

(

)

p = ( µ + γ ) aµ h2 + bβ h − αβ , q = ( µ + γ ) ( 2 ahµ + bβ ) − βΛ ( bh − α ) , r = aµγ − Λbβ + aµ 2 , and µS* = Λ − ( µ + γ ) I , am* = b −

(

α I*   . 1 + hI *  

)

The basic reproduction number of the model is 0 = Λbβ  µ a ( µ + γ ) . The author has proved that for 0 < 1, E1 is globally stable and for 0 > 1, E* is globally stable [36]. From numerical simulations, the author has shown that the mobility response does not affect the basic reproduction number 0 that characterizes the invasion threshold, but reduces or removes the infection or epidemic peaks which are very helpful in controlling influenza outbreaks. Charu et al. [47] studied the human mobility and spatial transmission of influenza, spatial and temporal dynamics of annual influenza epidemics in the United States over eight seasons, leveraging uniquely spatially resolved medical claims data on outpatient influenza-like-illnesses through active research collaboration with a data-warehouse company. System (3.29) has two equilibrium points: (i) E0 ( 0, 0, b a ) and (ii) E1 ( aγ ) ( bβ ) , 0,  b a . Since system (3.29) is a three-dimensional (3D) autonomous system of differential equations, the natural approach would be to find equilibria and linearize about each equilibrium point to determine stability. However, since every point with I = 0 is an equilibrium point, the system (3.29) has a line of equilibria, and this approach is not applicable (the linearization matrix at each equilibrium has a zero eigenvalue). The variational matrix at E1 ( aγ ) ( bβ ) , 0, b a has eigenvalues 0, 0, and −b. The author has shown that mobility responses to infection risk reduce significantly the transmission of influenza and a slighter mobility response can save a good fraction of susceptible population. Now, we introduce diffusion to study its effect on the dynamics, in one space dimension in Wang’s model [279]. The diffusion model considered is the following:

Long time scale: ∂S β mI  = Λ − µS − S + δ 1∆S, ∂t 1 + hI β mI ∂I  = S − ( µ + γ ) I + δ 2 ∆I, ∂t 1 + hI

(3.30)

αI  ∂m  = m  b − am −  + δ 3 ∆m.  ∂t 1 + hI 

Short time scale: ∂S β mI  = − S + δ 1∆S, ∂t 1 + hI

β mI ∂I =  S − γ I + δ 2 ∆I , ∂t 1 + hI

(3.31)

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Spatial Dynamics and Pattern Formation in Biological Populations

˛m ˜I  ˇ   =   m  b − am −  + ° 3 m, ˘ ˛t 1 + hI  with the initial conditions S ( x , 0 ) ˛ 0, I ( x , 0 ) ˛ 0, m ( x , 0 ) ˛ 0, for all x ˛[ 0, R ] and no-fux boundary conditions ˜S ˜I ˜m = = = 0. ˜x 0,  R ˜x 0, R ˜x 0,  R

(3.32)

For studying the linear stability of the spatial model (3.30), perturb it as S = S* + a exp ( ˜ k t + ikx ) , I = I * + b exp ( ˜ k t + ikx ) , m = m* + c exp ( ˜ k t + ikx ) , where a, b, and c are suffciently small constants, k is the wave number, and ˜ k is the wavelength. The system is linearized about the nontrivial interior equilibrium point E S* ,  I * , m* . The characteristic equation of the linearized system is given by

(

)

λ 3 + s1λ 2 + s2 λ + s3 = 0, where s1 = A1 + (  δ 1 + δ 2 + δ 3 ) k 2 ,

{

}

s2 = A2 − a11 (˜ 2 + ˜ 3 ) + a22 (˜ 3 + ˜ 1 ) + a33 (˜ 1 + ˜ 2 ) k 2 + (˜ 1˜ 2 + ˜ 2˜ 3 + ˜ 3˜ 1 ) k 4 , s3 = A3 + k 2 {( a22 a33 − a23 a32 )˜ 1 + ( a11a33 − a31a13 )˜ 2 + (a11a22 − a12 a21 )˜ 3 } − k 4 ( a11˜ 2˜ 3 + a22˜ 1˜ 3 + a33˜ 1˜ 2 ) + k 6 (˜ 1˜ 2˜ 3 ) ,  with   a11 = − µ −

a22 =

° I * m* ° S* m* ° S* I * ° I * m* , a12 = − , a13 = − , a21 = , 2 * * 1 + hI 1 + hI 1 + hI * 1 + hI *

(

° S* m*

(1 + hI )

* 2

− ( µ + ˛ ) , a23 =

a33 = b − 2am* −

)

° S* I * ˝ m* , a31 = 0,  a32 = − * 1 + hI 1 + hI *

(

)

2

,

˝ I* , A1 = − ( a11 + a22 + a33 ) 1 + hI *

A2 = a22 a33 − a13 a31 − a23 a32 + a11a33 + a11a22 − a12 a21 , A3 = −a11 ( a22 a33 − a23 a32 ) + a21 ( a12 a33 − a13 a32 ) − a31 (a12 a23 − a13 a22 ). Now, Re ( ˜ ) < 0 provided that s1 > 0,  s2 > 0, s3 > 0, and s1s2 − s3 > 0, (R–H criteria), which are therefore the conditions for the steady-state to be stable. If any one or more of the three inequalities are not satisfed, then spatial patterning will be observed. We performed numerical simulations for the model system (3.28) using the parameter values ˝ = 0.5,  µ = 0.00005,  ° = 0.00025, h = 0.01, ˛ = 0.2, a = 1,  b = 2, and ˜ = 0.1. The equilibrium

Modeling Virus Dynamics in Time and Space

141

point is obtained as (463.6, 2.3835, 1.7672). The coefficients of the characteristic equation are s1 ( 0 ) = 1.7729, s2 ( 0 ) = 0.055829, s3 ( 0 ) = 0.000366, and [s1s2 − s3 ]( 0 ) = 0.0986. Therefore, the temporal model system (3.28) is stable. Now, for the spatial model (3.30), the values of the diffusion coefficients are chosen as δ 1 = 0.5, δ 2 = 0.05, and δ 3 = 0.8. The conditions s1 ( k ) > 0,  s2 ( k ) > 0, s3 ( k ) > 0, and [s1s2 − s3 ]( k ) > 0 are satisfied. Hence, no Turing instability occurs for this set of parameter values. Plots of s3 ( k ) and [s1s2 − s3 ]( k ) are given in Figure 3.9. Plots of time series at the fixed point in space x = 1000 are plotted in Figure 3.10. For plotting the figure, the initial conditions are taken as S0 ( x ) = 463.6 + C, I 0 ( x ) = 2.3835 + C, and m0 ( x ) = 1.7672 + C, where C = 0.01 cos  ( 2 πx 20 ) . For the model system (3.29), the equilibrium point is E1 ( aγ ) ( bβ ) , 0, b a. The parameter values that differ from the previous set are Λ = 0, µ = 0, and  β = 0.00023. The equilibrium point is obtained as E1 ( 434.783, 0, 2 ). For the spatial model (3.31), the values of the diffusion coefficients are chosen as δ 1 = 0.5, δ 2 = 0.05, and δ 3 = 0.8. For this model, all the eigenvalues are negative: −δ 1k 2 ,  −δ 2 k 2 , −(b + δ 3 k 2 ) . Plots of time series at the fixed point in space x = 1000 are plotted in Figure 3.11 for the model system (3.31). For plotting the figure, the initial conditions are taken as S0 ( x ) = 434.783 + C ,  I 0 ( x ) = 0.0 + C , and m0 ( x ) = 2.0 + C, where C = 0.01 cos  ( 2 πx 20 ) . It was found that diffusion has an appreciable influence on the spatial spread of influenza. The epidemic propagates in the form of nonchaotic and chaotic waves as observed in H1N1 incidence data of positive tests in 2009 in the United States. The first wave is nonchaotic and the other wave is chaotic as the influenza cases are distributed in a non-Maxwellian way from the 32nd week onwards. Rajatonirina et al. [220]

(

)

FIGURE 3.9 Plots of s3(k) and [s1s2 − s3] (k).

FIGURE 3.10 Time series for the spatial model (3.30) at x = 1000 showing stable dynamics.

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 3.11 Time series for the spatial model (3.31) at x = 1000 showing stable dynamics.

detected waves of the H1N1 infuenza pandemic in laboratory tests carried out simultaneously in several cities of the South African country Madagascar in 2009. These waves were the result of an interaction between interpersonal contact and human movement. The shift in the time of occurrence between dominant H1N1 waves in different cities confrms the spatiotemporal nature of these waves. Gog et al. [91] observed that the determinants of infuenza spatial spread are not fully understood, in part due to the insuffcient geographic resolution of incidence data. The authors concluded that the 2009 pandemic autumn wave spread slowly because transmissibility of the infuenza virus was relatively low, as children (who travel long distances far less than adults) were the predominant source of infection. Aleta et al. [3] studied the crucial role of hosts’ mobility on the ecological dynamics of rapidly mutating pathogens, opening the path for further studies on disease ecology in the presence of a complex and heterogeneous environment. For such diseases, multiple biological, environmental, and population-level mechanisms determine the dynamics of the outbreak, including pathogen’s epidemiological traits (e.g., transmissibility, infectious period, and duration of immunity), seasonality, interaction with other circulating strains and hosts’ mixing, and spatial fragmentation.

3.3 Susceptible-Infected-Susceptible (SIS) Models SIS models are those models in which the infective individuals return to the susceptible class on recovery because the disease confers no immunity against re-infection. It indicates that the passage of individuals is from the susceptible class to the infective class and then back to the susceptible class. Such models are appropriate for most diseases transmitted by bacterial agents, and most sexually transmitted diseases (including gonorrhea but not AIDS, from which there is no recovery). One important way in which SIS models differ from SIR models (that is caused by a virus and not by bacteria) is that in the former, there is a continuing fow of new susceptibles, namely recovered infective populations. The simplest SIS model of Kermack and McKendrick is given by [139] dS dI = − ˜ SI + ° I, = ˜ SI − ° I. dt dt

(3.33)

Modeling Virus Dynamics in Time and Space

143

In this model, the recovered members return to the class S at a rate ˜ I instead of passing to the class R. The total population S + I = N is constant. The model can be reduced to a single differential equation as (Brauer and Castillo-Chavez, [29]) dI I  = ˜ ( N − I ) I − ° I = ˘ˇ ˜ ( N − I ) − °  I = rI  1 −  ,  dt K

(3.34)

which is a logistic equation, where r = ( ˜ N −  ° ) and K = ˇN ˘ − (˜ /° ) .  If ( ˜ N −  ° ) < 0,  then all the solutions of the model (3.34) with nonnegative initial values except the constant solution I = ˘ˇ K − ( ˜ /° )  , approach to zero as t ˜ °, while if ( ˜ N −  ° ) > 0, all solutions with nonnegative initial values except the constant solution I = 0, approach to the limit ˆK ˇ − (˜ /° ) ˘ > 0 as t ˜ °. Thus, there is always a single limiting value for I, but the value of the quantity [ ˜ K/° ] determines which limiting value is approached, regardless of the initial state of the disease. The dimensionless quantity [ ˜ K/° ] is called the basic reproduction number or contact number for the disease. If [ ˜ K/° ] < 1, the infection dies out in the sense that the number of infective individuals approaches zero. The model has the following equilibrium points: DFE E0 ( K, 0 ) and EE E1 (˜ ° , K − ˜ ° ) . Here, the model (3.33) is reduced to a single equation (3.34) by assuming that the total population (S + I) is constant. If there are deaths due to the disease, this assumption is violated, and it would be necessary to use a 2D system as a model. It was proved that for model (3.33), if the basic reproduction number ˛˝ ˜ K ° ˙ˆ ˇ 1, then the solution ( S ( t ) ,  I  ( t )) approaches the DFE E0 ( K, 0 ) , while if ˛˝ ˜ K ° ˙ˆ > 1, a unique EE exists and it is globally asymptotically stable (GAS). The authors [29] considered a model for a disease from which infective individuals recover with no immunity against re-infection and that includes births and deaths, as dI dS = ˇ ( N ) − ˜ ( N ) SI − µS + f˛ I, = ˜ ( N ) SI − ˛ I − µ I. dt dt

(3.35)

Here, ˛ ( N ) describes the density-dependent birth rate per unit time, µ is the proportional death rate in each class, ˜ is the rate of departure from the infectives class through recovery or disease death, and f is a fraction of infective individuals recovering with no immunity against re-infection. In this model, if f < 1, the total population size is not constant, and K represents carrying capacity or maximum population size. Adding the equations in (3.35) and using S + I = N, we obtain dN = ˆ ( N ) − µ N − (1 − f ) ° I. dt The authors carried out the analysis of the SIS model in the special case f = 1, so that N is a constant K. The system (3.35) is asymptotically autonomous and its asymptotic behavior is the same as that of the single differential equation dI = ˜ ( K ) I ( K − I ) − (° + µ ) I, dt which is a logistic equation that can be easily solved analytically by separation of variables or qualitatively by an equilibrium analysis. It is found that I ˜ 0, if K˜ ( K ) < (° + µ ) or 0 < 1 and I ˜ I ˛ > 0, with

144

Spatial Dynamics and Pattern Formation in Biological Populations

I∞ = K −

 α+µ 1  = K1− , if Kβ ( K ) > (α + µ ) , or 0 > 1.  β (K) 0 

The EE exists if 0 > 1, and it is always asymptotically stable. If 0 < 1, the system has only the DFE which is asymptotically stable. Carrero and Lizana [45] examined the geo-temporal evolution of a population in an SIS epidemiological model to describe a possible mechanism for the existence of endemic geographical foci during the disease. They derived the conditions under which the intrinsic epidemiological parameters and diffusion destabilize the homogeneous EE, giving rise to nonhomogeneous steady-state solution or solutions. They proposed the following system of reaction-diffusion equations subject to Neumann boundary conditions as a model for the spatial spread of the disease:

∂ S ( x , t ) =   DS ∆S − β Sq I p  − bS + γ I + a ( S + I ) , ∂t

(3.36a)

∂ I ( x , t ) =   DI ∆I + β Sq I p  − (α + b + γ ) I , ∂t

(3.36b)

∂S ∂ I = = 0 on ∂ Ω, Ω ∈  n , (t ,  x) ∈  + × Ω, ∂η ∂η

(3.36c)

where η is the outward normal vector to ∂ Ω, p ≥ 1 and q ≥ 1 are constants describing the incidence rate of the disease, a, b, β ,  γ , and α represent the birth rate, death rate, transmission rate, recovery rate, and the mortality caused by the disease respectively. The infective individuals can die from the disease with a disease-induced mortality rate α I, where 1 α is the life expectancy of an infective individual. The authors considered the nonlinear incidence rate of the disease as β Sq I p , which was first considered by Severo [245]. The temporal model has two equilibrium points. The equilibrium point E0 ( 0, 0 ) is a saddle point. The unique EE point S* ,   I * is given by

(

)

α +b+γ  p = 1: S =  β  

1/q

, I = AS, A =

α +b+γ  p > 1: S =   β A p−1 

a−b > 0. α +b−a

(3.37a)

1/( q + p −1)

(

, I = AS.

(3.37b)

)

The Jacobian matrix of the system evaluated at S* ,   I * is given by  a11 J=  a21

a12 a22

  , where a11 = fS = − AA1q + a − b ,   a12 = f I = − pA1 + a + γ ,  

a21 = gS = AA1q , a22 = g I = ( p − 1) A1 , A =

a−b ,  A1 = α + b + γ > 0. α +b−a

(

)

We can easily determine the conditions such that the EE point S* ,   I * of the model is asymptotically stable. (Problem 3.3, Exercise 3).

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Modeling Virus Dynamics in Time and Space

If trace ( J ) = 0, at q = q* , then the system undergoes a Hopf bifurcation. Using the eigenvalue analysis, the Jacobian of the system (3.36a) and (3.36b) is obtained as ˛ J=˙ ˙ ˝

fS − DS k 2

fI

gS

g I − DI k 2

ˆ ˘. ˘ ˇ

(3.38)

The characteristic equation in the spatial case is given by

˜ 2 − ˜T + ˝ = 0,

(3.39)

where T = trace ( J ) = fS + g I − k 2 ( DS + DI ), ˛ = det ( J ) = fS g I − f I gS − k 2 ( fSDI + g I DS ) + k 4 ( DI DS ) . The conditions for yielding Turing patterns were derived earlier. The conditions are given by (see equation 3.8) (i) fS + g I < 0, (ii) ( fS g I − f I g s ) > 0, (iii) ( fSDI + g I DS ) > 0, (iv) ( fSDI + g I DS ) > 4DSDI ( fS g I − f I g s ). 2

We have performed numerical simulations for the model (3.36a and b) using the parameter values ˜ = 0.06, ° = 0.0056, ˛ = 0.04, a = 0.05, b = 0.006, p = 2, q = 1,  DS = 7 and DI = 0.7. We obtain S* ,  I * = ( 2.62357, 7.21482 ), kc2 = 0.0580358,  ˝ kc2 = −0.00717594 < 0, for 0.0197673 < k 2 < 0.0963043. For these parameter values, conditions (i)–(iv) are satisfed for obtaining Turing patterns. The plot of ˛, k 2 is given in Figure 3.12. To illustrate the dispersion relation, the variation of the real part of the characteristic value of the model system as DS increases is plotted in Figure 3.13. From Figure 3.13, we observe that as the value of the diffusion coeffcient for the susceptible population is increased, Turing instability appears. For DS = 7, and DS = 11, diffusion-induced instability is obtained, but for DS = 5 and DS = 3, the system shows stable behavior both in temporal and spatial cases. Using MATLAB 7.5, we have also solved the 2D reaction-diffusion model (3.36a) and (3.36b) using a fnite-difference technique with zero fux boundary conditions. The initial population densities are taken as S ( x ,  y , 0 ) = 2.62357 + 0.01 × r and n,  I ( x , y , 0 ) = 7.21482 + 0.01 × r and n, which have been perturbed randomly. The mesh lengths are taken as ˜x = ˜y = 0.1, ˜t = 0.01. The parameter values are taken the same as in plotting Figure 3.12.

(

)

( )

(

FIGURE 3.12 Plot of ˜ vs. k2.

)

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 3.13 Plot of the real part of the characteristic value as DS is increased.

FIGURE 3.14 Plot of susceptible and infected populations after 20,000 iterations. The parameter values used are the same as in Figure 3.12.

The plot of susceptible and infected populations after 20,000 iterations, which is at t = 200, is given in Figure 3.14. From the fgure, it is observed that irregular spotted patterns prevail over the whole domain. Allen et  al. [5,6] and Deng and Wu [62] generalized the model (3.33) to an epidemic SIS reaction-diffusion model with frequency-dependent interaction and mass-action type nonlinearity respectively. Allen et al. [5] studied the SIS reaction-diffusion model1

1

∂S β SI = ds ∆S − + γ I , x  Ω, t > 0 ∂t S+ I

(3.40a)

β SI   ∂I = dI ∆ I + − γ I ,  x    Ω,  t > 0 ∂t S+ I

(3.40b)

Allen et al. [5]: “Copyright ©2007 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved.”

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Modeling Virus Dynamics in Time and Space

where S ( x ,t ) and I ( x ,t ) denote the densities of the susceptible and infected individuals at location x and time t; dS and dI are positive diffusion coefficients for the susceptible and infected populations; and β ( x ) and γ ( x ) are positive Hölder-continuous functions on Ω that represent the rates of disease transmission and recovery at x, respectively. Because SI ( S + I ) is a Lipschitz continuous function of S and I in the open first quadrant, the definition can be extended to the entire first quadrant by defining it to be zero when either S = 0 or I = 0. The problem was considered under the no-flux boundary conditions ∂S ∂I = = 0,  x ∈ ∂Ω, t > 0. ∂n ∂n

(3.41)

The authors dealt with the existence, uniqueness, and asymptotic properties of the steady state. They define a domain as high (low) risk if the average of the transmission rates is greater (less) than the average of the recovery rates. The model studies the phenomena in a continuous-time and continuous-space SIS model that includes both low-risk and high-risk sites. The authors showed that when susceptible and infected individuals move between patches, an EE is reached in every patch. But if the movement pattern is changed so that only infected individuals disperse between the patches, a surprising result occurs. The disease does not persist in any patch at equilibrium. Moreover, at equilibrium, all low-risk patches contain susceptible, and in some cases, high-risk patches can also contain susceptible. This is equivalent to the local reproduction number being less than (or greater than) one, respectively. It was assumed that initially there is a positive number of infected individuals, that is, A1.

∫ I(x, 0)dx > 0, with S(x, 0) ≥ 0 and I ( x, 0) ≥ 0 for x ∈Ω. Ω

By the maximum principle [215], both S ( x ,t ) and I ( x ,t ) are bounded and positive for x  ∈  Ω and t   ∈ ( 0,  Tmax ) ,where Tmax is the maximal existence time for solutions of (3.40) and (3.41). Hence, it follows from the standard theory for semilinear parabolic systems that Tmax = ∞ and that a unique classical solution ( S, I ) of (3.40) and (3.41) exists for all time [104]. Let N=

∫ S( x, 0) + I ( x, 0) dx,

(3.42)

Ω

be the total number of individuals in Ω at t = 0. Summing (3.40a) and (3.40b) and then integrating over Ω, we obtain

∂   ∂t

∂

∫ (S + I ) = ∫∆ ( d S + d I ) =   ∫ ∂ n  (d S + d I ) = 0,   t > 0. s

Ω

I

S

I

∂Ω

Ω

This result implies that the total population size is constant

∫ (S + I ) dx = N , t ≥ 0. Ω

(3.43)

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Spatial Dynamics and Pattern Formation in Biological Populations

If the local disease transmission rate β ( x ) is lower than the local disease recovery rate γ ( x ) , then x is defined as a low-risk site. A high-risk site is defined similarly. Define H − = { x   ∈ Ω:  β ( x ) < γ ( x )} and H + = { x   ∈  Ω:  β ( x ) > γ ( x )} , as the sets of low and high-risk sites, respectively. Ω is a low-risk domain if

∫ β dx < ∫ γ dx, and a high-risk domain if ∫ β dx ≥ ∫ γ dx. The authors defined Ω

Ω

Ω

Ω

the local reproduction number as 0 ( x ) = β ( x )/γ ( x ). Then, 0 ( x ) < 1 for low-risk sites x   ∈ H − , and 0 ( x ) > 1 for high-risk sites x ∈ H + . If there is no movement, the disease can persist at high-risk sites but not at low-risk sites. A2. It was assumed that β − γ changes sign on Ω, that is, H − and H + are nonempty. A3. By continuity, the set H 0 = { x ∈ Ω:  β ( x ) = γ ( x )} is also nonempty and it consists of finitely many disjoint C1 surfaces.

(

)

(

)

The authors investigated the equilibrium solutions of (3.40), that is, the solution of the elliptic problem  β  SI + γ I = 0, x ∈Ω, dS ∆S − S+ I

(3.44a)

 β  SI dI ∆I +  − γ I = 0, x   ∈  Ω, S + I

(3.44b)

with the boundary conditions

∂ S ∂ I = = 0, x  ∈  ∂Ω. ∂n ∂n Here, S ≥ 0 and I ≥ 0 denote the density of susceptible and infected individuals, respectively, at equilibrium. Since the total population size is constant, the additional condition

∫ (S + I ) dx = N ,

(3.45)

Ω

was imposed. By definition, a DFE is a solution in which I ( x ) = 0 for every x  ∈ Ω and is denoted by S , 0 . An EE is a solution in which I ( x ) > 0 for some x  ∈  Ω and is denoted by (S , I). The following theorem gives one of the authors’ main results.

( )

Theorem 3.2 [5] Suppose that (A1) and (A2) hold, and N is fixed. There exists a unique DFE given by Sˆ , 0 = ( N Ω , 0 ) . Let the basic reproduction number be defined as

( )

  R0 = supϕεH 1 ( Ω),  ϕ ≠ 0    βϕ 2  Ω

∫

   2  dI    ∇ϕ + γϕ 2   .    Ω 

∫

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Modeling Virus Dynamics in Time and Space

If 0 < 1, then the DFE is GAS, and if 0 > 1, then it is unstable. The above variational characterization of 0 is similar to the next-generation approach for heterogeneous populations that occupy a continuous spatial habitat [64,65]. Observe that 0 does not depend on the diffusion coeffcient dS for susceptibles. The authors have also shown the following results: (i) 0 is a monotone decreasing function of the diffusion coeffcient dI for infected individuals. (ii) As dI decreases and becomes arbitrarily small, tends to its maximum value max ( ˜ ( x ) ° ( x )) : x ˇΩ . (iii) As dI increases and becomes arbitrarily large, 0 tends to the ratio of the average transmission rate and the average recovery rate. (iv) An equivalent characterization for the stability of the DFE in terms of dI rather than on 0 was developed. (v) DFE is stable in the low-risk domain if and only if dI takes a value above a certain threshold value, but in the high-risk domain, DFE is always unstable. (vi) If EE exists, then it is unique with S > 0,  I > 0, and it exists if and only if the DFE is unstable ( 0 > 1) . Allen et al. [6] proposed a spatial SIS model to capture the impact of spatial heterogeneity of environment and the movement of individuals on the persistence and existence of disease and conjectured that if the unique EE exists, it should be GAS among all initial data satisfying (A1) and equation (3.42); otherwise, the DFE should be GAS for such initial data. Peng and Liu [209] investigated the SIS epidemic reaction-diffusion model studied by Allen et al. [6] and established some results regarding global attractiveness and asymptotic stability of the DFE and EE in two special cases: (i) The susceptible and infected individuals migrate with the same speed, dS = dI . Then, the epidemic disease will eventually die out if the habitat is a low-risk one. EE is GAS in the high-risk domain and DFE is GAS in the low-risk domain. (ii) The rate of disease transmission is proportional to the rate of disease recovery, ˜ ( x ) = r° ( x ) for some positive constant r ˛( 0,  ˝ ) and every x ˜Ω. Then, if r ˜ 1, then the epidemic disease will be completely extinct and it exists at any location and at any time if r > 1. In addition, if both ˜ and ˜ are positive constants, then EE is GAS. Their results provide potential applications to effectively control the spread of an epidemic disease. (It seems very necessary to create a low-risk habitat for the population.) Peng and Zhao [210] considered an SIS reaction-diffusion epidemic model, where the rates of disease transmission and recovery are assumed to be spatially homogeneous and temporally periodic and the total population is constant. Tuners and Martcheva [265] introduced a two-train spatially explicit SIS epidemic model with space-dependent transmission parameters and addressed the issues like whether the presence of spatial structure would allow the two strains to coexist, as the corresponding spatially homogeneous model leads to competitive exclusion. Numerically, they confrmed the stability of the coexistence equilibrium and investigated various competition scenarios between the strains. Zhao et al. [313] have taken into account the effect of a randomly fuctuating environment and studied the extinction and persistence of the stochastic SIS model with vaccination. Deng and Wu [62] considered an epidemic reaction-diffusion model and assumed that Ω is a bounded domain in  m with smooth boundary ˜ Ω. The authors studied the existence of the DFE and EE and their global attractivity. The authors have shown that there exists a unique reproduction number 0 . S ( x , t ) and I ( x ,  t ) are respectively the densities of susceptible and infected individuals at location x and time t. The individual populations move randomly in the domain with diffusion rates dS and dI respectively. It was assumed that all the infected individuals at the same location have the same rate for recovery and become susceptible immediately. Then, the model can be formulated as [62]

{

}

˜S = − ° ( x ) SI + ˛ ( x ) I + dS S, ˜t

(3.46a)

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Spatial Dynamics and Pattern Formation in Biological Populations

∂I = β ( x ) SI − γ ( x ) I + dI ∆I , x ∈ Ω, t > 0, ∂t

(3.46b)

∂S ∂ I = = 0, x ∈∂Ω, t > 0, ∂n ∂n where the disease transmission rate β ( x ) describes the effective interaction between susceptible and infected individuals and γ ( x ) represents the recovery rate of the infected individuals at location x. Both β and γ are positive Hölder-continuous functions in Ω. The initial data satisfies the following hypothesis: S ( x , 0 ) and I ( x , 0 ) are nonnegative continuous functions in Ω, and initially, the number of infected individuals is positive, I ( x , 0 ) dx > 0. Let S ( x , 0 ) + I ( x , 0 )  dx ≡ N be the total number of individuals at time

∫

Ω

∫

Ω

t = 0. In the absence of diffusion, adding the equations (3.46a) and (3.46b), and integrating ∂ over the domain Ω, we find that ( S + I ) dx = 0, t > 0, which implies that the total popu∂t Ω lation size N is constant. Let S ( x ) and I ( x ) respectively be the densities of susceptible and infected individuals at location x. The functions also satisfy the constraint

∫

∫ (S  + I ) dx = N.

(3.47)

Ω

The equilibrium points of the model system (3.46) are the nonnegative solutions of the semilinear elliptic system

with boundary conditions

− β S I + γ I + dS ∆S = 0,

(3.48a)

β S   I − γ I + dI ∆ I = 0 ,

(3.48b)

∂S ∂ I = = 0,  x ∈∂ Ω. The model has two equilibrium points: ∂n ∂n

i. A DFE is a solution of (3.48) with I ( x ) = 0 for all x ∈∂Ω. Let Ω be the measure of Ω. For any DFE, by (3.48a), we have ∆S = 0. Then, by the maximum principle and the boundary condition ∂S/ ∂n = 0, S must be a constant in Ω. Then, it follows from (3.47) that S = N Ω . ii. EE (S ,  I ) is a nonnegative solution of the system (3.48). dSS + dI I = K, x ∈Ω,

(

)

(3.49a)

dI ∆I + I β S − γ = 0,  x ∈ Ω,

(3.49b)

∂S ∂I = = 0,  x ∈∂ Ω, ∂η   ∂η  

(3.49c)

∫ (  S +  I   ) dx = N ,

(3.49d)

Ω

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Modeling Virus Dynamics in Time and Space

where K is some positive constant that is independent of x ∈Ω. From (3.49a), we get S = K − dI I dS . Substituting it into (3.49d), we find

(

)

K=

 1   dS N − ( dS − dI ) Idx  . Ω  Ω  

∫

It then follows from (3.49a) that S=

 d  1 1  d   N −  1 − I  Idx  − I I . K − dI I =  Ω dS  dS  dS Ω  

(

)

∫

(3.50)

Substituting S into (3.49b), we obtain    β  d   dI ∆ I + I   N − 1− I    Ω  dS   

∫

Ω

  β dI   Idx − I − γ = 0.  dS   

Taking the normal derivative of both sides of (3.50) and using the condition ∂ I ∂ n = 0, it can be seen that ∂ S ∂ n = 0, which verifies (3.49c). Integrating both sides of (3.50) over Ω, (3.49d) is obtained. Applying the Laplace operator to both sides of (3.50), it is found that dS ∆S + dI ∆I = ∆ dSS + dI I = 0. Since ∂ dSS + dI I /∂ n = 0, the maximum principle implies that dSS + dI I = constant. In view of (3.49d), this constant must be positive, which yields (3.49a). Using a variational formula, the authors defined the reproduction number as

(

)

(

)

  2  ( N Ω ) βϕ dx    Ω :  ϕ ∈ H 1 ( Ω ) , ϕ ≠ 0  ,  ϕ 2 dx = 1. 0 = sup  2 2  dI ∇ϕ + γϕ dx  Ω  Ω 

∫(

∫

)

∫

The DFE is stable for 0 < 1 and unstable for 0 > 1. When 0 > 1, there exists a unique EE. The authors have shown that for the case when the disease transmission and recovery rates are constants or the number of susceptible individuals is equal to the diffusion rate of the infected individuals, then the DFE is globally attractive if 0 ≤ 1, and the EE is globally attractive if 0 > 1. Cui and Lou [57] studied the effects of diffusion and advection for an SIS reaction-diffusion model in heterogeneous environments. The model considered is

β ( x ) SI ∂S + γ ( x ) I, 0 < x < L, t > 0 = dsSxx − qSx − ∂t S+ I

(3.51a)

β ( x ) SI  ∂ I − γ ( x ) I , 0 < x < L, t > 0 = dI I xx − qI x + ∂t S+ I

(3.51b)

dsSx − qS = 0,   dI I x − qI = 0, x = 0, L ;   t > 0,

(3.51c)

S ( x , 0 ) = S0   ≥ 0,   I ( x , 0 ) = I 0   ≥ 0,  0 < x < L,

(3.51d)

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Spatial Dynamics and Pattern Formation in Biological Populations

where L is the size of the habitat and q is the effective speed of the current or advection rate. x = 0 and x = L are the upstream and downstream ends respectively. SI ( S + I ) is a Lipschitz continuous function of S and I in the open first quadrant. Its definition can be extended to the closure of the first quadrant by setting it to be zero when S = 0, or  I = 0. Authors assumed that L

∫I ( x, 0) dx > 0, with S( x, 0) ≥ 0 and I ( x, 0) ≥ 0 for x ∈ (0, L). 0

By the maximum principle [215], both S ( x ,t ) and I ( x , t ) are positive for x   ∈[ 0,  L ] and t ∈ ( 0, Tmax ) , where Tmax is the maximal existence time for the solution of (3.51), and are bounded on [ 0,  L ] × ( 0,  Tmax ) . It follows from the standard theory for semilinear parabolic systems that Tmax = ∞, and the system (3.51) admits a unique classical solution for all time [104]. The equilibrium solutions of (3.51) are the nonnegative solutions of the system  β ( x ) SI dsS xx − qS x −   + γ ( x ) I = 0, 0 < x < L, S+ I

(3.52a)

β ( x ) S I − γ ( x ) I = 0, 0 < x < L, S + I

(3.52b)

dI Ixx − qIx +

dsS x − qS = 0,   dI   Ix − qI = 0,     x = 0,  L.

(3.52c)

Here, S ( x ) and I ( x ) denote the density of susceptible and infected individuals respectively, on x ∈[ 0,  L ] , at equilibrium. The total population satisfies the condition L

∫ (S ( x) + I ( x )) dx = N , S ( x ) ≥ 0 I ( x) ≥ 0.

(3.52d)

0

A DFE is a solution of (3.52) in which I ( x ) = 0 for every x ∈( 0,  L ) and an EE is a solution in which I ( x ) > 0 for some x ∈( 0,  L ). The DFE is given by S , 0 , where S = qNe( qx/dS )  dS (e( qL/dS ) − 1)  , which is spatially inhomogeneous, and an EE is given by (S , I). The basic reproduction number is defined as R0 ( dI ,  q ) = supϕεH 1 ( Ω), ϕ ≠ 0 {Q1 Q2 } ,

( )

L

where Q1 =

∫ 0

β ( x ) e( qx dI )ϕ 2 dx , Q2 = dI

L

∫ 0

qx d e( I )ϕ 2 dx + x

L

∫ γ ( x ) e(

qx dI )

ϕ 2 dx.

0

If 0 < 1, then the DFE is GAS, but if 0 > 1, then it is unstable. Cao et al. [38] investigated the basic features of an SIS-type infectious disease model with varying population and vaccination in the presence of environmental noise. By applying the Markov semigroup theory, the authors proposed a stochastic reproduction number, which can be seen as a threshold parameter useful in identifying the stochastic extinction and persistence.

Modeling Virus Dynamics in Time and Space

153

3.4 Susceptible-Infected-Removed (SIR) Models SIR models confer immunity against re-infection, to indicate that the passage of individuals is from the susceptible class to the infective class to the removed class. Removal is carried out through isolation from the rest of the population, through immunization against infection, through recovery from the disease with full immunity against re-infection (as in measles), or through death caused by the disease (as in plague and rabies). Epidemics are usually diseases of the SIR type and are caused by viruses. Kermack and McKendrick [138–140] were the pioneers in the feld of mathematical epidemiology who studied the deterministic structured population models for the spread of infectious diseases. An excellent exposition of their work was given by Inaba [126]. In 1927, they developed and studied the SIR model with variable (duration-dependent) infectivity. They assumed that the infection rate depends on the duration of the infected and infectious status and the infection occurs only once in the lifetime of an individual. If the infectivity is a constant, then the model reduces to a simple ordinary differential equation (ODE) model. Further studies of the variable infectivity model were made by Metz [190] and Diekmann [63]. The importance of the variable infectivity model is now well recognized as it can model epidemics with a long incubation period and variable infectivity such as HIV/AIDS epidemics [261]. Kermack and McKendrick developed more complex models in 1932 and 1933, [139,140]. In these models, they developed duration-dependent epidemics, where the transmission rate depends on both the durations of the infected host and susceptible host. In this model, the recovered individuals can be re-infected repeatedly. Their models have attracted attention recently as new types of epidemics are emerging because of genetic changes in the viruses. Owing to these genetic changes, host immunity becomes less effective and vaccinations cannot control them. Inaba [126] reformulated the Kermack and McKendrick variable infectivity model as a nonlinear age-dependent population dynamic. The author proved the existence, uniqueness, and local stability of the endemic steady state and also showed that Pease’s evolutionary epidemic model [208] is a special case of their model. We briefy discuss the Kermack and McKendrick’s SIR ODE model [138]. The model proposed to describe the spread of diseases is given by dS dI dR = − ˜ SI, = ˜ SI − ° I, . = ° I dt dt dt

(3.53)

S ( 0 ) = S0 ,  I ( 0 ) = I 0 ,  R ( 0 ) = R0 ˝ 0. Here, ˜ is the transmission rate (per capita) and ˜ is the mean recovery rate (1/˜ is the mean infectious period). The constants ˜ and ˜ give the transition rates (probabilities) between compartments, and their ranges are 0 ° ˜ ° 1 and 0 ° ˜ ° 1 respectively. The model is based on the following assumptions: (i) An average member of the population makes contact suffcient to transmit infection with ˜ N others per unit time, where N represents the total population size (mass action incidence). (ii) Infective individuals leave the infective class at the rate ˜ I per unit time. (iii) There is no entry into or departure from the population, except through death from the disease. It means that the time scale of the disease is much faster than the time scale of births and deaths, so that the demographic effects on the population may be ignored. (iv) There are no disease deaths and the total population size S + I + R = N is constant. Kermack and McKendrick [138] replaced the equation (3.53) with the following two equations:

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Spatial Dynamics and Pattern Formation in Biological Populations

dS ° dR = ˜ ( N − S − R ) , and = − S. dR ˜ dt Solving the second equation, we get S = S0 e − ° R ˛ . Thus,

(

)

dR = ˜ N − S0 e − ˆ R ˜ − R . dt Since an explicit form of the solution is not possible, expand the exponential term in powers of ˜ R ° ,  and assume that ˜ R °  1. Retaining up to second-order terms, one obtains   ˆ° S ° 2R2  dR = ˜  N − S0 + ˇ S0 − 1  R − 0 2  . dt 2˜ ˘˜   

(3.54)

N − S0 = I 0 , where I 0 is small. Although ˜ R °  1, its square may not be small compared to 2 ˙˝(˜ ° )S0 − 1ˆˇ R. Therefore, the third term in R is also considered. The solution of the equation (3.54) is R=

˜2 ° 2S0

ˇ −q    ° ˜ t − ˛   S0 − 1 + − q tanh  ˘ 2    ˜ 1/2

( ° S0 ˛ ) − 1 ,  and

2 ˙ ˜ ˘ ˜2  − q = ˇ S0 − 1 + 2S0 I 0 2  . where ˜ = tanh °  −q  ˆ °  2˜ ˙ ˜˘ Furthermore, at the end of the epidemic, R = S0 −  , where I 0 has been neglected. ° S0 ˆˇ ° I 0, the initial number of infected cases, is usually small compared to S0. If I 0 is neglected, S0 is identical with N. If S0 = ˜ ° , then no epidemic will take place. If, however, N slightly exceeds this value, then a small epidemic may occur. If we write N = (˜ ° ) + n, its magnitude is 2˜ n ( ° N ) or 2 ˝˙ n − n2 N ˆˇ . In this sense, the population density N 0 = ˜ ° may be considered as a threshold density of the population for an epidemic. No epidemic can occur unless the population density exceeds this value, and if it exceeds the threshold value, then the size of the epidemic will be, to a frst approximation, equal to 2n, which is twice the excess. At the end of the epidemic, the population density will be just as far below the threshold density, as initially it was above it. From the second equation of (3.53), when dI dt = 0, that is, when   S =  ˜ ° , the unaffected population is reduced to its threshold value. Therefore, we can say that once the population is below this value, any particular infected individual has more chance of being removed by recovery or by death than of becoming a source of further infection, and so the epidemic commences to decrease. In small epidemics, the plot of I is symmetric about the maximum. This symmetry exists for I as a function of t, and consequently also for dR dt. On the other hand, no such symmetry is obtained in the curve for the case of incidence, that is of ( dS dt ) = − ˜ SI. This is true, since ˘  I is symmetric and S = exp  − ° 2 ˛ Idt  .   Capasso and Serio [41] generalized the Kermack and McKendrick epidemic model (3.53) with R ( 0 ) = R0 = 0, by introducing an interaction term in which the dependence upon the −1

(

)

(

)˜

Modeling Virus Dynamics in Time and Space

155

number of infective individuals occurs via a nonlinear bounded function, which may take into account saturation phenomena for large numbers of infective individuals. The modified model is taken as dI dR dS = g ( I )S − γ I , = − g ( I ) S, = γ I , t > 0, dt dt dt with initial conditions S ( 0 ) = S0 ≥ 0, I ( 0 ) = I 0 ≥ 0, R ( 0 ) = 0 such that S0 + I 0 = N, where N is the total size of the population. Here g:  + →  + is a continuous bounded function, which takes into account the saturation phenomena. It was assumed that (i) g ( x ) ≥ 0, ∀x ∈ + , (ii) g ( 0 ) = 0, (iii) ∃  c ∈  + − {0} s.t., ∀x ∈  + :  g ( x ) ≤ c , (iv) g ′:  R+ → , the derivative of g exists and is bounded on any compact interval of  + , with g′ ( 0 ) > 0, and (v) ∀x ∈  +   g ( x ) ≤ g ′ ( 0 ) x , where  + = [ 0,  +∞ ). A diffusive analog of the model (3.53) with a modification in the interaction term was studied by Capasso [39]. The author considered the following diffusive epidemic model with the removal of infective individuals, in which the interaction term depends on the density of infection via a smooth bounded map:

∂ S ( x ,t ) ∂ 2S = d1 2 − a  I (., t )  ( x ) S ( x , t ) , ∂t ∂x ∂ I ( x ,t ) ∂2I = d2 2 + a  I (., t )  ( x ) S ( x , t ) − γ I ( x , t ) with t ≥ 0,  x ≤ L, ∂t ∂x subject to zero flux boundary conditions

∂S ∂S ∂I ∂I ( − L, t ) = ( L, t ) = 0, ( − L, t ) = ( L, t ) = 0, ∂x ∂x ∂x ∂x and initial conditions S ( x ;  0 ) = S0 ( x ) ≥ 0, I ( x ;  0 ) = I 0 ( x ) ≥ 0, but not identically zero, γ is the removal rate of infective individuals due to recovery or death from epidemic disease. The positive constants d1 and d2 are the diffusion coefficients while the interaction term of infective individuals with susceptibles is given by a [ I ] S. The author derived upper bounds for the number of susceptible and infective population. In many infectious diseases, such as in the case of measles, there is an arrival of new susceptible individuals into the population. For this type of situation, deaths must be included in the model. By considering a population characterized by a death rate µ equal to the birth rate, the epidemic model is given by [29,30,58,198], dS dI dR = β SI − (γ + µ ) I , = γ I − µR. = − β SI + µ ( N − S) , dt dt dt

(3.55)

with the initial conditions S ( 0 ) = S0 ,  I ( 0 ) = I 0 , R ( 0 ) = R0 , where S + I + R = N . This is the classic endemic model which considers vital dynamics such as births and deaths [110]. Both the simple SIR model ( µ ≠ 0 ) (represented by model system (3.53)) and the SIR model with vital dynamics ( µ ≠ 0 ) (given in model system (3.55)) are 2D dynamical systems in the S + I + R = N invariant plane. The dynamics of both systems is simple and well understood, including the bifurcation that takes place at β N = (γ + µ ) . For µ = 0, I = 0 is

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a line of degenerate equilibria, for all parameter values. For the model system (3.55), the time evolution of S + I + R trajectories with initial conditions out of the invariant plane tend exponentially fast to the invariant plane – there can be no attractor or invariant set, chaotic or otherwise, outside the invariant plane. There is no chaotic dynamics for µ ≠ 0 outside the S + I + R = N invariant plane. For β < 0, S = N, I = 0, and R = 0 is a global attractor – a stable node, while for β > 0, S* =

 γ + µ * µ  Nβ γ ,I =  − 1 , and R * = I * , β β γ +µ µ 

is a global attractor – a stable node/focus. It is interesting to note that the model systems (3.53) and (3.55) represent a modified 3D competitive Lotka–Volterra-type models [29,58,198]. These systems can also be related to the so-called T-systems introduced by Tigan [262], which have the form dy dR dx = ( c − a ) x − axz,  = xy − bz. = a( y − x) , dt dt dt This system is chaotic for a = 2.1, b = 0.6, and c = 30. Mathematical properties of the T-system were studied by several authors [263,271,307]. Harko et al. [98] obtained the exact analytical solutions of the SIR models (3.53) and (3.55) in terms of a parameter. Equations in (3.55) are reduced to an Abel-type equation, and the general solution is obtained using an iterative method. Briefly, we discuss their work by considering the general solution of (3.53), dS dI dR = − β SI, = β SI − γ I , = γ I, dt dt dt with the initial conditions S ( 0 ) = S0   ≥ 0,  I ( 0 ) = I 0   ≥ 0, R ( 0 ) = R0   ≥ 0 and satisfying the condition S + I + R = N . Let prime denote a derivative with respect to t. The first equation gives (S′ S) = − β I. By differentiating the first equation with respect to t and substituting the above expression for I, we obtain 2 S′′  dI 1  S′  =   −  .   dt β  S S

Using the second equation and then the first equation, we obtain 2

S′ S′′  S′  −  +γ − β   S′ = 0. S  S S

(3.56)

From the first and third equations, we obtain

γ S′ dR =− . β S dt

(3.57)

The solution of this equation is S = ke −( β R )/γ , k > 0. Using the initial conditions, we get k = S0e ( β R0 )/γ . By differentiating, we obtain

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S˛ = −

k˜ R˛e − ( ˜ R )/° . °

(3.58)

Now, by differentiating equation (3.57) with respect to t, we obtain R˛˛ = −

2 ˜  S˛˛ ˆ S˛    − ˘ˇ   . °S S 

(3.59)

Substituting the expressions from (3.56)–(3.58) into (3.59), we obtain the equation describing the spread of a nonfatal disease in a given population as R˛˛ = R˛ ˆˇ k˜ e − ( ˜ R )/° − ° ˘ .

(3.60)

Introduce a variable transformation u ( t ) = e − ( ˝ R )/˙ . Equation (3.60) reduces to the equation 2

u

d 2 u ˆ du  du − ˘  + (˜ − k° u) u = 0. 2 ˇ  dt dt dt

(3.61)

Using the initial conditions, we get u ( 0 ) = u0 = e − ( ˝ R0 )/˙ .

dt A further variable transformation ˜ ( u) defned by ˜ = reduces this equation to a du Bernoulli-type equation d˜ 1 + ˜ = (° − k˛ u)˜ 2 . du u

(3.62)

The general solution of this equation is given by

˜=

1 , u ( C − ° ln u + k˛ u) 

(3.63)

where C is a constant of integration. Integrating dt = ˜ du, we obtain the integral representation of the time as u

t − t0 =

d°

˜ ° (C − ˛ ln ° + k˝° ) ,

(3.64)

u0

where t0 is an arbitrary integration constant. We may choose t0 = 0, without loss of generality. Hence, the exact solution of the model system (3.53) is given in a parametric form. We have S = ku . By differentiating u ( t ) = e −( ˝ R )/˙ , we get du = e − ( ˜ R )/° dt

ˆ ˜ dR  ˆ ˜  ° C 1 ˘ˇ − ° dt  = ˘ˇ − °  ° uI , I = − u˜˛ = ˜ ln u − ku − ˜ .

From u ( t ) = e −( ˝ R )/˙ , we get ln u = − ( ˜ R ) ° , or R = − (˜ ° ) ln u. The expressions for S, I, and R give the parametric form of the solution where u is taken as a parameter. Now, by adding we obtain S + I + R = − ( C ˜ ) = N. We have C = − ˜ N , and hence C is a negative integration

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constant. The solution describes the dynamical solution of the SIR system for any given set of initial conditions, S0, I 0 ,  R0, and for arbitrary values of ˜ and ˜ . The numerical values of u0 and C are determined by the model parameters and the initial conditions. The authors found that changes in the numerical values of the initial conditions and/or the rate parameters did not affect the validity of the solution. The epidemic models given by equations (3.53) and (3.55) were investigated numerically by several authors. Some of the methods used by the authors are the following: the Adomian decomposition method [24], variational iteration method [218], homotopy perturbation method [219], and differential transformation method [21]. A stochastic epidemic-type model with enhanced connectivity was analyzed by Williams [288], and an exact solution of the model was obtained. With the use of a quantum mechanical approach, the master equation was transformed via a quantum spin operator formulation. The time-dependent density of infected, recovered, and susceptible populations for random initial conditions was calculated exactly. A stochastic model of infection dynamics based on the SIR model, where the distribution of the recovery times can be tuned, interpolating between exponentially distributed recovery times, as in the standard SIR model, and recovery after a fxed infectious period, was investigated by Black [27]. For large populations, the spectrum of fuctuations around the deterministic limit of the model was obtained analytically. 3.4.1 SIR Models with Vital Dynamics If we assume that the birth and death rates of a population are equal during the epidemic period of a disease and that there is no disease-induced death, then the population size in a closed environment is constant. Consider the model system (3.55) with equal birth and death rates with a constant population size. An example of this model is the transmission of whooping cough [77,100,134], which is one of the most serious childhood diseases and the cause of more deaths in children than any other infectious disease apart from measles. Chinviriyasit and Chinviriyasit [49] modeled the transmission of whooping cough by a spatial version of the SIR model (3.55). In the following, we discuss briefy their work. Let Ω be a bounded domain in R n with a smooth boundary ˜ Ω and ˜ be the outward unit normal vector on the boundary. The equations governing the model are given by [49]

˜S = − ° SI + µ ( N − S) + ˝ S, ˜t

(3.65a)

˜I = ° SI − (˛ + µ ) I + ˙I, ˜t

(3.65b)

˜R = ° I − µ R + ˝˘R , z Ω,  t > 0, ˜t

(3.65c)

with homogeneous Neumann boundary conditions

˜˙ S = ˜˙ I = ˜˙ R = 0,  z ˛˜ Ω, t > 0,

(3.66)

S ( z, 0 ) = S0   ˝ 0,  I ( z, 0 ) = I 0 ˝ 0,  R ( z, 0 ) = R0 ˝ 0.

(3.67)

and initial conditions

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The spatial propagation of individuals is modeled by diffusion coeffcients ˜ S ° 0, ˜ I ° 0, and ˜ R ° 0 for the susceptibles, infected, and recovered, respectively. Infected individuals are assumed not to be affected by the disease in their mobility, thus ˜ S =  ˜ I =  ˜ R = ˜ . The homogeneous Neumann boundary conditions imply that the system is self-contained and there is no infection across the boundary. By the maximum principle [215], the populations S ( z,  t ) ,  I ( z,  t ), and R ( z,  t ) are positive for z ˜° and t ˛( 0, Tmax ), where Tmax is the maximal existence time for solutions of the system (3.65). Then S ( z,  t ) ,  I ( z,  t ), and R ( z,  t ) are bounded on ˛ × ( 0, .Tmax ) Hence, it follows from the standard theory for a semilinear parabolic system that Tmax = ° and the system (3.65) admits a unique classical solution S ( z,  t ) ,  I ( z,  t ), and R ( z,  t ) for all time. As in the study of Allen et al. [6], defne N=

˜ ˆˇS( z, 0) + I ( z, 0) + R ( z, 0)˘ dz > 0, for all t > 0

(3.68)

Ω

to be the total number of individuals in Ω at t = 0. Adding the three equations in (3.65a–c) and then integrating over Ω, one obtains

° °t

˜ [S + I + R]dz = ˜ ˛˘ [S + I + R] dz = 0. Ω

Ω

This implies that the total population size is constant, that is,

˜ [S + I + R] dz = N, for all t > 0. Ω

In this case, the model system (3.65) can be reduced to the two equations [49] St = − ˜ SI + µ ( N − S) + ˛S,

(3.69a)

It = ˜ SI − (° + µ ) I + ˝I,

(3.69b)

and R is determined from R ( z,  t ) = N − S ( z,  t ) − I ( z, t ) . The model system (3.69) has DFE E0 ( N , 0 ) and an EE E* S* , I * , where

(

)

S* =

˜ +µ * µN µ , I =   −  . ˛ ˜ +µ ˛

EE exists provided the contact rate ˜ of the infected population is large. Let µ1 < µ2 < µ3 < … be the eigenvalues of the operator –˜ on Ω with the homogeneous Neumann boundary 2 condition (see [207]). Let V = u = ( S,  I ) ˆ ˇC Ω  | S =  I = 0 on Ω and Vi be the invari˘  ˛ ant subspace of V for a given eigenvalue µi , so that ˜i=1 Vi . * *   −˛S  ˜˘ 0   −µ − ˛ I . Let L =  +  * * ˜˘   ˛I ˛ S − (˝ + µ )   0  Then, the linearization of the system (3.69) is ut = Lu, and since Vi is invariant under the operator L for each i ° 1, ˜ is an eigenvalue of L on Vi , if and only if it is an eigenvalue of the matrix

{

( )

}

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Spatial Dynamics and Pattern Formation in Biological Populations

 −αµi L= 0 

  −µ − β I* 0  + −αµi   βI* 

− β S*

β S* − (γ + µ )

 . 

The characteristic equation is given by

( λ + αµ + µ + β I )( λ + αµ + µ + γ − β S ) + β S I i

*

i

*

2 * *

= 0.

The eigenvalues for E0 ( N, 0 ) are − (αµi + µ ) and − (αµi + µ + γ − β N ) . Let r0 =  β N ( µ + γ )  . If r0 < 1, that is, β N < ( µ + γ ) , then both the eigenvalues are negative and the DFE is locally asymptotically stable. The authors have also shown that the DFE is also GAS. For E* S* ,  I * , the characteristic equation becomes

(

)

( λ + αµi )2 + A ( λ + αµi ) + B = 0, A = ( βµ N ) ( µ + γ ) , B = βµ N − µ ( µ + γ ). Again, let r0 =  β N ( µ + γ )  . Then, A = µr0 and B = µ ( µ + γ )[ r0 − 1]. The EE is locally asymptotically stable if A > 0 and B > 0. Now, A > 0 is true always. If r0 > 1, then B > 0, and the EE E* S* ,  I * is locally asymptotically stable. The authors have also shown that the EE is also GAS with nontrivial initial functions. The expressions that we have derived in the above analysis are in a different form from the expressions given by the authors. However, the results in both cases are the same. For numerical simulations, the authors considered the model system (3.69) under the following initial and boundary conditions

(

)

S ( z, 0 ) = S0 = 325000 z,  0 ≤ z ≤ 0.5 = −325000 ( z − 1) , 0.5 ≤ z ≤ 1 I ( z, 0 ) = I 0 = 7500 z,  0 ≤ z ≤ 0.5 = −7500 ( z − 1) , 0.5 ≤ z ≤ 1,

∂ S ( 0,  t ) ∂ I ( 0,  t ) ∂ S ( 1,  t ) ∂ I ( 1,  t ) = = 0, = = 0,  t > 0, z ∈[ 0, 1] , ∂z ∂z ∂z ∂z The authors studied the following: (i) Effect of the diffusive rate α , which reveals the dynamical behavior of whooping cough (see also [204]). (ii) Effect of the transmissibility coefficient β . Numerical results have shown that the dynamics of whooping cough depends on both the diffusion rate and the contact rate. The authors suggested that reducing the contact rate for susceptible humans is a good policy to control the spread of whooping cough. If the contract rate is large, whooping cough may become a pandemic. The dynamics of whooping cough depends on both the contract and diffusion rates. Duncan et  al. [77] analyzed the annual deaths from whooping cough epidemics in London during 1701–1812 and concluded that the evolution of the whooping cough epidemics is consistent with a linearized model where the dynamics of the system is dependent on the population size and on susceptibility, which is directly associated with malnutrition.

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161

3.4.2 SIR Models with Treatment Rate Wang and Ruan [282] proposed an epidemic temporal model with a constant removal rate of infective individuals to understand the effect of limited resources for the treatment of infective individuals on the disease spread. They introduced a constant treatment that simulated a limited capacity for treatment in an SIR model and showed that the model exhibits various bifurcations including saddle-node bifurcation, subcritical Hopf bifurcation, and homoclinic bifurcation. Liu and Jin [167] considered the spatial version of the model proposed by Wang and Ruan [282] and studied how diffusive contacts and diffusive movements affect the formation of labyrinthine spatial patterns in two dimensions. The considered model with bilinear incidence rate is [167]2

˜S = A − dS − ° SI + Dsˆ 2S, ˜t

(3.70a)

˜I = ° SI − ( d + ˛ ) I − h ( I ) + Di 2 I, ˜t

(3.70b)

˜R = ° I + h ( I ) − dR + Dr ˘ 2 R, ˜t

(3.70c)

where Ds ,  Di , and Dr are the diffusion coeffcients of susceptible, infective, and recovered populations respectively. A is the recruitment rate of the population (such as the growth rate of an average population size, a recovered individual becomes susceptible, immigrant, and so on), d is the natural death rate of the population, ˜ is the natural recovery rate of the infective individuals, and ˜ is a measure of transmission effciency of the disease from a susceptible individual to an infective individual. Here, h ( I ) = r,when I > 0; it is equal to 0, when I = 0; it is the removal rate of infective individuals due to treatment. Authors assumed that the treated infective individuals become recovered when they are treated in treatment sites. Wang [277] modifed the constant treatment to h ( I ) = rI, 0 ˝ I ˝ I 0 , and rI 0 when I > I 0 , which means that the treatment rate is proportional to the number of the infective individuals before the capacity of treatment was reached, and then it took its maximum value rI 0 . Several authors adopted this staged treatment function and explored the dynamics of some epidemic models with standard incidence rate [122] and general incidence rate [311]. Zhang and Liu [310] introduced a continually differentiable saturated treatment function h ( I ) = aI ( 1 + bI ) , where a and b are positive constants to characterize the saturation phenomenon of limited medical resources. Here, the quantity (a/b) models the maximal supply of medical resources per unit time and 1 ( 1 + bI ) describes the reverse effect of the infected individuals being delayed for treatment, which plays an important role in the spread of the infectious disease. In the saturated treatment function, the limit on the medical resources (a/b) and the effciency of the supply of available medical resources 1/( 1 + bI ) are dependent. To better understand their effects on the spread of infectious diseases, it is more reasonable to modify it to the form h ( I ) = ˜ I (° + I ) , where ˜ ° 0 represents the maximal medical resources supplied per unit time and ˜ > 0 is the half-saturation constant, which measures the effciency of the medical resource supply in the sense that, if ˜ is smaller, then the effciency is higher. In order to understand how the limited medical resources and the effciency in supply affect the transmission of infectious diseases, Zhou and Fan [314] designed an SIR model with this modifed form of treatment 2

Liu and Jin [167]: “© SISSA Medialab srl. Reproduced by permission of IOP Publishing. All rights reserved.”

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Spatial Dynamics and Pattern Formation in Biological Populations

rate. Their study revealed that, with varying amount of medical resources and the effciency in supply, the target model admits both backward bifurcation and Hopf bifurcation. Equations (3.70a) and (3.70b) are independent of equation (3.70c). Hence, the authors [167] considered the reduced model (3.70a) and (3.70b). The equilibrium points are given by E1 ( S1 ,  I1 ) and E2 ( S2 ,  I 2 ) , where d  A d  A I1 = R1 − R12 − 4 H  , S1 = ;I = , and R + R12 − 4 H  , S2 =   2λ  ( d + λ I1 ) 2 2 λ  1 ( d + λ I2 ) λA λr ,  H = . R1 = R0 − 1 − H ,  R0 =  d ( d + γ )   d ( d + γ )  Positive solutions exist only if R0 − 1 − H > 0, that is, d ( d + ˜ ) + ° ( r − A ) < 0. (Problem 3.4, Exercise 3). There is a Turing space in the system at the attracting positive equilibrium point E2 ( S2 , I 2 ), but at E1 ( S1 , I1 ), there is no Turing space (see Qian and Murray [216]). The conditions for yielding Turing patterns for the system (3.70a) and (3.70b) are given by (see equation 3.8) (i) fS + g I < 0, (ii) ( fS g I − f I g s ) > 0, (iii) ( fS DI + g I DS ) > 0, (iv) ( fS DI + g I DS ) > 4DS DI ( fS g I − f I g s ), 2

(3.71)

where fS = − d − ˜ I 2 , f I = − ˜ S2 , gS = ˜ I 2 , g I = − ( d + ˜ ) + ° S2 , and k 2 = k x2 + k y2 . (Problem 3.5, Exercise 3). We have performed numerical simulations using the parameter values A = 3,  d = 0.2,  r = 0.5,   ˜ = 0.5, ° = 0.8, Ds = 0.05,  and Di = 0.0005. We obtain E1(S1, I1) = (12, 0.1), E2 (S2, I2) = ( 2.5, 2 ), kc2 = 238,  H kc2 = −0.4661 < 0, for 101.457 < k 2 < 374.543. For these parameter values, conditions (i)–(iv) are satisfed for obtaining Turing patterns. The plot of H k 2 , k 2 is given in Figure 3.15a. To illustrate the dispersion relation, the variation of the real part of the characteristic value of the model system as d increases is plotted in Figure 3.15b. From Figure 3.15b, we observe that as we increase the value of the natural death rate of the population d, we obtain Turing instability. For d = 0.3 and d = 0.2, we obtain diffusion-induced instability, but for d = 0.05, the system shows stable behavior both in temporal and spatial cases. A good alternative for this model is a modifed model with a standard incidence rate ˜ SI ( S + I ) (see [84,294]). Another model with a saturated incidence rate ˜ SI ( 1 + ° S) was

( )

FIGURE 3.15A Plot of H(k2) vs. (k2).

( ( ) )

Modeling Virus Dynamics in Time and Space

163

FIGURE 3.15B Plot of the real part of the characteristic value as parameter d is increased. (a) Black d = 0.3, (b) white d = 0.2, and (c) grey d = 0.05.

introduced by Anderson and May [10]. The saturation factor ˜ plays a role in epidemical control. The inhibition effect due to the saturation factor ˜ results due to the preventive measure to control the spread of the epidemic [147,296]. Li et al. [163] proposed an SIR model with a nonlinear incidence rate ˜ SI ( 1 + ° I ), which is of saturated type and refects the “psychological” effect or the inhibition effect [41]. A number of authors used Beddington– DeAngelis-type incidence rate (˜ SI ) ( 1 + ° S + ˛ I ) [70,82,124,130,131]. Lotf et  al. [176] and Dubey et  al. [72] have used monotone nonlinear incidence rates (˜ SI ) ( 1 + ° S + ˛ I + ˝ SI ) and (˜ SI ) ˇ˘( 1 + ° S) ( 1 + ˛ I )  respectively. Dubey et al. [70] investigated the dynamics of an SIR model with Beddington–DeAngelistype incidence rate and Holling type II treatment rate. The incidence rate was taken as f ( S,  I ) = ˜ SI ( 1 + ° S + ˛ I ) . For our discussion, we propose to extend their model to include diffusion and study its dynamics. Consider the diffusive model system as

˜S ° SI = A − dS − + Ds˘ 2S, ˜t 1 + ˛S + ˝ I

(3.72a)

˜I ° SI = − ( d + d1 + d2 ) I − h ( I ) + Di 2 I, ˜ t 1 + ˛S + ˝ I

(3.72b)

˜R = d2 I + h ( I ) − dR + Dr ˇ 2 R, ˜t

(3.72c)

where the treatment term is assumed as h ( I ) = aI ( 1 + bI ) , and a and b are positive constants, to take into account the resource limitations [157,312,314]. The susceptible population is assumed to be recruited at a constant rate A and d is the natural death rate of the population in each class. d1 and d2 are respectively the death rate of infected individuals due to infection and the natural recovery rate of infected individuals due to immunity. Here, ˜ ,  ° , and ˜ are the transmission rate, measures of inhibition effect such as a preventive measure taken by susceptible individuals, and inhibition effects such as treatment with respect to infective individuals respectively. Equations (3.72a) and (3.72b) are independent of R. Hence, it is suffcient to consider the reduced system

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Spatial Dynamics and Pattern Formation in Biological Populations

˜S ° SI = A − dS − + Ds˘ 2S, ˜t 1 + ˛S + ˝ I

(3.73a)

˜I ° SI = − d3 I − h ( I ) + Di 2 I , d3 = d + d1 + d2 . ˜ t 1 + ˛S + ˝ I

(3.73b)

First, consider the temporal model of Dubey et  al. [70]. The authors have shown that all solutions  of the temporal model are nonnegative and bounded in Ω = ( S,  I ) ˆR+2 : 0 < S + I < ( A/d ) . The temporal model of (3.73a) and (3.73b) has two equilibrium points:

{

}

i. A DFE point E0 ( S0 , 0 ) , where S0 = A/d. At DFE E0 , the reproduction number 0 is obtained as 0 = ( A˜ ) ˘(° 3 + a )( d + A˛ )  . The DFE is locally asymptotically stable if 0 < 1. If 0 > 1, the DFE is a saddle point with stable manifold locally in the S-direction and unstable manifold locally in the I-direction. The authors derived the conditions for the DFE to be GAS. Furthermore, the DFE changes its stability from stable to unstable at 0 = 1 and there exists a positive equilibrium as 0 crosses 1. Hence, the solution undergoes transcritical bifurcation at 0 = 1. ii. Endemic equilibrium point E1 ( S1 ,  I1 ): Solving the second equation, S1 can be written in terms of I1 as ˘d3 ( 1 + bI1 ) + a ( 1 + ˜ I1 ) S1 =  . (1 + bI1 )(° − ˛ d3 ) − ˛ a Since S1 > 0, we require ˜ > ° (d3 + a). The frst equation is quadratic in S1 . Substituting the expression for S1 in this equation, a cubic equation is obtained for I1 . The authors obtained the conditions such that this equation has a positive root. Furthermore, they have also obtained the conditions under which E1 ( S1 ,  I1 ) is locally asymptotically stable, GAS, unstable, a saddle point, and exhibits Hopf bifurcation. Next, consider the diffusive system (3.73a) and (3.73b). To determine under what conditions Turing instabilities occur, we test how perturbation of a homogeneous steadystate solution behaves in the long-term limit. Choose the perturbation functions as sˆ = exp ˆˇ k x x + k y y i + ˜ k t ˘ , iˆ = exp ˆˇ k x x + k y y i + ˜ k t ˘ . Substituting S = S1 + sˆ and I = I1 + iˆ in

(

)

(

)

system (3.73a) and (3.73b) and linearizing the equations about E1 ( S1 ,  I1 ) , we obtain the characteristic equation of the resulting system as

µk2 − trace ( J sp ) µk + det ( J sp ) = 0,

( )

where trace J sp = trace ( J ) − ( Ds + Di ) k 2 , and

( )

det J sp = det ( J ) − ( j11Di + j22 Ds ) k 2 + Ds Di k 4 . j11 = − d −

(

) +° I )

˜ I* 1 + ° I*

(1 + ˛ S

*

,  j12 = −

) +° I )

,  j21 =

(1 + ˛ S ˜ S (1 + ° S ) a =  −d − (1 + ° S + ˛ I ) (1 + bI ) * 2

*

j22

(

˜ S* 1 + ˛ S* *

* 2

*

*

* 2

3

* 2

.

(

) +° I )

˜ I* 1 + ° I*

(1 + ˛ S

*

* 2

,

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Modeling Virus Dynamics in Time and Space

( )

Since trace(J ) >   trace J sp , we obtain that for saddles and attractors (both with respect to the temporal model), a change of stability coincides with a change in the sign of det J sp . A change in the sign of det J sp occurs when k−2 < k 2 < k+2. . If both k−2 and k+2. exist and have positive values, they limit the range of instability for local stable equilibrium. We refer to this range as the Turing space (or Turing region, see Figure 3.16). With respect to homogeneous perturbations, E1 ( S1 ,  I1 ) is stable at frst, but as k 2 increases, one eigenvalue changes its sign (when k 2 arrives at k−2 ) and the instability occurs. The instability exists until k 2 reaches k+2 . When k 2 > k+2, the system returns to stability again. Thus, the Turing space is bounded. Numerical simulations are done using the parameter values A = 5,   d = 0.2,  ˜ = 0.04,  ° = 0.03,  ˛ = 0.05,  d1 = 0.058,  d2 = 0.002,  a = 0.3,  b = 0.08, Ds = 1 and Di = 0.0001. We obtain E0 ˛˝( A d ) , 0 ˙ˆ = ( 25, 0 ) , E1 ( S1 , I1 )   = ( 23.9296, 0.388559 ),

( )

( )

( )

( )

kc2 = 12.949, and H kc2 = −0.014 < 0, for 0.93087 < k 2 < 24.9671. The plot of H k 2 vs. k 2 is given in Figure 3.16 for different values of Ds = 1 ( green ) , 1.5 ( red ), and 2 (black). Turing patterns for the susceptible population are plotted in Figure 3.17, at the iteration steps 300, 600, 900, and 5000. It is observed that the density of susceptible population increases as we increase the number of iterations and is distributed in the whole domain. The numerical results capture some key features of the complex variations. The steady spatial patterns indicate the persistence of the epidemic in the space. Li et al. [157] introduced a saturated treatment and logistic growth rate into an SIR epidemic model with bilinear incidence. The treatment function which describes the effect of delayed treatment when medical facilities are limited and the number of infected individuals is large enough is taken as a continuously differentiable function. The model proposed by them is as follows:

FIGURE 3.16 Plot of H(k2) vs. (k2).

˜S S˘ ˙ = rs ˇ 1 −  − ° SI + Ds 2S, ˆ ˜t K

(3.74a)

˜I = ° SI − ( d + d1 + d2 ) I − h ( I ) + Di˘ 2 I, ˜t

(3.74b)

˜R = d2 I + h ( I ) − dR + Dr ˇ 2 R, ˜t

(3.74c)

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 3.17 Turing patterns for the susceptible population at iteration steps (a) 300, (b) 600, (c) 900, and (d) 5000.

where h ( I ) = aI ( 1 + bI ) . Using the bifurcation theory, the authors have shown that the temporal model system exhibits backward bifurcation, Hopf bifurcation, and Bogdanov – Takens bifurcation. They suggested that to eradicate the disease, we should raise the effciency and enlarge the capacity of treatment. That is, we should improve our medical technology and invest in medicines, beds, and so on to give patients timely treatment. Noble [203] applied the reaction-diffusion theory to study the spread of plague in Europe in the middle of the 14th century. Using the linear theory of semigroups, Saccomandi [235] investigated the existence and uniqueness of the solution for an SIR model with spatial inhomogeneity, nonlocal interactions, and an open population. Xu and Ma [296] studied an SIR epidemic model with nonlinear incidence rate and time delay. Ram Naresh et al. [200] formulated an SIR model to study the role of nonlinear incidence rate and time delay in a logistically growing time-delayed model with variable population size. Wu and Zou [293] and Zou and Wu [315] studied the existence of traveling wave fronts for delayed

Modeling Virus Dynamics in Time and Space

167

RD systems with reaction terms satisfying the quasi-monotonicity or exponential quasimonotonicity condition. The well-known monotone iteration techniques for elliptic systems with advanced arguments were studied by Leung [151] and Murray [198]. Ge and He [89] and Ge et al. [90] used the iteration technique developed by Wu and Zou [293] to investigate the existence of traveling wave solutions for a two-species predator–prey system with diffusion terms and stage-structure, respectively. Li et al. [162] investigated the existence of traveling wave solutions of a class of delayed RD systems with two equations in which reaction terms satisfy weak quasi-monotonicity and weak exponential quasimonotonicity conditions respectively. Sazonov et al. [241,242] studied the problem of traveling waves in the SIR model. Hu et al. [123] analyzed an SIR epidemic model to study the effect of limited resources for the treatment of patients in the public-health system, which could occur when there is a very large number of patients but the medical facilities are insuffcient, or the number of beds is limited, or the number of health-care workers is insuffcient. Rivero-Esquivel et  al. [225] studied the stability and bifurcation analysis of an SIR epidemic model with nonlinear incidence rate, vertical transmission, vaccination for the newborns, the capacity of treatment that takes into account the limitedness of the medical resources, and the effciency of the supply of available medical resources. Under some conditions, the authors proved the existence of backward bifurcation, stability, and the direction of Hopf bifurcation. They also explored how the mechanism of backward bifurcation affects the control of the infectious disease. Yang et  al. [299] studied the existence of traveling waves in an SIR epidemic model with nonlinear incidence rate, spatial diffusion, and time delay. The authors discussed the local stability of a disease-free steady state and an endemic steady state under homogeneous Neumann boundary conditions. By using the cross-iteration method and Schauder’s fxed point theorem, the problem of the existence of traveling waves was reduced to the existence of a pair of upper-lower solutions [125]. They constructed a pair of upper-lower solutions to the general model and showed the existence of a traveling wave connecting the disease-free steady state and the endemic steady state. They carried out numerical simulations to illustrate the results. The authors considered the following delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

˛ S ( x ,  t ) I ( x ,  t − ˝ ) ˜S = A − µ1S ( x ,  t ) − + DS S, ˜t 1 + ˙ I ( x, t − ˝ )

(3.75a)

˜ I ° S ( x ,  t ) I ( x ,  t − ˛ ) = − ( µ2 + ˆ ) I ( x ,  t ) + DI I , ˜t 1 + ˝ I ( x, t − ˛ )

(3.75b)

˜R = ° I ( x ,  t ) − µ3 R ( x ,  t ) + DR R, ˜t

(3.75c)

with the initial condition S ( x ,  t ) = ˜ 1 ( x ,  t ) , I ( x , ,t ) = ˜ 2 ( x ,t ) R ( x ,  t ) = ˜ 3 ( x ,  t ) , t ˘[ −° , 0 ] ,   x ˘Ω. The homogeneous Neumann boundary conditions are

˜ S ( x ,  t ) ˜ I ( x ,  t ) ˜ R ( x , t ) = = = 0,  t ˆ 0,  x ˇ˜ Ω. ˜° ˜° ˜°

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Spatial Dynamics and Pattern Formation in Biological Populations

The positive constants DS ,  DI, and DR are the diffusion rates of susceptible, infected, and removed populations, respectively. Ω is a bounded domain in R n with smooth boundary ˜ Ω.The functions ˜ i ( x ,  t ) ( i = 1,  2,  3 ) are nonnegative and Hölder continuous and satisfy ˜° i ˜˛ = 0 in ( −˝, 0 ) × Ω. The authors assumed that DS = DI = DR = D and studied the existence of traveling wave solutions for the reaction-diffusion equations with delays using the technique of Atkinson and Reuter [16]. The nonlinear reaction terms of the system (3.75a)– (3.75c) do not satisfy either the quasi-monotonicity or the exponential quasi-monotonicity conditions. Therefore, the method of upper-lower solutions and associated monotone iteration scheme developed by Wu and Zou [293] cannot be used to study the existence of traveling wave solutions of the system (3.75a)–(3.75c). To investigate the existence of traveling wave solutions of the system, they considered the existence of traveling wave solutions of a general delayed reaction-diffusion system

˜u ˜2u = f1 ( ut ( x ) ,  vt ( x ) ,  wt ( x )) + D 2 , ˜t ˜x

(3.76a)

˛v ˛2 v = f2 ( ut ( x ) ,  vt ( x ) ,  wt ( x )) + D 2 , ˛t ˛x

(3.76b)

˛w ˛2 w = f3 ( ut ( x ) ,  vt ( x ) ,  wt ( x )) + D 2 . ˛t ˛x

(3.76c)

Substituting u ( t, x ) = ˜ ( x + ct ) , v ( t ,  x ) = ° ( x + ct ) , and w ( t ,  x ) = ˛ ( x + ct ) in (3.76a)–(3.76c), where ( x + ct ) is the traveling wave coordinate, the authors studied the resulting system and showed that partial quasi-monotonicity conditions are satisfed and there exists a traveling wave solution. They derived the upper and lower solutions of the system and then applied the Schauder fxed point theorem. Consider now the local stability of the system (3.75a)–(3.75c). The model system has a DFE E0 ( A/µ1 , 0, 0 ) and the unique endemic steady state E* S* ,  I * ,  R* , where

(

I* =

)

A˜ − µ1 ( µ2 + ˛ ) ˛ * * 1 , R * = I , S = ( µ2 + ˛ ) ( 1 + ˝µ1 ) , and A˜ > µ1 ( µ2 + ˛ ) . µ3 ˜ ( µ2 + ˛ )( ˜ + ˝µ1 )

The basic reproduction number was obtained as 0 = A˜ ˘ µ1 ( µ2 + ˛ ) .  The reproduction number represents the average number of newly infected cells generated from one infected cell at the beginning of the infectious process. Performing eigenvalue analysis, they showed that (i) for 0 < 1, the DFE is locally asymptotically stable and for 0 > 1, the DFE is unstable; (ii) for 0 > 1, the endemic steady state E* S* ,  I * ,  R* exists and is asymptotically stable for all ˜ > 0. For the general system (3.76a)–(3.76c), the authors showed that a pair of upper-lower solutions exist under certain conditions. Hence, the general system has a traveling wave solution. These results were then used to study the existence of traveling wave solutions of the system (3.75a)–(3.75c). They have proved that for 0 > 1, and for all ˜ ° 0, the system has a traveling wave solution with speed c > c * , connecting the DFE E0 ( A µ1 , 0, 0 ) and the infected steady state E* S* ,  I * ,  R* . Kim et al. [145] investigated the behavior of positive solutions of a spatial SIR epidemic model with free boundary in a radially symmetric domain, which describes the spreading frontier of the disease. The existence and uniqueness of the global solution were proved by the contraction mapping theorem and Hopf lemma was used to prove the monotonicity of

(

(

)

)

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Modeling Virus Dynamics in Time and Space

the free boundary. Chen et al. [48] studied an SIR epidemic model with infection age and saturated incidence and established threshold dynamics by applying fluctuation lemma and Lyapunov functional. Wu et  al. [292] proposed a spatial SIR model in combination with random dispersal and nonlocal dispersal. The authors established the existence and nonexistence of the traveling wave solutions connecting the DFE and the EE. The main difficulties in proving the results lie in the fact that the semiflow generated here does not admit the order-preserving property and the solutions lack regularity. They used an iteration scheme to construct a pair of upper and lower solutions and then applied the Schauder fixed-point theorem and polar coordinates transform to study the threshold dynamics of the model. Random diffusion is essentially a local behavior, which depicts the movements of species between neighboring spatial locations. Standard convolution with space variable sufficiently describes the long-distance dispersal (non-local dispersal) of some organisms [132]. The authors considered a basic SIR model with nonlinear incidence rate, treatment rate, and introduced the random dispersal and nonlocal dispersal into it. The model is given by [145]

∂S = Ds ( K * S) ( t ,  x ) − S ( t ,  x )  + A − µS − h ( S) g ( I ) , ∂t

(3.77a)

∂I ∂2I = Di 2 + h ( S) g ( I ) − µ I − d ( I ) , ∂t ∂x

(3.77b)

∂R = Dr ( K * R ) ( t ,  x ) − R ( t ,  x )  + d ( I ) − µR , ∂t

(3.77c)

where t > 0 and x ∈ ,  Ds , Di ,  Dr > 0 are diffusion coefficients, K ( r ) represents the probability distribution of rates of dispersal over distance r, and ( K * u)( t ,  x ) =

∫ K ( y ) u(t,  x − y ) dy. 

A and  µ are the recruitment rate and natural death rate of the population respectively and d ( I ) is the treatment rate of infective individuals. A constant entering flux A as the external supply is more reasonable to describe slow-moving diseases such as malaria, cholera, typhoid, hepatitis A, and so on. The appearance of external supplies, the loss of order-preserving property, and lack of regularity of solution make the mathematical analysis very difficult. To be biologically feasible, the authors assumed that the functions h ( S) ,  g ( I ), and d ( I ) are nonnegative and differentiable and satisfy the following assumptions: A1. For all S > 0, I > 0: h ( 0 ) = g ( 0 ) = d ( 0 ) = 0,  d′ ( 0 ) ≥ 0 ;  h′ ( S) > 0,  g ′ ( I ) > 0, g ′′ ( I ) ≤ 0, d′′ ( I ) ≥ 0.

A2. There exists I 0 > 0 such that h ( A µ ) g ( I ) − µ I − dI < 0 for all I > I 0 .

A3. h′ ( S) is bounded in S ∈  0, ( A µ )  , g ′′ ( I ) , d′′ ( I ) are bounded in I ∈[ 0, I 0 ]. νy  A4. K ∈ C1 (  ) ,  K ( x ) = K ( − x ) ≥ 0, K ( x)dx = 1 and K ( y ) e dy < +∞  for any ν > 0.

∫



∫



Authors combined the method of Schauder’s fixed-point theorem with upper-lower solutions and showed that if the basic reproduction number satisfies the condition 0 > 1, then for each c ≥ c*, the system admits traveling waves. The nonlocal dispersal may cause the epidemic waves to oscillate more frequently. Improving the treatment rate (decreasing 0 ), controlling the wave speed c, and reducing the dispersal range can restrict further outbreaks of epidemic effectively.

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Spatial Dynamics and Pattern Formation in Biological Populations

3.5 Susceptible-Infected-Removed-Susceptible (SIRS) Models Consider an endemic model (a model used for studying diseases over longer periods, during which there is a renewal of susceptibles by births or recovery from temporary immunity), in which people move cyclically among compartments corresponding to three epidemiological states. When there is adequate contact of a susceptible (S) individual with an infective (I) individual, the disease transmission occurs, and then S enters the compartment I of infective individuals, who are infectious in the sense that they are capable of transmitting the infection. When the infectious period ends, the individual enters the recovered compartment R. When the infection-acquired immunity wanes, the individual moves back to the susceptible class. Based on the order of fow through the epidemiological compartments, the model is classifed as an SIRS model. In the SIR model, it is assumed that the immunity received by recovery from the disease is permanent. This may not always be true, since there may be a gradual loss of immunity with time. Temporary immunity can be included in an SIRS model in which a rate of transfer from R to S is added to an SIR model. The basic SIRS model is described by a system of ODEs for the evolution of an infectious disease in a well-mixed and closed population. The model defned by Brauer and Castillo-Chavez [29] is given by dI dR dS = − ˜ SI + ° R, = ˜ SI − ˛ I, and   = ˛ I − ° R, dt dt dt

(3.78)

where the infection rate ˜ , recovery rate ˜ , and the immunity loss rate ˜ are positive constants. The basic reproduction number is defned by 0 = ˜ ° , which gives the average number of infective individuals produced by a single infective individual introduced into a completely susceptible population. It was shown by many authors [29,108,260] that if 0 < 1, then every nonnegative solution ( S ( t ) , I ( t ) , R ( t )) lying on the plane S + I + R = 1 approaches the DFE (1, 0, 0) as t ˜ °, implying that the disease is eventually eradicated; while if 0 > 1, then ( S ( t ) , I ( t ) ,  R ( t )) with I ( 0 ) > 0 approaches the EE S* ,  I * ,  R * given by

(

S* =

˙ 1 1 ˘ ,  I * = ˇ 1 − ˆ 0 0 

)

˜˘ ˜ * ˙ * ˇˆ 1 +  , R = I . ° °

For the case R = 0, the fnal size formula and the severity of an epidemic for the model can easily be derived (Problem 3.6, Exercise 3). In an SIRS model, a common assumption is that the movement out of compartment R, back to the compartment S is described by ˜ R, which corresponds to an exponentially distributed waiting time in R, and the movement out of compartment I is governed by ˜ I , which corresponds to an exponentially distributed waiting time in I. With exponential waiting time in both R and I, it was shown that when the disease remains endemic, all solutions approach the EE, that is, there does not exist any periodic solution [106]. Ruan and Wang [232] studied the global dynamics of the SIRS epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action. The bifurcation analysis shows that the system undergoes a Bogdanov–Takens bifurcation at degenerate equilibrium, which includes a saddle-node, Hopf, and homoclinic bifurcation. Jin et al. [128] investigated the global dynamics of an SIRS model with a nonlinear incidence rate. They had also investigated the backward bifurcation, Hopf bifurcation, and Bogdanov-Taken

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Modeling Virus Dynamics in Time and Space

bifurcation and obtained the criteria for Hopf bifurcation and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease. Sun et al. [255] investigated the spatial version of the epidemic model proposed by Ruan and Wang [232], which includes the behavioral changes and crowding effect of the infective individuals. Their results reveal that the spatiotemporal chaos is induced by the breakup of waves (the breakup of spiral waves is from the core and target waves from the far-field and both waves become irregular patterns at last). Ai and Albashaireh [2] studied the traveling wave front solutions for two RD systems, which were derived as diffusion approximations of two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. The existence of these solutions was proved by a shooting argument combined with LaSalle’s invariance principle, and their uniqueness by a geometric singular perturbation argument. Li et al. [161] considered a distributed contact spatial analog of the basic SIRS endemic model in one space dimension, assuming that the density of the population is constant on ( −∞, ∞ ). Let S ( t , x ) and I ( t , x ) be the relative densities of the susceptible and infective population at time t and position x. The kernel K ( t , x ) is analogous to the contact rate β , so it is the number of adequate contacts of an infective individual at position y with a susceptible individual at position x. The distributed-contacts spatial analog of the basic SIRS endemic model [110] is given by the initial value problem [161] ∞

∂S = − K ( x , y ) I ( t ,  y ) dy  S ( t ,  x ) + δ R ( t ,  x ) , ∂t

∫

(3.79a)

−∞

∂I = ∂t

∞

∫ K ( x, y ) I (t,  y ) dy S(t,  x ) − γ I (t,  x ) ,

(3.79b)

−∞

∂R =  γ I ( t ,  x ) − δ R ( t ,  x ) , ∂t

(3.79c)

S ( 0,  x ) = s0 ( x ) ≥ 0, I ( 0,  x ) = i0 ( x ) ≥ 0, R ( 0,  x ) = r0 ( x ) ≥ 0. Here, 1 δ is the average period of temporary immunity and 1 γ is the average infection period with δ > 0,  γ > 0. The authors assumed that the kernel and the initial conditions are smooth, so that unique solutions of model system (3.79a)–(3.79c) exist and remain nonnegative for all t > 0 [97]. Assuming that the distribution of adequate contacts depends only on the distance from position y to x, the kernel is taken as K ( x , y ) = k ( x − y ) . The kernel is continuous and symmetric with k ( − y ) = k ( y ) , and contacts are local, so that k ( y ) = 0 for y ≥ ε , where ε is some small number. When δ = 0, model (3.79a)–(3.79c) reduces to the Kendall model [137], which was studied by several authors (Aronson [15], Barbour [20], Brown and Carr [32], and Mollison [193]). Aronson [15] showed that the minimal wave speed is the asymptotic speed of propagation of disturbances from the steady state of the model. Following the approaches of Bailey [18] and Hoppensteadt [118], Li et al. [161] derived a diffusion approximation to the model (3.79a)–(3.79c), assuming that the function I ( t , y ) does not change very much over the set of radius ε , so that the fourth derivative of I with respect to y is ο ( 1) on such a set. By changing the variables in the integral, one obtains ∞

∫

k ( x −   y ) I ( t ,  y ) dy =

−∞

∞

∫ k ( z) I (t,  x − z) dz.

−∞

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Spatial Dynamics and Pattern Formation in Biological Populations

Consider Taylor’s series approximation I ( t, x − z ) = I ( t, x ) − z

(

)

4 * ∂ I ( t, x ) z 2 ∂ 2 I ( t , x ) z 3 ∂ 3 I ( t , x ) z 4 ∂ I t , x − + , + 6 ∂x 2 ∂ x2 ∂ x3 24 ∂ x 4

where x − z < x * < x. By substituting the expansion into the second integral, using the property that the kernel is symmetric, and by simplifying, one obtains ∞  1 k ( z ) I ( t ,  x ) dz +  k ( z ) z 2 dz I xx ( t , x ) 2  

∞

∞

−∞

−∞

∫ k ( x −   y ) I (t,   y ) dy = ∫

∫

−∞

(

)

∞  ∂ 4 I t, x * 1  +  k ( z ) z 4dz . 24  ∂ x4 

∫

−∞

( )

Since the fourth derivative of I with respect to y is of ο ( 1) , the third term is of ο ε 4 and hence it may be neglected. The approximation becomes ∞

∫ k ( x −   y ) I (t,  y ) dy ≈ β I (t,  x ) + DI

xx

( t,  x ) ,

−∞

∞

β=

∫

∞  1 k ( z ) dz, D =  k ( z ) z 2 dz . 2  

∫

−∞

−∞

Since S ( t , x ) = 1 − I ( t ,  x ) − R ( t , x ) , the diffusion approximation of the model (3.79a)–(3.79c) becomes

∂I = ( β I + DI xx ) ( 1 − I − R ) − γ I , I ( 0,  x ) = i0 ( x ) ≥ 0, ∂t ∂R =  γ I − δ R , R ( 0,  x ) = r0 ( x ) ≥ 0. ∂t

(3.80a) (3.80b)

The authors [161] determined traveling wave front solutions of the system (3.80a) and (3.80b) of the form ( I ( x , t ) ,  R ( x ,  t )) = ( I ( z ) , R ( z )) ,  z = x + ct that move with constant speed

(

)

c > 0 and connect the DFE E0 ( 0, 0 ) and EE E I * ,  R* at z = ±∞, respectively. The system (3.80a) and (3.80b) can be written as the system of first-order equations as I ′ = v , v′ =

βI cv + γ I − , cR′ = γ I − δ R, D(1 − I − R ) D

(3.81)

where the differentiation is with respect to z, and I ( z ) > 0,  R ( z ) > 0, I ( z ) + R ( z ) < 1, ∀   z ∈( −∞,  ∞ ) . The conditions become ( I ,  v , R )( −∞ ) = ( 0, 0, 0 ) ,  ( I , v ,  R )( ∞ ) = E = I * ,  0,   R* . The equilibrium points of the system (3.81) are the DFE (0, 0, 0) and the EE

(

  − 1 I* =  0  0 

 1  ,  v = 0,  R* = qI * ,  1 + q 

)

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Modeling Virus Dynamics in Time and Space

where 0 = ˜ ° is the basic reproduction number, and q = ˜ ° is the quotient of the average period 1 ˜ of temporary immunity and the average infection period 1 ˜ . If 0 > 1, then the EE is in the epidemiologically realistic region. The linearized system of the system (3.81) about the origin has the characteristic equation ˇ˘ ˜ + (° c )  ˇ˘ D˜ 2 − c˜ + ˛ ( 0 − 1)  = 0. The roots are ˜1 = −° c ,  ˜2, 3 =  c ± c 2 − 4˛ D ( 0 − 1)  ( 2 D ) . When 0 > 1, the signs of the   coeffcients in the quadratic in ˜ are + , − , +. When 0 > 1 and c ˝ cmin = 2 D˜ ( 0 − 1) , two of the eigenvalues are positive. Therefore, the origin has a 2D unstable manifold. When c ˜ cmin , there is a repulsive direction away from the origin. The quantity cmin is the least value for the minimal wave speed. The phase plane approach [199] shows that the EE has an attractive direction in the region in which v > 0 and I is less than the endemic value, so that a heteroclinic orbit joining the origin and the EE would correspond to a traveling wave solution. The authors have also shown that periodic solutions do not arise around the EE by the Hopf bifurcation. To show this, they consider that I ( x ,  t ) = f ( x + ct ) = f (˜ ) , and R ( x ,  t ) = g ( x + ct ) = g (˜ ) with ˜ = x + ct and c is the traveling wave speed. If there is a periodic solution with f (˜ + P ) = f (˜ ) and g (˜ + P ) = g (˜ ) , around the EE in the region with f > 0 and g > 0 for some P > 0, then it corresponds to a periodic traveling wave solution. The Jacobian at the EE ( I ,  v ,  R ) is given by ˇ 0   J= A   ˜ /c ˘

1 ( 0c ) D 0

0 A − (° c )

   , where A = ˜ 0 ( 0 − 1) > 0.  D(1 + q )  

The characteristic equation is given by ˘ °   c2  ˘ c˜ 3 +  ° − 0  ˜ 2 −  A + 0  c˜ − A (° + ˛ ) = 0.  D  D   Substituting ˜ = iµ , and separating the real and imaginary parts, we get ˇ 0 c 2  2 0˜  ˇ 2 ˘ ˜ − D  µ + A (˜ + ˛ ) = 0, cµ ˘ µ + A + D  = 0. There are no nonzero real roots for µ, so that the characteristic equation has no pure imaginary roots. Thus, the Hopf bifurcation cannot occur and hence, periodic traveling wave solutions of the model do not arise by the Hopf-bifurcation. Periodic traveling infection wavefronts do not arise by the Hopf bifurcation for the spatial analog of the usual SIRS endemic model with exponential waiting times in I and R since the nonexistence of periodic traveling waves in the spatial SIRS model is connected to the nonexistence of periodic solutions in the analogous nonspatial SIRS endemic model. Now, we present the analysis of the following SIRS model of Ruan and Wang [232] and Sun et  al. [255], which investigates the spatiotemporal complexity of a spatial epidemic

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Spatial Dynamics and Pattern Formation in Biological Populations

model with nonlinear incidence rate and also includes the behavioral changes and crowding effect of the infective individuals.3 dS kSI l = B − dS − + ° R, dt 1 + ˜Ih

(3.82a)

dI kSI l = − ( d + ° ) I, dt 1 + ˜ I h

(3.82b)

dR = ˜ I − ( d + ° ) R, dt

(3.82c)

where B is the recruitment rate of the population, d is the death rate of the population, ˜ is the recovery rate of infective individuals, and ˜ is the rate of removed individuals who lose immunity and return to the susceptible class. ˙ˆ kSI l 1 + ˜ I h ˇ˘ is the nonlinear incidence rate proposed by Liu et al. [170], where kI l measures the infection force of the disease and 1 1 + ˜ I h measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. Particular cases of the model (3.82a)–(3.82c) were studied by many authors (for ˜ = 0, by Hethcote and van den Driessche [115], Liu et al. [169], Hethcote et al. [113]; for saturated mass action by Capasso and Serio [41], Busenberg and Cooke [34], Hethcote et al. [112]). Ruan and Wang [232] and Sun et al. [255] studied the case l = h = 2. Adding the three equations in (3.82a–c), we obtain

(

(

)

)

dN = B − dN ,  N = S + I + R. dt N ( t ) ˛ constant , as t ˜ °. Following the work of Lizana and Rivero [171], the authors assumed that the population is in equilibrium and investigated the behavior of the system on the plane S + I + R = N 0 > 0. The reduced system becomes dI kI 2 ( N 0 − I − R ) = − ( d + ° ) I, dt 1 + ˜ I2

(3.83a)

dR = ˜ I − ( d + ° ) R. dt

(3.83b)

Rescale (3.83a) and (3.83b) using the transformation X=

k I, Y = d + ( ˜)

k R,  ° = ( d + ˜ ) t. d + ( ˜)

Using I , R , and t again as the new variables, the following equations are obtained dI I 2 ( A − I − R ) dR = − mI, = qI − R, 2 dt 1 + pI dt 3

(3.84)

Sun et al. [255]: “© SISSA Medialab srl. Reproduced by permission of IOP Publishing. All rights reserved.”

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where p =

˜ (d + ° ) ˛ k d +˛ , A = N 0 , m = , and q = . k d +° d +° d +°

Model (3.84) has the following equilibrium points: (i) DFE (0, 0). (ii) EE points ( I1 ,  R1 ) and ( I 2 ,  R2 ), where R1 = qI1 , R2 = qI 2 , and I1,2 are the roots of the quadratic equation ( mp + q + 1) I 2 − AI + m = 0. Two positive roots exist when A2 > 4m ( mp + q + 1) and the roots are given by I1 =  

A − A 2    − 4m ( mp + q + 1) 2 ( mp + q + 1)

, I 2 =  

A + A 2   − 4m ( mp + q + 1) 2 ( mp + q + 1)

.

Sun et al. [255] investigated the following spatial model with both diffusion coeffcients equal to 1 and focused on the positive equilibrium point ( I 2 ,  R2 ). The equilibrium point ( I1 ,  R1 ) is a saddle point.

˜ I I 2 ( A − I − R) ˜R = − mI + ˇ 2 I, = qI − R + ˇ 2 R. ˜t 1 + pI 2 ˜t

(3.85)

Using the eigenvalue analysis, the Jacobian of the system (3.85) is obtained as ˙ a11 − k 2 J (k) = ˇ ˇ a21 ˆ where a11 =  

(

)

2 ( A − R ) I − 3 + pI 2 I 2

(1 + pI )

2 2

−m=

a12 = −  

˘   

a12 a22 − k

2

(

2 ( A − R − I ) I − I 2 1 + pI 2

(1 + pI )

2 2

) − m = 2m − I 1 + pI

2

2

− m,

I2 , a21 = q, a22 = −1, 1 + pI 2

and k 2 = k x2 + k y2 are wave numbers. The characteristic equation is given by ˜ 2 − trk ˜ + ˝ k = 0, where trk = ( a11 + a22 ) − 2 k 2 = tr0 − 2 k 2 ,   ∆ k = ( a11 a22 − a12 a21 ) − k 2 ( a11 + a22 ) + k 4 = ∆ 0 − k 2 ( a11 + a22 ) + k 4 , Turing bifurcation occurs when 2Re ( ˜ ( k )) = trk = tr0 − 2k 2 = 0, and Im ( ˜ ( k )) = 4 ˙ k − trk2 = 0, at k = kT ° 0. The conditions give the wave number kT as kT2 = ° 0 , where ˜ 0 = (a11 a22 − a12 a21 ). Equating ˜ 0 = 0, we get −

qI 2 2m − I 2 +m+ = 0. 2 1 + pI 1 + pI 2

(

)

From ( mp + q + 1) I 2 − AI + m = 0, we get m 1 + pI 2 = I ˆˇ A − ( 1 + q ) I ˘ . By substituting in the above equation and simplifying, the relation IA − 2m = 0 is obtained. Again, by substituting for I and simplifying, we obtain

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[A 2 − 4m ( mp + q + 1)] + A A 2    − 4m ( mp + q + 1) = 0. Hence, q =   qT =

A 2 − 4m2 p − 4m . 4m

(3.86)

The Hopf bifurcation occurs when Im ( ˜ ( k )) ˝ 0, Re ( ˜ ( k )) = 0, at k = 0. The conditions are obtained as a11 + a22 = 0, and (a11a22 − a12 a21 ) ° 0. From the above, it is easily verifed that (a11a22 − a12 a21 ) > 0, since A 2 > 4m ( mp + q + 1) . a11 + a22 = 0 gives that

(

)

2m − I 2 − ( m + 1) 1 + pI 2 = 0, ( m − 1) − I 2 ˆˇ1 + ( m + 1) p˘ = 0, 2

 A + A 2 − 4mP  ( m − 1) − ˙1  = 0, ˆ + ( m + 1) p ˘ˇ  2P   4P 2 ( m − 1) − 2A 2 + 4mP = 2A A 2 − 4mP , P = 1 + q + mp. ˙ˆ1 + ( m + 1) p ˇ˘ By squaring both sides and simplifying (P gets canceled two times), we get 2

˙ P ( m − 1) ˘ A 2 ( m − 1) = . + m  ˇ 1 + ( m + 1) p  ˆ 1 + ( m + 1) p By simplifying, we get qH =

1 ˆ A m−1ˇ

( m − 1)( 1 + p + mp ) + 1 − 2m − 2m2 p ˘ .

(3.87)

The expressions that we have derived above are in different forms from those derived by the authors but the results are the same. The authors [255] performed computations to study the dynamics of the model. For the parameter values p = 0.006, A = 20.478172, m = 5, and q = 8, the authors obtained k =   kT = 5.550598463, and the corresponding eigenvalues of nonspatial and spatial models are obtained respectively as 0.9349999996 × 10−7 ± 5.550598463i (for k = 0) and −5.550598369 ± 5.550598463i. For these values, the authors obtained a limit cycle. The authors have presented the bifurcation diagram for p = 0.006 and A = 20.478172. We have plotted the bifurcation diagram for p = 0.005, A = 25.0 in Figure 3.18 and found that the bifurcation lines separate the parametric space spanned by m and q, into three distinct domains. Domain I, located below the two bifurcations lines, corresponds to systems with homogeneous equilibria, which is unconditionally stable. Domain II contains pure Hopf bifurcation instabilities, which can be destabilized by a homogeneous perturbation. In domain III, both Hopf and Turing instabilities occur. Here, codimension-two bifurcation does not occur. The authors [255] have also proved the following results (analytically and computationally): (i) For the values of the parameters which produce a stable limit cycle in the nonspatial model, spiral and target waves can emerge in the spatial model. (ii) In the spatial model, two different types of breakup of waves, the breakup of spiral waves and target waves, were observed. The authors observed that the breakup of spiral waves

Modeling Virus Dynamics in Time and Space

177

FIGURE 3.18 Bifurcation diagram for the model system (3.85) with p = 0.005 and A = 25.0.

is from the core and the breakup of target waves is from the far-feld. Both types of waves become irregular over time and produce spatiotemporal chaos. They concluded that diffusion can form spiral waves, target waves, or spatial chaos of high population density in the epidemic model. To study the impact of intervention strategies on the spread of infectious diseases such as polio, tetanus, diphtheria, measles, hepatitis, infuenza, chickenpox, mumps, rubella, AIDS, and others, Wendi Wang [277] proposed an SIRS model and found that intervention strategies decrease endemic levels and tend to simplify the dynamical behavior of the disease. Furthermore, the author observed the following: (i) For a saturated infection force, the model may admit a stable DFE and a stable EE at the same time. (ii) If the recovery rate is varied, the boundaries of the region for the persistence of the disease undergo a change from a separatrix of a saddle to an unstable limit cycle. (iii) If the inhibition effect from behavioral changes is weak, then two limit cycles were obtained and bifurcations take place as the population size changes. Wang [278] modifed the SIRS epidemic model (3.82a)–(3.82c) by including the adaption of behavior of individuals under intervention policies and considered a general form of nonlinear incidence, H ( I ) = ˜ I f ( I ) (termed as infectious force). Here, 1 f ( I ) represents the effect of intervention strategies on the reduction of contact coeffcient ˜ . In the absence of intervention strategies, that is f ( I ) = 1, the incidence rate reduces to the well-known bilinear transmission rate ˜ SI. The author has assumed that the population size is a constant S + I + R = N , and to ensure a nonmonotonic infection force, the following assumptions were made: A1. f ( 0 ) > 0, and f ˛ ( I ) > 0, for I > 0. ˇ ˇ A2. There exists a ˜ > 0, such that ˛I ˝ f ( I ) ˙ˆ > 0 for 0 < I ˛ ˜ , and ˛˝ I f ( I ) ˙ˆ < 0 for I > ˜ . Under the above assumptions, the system reduces to [278] dI ˜I dR = ( N − I − R ) − ( d + ° ) I, = ° I − ( d + ˛ ) R. dt f ( I ) dt

(3.88)

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Spatial Dynamics and Pattern Formation in Biological Populations

Cai et al. [35] investigated the effect of environmental fuctuations introduced in infectious force on the disease’s dynamics by studying the global dynamics of a general SIRS model with infection force under intervention strategies in both deterministic (without noise) and the corresponding stochastic version (with noise). Li et  al. [152] studied the spread dynamics of a stochastic SIRS epidemic model with nonlinear incidence and varying population size, which is formulated as a piecewise deterministic Markov process. Authors have established the threshold dynamics of the disease extinction and persistence for the system. The authors concluded that the disease can be eradicated almost surely if 0 < 1, while the disease persists almost surely if 0 > 1. Ma et al. [178] introduced spatial effects in the model (3.88) to estimate the propagation speed of the traveling wave. The diffusive model is written as ˙I ˜I ˙2 I = ( N − I − R ) − ( d + ° ) I + dI 2 , ˙t f ( I ) ˙x

(3.89a)

˙R ˙2 R = ˜ I − ( d + ° ) R + dR 2 , ˙t ˙x

(3.89b)

˜I ˜R = = 0, t > 0,  x   ˝˙Ω, I ( 0,  x ) = I 0 ( x ) ˇ 0,  R ( 0, x ) = R0 ( x ) ˇ 0,  ˘   x ˝ Ω, ˜n ˜n where dI > 0 and dR > 0 are the diffusion coeffcients of I and R respectively. f ( I ) is an increasing function of I with f ( 0 ) > 0. The authors made the following assumptions on the function f ( I ), which are different from the assumptions made by Wang [278]: A1. The infection force I f ( I ) satisfes ( I f ( I ))˛ > 0 for I > 0, which is reasonable as many endemic diseases extend along with the increase of infectious individuals. A2. f ( I ) satisfes the Lipschitz condition, that is, there exists a constant L > 0 such that f ( I1 ) − f ( I 2 ) ˝ L I1 − I 2 for all I1 ,  I 2 ˜ Ω. Model system (3.89a)–(3.89b) has the basic reproduction number 0 = ˜ N ˇ˘( d + ° ) f ( 0 )  . The model admits two equilibrium points: (i) DFE if 0 < 1. (ii) If 0 > 1, a unique endemic steady state E* I * ,  R * exists where R* = ˜ I * ( d + ° ) , and I * is the positive root of

(

)

˜

( d + ˜ ) f ( I ) + ° ( PI − N ) = 0, P = 1 + d + ˛ . By direct computations, we obtain

( )

I * + R* = N − Qf I * < N − Qf ( 0 ) = Qf ( 0 )( 0 − 1) , Q = ( d + ˜ ) ° . N (d + ˜ ) N° := M1 , 0 < :R* < = M2 , and M1 + M2 = N. ° + d +˜ ° + d +˜ The authors also proved the following results for the system (3.89a) and (3.89b) (analytically and computationally): (i) If 0 < 1, then the DFE E0 ( 0, 0 ) is locally asymptotically stable and E0 is GAS for 0 ˜ 1. (ii) If 0 > 1, then the EE E* I * ,  R * is locally and globally asymptotically stable. (iii) Under the quasi-monotonicity assumption, combining the That is, 0 < I *
c * , the system admits a traveling wave solution  Then for every * with speed c connecting (0, 0) and I ,  R* . For computational purposes, the authors used the values of the parameters as dI = 0.04, dR = 0.02, ˜ = 0.8, N = 2.1, d = 0.4, ˜ = 0.3, ˜ = 0.1, f ( I ) = 1 + ˜ I, and ˜ = 0.5. In this case, 1 < R0 = 2.4 < 2.667, and a traveling wave exists. The initial conditions are assumed as the piecewise functions

{

(

}

)

˘E* , x ˆ[ −20, 0 ) ( I ( 0, ,x )   R ( 0, x )) =  E0 , x ˆ[ 0, 20 ]

˜I ˜R = = 0,  t > 0,  x = 0,  20. In this ˜n ˜n case, the authors obtained a traveling wave solution as given in Figure 3.19. and the Neumann boundary conditions are taken as

3.6 Susceptible-Exposed-Infected-Recovered (SEIR) Models SIR models assume that the disease has no latent period, so that infected hosts instantaneously become infectious. For many human diseases such as hepatitis B, Chagas disease, and AIDS, the infected hosts stay in a latent period before becoming infectious [13,111]. The latent hosts form an additional exposed (E) class. It was assumed that only susceptible populations are affected by the infectious populations. Since recovery does not give immunity, individuals move from the susceptible-exposed-infectious class to the susceptible class upon recovery when the temporary immunity disappears. In many infectious diseases, there is an exposed period after the transmission of infection from susceptibles

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Spatial Dynamics and Pattern Formation in Biological Populations

to potentially infective members but before these potential infective individuals develop symptoms and can transmit infection. To incorporate an exposed period with the mean exposed period 1 κ , Brauer and Castillo-Chavez [29] added an exposed class E and considered the following SEIR model with infectivity in the exposed stage dE dI dR dS = − β S ( I + ε E ) ,  = β S ( I + ε E ) − κ E,  = κ E − α I ,  = α I , dt dt dt dt

(3.90)

  S ( 0 ) = S0 ,  E ( 0 ) = E0 ,  I ( 0 ) = I 0 ,  R ( 0 ) = R0 . The total population size is N = S + E + I + R. The disease states are E and I. The model has a DFE point at (N, 0, 0, 0). The basic reproduction number is defined as the spectral radius of the matrix FV −1 , where F is the rate at which secondary infections increase (newly infected terms) at the DFE and V is the rate at which disease progresses (death and recovery decrease; i.e., the total outflow from infected compartments E and I) at the DFE. It is  ε 1 given by 0 = ρ( K L ) = β N  +  (Problem 3.7, Exercise 3). Since the class R does not have κ α  any effect on the dynamics of S, E, and I classes, it is sufficient to consider the model dS dE dI = β S ( I + ε E ) − κ E,   = κ E − α I . = −βS( I + ε E) , dt dt dt

(3.91) ∞

∫

Adding the equations and integrating from 0 to ∞ , we obtain N − S∞ = α I ( s ) ds. Integrating the last equation of (3.91), we obtain 0 ∞

∫

∞

∫

κ E ( s ) ds  = α I ( s ) ds − I 0 . 0

0

Dividing the first equation of (3.91) by S and integrating from 0 to ∞ , we obtain

log

∞ ∞  ∞  S0  I ( s ) ds + ε  α I ( s ) ds −   I 0    =   E s = β I s + ε ds   β  ( ) ( )    S∞ κ    κ 0 0  0

∫

∫

∫

∞

S εα  εβ I 0 εβ I 0 = β 1 + I ( s ) ds   −   = 0 1 − ∞  − . N κ  κ κ  

∫

(3.92)

0

Equation (3.92) is called the final size relation, which gives a relationship between the basic reproduction number and the size of the epidemic. The term 1 − ( S∞ N )  is called attack rate or attack ratio. If I ( 0 ) = 0, then the final size relation has the same form as for the SIR model. The quantities S0 and S∞ may be estimated by measurements of immune responses in blood samples before and after an epidemic. This estimate, however, is a retrospective one, which can be derived only after the epidemic has run its course. Urabe et  al. [268] investigated the spatial SEIR model with stochastic mobility of individuals on the 2D square lattice with the periodic boundary condition to reveal how heterogeneous spatial

Modeling Virus Dynamics in Time and Space

181

mobility infuences the fnal epidemic size. The authors found that the distance that each infected individual move during the latent and infectious periods plays an important role in the fnal epidemic size. London and Yorke [174] and Dietz [68] formulated several models to simulate yearly outbreaks of chickenpox and mumps and the biennial pattern of measles in which the contact rate was assumed to vary seasonally and in the absence of latent period, which indicate the possible existence of periodic solutions having periods one and two years for suitable values of the parameters and longer periods of three, four, and six years were also conjectured. Schwartz [243] numerically found the co-existence of periods one, two, and three years for measles parameters incorporating latency. Using perturbation methods, Grossman et al. [95] formally obtained periodic solutions of a period of two years in the Dietz model [68]. Following Dietz [68], Schwartz and Smith [244] made the following assumptions and designed an SEIR model: (i) Population size is constant and it consists of susceptible, infected but not yet infectious, infectious, and immune individuals. (ii) The disease is not lethal; birth and death rates are constant, equal, and equal to µ. (iii) The population is homogeneous and uniformly mixing. (iv) An exposed individual’s probability of becoming infectious in a specifed time interval is independent of time after initial contact and hence the probability of still remaining in the exposed class at time ˜ after initial contact is e −˜° , where 1 ˜ is the mean latent period. (v) After an individual enters the infectious class, the probability of that individual recovering at time ˜ is given by e −˜° , where 1 ˜ is the mean infectious period, and ˜ is called the recovery rate. (vi) Recovered individuals are permanently immune. The model is taken as [244] dE dS = µ ( 1 − S) − ° IS , = ° IS − ( µ + ˛ ) E, dt dt dI dR = ˜ E − ( µ + ˛ ) I,  = ˛ I − µR. dt dt

(3.93)

with S + E + I + R = 1. The model has two steady states: (i) (1, 0, 0, 0) and (ii) E ( S0 ,  E0 , I 0 ,  R0 ) ,  where S0 =

( µ + ° ) ( µ + ˛ ) ,  I ˝°

0

=

 ( µ + ˛ ) I ,  R = ˛ I , Q = 1 = µ 1 ˝° − 1 ,  E0 =   > 1. 0 0 0   ˝  S0 ° µ S0 ( µ + ° ) ( µ + ˛ )

Q is called the reproductive rate for the infection; i.e., the number of secondary cases produced by a single infectious individual in a population of susceptibles in one infectious period. The authors (Schwartz and Smith [244]) proved the existence of an infnite number of stable subharmonic solutions in an SEIR model that incur permanent immunity with seasonal variations in the contact rate. Anderson and May [11] estimated the value of Q for mumps, chickenpox, and measles as 7, 9, and 16 (approximately) respectively. It was shown that the EE E ( S0 ,  E0 , I 0 ,  R0 ) is asymptotically stable and trivial equilibrium is unstable for Q > 1. Olsen and Schaffer [205] reviewed the dynamics of chickenpox and measles, compared their incidence patterns for the real-world infections with the SEIR model, and suggested that measles epidemics are inherently chaotic. Conversely, the extent to which chickenpox outbreaks approximate a yearly cycle depends inversely on the population size. Bauch and Earn [22] explained the incidence pattern of four childhood diseases (measles, chickenpox, rubella, and whooping cough) using the following SEIR model and showed that transient dynamics is the source of the nonresonant peaks and identifcation

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Spatial Dynamics and Pattern Formation in Biological Populations

of the periods of the attractors of the model correctly predicts the resonant power spectral density peaks but fails to predict the nonresonant peaks, dS dE = ˜ − µS − ˛ IS ,  = ˛ IS − ( µ + ˝ ) E, dt dt dI dR = ˜ E − ( µ + ˛ ) I,  = ˛, I − µR dt dt

(3.94)

where ˜ is the birth rate, µ is the per capita death rate, ˜ is the mean transmission rate, 1 ˜ is the mean latent period, and 1 ˜ is the mean infectious period. Seasonally varying transmission rates are specifed according to school-term dates, with ˜ being high when school is in session and low otherwise [79] (the case R = 0 is discussed in Problem 3.8, Exercise 3). Heesterbeek and Metz [103] derived an expression for the saturating contact rate of individual contacts in an epidemiological model as C ( N ) = bN ˙ˆ1 + bN + 1 + 2bN ˇ˘, where b > 0 is the saturating contact rate of individual contacts in a population that mixes randomly. For small N ,  C ( N ) ˛ bN , whereas for large N ,  C ( N ) ˛ 1. Zhang and Ma [305] studied the global dynamics of an SEIR model with the saturating contact rate defned by C ( N ) .  The demographic structure used in this SEIR model assumes the following: (i) Recruitment at a constant rate A into the population. (ii) Natural deaths occur at a rate proportional to the population size, so that the death rate term is µ N , where µ > 0 is the death rate constant. The model is taken as [305] dS a SI dE a0SI = A − 0 − µS, = − ( ° 0 + µ ) E, dt h( N ) dt h ( N ) dI dR = ˜ 0E − (° 0 + µ ) I − ˝ 0 I, = ° 0 I − µR, dt dt

(3.95)

S ( 0 ) > 0,  E ( 0 ) ˝ 0,  I ( 0 ) ˝ 0,  R ( 0 ) ˝ 0, where a0 = ˜ b and h ( N ) = 1 + bN + 1 + 2bN . The parameters ˜ , ° 0 , and ˛ 0 are all positive constants, ˜ 0 is a nonnegative constant and represents the disease-related death rate, ˜ is the probability per unit time of transmitting the infection between two individuals taking part in a contact. ˜ 0 is the rate constant for recovery, so that 1 ˜ 0 is the mean infective period. Also, ˜ 0 is the rate constant at which the exposed individuals become infective, so that 1 ˜ 0 is the mean latent period. The recovered individuals are assumed to acquire permanent immunity, so that there is no transfer from the R class back to the S class. The positive constant A µ represents a carrying capacity, or maximum possible population size. The total population size is N ( t ) = S ( t ) + I ( t ) + E ( t ) + R ( t ) . Setting µdt = d° , (3.95) reduces to the equations dS A aSI dE aSI = − − S, = − ( 1 + ˛ ) E, d˜ µ h ( N ) d˜ h ( N ) dR dI = ° E − ( 1 + ˛ + ˝ ) I, = ˛ I − R, d˜ d˜

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Modeling Virus Dynamics in Time and Space

where aµ = a0 ,  °µ = ° 0 , ˛µ = ˛ 0 , and ˝µ = ˝ 0 . Substituting S ( t ) = N ( t ) − I ( t ) − E ( t ) − R ( t ) , the equations simplify as dE a ( N − E − I − R ) I dI = − ( 1 + ° ) E, = ° E − ( 1 + ˛ + ˝ ) I, d˜ h( N ) d˜ dR dN A = ° I − R, = − N − ˝ I. d˜ d˜ µ

{

(3.96)

} )

The system is considered in the closed set ˆ = ( E,  I ,  R ,  N ) ˘R+4 : 0  E + I + R  N  A µ . The model has two steady states: (i) DFE P0 ( 0, 0, 0,  A µ ) and (ii) the EE P* E* , I * , R* ,  N * where I* =

(

 1+˛ +˜ * 1ˇA − N *  , E* = I , and R* = ˛ I * .  ˜˘µ ˝ 

Substituting the values of E* , I * , and R * into the frst equation of (3.96) and taking h ( N ) = 1 + bN + 1 + 2bN , we obtain   A F ( N )  ( a − ˜ b )°˛ N −  a (°˛ − ˜˝ ) + ˜°˛  − ˜°˛ 1 + 2bN = 0, µ  

(3.97)

where ˜ = ( 1 + ° + ˛ ) and ˝ = 1 + ˙ . a˜ A a˜ A a Defne 0 =  = . µ˛˝ h ( A µ ) ˙˜ bA ˙ b   A We fnd F ( 0 ) = −  a (˜° − ˛˝ ) + ˛˜°  − ˛˜° < 0 and µ   F ( A µ ) = °˛˝ h ( A µ ) ( 0 − 1) . When 0 > 1, F ( A/µ ) > 0, and the graph of F crosses the axis. Equation (3.97) has at least one root N * ˝( 0, .A µ )   Now dF ˛b ˇ = ˜°  a − ˛ b − dN   1 + 2bN ˘

 , 

which shows that F ( N ) is decreasing if N < q, and increasing if N > q , where

{

}

2 q = ˘ a ( 2˜ b − a ) 2b ( a − ˜ b )  . Together with F ( 0 ) < 0, it can be claimed that the root   N * ˝( 0,  A µ ) of (3.97) is unique. Busenberg and Cooke [33] studied several diseases that transmit both vertically and horizontally and gave a comprehensive survey of the formulation and mathematical analysis of compartmental models that also incorporate vertical transmission. For human and animal diseases, horizontal transmission typically occurs through direct or indirect physical contact with infectious hosts or through disease vectors such as mosquitos, ticks, or other biting insects. Vertical transmission can occur through transplacental transfer of disease

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Spatial Dynamics and Pattern Formation in Biological Populations

agents. Among insects or plants, vertical transmission occurs often through eggs or seeds. Li et al. [158] established the global stability of an SEIR model with varying total population size. Li et al. [160] have also studied the global dynamics of an SEIR epidemic model with vertical transmission in a constant population and incidence term in the form of bilinear mass action. The global dynamics is completely determined by the basic reproduction number 0 ( p , q ), where p and q are fractions of infected newborns from the exposed and infectious classes respectively. If 0 ˜ 1, the DFE is globally stable and the disease always dies out and if 0 > 1, a unique EE exists and is globally stable in the interior of the feasible region, and the disease persists at an EE state if it initially exists. Li et al. [156] analyzed the global stability of an SEIR epidemic model considering constant immigration and infectious force in exposed, infected, and recovered class. Li and Muldowney [159] considered the global dynamics of an SEIR model with a nonlinear incident rate using the geometric method. Greenhalgh [93] considered an SEIR model with density-dependent death rate and constant infection rate. They found that the system possesses three equilibrium points: (i) At the frst equilibrium point, the population becomes extinct. This is always possible and the equilibrium is locally stable precisely when the birth rate is less than the minimum death rate. (ii) At the second equilibrium point, the population size maintains itself at a constant steady level and the disease is eradicated. This equilibrium is locally stable provided a certain threshold is not exceeded and locally unstable if this threshold is exceeded. (iii) At the third equilibrium point, the disease is possible and regulates the population size. This equilibrium exists if the threshold value is exceeded and may be either locally stable or unstable. The author has numerically found that cycles of disease incidence with varying amplitudes occur at different parametric values. Porter and Oleson [213] introduced a path-specifc (PS) SEIR model to allow for latent and infectious times of infectious diseases (e.g., mumps), which follow a nonexponentially distributed pattern, while still respecting the population-level mixing structure of the data. In other words, the PS SEIR model provides a method to analyze disease spread at the population level while simultaneously having the capability to handle specifc paths of individuals through the latent and infectious times. To determine the effect that a spatially indexed population has on the spring break effect and to assess the effect of incorporating explicit spatial dependence into the PS SEIR model for this epidemic, Porter and Oleson [214] have developed a spatial analog of their PS SEIR model [213] that allows epidemic data collected over a lattice to be analyzed while accounting for nonexponentially distributed latent and infectious times. The model smooths the spatial transmission over multiple conduits for sparse counts of new infections, which commonly occur when the data is collected over fne spatial and temporal partitions. Using bilinear incidence in the force of infection corrected by the infectivity factor, Röst and Wu [230] derived the following SEIR model with distributed infnite delay when the infectivity depends on the age of infection a at the current time t, that is, I ( t ) =

˜

˙

i ( t,  a ) da,

where i ( t ,  a ) is the density of the infected individuals with respect to the age of infection a at the current time t. They have introduced a kernel function 0 ˛ k ( a ) ˛ 1 to represent the infectivity as per the age of infection a. The classical SEIR epidemic model is modifed as ˘

dS = ˆ − ° S k ( a ) i ( t ,  a ) da − dS, dt

˜ 0



dE = ° S k ( a ) i ( t ,  a ) da − ( d + µ ) E, dt

˜ 0

0

Modeling Virus Dynamics in Time and Space

dI dR = µE − ( d + ° + r ) I, = rI − dR , dt dt

185

(3.98)

where ˜ is the constant recruitment rate, ˜ is the baseline transmission rate, d is the natural death rate, ˜ is the disease-induced death rate, 1 µ is the average latency period, and 1 r is the average infectivity period. The density of infected individuals i ( t ,  a ) satisfes the following evolution equations:

˜  ˇ˜ ˘ +  i ( t ,  a ) = ( d + ° + r ) i ( t , a ) , ˜t ˜ a i ( t ,  a ) = µE ( t ) . The solution of the evolution equations is i ( t ,  a ) = i ( t − a, 0 )  e −( d+ˆ + r )a = µE ( t − a ) e −( d+ˆ + r )a . Then, the model system (3.98) becomes 

dS = ˇ − ° S k ( a ) µE ( t − a ) e −( d+ + r )a da − dS, dt

˜ 0



dE = ° S k ( a ) µE ( t − a ) e −( d+ + r )a da − ( d + µ ) E, dt

˜ 0

dR dI = µE − ( d + ° + r ) I, = rI − dR , dt dt

(3.99)

The authors considered the varying infectivity of the infected individuals as a function of the age of infection and applied a permanence theorem for infnite-dimensional systems. They showed that a unique EE exists, which is locally asymptotically stable and the disease is always present when the basic reproduction number satisfes the condition 

°µˇ 0 = k ( a ) e − ( d + + r ) a da > 1. d (d + µ )

˜ 0

Tang and Wu [257] extended the above model and investigated the asymptotic behavior of the positive solution of a diffusive SEIR epidemic model with varying infectivity, distributed and infnite delay in a bounded domain, which describes the transmission of diseases such as tuberculosis and HIV/AIDS. They obtained suffcient conditions for local and global asymptotical stability using spectral analysis and comparison arguments. The spatial model is described by the following equations [257] 

°S = ˘ − ˛ S k ( a ) µE ( t − a ) e −( d+ + r )a da − dS + D1 S, °t

˜ 0

(3.100)

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Spatial Dynamics and Pattern Formation in Biological Populations



dE = ° S k ( a ) E ( t − a ) e − ( d + + r ) a da − ( d + µ ) E + D2 E, dt

(3.101)

dI = µE − ( d + ° + r ) I + D3 ˘I, dt

(3.102)

dR = rI − dR + D4 ˝R , x ˙Ω, t > 0. dt

(3.103)

˜ 0

The equations are solved under homogeneous Neumann boundary conditions and the initial conditions are given by prescribed nonnegative values. The initial functions ˜i are nonnegative, Hölder continuous, and satisfy the condition ( ˙˜i ˙° ) = 0 on the boundary. The diffusion coeffcients di  ( i = 1, 2, 3, 4 ) are positive. The model system (3.100)–(3.103) has the DFE P0 ( S0 , 0, 0, 0 ) ,  where S0 = °/d. Furthermore, if the basic reproduction number 0 satisfes the condition 



°µˆ °µS0 0  k ( a ) e −( d+ + r )a da = k ( a ) e −( d+ + r )a da > 1,  d (d + µ ) (d + µ )

˜

˜

0

0

(

)

then, the system has a unique EE P* S* , E* ,  I * ,  R* , where  ˆ  1  ˆµ 1  S , S* = , 0 E* = 1− ,  I * = 1−    0 ( d + µ )  0  ( d + µ ) ( d + ° + r )  0  R* =

˘ µrS0 1  1− .  ( d + µ ) ( d + ° + r )  0 

Let 0 < ˜ 1 <  ˜ 2 < … <  ˜ n < …  be the eigenvalues of the operator −° on Ω with the homogeneous Neumann boundary condition, and E (˜ i ) be the eigenspace corresponding to ˜ i in C1 Ω . Let D = diag ( D1 ,  D2 ,  D3 ,  D4 ) , Z = ( S,  E,  I ,  R ) , and Z = D˝Z + G Pˆ Z , where

()

( )

 − d − hE 0  hE 0 G( Pˆ )Z =   0   0      +     

0 −(d + µ )

0 0

µ 0

−(d + ˛ + µ ) r



− ˝µS

0

˜k ( a)E ( x, t − a) e

− (d +˛ + r ) a

da

0



˜

˝µS0 k ( a ) E ( x ,  t − a ) e − ( d +˛ + r ) a da 0

0 0

  S( x , t)      E( x , t)  0   I ( x , t)    −d   R( x , t)  0 0

     .     

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Modeling Virus Dynamics in Time and Space

(

Let Pˆ = S0 ,  E 0 ,  I 0 ,  R 0 (3.100)–(3.103), and

)

represent any feasible uniform steady state of the system ˘

h = °µS

0

˜k ( a ) e (

− d +ˇ + r ) a

da.

0

Linearization of the system (3.100)–(3.103) about Pˆ is of the form Zt = Z. ˜ is an eigenvalue of , if it is an eigenvalue of the matrix −˜ i D + G Pˆ . The characteristic equation of −˜ i D + G ( P0 ) takes the form

()

   ( ° + ˛ i D1 + d )  ° + ˛ i D2 + ( d + µ ) − ˙µS0 k ( a ) e − (° + d +ˆ + r ) a da    0  

˜

{

}

° + ˛ i D3 + ( d + ˆ + r ) ( ° + ˛ i D4 + d ) = 0.

(3.104)

For any i ˜ 1,  equation (3.104) has three negative real roots

˜1 = − (° i D1 + d ) , ˜2 = − (° i D3 + ( d + ˛ + r )) , ˜3 = − (° i D4 + d ) . The other root is determined by 

˜

g i ( ° ) = ° + ˛ i D2 + ( d + µ ) − ˙µS0 k ( a ) e −( ° + d+ + r )a da = 0. 0

For 0 > 1, and i = 1, we have for nonnegative real ˜ 

˜

g1 ( 0 ) = ( d + µ ) − ˛µS0 k ( a ) e −( d+ + r )a da = ( d + µ ) ( 1 − 0 ) < 0, and g1 (  ) = . 0

Hence, the equation g i ( ˜ ) = 0 has a positive real root. Therefore, there is a characteristic root ˜ with a positive real part in the spectrum of . This implies that P0 is unstable if 0 > 1. Stability of EE point P*: The characteristic equation of −˜ i D + G P* takes the form

( )

( ˜ + ° i D1 + 0d )  ˜ + ° i D2 + ( d + µ ) − S˜  ( ˜ + ° i D3 + d + ˝ + r )  ( ˜ + ° i D4 + d ) + S˜ ( 0 − 1) d ( ˜ + ° i D3 + d + ˝ + r )( ˜ + ° i D4 + d ) = 0, where Sˇ = °µS*

˜



0

(3.105)

k ( a ) e −( ˇ + d+˘ + r )a da. For any i ˜ 1, equation (3.105) always has the two nega-

tive roots ˜1 = − (° i D3 + d + ˛ + r ) and ˜2 = − (° i D4 + d ) .

Denote ˜ i ( ° ) = ( ° + ˛ i D1 + 0d ) ( ° + ˛ i D2 + ( d + µ ) − S° ) + S° ( 0 − 1) d = 0.

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Spatial Dynamics and Pattern Formation in Biological Populations

Now, S0 = d + µ and ˜ i ( 0 ) = (° i D1 + 0d ) ° i D2 + ( d + µ ) ( 0 − 1) d > 0, for any i ˜ 1 and 0 > 1. Suppose that ˜ is a root of ˜ i ( ° ) . If Re ˜ ° 0, then e − ° a ˜ 1 for any a ˜ 0. Hence, S° ˆ ( d + µ ) ˆ ° + d + µ + ˛ i D2 ,  and

˜ + d + ° i D1 < ˜ + ( 0 − 1) d + d + ° i D1 = ˜ + 0d + ° i D1 . But, the equation ˜ i ( ° ) = 0 gives that S˜ ( ˜ + d + ° i D1 ) = ( ˜ + d + µ + ° i D2 ) ( ˜ + 0d + ° i D1 ). This leads to a contradiction. Therefore, all roots of the characteristic equation (3.105) have negative real parts for any i ˜ 1, if 0 > 1. Moreover, the real parts of all roots are less than a negative constant. Hence, if 0 > 1, the EE P* is locally asymptotically stable for all a ˜ 0, and the DFE P0 ( S0 , 0, 0, 0 ) is unstable. If 0 1. If 0 < 1, then c > 0. There exist two EE points, if b  0. Now, b  0 gives the condition 2 R1* − R1*2 < 0 < 1. The intersection of these two conditions is the second condition. Authors have also studied the long-time behavior of the solutions and in particular, absorbing sets in the phase space were determined. By using a Rionero-Lyapunov function, the nonlinear asymptotic stability of EE was investigated. Dubey et al. [71] proposed an SEIR epidemic model with two different removal rates to account for different treatment capacities of the community. In any outbreak of a disease, its treatment capacity is first very low and then grows slowly with the improvement of the hospital’s capabilities, availability of effective drugs, and so on. Furthermore, when the number of infected individuals is very large, the treatment capacity reaches to its maximum due to limited treatment facilities. This condition pertains to newly emerging diseases whose treatment is very limited. The considered model is [71]

dS dE = A − δ 0S − α SI , = − (δ 0 + δ 1 ) E + α SI , dt dt



dI = − (δ 0 + δ 2 ) I + δ 1E − δ 3 I − h ( I ) , dt



dR = δ 2 I − δ 0 R + h ( I ) , (3.107) dt

with S ( 0 ) > 0,  E ( 0 ) ≥ 0,  I ( 0 ) ≥ 0,  R ( 0 ) ≥ 0. The variables and parameters have the following meanings. Susceptibles are recruited into the population at a constant rate A and δ 0 is the natural death rate for the population in all the four classes. The susceptibles become infected on contact with infected individuals. This interaction is considered to be of

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Spatial Dynamics and Pattern Formation in Biological Populations

mass-action type. α is the rate at which a susceptible individual is exposed to the infection. Upon infection, the susceptible individuals move to the exposed class, and only after latency do they become infected and move to the infectious class. δ 1 is the rate at which an individual leaves the exposed class and becomes infective, that is, 1 δ 1 is the latency period. δ 2 is the natural recovery rate of the infectious individuals and 1 δ 2 is the infectious period. δ 3 is the death rate of the infectious individual due to infection. Removal of an infectious individual is also made by treatment using the removal rate function. h ( I ) is the recovery rate of the infectious subpopulation through treatment, which is taken as belonging to the following two types: Holling type III and IV. It was also assumed that after recovery, the individuals become immunized, and hence, they are no longer susceptible to it. This happens because acquired immune response leads to the development of immunological memory and therefore an individual is not infected from the same disease again and does not enter the susceptible population. Since the removed class R does not have any effect on the dynamics of S, E, and I classes, the reduced system is taken as [71] dS = A − δ 0S − α SI, dt dE = − (δ 0 + δ 1 ) E + α SI, dt dI = − (δ 0 + δ 2 ) I + δ 1E − δ 3 I − h ( I ) , dt

(3.108)

with S ( 0 ) > 0,   E ( 0 ) ≥ 0,  I ( 0 ) ≥ 0. The following choices of the treatment rates were considered: (i)  h ( I ) = (ii)  h ( I ) =

βI2 ,  I ≥ 0,   β ,  γ > 0, with h ( 0 ) = 0,  h′ ( I ) > 0. 1 + γ I2

(

βI , I ≥ 0,  β ,  a, b > 0, with h ( 0 ) = 0,  h′ ( I ) > 0,  if I 2 < ab. I a +I+b

)

2

The authors have shown that all solutions of the temporal model (3.108) are nonnegative and bounded in Ω = ( S,  E,  I ) ∈ R+3 : S > 0, E ≥ 0, I ≥ 0, S + E + I ≤ S0 = ( A δ 0 ) . The temporal model has two equilibrium points: (i) disease-free equilibrium (DFE) point P0 ( S0 , 0, 0  ) , where S0 = ( A δ 0 ) , and (ii) EE point P* S* , E* ,   I * , where

{

}

(

S* =

)

A Aα I * * , E = , and I * is the positive root of the equation δ 0 + α I* (δ 0 + δ 1 ) δ 0 + α I *

(

)

(δ

(

0

)

{( )

)

}

+ α I * (δ 0 + δ 1 ) h I * + I * (δ 0 + δ 2 + δ 3 ) −  δ 1 Aα I * = 0.

The authors obtained the conditions such that this equation has a positive real root and the EE point P* S* , E* ,  I * exists. Furthermore, they have also obtained the conditions under which the EE point P* S* , E* ,  I * is locally asymptotically stable and GAS with Holling type II and IV type treatment rates. Now, we consider the spatial version of model (3.108) as

(

)

(

)

Modeling Virus Dynamics in Time and Space

191

˜S ˜ 2S = A − ° 0S − ˛ SI + Ds 2 , ˜t ˜x ˜E ˜ 2E = − (° 0 + ° 1 ) E + ˛ SI + De 2 , ˜t ˜x ˜I ˜2I = − (° 0 + ° 2 ) I + ° 1E − ° 3 I − h ( I ) + Di 2 , ˜t ˜x

(3.109)

where Ds ,  De ,  and Di are the diffusion coeffcients of susceptible, infective, and recovered populations respectively, and h ( I ) = ˜ I ( I + b ) ; ˜ > 0,  b > 0. The initial conditions and nofux boundary conditions are taken as S ( x, 0 ) ˙ 0,  E ( x, 0 ) ˙ 0,  I ( x, 0 ) ˙ 0, x ˆ[ 0, R ],

˜S ˜E ˜I = = = 0. ˜ x 0,  R ˜ x 0,  R ˜ x 0,  R For studying the linear stability of the spatial model (3.109), it is perturbed as S = S* + a exp ( ˜ k t + ikx ) , E = E* + b exp ( ˜ k t + ikx ) , I = I * + c exp ( ˜ k t + ikx ) , where a, b, and c are suffciently small constants, k is the wave number, and ˜ k is the wavelength. The system is linearized about the nontrivial interior equilibrium point E , S* E* , I *  . The characteristic equation of the linearized system is given by (* is dropped for convenience)

(

)

˜ k 3 + °1˜ k 2 + °2 ˜ k + °3 = 0, where ˜1 = A1 + ( Ds + De + Di ) k 2 ,

˜2 = A2 − { a11 ( De + Di ) + a22 ( Di + Ds ) + a33 ( Ds + De )} k 2 + ( Ds De + De Di + Di Ds ) k 4 , ˜3 = A3 + k 2 {( a22 a33 − a23 a32 )Ds + (a11a33 − a31a13 )De + (a11a22 − a12 a21 )Di } − k 4 ( a11De Di + a22 Ds Di + a33 Ds De ) + k 6 ( Ds De Di ) , with a11 = −˜ 0 − ° I ,  a12 = 0,  a13 = −° S,  a21 = ° I, a22 = − (˜ 0 + ˜ 1 ) , a23 = ° S,  a31 = 0, ,a32 = ˜ 1 a33 = − (˜ 0 + ˜ 2 + ˜ 3 ) − hˇ ( I ) , A1 = − ( a11 + a22 + a33 )

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Spatial Dynamics and Pattern Formation in Biological Populations

A2 = a22 a33 − a13 a31 − a23 a32 + a11a33 + a11a22 − a12 a21  , A3 = −a11 ( a22 a33 − a23 a32 ) + a21 ( a12 a33 − a13 a32 ) − a31 ( a12 a23 − a13 a22 ). From the Routh-Hurwitz criterion, Re ( λ ) < 0, if ρ1 > 0,   ρ2 > 0,   ρ3 > 0, and ρ1 ρ2 − ρ3 > 0. Using these inequalities, we can obtain the conditions for the steady state to be stable. If any one or more of the three inequalities are not satisfied, then spatial patterning will be observed. We have performed numerical simulations for the model (3.109) using the parameter values A = 1.7, δ 0 = 0.02,   β = 0.3,  δ 1 = 0.2,  δ 2  = 0.025,  δ 3 = 0.03,  b = 1, and α = 0.003. The equilibrium point is obtained as (40.9763, 4.00215, 7.16245). We obtain ρ1 ( 0 ) = 0.34099,   ρ2 ( 0 ) = 0.00533041,   ρ3 ( 0 ) = 0.000233923,  and [ ρ1 ρ2 − ρ3 ]( 0 ) = 0.00158369. Therefore, the temporal model (3.108) with Holling type II treatment rate is stable. Now, for the spatial model (3.109), the values of the diffusion coefficients are chosen as Ds = 0.03,  De = 0.001, and Di = 0.0001.  We obtain ρ1 k 2 > 0,   ρ2 k 2 > 0, [ ρ1 ρ2 − ρ3 ] k 2 > 0 , and ρ3 k 2 < 0 in the range (1.14018, 63.2737). Hence, Turing instability occurs for this set of parameter values. Plots of ρ3 k 2 vs. k 2 are given in Figure 3.20 for Ds = 0.01,  0.02, and 0.03. From the figure, we observe that as Ds decreases, the system approaches stability. Space series plots at the fixed time t = 200 are plotted in Figure 3.21a for the model system (3.109). The corresponding spatiotemporal plots showing the oscillatory dynamics are plotted in Figure 3.21b. For plotting the figure, the initial conditions are taken as S0  ( x ) = 40.9763 + C, E0  ( x ) = 4.00215 + C , and I 0  ( x ) = 7.16245 + C, where C = 0.01 cos ( 2 πx 20 ) .

( )

( )

( ) ( )

FIGURE 3.20 Plot of ρ 3(k2) vs. (k2).

FIGURE 3.21a Space series for model system (3.109).

( )

( )

Modeling Virus Dynamics in Time and Space

193

FIGURE 3.21b Spatiotemporal plot for model system (3.109).

Wang et al. [284] obtained the asymptotic solution of an SEIR epidemic model employing the method of perturbation and analyzed its dynamical behavior using the conventional qualitative and stability theory. Ruan [231] gave a review of the models, results, and simulations for studying the transmission of rabies in China. The author frst constructed a basic SEIR type model for the spread of rabies virus among dogs and from dogs to humans and used the model to simulate the human rabies data in China from 1996 to 2010. Then, the basic model was modifed by including both domestic and stray dogs and the model was applied to simulate the human rabies data from Guangdong Province, China. To study the seasonality of rabies, he further proposed an SEIR model with periodic transmission rates and employed the model to simulate the monthly data of human rabies cases reported by the Chinese Ministry of Health from January 2004 to December 2010. To understand the spatial spread of rabies, they added diffusion to the dog population in the basic SEIR model to obtain a reaction-diffusion equation model and determined the minimum wave speed connecting the DFE to the EE. Finally, to investigate how the movement of dogs affects the inter-provincial spread of rabies in Mainland China, they proposed a multipatch model to describe the transmission dynamics of rabies between dogs and humans

194

Spatial Dynamics and Pattern Formation in Biological Populations

and used a two-patch submodel to investigate the rabies virus clades lineages and to simulate the human rabies data from Guizhou and Guangxi, Hebei and Fujian, and Sichuan and Shaanxi, respectively. Khan et al. [143] studied the complex dynamics of an SEIR epidemic model (for those diseases that transmit between humans through contact and where some effective medical treatment for cure is available) with saturated incidence rate, treatment function, and optimal control. They considered the treatment function as a combination of infected individuals and available treatment control, with an addition in saturation form. They noticed the phenomena of backward bifurcation occurring due to both nonlinear treatment and incidence rate at 0 = 1. As most real-world problems are not deterministic, incorporating stochastic effects into the model gives us a more realistic way of modeling epidemic diseases. Yang et al. [300] included stochastic perturbations into SIR and SEIR epidemic models with saturated incidence and investigated their dynamics in terms of the basic reproduction number. The long-time behavior of the two stochastic systems was studied. Mainly, they used the stochastic Lyapunov functions to show that under some conditions, the solution has the ergodic property for 0 > 1, and exponential stability for 0 ˜ 1. The authors also proved that the SIR model has the ergodic property as the fuctuation is small, where the positive solution converges weakly to the unique stationary distribution. Yuan et al. [304] studied a class of multigroup epidemic deterministic models of SIR and SEIR types with bilinear incidence, and the global stability of their unique EE was proved by using the global Lyapunov function and graph theory. Yang and Mao [301] considered a class of multigroup SEIR epidemic models with stochastic perturbations. By the method of stochastic Lyapunov functions, they studied the asymptotic behavior in terms of the intensity of the stochastic perturbations and the reproduction number. Witbooi [289] studied a model with independent stochastic perturbations for a disease of the SEIR type and proved a theorem on almost sure exponential stability of the DFE, which shows that the stochastic perturbation enhances the stability of the DFE. This means that starting with a certain deterministic compartmental model, the differential equations are perturbed by mutually independent white noise terms. In particular, for this type of model, the total population size itself is perturbed directly by white noise. Examples of such models can be found in many works [300,301,304]. 3.6.1 Influenza Model Revisited We return to the problem of modeling the infuenza epidemics from Section 3.2.3, in which we discussed the SI model. The simplest scheme that can be considered to model infuenza spread is a deterministic homogeneous SEIR model. In this model, it was assumed that individuals who have been infected go frst into a latent, then to infectious, and then to the recovered stage. In infuenza, the latent period and the incubation period are considered to be the same. During this stage, there is a very low level of infectivity. On the other hand, in the infectious stage, there may be a very high level of infectivity. Massad et al. [182] proposed a model for infuenza transmission and applied it to reanalyze the impact of the pandemic 1918 fu strain in the city of São Paulo, Brazil. Latent and infectious periods are assumed to be constants with the susceptible populations being affected by the infectious as well as exposed populations. The authors also assumed that all newborns are susceptible and the model differs from other models by considering both the latent and infected as infective population. Therefore, the model’s variables are susceptible S(t), latent

Modeling Virus Dynamics in Time and Space

195

and infective E(t), infected and infective I(t), and removed R(t). The model’s dynamics is described by the following system of equations [182]:

dS N  E+I  = −β S  − µS + rN  1 −  ,  N   dt K



dE  E+I = βS − ( µ + σ + κ ) E,  N  dt



dI = σ E −   ( µ + α + γ ) I , dt



dR = κ E + γ I − µ R , (3.110) dt

where N = S + E + I + R is the total population and the parameters are β , the contact rate; µ, the natural mortality rate; r, the birth rate; σ −1 , the incubation period; κ and γ , the recovery rate for both latent and infected compartments; α , the disease-induced mortality rate; and K, the carrying capacity of the population. The main transmission route is by direct contact through contaminated hands, surfaces, or close contacts with infective individuals [175]. The basic reproduction number is defined as R0 = β ( µ + σ + α + γ ) ( µ + σ + κ ) ( µ + α + γ )  . This model can also be applied to compare the probable impact of chemotherapy, isolation/quarantine, and other preventing transmission measures as control strategies. Samsuzzoha et  al. [236] studied the above SEIR epidemic model (3.110) with diffusion and solved the system numerically using the operator splitting method under four different initial conditions. Jaichuang and Chinviriyasit [127] studied the above SEIR epidemic model (3.110) with diffusion. The resulting RD system was solved by a finite difference technique, which is first-order accurate in time and second-order accurate in space. The diffusive SEIR model was taken as

∂S N ∂ 2S E+ I  − + − + 1 , = −βS  µ S rN d 1    N  ∂t K ∂ x2



∂E ∂ 2E E+ I − ( µ + σ + κ ) E +   d2 2 , = βS   N  ∂t ∂x



∂I ∂2I = σ E −  ( µ + α + γ ) I +   d3 2 , ∂t ∂x



∂R ∂2R . (3.111) = κ E + γ I − µ R + d4 ∂t ∂ x2

The initial conditions and boundary conditions are taken as [146]

S ( x , 0 ) = S0 ,  E ( x , 0 ) = E0 ,  I ( x , 0 ) = I 0 ,  R ( x , 0 ) = R0 ; − L ≤ x ≤ L.

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Spatial Dynamics and Pattern Formation in Biological Populations

∂ S ( − L, t ) ∂ E ( − L, t ) ∂ I ( − L, t ) ∂ R ( − L, t ) = = = = 0, ∂x ∂x ∂x ∂x ∂ S ( L, t ) ∂ E ( L, t ) ∂ I ( L, t ) ∂ R ( L, t ) = = = = 0. ∂x ∂x ∂x ∂x The authors have taken the initial subpopulations for simulations as S ( x , 0 ) = 0.96 exp −10 x 2 ,  E ( x , 0 ) = 0,  I ( x , 0 ) = 0.04 exp −100  x 2 ,  R ( x , 0 ) = 0; − 2 ≤ x ≤ 2. The initial proportions of susceptible and infected individuals are concentrated at the origin in which the proportion of susceptible individuals is greater than the infected individuals and the proportions of all populations are spread throughout the domain [−2,2], as time increases. Samsuzzoha et al. [237] formulated a diffusive epidemic model for H1N1 influenza with a variable transmission coefficient and numerically showed that the system supports the existence of sustained and damped oscillations depending on initial population distributions, the disease transmission rates (constant as well as variable), and diffusion. The authors tried to understand the role of the variables as well as the constant transmission coefficient on the disease dynamics by considering different cases with emphasis on diffusion and initial population distribution. The following assumptions were made: (i) All newborns are susceptible and only susceptible populations are affected by the infectious populations. (ii) The mortality rate for infective individuals in the population is greater than the natural mortality rate. Since recovery does not give immunity, individuals move from the SEI class to the susceptible class upon recovery when the temporary immunity disappears. The model consists of the following equations:

(

)

(

β IS dS =− − µS + rN + δ R , dt N dI = σ E −  ( µ + α + γ ) I , dt

)

dE β IS = − ( µ + σ + κ ) E, dt N dR = κ E + γ I − µR − δ R, dt

(3.112)

where β is the transmission coefficient of the disease, µ the natural mortality rate, r the birth rate, σ −1 the incubation period, κ and γ the recovery rate for both exposed and infected populations, α the disease-induced mortality rate (α > µ ), δ −1 the loss of immunity period, and S + E + I + R = N. Introducing the nondimensional variables s=

S E I R ,  e = ,  i = ,  r1 = , N N N N

the nondimensional equations are obtained as (after some manipulations) (renaming the transformed variables again as S, E, I, and R) [237] dS = (α − β ) IS − rS + δ R + r = f1 ( S, E,  I , R ) , dt dE = β IS − (σ + κ + r ) E + α IE = f2 ( S, E,  I , R ) , dt

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Modeling Virus Dynamics in Time and Space

dI = σ E −  (α + γ + r ) I + α I 2 = f3 ( S,  E,  I ,  R ) , dt dR = κ E + γ I − ( r + δ ) R + α IR = f 4 ( S, E,  I , R ) , S + E + I + R = 1. dt

(3.113)

The system with diffusion is taken as

∂S ∂ 2S ∂E ∂ 2E = f1 ( S, E,  I , R ) + d1 2 , = f2 ( S,  E,  I ,  R ) + d2 2 , ∂t ∂ x ∂t ∂x ∂I ∂2I ∂R ∂2R = f3 ( S, E,  I , R ) + d3 2 , = f 4 ( S,  E,  I ,  R ) + d4 , ∂t ∂x ∂t ∂ x2 S + E + I + R = 1,

(3.114)

where d1 , d2 , d3 , and d4 are diffusion coefficients. The domain is taken as [ −2,  2 ] , and the boundary conditions are taken as

∂S ∂E ∂ I ∂R = = = = 0, at x = −2, and 2. ∂x ∂x ∂x ∂x Three types of initial conditions were considered. Disease-free equilibrium point: The model has DFE at (1, 0, 0, 0). The Jacobian matrix at the DFE (1, 0, 0, 0) is given by  −r   0 J=  0  0 

0

(α − β )

0

− (σ + κ + r )

β

0

σ

− (α + γ + r )

0

κ

γ

− (δ + r )

   .   

Trace ( J ) = − ( 4r + σ + κ + α + γ + δ ) < 0. The basic reproduction number is defined as 0 =

βσ

(α + γ + r ) (σ + κ + r )

.

det ( J ) = r (σ + κ + r ) (α + γ + r ) (δ + r ) ( 1 − 0 ) > 0, for 0 < 1.

(

)

Hence, the DFE (1, 0, 0, 0) is stable for 0 < 1. The EE point P* S* , E* ,  I * , R* exists (Problem 3.9, Exercise 3). The authors linearized the system about the equilibrium point P*. It was assumed that a Fourier series solution of form

∑ (∗) e

λt

tional matrix of the system is given by

cos ( kx ) exists for the linearized system. The varia-

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Spatial Dynamics and Pattern Formation in Biological Populations

˛ a − d k2 ˙ 11 1 ˙ a21 V=˙ a31 ˙ ˙ a41 ˝

a12

a13

a22 − d2 k 2

a23

a32

a33 − d3 k

a42

a43

a14 a24 2

a34 a44 − d4 k 2

ˆ ˘ ˘ ˘. ˘ ˘ ˇ

where a11 = (˜ − ° ) I * − r ,  a12 = 0,  a13 = (˜ − ° ) S* ,  a14 = ˛ , a21 = ˜ I * ,  a22 = −t1 + ° I * ,  a23 = ˜ S* + ° E* , a24 = 0, a31 = 0,  a32 = ˜ ,  a33 = −t2 + 2° I * ,  a34 = 0, a41 = 0,  a42 = ˜ ,  a43 = ° + ˛ R* , a44 = − ( r + ˝ ) + ˛ I * . The expression given here for a43 differs from the expression given by the authors as a43 = ˜ + ° r. The characteristic equation of V is given by ˜ 4 + p1˜ 3 + + p2 ˜ 2 + p3˜ + p4 = 0, where p1 ,  p2 ,  p3 , and   p4 are suitably defned. The Routh-Hurwitz criterion was applied to study the stability. The system is stable when [150] p1 > 0,   p2 > 0,  p3 > 0,   p4 > 0,  and   p1 p2 p3 − p32 − p12 p4 > 0. Three different types of initial conditions were considered. Detailed computations were made by the authors to study the effect of the transmission coeffcient ˜ , when (i) ˜ = a constant; (ii) ˜ = ˜ ( t ) ; and (iii) ˜ = ˜ ( x ) . Numerical studies were made using an operator splitting method with forward and central differences. The step lengths were taken as ˜x = 0.2, and ˜t = 0.025 days. The values of diffusion coeffcients were taken as d1 = 0.05, d2 = 0.025, d3 = 0.001, and d4 = 0.0. The authors concluded that in the absence of diffusion, oscillations leading to a steady state are produced with all the three types of initial conditions, when ˜ is taken to be a constant or a function of x. It indicates recurring of the disease initially and becoming steady over time. The infection is prevailing at a low level. When ˜ is a function of t, oscillations leading to limit cycle are produced with all the three types of initial conditions. This implies that the disease keeps recurring with the same intensity. Furthermore, in the absence of diffusion in the system, the disease always prevails. It prevails with lesser intensity when ˜ is taken to be a constant or a function of x. The disease keeps recurring with full intensity when ˜ is taken to be varying with time t. Diffusion in the system plays an important role in the spread of the disease along with the transmission coeffcient and initial distribution of the population. To explore the impact of vaccination as well as diffusion on the transmission dynamics of infuenza, Samsuzzoha et al. [238] developed a vaccinated diffusive compartmental epidemic model. The authors obtained the basic reproduction numbers with and without vaccination and investigated the sensitivity analysis of the reproduction numbers based on parameters involved in the system. The combined effect of the vaccine effcacy and vaccination rate was studied to determine the criteria for the control of the infuenza epidemic. It was shown that higher levels of vaccine effcacy and vaccination rate led to a decrease in the epidemic size. The authors also showed that an accurate estimation of the effciency

199

Modeling Virus Dynamics in Time and Space

of vaccines is necessary to control the spread of infuenza. The model is governed by the equations

˜S ˜ 2S ˜V ˜ 2V = f1 ( S, V ,  E,  I ,  R ) + d1 2 , = f2 ( S, V , E , I , ,R ) + d2 ˜t ˜ x ˜t ˜ x2 ˜E ˜ 2E ˜ I ˜2I = f3 ( S, V ,  E,  I ,  R ) + d3 2 , = f 4 ( S, V , E , I , R ) + d4 2  , ˜t ˜ x ˜t ˜x ˜R ˜2R = f 5 ( S, V ,  E,  I ,  R ) + d5 , ˜t ˜ x2

(3.115)

where V denotes the vaccinated population, and f1 ( S, V ,  E,  I ,  R ) = − ˜˜EES − ˜˜ I IS + ° IS − ˛S − rS + ˝ R + ˙ V + r, f2 ( S, V ,  E,  I ,  R ) = − ˜˜E ˜V EV − ˜˜ I ˜V IV + ° IV + ˛S − rV − ˝ V, f3 ( S, V ,  E,  I ,  R ) = ˜˜EES + ˜˜ I IS + ˜˜E ˜V EV + ˜˜ I ˜V IV + ° IE − t1E, f 4 ( S, V ,  E,  I ,  R ) = ˜ E − t2 I + ° I 2 , f  5 ( S, V ,  E,  I ,  R ) = ˛ E + ˝ I − ( r + ˙ ) R + ° IR , where t1 = ˜ + ° + r ,  t2 = ˛ + ˝ + r;  ˙ denotes the contact rate, ˜E  and   ˜ I represent the ability to cause infection by the exposed and infected individuals (0 ° ˜E ,  ˜ I ° 1) respectively, and ( 1 − ˜V ) is the factor by which the vaccine reduces infection (0 ° ˜V ° 1). The domain is taken as °−2,  2˝˙ , and the boundary conditions are taken as ˛

˜S ˜V ˜E ˜ I ˜ R = = = = = 0, at x = −2, and 2. ˜ x ˜t ˜ x ˜ x ˜ x

(3.116)

Three types of initial conditions were considered. Disease-free equilibrium: Denote t3 = r + ˜ + ° . The model (3.115) has DFE at E 0(S0 , V 0 , 0, 0, 0), where S0 = ( r + ˜ ) t3 , V 0 = ° t3 . The DFE is asymptotically stable under certain conditions (Problem 3.10, Exercise 3) Endemic equilibrium: P* S* , V * ,  E* , I * ,  R* . The model system (3.115) is linearized about the EE, to obtain small perturbations s ( x ,  t ) ,  v ( x ,  t ) ,  e ( x ,  t ) ,  i ( x ,  t ), and r ˛ ( x ,  t ) [239]. The linearized system is given by

(

)

˜s ˜2s = a11s + a12 v + a13 e + a14 i + a15 r ˝ + d1 2 , ˜t ˜x ˜v ˜2v = a21s + a22 v + a23 e + a24 i + a25 r ˝ + d2 2 , ˜t ˜x

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Spatial Dynamics and Pattern Formation in Biological Populations

∂e ∂2e = a31s + a32 v + a33 e + a34 i + a35 r ′ + d3 2 , ∂t ∂x ∂i ∂2i = a41s + a42 v + a43 e + a44 i + a45 r ′ + d4 2 , ∂t ∂x ∂ r′ ∂ 2 r′ = a51s + a52 v + a53 e + a54 i + a55 r ′ + d5 , ∂t ∂ x2

(3.117)

where a11 = − ( r + φ ) + (α − ββ I ) I * − ββE   E* ,  a12 = θ ,  a13 =   − ββE   S* , a14 = (α − ββ I ) S* , a15 = δ , a21 =  φ , a22 = − ( r + θ ) + (α − ββV β I ) I * − ββV β EE* ,

(

)

a23 =   − ββV βE  V * , a24 = (α − ββV β I ) V * , a31 = β βE   E* + β I I * ,

(

)

(

)

a32 = ββV βE   E* + β I I * ,   a33 = −t1 + α   I * + ββE S* + βV  V * , a34 = α   E* + ββ I (S* + βV  V * ), a43 = σ , a44 = −t2 + 2α   I * , a53 = κ , a54 = α   R* + γ ,  a55 = − ( r +  δ ) + α   I * . All the remaining coefficients are zero. Assume that a Fourier series solution exists for (3.117), of the form s ( x , t ) =

∑s e k

λt

cos ( kx ) ,  v ( x ,  t ) =

k

  i ( x , t ) =

∑i e k

λt

∑v e k

λt

cos ( kx ) , e ( x , t ) =

k

cos ( kx ) ,  r ′ ( x ,  t ) =

k

∑r e k

λt

∑e e k

λt

cos ( kx ) ,

k

cos ( kx ) ,

k

where k = ( nπ 2 ) , ( n = 1, 2, 3,…) is the wave number for the node n. By substituting the values of s,  v ,  e ,  i, and r′ in equation (3.117) and simplifying, the variational matrix for the transformed equation is obtained as  a11 − d1k 2   a21  ∗ a31 J =  0   0 

a12 a22 − d2 k a32

a13 2

a23 a33 − d3 k

2

a14

a15

a24

0 0

a34

0

a43

a44 − d4 k

0

a53

a54

2

The characteristic equation of the variational matrix J * is given by

λ 5 + p1λ 4 + p2 λ 3 + p3λ 2 + p4λ + p5 = 0,

0 a55 − d5 k 2

    .    

201

Modeling Virus Dynamics in Time and Space

where the expressions for p1 ,  p2 ,  p3 ,  p4, and p5 are given in the work of Samsuzzoha et  al. [236]. The Routh-Hurwitz criterion for stability requires pi > 0,  i = 1, 2, 3, 4, 5;  p1 p2 p3 − p32 − p12 p4 > 0 and (p 1 p4 − p5 )  p 1 p2   p3 − p32 − p12   p4 − p5

( p1   p2 −   p3 )

(

2

)

− p1   p52 = g ( k ) > 0.

The authors determined the frst excited mode of oscillations n, just by expressing g ( k ) in terms of the number of contacts ˜ , between susceptible and infected populations per unit time. They concluded the following: (i) For higher values of the bifurcation parameter ˜ , the spatial model system destabilizes as compared to the temporal system. (ii)  With an increase in the rate of vaccination, a higher value of ˜ destabilizes the system in the absence of diffusion. (iii) In the presence of diffusion, a lower value of ˜ destabilizes the system with the increase in the rate of vaccination. Therefore, the contact parameter ˜ plays the main role in the spread of disease. Its value must not exceed the bifurcation point to make the system unstable. (iv) Diffusion in the system helps in stabilizing the system, thus reducing the chances of an outbreak of disease beyond control. (v) Measure of vaccination effcacy is essential before the implementation of a mass vaccination program. Avian infuenza is a zoonotic disease caused by the transmission of the avian infuenza A virus, such as H5N1 and H7N9, from birds to humans and was frst reported in Hong Kong in 1997. The avian infuenza A H5N1 virus has caused more than 500 human infections worldwide with a nearly 60% death rate. The four outbreaks of the avian infuenza A H7N9 in China from March 2013 to June 2016 have resulted in 580 human cases including 202 deaths with a death rate of nearly 35%. Liu et al. [168] constructed two avian infuenza bird-to-human transmission models with different growth laws of the avian population, one with logistic growth and the other with the Allee effect, and analyzed their dynamical behavior. They obtained a threshold value for the transmission rate from infective avian to susceptible avian, for the prevalence of avian infuenza and discussed the local and global asymptotical stability of each equilibrium point of these systems. Zhang [309] studied the long-time behavior of a stochastic avian–human infuenza epidemic model with logistic growth for the avian population proposed by Liu et al. [168] for the deterministic case and studied how the noise affects the dynamics of the deterministic system. This model describes the transmission of avian infuenza among the avian population and the human population in random environments. Comparing the stochastic avian-human infuenza epidemic model with its corresponding deterministic model, the author has found a critical condition 0S < 0 , which means that environmental white noise is helpful for control of the disease.

Exercise 3 3.1

Derive the conditions for emerging cross-diffusion-induced Turing patterns for the spatial epidemic model (3.9). 3.2 Obtain the conditions for which the Hopf bifurcation or Turing bifurcation occurs in the epidemic model (3.10). Find the critical value of the bifurcation parameter ˜ . 3.3 Determine the conditions such that the endemic equilibrium point S* ,  I * of the model (3.36) is asymptotically stable.

(

)

202

Spatial Dynamics and Pattern Formation in Biological Populations

3.4 Obtain the endemic equilibrium points E1 ( S1 ,  I1 ) and E2 ( S2 , I 2 ) of the reduced model (3.70a) and (3.70b). 3.5 Derive the conditions under which Turing instabilities occur in the system (3.70a) and (3.70b). 3.6 Obtain the fnal size formula and the severity of an epidemic for the following model [29] dS dI = ˜ SI − ° I.  = − ˜ SI, dt dt 3.7 Find the basic reproduction number of the model (3.90). 3.8 Consider the variant of the model system (3.94), with R = 0. dS dE dI = µ − µS − ° IS ,  = ° IS − ( µ + ˛ ) E,  = ˛ E − ( µ + ˝ ) I . dt dt dt Find the equilibrium points and prove that the DFE E0 ( 1, 0, 0 ) is globally asymptotically stable in ˜ if 0 ˜ 1. 3.9 Show the existence of endemic equilibrium point P* S* ,  E* ,  I * , R* for the model (3.113). 3.10 Derive the conditions under which the DFE of the model (3.115) is asymptotically stable.

(

)

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4 Modeling the Epidemic Spread and Outbreak of Ebola Virus

4.1 Introduction The 2014-2016 Ebola Virus Disease (EVD) epidemic in West Africa was one of the largest ever recorded virus disease, which caused 28,646 cases and 11,323 deaths as of March 30, 2016 [99], representing a fundamental shift in Ebola epidemiology with unprecedented spatiotemporal complexity [34]. The outbreak was primarily in the contiguous countries of Guinea, Sierra Leone, and Liberia, which experienced widespread and intense transmission [99,100,101]. Interventions included quarantine, case isolation, additional treatment centers, border closures, lockdowns, and restricting travel within a region [102]. The outbreak has sparked an impetus in EVD modeling [29,77,83,104]. The spread between contiguous countries in the outbreak highlights the spatial element to its proliferation [99]. However, as previous EVD outbreaks were more localized than the 2014–2016 epidemic [80], there is little historical data on the geospatial spread of Ebola virus. Mobility data (which may help in the spatial EVD modeling) is limited although some studies have highlighted its usefulness and extrapolated based on the mobility data from other regions [51,98]. On May 14, 2018, WHO has declared the following: (i) Ebola virus has appeared in Democratic Republic of Congo, (ii) 39 cases of Ebola are reported between April 4 and May  13, 2018, and (iii) 393 people have been identifed who had contacts with the Ebola patients and they were being followed up to contain the virus [54]. 4.1.1 Source and Symptoms The main source of the Ebola virus is considered to be possibly fruit bats of the Pteropodidae family. It is believed that the virus is transmitted through monkeys, gorillas, and chimpanzees [59]. The disease is transmitted through a direct contact with an infected person or animal via the skin, the blood, or bodily fuids [25,73]. There is evidence to suggest that the population groups at a higher risk of infection include healthcare workers and relatives who may have come in contact with a patient and people physically involved in the burial process of an infected individual who has died from the disease [73]. It is worth noting that a recovered individual may not actually spread the virus. However, the Ebola virus has been found to remain in the semen for up to 3 months. Therefore, abstinence from sex with a recovered individual is recommended for at least 3 months [25]. Throughout its history, it has been observed that the Ebola virus and its strains cannot naturally transmit through the air, water, or food unlike infuenza or diarrheal diseases [59,73]. Furthermore, Ebola virus does not infect individuals during the incubation period that is, 2–21 days [59]. 215

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The common symptoms of EVD are fever, myalgia, malaise, sore throat, chest pain, red eyes, hiccups, rash, weakness, severe headache, joint and muscle pain, diarrhea, vomiting, stomach pain, dehydration, dry and hacking cough, and loss of appetite. These symptoms typically start from 2 days to 3 weeks after acquiring the EVD. As the infection spreads, the body undergoes severe blood loss and coagulation abnormalities. Ultimately, the liver, kidney, and microvascular endothelial cells (capillary walls) get infected, leading to the compromise of vascular integrity. If not diagnosed and treated, death usually occurs during the second week of symptoms and is usually due to massive blood loss [59]. Diagnosis of EVD is diffcult during the frst few days of the incubation period as the early symptoms are often seen in a number of other diseases such as malaria or typhoid. If an individual comes in contact with an infected person, he or she must be tested to confrm infection or not infected, using laboratory tests including antigen-capture enzyme-linked immunosorbent assay (ELISA) testing, IgM ELISA, polymerase chain reaction (PCR), and virus isolation. For infected individuals who are thought to be possible infection carriers, testing of IgM and IgG antibodies is done [25]. Good supportive clinical care and the infected individual’s immune response are the primary factors for Ebola recovery. Individuals who recover from EVD develop antibodies that last for at least 10 years [25], and they may still experience weakness, fatigue, headaches, hair loss, hepatitis, sensory changes, and infammation of organs [46,59]. 4.1.2 Transmission and Control of Epidemics Clinical progression of EVD includes two broad stages such as early and late infection [75]. In the frst stage, ~5–7 days, symptoms include fever, weakness, headache, muscle/joint pain, diarrhea, and nausea [24,75]. In some patients, the disease progresses to a second stage, with symptoms including hemorrhaging, neurological symptoms, tachypnea, hiccups, and anuria [21,75]. Mortality rates are higher among those exhibiting second-stage symptoms [21,75]. EVD is transmitted through a direct contact with an infected individual [38]. Transmission risk factors include contact with bodily fuids, close contact with a patient, needle reuse, and contact with cadavers, often prepared for burial by the family of the deceased [38,40,85]. Once an individual is infected by Ebola virus, his or her chances of recovery can be increased through pharmaceutical interventions that consist of providing intravenous fuid and balancing electrolytes, maintaining oxygen status and blood pressure, and treating other infections if they occur [27]. In the context of the spread of Ebola virus, the following works support the assumption that humans can also be infected through the contaminated environment: (i) Piercy et al. [78] established the survival of floviruses in liquids, surfaces, and glasses. (ii) Bibby et al. [17] demonstrated the persistence of Ebola virus in the environment. (iii) Environmental contamination was also evidenced by Youkee et al. [106]. (iv) Chowell and Nishiura [30] reported that human epidemics took off not only by direct contact via bodily fuids but also by indirect contact with contaminated surfaces. (v) It was observed by Francesconi et al. [45] that an individual contracted the Ebola virus in Uganda after using a blanket previously belonging to a positive case. Further, the consumption of contaminated bush meat in Africa may also contribute in the spread of Ebola virus [66,67]. Some of the Ebola outbreaks have lasted for more than 2 years, similar to the Western Africa outbreak. During this time, there might be new births or infow of susceptible individuals from other places as well as natural deaths, which allow a demographic process to take place, as studied in [3,57].

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Some of the strategies suggested for controlling the epidemic are given as follows [35,103]: i. Establishment of a committee to take charge of the overall coordination of operations and notifying the local, regional, national, and international partners about the epidemic. ii. Collaboration and work with the media. iii. Interruption of transmission routines through identifcation of sources of infection in the human population and prevention of renewed primary infection, active case fnding, contact tracing, monitoring of viral persistence, hospitalization or home care with quarantine, implementation of barrier nursing to protect healthcare workers, safe burials, and psychological assistance to patients and their families. iv. Establishment of social mobilization and health education program to listen and address public concerns. v. Assessment of the global health risk. Application of such control measures certainly lead to the damping of the actual EVD epidemic. In contrast, cultural beliefs can be a barrier to good implementation of control strategies in the affected countries [20]. There is a lot of literature on the Ebola epidemic viruses. In this chapter, we have included only fve models that we thought that they represent the issues suffciently. We do not claim to have covered the topic in totality.

4.2 Formulation of Ebola Epidemic Models After the outbreak of Ebola virus, many data-driven epidemic models for Ebola virus spread have been proposed. However, a few of them studied the spatiotemporal behavior of the disease at the global scale. Chowell et al. [29] used epidemic modeling and data from the two Ebola outbreaks in Congo and Uganda in 1995 and 2000, respectively, and estimated the reproduction number 0 in the absence of control interventions. They ftted the data to a simple deterministic (continuous time) SEIR epidemic model. To estimate the epidemic parameters, the least-square ftting was used. The ftted model was then used to estimate 0 and quantify the impact of intervention measures on the transmission rate of the disease. The ftted model was taken as an expected value of a Markov process, and multiple stochastic realizations of the epidemic were used to estimate a distribution for the fnal epidemic size. They had also studied the sensitivity of the fnal epidemic size to the timing of interventions and performed an uncertainty analysis on 0 to account for the high variability in disease-related parameters in the model. The transmission process is modeled by the following system of equations [7,18]:

˜ SI dE ˜ SI dI dR dS = ° I, =− , = − kE, = kE − ° I, dt dt dt N dt N

dC = kE, dt

(4.1a, b, c, d, e)

where S(t), E ( t ) , I ( t ) , and R ( t ) denote the number of susceptible, exposed, infectious, and removed individuals at time t. C ( t ) is not an epidemiological state but serves to keep track

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of the cumulative number of Ebola cases from the time of onset of symptoms. The fnal epidemic size is Z = C (T ), where T = min{t > 0, E ( t ) + I ( t ) = 0}, and its empirical distribution can be computed via Monte Carlo simulations [82]. N is the total effective population size, and ˜ is the transmission rate per person per day. Exposed individuals undergo an average incubation period (assumed asymptotic and uninfectious) of ( 1/k ) days before processing to the infectious class I. Infectious individuals move to the R-class (dead or recovered) at the per capita rate ˜ . The infectious period (1/˜ ) can be obtained by linearizing equations (4.1b) and (4.1c) about the DFE (N, 0, 0, 0) with S = N. N is the total effective population size. ˙ −k ˜ ˘ The corresponding Jacobian is given by J = ˇ  and the characteristic equation ˇˆ k −°  is ˜ 2 + ( k + ° ) ˜ + (° − ˛ ) k = 0, where the early-time and per capita free growth ˜ is essen2 tially the dominant eigenvalue. The eigenvalues are real when ( k + ˜ ) > 4 (˜ − ° ) . Solving the characteristic equation for ˜ in terms of other parameters, the expression for 0 is obtained as 0 =

˛2 + (k + ° )˛ ˜ = 1+ . ° k°

Walsh et al. [95] studied the Zaire strain of Ebola virus (ZEBOV) that has emerged repeatedly in human populations in central Africa and caused massive die-offs of gorillas and chimpanzees. ZEBOV outbreaks showed a distinct spatiotemporal pattern (wave like), both over the entire period since 1976 and during shorter time intervals. Their analysis suggests that ZEBOV has spread across the region rather than being long persistent at each outbreak locally. In another study, Chowell and Nishiura [30] carried out a comparative review of mathematical models of the spread and control of Ebola. It is crucial to collect spatiotemporal data on population behaviors, contact networks, social distancing measures, and education campaigns. Datasets comprising demographic, socioeconomic details, contact rates, and population mobility estimates in the region (commuting networks, air traffc, etc.) need to be integrated and made publicly available to develop highly resolved transmission models, which could guide control strategies with greater precision in the context of the EVD epidemic in West Africa [31]. To estimate the parameters from daily incidence and mortality time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995, Lekone and Finkenstädt [65] developed a stochastic discrete-time SEIR model and introduced Markov chain Monte Carlo methods for inference to explore the posterior distribution of the parameter and its estimation. Consider a time interval (t, t + h), where h represents the length between the time points at which measurements are taken. Normally, h is taken as 1 day. B(t) denotes the number of susceptible individuals who become infected, C(t) denotes the number of cases by date of symptom onset, and D(t) denotes the number of cases who are removed (die or recover) from the infectious class during that time interval. The binomial random variables B(t), C(t), and D(t) are assumed to be independent. Furthermore, ˜ * denotes the time point when the epidemic goes extinct, that is, the frst time point at which there is no exposed or infectious individuals in the population. S(t), E(t), I(t), and R(t) denote the number of susceptible, exposed, infectious, and removed individuals in the population at time t, respectively. Authors have used a discrete-time approximation to the stochastic continuous time SEIR model [47]. The discretized stochastic SEIR model is defned as

Ebola Virus: Modeling of Spread and Outbreak

S(t + h) = S(t ) − B (t ) , E (t + h) = E (t ) + B (t ) − C (t ) , I ( t + h ) = I ( t ) + C ( t ) − D ( t ) , S ( t ) + E ( t ) + I ( t ) + R ( t ) = N,

219

(4.2)

with initial conditions S ( 0 ) = s0 , E ( 0 ) = e0 , I ( 0 ) = a, and B ( t ) ~ binom ( S ( t ) , P ( t )) , C ( t ) ~ binom ( E ( t ) , pC ) , D ( t ) ~ binom ( I ( t ) , pR ) ,

(4.3)

are random variables with binomial (n, p) distributions with probabilities ˇ ˜ (t )   ) , pR = 1 − exp ( −° h ) . P ( t ) = 1 − exp  − hI ( t )  , pC = 1 − exp ( −nh ˘ N 

(4.4)

The parameters ˜ ( t ) , 1/n , and 1/˜ are time-dependent transmission rate, mean incubation period, and mean infectious periodic, respectively. The binomial distributions (4.3) result from summation over the individual Bernoulli trials assuming that they are independent and identical for all members of a compartment. Note that the compartment-specifc exponential rates ˙˝ ˜ ( t )/N ˇˆ I ( t ), n and γ for the susceptible, exposed, and infectious compartments, respectively, lead to the probabilities of staying in a compartment as specifed in (4.4) [72]. It follows that the exponential distribution of the incubation and the infectious period is approximated by the corresponding geometric distribution with means 1/pC and 1/pR, respectively. The population size N remains constant, and that individuals mix homogeneously. To account for the control intervention, the transmission parameter ˜ ( t ) is taken as a constant up to the time point when the control measures are introduced, and after that, it decays exponentially, that is ˜ ( t ) = ˜ , t < t* and ˜ e −q(t − t* ) , for t ˜ t* , where t* is the time point at which control measures are introduced. ˜ is the initial transmission rate, and q > 0 is the rate at which ˜ ( t ) decays for t > t* . Note that the intervention does not affect γ unless the disease is curable, which is not the case for Ebola. Chowell et al. [29] defned the time-dependent effective reproductive number as the number of secondary cases per infectious case at time t, that is, 0 ( t ) = ˜ ( t ) S ( t )/(° N ). Since S(t) ≈ N, it follows that 0 ( t ) ˙ ˜ ( t )/° is a function proportional to the time-varying transmission rate. The time point at which 0 ( t ) < 1, indicates when control measures have become effective in controlling the epidemic. The epidemic model specifed in (4.2)–(4.4) together with the contact rate ˜ ( t ) has parameter vector ˙ = {˜ , q , n , ° }, which can be estimated from the knowledge of initial conditions, population size, and observation of {B, C, D} or a subset thereof. The temporal evolution of the effective 0 ( t ) is then derived from the estimated parameters. A mathematical model for the spread of Ebola hemorrhagic fever epidemic taking into account transmission in different epidemiological settings (illness in the community, hospitalization, and traditional burial) was developed by Legrand et al. [64], where a relationship between hospitalization rate and epidemic size was considered. The following model was considered by the authors: dS ˙ ˜ SI ˜ SH ˜ F SF ˘ = −ˇ I + H + , ˆ N dt N N  dE ˙ ˜ I SI ˜ H SH ˜ F SF ˘ =ˇ + +  − ° E, dt ˆ N N N 

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Spatial Dynamics and Pattern Formation in Biological Populations

dI = ˜ E − (° h˛ 1 + ° i (1 − ˛ 1 ) ( 1 − ˝ 1 ) + ° d (1 − ˛ 1 )˝ 1 )I , dt dH = ˜ h° 1I − (˜ dh˛ 2 + ˜ ih (1 − ˛ 2 ))H , dt dF = ˜ d (1 − ° 1 )˛ 1I + ˜ dh˛ 2 H − ˜ f F, dt dR = ˜ i (1 − ° 1 ) ( 1 − ˛ 1 ) I + ˜ ih ( 1 − ˛ 2 ) H + ˜ f F. dt

(4.5)

S, E, I, H, F, and R are the number of susceptible, exposed, infectious, hospitalized, cases who are dead but not yet buried, and individuals removed from the chain of transmission respectively. ˜ I , ˜ H , and ˜ F are transmission rates in the community before interventions, at the hospitalization and during traditional burial, respectively. ˜ 1 is computed in order that ˜ % of infectious cases are hospitalized. ˜ 1 and ˜ 2 are computed in order that the overall case-fatality ratio is ˜ . The inverse of the mean duration of the incubation −1 period is ˜ . The mean duration from the symptom onset to hospitalization is ˜ h−1; ˜ dh is the −1 mean duration from hospitalization to death; and ˜ i is the mean duration of the infectious period for survivors. The mean duration from hospitalization to end of infectiousness for survivors is ˜ ih−1 , and ˜ −1 f is the mean duration of the infectious period between death and burial. Transmission coeffcients are expressed in terms of per week. Following the method described by van den Driessche and Watmough [93], the authors [64] determined the expression for 0 , which can be written as the sum of three terms relative to transmission in the community, during hospitalization, and during traditional burial. The expression was derived as 0 = 0I + 0H + 0F . =

˜1 ° h˛ 1˜ H ˝˜ + + F,  (° dh˝ 2 + ° ih (1 − ˝ 2 ))  ° f

(4.6)

˝ 1 ˝ 1 1ˇ 1ˇ −  , and where ˜ ih = 1/ ˆ −  , ˜ dh = 1/ ˆ ˙˜i ˜h˘ ˙˜d ˜h˘ ˆ = ˜ h° 1 + ˜ d (1 − ° 1 )˛ 1 + ˜ i (1 − ° 1 ) ( 1 − ˛ 1 ) . Ndanguza et al. [76] performed the statistical data analysis of the 1995 Ebola outbreak in the Democratic Republic of Congo using two sets of data (onset and death data) and showed that the model fts the observed onset Ebola data at 99.95% and the observed death data at 98.6%. Results obtained from both approaches were contrasted and compared. Fisman et al. [42] used “incidence decay with exponential adjustment” (IDEA) model to evaluate epidemic dynamics. This model describes epidemic processes both exponential growth and simultaneous decay, brought about by behavioral change, public health interventions, increased immunity in the population, or any other dynamic change that slows disease transmission. The IDEA model can be parameterized by ftting to either

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Ebola Virus: Modeling of Spread and Outbreak

incidence or cumulative incidence data and requires no assumptions regarding immune status in the population. It provides reasonably accurate projections about epidemic size and duration (in the absence of change in control efforts) based on pre-peak epidemic data when 0 is low or moderate. Numerical simulations suggest that the model can identify multiwave epidemics or abrupt changes in control, based on sudden changes in the value of the control parameter d (as described below) from generation to generation [42]. t Fisman et al. [43] used a simple, two-parameter mathematical model It = 0 (1 + d)t , where t is scaled in generation time, d a control parameter that causes incidence to decay, and It represents incident cases in a given generation. The model characterizes epidemic growth patterns in West Africa, which can be used to evaluate the degree to which the epidemic is being controlled and to assess the potential implications of growth patterns for epidemic size. In the absence of control, incident case counts grow to the power of  t. However, when control is present, the effective reproduction number is reduced by a power of t, causing transmission to slow and stop even when the absolute value of d is small. The best ft parameter values were estimated by ftting. The authors have identifed preferred parameter values as those that minimize the root mean-squared distance between model estimates and empirical data, but other approaches (e.g., Bayesian maximum likelihood approaches) are also possible. Camacho et al. [22] used the model given by Legrand et al. [64] to analyze the temporal dynamics of Ebola and suggested that changes in the behavior caused a signifcant reduction in both hospital-to-community and within-community transmission. Camacho et al. [22] depicted the potential for large EVD outbreaks by ftting a mathematical model to time series and estimating the epidemiological factors responsible for disease transmission. Authors assumed that the susceptible hosts in the community could become infected in three different ways at the rates given by the following: (i) ˜ i ( t ) , person-to-person transmission from an infectious host in the community; (ii) ˜ d ( t ) , from a dead but not buried patient during a traditional funeral ceremony; and (iii) ˜ h ( t ) , hospital transmission via syringe during outpatient visits. They assumed that individuals start off susceptible to infection (S). Upon infection they enter an incubation period (E) and then become symptomatic and infectious in the community (I). They had assumed that the latent and incubation periods are equivalent. After this point, they either enter a recovered state (R), remain infectious, and go into hospital (H) or die and remain infectious (D) until buried (B). The E compartment is split according to the route of transmission to keep track whether a case was infected via contaminated syringes at the hospital (Eh) or by personto-person contact (Epp) with either an infective in the community or a dead but not buried case. Following hospitalization, infectious hosts also move either into the recovered or into the dead compartment. As time went on, the population became very suspicious and did not touch the corpses anymore, not even to bury them [19]. Therefore, the authors used time-dependent smooth decreasing functions for ˜ i ( t ) , ˜ d ( t ), and ˜ h ( t ) as [29,65,76]:

(

(

)

(

)

˜ i ( t ) = ˜ i 1 − ° pp˛ ( t, ˝ pp ,tpp ) , ˜ d ( t ) = ˜ d 1 − ° pp˛ ( t, ˝ pp , tpp ) , ˜ h ( t ) = ˜ h ( 1 − ˛ ( t, ˝ h ,th )) t 0, E ( 0 ) > 0, I ( 0 ) > 0, R ( 0 ) > 0. All the parameters in the model system are positive. A is the recruitment rate of susceptible population, ˜ is the transmission rate, 1/˜ is an average time span of infected individuals in exposed class, ˜ is the rate at which infected individuals are recovered, µ is the natural death rate, and f is the fatality rate. Analysis of equilibrium points: The model has two equilibrium points. The diseasefree equilibrium (DFE) point is given by E0 = ( A/µ , 0, 0, 0 ) . The endemic equilibrium point E1 S* , E* , I * , R* is given by

(

)

I* =

A ( ˜° − t1t2 ) * A °˛ ( 1 − f ) + µ (˛ + ° + µ )  ( 1 − f )˛ I * , t , S = , E* = 1 I * , R* = ˜ ˛ f t t µ° ˜ ˛ ° µ − − f ( )12 ( )

227

Ebola Virus: Modeling of Spread and Outbreak

where t1 = (˜ + µ ) and t2 = (˜ + µ ) . Thus, unique endemic equilibrium exists if ˜° > t2t1 , ˜ > f° , and f < 1 hold (Problem 4.2, Exercise 4). Calculation of the basic reproduction number 0 : Following van den Driessche and Watmough [93], the basic reproduction number can be derived. Let x = ( E, I ). Then, from (4.13), we get ˙ ˜ SI dx ˇ = F − V , where F = ˇ N dt ˆˇ 0

˘ ˙ t2E   , V = ˇ −° E + t1I ˇˆ 

˛ 0 F1 = Jacobian of F at DFE = ˙ ˙˝ 0

˘ . 

˜ ˆ ˘, 0 ˘ˇ 0 ˇ . t1 ˘

˝ t2 V1 = Jacobian of V at DFE = ˆ ˆ˙ −˜

The next-generation matrix for the epidemic model is given by K = F1V1−1 =

1 t1t2

˝ ˜° ˆ ˆ˙ 0

˜ t2 0

ˇ . ˘

The basic reproduction number 0 is defned as the spectral radius of the next-generation matrix. The value of 0 is obtained as 0 = ˜° /( t1t2 ) . For the values of the parameters used for the simulation in the later part of this section: ˜ = 0.1751, µ = 0.001, ˛ = 0.08, ˝ = 0.27, f = 0.74, the authors obtained the value of 0 as 0 = 1.5143. Stability analysis: The model system (4.13) can be written in the form U = F (U, ˜ ) , where T U = ( S, E, I , R ) . The Jacobian matrix of the system is given by ° ˝ ˝ J=˝ ˝ ˝ ˝ ˛ where a11 = − µ − a21 =

a11 a21

a12 a22

a13 a23

a14 a24

a31 a41

a32 a42

a33 a43

a34 a44

˙ ˇ ˇ ˇ, ˇ ˇ ˇ ˆ

(4.14)

° I ( N − S) °S( N − I ) ° SI ° SI , a12 = 2 , a13 = − , a14 = 2 , N N N2 N2

˜ I ( N − S) ˜S( N − I ) ˜ SI ˜ SI , a22 = − 2 − t2 , a23 = , a24 = − 2 , N N N2 N2 a31 = 0, a32 = ˜ , a33 = −t1 , a34 = 0, a41 = 0, a42 = 0, a43 = ˜ ( 1 − f ) , a44 = − µ.

At the DFE, E0 = ( A/µ , 0, 0, 0 ) , the nonzero elements of the variational matrix denoted by J 0 are given by a11 = − µ , a13 = − ˜ , a22 = −t2 , a23 = ˜ , a32 = ˜ , a33 = −t1 , a43 = ˜ ( 1 − f ) , a44 = − µ.

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Spatial Dynamics and Pattern Formation in Biological Populations

2 The eigenvalues of J 0 are − µ , − µ , and − ( t1 + t2 )  Q  /2, where Q = ( t1 + t2 ) − 4t1t2 ( 1 − 0 ). If 0 < 1, then all the four eigenvalues of J 0 are negative or two of the eigenvalues are negative and two have negative real parts. If 0 > 1, then three eigenvalues of J 0 are negative and one eigenvalue is positive. Hence, DFE is locally asymptotically stable if 0 < 1 and unstable if 0 > 1. If 0 < 1, all the nearby trajectories of the system states, starting from various initial conditions approach E0 . To maintain disease-free situation, the basic reproduction number 0 must be made less than one. This can be achieved through (i) a decrease in the probability of disease transmission rate β or (ii) increase in natural death rate µ or (iii) an increase in the recovery rate γ by medication or control measures. For studying the local asymptotic stability of the endemic equilibrium E1 S* , E* , I * , R* , the authors used a method based on central manifold theory, taking β as a bifurcation parameter [23]. The authors have shown that at 0 = 1, DFE E0 = ( A/µ , 0, 0, 0 ) is a nonhyperbolic equilibrium point. One of the eigenvalues of J 0 is zero when det ( J 0 ) = µ 2t1t2 ( 1 − 0 ) = 0, that is, when 0 = 1. This gives β = β c = t1t2 /σ = (γ + µ ) ( µ + σ ) /σ . The other three eigenvalues are given by − µ , − µ , − ( t1 + t2 ) = − (γ + 2 µ + σ ) . The eigenvalues of J0 are a simple zero, and the other three are real and negative. Hence, for 0 = 1, that is for β = β c, the disease-free equilibrium E0 = ( A/µ , 0, 0, 0 ) is a nonhyperbolic equilibrium point. Now, denote ν = (ν 1 , ν 2 , ν 3 , ν 4 ), and T w = ( w1 , w2 , w3 , w4 ) as the left and the right eigenvectors associated with the zero eigenvalue, respectively, such that ν ⋅ w = 1. The components of the eigenvector w are given by

(

w1 = −

)

µ t1t2 t1µ , w3 = , w4 = 1. , w2 = (1 − f )γσ (1 − f )γ (1 − f )γσ

Also, the components of the eigenvector v are given by

ν 1 = 0, ν 2 =

(1 − f )γσ , ν µ ( t1 + t2 )

3

=

γ ( 1 − f ) t2 , ν 4 = 0. µ(t1 + t2 )

v and w satisfy the equalities v ⋅ J 0 = 0 and J 0 ⋅ w = 0. Note that fi denotes the right-hand side parts of model system (4.13). The authors used the following results from Castillo-Chavez and Song [23] Consider the system dx = f ( x,φ ) , f :  n ×  →  n , f ∈2  n ×  . dt

(

)

(4.15)

The equilibrium point 0 satisfies f ( 0, φ ) ≡ 0 for all φ . Theorem 4.1 (Castillo-Chavez and Song [23]) Assume the following:  ∂f 1. A = Dx f ( 0, 0 ) =  i ( 0, 0 ) is the linearized matrix of the system (4.15) around the  ∂ xj  equilibrium point 0 with parameter φ evaluated at 0. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts; 2. Matrix A has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue.

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Ebola Virus: Modeling of Spread and Outbreak

Let f k be the kth component of f and n

a=

˜

n

v k wi w j

k , i , j =1

° 2 fk ° 2 fk ( 0, 0 ) , b = vk wi ( 0, 0 ). ° xi ° x j ° xi ° ˛ k , i= 1

˜

Then, the local dynamics of system (4.15) around x = 0 is totally determined by a and b as follows: i. a > 0, b > 0: When ˜ < 0 with ˜  1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium. When 0 < ˜  1, 0 is unstable, and there exists a negative and locally asymptotically stable equilibrium. ii. a < 0, b < 0: When ˜ < 0 with ˜  1, 0 is unstable. When 0 < ˜  1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium. iii. a > 0, b < 0: When ˜ < 0 with ˜  1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium. When 0 < ˜  1, 0 is stable, and a positive unstable equilibrium appears. iv. a < 0, b > 0: When ˜ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable. Bifurcation analysis: For the above system, the coeffcients a and b are defned as 4

a=

˜

4

v k wi w j

k , i , j =1

° 2 fk ° 2 fk (E0 , ˛ c ) , b = vk wi (E0 , ˛ c ) , ° xi ° x j ° xi ° ˛ k , i= 1

˜

which can be explicitly computed. Here, x1 = S, x2 = E, x3 = I , x4 = R. a and b are obtained as a = 2 v2 w 2 w 3

˜ 2 f2 ˜2 f (E0 , ° c ) + v2 w32 22 (E0 , ° c ) , ˜ E˜ I ˜I

and b = v2 w3

Also,

˜ 2 f2 (E0 , ° c ). ˜I˜°

˜ 2 f2 ° ˜2 f 2 ° ˜ 2 f2 (E0 , ° c ) = − , 22 (E0 , ° c ) = − , (E0 , ° c ) = 1. S ˜I S ˜I˜° ˜ E ˜I

Substituting in b and a, we obtain b = ˜ /(t1 + t2 ) > 0, and ˆ 2˜  a = v2 w 3 ˘ − ( w2 + w3 ) < 0, since v2 > 0, w2 > 0, w3 > 0. ˇ S  Since a < 0 and b > 0, at ˜ = ˜ c, a transcritical bifurcation occurs at 0 = 1 (see Figure 4.1) and the unique endemic equilibrium is locally asymptotically stable for 0 > 1. This situation is not favorable for West Africa as the epidemic condition for Ebola will sustain for lifetime because the equilibrium E1 will act as a sink and will attract nearby solutions as t ˜ ° [92].

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 4.1 Bifurcation diagram for infected population as a function of R0 (Reproduced with permission from Upadhyay, R. K., Roy, P. 2016. Deciphering dynamics of recent epidemic spread and outbreak in West Africa: The case of Ebola virus. Int. J. Bif. Chaos 26(9), 1630024, 1–25, [92], and World Scientifc Publishing Company. Copyright 2016).

Bifurcation diagram of system (4.13) is presented in Figure 4.1. The values of the parameters used for simulation are A = 5.2, ˜ = 0.1751, µ = 0.001, ˛ = 0.08, ˝ = 0.27, f = 0.74. In the fgure, successive variations of the infected populations are taken in the range 0 < I ° 600 as a function of 0 ˛[ 0, 2 ]. When the reproduction number 0 < 1, the system remains in the disease-free state, and the endemic state sets in as 0 crosses the value 1. The diagram also displays that the model system experiences transcritical bifurcation at 0 = 1 and backward bifurcation (transcritical bifurcation in opposite direction) at some higher value of 0 . Bifurcation analysis displays very rich and complex dynamics, presenting various sequences of period-doubling bifurcations leading to chaotic dynamics and sequences of period-halving bifurcation leading to limit cycles. Sensitivity analysis: In order to determine how best, human morbidity and mortality can be reduced due to Ebola virus, it is necessary to know the relative importance of the different parameters responsible for its transmission and prevalence. Initially, the disease transmission and prevalence are directly related to the basic reproduction number 0 , and the endemic equilibrium point E1 is related to the magnitude of I * . It is important to calculate the sensitivity indices of the reproduction number 0 and the endemic equilibrium point E1 with respect to the parameters in the model. These indices indicate how crucial each parameter is for the disease transmission and prevalence. In order to fnd the sensitivity indices, the authors used a method similar to the methodology as suggested by Chitnis et al. [28]. Definition 4.1 [28] The normalized forward sensitivity index of a variable u that depends differentiably on a °u p . parameter p is defined as ˜ pu = °p u

Ebola Virus: Modeling of Spread and Outbreak

231

Sensitivity indices of 0 : The aim is to determine the uncertainty of 0 based on the uncertainty of the input. Since ˜ is a parameter in the model, ˜ is used in its place in the defnition of sensitivity index. Values of some of the parameters in the model may change on the basis of natural history of Ebola virus. The values of these parameters need a statistical analysis to examine the impact of their uncertainties. 0 = ˜° /( t1t2 ) depends on the parameters ˜ , ° , µ , and ˜ . We can derive an analytical expression for the sensitivity of 0 ° 0 p as ˜ p0 = for each parameter. ° p 0 We obtain the sensitivity indices as follows:

˜˛0 = ˜ µ0 =

° 0 ˝ µ ° 0 ˛ = 1, ˜˝0 = = , °˝ 0 t2 ° ˛ 0

µ ( t1 + t2 ) 0 ° 0 ˝ ˝ ° 0 µ =− , ˜˝ = =− . °˝ 0 ° µ 0 t1 t1t2

For the values of the parameters used for the simulation, ˜ = 0.1751, µ = 0.001, ˛ = 0.08 (see Althaus [5]), the following sensitivity indices are obtained:

˜˝0 = −0.994321, ˜ µ0 = −0.018024, ˜ˆ0 = 0.012345. As expected, the transmission rate ˜ is the most infuential parameter. The sensitivity indices with respect to the other three parameters depend on the numerical values chosen for these parameters. The most infuential parameter in determining the value of 0 is the transmission rate ˜ . Since ˜˝0 = +1, decreasing (or increasing) ˜ by 10%, decreases (or increases) the value of 0 by 10%. Similarly, the second most infuential parameter is ˜ . Variation of 0 as a function of (a) transmission rate ˜ and (b) recovery rate ˜ is plotted in Figure 4.2. Sensitivity indices of endemic equilibrium E1: Sensitivity analysis of the state variables at endemic steady state with respect to the six parameters ˜ , ° , µ , ˝ , f , A was also

FIGURE 4.2 Variation of R0 as a function of (a) transmission rate, β (b) recovery rate, γ. (Reproduced with permission from Upadhyay, R. K., Roy, P. 2016. Deciphering dynamics of recent epidemic spread and outbreak in West Africa: The case of Ebola virus. Int. J. Bif. Chaos 26(9), 1630024, 1–25, [92], and World Scientifc Publishing Company. Copyright 2016).

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Spatial Dynamics and Pattern Formation in Biological Populations

° Ej p , where E j are the state ° p Ej variables S* , E* , I * , R* , and p is one of the six parameters. For example, consider the sensitivity indices with respect to the parameter ˜ . The following values are obtained: E

performed. The sensitivity indices are defned as ˜ p j =

˜ I* A  t1t2 − ˛ f˝  °  ˜ I*  I* , , = ˙ =   ° I *  ˜ °  ˜ ° t1t2  ( ° − ˛ f )2  *

˜°S = −

° ( 1 − f )ˆ  ˙ I *  * * ° t1t2  ˙ I *  °t  ˙ I *  . , ˜°E = 1*  , ˜°R = *    µR* µ˝ S  ˙ °  ˝E  ˙°   ˙ ° 

For the values of the parameters used in the simulation, ˜ = 0.1751, µ = 0.001, ˛ = 0.08, ˜ = 0.27, f = 0.74, A = 5.2 (see Althaus [5]), the following results are obtained: S* = 1804.378819, E* = 41.92124915, I * = 19.04429263, R * = 867.0104663. *

*

*

*

˜˝I = ˜˝E = ˜˝R = 1.0217, ˜˝S = −1.92272. Similarly, sensitivity indices with respect to the other parameters can be obtained. The  indices are given in Table 4.1. Variation of I * as a function of (i) transmission rate ˜ , and (ii) recovery rate ˜ is plotted in Figure 4.3. Now, assume that the population performs active movements in x and y directions, which are biologically relevant. The factors contributing to the geographical spread of Ebola virus in West Africa are mainly (i) the movement of individual populations (including patients) who are not infected with Ebola virus, seeking and providing assistance in healthcare facilities, (ii) the movements of individuals taking care of patients infected with Ebola virus not admitted to hospital, and (iii) the population attending the traditional funeral ceremony and involving in social rituals. 4.3.1 Spatial SEIR Ebola Epidemic Model Merler et al. [71] noted that the movement and mixing of patients infected with Ebola virus and those not infected in hospitals at the early stage of the epidemic was a sufficient driver of the reported pattern of spatial spread. Hornsby [52] mentioned that a spreading TABLE 4.1 Sensitivity Indices of the State Variables of the Endemic Equilibrium with Respect to the Parameters Parameter p Β Γ Μ Σ A F

˜ pS* −1.92272 1.90194 −0.934046 −0.0451749 1 −1.73645

˜ pE*

˜ pI *

˜ pR*

1.0217

1.0217

1.0217

−1.01066 −0.0473925 −0.963649 1 0.922721

−2.00498 −0.0530711 0.0363509 1 0.922721

−1.00498 −1.05307 0.0363509 1 0.922721

Source: Reproduced with permission from Upadhyay, R. K., Roy, P. 2016. Deciphering dynamics of recent epidemic spread and outbreak in West Africa: The case of Ebola virus. Int. J. Bif. Chaos 26(9), 1630024, 1–25, [92], and World Scientifc Publishing Company. Copyright 2016.

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Ebola Virus: Modeling of Spread and Outbreak

FIGURE 4.3 Variation of R0 as a function of (a) transmission rate, β (b) recovery rate, γ. (Reproduced with permission from Upadhyay, R. K., Roy, P. 2016. Deciphering dynamics of recent epidemic spread and outbreak in West Africa: The case of Ebola virus. Int. J. Bif. Chaos 26(9), 1630024, 1–25, [92], and World Scientifc Publishing Company. Copyright 2016).

phenomenon through geographic space is considered in many diverse felds such as the spread of infectious diseases, growth of an urban center, the spread of forest fres, movement of currencies, and ripple effects in the natural world. Disease diffusion refers to the spread of a disease into new locations. It occurs when incident of a disease spreads out from an initial source. To represent the characteristics of the spread, Cliff [32] classifed spatial diffusion into four basic categories: (i) relocation, (ii) hierarchal, (iii) expansion, and (iv) contagious. Contagious diffusion is the spread of an infectious disease through the direct contact of individuals with those infected. The process is strongly infuenced by distance because nearby individuals or regions have a much higher probability of contact than remote individuals or regions. The spread of Ebola virus belongs to this category of diffusion, that is, human population is responsible for the spatial spread of the disease. In the proposed model, the movements are taken as random and uniformly distributed in all directions. The model (4.13) is modifed as [92]:

˜S = ˜t ˜E = ˜t ˜I = ˜t ˜R = ˜t

f1 ( S, E, I , R ) + D1ˆ 2S,

(4.16a)

f2 ( S, E, I , R ) + D2ˆ 2E,

(4.16b)

f3 ( S, E, I , R ) + D3ˆ 2 I,

(4.16c)

f 4 ( S, E, I , R ) + D4ˆ 2 R,

(4.16d)

where ˜ 2 is the Laplacian operator. The boundary conditions are taken as

( n ˝ ˙ ) S = ( n ˝ ˙ ) E = ( n ˝ ˙ ) I = ( n ˝ ˙ ) R = 0,

for ( x , y,t ) ˇ˜ ˘ × I,

and initial conditions are taken as S ( x , y , 0 ) = S0 ( x , y ) > 0, E ( x , y , 0 ) = E0 ( x , y ) > 0, I ( x , y , 0 ) = I 0 ( x , y ) > 0, R ( x , y , 0 ) = R0 ( x , y ) > 0.

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Spatial Dynamics and Pattern Formation in Biological Populations

The kinetic functions fi ( S, E, I , R ) , i = 1, 2, 3, 4 are as defined in (4.13) and (D1 , D2 , D3 , D4 ) are the diffusion coeffcients of susceptible, exposed, infected and recovered population, respectively. The initial data S0 ( x , y ) , E0 ( x , y ) , I 0 ( x , y ) , and R0 ( x , y ) is formed of nonnegative continuous bounded functions in the habitat ˜ °R 2 , which represents the domain in which populations inhabit. The homogeneous Neumann boundary conditions signify that the system is self-contained, and there is no population flux across the boundary ˜ °. The spatial model system (4.16) is linearized about the nontrivial interior equilibrium point E1 S* , E* , I * , R* , in order to obtain small perturbations s ( x , y , t ) , e ( x , y , t ) , i ( x , y , t ), and r ( x , y , t ) . The linearized system in two dimensions is obtained as follows:

(

)

˝ ˜2s ˜2s ˇ ˜s = a11s + a12 e + a13 i + a14 r + D1 ˆ 2 + 2  , ˜t ˜y ˘ ˙ ˜x ˝ ˜2e ˜2e ˇ ˜e = a21s + a22 e + a23 i + a24 r + D2 ˆ 2 + 2  , ˜t ˜y ˘ ˙ ˜x ˝ ˜2i ˜2i ˇ ˜i = a31s + a32 e + a33 i + a34 r + D3 ˆ 2 + 2  , ˜t ˜y ˘ ˙ ˜x ˝ ˜2r ˜2r ˇ ˜r = a41s + a42 e + a43 i + a44 r + D4 ˆ 2 + 2  . ˜t ˜y ˘ ˙ ˜x

(4.17)

The values of the coeffcients are given in (4.14). Assume that a Fourier series solution exists for (4.17), which is of the form s( x , y , t) =

˜s exp ( ° t + i ( k u + k v)) , e(x, y, t) = ˜e exp ( ° t + i ( k u + k v)) , k

k

u

v

k

k

i( x , y , t) =

k

u

v

k

˜i exp ( ° t + i ( k u + k v)) , r ( x, y, t ) = ˜r exp ( ° t + i ( k u + k v)) , k

k

u

v

k

k

k

u

v

k

where ku and k v are the components of wave number k = ku2 + k v2 along x and y directions, respectively, and ˜ k is the wave length. Substituting the values of s, e , i, and r in (4.17), the variational matrix for the transformed equation is obtained as ˛ a11 − D1k 2 ˙ ˙ a21 * J =˙ a31 ˙ ˙ a41 ˝

a12 a22 − D2 k

a13 2

a14

a23

a32

a33 − D3 k

a42

a43

a24 2

a34 a44 − D4 k 2

The characteristic equation of the variational matrix J * is given by

˜ 4 + °1˜ 3 + °2 ˜ 2 + °3 ˜ + ° 4 = 0,

ˆ ˘ ˘ ˘. ˘ ˘ ˇ

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Ebola Virus: Modeling of Spread and Outbreak

where the expressions for ˜1 , ˜2 , ˜3, and ˜ 4 are as given in Upadhyay and Roy [92]. Routh-Hurwitz criterion for stability requires ˜i > 0, i = 1, 2, 3, 4 and ˜1 ˜2 ˜3 − ˜32 − ˜12 ˜ 4 > 0. For the values of the parameters satisfying these conditions, the endemic equilibrium point  E1 S* , E* , I * , R* is asymptotically stable. Turing instability occurs when Im ( ˜ ( k )) = 0, Re ( ˜ ( k )) = 0 at k = kT ° 0. Hence, at least one of the following inequalities is to be violated for some k > 0

(

)

( ) ( ) ( ) ( ) ˜ ( k ) ˜ ( k ) ˜ ( k ) − ˜ ( k ) − ˜ ( k ) ˜ ( k ) > 0. ˜1 k 2 > 0, ˜2 k 2 > 0, ˜3 k 2 > 0, ˜ 4 k 2 > 0, 1

2

2

2

3

2

2 3

2

2 1

2

4

2

It is very difficult to find the restrictions on the parameters analytically. Simulations were done to study the Turing instability. Consider the data that was used earlier: A = 5.2, ˜ = 2.7, µ = 0.001, f = 0.74, ˛ = 0.1751, and ˝ = 0.08. The coeffcients in the dispersion relation satisfy the inequalities:

˜1 ( 0 ) ˝ 0.261 > 0, ˜2 ( 0 ) ˝ 0.001 > 0, ˜3 ( 0 ) ˝ 0.000015 > 0, ˜ 4 ( 0 ) ˝ 1.4 × 10−8 > 0, ˜1 ( 0 ) ˜2 ( 0 ) ˜3 ( 0 ) − ˜32 ( 0 ) − ˜12 ( 0 ) ˜ 4 ( 0 ) ˙ 2.67 × 10−9 > 0.

(

)

Therefore, E1 S* , E* , I * , R* is temporally stable. Now, consider the presence of diffusion, with diffusion coeffcients taken as D1 = 5, D2 = 0.0005, D3 = 10−5 , D4 = 5. The system may eventually go to a nonconstant positive steady state. The model system in one dimension was solved numerically using the central fnite differences in space and a semiimplicit time stepping method. Space-time versus population densities are plotted to exhibit the spatiotemporal dynamics of the model system. The parameter values used are the following: ˜ = 0.27, ° = 0.1751, µ = 0.001, ˝ = 0.08, f = 0.74, A = 5.2 with D1 = 15, D2 = 0.0005, D3 = 10−5 , D4 = 5. Spatiotemporal patterns in (i) exposed population, (ii) infected population: (a) without control measure and (b) with control measure (m = 0.003), are plotted in Figure 4.4. (The authors [92] presented the plots for m = 0.0023.) From Figure 4.4a, it can be observed that the spatial distribution of infected and exposed population has erratic density distribution in the system without control measure. In such system, both infected and exposed classes experience a spatiotemporal chaos in the initial stage. In contrast, in the system with control measures, exposed and infected density varies regularly in both space and time direction. After control measures are introduced, the transmission rate ˜ ( t ) = ˜ e m(t −˙ ) is assumed to decay exponentially at rate m and is the time at which interventions start [5,65]. The simulation study of the model system reveals that control measures can effectively reduce the spatiotemporal complexity and shows how biological processes can afect spatiotemporal pattern formation and propagation of waves. System in two dimensions: The Ebola virus began its journey in Guinea in December 2013 and then spread to Sierra Leone and Liberia. Considering this aspect, the authors employed the initial conditions (by assuming that they include scattered groups of infective population found at three different locations in West Africa) as follows (see Hu et al. [55]): S ( x , y , 0 ) = S* , E ( x , y , 0 ) = E* ,

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 4.4 Spatiotemporal patterns in (i) exposed population, (ii) infected population: (a) without control measure and (b) with control measure (m = 0.003).

ˇ   I* ,  I ( x, y , 0) = ˘    0.0001,  ˇ    R* , R ( x, y , 0) = ˘    0, 

if ( x − 100 ) + ( y − 50 ) < 200, 2

2

( x − 50 )2 + ( y − 100)

2

( x − 150 )2 + ( y − 100 )

< 200,

2

< 200,

2

< 200,

otherwise. if

( x − 100 )2 + ( y − 50 ) ( x − 50 )

2

( x − 150 )

2

+ ( y − 100 ) < 200, 2

+ ( y − 100 ) < 200,

otherwise.

2

Ebola Virus: Modeling of Spread and Outbreak

237

which indicate the uniform distribution of susceptible and exposed classes throughout the domain and availability of infective and recovered population at three distinct places (Guinea, Sierra Leone, and Liberia) and very less at other locations. The model system was solved numerically in 2D spatial domain   ˛ = ˆˇ 0, 200 ]×[ 0, 200 ˘, with time and space steps taken as t = 0.001, h = 0.25, respectively. In the numerical simulations, diferent types of Turing patterns are observed and attributed to a large variety of symmetry properties realized by different values of diffusion coeffcients. Snapshots of pattern formations for the time evolution of different classes of population at time t = 200 days are given in Figure 4.5. From the fgure, it is concluded that starting with a homogeneous state E1 = (1804.38, 41.9212,19.0443, 867.01), initial perturbation leads to the formation of circular patterns in all classes except for the recovered population. In this case, it was observed that initially (approximately at the end of March 2014), the infective and exposed populations are concentrated at three main locations (Guinea, Liberia, and Sierra Leone). As the time level is increased from t = 10 days to t = 100 days, it is observed that the maximum density of susceptible and recovered population is moving to the left of the domain [92]. It is also observed that density of infective and exposed populations increases with time. For the time level at t = 200 days, recovered population is now observed to be shifted toward the right of the domain and the infective and exposed population density increases with respect to the initial cases (at t = 10 days). The simulation results match with the observations made by the UN’s Ebola coordinator Nabarro, who told Agnese France-Presse on January 29, 2015, that “The number of cases is decreasing week by week and getting to

FIGURE 4.5 Snapshots of pattern formations for the time evolution of different classes of population at time t = 200 days.

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Spatial Dynamics and Pattern Formation in Biological Populations

zero in many places … but we still see occasional fare ups and we still see some surprises with new cases out of our contact lists.” According to the WHO fgures released in Geneva, 99 new cases were confrmed in the week up to January 2015, the frst time the fgure has dropped below 100 since the end of June 2014. The three nations have been devastated by the outbreak that began in December 2013, but all have seen recent signs that the virus is on the wane, with the number of new cases dropping weekly. The simulated results and other recent published works suggest that the worst outbreak of the dreaded disease may be coming under control (The authors [92] presented the snapshots of pattern formation for the time evolution of diferent classes of population, for t = 50, 100, and 300.)

4.4 Model 2: Ebola Epidemic SEIRHD Model Roy and Upadhyay [86] modifed the model given in Section 4.3 by considering three different transmissions routes and called it an SEIRHD Ebola epidemic model. The following assumptions are made: i. The population is interacting homogeneously and well mixing [9,71,90]. ii. Under high risk, the total human population (N) is divided into six mutually exclusive classes, namely a susceptible class of size S (those at risk of contracting the disease), exposed class of size E (infected but not yet infectious), infected class of size I (showing symptoms of Ebola and capable of transmitting the disease), recovered class of size R (infectious people who have cleared or recovered from Ebola), hospitalized class of size H and dead population of size D. iii. The susceptible population increases through the recruitment by new sexually active individual, migration, and decreases due to direct contact with infected individuals and natural death. iv It is also assumed that the susceptible host in the community could become infected in three different ways: (a) disease can be transmitted by contact between an infected and a susceptible individual in the community at a rate ˜ i ( t ), (b) hospital transmission via syringe or visiting loved ones in healthcare center at a rate ˜ h ( t ), and (c) from a dead individual who is not properly buried at a rate ˜ d ( t ). v. The exposed period begins when Ebola virus enters into the healthy individual. The exposed individual then starts to show symptoms of Ebola infection and moves to class I. The infective class is decreased through recovery from infection, hospitalization, and by natural death. The intervention strategies to control the spread of Ebola includes surveillance, removal of infected individuals from the general population, and place them in a setting that can provide both isolation and dedicated care, awareness about the disease, use of strict barrier nursing techniques and rapid burial of patients who die from the disease. As time passed, population became more aware, and the following intervention strategies were practiced: (i) avoiding healthcare facilities in West Africa where Ebola people are being treated, (ii)  avoiding funeral or burial rituals that require handling the body of someone who has died from Ebola, and (iii) avoiding mixing with the people having the

239

Ebola Virus: Modeling of Spread and Outbreak

symptoms of Ebola. Keeping these issues in mind, the transmission rates were assumed as ˜ i ( t ) = ˜1e −m1 (t −˙ ), ˜ h ( t ) = ˜ 2 e −m2 (t −˙ ) , and ˜ d ( t ) = ˜ 3 e −m3 (t −˙ ), which decay exponentially at rates m1 , m2 , m3 respectively, and τ is the time at which interventions start [5,65]. The factors that contributed to the geographical spread of Ebola in Sierra Leone are the same as discussed in Ebola epidemic model 1. In the proposed model, diffusion terms are included to take care of random movements that are uniformly distributed in all directions. The transmission process of Ebola virus is governed by the following set of nonlinear partial differential equations [86].

˜S = f1 ( S, E, I , R , H , D ) + D1ˆ 2S, ˜t

(4.18a)

˜E = f2 ( S, E, I , R , H , D ) + D2ˆ 2E, ˜t

(4.18b)

˜I = f3 ( S, E, I , R , H , D ) + D3ˆ 2 I, ˜t

(4.18c)

˜R = f 4 ( S, E, I , R , H , D ) + D4ˆ 2 R, ˜t

(4.18d)

˜H = f5 ( S, E, I , R , H , D ) + D5ˆ 2 H, ˜t

(4.18e)

˜D = f6 ( S, E, I , R , H , D ) + D6ˆ 2 D, ˜t

(4.18f)

with boundary conditions

( n ˛ ˝ ) S = ( n ˛ ˝ ) E = ( n ˛ ˝ ) I = ( n ˛ ˝ ) R = ( n ˛ ˝ ) H = ( n ˛ ˝ ) D = 0,

for ( x , y ,t ) ˆˇ˘ × I,

and initial conditions S ( x , y , 0 ) = S0 ( x , y ) > 0, E ( x , y , 0 ) = E0 ( x , y ) > 0, I ( x , y , 0 ) = I 0 ( x , y ) > 0, R ( x , y , 0 ) = R0 ( x , y ) > 0, H ( x , y , 0 ) = H 0 ( x , y ) > 0, D ( x , y , 0 ) = D0 ( x , y ) > 0. The kinetic functions fi ( S, E, I , R , H , D ) ; i = 1, 2, 3, 4, 5,6 are defined as f 1 ( S, E , I , R , H , D ) = A − f2 ( S, E, I , R , H , D ) =

S ( ˜ i I + ˜ h H + ˜ d D) − µS, N

S ( ˜ i I + ˜ h H + ˜ d D) − (° + µ ) E, N

f3 ( S, E, I , R , H , D ) = ˜ E − (° + ° h + µ ) I, f 4 ( S, E, I , R , H , D ) = ( 1 − f )˜ I + r1H − µR, f5 ( S, E, I , R , H , D ) = ˜ h I − ( d1 + r1 ) H , f6 ( S, E, I , R , H , D ) = d1H + f˜ I − ° D,

240

Spatial Dynamics and Pattern Formation in Biological Populations

where ˜ i , ˜ h , ˜ d are functions of t, and N = S + E + I + R + H + D. Analysis of equilibrium points: The model has two equilibrium points: i. Disease-free equilibrium (DFE) point: E0 ( A/µ , 0, 0, 0, 0, 0 ) . ii. Endemic equilibrium point is given by E1 S* , E* , I * , R* , H * , D* , where (Problem 4.3, Exercise 4)

(

)

E* = ( t1/˜ ) I * , t1 = ° + ° h + µ. H * = t2 I * , t2 =

t3 =

°h . D* = t3 I * , d1 + r1

(

)

1 1 d1t2 + f° ) , R * = t4 I * , t4 = ( (1 − f )° + r1t2 , ˜ µ

S* = t7 I * , t7 =

(˜ + µ ) t1t6 , I * = A (t6 + t7 ) , ˜ t5 − (˜ + µ ) t1 µt7 ( t6 + t7 ) + t5t7

t5 = ˜ i + ˜ ht2 + ˜ dt3 , t6 = 1 + ( t1/° ) + t2 + t3 + t4 . Let x = ( E, I , H , D ) . Then, from the above Ebola epidemic model, it follows:   (µ + ˛ )E    −˛ E + t1I   , and V =   −˝ h I + ( d1 + r1 ) H   − d H + f˝ I + ˙ D  (1 )  

 S ( ˜i I + ˜ h H + ˜d D)  N  dx = F − V, where F =  0 dt  0  0 

   .   

We obtain ˛ ˙ F1 = Jacobian of F at DFE = ˙ ˙ ˙ ˝

0 0 0 0

˘ µ +°   −° V1 = Jacobian of V at DFE =  0  0  

˜i 0 0 0

˜h 0 0 0

˜d 0 0 0

0 t1 −˛ h

0 0 d1 + r1

− f˛

−d1

ˆ ˘ ˘, ˘ ˘ ˇ    .  ˝   0 0 0

The basic reproduction number, 0 , is defned as the spectral radius of the next-generation matrix K = F1V1−1 . The basic reproduction number is obtained as 0 =

˜ ° ( ˝ i + ˝ ht2 ) + ˝ d ( d1t2 + f˙ ) . ° t1 ( µ + ˜ )

(

)

241

Ebola Virus: Modeling of Spread and Outbreak

For the values of the parameters used for the simulation in the later part of this section A = 555, ˜ = 0.8783, µ = 0.00004, ˜ h = 0.5, ˛ = 0.06, ˝ = 0.5, ˙ i = 0.48, ˙ h = 0.295, d1 = 0.5, ˙ d = 0.75, r1 = 0.5366, 0 was computed as 0 = 1.1815, [86]. Both DFE and endemic equilibria exist with E0 = 1.3875 × 107 , 0, 0, 0, 0, 0 and E1 = ( 1.02497 × 107 , 2415.25,105.137,1.85767 × 106 , 50.7125,141.208 ) . For ˜ = 1.8783, with other parameters having the same values, 0 was computed as 0 = 0.993561. In this case, only DFE E0 = 1.3875 × 107 , 0, 0, 0, 0, 0 exists and no endemic equilibrium exists. The model with constant transmission rate has no endemic equilibrium if 0 ˜ 1, and the unique endemic equilibrium exists if 0 > 1. Stability analysis: The DFE of the model system (4.18) is locally asymptotically stable. Numerically, it was shown that the endemic equilibrium point E1 = ( 1.02497 × 107 , 2415.25, 105.137,1.85767 × 106 , 50.7125,141.208 ) is asymptotically stable (Problem 4.4, Exercise 4).

(

)

(

)

4.4.1 Sensitivity Indices of 0 The most infuential parameters in determining the value of 0 are the transmission rate of the dead ˜ d and the burial rate of dead ˜ . The authors obtained the sensitivity indices as the following:

˜˛d0 =

˛ d ( f˝ + t2 d1 ) ° 0 ˛ d = , ° ˛ d 0 ˛ d ( f˝ + t2 d1 ) + ˙ ( ˛ i + t2 ˛ h )

˜˛0 =

˝ d ( f˙ + t2 d1 ) ° 0 ˛ =− . °˛ 0 ˝ d ( f˙ + t2 d1 ) + ˛ ( ˝ i + t2 ˝ h )

For the values of the parameters used for the simulation, ˜ = 0.1783, ˜ h = 0.5, ° = 0.5, ˜ i = 0.48, ˜ h = 0.295, d1 = 0.5, ˜ d = 0.75, r1 = 0.5366, f = 0.49, it was found that ˜˛d0 ° 0.4419, ˜˝0 ° −0.4419. This implies that if ˜ d is increased (decreased) by 10%, then the value of 0 is increased (decreased) by 4.419%. If ˜ is increased (decreased) by 10%, then the value of 0 is decreased (increased) by 4.419%. Sensitivity indices with respect to the other parameters depend on the numerical values chosen for these parameters. After the person dies, the bodily contact continues to spread infection as people come by to express their grief and offer condolences. Sensitivity analysis of the temporal model system (4.18) suggests that control efforts should be focused on avoiding funeral or burial rituals that require handling the body of someone who has died from Ebola. Therefore, it is necessary to explain the reasoning behind safe burial practices and get the local population on safer side. These fndings underscore the importance of rapid contact tracing and quarantine of symptomatic individuals. The analysis also points the need for more hospitalization and to make sure every infected individual in that zone is quickly isolated and receives professional care. System in two dimensions: The authors employed an initial condition by assuming that groups of infective populations are mainly found in Sierra Leone in West Africa. Since the authors were mainly concerned with the spread dynamics at Sierra Leone, they considered a circular domain within a rectangular domain representing this country surrounded by neighboring countries. The following initial conditions were considered [55,86]: S ( x , y , 0 ) = S* , E ( x , y , 0 ) = E* ,

242

Spatial Dynamics and Pattern Formation in Biological Populations

ˇ I* ,  I ( x, y , 0) = ˘  0.0001,

if ( x − 100 ) + ( y − 100 ) < 400,

ˇ  R* , R ( x, y , 0) = ˘  0.001,

if ( x − 100 ) + ( y − 100 ) < 400,

ˇ  H* , H ( x, y , 0) = ˘  0.001,

if ( x − 100 ) + ( y − 100 ) < 400,

ˇ  D* , D ( x, y , 0) = ˘  0.001,

if ( x − 100 ) + ( y − 100 ) < 400,

2

2

otherwise, 2

2

otherwise, 2

2

otherwise, 2

2

otherwise,

which indicate the uniform distribution of susceptible and exposed classes throughout the domain and concentrated availability of all other classes at Sierra Leone and very less at other locations. The model system was solved numerically in a 2D spatial domain, ˛ = [ 0, 200 ] × [0,200], with time and space steps taken as t = 0.001, h = 0.25, respectively. Numerical simulations were done using the parameter values A = 555, ˜ = 0.1783, µ = 0.00004, ˛ = 0.06, ˝ = 0.5, ˙1 = 0.48, ˙ 2 = 0.295, d1 = 0.5, ˙ 3 = 0.75, r1 = 0.5366, and f = 0.49. The values of the diffusion coeffcients were taken as D1 = 15, D2 = 0.5, D3 = 10−5 , D4 = 15, D5 = 0, D6 = 0. The endemic equilibrium point is obtained as E1 6.17891 × 106 , 5127.31, 453.516, 3.96555 × 106 , 218.752, 297.997 . Spatiotemporal patterns in (i) infected population and (ii) dead population: (a) without control measure and (b) with control measure (m1 = 0.0085, m2 = 0.009, m3 = 0.0012) are plotted in Figure 4.6. The spatiotemporal patterns refect the effect of control measures taken at both spatial and temporal scales. Diferent types of spatiotemporal patterns are observed and attributed to a large variety of symmetry properties realized by different values of diffusion coeffcients. From Figure 4.6a, it was observed that the spatial distributions of infected and dead populations have erratic density distribution in the system without control measures. In such system, both infected and dead classes experience a spatiotemporal chaos in the initial stage. In contrast, in the system with control measures, infected and dead population densities vary regularly in both space and time. Therefore, control measures effectively reduce the spatiotemporal complexity, and the simulation exhibits how biological processes affect spatiotemporal pattern formation and propagation of waves. Controlling the spread of the disease will also require stronger district surveillance and epidemiology, contact tracing, and burial teams. Nonessential travel must be avoided to control the outbreak. The spread can be prevented in the following two ways: (i) protecting people who may be planning to travel to the affected areas and (ii) enabling the government to respond most effectively to restrict this outbreak. It was found that the outbreaks were advancing in a wavy pattern. Hence, the authorities have to put in efforts to get ahead of the wave with safe and reliable interventions. A reaction-diffusion model of this type that takes into account transmission in different social contexts with control measures may provide useful estimates of infection spread.

(

)

Ebola Virus: Modeling of Spread and Outbreak

243

FIGURE 4.6 Spatiotemporal patterns in (i) infected population and (ii) dead population: (a) without control measure and (b)  with control measure (m1 = 0.0085, m2 = 0.009, m3 = 0.0012). (Reproduced with permission from Roy, P., Upadhyay, R. K. 2017. Spatiotemporal transmission dynamics of recent Ebola outbreak in Sierra Leone, West Africa: Impact of control measures. J. Biol. Sys. 25(3), 369–397, [86]; and World Scientifc Publishing Company. Copyright 2017).

4.5 Model 3: Ebola Epidemic SEIORD Model and Its Extension Djiomba Njankou and Nyabadza [35] studied an optimal control [79] model for Ebola virus disease by considering three control measures, namely educational campaigns, active casefnding, and pharmaceutical interventions. The dynamics of Ebola disease is described by a seven-compartment deterministic model (susceptible S, exposed E, infected asymptomatic I a , infected symptomatic I s, hospitalized O, recovered R, and dead individuals D). The total population involved in the EVD transmission is given by, N = S + E + I a + I s + O + R + D. Three control strategies were implemented to limit the number of individuals entering the

244

Spatial Dynamics and Pattern Formation in Biological Populations

exposed and dead classes. Some individuals may be EVD seropositive but never develop symptoms because of a low viral load [12]. Dead bodies are highly infectious and thus contribute to the force of EVD infection together with the fuids of infectious individuals [83]. The dynamics of EVD within the population is described by the following system of equations [35]: dS dE dI = ˜ − ( ° + µ ) S, = ° S − (˝ + µ ) E, a = p˝ E − ( µ + ˙ + ˆ 1 ) I a , dt dt dt dI s = ( 1 − p )˜ E + ° I a − (˛ 1 + ˝ + µ + ˆ 2 ) I s , dt dR dO = ˜ I s − ( µ + ˛ 3 + ˝ 2 ) O, = ˛ 1I a + ˛ 2 I s + ˛ 3O − µR, dt dt dD = ˜ 1I s + ˜ 2O − ° D. dt

(4.19)

The parameters are defned as follows: ˜ : rate at which susceptible individuals enter the heterogeneous population. ˜ : rate at which exposed individuals develop symptoms after 2–21 days [74]. p: proportion of infected asymptomatic individuals in compartment I a . (1 − p): proportion of infected symptomatic individuals in compartment I s [ 68 ]. ˜ : rate at which asymptomatic individuals develop symptoms later [50]. ˜ : per capita rate at which infected individuals are isolated and treated in a quarantined in a hospital [83]. ˜ 1 : rate at which asymptomatic EVD seropositive individuals can recover. ˜ 2 , ˜ 3 : rate at which symptomatic and hospitalized individuals can either recover. ˜ 1 , ˜ 2 : rate at which symptomatic and hospitalized individuals die. µ : rate at which individuals die naturally. ˜ : rate at which dead bodies are disposed. Force of infection was assumed as ˜ ( t ) = c° ( I s ( t ) + ˛D ( t ))/N ( t ), ˜ > 1. All the model parameters are positive. The equations are solved under homogeneous Neumann boundary conditions, and the initial conditions are given by prescribed non-negative values. Denote Q1 = (˜ + µ ) , Q2 = ( µ + ˛ + ˝ 1 ) ,Q3 = (˙ 1 + ˆ + µ + ˝ 2 ) Q4 = ( µ + ˝ 3 + ˙ 2 ) ,

˜1 =

˙ ( 1 − p ) + ˆ˜ 3 p˙ ° ˛ + ˛ 2˜ 1 1 ˇ + ˇ 3˜ 1 + ˇ 1˜ 3˜ 5 ,˜ 2 = 1 ,˜ 3 = ,˜ 4 = ,˜ 5 = ,˜6 = 2 , Q4 ˝ Q2 Q3 ˜4 µ ˜1 = ° c ( 1 + ˛˝ 2 ) , ˜ 2 = ˝ 5 ( 1 + ˝ 3 ) + 1 + ˝ 1 + ˝ 6 + ˝ 2 .

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Ebola Virus: Modeling of Spread and Outbreak

The model has the following equilibrium points: i. Disease-free equilibrium (DFE) point is given by E0 = ( ˙/µ , 0, 0, 0, 0, 0, 0 ) . The basic reproduction number 0 computed using the next-generation method was obtained as follows: 0 =

˜ c° t1t2 , t1 = p˝ + ( 1 − p ) Q2 , t2 = ( ˛Q4 + ˙ (Q4ˆ 1 + ˇˆ 2 ) ) . ˛Q1Q2Q3Q4

(

)

(4.20)

Choosing a suitable Lyapunov function and applying LaSalle invariance principle, the authors have shown that the DFE is globally asymptotically stable for 0 < 1. ii. Endemic equilibrium (EE) point is given by E = S , E , I a , I s , O, R , D , where O = ˜ 1 I s , D = ˜ 2 I s , R = ˜ 6 I s , I a = ˜ 3˜ 5 I s , E = Q2 /( P˜ ) I a = ° 5 I s , I s = ˙ ( 0 − 1)/˜ * , S = ˜° 2 /˛ * , and ˜ * = Q1˜ 5 ( 0 − 1) + µ˛ 2 . EE is locally asymptotically stable for 0 > 1 (Problem 4.5, Exercise 4.1).

(

)

(

)

The authors defned three time series controls as u1 ( t ) , u2 ( t ), and u3 ( t ) for t ˛[ 0, T ] , where T is the time duration of the interventions [44], to represent the educational campaigns, active case fnding, and pharmaceutical interventions, respectively. The overall objective is to minimize the number of EVD cases and the cost of control strategies involved. The authors have shown that there exists an optimal control triplet u1* , u2* , u3* and corresponding state solutions that minimize the objective functional. From the numerical solutions, it was observed that in the presence of control measures, an initial increase in the number of EVD-infected individuals was observed followed by a rapid decrease in the number of cases that ends at the Ebola-free stage. Djiomba Njankou and Nyabadza [36] studied the potential impact of limited hospital beds on EVD dynamics. After exposure to Ebola virus, susceptibles become infected and can transmit the disease. After c contacts with susceptibles, an infectious individual can transmit the disease with a probability ˜ . The effective transmission rate of the disease is c˜ . The force of infection is taken as ˜ ( t ) = c° ( I ( t ) + ˛1H ( t ) + ˛2 D ( t ))/N ( t ) , where N ( t ) = S ( t ) + I ( t ) + H ( t ) + R ( t ) + D ( t ) , for all t ˜ 0. The hospitalized individuals are assumed to be infectious, but with a lower infectivity than individuals in class I, because of the controlled environment in which they are isolated. Hence, it is assumed that 0 < ˜1 < 1. Dead bodies of EVD deceased are highly infectious, and this high infectivity is represented by the factor ˜2 > 1. The fow between the different compartments is governed by the following system of differential equations [36]:

(

(

(

)

)

)

dS = µ N ( t ) − ° + µ S ( t ) , dt

(4.30a)

dI  = ˜ S ( t ) − ° ( b ( t ) , I ( t )) I ( t ) − Q1I ( t ) , dt

(4.30b)

dH = ˜ ( b ( t ) , I ( t )) I ( t ) − Q2 H ( t ) , dt

(4.30c)

dR = ˜ I ( t ) + ° H ( t ) − µR ( t ) , dt

(4.30d)

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Spatial Dynamics and Pattern Formation in Biological Populations

dD = ˜ 1I ( t ) + ˜ 2 H ( t ) − °D ( t ) , dt

(4.30e)

db ( t ) b (t )  ˆ = ˝˘1− b (t ) , ˇ dt bmax 

(4.30f)

where Q1 = µ + ° + ˛ 1 and Q2 = µ + ° + ˛ 2 . Authors have used a nonlinear hospitalization rate and formulated the rate at which the time-dependent number of available beds evolves. Control functions for such models are mostly functions of time appearing as coeffcients in the model [60]. To capture the impact of limited hospital beds represented by the time-dependent number of newly available beds b(t), the authors introduced a per capita dynamic hospitalization rate as ˘ b (t )  ˜ ( b ( t ) , I ( t )) = µ0 + ( µ1 − µ0 )  ,  b ( t ) + I ( t )  where µ0 and µ1 represent the minimum and the maximum per capita hospitalization rates respectively, because of suffcient and insuffcient resources [1]. At t = 0, S ( 0 ) > 0, I ( 0 ) > 0, H ( 0 ) > 0, R ( 0 ) > 0, D ( 0 ) > 0, and b ( 0 ) > 0. To nondimensionalize the model (4.30), set s (t ) =

S(t ) I (t ) H (t ) R (t ) D(t ) , i (t ) = , h (t ) = , r (t ) = , d (t ) = . N (t ) N (t ) N (t ) N (t ) N (t )

The rate of change in the number of available beds is formulated as db ( t ) b (t )  ˆ = ˝˘1− b ( t ) , t  0, where ˝ is the growth rate. ˇ dt bmax  Since the recovered individuals do not contribute to EVD transmission, the dynamics of r ( t ) is decoupled from other variables in the closed system. Therefore, the model in nondimensional form is given by [36] ds = µ − ( ° (t ) + µ ) s (t ) , dt

(4.31a)

di = ˜ ( t ) s ( t ) − ° ( b ( t ) , i ( t )) i ( t ) − Q1i ( t ) , dt

(4.31b)

dh = ˜ ( b ( t ) , i ( t )) i ( t ) − Q2 h ( t ) , dt

(4.31c)

dd = ˜ 1i ( t ) + ˜ 2 h ( t ) − ° d ( t ) , dt

(4.31d)

db ( t ) b (t )  ˆ = ˝˘1− b (t ) , ˇ dt bmax 

(4.31e)

  b (t ) where ˜ ( t ) = c° ( i ( t ) + ˛1h ( t ) + ˛2 d ( t )) , ˝ ( b ( t ) , i ( t )) = µ0 + ( µ1 − µ0 )  .  ( b ( t ) + i ( t )) 

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Ebola Virus: Modeling of Spread and Outbreak

Equilibrium points and reproduction number: At equilibrium, equation (4.31e) gives b = 0 or b = bmax . For b = 0, disease-free equilibrium is E01 = ( 1, 0, 0, 0, 0 ). Endemic equilibrium point is given by E*1 s* , i* , h* , d * , 0 , where

(

s* =

)

(

)

µ Rh0 − 1 1 * µ0 i* * (Q2° 1 + µ0° 2 ) i* * , i = , h = ,d = . Rh0 Q2 ˛Q2 ( µ0 + Q1 ) Rh0  ( ° + ˛2˝ 1 ) Q2 + ( °˛1 + ˛2˝ 2 ) µ0  Rh0 = ˜ c  . °Q2   (Q1 + µ0 )  

When Rh0 > 1, i* > 0. In this case, the existence of E*1 is driven by Rh0, which is considered as the reproduction number at the minimum hospitalization rate µ0 . In the absence of beds for hospitalization of EVD patients, the existence of endemic equilibrium point depends on the reproduction number at the minimum hospitalization rate. For b = bmax , the disease-free equilibrium is obtained as, E02 = ( 1, 0, 0, 0, bmax ) , with the maximum number of beds. Endemic equilibrium point is given by E*2 s* , i* , h* , d * , bmax , where

(

s* =

µ°Q2 , c˛ ( ° + ˝2˙ 1 ) Q2 + ˆ * ( °˝1 + ˙ 2˝2 ) i* + µ°Q2

h* =

Q2˙ 1 + ˙ 2ˆ * i* * bmax µ1 + i* µ0 i*ˆ * * ,d = ,ˆ = , Q2 °Q2 bmax + i*

(

)

)

(

)

(

)

where i* is the solution of the cubic polynomial

( )

a0 i *

3

( )

+ a1 i*

2

+ a2 i* + a3 = 0,

(4.32)

with a0 = ˜Q2 (Q1 + µ0 ) Rh0 , 2

a1 = c˜ ( °˛1 + ˝ 2˛2 ) ( bmaxQ1µ1 + µ0 ( − µ + bmaxQ1 + 2bµ1 ) ) + Q2 (Q1 (( 2bmax c˜ + µ )

(

)

+ 2bmax c˜˝ 1˛2 ) + ° −c˜µ + ( bmax c˜ + µ ) µ0 + bmax c˜µ1 + c˜˝ 1˛2 (− µ + bmax ( µ0 + µ1 )))

( (

)

))

(

= °Q2 (Q1 + µ0 ) µ 1 − Rh0 + bmax (Q1 + µ1 ) Rh0 + Rh ,

(

(

))

a2 = ˜Q2bmax µ (Q1 + µ1 ) ( 1 − Rh ) + (Q1 + µ0 ) 1 − Rh0 + bmax (Q1 + µ1 ) Rh , 2

2 a3 = µ°Q2bmax (Q1 + µ1 ) (1 − Rh ) ,

where Rh is the reproduction number derived below. The expressions for the coeffcients a0 , a1 given here (the coeffcients are computed using Mathematica) differ from the expressions derived by the authors (in a private communication, the authors are informed).

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Spatial Dynamics and Pattern Formation in Biological Populations

Irrespective of the above comment, the authors’ discussions and conclusions based on the numerical computations are all valid. We have a0 > 0 and a1 > 0, since Rh0 ˜ 1. The polynomial cannot have three positive roots. Also, if a2 > 0 and a3 > 0, then there is no positive root and hence no endemic equilibrium exists. If Rh > 1, then a3 < 0, and the system has a unique endemic equilibrium point irrespective of the sign of a2 . If Rh < 1, then a3 > 0, and the system may have two endemic equilibrium points if a2 < 0. The reproduction number Rh was derived as   (˝ 2˙2 + ˛˙1 )   , R = R + R + R , 1 1 Rh = ˜ c  + ˝ 1˙2 + µ1 I H D    h Q2  Q1 + µ1 ˛ (Q1 + µ1 )  

˜c ( ˜ cµ1˛1 ) , R = ˜ c°2  ˙ + µ1˙ 2  , and R , R , R are, , RH = I H D Q Q1 + µ1 ( 1 + µ1 ) Q2 D ˛ (Q1 + µ1 )  1 Q2  respectively, the contributions of the infected, hospitalized, and deceased individuals to EVD transmission. The DFE and endemic equilibrium points of the model (4.31) are asymptotically stable (Problem 4.6, Exercise 4). Based on the above results, the authors have concluded the following: (i) the number of beds available in Ebola treatment units (ETUs) is a determining factor in the disease control. It was shown that when there are no beds or only a limited number of beds are available, controlling EVD is more diffcult because of the presence of a backward bifurcation. (ii) When the maximum capacity of an ETU is increased, a forward bifurcation appears, and reducing the reproduction number to values less than 1 will stop EVD. (iii) As part of the model validation, the model was ftted to data from Liberia and Sierra Leone and compared. Authors recommended a timely supply of beds into Ebola healthcare units and highlighted the need for upgraded infrastructures in countries affected by the last EVD outbreak. (iv) Numerical simulations show that timely supply of beds in ETU contributes to the reduction of the number of individuals infected by Ebola virus and limits the spread of the disease.

where RI =

4.6 Model 4: Ebola Epidemic SEIRD Model with Time Delay Weitz and Dushoff [97] modeled the post death transmission of Ebola virus from the dead to the living and planning of control efforts leading in the long term to the control and elimination of the outbreak of Ebola. Motivated by their model, Al-Darabsah and Yuan [4] proposed a time-delayed model for the transmission of Ebola in humans and considered the transmission of infection to the living humans from the infected corpses, in which the latent period of Ebola is incorporated. The total population N ( t ) is divided into susceptible S ( t ), exposed E ( t ), infectious I ( t ), recovered R ( t ) , and infected corpses who are nonetheless infectious D ( t ). Let ˜ be the latent period, c the probability of transmission of infection from an infectious human to a susceptible individual when a contact occurs, and d the probability of transmission of infection from an infectious corpse to a susceptible individual. Then, the total number of new infected individuals at time t is given by c

S(t ) I (t ) S(t ) D(t ) S(t ) +d = ( cI (t ) + dD (t )). N (t ) N (t ) N (t )

249

Ebola Virus: Modeling of Spread and Outbreak

If µ is the natural death rate, then the probability that an individual survives in the latent period [ t − τ , t ] is e − µτ . Therefore, the total number of individuals surviving the latent period τ and becoming infectious at time t is c

S ( t − τ ) I ( t − τ ) − µτ S ( t − τ ) D ( t − τ ) − µτ S ( t − τ ) e +d e = ( cI (t − τ ) + dD (t − τ )) e − µτ . N (t − τ ) N (t − τ ) N (t − τ )

In the following, we present the work of Al-Darabsah and Yuan [4]. The governing equations are given by dS ( t ) S(t ) = Λ− ( cI (t ) + dD (t )) − µS(t ) , dt N (t ) dE ( t ) S ( t ) = ( cI (t ) + dD (t )) − NS((tt−−ττ)) ( cI (t − τ ) + dD (t − τ )) e − µτ − µE (t ) , dt N (t ) dI ( t ) S ( t − τ ) = ( cI (t − τ ) + dD (t − τ )) e − µτ − ( ρ + µ + δ ) I (t ) , dt N (t − τ ) dR ( t ) dD ( t ) = ( µ + δ ) I (t ) − γ D(t ) , = ρ I ( t ) − µR ( t ) , dt dt

(4.33)

where N ( t ) = S ( t ) + E ( t ) + I ( t ) + R ( t ) + D ( t ) , all the parameters are positive, Λ is the recruitment rate, δ is an additional death rate due to infection by Ebola, ( 1/ρ ) is the average duration of the infectious period with recovery rate ρ , and the average period of infectiousness after death in human corpses is ( 1/γ ) .

(

)

Denote C := C [ −τ , 0 ) ,  . For φ = (φ1 , φ2 , φ3 , φ4 , φ5 ) ∈ C, define  φ = 5

5

∑ φ 

i ∞

where

i=1

 φi ∞ = max φi (θ ) . θ ∈[− τ ,0 ]

Then, C is a Banach space and C + = {φ ∈ C : φi (θ ) ≥ 0 ; ∀i ∈{1, 2, 3, 4, 5} , θ ∈[ −τ , 0 ]} is a normal cone of C with nonempty interior in C. For any given continuous function u = ( u1 , u2 , u3 , u4 , u5 ) : ( −τ , ζ ) →  5 with ζ > 0, define ut ∈ C for each t ≥ 0, by ut (θ ) = ( u1 ( t + θ ) , u2 ( t + θ ) , u3 ( t + θ ) , u4 ( t + θ ) , u5 ( t + θ )) for all θ ∈[ −τ , 0 ]. The initial data for system (4.33) is taken in the form   χ = φ ∈C + :  

   φ1 (θ ) ( cφ3 (θ ) + dφ5 (θ )) µθ    φi (θ ) > 0, ∀θ ∈[ −τ , 0 ] , φ2 ( 0 ) =  e dθ , 5    φ  i=1 i (θ ) −τ     i=1 5

∑

0

∫

∑

where the form of φ2 ( 0 ) is derived below. From the second equation in system (4.33), we have  S(t )    dE ( t ) e µt  + µE ( t ) = e µt  cI ( t ) + dD ( t )) (  dt   N (t )   S(t − τ )  − e µ (t−τ )  cI ( t − τ ) + dD ( t − τ )) . (  N (t − τ ) 

250

Spatial Dynamics and Pattern Formation in Biological Populations

By integrating on both sides of the equation on (0, t), we get t

e µt E ( t ) − E ( 0 ) =

˜ 0 t

=

t

 S (° )  S (° − ˛ ) µ (° − ˛ )  µ°   N (° ) {cI (° ) + dD (° )} e  d° −  N (° − ˛ ) {cI (° − ˛ ) + dD (° − ˛ )} e  d°

˜ 0

 S (° )

˜  N (° ) {cI (° ) + dD(° )} e

µ° 

0

t

=

˜

t− ˛ 0

Hence, if E ( 0 ) =

t− ˛

 d° −

 S (° ) µ°   N (° ) {cI (° ) + dD (° )} e  d° − S (° )

˜ N (° ) ( cI (° ) + dD(° )) e

µ°

 S (° )

µ° 

 S (° )

µ° 

˜  N (° ) {cI (° ) + dD(° )} e

−˛

0

˜  N (° ) {cI (° ) + dD(° )} e

 d°

−˛

 d° .

d° , then

−ˇ

t

E (t ) =

S (° )

˜ N (° ) ( cI (° ) + dD(° )) e

− µ ( t−° )

d° .

t− ˇ

Note that when N ( 0 ) > 0, the total population N ( t ) is positive for all t ˜ 0. Analysis of the equilibrium points and the basic reproduction number 0 : The disease-free equilibrium, E0 = ( ˙/µ , 0, 0, 0, 0 ) always exists for all values of the parameters. Linearizing (4.33) about E0 , the following equations are obtained [4]: dS ( t ) = −cI ( t ) − dD ( t ) − µS ( t ) , dt

(4.34a)

dE ( t ) = cI ( t ) + dD ( t ) − {cI ( t − ˜ ) + dD ( t − ˜ )} e − µ˜ − µE ( t ) , dt

(4.34b)

dI ( t ) = {cI ( t − ˜ ) + dD ( t − ˜ )} e − µ˜ − t1I ( t ) , dt

(4.34c)

dR ( t ) = ˜ I ( t ) − µR ( t ) , dt

(4.34d)

dD ( t ) = ( µ + ° ) I ( t ) − ˛ D ( t ) , t1 = ( ˝ + µ + ° ) . dt

(4.34e)

To calculate the basic reproduction number, the authors consider the equations for the diseased classes in system (4.34), which can be rewritten as d Y ( t ) = AY ( t − ˜ ) − BY ( t ) , dt ˝ E (t ) ˆ where Y ( t ) = ˆ I ( t ) ˆ ˆ˙ D ( t )

ˇ ˝ 0  ˆ , A = ˆ 0  ˆ 0 ˘ ˆ˙

− ce − µ ce − µ 0

−de − µ de − µ 0

(4.35) ˇ  − µ =e  ˘

˝ 0 ˆ ˆ 0 ˆ 0 ˙

−c c 0

−d d 0

ˇ  ,  ˘

251

Ebola Virus: Modeling of Spread and Outbreak

 µ  and B =  0   0

−c t1

−(µ + ° )

−d   0 .  ˛  

Let 0 = ( y1 , y 2 , y 3 )T be the number of classes E ( t ) , I ( t ), and D ( t ) at t = 0, then, from equations (4.35), the distribution of the remaining population of classes E ( t ) , I ( t ), and D ( t ) at time t > 0 is given by  ( t ) = e − Bt0 . Since B is nonsingular, the total number of newly infected individuals is ˇ

ˇ

˜

˜

 = A ( t − ° ) dt = Ae − B (t−° )0 dt = AB −10 . °

°

The next infection operator is

M 0 = AB

−1

˘ 0 e − µ  =  0 ˜ t1  0 

−t2 t2 0

−dt1 dt1 0

   , t2 = c˜ + d ( µ + ˛ ) .  

Following the work of Lai and Zou [61], the basic reproduction number 0 for system (4.33) is defned by the spectral radius of the matrix M 0 . The value of 0 is obtained as follows: 0 =

(

)

c˜ + d ( µ + ˛ ) e − µ d ( µ + ˛ ) e − µ e − µ t2 ce − µ = = + . ˜ t1 ˜ (˝ + µ + ˛ ) ˝ + µ + ˛ ˜ (˝ + µ + ˛ )

(4.36)

Biologically, e − µ° is the survival rate of the infected individuals in the latent period, ( µ + ° )/˛ is the number of the deaths occurring from one infected individual and 1/( ˜ + µ + ˛ ) is the average period of infected individual to recover or survive or die. Therefore, ce − µˇ /( ˜ + µ + ˛ ) is the number of newly infected individuals that arise from one infectious individual and d ( µ + ° ) e − µ˘ / ˛ ( ˝ + µ + ° ) is the number of newly infected individuals that arise from one infectious corpse. From the epidemiological point view, if the basic reproduction number 0 > 1, then each individual is causing more than one infection and the disease will persist, while when 0 < 1, the disease will die out. Therefore, to reduce the spread of the disease, we need to restrict the value of 0 below 1 by controlling the related parameters. For example, for the fxed parameters c, µ , ° , ˛, and ˜ , frst, notice that (˜ 0 /˜° ) = − µ0 < 0 for all ˜ ° 0, implying that the reproduction number decreases as the latent period increases. Hence, we may try to extend the duration of latent period to slow ˜ 0 e − µ ( µ +˙ ) Ebola spread by prescription drugs or control measures. When = > 0, 0 ˜d ° (˛ + µ + ˙ ) increases as the value of d increases. Therefore, we need to keep the transmission rate d small to reduce the spread of Ebola. One of the control measures is to dispose the human remains either by cremation or burial. From the above, the authors show that 0 < 1, when 1  c˛ + d ( µ + ˝ )  ˜ > ln  . µ  ˛ ( ˙ + µ + ˝ ) 

(

)(

)

252

Spatial Dynamics and Pattern Formation in Biological Populations

4.6.1 Existence of Endemic Equilibrium and Stability Analysis Theorem 4.2

(

)

If 0 > 1, then system (4.33) has a unique endemic equilibrium point E1 = S* , E* , I * , R* , D* with 0 < S* < ( ˙/µ ) , E* > 0, I * > 0, R * > 0 and D* > 0 [4]. Proof: Let the positive equilibrium point be E1 S* , E* , I * , R* , D* . From the fourth (4.33d) and ffth (4.33e) equations, we get

(

)

(

)

R* = ( ˜/µ ) I * and D* = ( µ + ˛ )/˝ I * .

(4.37)

The remaining three equations become

( ) (1 − e ) g (S , E , I ) − µE = 0, e g ( S , E , I ) − t I = 0, t = ˜ + µ + ˛ , S ( cI + dD ) µt S I g (S , E , I ) = = S +E +I +R +D µ° ( S + E ) + t I ˝ − g S* , E* , I * − µS* = 0, − µˆ

− µ

*

*

*

*

*

*

*

*

*

*

*

1

*

*

2

*

*

(4.38b) (4.38c)

1

*

*

(4.38a)

*

*

* *

*

.

(4.38d)

˛ − µS* . t1e µ˙

(4.39)

*

3

*

t2 = ˜ c + d ( µ + ˛ ) , t3 = µ 2 + µ (˜ + ˛ ) + ˜˝. From the frst (4.38a) and third (4.38c) equations, we obtain I * =

If S* ° ˛/µ , then I* ≤ 0, that is, the positive equilibrium does not exist. For the positive equilibrium point E1 to exist, the condition is 0 < S* < ˛/µ. From the second (4.38b) and third (4.38c) equations, we get

(

)( ˆ − µS )/µ.

)

(

E* = 1 − e − µˇ

*

(4.40)

In the third equation (4.38c), let

( )

(

)= t e

µS* ˆ − µS* t4

) (

)

G S* = g S* , E* , I * − t1e µˇ I * = g S* , E* , I * − ˆ − µS* .

(

*

*

Now, g S , E , I

*

1

µ

( (° µ (S

*

) = + E ) + t I ) ° µt e *

3

*

1

(

)

µS* ˆ − µS* t4 µ

(

S + ° t1 (e *

µ

)(

− 1) + t3 ˆ − µS*

)

.

˘  AS* Therefore,G S* = ˙ − µS*  − 1 = 0, *  ( B − C ) S + D1 

( ) (

)

(

)

where A = µt2 > 0, B = ˜ µt1e µ˙ > 0, C = µt4 > 0, D1 = ˝t4 > 0, t4 = ˜ e µ˘ − 1 t1 + t3 . Since ° − µS* > 0, that is, 0 < S* < ˛/µ ; we obtain AS* D1 . = 1, or S* = * B − C S + D A − B+C ( ) 1

(4.41)

253

Ebola Virus: Modeling of Spread and Outbreak

A − B + C = µt2 − °µt1e µ + µt4

(

(

)

= µ t2 − ° t1e µ + ° e µ − 1 t1 + t3

)

( (

)

)

= µ ( t2 − ° t1 + t3 ) = µ ° t1 e µ 0 − 1 + t3 > 0. Therefore, when 0 > 1; S* > 0 and is obtained from (4.41). The other components E* , I * , D* , R* are, respectively, obtained from (4.40), (4.39), and (4.37). Hence, unique endemic equilibrium exists that can be written in terms of 0 . Furthermore, we have N * = S* + E* + I * + R * + D* = 0S* . Stability analysis: Authors have established the following: (i) For 0 < 1, the DFE E0 is locally as well as globally asymptotically stable and the disease dies out. (ii) The unique endemic equilibrium E1 is locally asymptotically stable under certain conditions and globally asymptotically stable in a special case when 0 > 1 (Problem 4.7, Exercise 4). When 0 = 1,E0 and E1 coalesce into one point. The authors demonstrated or computed the following: (i) assuming the parameter values in the specifed ranges as given in Table 1 of the research publication of Al-Darabsah and Yuan [4], they performed numerical simulations. The values of ˜ and µ were calculated by using the life expectancy and the total human population, respectively, where µ = 1/ ( life expectancy × 365 ) and ° = µ × (total human population). For the set of the parameter values ˆ = 554.80, c = 0.05, d = 0.2, µ = 0.00047, ° = 0.13, ˛ = 0.5, ˝ = 5.0, ˙ = 0.05, 0 was obtained as 0 = 0.567 < 1, implying that the DFE, E0 1.18 × 107 , 0, 0, 0, 0 is globally asymptotically stable. (ii) Taking the values of d as d = 0.5007, 0.75, they have demonstrated the infuence of the transmission rate d on the dynamics and calculated the values of 0 as 1 and 1.3608, respectively, at which the DFE and EE are locally asymptotically stable. It was found that the endemic equilibrium is globally asymptotically stable when µ =  ° = 0.125 and 0 = 1.377. The values of the other parameters were taken as ˙ = 362.84, c = 0.25, d = 0.71, ˜ = 0.084, ° = 0.5, ˛ = 10, ˝ = 0.09. (iii) The authors compared the model predictions with real data of Ebola infectious/death cases in Guinea from January 26, 2015 to December 29, 2016. The simulation results have good agreement with the real data. The infuence of increase in the latent period from ˜ = 5 to 12 and then to 21 was studied, keeping the values of all the parameters and initial conditions same. (iv) Finally, the authors have discussed the effect of climate change, like seasonality, on Ebola transmission by taking a different set of parameter values as ˆ = 189.75, µ = 0.000044, c = 0.15, d = 0.45, ° = 0.125, ˛ = 0.17, ˝ = 6, ˙ = 0.049, 0 = 2.763. The endemic equilibrium point E1 S* , E* , I * , R* , D* = ( 595000, 981.28, 939.57,1046300,691.1) exists with constant transmission rates and is locally asymptotically stable. A positive periodic solution exists with periodic time-dependent transmission rates c1 ( t ) = 0.15 ( 1 + sint ) , d1 ( t ) = 0.45 ( 1 + cos2t ) and the infectious population oscillates around the infectious values I * and D* . As the temperature changes seasonally, the number of infectious cases varies. Authors have taken different values of periodic time-dependent transmission rates and obtained the same results that the infectious population oscillates around the infectious values that arise from constant transmission rates. In summary, the authors have concluded the following: (i) the DFE is globally stable and disease dies out when 0 < 1. (ii) The infection always persists and a unique endemic equilibrium exists when 0 > 1. (iii) The endemic steady state is locally asymptotically stable under certain conditions and globally asymptotically stable in a special case of the model. (iv) Basic reproduction number 0 is a decreasing function of ˜ and increasing function of d, the

(

(

)

)

254

Spatial Dynamics and Pattern Formation in Biological Populations

probability of transmission of infection from an infectious corpse to a susceptible individual. This implies that to reduce the spread of the Ebola, one need to extend the duration of latent and/or dispose the human remains by cremation or burial. (v) Numerically, they have demonstrated that sensitivity of 0 changes the number of infectious individuals. (vi) The model predicts stabilization of the number of Ebola infectious or death cases.

4.7 Model 5: General Ebola Transmission Model for Population in a Community Agusto et al. [3] extended the Ebola transmission models studied by Chowell et al. [29] and Althaus [5] by including the following: (i) the dynamics of healthcare workers, (ii) the interaction between susceptible individual in a community and infected individuals in a hospital, through visits, (iii) the effect of traditional (cultural) belief systems and customs that aid EVD transmission (such as the handling of corpses during traditional burial practices, etc.). An important thing to consider is the mistrust of members of the community for authority and fear against seeking medical care (for fear of being quarantined and/ or acquiring infection during quarantine). The proposed model also extends the model considered by Legrand et al. [64], by incorporating epidemiological compartments for healthcare workers and members of the general public who visit family members and/or acquaintances in hospitals, in addition to also including the role of traditional belief systems and customs on EBOV transmission dynamics. The authors have designed a general Ebola transmission model for the transmission dynamics of EVD in a population stratifed into two epidemiological settings—those in the community and those within the healthcare system. The equations for the dynamics of healthcare workers (those in hospitals or healthcare facilities in general and healthcare workers who return to the community at the end of their shift) are similarly derived. The model system consists of 21 differential equations that were used to study the 2014 Ebola outbreaks in Guinea. It is used to assess the population level impact of basic (nonpharmaceutical) public health control measures (such as proper handling of Ebola-infected patients and Ebola-deceased patients, limiting the duration of family visits to healthcare facilities to see infected loved ones, etc.). They have also studied the special case where no public health interventions (that is, no basic anti-Ebola control measures and/or disease management in the healthcare settings) are implemented in the community. In the absence of such interventions, the above model consisting of 21 differential equations reduces to the following system of 7 differential equations: SC ( t ) = ˆC − ˜C ( ICE , ICN , DC ) SC ( t ) − µH SC ( t ) ,

(4.42a)

E C ( t ) = ˜C ( ICE , ICN , DC ) SC ( t ) − (° C + µ H ) EC ( t ) ,

(4.42b)

ICE ( t ) = ˜ CEC ( t ) − (° C + µ H ) ICE ( t ) ,

(4.42c)

ICN ( t ) = ˜ C ICE ( t ) − (° C + µ H ) ICN ( t ) ,

(4.42d)

R C ( t ) = h˜ C ICN ( t ) − µ H RC ( t ) ,

(4.42e)

255

Ebola Virus: Modeling of Spread and Outbreak

 C ( t ) = ( 1 − h )γ C ICN ( t ) − δ C DC ( t ) , D

(4.42f)

C D ( t ) = δ C DC ( t ) ,

(4.42g)

βCφC ( ICE + ICN + τ C DC ) is the infection rate of the disease in SC + EC + ICE + ICN + RC + DC the community. The meanings of all other parameters are defined in [3]. The associated basic reproduction number 0 is given by

where λC ( ICE , ICN , DC ) =

0 =

(

)

(

)

βCφCσ C δ C α C + k3 + τ Cα Cγ C 1 − h  , k1k2 k3δ C

where k1 = σ C + µH , k2 = α C + µ H , k3 = γ C + µH . EBOV can be effectively controlled in the community if 0 < 1. Agusto [2] modeled the transmission dynamics of the EVD by extending the compartment framework of the preintervention model (4.42) by including the (i) relapse of recovered individuals; (ii) reinfection of recovered individuals; and (iii) two infected and recovered classes. The total population, N ( t ) at time t is split into mutually-exclusive sub-populations of individuals denoted by S ( t ) , E ( t ) , ( IE ( t ) , I L ( t )) , ( R1 ( t ) , R2 ( t )) , D ( t ) , the susceptible, exposed, symptomatic individuals in the early and late-stage of EVD infection, recovered and immune individuals, and Ebola-infected deceased individuals, respectively. The author has included two recovered classes and and assumed that individuals in R1 ( t ) class can experience reinfection since the possibility of reinfection is a function of host immunity and the viral load to which an individual is exposed [69]. Further, it was also assumed that these individuals are capable of transmitting the virus since the viruses can persist after recovery in parts of the body and can spread through sex or other contracts with semen [26]. The author had also assumed that the recovered individuals in the R2 ( t ) class experience lifelong immunity. Studies from previous Ebola outbreaks show that antibodies to the virus could still be detected 10 years after recovery in the disease survivors [26]. The total population N ( t ) is given by N ( t ) = S ( t ) + E ( t ) + IE ( t ) + I L ( t ) + R1 ( t ) + R2 ( t ) + D ( t ) . The model is governed by the following system of ODEs [2]: S ( t ) = Π − λ ( IE , I L , R1 , D ) S ( t ) − µS ( t ) ,

(4.43a)

E ( t ) = λ ( IE , I L , R1 , D ) S ( t ) + ελ ( IE , I L , R1 , D ) R1 ( t ) − (σ + µ ) E ( t ) ,

(4.43b)

IE ( t ) = σ E ( t ) − (α + µ ) IE ( t ) + ρ R1 ( t ) ,

(4.43c)

IL ( t ) = α IE ( t ) − (γ + µ ) I L ( t ) ,

(4.43d)

R 1 ( t ) = hγ I L ( t ) − ( ρ + ξ + µ ) R1 ( t ) − ελ ( IE , I L , R1 , D) R1 ( t ) ,

(4.43e)

R 2 ( t ) = ξ R1 ( t ) − µR2 ( t ) ,

(4.43f)

 ( t ) = ( 1 − h )γ I L ( t ) − δ D ( t ) , D

(4.43g)

256

Spatial Dynamics and Pattern Formation in Biological Populations

° ( IE + I L + ˛ 1R1 + ˛ 2 D) is the infection rate of the disease. ˜ 1 and S + E + IE + I L + R1 + R2 + D ˜ 2 are the modifcation parameters that account for the assumed reduced infectiousness of the recovered individuals in the R1 class and the Ebola-infected deceased individuals (in comparison to the living individuals with Ebola symptoms). The parameter ˜ is the effective contact (transmission) rate, ˜ is the recruitment rate, and µ is the natural death rate. The other parameters ˜ , (° , ˛ , ˝ ) , ˙ , ˆ , h, and ˜ represent infection reactivation rate, progression rate of symptomatic, early-symptomatic, recovered individuals to immune classes, recovery rate of symptomatic individuals, reinfection modifcation parameters, fraction of symptomatic individuals who recovered and burial rate of Ebola-deceased individuals, respectively. Equilibrium points: The DFE of the model (4.43) is given by E0 = ( ˙/µ , 0, 0, 0, 0, 0, 0 ) . The DFE is asymptotically stable under certain conditions (Problem 4.8 in Exercise 4). The basic reproduction number of the model (4.43) is obtained as [2] where ˜ ( IE , I L , R1 , D ) =

(

(

°˛ k 4˝ ( k3 + ˙ ) + ˙ˆ (˝ hˇ 1 + ( 1 − h ) k 4ˇ 2 )

)

0 = ˜ FV −1 =

k1˝ ( k2 k3 k 4 − h˙ˆ˜ )

).

Here, 0 means the average number of Ebola cases that arose due to a typical Ebolainfected individual (living or dead) introduced into a completely susceptible human population. EBOV can be effectively controlled in the community if 0 < 1 can be maintained. Let E1 = S* , E* , IE* , I L* , R1* , R2* , D* be the endemic equilibrium point.

(

Let ˜ * =

(

)

° I + I + ˛ 1 R + ˛ 2 D* * E

* L

* 1

)

be the associated force of infection for Ebola at S + E + I + I + R + R + D* endemic steady-state E1. Setting the right-hand sides of the equations of the model to zero, we obtain (in terms of ˜ * ) *

S* =

I L* =

*

* E

* L

* 1

* 2

( (

) )

)

(

) )

˜ *  k2 k3 k 4 + ˛˜ * − h˝˙ˆ ˜ * k3 k 4 + ˛˜ * ˇ  * * , E = , , I = E ˜* + µ K ˜* + µ K ˜* + µ

(

) K (˜ + µ )

˜ *˝ k 4 + ˛˜ * ˇ *

(

(

,

(

)

˜ * ( 1 − h )°˛ k 4 + ˇ˜ * ˝ ˜ * h°˛˝ ˜ * h°˛ˆ˝ * * R = , R2 = , D = , ˘ K ˜* + µ K ˜* + µ µK ˜ * + µ * 1

(

(

)

)

(

(

)

(

)

)

where K = k1k2 k3 k 4 + ˜° * − h˛˝ k1 ˙ − ˜ˆ° * . Note that ˜ * > 0. For the existence of E1 , we require h < 1. This is true as h is the fraction of symptomatic individuals who have recovered. The expressions for the elements of E1, given here (obtained by using Mathematica), differ from the expressions derived by the authors except for S* (in a private communication, the author is informed). The following analysis is done using the expressions derived above. Substituting these expressions into ˜ * and simplifying, we obtain the quadratic equation [53]: a2 (˜ * )2 + a1˜ * + a0 = 0,

(4.44)

257

Ebola Virus: Modeling of Spread and Outbreak

where a2 = µ° ( k3˛ + k2 k3 + ˝˛ )˙ + ˝ˆ ( 1 − h )˛  , a1 = µ° k3 k 4 ( k2 + ˛ ) + µ˝˛ k 4 (˙ (1 − h) + ° ) + h˝˙° (˛ˆ − µˇ ) + µ° ( h˝˙˛ (1 − ˘ ) + k1k2 k3˘ ) − ˛µ (˝°˘ + k3°˘ + (1 − h) 2˙˝˘ ) , a0 = k1µ° ( k2 k3 k 4 − h˛˝˙ ) ( 1 − 0 ) . Since ˜ * > 0, we need to determine the conditions under which the equation (4.44) has one or both positive roots. Since h < 1, we fnd that a2 > 0. Also, ( k2 k3 k 4 − h˜°˛ ) > 0, as h < 1, a0 > 0 for 0 < 1, and a0 < 0 for 0 > 1. For 0 < 1, if a1 > 0, then the signs of the coeffcients in (4.44) are +, +, +. Thus, there are no positive roots, and the equation has negative roots or a complex pair with negative real parts. For 0 < 1, if a1 < 0, then the signs of the coeffcients in (4.44) are +, −, +. Thus, the equation has two positive roots or a complex pair with positive real parts. For 0 > 1, if a1 > 0, then the signs of the coeffcients in (4.44) are +, +, −. Again, for 0 > 1, if a1 < 0, then the signs of the coeffcients in (4.44) are +, −, −. Thus, in these two cases, there is always a positive root. Therefore, the Ebola epidemic model (4.43) has at least one positive endemic equilibrium point for 0 > 1 and may have zero or two positive endemic equilibria for 0 < 1. The existence of multiple endemic equilibria for 0 < 1 gives the possibility of bifurcation phenomenon to occur, where the DFE and a stable EE coexist. Bifurcation analysis: Using the method based on central manifold theory of CastilloChavez and Song [23] and taking ˜ = ˜ * as the bifurcation parameter, the authors have shown that the model system (4.43) undergoes a simple transcritical bifurcation at 0 = 1. k1° ( k2 k3 k 4 − h˛˝˙ ) For 0 = 1, ˜ is determined as ˜ = ˜ * = . °ˆ ( k 4 ( k3 + ˛ ) + h˛˝ˇ 1 ) + ( 1 − h ) k 4˛˝ˆˇ 2 The Jacobian matrix J 0 of the system (4.43) linearized about (E0 , ˜ * ) is given by  −µ   0   0 J0 =  0   0   0  0 

0

−° *

−° *

− ° *˛ 1

0

− k1

°* 0 − k3 hˇ

° *˛ 1 ˙ 0 − k4

0

˝ 0 0

°* −k2 ˆ 0

0 0

0

0

0

˘

−µ

0

0

( 1 − h )ˇ

0

0

0

− ° *˛ 2 

° *˛ 2

0

. 0

0

0

− 

Zero is a simple eigenvalue of J 0 and the corresponding right eigenvector w = ( w1 , w2 , w3 , T w4 , w5 , w6 , w7 ) and the left eigenvector ˜ = (˜ 1 , ˜ 2 , ˜ 3 , ˜ 4 , ˜ 5 , ˜ 6 , v7 ) satisfy ˜ ° w = 1. The components of the eigenvector w are given by w1 = −w * /µ , w2 = w * /k1 , w * = ° ( w3 + w4 + ˛ 1w5 + ˛ 2 w7 ) , w3 > 0, w4 = ˜ w3 / k3 , w5 = h° w4 / k 4 , w6 = ˛ w5 / µ , w7 = ( 1 − h )° w4 / ˙ .

258

Spatial Dynamics and Pattern Formation in Biological Populations

The components of the eigenvector v are given by v1 = 0, v6 = 0, v2 = v3˜ / k1 , v3 > 0,

(

)

v4 = − ˜ ( v1 − v2 ) + ° ( hv5 + ( 1 − h ) v7 ) / k3 , v5 = ( − ˜° 1 ( v1 − v2 ) + ˛ v3 + ˝ v6 ) / k 4 , v7 = − ˜° 2 ( v1 − v2 ) / ˙ . v and w satisfy v ˜ J 0 = 0 and J 0 ˜ w = 0. For the above system, the coeffcients a and b are defned as 7

a=

˜

v k wi w j

k ,  i ,  j =1

7

˜

ˆ2 f k ˆ2 f k E0 , ° * , b = v k wi E0 , ° * , ˆ xi ˆ x j ˆ x i ˆ° k ,  i = 1

(

)

(

)

which can be explicitly computed as a=−

˜µ ( v2 + v3 + v4 + v5 + v7 ) 

 ( w2 ( w3 + w4 + w5˛ 1 + w7˛ 2 ) + w5˝ ( w3 + w4 + 2w5˛ 1 + w7˛ 2 )) < 0, b=

˜ v3 w3 ( k 4 ( k3 + ° )˛ + h°˝˛˙ 1 + ( 1 − h ) k 4°˝˙ 2 ) > 0. k1k3 k 4˛

At ˜ = ˜ * , the Ebola epidemic model (4.43) undergoes forward bifurcation at 0 = 1, whenever the inequality a < 0 holds. The expression for a given here differs from the expression derived by the author. (In a private communication the author is informed.) We fnd that a < 0 always. Therefore, it does not undergo backward bifurcation. Further, we fnd that the reinfection parameter (˜ ) does not infuence the sign of a . Numerical simulations: For the sake of completeness, we have performed numerical simulations choosing the same parameter values as taken by the author except for the value of ˜ , as ˇ = 400, ˜ = 0.3045, µ = 0.00004, ˛ 1 = ˛ 2 = 0.21, ˝ = 0.02,  ˙ = 0.5239, ˆ = 0.5472, h = 0.48, ˜ = 2, ° = 0.5366, ˜ = 0.5, ° = 0.1, to verify the conclusions about bifurcation and time series analysis. For this parameter set, we obtain E0 107 , 0, 0, 0, 0, 0, 0 , E1 ( 4.31902 × 106 , 433.911, 451.473, 460.357, 986.916, 2.46729 × 106 , 256.909 ) and 0 = 1.57151. For ˜ = 0.1045, we obtain 0 = 0.539318, and the eigenvalues corresponding to the DFE are obtained as −0.686378  0.207691i, −0.570706, −0.225166, −0.0592319, −0.00004, −0.00004. We obtain

(

)

A1 = 2.22786, A2 = 1.86374, A3 = 0.688466, A4 = 0.100774, A5 = 0.00391422, A1 A2 A3 − A32 − A12 A4 ˝ 1.88 > 0,

( A1 A4 − A5 )( A1 A2 A3 − A32 − A12 A4 ) − ( A1 A2 − A3 )2 − A1 A52 ˝ 0.3687 > 0. The Routh-Hurwitz criteria are satisfed. For 0 = 1.57151 > 1, the eigenvalues corresponding to the EE are obtained as −0.734821  0.23635i, − 0.000032523  0.00211926i, −0.571176, − 0.187194, −0.0000399887.

Ebola Virus: Modeling of Spread and Outbreak

259

Hence, the EE is locally asymptotically stable. The values of bifurcation coeffcients are obtained as a = −1.93896 × 10−6 and b = 2.59844 showing forward bifurcation. (This is in variation to the conclusion drawn by the author [2].) The bifurcation diagrams are plotted in Figure 4.7, and the time series are plotted in Figure 4.8. These fgures show the occurrence of transcritical bifurcation, which bifurcates at 0 = 1. Agusto [2] has also studied the model system (4.43) numerically and concluded the following: (i) the total number of new cases of Ebola-infected individuals increases with increasing values of relapse and reinfection parameters. (ii) If the reinfection rate is greater than relapse rate and vice versa, then reinfection will lead to more cases. (iii) The high-effectiveness level is more effective than either the moderate- or low-effectiveness levels with respect to transmission rate (˜ ) and recovery rate (˜ ). The plots of bifurcation diagrams show the transcritical bifurcation at ˜ = ˜ * . From Figure 4.8a, we observe that as the infection reactivation rate (˜ ) and reinfection modifcation parameter (˜ ) increases, population of symptomatic individuals in the

FIGURE 4.7 Bifurcation diagrams for all the infected class for the given set of parameters in the text.

260

Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 4.8 Time series of the model system (4.43). (a) Effect of reinfection and relapse on early infected class. (b) Effect of transmission and recovery rate on early infected class. (c and d) Effect on recovery class R1 with and without reinfection and relapse.

early stage of EBOV infection increases; however, the infection peak time decreases with increasing values of ˜ and ˜ . Figure 4.8b depicts the symptomatic individuals in the early stage of EBOV infection generated using the three basic public health control effectiveness levels (i.e., low, moderate, and high) over a 1,000-day period (the end of simulation period). To assess the impact of incorporating disease reinfection and relapse on Ebola transmission dynamics, the dynamics of the model (4.43) is compared with the version that does not include disease reinfection (that is, model (4.43) with ˜ = 0) as given in Figure 4.8c and relapse (that is, model (4.43) with ρ = 0) as given in Figure 4.8d. These simulations show that Ebola outbreaks in the presence of disease relapse and reinfections are controllable using basic public health control measures, such as the moderate and high-effective levels of the control strategies described above. Berge et al. [16] formulated a model to explain the recurrence and persistence of EVD outbreaks in Africa due to consumption of contaminated bush meat, funeral practices, and the environmental contamination. Berge et al. [15] extended the above model to incorporate vaccination and change of behavior for self-protection of susceptible individuals. Educational interventions, standard incidence, and mass action principle were included in the formulation of the force of infection. Application of both types of interventions are very

261

Ebola Virus: Modeling of Spread and Outbreak

important to have a smaller number of infectious individuals that minimize the severity of EVD in the population. Berge et al. [14] also proposed a deterministic mathematical model for the transmission dynamics of EVD in a complex Ebola virus life ecology. Model tries to capture the essential patterns of the disease evolution as a three-cycle transmission process. Coltart et al. [33] reviewed and discussed the key successes, failures, and lessons learned from the evolution of the 2013–2016 West African outbreaks and aims to summarize the events by country in chronological order including epidemiological parameters and implementation of outbreak containment strategies. The landscape of previous Ebola outbreaks was described to highlight differences and summarize theories about the origins of the 2013–2016 outbreaks, as well as the factors that led to outbreak propagation.

Exercise 4 4.1 Show that the endemic-free state EFE for the model (4.11) is always unstable as long as ˜ is positive. 4.2 Obtain the endemic equilibrium point E1 S* , E* , I * , R* , for the Model (4.13). 4.3 Obtain the endemic equilibrium point E1 S* , E* , I * , R* , H * , D* of the temporal model of the system (4.18). 4.4 Show that the DFE of the temporal model of the system (4.18) is locally asymptotically stable. Numerically test the asymptotic stability of the endemic equilibrium point E1 = 1.02497 × 107 , 2415.25,105.137,1.85767 × 106 , 50.7125,141.208 . 4.5 Establish the linear stability of the endemic equilibrium point for the model system (4.19). 4.6 Discuss the asymptotic stability of the DFE and endemic equilibrium point of the model (4.31). 4.7 Discuss the asymptotic stability of the disease-free equilibrium point E0 of system (4.33). Find the conditions under which the endemic equilibrium is asymptotically stable. Also, plot the time series for the following parameter set ˙ = 600, c = 0.05, d = 0.5, µ = 0.00047, ° = 0.23, ˛ = 0.5, ˝ = 5.0.

(

(

(

)

)

)

4.8 Find the conditions under which the DFE of the model (4.43) is asymptotically stable.

References 1. Abdelrazec, A., Bélair, J., Shan, C., Zhu, H. 2016. Modeling the spread and control of dengue with limited public health resources. Math. Biosci. 271, 136–145. 2. Agusto, F. B. 2017. Mathematical model of Ebola transmission dynamics with relapse and reinfection. Math. Biosci. 283, 48–59. 3. Agusto, F. B., Teboh-Ewungkem, M. I., Gumel, A. B. 2015. Mathematical assessment of the effect of traditional beliefs and customs on the transmission dynamics of the 2014 Ebola outbreaks. BMC Med. 13(1), 96, 1–17.

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5 Modeling the Transmission Dynamics of Zika Virus

5.1 Introduction The rapid spread of Zika virus has gained a lot of attention among the public and the Governments. Zika virus infection can lead to some serious neurological complications like the following: (i) among the newly born babies, it may lead to a birth defect where a baby’s head and/or brain is smaller than expected. This condition is termed microcephaly. (ii) It may lead to a neurological disorder that could lead to paralysis and death, which is called Guillain-Barre syndrome [76]. Since the effects of contracting Zika virus in the frst few days are similar to those of dengue fever, it was thought of as mild dengue. World Health Organization (WHO) declared the re-emergence of Zika virus as a Public Health Emergency of International Concern on February 1, 2016 [56]. WHO called on the global research and product development (R&D) communities to prioritize the development of preventative and therapeutic solutions [111]. In response, the R&D communities produced 45 vaccine candidates. Several candidates of vaccines have advanced beyond pre-clinical studies in animals and entered phase 1 human trials [41,42]. Of those, two vaccines have entered phase 2 trials [112,113]. Zika is a single-stranded RNA favivirus and belongs to the Flaviviridae family [68]. The virus is genetically related to many other viruses that are responsible for encephalitis, chikungunya, dengue, Japanese encephalitis, West Nile fever, and yellow fever in humans [48,70,105]. The main vectors (mosquitoes) that are responsible for the transmission of Zika virus are Aedes aegypti and Aedes albopictus [6]. These mosquitoes are mostly found in tropical and subtropical regions [91]. Zika virus was frst discovered in rhesus monkeys in 1947 while researchers were studying yellow fever in Zika Forest, Uganda. It was isolated from Aedes africanus mosquitoes in the subsequent year [36]. In 1952, Zika virus was frst isolated from a captive rhesus monkey stationed in the Zika forest near Entebbe, Uganda [35]. The virus was also isolated from a batch of wild mosquitoes. No virus was recovered from tissues other than the brains of infected mice. Zika virus is a relatively unstable agent in suspensions [35]. In 1954, MacNamara [71] reported three cases of Zika infection among humans during an epidemic of jaundice (suspected of being yellow fever) in Eastern Nigeria. One of the patients was infected by Zika virus. The other two patients exhibited a rise in the titer of serum antibodies against this virus. In 1954, the frst human isolation was recorded in Nigeria [71,91]. For decades, the viral infection was occasionally reported in Africa and Southeast Asia [48,51]. Transmission and outbreak: One of the earliest experimental studies on the transmission of Zika virus was carried out in 1956, by Boorman and Porterfeld [18]. Since a supply of A. africanus was not available, the authors carried out experiments with A. aegypti. 267

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They have used an artifcial method to infect these A. aegypti mosquitoes with Zika virus. They found that initially for about 10 days, the virus was not detectable, but thereafter, the virus was detected at high levels. The infected mosquitoes were used to perform transmission studies among (i) rhesus monkeys; (ii) infant mice; and (iii) backfeeding into clean blood. A rhesus monkey was infected with Zika virus by the bites of three mosquitoes, 72 days after it was fed an infected blood meal. In 2007, the frst large outbreak was reported on Yap Island, Federated States of Micronesia [39]. In this epidemic, 49 confrmed cases were reported. During the outbreak, it was estimated that up to 73% of the residents of the Yap Island might have been infected [39,63]. In 2013, the largest outbreak of Zika virus was recorded in French Polynesia, a French territory located in the South Pacifc [24,25,81]. During this epidemic, 19,000 suspected cases were reported [25]. During the same epidemic, the frst evidence of the GuillainBarre syndrome related to Zika virus was observed [24]. In 2014, the virus dispersed from French Polynesia to many countries in the Pacifc Ocean. During this year, several cases were reported in Easter Island in Chile [103]. In particular, the virus seems to have been established in Latin American countries [97]. In 2015, the transmission of the virus was frst confrmed in Brazil [114]. The Brazilian Ministry of Health estimated the number of suspected cases between 440,000 and 1,300,000 that year. Further, during the Zika infection, the unusual incidence of microcephaly in newborn infants [32,56,78] and also some neurological disorders including the Guillain-Barré syndrome [8,33] were reported. WHO reported that by March 10, 2017, vector-borne Zika virus transmission was found in 84 countries, territories, or subnational areas [110]. A study published in ‘The Lancet’ reports that India is at high risk for the spread of Zika virus as it hosts a large number of tourists from the areas where there is an active circulation of the virus [80]. Most of Asia’s population lives in areas that are suitable for Zika virus transmission [77]. Other most vulnerable countries include China, the Philippines, Indonesia, Nigeria, Vietnam, Pakistan, and Bangladesh [15]. Globally, Zika virus has the potential to spread across all continents. Most cases show no symptoms, but when present, they are usually mild and can resemble dengue fever [19]. Symptoms generally last 1, ( d > 1) , then there is an outbreak of Zika (dengue), whereas the number of Zika (dengue) infections will directly decrease to zero if z < 1, ( d < 1) . Now, we briefy present the mathematical models that model the transmission dynamics of Zika virus, studied in the following sections. Model 1 is an SIR model for transmission of ZIKV to humans, which incorporates some optimal control strategies for the prevention, treatment, and use of an insecticide. The temporal model is analyzed for the following: equilibrium points, calculation of basic reproduction number 0 , global stability of the equilibrium points using non-linear Lyapunov functions, and optimal control analysis. Model 2 is an SEIR Zika epidemic transmission model. It is a modifcation of the SIR model. In this model, human-to-human infection and the vector (mosquito)-tohuman transmission are considered. The temporal model is analyzed for the equilibrium points. Bifurcation analysis showing the transcritical bifurcation and the optimal control analysis are presented. Model 3 is a deterministic SEIR model that investigates the transmission dynamics of Zika virus including the horizontal and vertical modes of transmission in both human and vector population. Analysis of the equilibrium points and the occurrence of the backward bifurcation phenomenon for a large vector population are presented in the temporal model of the system. Model 4 considers the transmission dynamics of ZIKV using a compartmental framework and analyzes the system that includes human vertical transmission of Zika virus, the birth of babies with microcephaly, and asymptotically infected individuals. The human population is divided into adults and newly born babies. Model 5 is a Zika virus SIR transmission model infuenced by human and vector mobility and human–mosquito interaction in space and time. Using the approach of Malinzi et al. [74] and Tireito et al. [102], the model system is investigated for the existence of traveling wave solutions. Model 6 is a criss-cross partial differential equation (PDE) model to understand how spatial heterogeneities of the vector and host populations infuenced the dynamics of the outbreak of Zika virus in Rio de Janeiro, Brazil. Finally, Model 7 is a Zika virus SEIR transmission model. The temporal model is analyzed for the stability of the equilibrium points and bifurcation. The diffusion model in two dimensions is analyzed and patterns are presented for the time evolution of different classes of population.

5.3 Model 1: Zika Virus SIR Transmission Model Bonyah and Okosun [17] formulated a mathematical model for transmission of Zika virus (ZIKV), which incorporated some optimal control strategies for prevention, treatment, and use of an insecticide. Pontryagin’s maximum principle was used to characterize the necessary conditions for optimal control of ZIKV. The authors explored an SIR model to examine the dynamics of transmission of ZIKV to humans [40]. The use of preventive and insecticide techniques to minimize the spread of the virus showed a reduction in both infected humans and mosquitoes. The human population at time t is given by N H (t) = SH (t) + I H (t) + RH (t), where SH (t), I H (t), and RH (t) respectively denote the number of individuals with Zika symptoms, infectious individuals, and individuals recovered from Zika. The overall vector (mosquito) population at time t is given by NV (t) = SV (t) + IV (t),

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where SV (t) and IV (t) respectively denote the susceptible and infectious mosquitoes. The transmission is governed by the following set of ODEs [17]: dSH = ˇ H − ( 1 − µ1 ) ° H SH ( IV + ˛ I H ) − µH SH , dt

(5.5a)

dI H = ( 1 − µ1 ) ° H SH ( IV + ˛ I H ) − ( µH + ˝ + ˙H µ2 ) I H , dt

(5.5b)

dRH = (˜ + °H µ2 ) I H − µ H RH , dt

(5.5c)

dSV = ˇV − ( 1 − µ1 ) °V SV I H − ( µV + ˛V µ3 ) SV , dt

(5.5d)

dIV = ( 1 − µ1 ) °V SV I H − ( µV + ˛V µ3 ) IV . dt

(5.5e)

All the model parameters are positive and the initial conditions are given by SH (0) > 0, I H (0) > 0, RH (0) > 0, SV (0) > 0, IV (0) > 0. The parameters in the model are defned as follows:

˜ H , ˜V : Transmission rates of ZIKV from humans to mosquitoes and from the vector (mosquitoes) to humans respectively. µH , µV : Natural death rates of host and vector respectively. ˜ H : Recruitment rate into susceptible population. ˜H : Recovery rate from treatment. ˜V : Recruitment rate into susceptible mosquito population. ˜V : Vector death rate from the use of an insecticide. 1/˜ : Average infectious period for humans. µ1 , µ2 , µ3 : (Control functions) Prevention, treatment, and insecticide controls respectively. These control functions are bounded and Lebesgue integrable. ˜ : Effective contact rate between infected and susceptible humans. Analysis of equilibrium points: The DFE point is given by E0 = ( ˙ H µ H , 0, 0, ˙V µV , 0 ) . Let us consider the case when the control measures are not employed. In that case, the EE point is given by E1 = SH* , I H* , RH* , SV* , IV* , where

(

RH* =

)

I H* ˛V ˇV ˜ I H* * ˇV ˛ S* I * ,SV = * , IV* = V V H = , I H ˛V + µV µH µV µV I H* ˛V + µV

(

SH* =

(

ˇ H µV I H* °V + µV

(

I ° H °V ˇV + µV ˛ I ° H + µ H * H

* H

( )

where I H* is a positive root of the equation a I H*

2

) )( I

* H

°V + µV

)

)

+ bI H* + c = 0, with

,

(5.6)

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Zika Virus: Modeling of Spread and Outbreak

(

)

a = ˜° H °V (˛ + µ H ) µV , c = µ H t3 µV2 − ° H  H °V V + ˜µV2 , t3 = (˛ + µH ) ,

(

)

b = t3  ˜V µ H µV + ˜ H ˛µV2 + ˜V ˘V  − ˛˘ H µV ˜ H ˜V . The solution given here differs from the solution given by the authors [17]. We analyzed the model using the above solutions. (In a private communication, the authors are informed of the results). We have a > 0. If c < 0, then the equation has always a positive root irrespective of the sign of b. (The signs of the coeffcients are then +, +, −, or +, −, −.) Calculation of the basic reproduction number 0 : The next-generation operator approach of Okosun and Makinde [73,85] was employed to determine the reproduction number. The matrices describing the new infections generated and the transition terms, F and V, are given by ˙ ˜ t1 F=ˇ ˇˆ t2

˙ t3 t1 ˘  and, V = ˇ 0  ˇˆ 0 FV −1 =

0 ˘ ˛H H ˛  , t2 = V V .  , t1 = µV  µH µV 

1 ˝ ° t1µV ˆ t3 µV ˆ t2 µV ˙

t1t3 ˇ . 0  ˘

The basic reproduction number 0 (the spectral radius of FV −1) is obtained as 0 =

= 1 =

2 =

1  ° t1µV + t1µV 4t2t3 + ° 2t1µV   2t3 µV 

(

)

°˛ H  H µV + ˛ H  H  4µ H ˛V V (˝ + µ H ) + ° 2 ˛ H  H µV2  2 µ H µV (˝ + µ H )

= 1 + 2 .

˜° H ˘ H 2° ˘ , 3 = V 2 V , 2 µ H (˝ + µ H ) ˜µV ˜H ˘H ° + µH

 ˜V ˘V ˝ 2˜H ˘H   µ µ 2 + 4µ 2 ° + µ  =  H V H( H )

21µH  ˜V ˘V ˝ 1  + = 1 ( 3 + 1 ) . ˝  µH µV2 2 µH 

˜ H ˘ H 21 (˛ + µH ) ˜ ˙ ˛µ  , t2 = V V = V 3 . = µH ˝ µV 2 0 depends on the infection rate of human and vector populations and is driven by the respective recruitment rates. The reduction of ZIKV is driven by the recovery rate and natural mortality of human and vector populations. Stability analysis: The Jacobian matrix of system (5.5) at the DFE E0 is obtained as We can write t1 =

JE0

 − µH  0  = 0  0   0

−° t1 −t3 + ° t1 ˛ −t2 t2

0 0 − µH 0 0

0 0 0 − µV 0

−t1 t1 0 0 − µV

    , t3 = (˛ + µH ) .   

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Spatial Dynamics and Pattern Formation in Biological Populations

Three of the eigenvalues of JE0 are − µ H , − µ H , − µV , and the remaining two eigenvalues are the solutions of the polynomial P( x) = x 2 + ˜1x + ˜0 = 0, where

˜1 = µV + t3 − ˛ t1 = ( µV + ˝ + µH ) − 21 (˝ + µH ) = µV + (˝ + µH ) ( 1 − 21 ) , ˜0 = t3 µV − t1 (˛µV + t2 ) = (˝ + µH ) µV −

21 (˝ + µH )  ˛µV 3   ˛µV +  = (˝ + µH ) µV 1 − 1 ( 2 + 3 )  . ˛ 2 

A suffcient condition that ˜1 > 0 and ˜0 > 0, is 1 ( 2 + 3 ) < 1. In this case, the equation has negative roots or a complex pair with negative real parts. Then, the DFE is asymptotically stable. Consider the Lyapunov function S ˘ °  ˙ S ˘ °  ˙ L(t) = µV ˆSH − SH0 − SH0 ln H0  + µV I H + H H ˆSV − SV0 − SV0 ln V0  + H H IV . SH  µH ˇ SV  µH ˇ

(5.7)

Taking the time derivative of (5.7) along the solutions of the system (5.5), we obtain  S − SH0  L(t) = µV  H    H − ° H SH ( IV + ˛ I H ) − µ H SH  + µV  ° H SH ( IV + ˛ I H ) − ( µ H + ˝ ) I H   SH  +

° H  H  SV − SV0  °H H [ °V SV I H − µV IV ].  [ V − °V SV I H − µV SV ] + µ H  SV  µH

(5.8)

Substituting the DFE solution, SH0 = ˛ H µ H and SV0 = ˛V µV in equation (5.8) and rearranging, we get

Lˇ(t) = − µV µH +

(S

H

− SH0

)

2

SH

(

)

2

(

)

2

)

2

SV − SV0 µ °  − V H H. µH SV

− I H ( µH + ˛ ) µV

 ° H  H  °V  V + ˝µV  IH  µH   µV

= − µV µH

(S

H

− SH0

)

2

SH

SV − SV0 µ °  − V H H. SV µH

− I H ( µH + ˛ ) µV

°  °  °   + IH  H H . V 2 V + ˝ H H  µV µH   µH = − µV µH

(S

H

− SH0 SH

)

2

−

(

0 µV ° H  H SV − SV . SV µH

− I H ( µH + ˛ ) µV

 21 ( µH + ˛ ) ˝ 3  + IH  + 21 ( µH + ˛ ) µV . 2 ˝  

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Zika Virus: Modeling of Spread and Outbreak

L′(t) = − µV µ H

= − µV µH

= − µV µH

= − µV µH

(S

H

− SH0

)

2

)

2

SV − SV0 µ β Λ − V H H. SV µH

)

2

SV − SV0 µ β Λ − V H H. SV µH

)

2

−

SH

(S

H

− SH0 SH

(S

H

− SH0 SH

(S

H

− SH0 SH

(

)

2

(

)

2

(

)

2

(

)

2

0 µV β H Λ H SV − SV . µH SV

SV − SV0 µ β Λ − V H H. µH SV

− I H ( µ H + γ ) µV + µV I H 1 ( µ H + γ ) ( 3 + 2 ) , − I H ( µ H + γ ) µV ( 1 − 1  ( 3 + 2 )) − I H ( µH + γ ) µV ( 1 − 1 + 2 ) ( 1 − 0 ) − I H ( µ H + γ ) µV ( 1 − 0 + 2 2 ) ( 1 − 0 ) ,

since 0 = 1 + 2 . L˛(t) < 0, if 0 < 1. L°(t) = 0, if SH = SH0 , SV = SV0 , I H = IV = 0. Hence, the largest compact invariant set ( SH , I H , RH , SV , IV ) ˛˝ : L˙(t) = 0 is the singleton set {E0 }. Thus, E0 is globally asymptotically stable under the above conditions, when all solutions of system (5.5) in R 5 are bounded [65]. Stability of the endemic equilibrium: Denote t4 = SH* ˜ H , t5 = SV* ˜V , t6 = ˜ H ° I H* + IV* , t8 = t6 + µ H , t9 = µV + I H* °V , t10 = t3 − ˜ t4 , and t7 = t9 + t10. The Jacobian matrix of the system (5.5) at the EE point E1 is given by

(

(

JE1

)

 −˜ ° I * + I * − µ H H V H   ˜ H ° I H* + IV*  = 0   0  0     −t8  t6  = 0  0   0 

(

)

−° SH* ˜ H

0

0

−t3 + ° SH* ˜ H

0

0

˝

− µH

0

−˜ S

0

− µV − I H* ˜V

˜V SV*

0

I H* ˜V

* V V

−° t4 −t10 ˝

0 0 − µH

0 0 0

−t4 t4 0

−t5

0

− µV − I H* ˜V

0

0

I ˜V

− µV

t5

* H

)

− ˜ H SH*   ˜ H SH*   0   0  − µV  

    .    

The characteristic equation of the Jacobian matrix JE1 is given by

(

)

q ( x ) = ( x + µ H )( x + µV ) x 3 + °12 x 2 + °11x + °10 = 0, where ˜12 = t8 + t9 + t10 = t7 + t8 , ˜11 = t7 t8 + t9t10 + ° t4t6 − t4t5 ,

˜10 = t8 [ t9t10 − t4t5 ] + t4t6 [° t9 + t5 ]. (The solution given here differs from the solution given by the authors [17]). Two of the eigenvalues of JE1 are − µ H , − µV , and the remaining three eigenvalues are the solutions of

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Spatial Dynamics and Pattern Formation in Biological Populations

the polynomial q ( x ) = 0. The roots of q ( x ) = 0 are negative or have negative real parts when the following Routh–Hurwitz criteria are satisfed: ˜10 > 0, ˜11 > 0, ˜12 > 0, ˜12˜11 − ˜10 > 0. From these conditions, we obtain the following suffcient conditions: t3 > σ t4 , t3t9 > t4 ( t5 + σ t9 ) , and A > B, where A = t4t8 ( t5 + σ t9 ) + ( t3 + t8 + t9 ) ( t8t9 + t3 ( t8 + t9 ) + σ t4t6 ) + σ t42 ( t5 + σ ( t8 + t9 )) , B = t4t6   ( t5 + σ t9 ) + t3t8t9 + ( t3 + t8 + t9 ) ( t4t5 + σ t4 (t8 + t9 )) + σ t4 ( t8t9 + t3 ( t8 + t9 ) + σ t4t6 ) . If these conditions are satisfed, then the EE point E1 is asymptotically stable. Global stability of the endemic equilibrium: Consider the following non-linear Lyapunov function to establish the global stability of the EE   ˝S ˇ ˝ I ˇ ˝ R ˇ V1 = SH − SH* − SH* ln ˆ H*  + ˜ 1  I H − I H* − ln ˆ H*   + ˜ 2 RH − RH* − RH* ln ˆ H*   ˙ SH ˘ ˙ IH ˘  ˙ RH ˘      ˝ S ˇ ˝ I ˇ + ˜ 3 SV − SV* ln ˆ V*   + ˜ 4  IV − IV* − IV* ln ˆ V*   . ˙ SV ˘  ˙ IV ˘   

(5.9)

Differentiating the above equation, simplifying, and setting ˜ 1 = ˜ 2 = ˜ 3 = ˜ 4 = 1, we obtain  S*     S*   I*  R*  I*  V1 = 1 − H  SH + 1 − H  IH + 1 − H  R H + 1 − V  SV + 1 − V  IV IH  RH  IV   SH     SV   =−

µH SH − SH* SH

(

µ +° ) − ( I ) (I 2

H

H

− I H*

H

)

2

−

µH RH − RH* RH

(

)

2

−

µV SV − SV* SV

(

)

2

−

µV IV − IV* IV

−

˝˛ H ˛ SH I H* − SH* I H SH I H − SH* I H* − v SV IV* − SV* IV SV I H − SV* I H* . SH I H SV IV

(

)

2

° ˛ I H − I H* RH − RH* − H SH I H* − SH* I H SH I v − SH* I v* RH SH I H

(

+

(

)(

)

)(

)

(

)(

(

)(

)

The equation can be written as

(

V1 = −b11 SH − SH*

( (S (S

− b45

V

V

2

(

− b22 I H − I H*

)

2

(

− b33 RH − RH*

) + b (S − S )( I − S )( I − I ) − b ( S − S )( I − I ) ,

− b55 IV − IV* − b24

)

2

12

H

* V

H

* H

* V

V

* V

* H

15

H

H

)

2

(

− b44 SV − SV*

) ( )( )( I − I ) +  b ( I

)

2

− I H* + b23 I H − I H* RH − RH*

− SH*

V

* V

25

H

)(

)

− I H* IV − IV*

)

)

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Zika Virus: Modeling of Spread and Outbreak

where b11 =

µH ° H I H* * ° I * S* µ + ˝ − ˛° H SH* µ + IV + ˛ I H* , b12 = H v H , b22 = H , b33 = H , SH SH I H I H SH IH RH

(

)

b44 =

µV °V I H* IV* µ ˝ ° ° I* + , b55 = V , b23 = , b15 = H I H* , b24 = V V , SV SV IV IV RH IH IV

b25 =

° H SH* °V SV* ° I * S* + , b45 = V H V . IH IV SV IV

Suffcient conditions for the quadratic form V1 to be negative defnite are the following: 2 2 2 b22 > 0 if µ H + ° > ˛˝ H SH* , b23 < b22b33 , b12 < b11b22 , b15 < b11b55 , 2 2 2 b25 < b22b55 , b24 < b22b44 , b45 < b44b55 .

(5.10)

Thus, the EE point whenever it exists is globally asymptotically stable under the conditions (5.10). Numerical simulations: To study the stability, numerical computations are performed using the following parameter values: ˜ H = 100, ˜ H = 0.0001, ˜ = 0.05, µH = 1 (365 × 60), ˛ = 1, 000, ˜V = 1, 000, ˜V = 0.00002, µV = 0.07142. The initial conditions are taken as (100,000, 2, 2, 2,000, 10). The DFE is obtained as E0 2.19 × 106 , 0, 0,1.4 × 10 4 , 0 with 0 = 0.932033. Local asymptotic stability of DFE is displayed in Figure 5.1. Now, if we take the values ˜ H = 0.0005, and ˜V = 0.0002, we obtain the EE as E1 (50969.3, 0.0976726, 2.13903 × 106 ,13992.2, 3.82772) with 0 = 6.579073. Local asymptotic stability of EE is displayed in Figure 5.2.

(

)

FIGURE 5.1 Local asymptotic stability of the DFE E0.

280

Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 5.2 Local asymptotic stability of the EE, E1 with β H = 0.0005 and βV = 0.0002.

5.3.1 Optimal Control Analysis The authors [17] used the Pontryagin’s maximum principle to determine the necessary conditions for the optimal control of the disease. The problem is to minimize the number of ZIKV-infected human hosts and the cost, which is involved in the mass treatment and insecticide controls. The objective function is taken as J ( µ1 , µ 2 , µ 3 ) =

F

ˇ

˜ ˘ AI

H

+ BIV +

0

d1 2 d2 2 d3 2  µ1 + µ2 + µ3  dt , 2 2 2  

(5.11)

where d1 , d2 , and d3 denote the weighting constants for prevention, treatment, and insecticide efforts, respectively. The costs of the prevention, treatment, and insecticide are non-linear and assume quadratic function forms. Optimal controls µ1* , µ2* , and µ3* are to be determined such that

(

)

J µ1* , µ2* , µ3* = min J ( µ1 , µ2 , µ3 ) , ˇ =

{( µ , µ , µ )|0 ˘ µ 1

2

3

i

}

˘ 1, i = 1, 2 .

The model system with control is dSH = ˇ H − ( 1 − µ1 ) ° H SH ( IV + ˛ I H ) − µH SH , dt dI H = ( 1 − µ1 ) ° H SH ( IV + ˛ I H ) − ( µH + ˝ + ˙H µ2 ) I H , dt

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Zika Virus: Modeling of Spread and Outbreak

dRH = (˜ + °H µ2 ) I H − µ H RH , dt dSV = ˇV − ( 1 − µ1 ) °V SV I H − ( µV + ˛H µ3 ) SV , dt dIV = ( 1 − µ1 ) °V SV I H − ( µV + ˛H µ3 ) IV . dt

(5.12)

The necessary conditions that an optimal solution must satisfy come from the maximum principle of Pontryagin et al. [92]. This principle converts (5.11)-(5.12) into a problem of minimizing pointwise the following Hamiltonian H , with respect to µ1, µ2, and µ3. H = AI H + BIV + + °IH + °SV

d1 2 d2 2 d3 2 µ1 + µ2 + µ3 + °SH {  H − ( 1 − µ1 ) ˛ H SH ( IV + ˝ I H ) − µH SH } 2 2 2

{(1 − µ ) ˛ S ( I + ˝ I ) − ( µ + ˙ + ˆ µ ) I } + ° {(˙ + ˆ µ ) I { − (1 − µ ) ˛ S I − ( µ + ˆ µ ) S } + ° {(1 − µ ) ˛ S I − ( µ 1

V

H H

1

V

V V H

H

H

V

H

V

3

2

V

H

IV

RH

1

H

V V H

2

}

H

− µ H RH

V

+ ˆV µ3 ) IV } , (5.13)

where ˜SH , ˜ IH , ˜RH , ˜SV , ˜ IV are the adjoint variables or co-state variables. The system of equations is found by taking the appropriate partial derivatives of the Hamiltonian (5.13) with respect to the associated state variables. We summarize the results of the authors [17]: Given optimal controls µ1* , µ2* , µ3* and solutions SH , I H , RH , SV , IV of the corresponding state system (5.11)-(5.12) that minimize J ( µ1 , µ2 , µ3 ) over U , there exist adjoint variables ˜SH , ˜ IH , ˜RH , ˜SV , ˜ IV satisfying the equation −

d˜i ˝H = , and the transversality conditions dt ˝xi

˜SH ( t f ) = ˜ IH ( t f ) = ˜RH ( t f ) = ˜SV ( t f ) = ˜ IV ( t f ) = 0, where   SH ° H ( IV + ˛ I H ) ( ˝ IH − ˝SH ) + SV °V I H ( ˝ IV − ˝SV )   µ1* = min 1, max  0,  , d1    

 ˘ °H I H ( ˛ IH − ˛RH )   µ2* = min 1, max  0,  , d2      ˘ °V ( SV ˛SV + IV ˛ IV )   µ3* = min 1, max  0,  . d3     Corollary 4.1 of Fleming and Rishel [45] gives the existence of an optimal control due to the convexity of the integrand of J with respect to µ1 , µ2 , and µ3 , a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the state variables. The differential equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function, evaluated at the optimal control.

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Spatial Dynamics and Pattern Formation in Biological Populations

Then, the adjoint equations can be written as [45]: −

d˜SH = µH ˜SH + ( 1 − µ1 ) ˛ H ( ˜SH − ˜ IH ) ( IV + ˝ I H ) , dt

−

d˜ IH = −A + ( µ2˛H + µ H + ˝ ) ˜ IH − (˝ + µ2˛H ) ˜RH dt + ˙ ( 1 − µ1 ) ˆ H SH ( ˜SH − ˜ IH ) + ( 1 − µ1 ) ˆV SV ( ˜SV − ˜ IV ) ,

−

d˜RH = µ H ˜ RH , dt

−

d˜SV = ( 1 − µ1 ) ˛V I H ( ˜Sv − ˜ IV ) + ( µV + ˝V µ3 ) ˜SV , dt

−

d˜ IV = − B + ( 1 − µ1 ) ˛ H SH ( ˜SH − ˜ IH ) + ( µV − ˝V µ3 ) ˜ IV . dt

Authors [17] have presented the simulation results of the model (5.12) for different cases of control mechanism including the case when all control mechanisms ( µ1 , µ2 , µ3 ) are used to optimize the objective function J. That is, all the prevention, treatment, and use of insecticide controls are optimized. A signifcant difference in the number of infected humans I H in the controlled cases and the cases without control can be observed from Figure 5.3a. The result depicted in Figure 5.3b suggests that this strategy is very effcient and effective in the control of the number of infected mosquitoes IV . Numerical results on the optimal controls suggest that the best strategy is to combine all controls, that is, preventive, treatment, and insecticides to control the disease.

FIGURE 5.3 Effect of all controls on virus transmission, with (a) β H = 0.2 and (b) βV = 0.09. [From Bonyah, E., Okosun, K. O. 2016. Mathematical modeling of Zika virus. Asian Pac. J. Trop. Dis. 6(9), 673–679 [17]. Copyright 2016. Reprinted with kind permission from Elsevier.]

Zika Virus: Modeling of Spread and Outbreak

283

5.4 Model 2: Zika Virus SEIR Transmission Model Bonyah et al. [16] modifed Model 1, described in Section 5.3, by incorporating the exposed population and called it an SEIR Zika epidemic model. The authors have considered the human-to-human infection as well as the vector (mosquito)-to-human transmission. The total human population N H (t) is taken as N H (t) = SH + EH + I H + RH , where SH (t), EH (t), I H (t), and RH (t) are the susceptible humans, exposed humans, infected humans, and recovered humans respectively. Similarly, the mosquito population NV (t) is partitioned as NV = SV + EV + IV , where SV (t), EV (t), and IV (t) are the susceptible vector, exposed vector, and infected vector respectively. The model system is given by [16]: dSH = ˇ H − ˜ H SH ( IV + ° I H ) − µ H SH , dt

(5.14a)

dEH = ˜ H SH ( IV + ° I H ) − ( µ H + ˝ H ) EH , dt

(5.14b)

dI H = ˜ H EH − ( µH + ˛ + ˝ ) I H , dt

(5.14c)

dRH = ˜ I H − µ H RH , dt

(5.14d)

dSV = ˝V − ˜V SV I H − µV SV , dt

(5.14e)

dEV = ˜V SV I H − ( µV + ˛ V ) EV , dt

(5.14f)

dIV = ˜ V EV − µV IV . dt

(5.14g)

All the model parameters are positive and the initial conditions are given by SH (0) > 0, EH (0) > 0, I H (0) > 0, RH (0) > 0, SV (0) > 0, EV (0) > 0, IV (0) > 0. The parameters are defned as follows:

˜ H , ˜V : Effective contact rate between susceptible humans and infected mosquitoes and the transmission rate from infected humans to susceptible vectors respectively. µH , µV : Natural mortality rates due to each subpopulation of human and vector groups respectively. ˜ H , ˜V : Recruitment of susceptible human and susceptible mosquito populations respectively. ˜: Effective contact rate between infected humans and susceptible humans that can result in infection. γ, η: Natural and treatment rates. ˜ H , ° V : Progression rates from exposed to infected human and mosquito populations respectively.

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Spatial Dynamics and Pattern Formation in Biological Populations

Let the total dynamics of the human population be given by N H˛ (t) = ˙ H − µ H N H − ° I H . Hence, N H° (t) + µ H N H ˝ ˙ H . Integrating the above inequality, the following inequality is obtained [14] 0 < N H ( SH , EH , I H , RH ) ˙

ˆH 1 − e − µH t + N H ( 0 ) e − µH t , µH

(

)

where N H ( 0 ) = ( SH ( 0 ) + EH ( 0 ) + I H ( 0 ) + RH ( 0 )) . As t ˜ °, we obtain 0 < N H ˛ ˝ H /µ H . The total dynamics of vector population is given by NV˝ ( t ) = ˆV − µV NV . The solution of this equation is NV ( SV ,EV , IV ) =

˙V 1 − e − µV t + NV ( 0 ) e − µV t , µV

(

)

where NV ( 0 ) = ( SV ( 0 ) + EV ( 0 ) + IV ( 0 )) . As t ˜ °, we obtain NV ( t ) = ˙V /µV . Also ˆ=

{(S

H

, EH , I H , RH , SV , EV , IV ) ˘R+7 |0  SH + EH + I H

+ RH   H µ H , and 0  SV + EV + IV  V /µV } which is positively invariant, dissipative and the global attractor is attained in ˜. Analysis of equilibrium points and the basic reproduction number 0 : The DFE point is given by E0 = ( ˙ H /µ H , 0, 0, 0, ˙V /µV , 0, 0 ). To obtain the basic reproduction number, form the matrices ˆ ˘ F=˘ ˘ ˘ ˇ

0 0 0 0

˜t1 0 t2 0

0 0 0 0

ˆ k1 t1  ˘  −°h 0  , V=˘ ˘ 0 0  ˘  ˘ˇ 0  0

0 k2

0 0

0 0

0 0

k3 −˛ V

0 µV

  .   

where t1 = ˜ H ˝ H /µ H , t2 = ˜V ˝V /µV , k1 = µ H + ° H , k2 = µH + ° + ˛ , and k3 = µV + ° V . Taking t5 = ˜ H ˙ H ° H , t6 = ˜V˛ V ˙V , the basic reproduction number of the model, which is the spectral radius of the matrix ˜ FV −1 , is given by

(

0 =

where 1 =

)

˜t5 ˜ 2t52 t5t6 + = 1 + 2 + 4 µ H2 k12 k22 µ H µV2 k1k2 k3 2 µ H k1 k 2

˜t5 , and 2 = 2 µ H k1 k 2

˜ 2t52 t5t6 t5t6 + = 12 + 4 µH2 k12 k22 µH µV2 k1k2 k3 µH µV2 k1k2 k3

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Zika Virus: Modeling of Spread and Outbreak

We fnd that 2 > 1 , and 0 > 2 1 . Endemic equilibrium: The EE point of the system is given by E1 = SH* , EH* , I H* , RH* , SV* , EV* , IV* , where

(

SH* =

)

° H I H* ˘ H ( k3 ˛µV t3 + t6 ) k3 ˘ H µV t3 ˝ I H* ˘V I H* °V ˘V * * * , , , EH* = , = , = S = E R H V V * k3 µV t4t3 + ° H I H t6 µH t3 k 3 t3 k1 k3 µV t4t3 + ° H I H* t6

(

IV* =

)

I H* t6 , t3 = µV + I H* °V , t4 = ˛° H I H* + µ H . µV k3t3

(

)

I H* ˜ 0, satisfies the equation aI H* 2 + bI H* + c = 0,

(5.15)

where a = ˜ k1k2 k3 ° H °V µV > 0, b = ˜ H ˜V ( k1k2° V V − k3 ˛ H ˝ H µV ) + k1k2 k3 µV ( ˛˜ H µV + µ H ˜V ) ,

(

c = k1k2 k3 µ H µV2 ( 1 − 0 )( 1 + 0 − 21 ) = k1k2 k3 µ H µV2 1 − 0*

)

and 0* = 02 + 21 ( 1 − 0 ). Note that 0 > 2 1 . We have a > 0. Now, c > 0 when 0 < 1, and c < 0, when 0 > 1. The signs of b and c decide the existence of a positive solution. Consider the case c > 0, that is 0 < 1 : (i) when b > 0, then the signs of the coeffcients are + , + , +. In this case, there is no positive root. (ii) When b < 0, then the signs of the coeffcients are + , − , +. In this case, there are two positive roots if b 2 − 4ac > 0. That is, two positive equilibrium points exist. For 0 = 1, c = 0, and if b < 0, then a unique positive solution I H* = −b/a exists. Now, consider the case c < 0, that is 0 > 1 : irrespective of the sign of b, there is a positive root, that is, a unique EE point exists. Using MATLAB 2013, we have simulated the system (5.14) taking the parameter values  as ˜ H = 0.0022, ° v = 0.0009, µ H = 0.01, ˝ v = 0.3, µ v = 0.003, ˇ v = 1.3, ˇ H = 0.4, ˙ = 0.11, ˜ = 0.029, ° = 0.0614799, for the two cases: (i) ˜ H = 0.00005. The value of 0 is obtained as 0 = 0.505787. (ii) ˜ H = 0.0005. The value of 0 is obtained as 0 = 1.59964. Initial conditions are taken as [ 100, 2, 3,1, 500, 5,70 ]. For this set of parameter values, we obtain E0 = ( 40, 0, 0, 433.333, 0, 0 ) , E1 = (16.719,19.0828, 0.231333,1.42223, 405.212, 0.278432, 27.8432). The time series is presented in Figure 5.4a for 0 < 1 and in Figure 5.4b for 0 > 1. Stability analysis: We can obtain the conditions on the coeffcients of the characteristic equation so that the DFE point E0 = ( ˙ H µ H , 0, 0, 0, ˙V µV , 0, 0 ) of the model system (5.14) is locally asymptotically stable (Problem 5.1, Exercise 5). The endemic equilibrium point E1 of the model system (5.14) is asymptotically stable for 0 > 1 (Problem 5.2, Exercise 5 [5]). The authors have derived the Jacobian matrix without considering RH and obtained a characteristic equation, which is of order six. We have derived the Jacobian matrix considering RH and obtained a characteristic equation of order seven. The results in both the cases are the same.

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 5.4 Time series for the model system (5.14): (a) 0 < 1 and (b) 0 > 1.

5.4.1 Bifurcation Analysis Using the center manifold theory [27,86], the authors [16] have established the backward bifurcation phenomenon by taking into account the transmission rate ˜ H as a bifurcation parameter. 0* = 1, if and only if

˜ H* = ˜ H =

k1k2 k3 µH µV2 .  H ˛ H k3 ˝µV2 + ˜V˙ V V

(

)

Denote the variables of the system as SH = x1 , EH = x2 , I H = x3 , RH = x4 ,SV = x5 ,EV = x6 , T IV = x7, and the vector of variables as x = ( x1 , x2 , x3 , x4 , x5 , x6 , x7 ) . The model can then be T reformulated in the form ( dx/dt ) = F ( x ), with F = ( f1 , f2 , f3 , f 4 , f5 , f6 , f7 ) , where dx1 =  H − ˜ H x1 ( x7 + ° x3 ) − µ H x1 , dt dx2 dx = ˜ H x1 ( x7 + ° x3 ) − k1x2 , 3 = ˝ H x2 − k2 x3 , dt dt dx4 dx = ˙ x3 − µ H x4 , 5 = V − ˜V x3 x5 − µV x5 , dt dt dx6 dx = ˜V x3 x5 − k3 x6 , 7 = ˆ V x6 − µV x7 . dt dt For the system (5.16), the Jacobian matrix evaluated at the DFE E0 is given by

(5.16)

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Zika Virus: Modeling of Spread and Outbreak

 − µH  0   0  0 J ( E0 ) =   0    0    0 

0

− °t1

0

0

0

−t1

−k1 ˛H

°t1 −k2 ˝

0

0

0

t1

0

0

0

0

0

0

0

0

− µv

0

0

0

0

−k3

0

0

0

˙v

− µv

µv k3 ( °t1 ˛ H − k2 k1 ) t1 ˛ H ˙ v

0 0

0 − µH

−

0

µv k3 ( °t1 ˛ H − k2 k1 ) t1 ˛ H ˙ v 0

     .        

(5.17)

The characteristic equation has a simple zero eigenvalue and the other eigenvalues are negative or have negative real parts (Problem 5.3, Exercise 5). Hence, the center manifold theorem [27] can be applied. For applying the theorem, we need to calculate the values of the parameters a and b as defined in the theorem. The right and left eigenvectors of   J (E0 ) T denoted respectively by w = ( w1 , w2 , w3 , w4 , w5 , w6 , w7 ) and v = ( v1 , v2 , v3 , v4 , v5 , v6 , v7 ) are obtained as w1 = − w6 =

w ° w ° ˛ w k k ˙° H  w 2 k1 , w2 > 0, w3 = 2 H , w4 = 2 H , w5 = 2 3  1 − , k2  µH k2 k2 µH ˝ V  t1

w2 µV  k1 ˙° H  ˙° H  k − , w7 = w 2  1 − ,  t1 ˝ V  t1 k2  k2 

and v1 = v4 = v5 = 0, v6 > 0, v2 =

v6 µV k3 vkkµ vk , v3 = 6 1 3 V , v7 = 6 3 . t1° V t1° V ˛ H °V

The values of the non-zero derivatives are ˙ 2 f1 ˙2 f2 ˙ 2 f1 ˙2 f2 =− = −˜H ° , =− = −˜H , ˙ x1 ˙ x3 ˙ x1 ˙ x3 ˙ x1 ˙ x7 ˙ x1 ˙ x7 ˙2 f5 ˙ 2 f6 ˙ 2 f1 ˙2 f2 =− = − ˜V , =− = − ( x7 + ° x3 ) , ˙ x3 ˙ x5 ˙ x3 ˙ x5 ˙ x1 ˙˜ H ˙ x1 ˙˜ H ˙2 f2 ˙ 2 f1 ˙2 f2 ˙ 2 f1 =− = − ° x1 , =− = −x1 . ˙ x3 ˙ ˜ H ˙ x3 ˙ ˜ H ˙ x7 ˙˜ H ˙ x7 ˙˜ H The coeffcients a and b are given by 7

a=

˜

7

v k wi w j

k , i , j =1

˜

ˆ2 f k ˆ2 fk (E0 , ° H ) , b = vk wi (E0 , ° H ). ˆ xi ˆ x j ˆ xi ˆ ° H k ,  i =1

Substituting the values of the derivatives, we obtain

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Spatial Dynamics and Pattern Formation in Biological Populations

a = v2 w1w3 ˜ H ° + v2 w1w7 ˜ H + v6 w3 w5 ˜V , =−

(

k3 v6 w22 k12 k22 ˜ H µ v − k1k2t1˜V µ H ˝ H + °t1˜ H ˜V  H ˝ H2

),t

1

k t ˜ ˙ vH 2 2 1 H

=

˜H H . µH

k1k3 v6 w2 µ v . t1˛ H˝ v

b = v2 ( w3 ˜ + w7 ) x1 =

We fnd b > 0. It follows from the center manifold theorem [27] that the system (5.14) undergoes forward transcritical bifurcation if a < 0, that is, under the condition k1k2 ˜V ˙ H ° H < k12 k22 µ v + ˝t1˜V ˙ H ° H2 . To test this conclusion, we performed the following computations. For example, for the values of the parameters ˜ H = 0.0022, ˜ H = 0.0002, ˜V = 0.0009, µH = 0.01, ˜ V = 0.3, µV = 0.003, ˜V = 1.3, ˜ H = 0.4, ˜ = 0.11, ˜ = 0.029, and ˜ = 0.0614799, the values of a and b are obtained as a = −0.000124 < 0 and b = 23.104 > 0. The bifurcation diagram generated by the MATCONT package (in MATLAB) is presented in Figure 5.5. It shows that the DFE is locally asymptotically stable for 0 < 1 and the unique EE is locally asymptotically stable for 0 > 1. This confrms the existence of a transcritical bifurcation in the system. The branch point indicates the changes in the stability of the equilibrium points. (The authors [16] have shown that backward bifurcation occurs for a > 0.) 5.4.2 Optimal Control Analysis To study the optimal control of the SEIR model of Zika virus, the authors [16] included time-dependent controls in the model and explored the appropriate optimal strategy for controlling the virus. We use the following three control variables: (i) u1 ( t ) represents the efforts on preventing Zika infections through bednets; (ii) u2 ( t ) represents the efforts on the treatment of Zika-infected individuals; and (iii) u3 ( t ) represents the efforts to control/ eliminate Zika-spreading mosquitoes through insecticides spray. Consider the objective functional as [62,87,92], J ( u1 , u2 , u3 ) =

tf

ˆ

˜ ˘ˇ BE 0

H

+ CI H + DEV + EIV +

a1 2 a2 2 a3 2  u1 + u2 + u3  dt , 2 2 2 

(5.18)

where B, C , D, E are the balancing cost factors due to scales and a1 , a2, and a3 denote respectively the weighting constants for making uses of bednets, which have the potential

FIGURE 5.5 Bifurcation diagram showing transcritical bifurcation in the system (5.14).

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Zika Virus: Modeling of Spread and Outbreak

to reduce the spread of the disease (prevention); effective treatment activities, which include the effcacy of the drugs and encouraging patients to take their drugs timely; and availability of insecticides for spraying against the mosquitoes. The costs associated with prevention, treatment, and insecticide are taken to be of the non-linear form. Optimal controls u1* , u2*, and u3* are to be determined such that

(

)

J u1* , u2* , u3* = min J ( u1 , u2 , u3 ) , ˆ = {( u1 , u2 , u3 )|0 ˇ ui ˇ 1, i = 1, 2, 3} . The model system with control is taken as dSH = ˇ H − ( 1 − u1 ) ˜ H SH ( IV + ° I H ) − µ H SH , dt dEH = ( 1 − u1 ) ˜ H SH ( IV + ° I H ) − ( µ H + ˝ H )EH , dt dI H dRH = ˜ H EH − ( µ H + u2˛ + ˝ ) I H , = u2˛ I H − µ H RH , dt dt

(5.19)

dSV = ˆV − ( 1 − u1 ) ˜V SV I H − u3 µV SV , dt dEV dI = ( 1 − u1 ) ˜V SV I H − (u3 µV + ˛ V )EV , V = ˛ V EV − u3 µV IV . dt dt The necessary conditions that an optimal solution must satisfy are obtained from the maximum principle of Pontryagin et al. [92]. This principle converts (5.18)-(5.19) into a problem of minimizing pointwise the following Hamiltonian H , with respect to u1 , u2, and u3 H = BEH + CI H + DEV + EIV +

a1 2 a2 2 a3 2 u1 + u2 + u3 + ˜SH {  H − ( 1 − u1 ) ° H SH ( IV + ˛ I H ) 2 2 2

{(1 − u ) ° S ( I + ˙ I ) − ( µ + ˆ ) E } + ˜ { ˆ E − ( µ + u ˇ + ˘ ) I } + ˜ {u ˇ I − µ R } + ˜ {  − ( 1 − u ) ° S I

− µ H SH } + ˜EH H

2

1

H

H H

RH

2

V

H

H

H

H

H

H

SV

H

V

IH

1

+ ˜EV {( 1 − u1 ) °V SV I H − (u3 µV +  V )EV } + ˜ IV { V EV − u3 µV IV } ,

H

H

V V H

− u3 µV SV } (5.20)

where ˜SH , ˜EH , ˜ IH , ˜RH , ˜SV , ˜EV , and ˜ IV constitute the adjoint variables or co-state variables. The solution of the system is obtained by appropriately taking partial derivatives of the Hamiltonian (5.20) with respect to the associated state variables. The result is summarized in the following theorem [16]: Theorem 5.1 Given optimal controls u1* , u2* , u3* and solutions SH , EH , I H , RH , SV , EV , IV of the corresponding state system (5.18) and (5.19) that minimize J ( u1 , u2 , u3 ) over Γ, adjoint variables ˜SH , ˜EH , ˜ IH , ˜RH , ˜SV , ˜EV , ˜ IV exist satisfying

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Spatial Dynamics and Pattern Formation in Biological Populations

−

d˜i ˝H = , dt ˝i

where i = SH , EH , I H , RH , SV , EV , IV ; with the transversality conditions

˜SH ( t f ) = ˜EH ( t f ) = ˜ IH ( t f ) = ˜RH ( t f ) = ˜SV ( t f ) = ˜EV ( t f ) = ˜ IV ( t f ) = 0,

(

 SH ˜ H IV ( °EH − °SH ) + I H ˛SH ˜ H ( °EH − °SH ) + SV ˜V ( °EV − °SV )  and u1* = min 1, max  0, a1  

)    

 ˇ ˜ I H ( ° IH − °RH )   u2* = min 1, max  0, , a2  ˘    ˙ µ S ° + µV EV °EV + µV IV ° IV ˘  u3* = min 1, max ˇ 0, V V SV  . a3  ˆ  Simulation results of a study [16] suggest that the best strategy to minimize the spread of Zika virus is to optimize all the three controls. The reduction of the disease can only be attained when attention is given to all the three controls. The activation of all the controls has a greater effect on minimizing the number of infected and exposed humans in the communities. Application of all the three controls is the best strategy to minimize the number of infected mosquitoes IV , which eventually can lead to the reduction of the spread of Zika virus. The control profles suggest that control u1 be kept at a maximum of 100% for about 40 days and gradually reduced to 25% and then kept the same during the entire 120 days period. The control u2 is maintained at 8% and then gradually decreased and maintained during the entire 120 days. The control u3 is kept at a maximum of 100% for 20 days and then decreased to 25%, which is maintained throughout the entire 120 days.

5.5 Model 3: Zika Virus SEIR Horizontal and Vertical Transmission Model Imran et al. [60] proposed a deterministic SEIR model to investigate the transmission dynamics of Zika virus including the horizontal and vertical modes of transmission in both humans and vectors. To consider a vertical transmission in the model, the authors made the assumption that a fraction of newborn individuals from parents in Eh and I h classes will be infected and thus remain in Eh class before becoming infectious [100]. Because of this vertical transmission, a fraction of susceptible individuals will enter the exposed class. Thus, the newborn individuals entering the Eh class are represented by ( p˛hEh + q˛h I h ). Similarly, the newborn individuals exiting the Sh class are represented by ( pBhEh + qBh I h ). In both cases, 0 ˜ p ˜ 1 and 0 ˜ q ˜ 1; p is the fraction of new babies from exposed individuals, and q is the fraction of newborn babies from infected individuals. The exposed population Eh (t) is depleted at the natural death rate µ h . Additionally, exposed individuals develop symptoms and move into the infected class I h at a rate ˜ . The infected

Zika Virus: Modeling of Spread and Outbreak

291

population I h (t) is decreased due to the natural death rate µ h , the disease-induced death rate ˜ I , and the recovery rate of infected individuals ˜ . The recovered population Rh (t) is decreased due to the natural death rate µ h. The susceptible human population Sh (t) has a constant recruitment rate ˜ h = Bh K h where K h is the carrying capacity and Bh is the natural birth rate [20]. Susceptible individuals get infected with Zika virus (due to contact with infected vectors) at a rate ˜ h and thus enter the exposed class Eh. The total human population N h (t) at time t is divided into four mutually exclusive classes: (i) susceptible humans Sh (t), (ii) exposed humans Eh (t), (iii) infected humans I h (t), and (iv) recovered humans Rh (t). It was assumed that individuals who recover from infection with a particular serotype of Zika virus gain lifelong immunity to it. Similarly, the total vector population N v (t) is divided into three mutually exclusive classes: (i) susceptible vectors Sv (t), (ii) exposed vectors Ev (t), and (iii) infected vectors I v (t). It is assumed that vectors (mosquitoes) infected with Zika virus never recover. The susceptible vector population Sv (t) has a constant recruitment rate ˜ v = Bv K v and a natural death rate µ v. A fraction of the offspring in Ev and I v classes will be infected and thus remain in Eh class before becoming infectious. Because of this vertical transmission, a fraction of susceptible individuals will enter the exposed class. Thus, the newborn individuals entering the Eh class are represented by ( rBvEv + sBv I v ), where 0 ˜ r ˜ 1 and 0 ˜ s ˜ 1, and similarly, these offsprings are exiting the Sv class, where r is the fraction of offsprings from exposed individuals and s is the fraction of offsprings from infected individuals. Susceptible vectors are infected with Zika virus (due to effective contact with infected humans) at a rate ˜ v and thus move to the exposed vector class Ev. The exposed vector class Ev (t) is depleted due to the natural death rate µ v. In addition, exposed vectors develop symptoms and move to the infected vector class I v (t) at a rate ˜ . Infected vectors, in addition to the natural death rate µ v, die at a disease-induced death rate ˜ v . The forces of infection are given by, ˜ h = Chv I v /N h and ˜ v = Chv I h /N h where Chv is the effective contact rate, that is, the rate at which mosquitoes/humans acquire infection from infected humans/mosquitoes. The transmission of Zika virus is governed by the following set of non-linear differential equations [60] dSh = ˝ h − pBhEh − qBh I h − ˜ hSh − µ hSh , dt

(5.21a)

dEh = ˜ hSh + pBhEh + qBh I h − K1Eh , dt

(5.21b)

dI h = ˜ Eh − K 2 I h , dt

(5.21c)

dRh = ˜ I h − µ h Rh , dt

(5.21d)

dSv =   ˝ v − rBvEv − sBv I v   − ˜ vSv − µ vSv , dt

(5.21e)

dEv = rBvEv + sBv I v + ˜ vSv − K 3Ev , dt

(5.21f)

dI v = ˜ Ev .−K 4 I v dt

(5.21g)

where K1 = ˜ + µ h , K 2 = ˛ + ˝ I + µh , K 3 = ˙ + µv and K 4 = µv + ˝ v .

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Spatial Dynamics and Pattern Formation in Biological Populations

Analysis of equilibrium points: The model has the DFE, N 0 ( ˝ h /µ h , 0, 0, 0,˝ v /µ v , 0, 0 ) . Let N 1 = Sh* , Eh* , I h* , Rh* , Sv* , Ev* , I v* be an EE of the model. Denote ˜ h* = Chv I v* /N h* , ˜ v* = Chv I h* /N h* . We obtain the equilibrium point as

(

)

Sh* =

m1ˇ h * K 2 ˜ h* ˇ h * °˜ h* ˇ h , Eh = , Ih = , t2 t2 t2

(5.22)

˛°˜ h* ˇ h * m2ˇ v * K 4 ˜ v* ˇ v * ˙˜ v* ˇ v Rh* = , Sv = , Ev = , Iv = . µht2 t3 t3 t3 where t2 = K1K 2 ˜ h* + m1µh , t3 = K 3 K 4 ˜ v* + m2 µ v , m1 = K1K 2 − (pK 2 + q˜ )Bh , m2 = K 3 K 4 − ( rK 4 + s° ) Bv . Hence, ˜ h* =

(

)

° Chv ˜ v* µh K1K 2 ˜ h* + m1µh  v

(˝˙˜ + ( m + (˙ + K ) ˜ ) µ )( K K ˜ * h

1

and ˜ v* =

* h

2

h

* 4 v

3

)

+ m2 µ v  h

(5.23a)

,

° Chv ˜ h* µh . ˝°˜ h* + m1 + (° + K 2 ) ˜ h* µh

(

(5.23b)

)

For the existence of the EE, we require K1K 2 > K 5 , and K 3 K 4 > K6 , where K 5 = Bh ( pK 2 + q˜ ) , K6 = ( rK 4 + ˜ s ) Bv (Problem 5.4, Exercise 5). Substituting (5.23b) into (5.23a), we obtain the quadratic equation

( )

a ˜ h*

2

+ b˜ h* + c = 0,

(5.24)

where

)(

(

) )

(

a = ˜° + ( K 2 + ° ) µ h K 3 K 4° Chv µh + m2 ˜° + ( K 2 + ° ) µh µv ˘ h

( (

(

) )

)

b = µ h m1 K 3 K 4° Chv µ h + 2m2 ˛° + ( K 2 + ° ) µ h µ v  h − K1K 2°˝ Ch2v µ h v ,

(

)

c = µ v m12 m2 µ h2˙ h 1 − 02 , where 0 is the basic reproduction number given by (Problem 5.5, Exercise 5) 02 = c =

2 Chv  v µh°˛ . µv h  K 2 ( µh − pBh ) + ° (˝ + ˙ I + µh − qBh )   K 4 ( µv − rBv ) + ˛ (˙ v + µv − sBv ) 

When p = 0 = q, and r = 0 = s, vertical transmission is not present in the model and 0 reduces to the basic reproduction number 0 = v for an SEIR model with vector population as given by Derouich and Boutayeb [34]. We fnd that a > 0. We have the following two cases: (i) 0 < 1 : in this case, c > 0. If b > 0, then the signs of the coeffcients are +, +, +. The equation has no positive root. If b < 0, then the signs of the coeffcients are + , − , +. The equation has two positive roots when b 2 − 4ac > 0. In this case, two EE points may exist. Otherwise, there is no positive root. (ii) 0 > 1 : in

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Zika Virus: Modeling of Spread and Outbreak

this case, c < 0. If b > 0, then the signs of the coeffcients are + , + , − ; when b < 0, the signs of the coeffcients are + , − , −. Hence, the equation has a positive root always. The unique EE is obtained from equation (5.22). If 0 = 1, then c = 0. If b < 0, then a positive solution exists for equation (5.24), which is given by ˜ h* = − ( b/a ) . In this case, a unique EE point exists. DFE N 0 ( ˝ h /µ h , 0, 0, 0,˝ v /µ v , 0, 0 ) is locally asymptotically stable if 0 < 1 and unstable if 0 > 1 (Problem 5.6, Exercise 5 [61]). The result implies that Zika virus can be eliminated from the population when 0 < 1, if the initial sizes of the subpopulations of the model (5.21) are in the basin of attraction of the DFE N 0 . The effective contact rate and vertical transmission can help to control the disease. The reproduction number 0 can be reduced and maintained at a value below unity, if the initial sizes of the subpopulations of the model are in the neighborhood of attraction of the DFE. Bifurcation analysis: Denote Sh = x1 , Eh = x2 , I h = x3 , Rh = x4 , Sv = x5 , Ev = x6 , I v = x7 . Let ˆf = ° f , , f ˝ denote the vector feld of the original model in terms of x˜s. Then, the model 7˙ i ˛ 1 system (5.21) reduces to the system dx1 = f1 = ˙ h − pBh x2 − qBh x3 − ( Chv x1x7 /N x ) − µ h x1 , dt dx2 dx = f2 = ( Chv x1x7 /N x ) + pBh x2 + qBh x3 − K1x2 , 3 = f3 = ˜ x2 − K 2 x3 , dt dt dx4 dx = f 4 = ˜ x3 − µ h x4 , 5 = f5 = ˇ v − rBv x6 − sBv x7 − ( Chv x3 x5 /N x ) − µ v x5 , dt dt dx6 = f6 = rBv x6 + sBv x7 + ( Chv x3 x5 /N x ) − K 3 x6 , dt dx7 = f7 = ˜ x6 − K 4 x7 , N x = x1 + x2 + x3 + x4 . dt

{

}

The DFE of the system is given by ˜ 0 = x1* , 0, 0, 0, x5* , 0, 0 , where x1* = ˛ h /µ h , and x5* = ˛ v /µ v . * Consider the case 0 = 1. Let Chv = Chv be a bifurcation parameter * Chv =

˙ h µ v m1m2 . °˛µh˙ v

The Jacobian of the matrix at the DFE is given by  −µ h   0   0  J (˜ 0 ) = 0   0   0  0 

− pBh

−qBh

0

0

0

− K1 + pBh

qBh

0

0

0

˛ 0 0

−K 2

0

0

0

˝ − j2

− µh 0

0 − µv

0 −rBv

0

j2

0

0

− K 3 + rBv

0

0

0

0

˙

*  −Chv  *  Chv  0  , 0  −sBv   sBv  −K 4  

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* ˛ v µ h /˛ h µ v . The Jacobian matrix of the linearized system has zero as a simple where j2 = Chv eigenvalue and the remaining eigenvalues have negative real parts. Hence, the center manifold theory of Castillo-Chavez and Song [27] can be used to analyze the dynamics of the system. Following their analysis, the right and left eigenvectors of the Jacobian matrix * corresponding to zero eigenvalue at Chv are given by w = ( w1 , , w7 ) and v = ( v1 ,, v7 ) , where

w1 = −

* C* K w C* ° w C* °˛ w7 Chv K 1 K 2 w7 , w2 = hv 2 7 , w3 = hv 7 w4 = hv m1µ h m1 m1 m1µ h

w5 = −

* ° j2 Bv ( s˝ + rK 4 )  w7  Chv Kw + , w6 = 4 7 , w7 > 0.   µv  m1 ˝ ˝ 

v1 = 0, v2 =

j2˜ ( K1 − pBh ) v7 ˜ v7 m2 v7 , v4 = v5 = 0, v6 = , v7 > 0. , v3 = C ( K 3 − rBv ) K 3 − rBv ( K3 − rBv ) m1 * hv

The values of non-zero derivatives are

˜ 2 f1 ˜ 2 f1 ˜ 2 f1 ˜ 2 f2 ˜ 2 f2 ˜ 2 f2 C* µ C* µ = = = hv h , = = = − hv h , ˝ h ˜ x2 ˜ x7 ˜ x3 ˜ x7 ˜ x4 ˜ x7 ˝h ˜ x2 ˜ x7 ˜ x3 ˜ x7 ˜ x4 ˜ x7 ˜ 2 f5 ˜ 2 f5 ˜ 2 f5 ˜ 2 f5 ˜ 2 f6 C* µ C* µ 2 C* µ = − hv h , = = = hv2 h˙v , = hv h , ˙ h ˜ x1 ˜ x3 ˜ x2 ˜ x3 ˜ x3 ˜ x4 ˙h µv ˙h ˜ x3 ˜ x5 ˜ x3 ˜ x5 2 * 2 ˜ 2 f6 ˜ 2 f6 ˜ 2 f6 f6 C * µ 2˙ ˜ 2 f5 2Chv 2C* µ 2 µ h˙ v ˜ = = = − hv 2 h v , = = − hv2 h˙v . 2 2 2 ˙h µv ˙ h µ v ˜ x3 ˙h µv ˜ x1 ˜ x3 ˜ x3 ˜ x2 ˜ x3 ˜ x4 ˜ x3

The values of the parameters a and b defned in the center manifold theory are obtained as 7

a =

˜

v k wi w j

k , i , j =1

=−

=

* µh Chv {( v2w7 (w2 + w3 + w4 ) − v6w3w5 )µvh + v6w3 ( w1 + w2 + 2w3 + w4 ) µhv } 2h µ v

* v7 w72Chv µh m1 ( rBv − K 3 ) µ v2h

+

(

)

 m2 ˆ˝ + (˝ + K 2 ) µh µv h * ˝ BvChv ( s˙ + rK 4 )  h + µh 

*2 ˝˙ Chv ( j2˝h + ˝ ( µh − ˇ I ) v )  ,  m1 

7

b=

° 2 fk ( v + v3 + v6 + v7 )( w3w5 + w2 w7 ) Chv* µh =− 2 h ° xi ° x j

˜

v k wi

k , j= 1

° 2 fk vw µ˘ v7 w7  m2 ˝˙ j2  > 0. = v 2 w7 + 6 3 h v = + * * K 3 − rBv  Chv m1  ° xi ° Chv µv ˘h

Since b is always positive, backward bifurcation occurs when a > 0. Therefore, 0 < 1 is a necessary condition but it is not suffcient to effectively control the spread of Zika virus in

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Zika Virus: Modeling of Spread and Outbreak

the population. In other words, efforts to bring 0 < 1, may fail to lead to effective control of ZIKV (due to the phenomenon of backward bifurcation). In such a backward bifurcation scenario, effective disease control is dependent on the initial sizes of the populations of the model. The global stability is not investigated in this case because of the existence of a backward bifurcation due to a large vector population. The authors [60] have shown numerically that the model exhibits the phenomenon of backward bifurcation when the stable DFE co-exists with two endemic equilibria (one of which suggested to be stable) when 0 < 1. To study the impact of the model parameters on the prevalence of Zika infection, uncertainty and sensitivity analyses were also performed. The authors concluded that the effective contact rates, the recovery rate of the infected individuals, and the birth rate of mosquitoes are the most infuential parameters. The authors [60] have also studied a modifed model system (5.21) by neglecting the disease-induced death rate (˜ I = ˜ v = 0). In this case, both the total host population and the vector population are asymptotically constant. The endemic state of the modifed model (5.21) is locally asymptotically stable for 0 > 1. The proof is based on the Krasnoselskii sublinearity trick [54,55]. Rewrite the system as [60]: dEh Chv I v = ( N h − Eh − I h − Rh ) + pBhEh + qBh I h − K1Eh , dt Nh dI h = ˜ Eh − (° + µ h ) I h , dt dRh = ° I h − µ h Rh , dt

(5.25)

dEv Chv I h = ( N v − Ev − I v ) + rBvEv   +   sBv I v   −   K3Ev , dt Nh dI v = ˝ Ev − µ v I v . dt

(

)

Linearizing the system (5.25) about the EE ˜ 1 = Sh° , Eh° , I h° , Rh° ,Sv° , Ev° , I v° , we obtain   C I°  dEh  Chv I v° C I° C S° = − − K1 + pBh  Eh +  − hv v + qBh  I h − hv v Rh + hv h I v , dt  N h Nh Nh   Nh  dI h = ˜ Eh − (° + µh ) I h , dt dRh = ° I h − µ h Rh , dt    C I°  C I° dEv ChvSv° = I h +  − hv h − K 3 + rBv  Ev +  − hv h + sBv  I v , dt Nh    Nh  Nh dI v = ˝ Ev − µv I v . dt The Jacobian of the system evaluated at ˜ 1 is

(5.26)

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Spatial Dynamics and Pattern Formation in Biological Populations

˘ − j5 − K1 + pBh  ˜   J= 0   0  0  where K 2° = ˜ + µh , j1 =

− j5 + qBh

− j5

0

j3

0

0

0

° j4

− µh 0

0 − j1 − K 3 + rBv

0 − j1 + sBv

0

0

˝

− µv

−K

° 2

   ,    

Chv I h° C I° C S° C S° , j5 = hv v , j3 = hv h , and j4 = hv v . Nh Nh Nh Nh

Write the solution of the model (5.26) in the form Z ( t ) = Ze˝ t , where Z = (Z1 , Z2 , Z3 , Z4 , Z5 ). Substituting Z ( t ) into (5.26), we obtain

˜ Z1 = ( − j5 + pBh − K1 ) Z1 + ( − j5 + qBh ) Z2 − j5 Z3 + j3 Z5 , ˜ Z2 = ° Z1 − (˛ + µh ) Z2 , ˜ Z3 = ˛ Z2 − µh Z3 ,

(5.27)

˜ Z4 = j4 Z2 + ( − j1 + rBv − K 3 ) Z4 + ( − j1 + sBv ) Z5 , ˜ Z5 = ˙ Z4 − µv Z5 . Rearrange the above system of equations as follows: frst, move the negative terms in the last four equations of (5.27) to the respective left-hand sides. Secondly, the last four equations are then re-written in terms of Z1. We obtain

˜ Z1 = ( − j5 + pBh − K1 ) Z1 + ( − j5 + qBh ) Z2 − j5 Z3 + j3 Z5 , Z2 =

˜ Z1 ˛˜ Z1 , Z3 = , + ° + ˛ + µh ° µ ( h ) (° + ˛ + µ h )

Z4 =

(° + µv ) j4˜ Z1 ,  (° + ˛ + µh ) (° + µv ) (° + j1 − rBv + K3 ) + ˙ ( j1 − sBv )

Z5 =

˙ (° + µv ) j4˜ Z1 . (° + µv ) (° + ˛ + µh ) (° + µv ) (° + j1 − rBv + K3 ) + ˙ ( j1 − sBv )

Substituting the values of Z2 , Z3 , Z5 in the frst equation of (5.27), we obtain

(1 + F1 (˜ )) Z1 = ( MZ )1 , (1 + F2 (˜ )) Z2 = ( MZ )2 , (1 + F3 (˜ )) Z3 = ( MZ )3 , (1 + F4 (˜ )) Z4 = ( MZ )4 , (1 + F5 (˜ )) Z5 = ( MZ )5 , where Fi (˜ ), i = 1,, 5 are positive functions of parameters and

(5.28)

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Zika Virus: Modeling of Spread and Outbreak

 0   ˜ / (° + µh )  0 M=  0   0 

qBh / ( K1 − pBh )

0

0

0

0

0

° / µh

0

0

C S / N h ( K 3 − rBv )  0

0

0

0

˝ / µv

° hv v

Chv Sh° /  N h ( K1 − pBh )    0  . 0  sBv / [ K 3 − rBv ]   0 

Note that the matrix M has non-negative entries. The notation ( MZ )i denotes the ith coordinate of the vector MZ. Defne F (˜ ) = min i 1 + Fi . It can be verifed that the equilibrium point ˜ 1 satisfes the equation ˜ 1 = M˜ 1. If Z is a solution of (5.28), then it is possible to fnd a minimal positive real number r such that Z ° r˜ 1 , [54,55]. To show that Re (˜ ) < 0, assume that Re (˜ ) ˝ 0, and consider the following two cases. Case 1: ˜ = 0 : In this case, equation (5.27) is a homogeneous linear system. It is easy to show that the determinant of this system is negative. It follows that the system has a unique solution Z ˜ 0, which corresponds to the disease-free steady state of the modifed model system (5.21). Case 2: ˜ ° 0 : By assumption, 1 + Fi (˜ ) > 1. Since r is a minimal positive real number, it follows that Z > r˜ 1/F (° ) , where F (˜ ) is minimal of 1 + Fi (˜ ) . From the second equation of (5.28), we have F (˜ ) Z2 ˝ rI h° , which contradicts Z > r˜ 1/F (° ) . Hence, Re (˜ ) < 0. Thus, all eigenvalues of the characteristic equation associated with the linearized system (5.26) have negative real parts. This implies the local asymptotical stability of the endemic state. Numerical simulations: To study the stability and bifurcation, numerical computations are performed using the set of parameter values as

µh = 0.000046, µv = 0.099, ° = 0.33, ˛ = 0.2, p = 0.050027, q = 0.050041, r = 0.05, s = 0.049, ˝ I = 0.001, ˝ v = 0.00001, Bh = 0.0005, Bv = 0.0005, ˇ h = 100, ˇ v = 1, 000, ˙ = 0.99. The initial conditions are taken as (100000,100, 300,10,10000, 50, 200). We obtain the DFE as N 0 = 2.17391 × 106 , 0, 0, 0,10101, 0, 0 , 0 = 0.65884, and Chv = 1.43. The local asymptotic stability of DFE is shown in Figure 5.6a. We obtain the EE as N 1 = ( 1.73284 × 106 , 61.4744,100.905, 438718,10089.5,1.04632,10.462 ) , Chv = 2.43, and 0 = 1.11957. The local asymptotic stability of EE is shown in Figure 5.6b. As discussed earlier, the epidemiological implication of the backward bifurcation phenomenon of the model (5.21) is that having 0 < 1 is only a necessary condition to effectively control the spread of Zika virus in the population. In other words, the efforts to bring 0 < 1 may fail to lead to effective control of ZIKV. In such a backward bifurcation scenario, effective disease control (0 < 1) is dependent on the initial sizes of the populations of the model. This effect is shown in Figure 5.7. One of the reasons for the occurrence of the backward bifurcation phenomenon in the model is a large vector population. If the vector population can be controlled, then backward bifurcation may be eliminated.

(

)

298

Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 5.6 Time series of model (5.21) converging to steady states. (a) DFE with Chv = 1.43, 0 = 0.65884 and vector (mosquito) population. (b) EE with Chv = 2.43 and 0 = 1.11957.

FIGURE 5.7 Backward bifurcation in model (5.26). [From Imran, M., Usman, M., Dur-e-Ahmad, M., Khan. A. 2017. Transmission dynamics of Zika fever: A SEIR based model. Diff. Eqs. Dyn. Sys. 1–24, [60], Copyright 2017, Springer Nature. Reprinted with kind permission from author and Springer Nature.]

Zika Virus: Modeling of Spread and Outbreak

299

5.6 Model 4: Zika Virus with Vertical Transmission Agusto et al. [2] modeled the transmission dynamics of ZIKV using a compartmental framework and analyzed the system of ODEs, which includes human vertical transmission of Zika virus, the birth of babies with microcephaly, and asymptotically infected individuals. The human population is divided into adults and newly born babies. The population of newly born babies is split into the following: susceptible SB (t), exposed EB ( t ), asymptomatic AB ( t ), symptomatic newly born without microcephaly I B ( t ), newly born with microcephaly I BM ( t ), and recovered newly born babies RB ( t ) . The total population of adults NW (t) at time t is split into the following mutually exclusive subpopulations of individuals: susceptible SW (t), exposed EW (t), asymptomatic AW ( t ), symptomatic IW ( t ), adults with microcephaly IWM ( t ), and recovered adults RW ( t ). The population of the mosquitoes is split into the following: susceptible female mosquitoes SV ( t ), exposed female mosquitoes EV ( t ), and infected female mosquitoes IV ( t ). The total population for each group is given as: N B ( t ) = SB ( t ) + EB ( t ) + AB ( t ) + I B ( t ) + I BM ( t ) + RB ( t ) , NW ( t ) = SW ( t ) + EW ( t ) + AW ( t ) + IW ( t ) + IWM ( t ) + RW ( t ) , NV ( t ) = SV ( t ) + EV ( t ) + IV ( t ) . The total human population is N H ( t ) = N B + NW . The full model is written as follows [2]: SBˆ ( t ) = ˘ B − q A ˘ B AW ( t ) − qI ˘ B IW ( t ) − qR ˘ B RW ( t ) − ˜B ( IV , N B ) SB ( t ) − (° + µB ) SB ( t ) , EBˇ ( t ) = ˜B ( IV , N B ) SB ( t ) − (° + ˛ B + µB ) EB ( t ) , ABˇ ( t ) = q A  B AW ( t ) + ( 1 − p )˜ BEB ( t ) − (° + ˛ B + µB ) AB ( t ) , I Bˇ ( t ) = qI  B IW ( t ) + p˜ BEB ( t ) − (° + ˛ B + µB ) I B ( t ) , I BM ˙ ( t ) = rqR ˇ B RW ( t ) − (˜ + µB ) I BM ( t ) , RBˆ ( t ) = ( 1 − r ) qR  B RW ( t ) + ˜ B AB ( t ) + ˜ B I B ( t ) − (° + µB ) RB ( t ) , SW ˆ ( t ) = ˜ SB ( t ) − °W ( IV , NW ) SW ( t ) − µW SW ( t ) , EW ˆ ( t ) = ˜W ( IV , NW ) SW ( t ) − (° W + µW ) EW ( t ) , AˆW ( t ) = ( 1 − p )˜ W EW ( t ) − (° W + µW ) AW ( t ) , IW ˆ ( t ) = p˜ W EW ( t ) − (° W + µW ) IW ( t ) , I˙WM ( t ) = ˜ I BM ( t ) − µW IWM ( t ) ,

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Spatial Dynamics and Pattern Formation in Biological Populations

RW ˆ ( t ) = ˜ RB ( t ) + ° W AW ( t ) + ° W IW ( t ) − µW RW ( t ) , SV˙ ( t ) = ˇ V − ˜V ( AB , I B , AW , IW , N B , NW ) SV ( t ) − µV SV ( t ) , EVˆ ( t ) = ˜V ( AB , I B , AW , IW , N B , NW ) SV ( t ) − ( µV + ˛ V ) EV ( t ) , IV˙ ( t ) = ˜ V EV ( t ) − µV IV ( t ) , where ˜W ( IV , NW ) =

(5.29)

°W bV IV ˛° b I , ˜ B ( IV , N B ) = B V V , NW NB

 IW + ˝W AW + ˛ ( I B + ˝B AB )  ˜V ( AB , I B , AW , IW , N B , NW ) = °V bV  , NW + ˛ N B   are the disease forces of infection rates. The parameters are defned as follows: ˜ B : Birth rate of newly born babies. p: Fraction of adults and newly born babies who are asymptomatic. (1 − p ): Remaining fraction of adults and newly born babies who are infectious. ˜ : Maturation rate. r , q A , qI , qR : Fractions of newly born babies who are infected and are having microcephaly. ( 1 − r ) : Remaining fraction of newly born babies who have microcephaly. ˜: Modifcation parameter that indicates that babies’ exposure rate is different from that of adults. ˜W , ˜ B : Transmission probabilities per contact of adults and newly born babies. ˜W , ˜B : Infectivity modifcation parameters in asymptomatic adults and newly born babies. ˜ W , ˜ B : Progression rate of exposed adults and newly born babies. ˜ W , ˜ B : Recovery rate of asymptomatic and symptomatic adults and newly born babies. µW , µB : Natural death rate of adults and newly born babies. ˜ V : Recruitment rate of mosquitoes. ˜V : Transmission probability per contact of susceptible mosquitoes. bV : Mosquito biting rate. ˜ V : Progression rate of exposed mosquitoes. µV : Natural death rate of mosquitoes. Infection in the asymptomatic individuals is assumed to be not high enough to infect the susceptible mosquitoes or is at the same level as for the infectious individuals, in which case, the modifcation parameters are taken as ˜B ° 0, ˜W ˛ 1. Zika virus is passed prenatally from a pregnant woman to her unborn fetus [79].

301

Zika Virus: Modeling of Spread and Outbreak

Depending on the timing of infection in the womb, newborn babies can also be infected from birth [12]. Thus, we assume that some babies are born with infected virus. The parameters q A , qI , qR represent the fractions of newly born babies who are infected due to vertical transmission. Therefore, the fraction ˛ B ( 1 − q A AW − qI IW − qR RW ) denotes the number of babies born healthy by infected and recovered mothers, and the remaining fractions q A ˜ B AW , qI ˜ B IW , qR ˜ B RW denote the number of babies born with infection. Even though there is suffcient evidence to conclude that intrauterine Zika virus infection is a cause of microcephaly [79], not all newly born babies are born with microcephaly, although they may have other congenital abnormalities [30]. Thus, it is assumed that some babies are born recovered from the virus. The quantity rqR ˜ B RW defnes the recovered babies born from recovered mothers while the remaining portion, ( 1 − r ) qR ˝ B RW are newly born babies who have microcephaly. Authors have analyzed the model system (5.29) in a biologically feasible reason ˜ 1 = ˜ H × ˜ V ˝ R+12 × R+3 with µ H = min { µB , µW } ˇ ˇ ˆ  ˆ  ˝ H = ˘ N B ( t ) , NW ( t ) : N H ( t ) = B  , ˝ V = ˘SB ( t ) , EV ( t ) , IV ( t ) : NV ( t ) = V . µH  µV    The region ˜ 1 is positively invariant for the model system (5.29) (Problem 5.7, Exercise 5). Analysis of equilibrium points: The DFE of the model is given by ˘ ˆB ˜ˆ B ˆ E0 =  , 0, 0, 0, 0, 0, , 0, 0, 0, 0, 0, V , 0, 0 µW (˜ + µB ) µV  ˜ + µB

  . 

The local asymptotic stability of E0 can be established using the next-generation operator method. The basic reproduction number 0 is given by 0 = V ( W + B ) , where V =

W =

˝ 2 ˜ BbV° B  ˙B ( 1 − p ) + p  SV* ˜V bV° V , , B = * k5 µV + ˝SB* k2 k3 SW

˜W bV° W * + ˛SB* k7 SW

(

(

)

)

 ( 1 − p )( k3 ˝W + ˛˝B q A  B ) p ( k3 + ˛qI  B )  +  . k 3 k6 k 3 k6  

(Problem 5.8, Exercise 5) The epidemiological quantity 0 gives the average number of ZIKV cases generated by a typical infected individual introduced into an entirely susceptible human population [7,38,53,104]. B and W respectively denote the number of secondary infections in newly born babies and adults due to introduction of one infectious mosquito. The expression for W denotes the combination of the effects from infections from newly born babies due to vertical transmission (mother-to-child infection), infections due to horizontal transmissions from adults, and infections from infants that have matured into adults. The expression for V denotes the number of secondary infections in mosquitoes resulting from a newly introduced infectious adult woman and newly born baby. ZIKV can be adequately controlled in the community of adults and newly born babies if the threshold quantity 0 can be reduced to and maintained at 0 < 1.

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Spatial Dynamics and Pattern Formation in Biological Populations

Using Theorem 2 given in the work of van den Driessche and Watmough [104], the following result was established. The DFE E0 is locally asymptotically stable if 0 < 1, and unstable if 0 > 1. Using the results of Castillo-Chavez et al. [26], it can be shown that the DFE is globally asymptotically stable. The following result gives the global asymptotic stability of the DFE [2]. The DFE E0 of the model (5.29) is globally asymptotically stable in ˜ 1 , for 0 ˜ 1, where * ˛ 1 = X ˙˛ : SB ˆ SB* , SW ˆ SW , SV ˆ SV* . (Problem 5.9, Exercise 5) In another study, Agusto et al. [3] developed a deterministic model that incorporates both vector-borne and sexual transmission routes considering both heterosexual and homosexual transmissions (human population was stratifed by gender). Using a compartmental modeling framework with susceptible, exposed, infected, and recovered males and females, and susceptible, exposed, and infected mosquito population, the authors have formulated and analyzed a system of 11 ODEs for transmission dynamics of Zika virus. The model has two equilibrium points, DFE and EE. The following theoretical and epidemiological fndings were obtained by the authors [3]:

{

}

i. The DFE is locally and globally asymptotically stable when the associated reproduction number (F ) is less than unity and unstable otherwise. ii. The model has a unique EE for the system with heterosexual only and homosexual male-only couples. These equilibria are locally asymptotically stable when their respective reproduction numbers are greater than unity. iii. In the presence of disease-induced mortality, the model exhibits the phenomenon of backward bifurcation, where the stable DFE co-exists with a stable EE when the associated reproduction number is less than unity. Once the disease spread takes hold, it will be very diffcult to eradicate. In particular, when backward bifurcation occurs, identical conditions (for example, same levels of mosquito control, same human sexual behavior) can lead to either disease extinction or endemicity depending on initial conditions. Consequently, once the disease goes out of control, more aggressive management may be required to reduce disease spread and return the system to a state below the bifurcation point. Another concern due to backward bifurcation is that disease risk and/or incidence could increase suddenly with even small changes in management strategies. In particular, if the system lies beyond this bifurcation point, then even a slight decrease in mosquito control efforts, or a slight increase in promiscuity could move the system into a regime where only the endemic state is stable, and the disease incidence levels are already quite high. iv. The study also points to the fact that including risky behavior involving multiple sexual partners particularly among male populations substantially increases the number of infected individuals in the population, contributing signifcantly to the disease burden in the community. The authors [3] have also shown that the disease burden in the community increases the infectivity of the asymptomatic individuals and the cumulative infections to mosquitoes from asymptomatic individuals increase as their infectivity level increases. Sensitivity analysis has shown that the most important parameters for ZIKV spread are the death rate of the mosquitoes (µV ), the mosquito biting and recruitment rates (bV and ˜ V ), the transmission probability per contact to mosquitoes and to adult humans (˜V and ˜W), and the

Zika Virus: Modeling of Spread and Outbreak

303

adult recovery rate (˜ W ). From the numerical results, the authors [3] have concluded that delaying conception reduces the number of cases of microcephaly, although it does little to pre-ZIKV transmission in the broader community. However, with aggressive mosquito control and personal protection, it is possible to reduce microcephaly and also prevent ZIKV transmission. Numerical simulations using mosquito control and personal protection indicate that personal protection is a better and more effective strategy than mosquito reduction in reducing the disease burden in the population. Sasmal et al. [96] studied the effect of sexual transmission by formulating fve mathematical compartmental models of Zika transmission with both transmission routes (vector-borne and sexual transmission). To make the models more realistic, heterogeneity in the sexual transmission was modeled in several ways. The authors ftted the fve different models with the available data, defned suitable parameters, and selected the most appropriate model, which describes the Zika virus outbreak in Colombia. For all the models, they estimated the reproduction numbers for direct (sexual) transmission, vector transmission, and determined the basic reproduction number (0 ). The results revealed that the sexual transmission contribution to 0 is highest [15.36% (95% CI: 12.83–17.4)] for the model, which stratifes each gender to high-risk and low-risk individuals in their sexual behavior. For this model, the estimated 0 is 1.89 (95% CI: 1.21–2.13), the direct transmission reproduction number is 0.42 (95% CI: 0.29–0.64), and the vector transmission reproduction number is 1.51 (95% CI: 1.23–1.87). Sensitivity analysis demonstrated that the value of 0 depends on three controllable parameters: the biting rate, the sexual transmission rate, and the average ratio of a mosquito to a human.

5.7 Model 5: Zika Virus SIR Transmission Model with Human and Vector Mobility Zika virus transmission dynamics is infuenced by human and vector mobility and human–mosquito interaction in space and time. Charles et al. [29] investigated the spatiotemporal transmission dynamics of ZIKV disease to account for human and vector mobility. The total human population at a spatial location x in one dimension and at time t given by N h ( t , x ) is divided into the following three compartments: (i) Sh ( t , x ) , the number of susceptible humans (humans capable of being infected), (ii) I h ( t , x ) , the number of infected humans, and (iii) Rh ( t , x ) , the number of recovered humans. The vector population at a spatial location x in one dimension and at time t given by N v ( t , x ) is divided into the following two compartments: (i) Sv ( t , x ) , the number of susceptible mosquitoes (mosquito population capable of being infected), and (ii) I v ( t , x ) , the number of the infectious mosquitoes. To formulate a spatiotemporal model on the transmission dynamics, the following assumptions were considered [29]: i. The susceptible humans and susceptible vectors are assumed to be recruited at constant rates ˜ h and ˜ v respectively. Susceptible humans acquire infection from an infectious bite of a mosquito via a force of infection ˜ = c° vh I v / (˛ + Sh ) , where ˜ vh is the vector–human probability of transmission upon a bite, c is the biting rate of vector, and ˜ is the half-maximal human saturation constant. The susceptible vectors acquire infection from infectious humans through blood feeding via a force of infection ˜ = c° hv I h /(˛ + Sv ) , where the transmission probability from

304

ii.

iii.

iv.

v.

Spatial Dynamics and Pattern Formation in Biological Populations

human to vector is ˜ hv and ˜ is the half-maximal vector saturation constant. Both are considered to be of Michaelis–Menten form to account for saturation of the human infection and of the vector infection [107]. The disease transmission from humans to humans is insignifcant to cause an epidemic [93]. Therefore, this mode of transmission was ignored in this model. The effective contact rate between susceptible humans and the infected vector is denoted by ˜ , (˜ = c° vh). The effective contact rate between infected humans and susceptible humans is denoted by ˜ (˜ = c° hv ). Both the human and vector populations are non-negative and bounded. The mobilities of susceptible humans, infected humans, susceptible vector, and infected vector are taken to be diffusive with diffusion constants D1 , D2 , D3 and D4 respectively. Natural mortality rates are assumed to occur in the human and vector populations at the rates µ h and µ v respectively. ZIKV disease-related deaths are rare and therefore such deaths are not considered in this model [89]. The incubation period of ZIKV is relatively short and therefore it is assumed that all the exposed individuals live to become infectious and therefore the exposed class is not considered. It was observed that infection with ZIKV confers permanent immunity upon recovery [47]. This recovery from the infection is assumed to take place at a rate ˜ . The diffusivity of the recovered humans is therefore ignored since they do not affect the transmission dynamics of ZIKV disease.

Under the above assumptions, the spatiotemporal model is represented by the following set of PDEs (Charles et al. [29]):

˜ Sh ° Sh I v ˜I ° Sh I v = h − − µ hSh + D1 2Sh , h = − ( µh + ˙ ) I h + D2 2 I h , ˜t ˛ + Sh ˜ t ˛ + Sh ˜ Rh ˜S ˆ Sv I h = ˙ I h − µ h Rh , v =  v − − µ vSv + D3 2Sv , ˜t ˜t ˇ + Sv

(5.30)

˜ I v ˆ Sv I h − µ v I v + D4 2 I v , = ˜ t ˇ + Sv with ( Sh , I h , Rh , Sv , I v ) ˝   5+ , and ˙ 2 =

˜2 . ˜ x2

The model was solved under zero fux boundary conditions so that the populations do not move across the boundary ˜ ° of the domain [ 0, +˙ ) × ˜ ˇ

˜ Sh ˜ I h ˜ Rh ˜ Sv ˜ I v = = = = = 0, ˜n ˜n ˜n ˜n ˜n where n is the outward normal to ˜ °. For x   ˜  ° and the space ( −˝ , +˝ ), ˜ is a bounded domain in ˜ °  5+ with a smooth boundary ˜ ° and t ˜ 0. Initial conditions are given by Sh ( 0, x ) = Sh0 ( x ) ˝ 0, I h ( 0, x ) = I h0 ( x ) ˝ 0, Rh ( 0, x ) = Rh0  ( x ) ˝ 0,

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Sv ( 0, x ) = Sv0 ( x ) ˝ 0, I v ( 0, x ) = I v0 ( x ) ˝ 0. The authors [29] have shown the following: (i) Under the above initial conditions, the solutions are non-negative in [ 0, +˝ ) for all t ˜ 0. (ii) The solutions are bounded in the region ˜ = ˜ h × ˜ v for all t ˜ 0. Model (5.30) can also be written in the form

˜ Sh − D1˙ 2Sh = F1 ( Sh , I h , Rh , Sv , I v ) , ˜t ˜ Ih ˜ Rh − D2˙ 2 I h = F2 ( Sh , I h , Rh , Sv , I v ) , = F3 ( Sh , I h , Rh , Sv , I v ) , ˜t ˜t ˜ Sv ˜I − D3˙ 2Sv = F4 ( Sh , I h , Rh , Sv , I v ) , v − D4˙ 2 I v = F5 ( Sh , I h , Rh , Sv , I v ) . ˜t ˜t The functions Fi ( Sh , I h , Rh , Sv , I v ) , i = 1, 2, 3, 4, 5, are continuously differentiable and satisfy the following: F1 ( 0, I h , Rh , Sv , I v ) = ˝ h ˙ 0, F2 ( Sh , 0, Rh , Sv , I v ) = 0, F3 ( Sh , I h , 0, Sv , I v ) = 0, F4 ( Sh , I h , Rh , 0, I v ) = ˝ v ˙ 0, and F5 ( Sh , I h , Rh , Sv , 0 ) = 0, for all t ˙ 0. Since ( Sh , I h , Rh , Sv , I v ) ˛ 0 with non-negative initial conditions, the solutions are positive. Analysis of equilibrium points: The DFE is given by E0 = ( ˙ h /µ h , 0, 0, ˙ v /µ v , 0 ) . The basic reproduction number 0 is defned as the average number of secondary infections produced by a single individual introduced into a fully susceptible population during an individual’s entire infectious period [7,69]. Using the next-generation matrix approach [104], 0 was derived as 02 =

˜ vh ˜ hv c 2  h  v . µv ( µh + ˛ ) (˝µh +  h ) ( µv˙ +  v )

The disease-free equilibrium E0 of the temporal model system (5.30) is locally asymptotically stable for 0 < 1 and unstable for 0 > 1, [29] (Problem 5.10, Exercise 5). The DFE E0 of the temporal model system (5.30) is also globally asymptotically stable (Problem 5.11, Exercise 5). The result implies that for small perturbations of the DFE, the solutions will eventually converge to the DFE whenever 0 < 1. Epidemiologically, it implies that if a few infectious individuals are introduced into a fully susceptible population, the disease would die out for 0 < 1, otherwise, the disease would spread [29]. The EE of the model is defned as the state at which ZIKV disease persists in the population. The EE E* = Sh* , I h* , Rh* , Sv* , I v* is given by

(

Sh* =

)

(

)

(

)

(

)(

)

Sv* + ° ( ˛ + µh ) −ˇ v + Sv* µv  * Sv* + ° ˇ v − Sv* µv 1  ,  ˇh +  , Ih = * kSv µh  kSv* 

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Spatial Dynamics and Pattern Formation in Biological Populations

Rh* =

I v* =

(S

* v

(

)((  k˙ S ( k S + ( S

)

(

+ ˜ ( ° + µh ) − v + Sv* µv * v

)(

)

* * ˜ * ˜ Sv + ˛ ˘ v − Sv µv Ih = , µh µh kSv*

* h v

h

* v

( ) ( + ˜ ) ( ° + µ ) ( − + S µ ))

+ ˝µh ) kSv* + Sv* + ˜ ( ° + µ h ) − v + Sv* µv h

v

* v

)) .

v

Note that I h* and Rh* are positive when ° v > µ vSv* . Values of the parameters are to be chosen such that Sh* > 0, and I v* > 0. Sv* is a positive solution of the cubic equation a1Sv* 3 + a2Sv* 2 + a3Sv* + a4 = 0,

(

(5.31)

)

where a1 = µ v ( λ + µ h ) kα − ( λ + µh ) µv > 0, if kα > ( λ + µh ) µv ,

(

)(

)

(

)

a2 = kα − ( λ + µh ) µv kΛ h − Λ v ( λ + µh ) + µv ( λ + µh )  k (αδ − ηµh ) − 2δµv ( λ + µh ) ,

( (

)

(

))

a3 = −δ ( λ + µh ) Λ v kα − 2 ( λ + µh ) µv + µv k ( Λ h + ηµh ) + δ ( λ + µh ) µv , a4 = δ 2 Λ v ( λ + µ h ) µ v > 0. 2

Note that a1 > 0 and a4 > 0. The discriminant of the cubic equation is given by ˜ = 18a1a2 a3 a4 − 4a23 a4 + a22 a32 − 4a1a33 − 27a12 a42 . If i. ˜ > 0, then the equation (5.31) has three distinct real roots. ii. ˜ = 0, then the equation (5.31) has multiple roots and all of its roots are real. iii. ˜ < 0, then the equation (5.31) has one real root and a pair of complex conjugate roots. We can also use the Descartes rule of signs to discuss the existence of positive real roots. The endemic equilibrium E* is locally asymptotically stable for 0 > 1, and unstable for 0 < 1, [29] (Problem 5.12, Exercise 5). 5.7.1 Existence of Travelling Wave Solutions Authors [29] investigated the existence of travelling wave solutions for the model (5.30), using the approach of Malinzi et al. [74] and Tireito et al. [102]. Defne z = x − vt , v > 0, where v is the speed of the propagating wave. Let Sh ( t , x ) = Sh ( z ), I h ( t , x ) = I h ( z ), Rh ( t , x ) = Rh ( z ) , Sv ( t , x ) = Sv ( z ) , and I v ( t , x ) = I v ( z ) . Then, the model (5.30) is transformed as D1S˝˝h + vS˝h + ˆ h − D2 Iˇˇh + vIˇh +

˜ Sh I v − µ hSh = 0, ° + Sh

˜ Sh I v − ( µ h + ˝ ) I h = 0, ° + Sh

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Zika Virus: Modeling of Spread and Outbreak

D3Sv′′ + vS′v + Λ v − D4 I v′′ + vI v′   +

κ Sv I h − µ vSv = 0, δ + Sv

κ Sv I h − µ v I v = 0, δ + Sv

where ' denotes the derivative with respect to the variable z. Express the above equations in the form Sh˛˛ + G1S˛h + F1 ( Sh , I h , Sv , I v ) = 0, I h˛˛ + G2 I h˛ + F2 ( Sh , I h , Sv , I v ) = 0, Sv˛˛ + G3Sv˛ + F3 ( Sh , I h , Sv , I v ) = 0, I v˛˛ + G4 I˛v + F4 ( Sh , I h , Sv , I v ) = 0,

(5.32)

   ˜S I  ˜ Sh I v where F1 = d1  ˘ h − − µ hSh  , F2 = d2  h v − ( µ h + ˝ ) I h  , ° + Sh    ° + Sh  ˘  ˘ ˜S I  ˜ Sv I h 1 v F3 = d3  ˙ v − − µ vSv  , F4 = d4  v h − µ v I v  , and di = , i = 1, 2, 3, 4. ,Gi =    ° + Sv  Di ° + Sv Di 8 Denote x1 = Sh°, x2 = I°, h x3 = Sv°, x 4 = I°, v and X = [ x1 , Sh , x2 , I h , x3 , Sv , x 4 , I v ] ˝ . Then, model (5.32) is transformed into the system of frst-order differential equations T

dX T = f ( X ) = [ −G1x1 − F1 , x1 , −G2 x2 − F2 , x2 , −G3 x3 − F3 , x3 , −G4 x4 − F4 , x4 ] , dz

(5.33)

with boundary conditions lim ( x1 , Sh , x2 , I h , x3 , Sv , x4 , I v ) = E 0 ,

z˝−ˆ

lim ( x1 , Sh , x2 , I h , x3 , Sv , x4 , I v ) = E* ,

z˝+ˆ

where E 0 is the DFE point and E* is the EE point. A traveling wave solution is a trajectory that joins E 0 and E* . Computing the Jacobian matrix of the model (5.32) and evaluating it at the DFE point E 0 = ( ˙ h /µ h , 0, 0, ˙ v /µ v , 0 ) , we obtain

JE0

˝ − v/D1 ˆ 1 ˆ ˆ 0 ˆ 0 =ˆ ˆ 0 ˆ 0 ˆ ˆ 0 ˆ 0 ˆ˙

µh /D1 0 0 0 0 0 0 0

0 0 − v/D2 1 0 0 0 0

0 0 t8 0 t9 0 −t10 0

0 0 0 0 − v/D3 1 0 0

0 0 0 0 µv /D3 0 0 0

0 0 0 0 0 0 − v/D4 1

t7 0 −t7 0 0 0 µv /D4 0

ˇ     ,      ˘

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Spatial Dynamics and Pattern Formation in Biological Populations

where t7 =

˜ h µ +˝ ˙ v ˙ v , t8 = h , t9 = , t10 = . D1 ( µh˛ +  h ) D2 D3 ( µvˆ +  v ) D4 ( µvˆ +  v )

The Jacobian matrix JE0 has eight real eigenvalues given by

˜1,2 =

− v ± v 2 + 4 ( µh + ˜ ) D2 − v ± v 2 + 4 µ h D1 , ˜3, 4 = 2D1 2D2

˜5,6 =

− v ± v 2 + 4 µv D3 − v ± v 2 + 4 µ v D4 , ˜7 ,8 = . 2D3 2D4

Four eigenvalues are positive and the other four eigenvalues are negative. Therefore, there exists a traveling wave solution of the model (5.30) with propagation speed v > 0. That is, there exists a traveling wave profle that connects the DFE to the EE. Biologically, this means that if infectious individuals are introduced into a fully susceptible population, then there would be formation of a transition zone of infectious individuals with a spread speed v, whenever 0 > 1 [29]. The authors have also performed the sensitivity analysis with respect to some key parameters involved in the derivation of 0 , which indicate that control strategies should target the reduction of the vector biting rate. Numerical simulations were carried out by the authors [29], (i) to graphically illustrate the long-term behavior of the model solutions, (ii) to study the effect of the transmission probabilities and the vector biting rate, and (iii) to determine the effects of control strategies such as the use of insecticide, treated mosquito bed and window nets, clearing of bushes near the homesteads, etc. The control strategies were found to signifcantly reduce the spread of ZIKV disease. From the simulations of the diffusion model, it was concluded that control strategies such as the effective use of treated mosquito bed and window nets and the use of an insecticide are effcient and effective in controlling the transmission of the disease. These control strategies were also identifed in the studies of Gao et al. [47] and Oluyo and Adeyemi [89].

5.8 Model 6: Zika Virus Transmission with Criss-Cross Interactions Model To understand how spatial heterogeneities of the vector and host populations infuenced the dynamics of the outbreak of Zika virus in Rio de Janeiro, Brazil, in both the geographical spread and the fnal size of the epidemic, Fitzgibbon, Morgan, and Webb [44] formulated the following criss-cross PDE model:

˜ H i (t, x, y ) = ˘  ° 1 ( x , y ) ˘H i ( t , x , y ) − ˛ ( x , y ) H i ( t , x , y ) ˜t

(

)

+ ˝ 1 ( x , y )  ( x , y ) Vi ( t , x , y ) ,

(5.34a)

˜ Vu ( t , x , y ) =   ° 2 ( x , y ) Vu ( t , x , y ) − ˛ 2 ( x , y ) H i ( t , x , y ) Vu ( t , x , y ) + ˝ ( x , y ) ˜t

(

{

)

}

{

}

× Vu ( t , x , y ) + Vi ( t , x , y ) − µ ( x , y ) Vu ( t , x , y ) + Vi ( t , x , y ) × Vu ( t , x , y ) ,

(5.34b)

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Zika Virus: Modeling of Spread and Outbreak

˜ Vi ( t , x , y ) = ˘  ° 2 ( x , y ) ˘Vi ( t , x , y ) + ˛ 2 ( x , y ) H i ( t , x , y ) Vu ( t , x , y ) ˜t

(

)

(

)

− µ ( x , y ) Vu ( t , x , y ) + Vi ( t , x , y ) Vi ( t , x , y ) ,

(5.34c)

with the following boundary and initial conditions ˛n H i = ˛n Vu = ˛n Vi = 0, X = ( x , y ) ˙˛ˆ,t > 0, H i ( X , 0 ) = H i0 ( X ) ˝ 0, Vu ( X , 0 ) = Vu0 ( X ) ˝ 0, Vi ( X, 0 ) = Vi 0 ( X ) ˝ 0, X = ( x , y ) ˙ ˆ. ˜ is a bounded domain in  2 with smooth boundary ˜°. The variables and parameters are defned as follows: H i ( t , x , y ) : Density of infected hosts at time t and position ( x , y ) ˛˝. Vu ( t , x , y ) : Density of uninfected vectors at time t and position ( x , y ) ˛˝. Vi ( t , x , y ) : Density of infected vectors at time t and position ( x , y ) ˛˝. ˛ ( x , y ) : Geographic density of the background population of uninfected and susceptible hosts in ˜. ˜ ( X ) : Loss rate of the infected host population (due to recovery or other removal). ˜ ( X ) : Breeding rate of the vector population. µ ( X ) : Loss rate of the vector population due to environmental crowding. ˜ 1 ( X ) : Transmission rate of infected hosts. ˜ 2 ( X ) : Transmission rate of uninfected vectors. ˜ 1 ( X ) : Diffusion rate of the infected hosts. ˜ 2 ( X ) : Diffusion rate of the infected vectors. The authors have proved that the model (5.34) is mathematically well posed and compared its properties with the following analogous ordinary differential equations model in the spatially independent case: dH i ( t ) = − ˜ H i ( t ) + ° 1˘Vi ( t ) , dt dVu ( t ) = −˜ 2 H i ( t ) Vu ( t ) + ° {Vu ( t ) + Vi ( t )} − µ {Vu ( t ) + Vi ( t )} Vu ( t ) , dt

(5.35)

dVi ( t ) = ˜ 2 H i ( t ) Vu ( t ) − µ {Vu ( t ) + Vi ( t )} Vi ( t ) . dt The basic reproduction number 0 = ˜ 1˜ 2 ˇ/( °µ ), which is interpreted as the average number of new cases generated by a single case at a given location X in ˜, is independent of the vector reproduction number ˜ . The size of the epidemic is proportional to ˜ . For 0 < 1, the steady states of (5.35) are S0 ( 0, 0, 0 ) and S1 ( 0, ˜ /µ , 0 ) , which are unstable and

310

Spatial Dynamics and Pattern Formation in Biological Populations

locally exponentially asymptotically stable respectively in +3 . If 0 > 1, then S0 ( 0, 0, 0 ) and S1 ( 0, ˜ /µ , 0 ) are unstable in +3 and there is another steady state in +3, given by  ˜ ( ˇ° 1° 2 − ˛µ ) ˜˛  ˜ ( 0 − 1) ˜˛ ˜˛ ( 0 − 1)  ˜ ( ˇ° 1° 2 − ˛µ )  S2  , , , , ,  = S2  ˛µ° 2 ˇ° 1° 2 ˇµ° 1° 2 °2 0 µ ˇ° 1° 2   which is proportional to ˜ and locally exponentially asymptotically stable in +3 . The authors have also modifed the model equations to take care of the effect of seasonality on the vector population, by assuming the following: (i) a breeding term of the form ˜ ( t , x , y ) that is dependent on time and (ii) in addition to the vector loss parameter µ ( x , y ) corresponding to carrying capacity, there is a time-independent vector loss term µ1 ( x , y ) corresponding to the average vector life-span 1/µ1 ( x , y ) . The authors numerically studied the following modifed model

˜ t H i =   {° 1 ( X ) H i } − ˛ ( X ) H i + ˝ 1 ( X )  ( X ) Vi , ˜ t Vu =   {° 2 ( X ) Vu } + ˛ ( t , X ) (Vu + Vi ) − ˝ 2 ( X ) H iVu − µ ( X ) (Vu + Vi ) Vu − µ1 ( X ) Vu , ˜ t Vi =   {° 2 ( X ) Vi } + ˛ 2 ( X ) H iVu − µ ( X ) (Vu + Vi ) Vi − µ1 ( X ) Vi , ˜ n H i = ˜ n Vu = ˜ n Vi = 0,

(5.36)

H i ( X , 0 ) = H i0 ( X ) ˝ 0,Vu ( X , 0 ) = Vu0 ( X ) ˝ 0,Vi ( X , 0 ) = Vi0 ( X ) ˝ 0. The authors found that the spatial distribution and fnal sizes of the 2015–2016 Zika virus outbreak in Rio de Janeiro are strongly dependent on the location and magnitude  of  local outbreaks at the beginning of the season. The authors have not studied in detail the dynamics of the model (5.34) except for its well-posedness. In a new study, Magal et al. [72] performed a detailed dynamical analysis of model (5.34) involving the extinction and persistence of ZIKV. Combining the arguments from the monotone dynamical system theory, persistence theory, and the theory of asymptotically autonomous semi-fows, authors proved the global stability of the positive steady state for a three-equation parabolic system when there is no clear Lyapunov-type functional. Cai et al. [22] investigated the effect of spatial heterogeneity on the extinction and persistence of Zika virus. The authors defned the basic reproduction number 0 for the spatial model system and proved that 0 can be used to govern the threshold dynamics of the ZIKV. Following Diekmann and Heesterbeek [37], Diekmann et al. [38], Van den Driessche and Watmough [104], and Wang and Zhao [109]; Cai et al. [22] identifed the basic reproduction number 0 for the model (5.34). Let V* ( X ) be the unique non-negative classical stationary solution of the problem −  {˜ 2 (X )V* } = V* ( ° (X ) − µ(X )V* ) , X 

˜ n V* = 0, X ˛˜ ˝. Linearizing model (5.43) about the disease-free steady state E1 = ( 0, V* (X ), 0 ) , the authors obtained the following model for the infection-related variables Hi and Vi:

311

Zika Virus: Modeling of Spread and Outbreak

∂ t H i   = ∇ ⋅ (δ 1 (X )∇H i ) − λ (X )H i + σ 1 (X )Λ(X )Vi , ∂ t Vi = ∇ ⋅ (δ 2 ( X ) ∇Vi ) − σ 2 ( X ) V*φ − ( µ ( X ) V* − β ( X )) V*φ , ∂ n H i = ∂ n Vi = 0, X ∈∂ Ω, t > 0.

(5.37)

Substituting H i (X , t) = e˛t˜ 1 (X), Vi (X , t) = e˛t˜ 3 (X) into (5.37) and canceling e˜t , we arrive at the eigenvalue problem    (˛ ( X )  ) − ˝ ( X ) 1 ˜° ( X ) =   ˙ 2 ( X ) V* 

˙ 1 (X )  (X )   (˛ 2 ( X )  ) − µ ( X ) V*



° (X ),



°n˜ = 0,

(

)

where ˜ = ˆ˙˜ 1 ˜ 3 ˇ˘  C  ,  2+ . Let  ( X , H i , Vi , Vu ) be the input rate of the newly infected T

hosts and  ( X , H i , Vi , Vu ) be the rate of transfer of hosts. Then, ˆ ˜ 1 ( X ) ˙ ( X ) Vi ˘  ( X , H i , Vi , Vu ) = ˘ ˜ 2 ( X ) H iVu ˘ 0 ˘ˇ

  ,  

 ˜ (X )H i  and  ( X , H i , Vi , Vu ) =  µ(X ) (Vu + Vi ) Vi   ˛ 2 (X )H iVu + µ(X ) (Vu + Vi ) Vu − ˝ (X ) (Vu + Vi ) At E1, the following results can be derived. ˆ 0 ˘ D( Hi ,Vi ,Vu ) ( X,E1 ) = ˘ ˜ 2 (X )V* ˘ 0 ˇ  ˜  and D( Hi ,Vi ,Vu ) ( X, E1 ) =  0  ˛ 2V* 

0   0 , 0 

˜ 1 (X)˙ ( X ) 0 0

0 µV*

µV* − ˝

  . 2 µV* − ˝  0 0

It follows that ˘ 0  F (X ) =   ˜ 2 ( X ) V*

˘ ° ˜ 1 (X ) ˇ (X )   ,V ( X ) =  0  0 

 0 . µ ( X ) V* 

  .  

312

Spatial Dynamics and Pattern Formation in Biological Populations

Let ˜: = (˜1 , ˜3 ) be the spatial distribution of initial infection and T ( t )˜ ( X ) be the solution semigroup generated by the following linear system

˜ t H i = ˇ ˘ (° 1 ( X ) ˇH i ) − ˛ ( X ) H i , ˜ t Vi = ˇ ˘ (° 2 ( X ) ˇVi ) − µ ( X ) V*Vi , ˜ n H i = ˜ n Vi = 0, X ˆ˜ ˇ, t > 0, H i ( 0, X ) = °1 ( X ) , Vi ( 0, x ) = °3 ( X ) , X ˆˇ. Thus, the spatial distribution of total new infected hosts caused by the initial infection ˙

˜

distribution ˜ ( X ) is F ( X ) T ( t )° ( X ) dt. 0

ˆ

()

ˆ

˜

˜

Define  ° ( X ) := F ( X ) T ( t )°dt = F .( X ) T ( t )°dt 0

0

 is a continuous positive operator. Following the works of the authors in references [37,38,104,109], the basic reproduction number 0 for the model (5.34) is defned by the spectral radius of . The authors [22] obtained the following results: 1. Model (5.34) has a unique disease-free steady state E1 ( 0,V* ( X ) , 0 ) ˛ , which is T globally asymptotically stable in  if 0 < 1 and  = u = ( H i , Vu , Vi ) ˙ , 0 ˆ H i  0 ˝ Vu ( X,.) + Vi ( X,.) ˝ M . If 0 > 1, then E1 is a uniform weak ( X,.) ˝ H, repeller for W0 and the model (5.43) admits at least one positive steady-state u* ( X ) = H i* ( X ) , Vu* ( X ) , Vi* ( X ) ˝W0 and there exists a ˜ > 0 such that for any ˜ °W0 , we have lim u ( X,t ) ˝ ˜ .

{

}

(

)

t˙ˆ

2. E0 = ( 0, 0, 0 ) is unstable. First, it can be shown that E0 = ( 0, 0, 0 ) is linearly unstable. Linearize the model (5.34) about the E0 to obtain

˜ t H i = ˘  (° 1 ( X ) ˘H i ) − ˛ ( X ) H i + ˝ 1 ( X )  ( X ) Vi , ˜ t Vu = ˇ ˘ (° 2 ( X ) ˇVu ) + ˛ ( X ) Vu + ˛ ( X ) Vi , ˜ t Vi = ˆ ˇ (° 2 ( X ) ˆVi ) , ˜ n H i = ˜ n Vu = ˜ n Vi = 0, X ˛˜ ˝, t > 0. Substituting ( H i , Vu , Vi ) = e˙t (˜1 ( X ) , ˜2 ( X ) , ˜3 ( X )) , and canceling e˜t , we obtain the following eigenvalue problem:    ˛ (X ) − ˝ (X ) (1 )   ˜° ( X ) = 0   0 

∂ nϕ ( X ) = 0, X ∈∂ Ω,

0   (˛ 2 ( X )  ) + ˆ ( X ) 0

˙ 1 (X )  (X )   ° (X ) , ˆ (X )    (˛ 2 ( X )  ) 

313

Zika Virus: Modeling of Spread and Outbreak

where ˜ = (˜ 1 , ˜ 2 , ˜ 3 ) . We fnd that zero is an eigenvalue with eigenfunction ˜ . Therefore, ( 0, 0, 0 ) is linearly unstable. Instability of E0 follows from this result (see [52]). Numerical simulations: The authors [22] applied the model (5.34) to the data of the 2015–2016 ZIKV epidemic in Rio de Janeiro, Brazil and performed numerical simulations to have some epidemiological insights. The cumulative number of the weekly reported cases for the Rio de Janeiro Municipality from November 1, 2015 through April 10, 2016 (during which time the reporting of cases became mandatory) was 25,400 [11,44]. More details about the background of the ZIKV outbreak in Rio de Janeiro can be found in the references [11,44]. Following Fitzgibbon et al. [44], Cai et al. [22] considered the 2D domain X = ( x, y ) ˆˇ = [ −25, 25 ] × [ −12,12 ]   2 . The parameter values are taken as T

˜ ( x , y ) = 1.0, ° 1 ( x , y ) = 5.0 × 10−7 , ° 2 ( x , y ) = 0.78, ˛ 1 ( x , y ) = ˛ 2 ( x , y ) = 0.2,

(

)

˛ ( x , y ) = 5000 1.08 + sin ( 0.02625ˆx ) cos ( 0.0585ˆy ) ,

(

)

µ ( x , y ) = 0.0015 1.008 + 717 × gauss ( 20, 30, x ) × gauss ( 0, 30, y ) , where gauss ( m, s, x ) is the probability density function in x of the normal distribution function with mean m and standard deviation s

(

˜ ( t , x , y ) = 330 emg t , µ , ˛ , ˝ = 330 ×

)

˝ ˝ exp  2 µ + ˝˛ 2 − 2t 2 2

(



) Erfc 

(

)

1  µ + ˝˛ 2 − t  ,  2˛

where Erfc is the complementary error function and µ = −2.0, ° = 5.5, ˛ = 0.2. It was found that 0 > 1, which gives the conclusion that the ZIKV outbreak will continue in Rio de Janeiro. For initial spatial distribution (44 weeks in 2015) of a very small number of cases located in a small region in the eastern subregion of the Municipality, they set H i ( 0, x , y ) = H i 0 × gauss ( x0 ,1.0, x ) × gauss ( y 0 ,1.0, y ) , Centered at ( x0 , y 0 ) , Vu ( 0, x, y ) = 100, Vi ( 0, x , y ) = 10H i ( 0, x , y ) , Centered at ( x0 , y 0 ) , with x0 = 15, y 0 = 0, H i0 = 10 (that is, the total number of infected people at time t = 0 is 10). Expansion of the evolution of the spatial distribution of infected people H i ( 0, x , y ) from the initial spatial distribution of about ten cases in week 44 in 2015 was obtained from the following data: 2015 Weak Number of cases

45 10

49 129

2016 3 1738

8 1895

13 1015

18 427

314

Spatial Dynamics and Pattern Formation in Biological Populations

After the 12th week of 2016, the ZIKV dispersed throughout the eastern region of the municipality. The numerical results agree qualitatively with the weekly reported data for Rio de Janeiro Municipality given in the references [11,44], with approximately 25,400 reported cases between November 1, 2015 (44th week, 2015) and April 10, 2016 (15th week, 2016). The simulated results of the model with seasonality (5.45) are presented as Examples 1 and 4 in the work of Fitzgibbon et al. [44].

5.9 Model 7: Zika Virus SEIR Transmission Model Dantas et al. [31] developed an epidemic model to describe the 2016 outbreak of Zika virus in Brazil. An inverse problem associated with the model was also formulated and solved. Kucharski et al. [64] have used a compartmental model to simulate vector-borne transmission dynamics. Both human and mosquito populations were modeled using a susceptibleexposed-infectious-removed (SEIR) framework. The model incorporated delays as a result of the intrinsic (human) and extrinsic (vector) incubation periods. Caminade et al. [23] developed a global 0 model for the transmission of the virus that explicitly includes two vector species and one host and considers the infuence of climate dynamically. They extended a recently developed two-vector mathematical framework for animal vectorborne diseases to ZIKV and parameterized the model using some published estimates. They have also derived a model using global observation-based historical climate data to derive global and seasonal estimates of 0 of ZIKV that describe transmission risk by one, the other, and both vectors where they co-occur. Olaniyi and Obabiyi [88] considered the transmission of ZIKV between human–mosquito and human–human populations. They found that increasing the recovery to a very high rate has a signifcant effect on reducing infection, and isolation of infected individuals also reduces the transmission of the ZIKV infection. Further, the rate of human-induced deaths of mosquitoes should be increased. This can be achieved by reducing the mosquito population through source reduction. Using the above works as motivation, Roy et al. [95] formulated a model for the transmission of Zika virus. The following assumptions were made in the derivation: i. The population is interacting homogeneously and well mixing. ii. Under high risk, the total human population (N h ) is divided into four mutually exclusive classes: a susceptible class (those at risk of contracting the disease, Sh), exposed class (infected but not yet infectious, Eh), infected class (showing symptoms of Zika disease and capable of transmitting the disease, I h), and recovered class (infectious people who have recovered from Zika virus, Rh ). N h = Sh + Eh + I h + Rh and the total vector population is N m = Sm + Em + I m, where both are assumed to be constant. iii. The infection force λ(t) is taken as ˜ ( t ) = pMI/N , where M is the contact number of one susceptible with all individuals per unit time. ( MI/N ) defnes the number of infectious contacts from those contacts, and p is the valid transmission probability under ( MI/N ) infectious contacts. It may be noted that the increase in the number of infectious cases may motivate people to use better protection measures. Thus, the transmission probability p is assumed to be a decreasing function of the infective number I.

315

Zika Virus: Modeling of Spread and Outbreak

p = p0 /( 1 + hI ), p0 > 0, h > 0. Then, ˜ ( t ) = (Mp0 I)/ ˆˇ( 1 + hI ) N ˘ . b. The contact number is M = k1mN where k1 is a constant of proportionality and m denotes the intensity of population mobility. Then, ˜ ( t ) = (k1mp0 I)/( 1 + hI ) . c. Defne a parameter ˜ h as ˜ h = mk1 p0 , where m is infuenced by the economic benefts of mobility and infection risk. Without epidemic infections, the mobility of the population is only determined by economic benefts [108].

iv. a.

Under the above assumptions, the model is governed by the following system of ordinary differential equations: dSh ˜ I = A − h m Sh − µ hSh = f1 ( Sh , Eh , I h , Rh , Sm , Em , I m ) , dt 1 + hI m dEh ˜ I = h m Sh − (° h + µ h ) Eh = f2 ( Sh , Eh , I h , Rh , Sm , Em , I m ) , dt 1 + hI m dI h = ˜ hEh − (° h + µ h ) I h = f3 ( Sh , Eh , I h , Rh , Sm , Em , I m ) , dt dRh = ( 1 − f )˜ h I h − µ h Rh = f 4 ( Sh , Eh , I h , Rh , Sm , Em , I m ) , dt

(5.38)

dSm = M − ˜ m I hSm − µmSm = f5 ( Sh , Eh , I h , Rh , Sm , Em , I m ) , dt dEm = ˜ m I hSm − (° m + µm ) Em = f6 ( Sh , Eh , I h , Rh , Sm , Em , I m ) , dt dI m = ˜ mEm − µm I m = f7 ( Sh , Eh , I h , Rh , Sm , Em , I m ) . dt Sh ( 0 ) > 0, Eh ( 0 ) > 0, I h ( 0 ) > 0, Rh ( 0 ) > 0,Sm ( 0 ) > 0, Em ( 0 ) > 0, I m ( 0 ) > 0. Analysis of equilibrium points: The system has a DFE point Q0 = ( A/µ h , 0, 0, 0, M/µm , 0, 0 ) . EE point Q* Sh* , Eh* , I h* , Rh* , Sm* , Em* , I m* exists. From the seventh equation, we get Em = ( µm /° m ) I m . M ( µm + ° m ) Adding the ffth and sixth equations and solving, we get Sm = − Im . µm °m

(

)

From the third and fourth equations, we get Rh =

( 1 − f )˜ h I

Adding the frst two equations and solving, we get Sh =

1 µh

µh

h

 (° h + µh ) ( µh + ˛ h ) I  . h A − ˛h  

, Eh =

(˜ h + µh ) I ˛h

h

.

316

Spatial Dynamics and Pattern Formation in Biological Populations

The solution of the system in I h , I m gives Ih = Im =

µm

(

t1 − t2 , ˜ m (° h + µh ) (˝ h + µh ) t3

t1 − t2 . A° h ° m˛ h + (˝ h + µ h ) ( ° h + hµh ) µm (˛ h + µ h ) (˛ m + µm )

)

t1 = AM˜ h ˜ m° m° h , t2 = µ h (˝ h + µ h ) µm2 (° h + µ h ) (° m + µm ) ,

(

))

(

t3 = M˜ h° m + µh µm2 + ( hM + µm ) ° m .

(

)

Substituting backward, we get the solution for Q* Sh* , Eh* , I h* , Rh* , Sm* , Em* , I m*   , where Sh* =

t4 + A˜ m° h ( hM° m + µm (° m + µm ) )

˜ m° ht3

Eh* =

t1 − t2 t1 − t2 , I h* = , ˜ m° h (° h + µh ) t3 ˜ m (˝ h + µh ) (° h + µh ) t3 Rh* =

Sm* =

, t4 = (˝ h + µ h ) µm2 (° h + µ h )(° m + µm ) ,

(1 − f )˜ h (t1 − t2 )

° m µh (˜ h + µh ) (˝ h + µh ) t3

,

(˜ h + µh ) (˛ h + µh ) t3 , ( hµh (˜ h + µh ) µm (˛ h + µh ) + ˝h ( A˝m˛ h + (˜ h + µh ) µm (˛ h + µh )))˛ m Em* =

t1 − t2 t1 − t2 , I m* = , t5 (˜ m + µm ) µmt5 (˜ m + µm )

(

)

t5 = A˜ h ˜ m° h + (˛ h + µ h ) ( ˜ h + hµ h ) µm (° h + µ h ) . We fnd Sh* > 0, Sm* > 0. EE point exists, when the conditions t1 > t2 , and f < 1 are satisfed. Further, no endemic equilibrium exists if 0 < 1, and a unique endemic equilibrium exists if 0 > 1 (Problem 5.13, Exercise 5). Stability analysis: At Q0 = ( A/µ h , 0, 0, 0, M/µm , 0, 0 ) , the Jacobian is given by  − µn  0   0   0 J0 =   0  0   0 

0

0

0

0

0

− ( µn + ˛ h )

0

0

0

0

− (˝ h + µn )

0

0

0

A˛ h 0

( 1 − f )˝ h

− µn

0

0

0

−M° m /µm

0

− µm

0

0 0

M° m /µm 0

0 0

0 0

−( µ m + ˛ m ) ˛m

−A° h /µ h   A° h /µ h   0   0 .  0  0   − µm 

317

Zika Virus: Modeling of Spread and Outbreak

The characteristic equation of J 0 is obtained as

( ˜ + µh )2 ( ˜ + µm )( ˜ 4 + a1˜ 3 + a2 ˜ 2 + a3˜ + a4 ) = 0, where a1 = ˜ h + 2 µ h + 2 µm + ˛ h + ˛ m , a2 = µh2 + µm ( µm + 2° h ) + ( µm + ° h ) ° m + ˛ h ( µh + 2 µm + ° h + ° m ) + µ h ( 4 µm + ° h + 2° m ) , a3 = µm ( 2 µh ( µ h + µm ) + ( 2 µ h + µm ) ° h + ˛ h ( 2 µ h + µm + 2° h ) ) + ( µ h ( µ h + 2 µm ) + ( µ h + µm ) ° h + ˛ h ( µ h + µm + ° h ) )° m , a4 = =

(

)

µh (° h + µh ) µm3 ( µh + ˛ h ) + −AM˝ h ˝ m˛ h + µ h (° h + µ h ) µm2 ( µ h + ˛ h ) ˛ m µ h µm

−AM˝ h ˝ m˛ h˛ m + µ h (° h + µ h ) µm2 ( µ h + ˛ h ) ( µm + ˛ m ) µ h µm

(

)

= µm (° h + µ h ) ( µ h + ˛ h ) ( µm + ˛ m ) 1 − 02 . We observe that three eigenvalues of J 0 are − µ h , − µ h , − µm . We find that a1 > 0, a2 > 0, and a3 > 0. If 0 < 1, then a4 > 0. In this case, all the coefficients are positive and there is no positive root. The remaining roots are negative or have negative real parts, if the coefficients satisfy the Routh–Hurwitz criteria a1 , a2 , a3 , a4 > 0, and a1a2 a3 > a32 + a12 a4 . Then, the DFE point Q0 , is locally asymptotically stable if 0 < 1 and unstable if 0 > 1. Stability of EE: At the EE point Q* Sh* , Eh* , I h* , Rh* ,Sm* , Em* , I m*   , the Jacobian is obtained as

(

 ˜ h I m* − µh  − 1 + hI m*   ˜ h I m*   1 + hI m*   0  * J = 0  0   0   0 

0 −t6

0 0

)

0

0

0

0

0

0

−

˜ h Sh*

(1 + hI ) * m

˜ h Sh*

(1 + hI )

* 2 m

˛h 0 0 0 0

−t7 t9 − ˜ m Sm*

˜ m Sm* 0

0 − µh 0 0 0

(

0 0

− µm + ˜ m I h*

˜ m I h* 0

where t6 = µ h + ° h , t7 = ˛ h + µ h ,t8 = µm + ° m , t9 = ( 1 − f )˛ h .

)

0 0 0

0 0 0

−t8

0

˛m

− µm

2

         ,      

318

Spatial Dynamics and Pattern Formation in Biological Populations

The characteristic equation of J * is given by

(

)

Q ( x ) ˙ ( ˜ + µh ) ( ˜ + µm ) ˜ 5 + b1˜ 4 + b2 ˜ 3 + b3 ˜ 2 + b4 ˜ + b5 = 0, where

((

)

b1 =

1 1 + hI m* t6 + t7 + t8 + ˜ m I h* + µ h + µm + I m* ˜ h , 1 + hI m*

b2 =

1 t8 ˜ h I m* + t8 ˜ m I h* + µ h 1 + hI m* + ( ˜ h + hµ h ) ˜ m I m* I h* + ˜ m µh I h* 1 + hI m*

)(

(

((

)

(

)(

)

)

)

((

)(

)

+ 1 + hI m* ( t8 + µ h ) + I m* ˜ h µm + t7 ˜ m I h* + µ h + µm + t8 1 + hI m* + I m* ˜ h

((

)(

)

))

)

+ t6 t7 + t8 + ˜ m I h* + µ h + µm 1 + hI m* + I m* ˜ h , ˇ    t7 t8 ˜ m I h* + µ h + ( t7 + t8 ) ˜ m µ h I h* + ( t8 µ h + t7 ( t8 + µ h ) ) µm   ˇ  * *  t8 ˜ m I h + t8 µ h + ˜ m µ h I h + ( t8 + µ h ) µm   1   +t7 1 + hI m* t8 + ˜ m I h* + µ h + µm + I m* ˜ h + h ˜ m I h* + µ h + µm b3 = +t 6  1 + hI m*    * *  *  ˘ +I m ( ˜ h + hµ h ) ˜ m I h + µm + t8 ˜ h + h ˜ m I h + µ h + µm   ˇ   ˇ ( ˜ h + hµ h ) ˜ m I h* + µm  * *    + I m  t8 ( ˜ h + hµ h ) ˜ m I h + µm + t7   *  ˘ + t8 ˜ h + h ˜ m I h + µ h + µm   ˘ ˘

(

)

(( (

(

)

(

)

(

(

1

(1 + hI )

* 2 m

)))

(

)))

(

(

)

(

)

))

(

)(

)

˘ t6t7 t8 µ h + ° m I h* + µm t6t7 t8 + ( t6t7 + ( t6 + t7 ) t8 ) µ h   ˘ ˘ ( t3 + t7 ) ( ° h + hµ h ) ° m I h* + µm    *     t7 t8 ( ° h + hµ h ) ° m I h + µm + t6    *    +t7 t8 ° h + h ° m I h + µ h + µm   +h I m* 2    ˘ ˘ ( t8 + t7 ) ( ° h + 2hµ h ) ° m I h* + µm   *  *   +I m  t7 t8 ( ° h + 2hµ h ) ° m I h + µm + t6      + t7 t8 ( ° h + 2hµ h ) + 2h ° m I h* + µm     * *   − ° h ° m˛ h˛ mShSm

(

b4 =

(

          ,          

(

)

(

( )

(

)

)

))

(

(

(

)

(

        ,              

))

319

Zika Virus: Modeling of Spread and Outbreak

b5 =

=

(

)(

)(

)

t6t7 t8 1 + hI m* µ h + I m* ( ° h + hµh ) ° m I h* + µm − ° h ° m µh˛ h˛ mSh* Sm*

(1 + hI )

* 2 m

)(

(

).

t6t7 t82 µh µm2 t6t7 hµh µm + ° h ( A° m˛ h + t6t7 µm ) R02 − 1

° h ( µmt3 ( A° m˛ h + t6t7 µm ) + AhM° m˛ h˛ m )

Two of the eigenvalues of J * are − µ h , − µm , and the remaining fve eigenvalues are the roots of the polynomial ˜ 5 + b1˜ 4 + b2 ˜ 3 + b3 ˜ 2 + b4 ˜ + b5 = 0. The roots of this polynomial are negative or have negative real parts, if the coeffcients satisfy the Routh–Hurwitz criteria

(

)

b1 , b2 , b3 , b4 , b5 > 0; b1b2b3 > b32 + b12b4 ,

(b1b4 − b5 )(b1b2b3 − b32 − b12b4 ) > b5 (b1b2 − b3 )2 + b1b52 . This gives the required conditions for the EE to be locally asymptotically stable, if 0 > 1. Bifurcation analysis: A method based on the central manifold theory was used by the authors to establish the local stability of the EE taking ˜ h as a bifurcation parameter [27]. At least, one of the eigenvalues of J 0 will be zero if and only if det ( J 0 ) = 0. This gives

˜ h = ˜ hc =

µh µm2 (˛ h + µh ) (˝ h + µh )(˝ m + µm ) . A 2 M˜ m˝ h˝ m

For example, for the values of the parameters A = 0.0154, µ h = 0.001, σ h = 0.2, h = 10.9, γ h = 0.64, f = 0.5, M = 0.07, β m = 0.116, µm = 0.028, σ m = 0.0833; two of the eigenvalues are nearly zero. The other five eigenvalues are real and negative and are given by −0.641043, −0.200082, −0.112865, −0.028, and −0.0273093. Hence, when β h = β hc ≈ 0.0053965 (at  0 = 1), the  DFE point Q0 = ( A/µ h , 0, 0, 0, M/µm , 0, 0 ) is non-hyperbolic. Let v = ( v1 , v2 , v3 , v4 , v5 , T v6 , v7 ) and w = ( w1 , w2 , w3 , w4 , w5 , w6 , w7 ) , denote respectively the left and the right eigenvectors associated with the zero eigenvalue, such that v ⋅ w = 1. The components of the eigenvectors w and v must satisfy the equalities v ⋅ J 0 = 0 and J 0 ⋅ w = 0. The following are the values of the arguments w1 = −

(˜ h + µh ) µm2 ( µh + ˛ h )( µm + ˛ m ) w7 , w M˝ m µh˛ h˛ m

w4 =

(1 − f )˜ h µm2 ( µm + ˛ m ) w7 , w

v5 = 0, v2 =

M˝ m µh˛ m

5

2

=

=−

(˜ h + µh ) µm2 ( µm + ˛ m ) w7 , w M˝ m˛ h˛ m

( µ m + ˛ m ) w7 , w ˛m

6

=

3

=

µm2 ( µm + ˛ m ) w7 , M˝ m˛ m

µ m w7 . v1 = 0, v4 = 0, ˛m

M˜ m° m v7 ° m v7 M˜ m° h° m v7 , v3 = , v6 = . µm + ° m (˛ h + µh ) µm ( µh + ° h )( µm + ° m ) (˛ h + µh ) µm ( µm + ° m )

The values of v7 and w7 are taken as arbitrary. Non-zero derivatives at DFE are computed as

˜ 2 f2 ˜ 2 f2 ˜ 2 f2 A 2hA° h ˜ 2 f6 = °h , =− , = °m , = . 2 ˜ x1 ˜ x7 ˜ x7 µh ˜ x5 ˜ x3 ˜ x7 ˜ ° h µ h

320

Spatial Dynamics and Pattern Formation in Biological Populations

Denote x1 = Sh , x2 = Eh , x3 = I h , x4 = Rh , x5 = Sm , x6 = Em , x7 = I m. The coeffcients a and b defned by (see [27]) 7

a=

˜

7

v k wi w j

k , i , j =1

° 2 fk ° 2 fk (Q0 , ˛ hc ) , b = vk wi (Q0 , ˛ hc ) , ° xi ° x j ° xi ° ˛ k , i= 1

˜

can now be explicitly computed. The values of a and b are given by a=− b=

 1  µm3 ( µm + ° m ) ˛ hc  2 2AhM˛ m° h° m + v w2 , µm +   µm  M° m µh  (˝ h + µh ) ( µh + ° h )( µm + ° m )   7 7

AM˜ m° h° m v7 w7 . µh (˝ h + µh ) µm ( µh + ° h ) ( µm + ° m )

It can be seen that a < 0 and b > 0 at ˜ h = ˜ hc. Hence, a forward bifurcation occurs at 0 = 1. Bifurcation diagrams were generated by taking the transmission rate ˜ h as the bifurcation parameter with the values of the other parameters taken from Table 5.1, (see Figure 5.8). The authors examined the variations of the infected and exposed classes in the ranges 0 < I h < 1500, and 0 < Eh < 5000, as a function of ˜ h which varies in the ranges 0 < ˜ h < 0.25 and 0 < ˜ h < 0.2 respectively (Figure 5.8a and b). For the system parameters given in Table 5.1, the partial rank correlation coeffcient (PRCC) sensitivity indices of the basic reproduction number 0 can be calculated (Problem 5.14, Exercise 5). 5.9.1 Model with Diffusion Diffusion terms can be included in the model (5.38), to take care of the movements that are taken as random and uniformly distributed in all directions. The diffusion model is written as

FIGURE 5.8 Bifurcation diagrams for (a) infected human population and (b) exposed human population as a function of control parameter βh.

321

Zika Virus: Modeling of Spread and Outbreak

TABLE 5.1 Variables, Their Defnitions, and Previously Published Estimates of Parameters Used for Numerical Simulations Parameter A βh μh σh H γh F M βm μm σm

Description

Value

Recruitment rate of infected humans Transmission rate (mosquito-to-human) Death rate of humans Incubation period of humans Precaution constant Recovery rate of humans Fatality Recruitment rate of infected mosquitoes Transmission rate (human-to-mosquito) Death rate of mosquitoes Incubation period of mosquitoes

0.0154 0.0885 0.001 0.2 10.9 0.64 0.5 0.07 0.116 0.028 0.0833

ˇ ° I  ˜ Sh = A −  h m  Sh − µ hSh + D1 2Sh , ˘ 1 + hI m  ˜t

˜ Eh  ° h I m  = Sh − (˛ h + µ h ) Eh + D2 2Eh , ˜ t  1 + hI m  ˜ Ih = ° hEh − (˛ h + µ h ) I h + D3 2 I h , ˜t ˜ Rh = ( 1 − f )° h I h − µ h Rh + D4 2 Rh , ˜t ˜ Sm = M − ° m I hSm − µmSm + D5ˇ 2Sm , ˜t ˜ Em = ° m I hSm − (˛ m + µm )Em + D6˘ 2Em , ˜t ˜ Im = ° m I m − µm I m + D7 ˇ 2 I m , ˜t with boundary conditions

( n ˛ ˝ ) Sh = ( n ˛ ˝ ) Eh = ( n ˛ ˝ ) I h = ( n ˛ ˝ ) Rh = ( n ˛ ˝ ) Sm = ( n ˛ ˝ ) Em = ( n ˛ ˝ ) Im = 0, and initial conditions Sh ( x , y , 0 ) = Sh0 ( x , y ) > 0,Eh ( x , y , 0 ) = Eh0 ( x , y ) > 0, I ( x , y , 0 ) = I h 0 ( x , y ) > 0, R ( x , y , 0 ) = Rh0 ( x , y ) > 0,Sm ( x , y , 0 ) = Sm0 ( x , y ) > 0, Em ( x , y , 0 ) = Em0 ( x , y ) > 0, I m ( x , y , 0 ) = I h0 ( x , y ) > 0.

(5.39)

322

Spatial Dynamics and Pattern Formation in Biological Populations

D1 , D2 , D3, and D4 are diffusion coeffcients of susceptible, exposed, infected, and recovered human populations respectively. D5 , D6 , and D7 are diffusion coeffcients of susceptible, exposed, and infected mosquito populations respectively. The initial data Sh0 ( x , y ) , Eh0 ( x , y ) , I h0 ( x, y ) , Rh0 ( x , y ) , Sm0 ( x , y ) , Em0 ( x , y ) , and I m0 ( x, y ) are non-negative continuous bounded functions in the habitat ˜ ° R 2 ; ˜ 2 is the two-dimensional Laplacian; n is an outward unit normal vector to the smooth boundary ∂Ω of the habitat Ω. The homogeneous Neumann boundary conditions signify that the system is self-contained and there is no population flux across the boundary ∂Ω. System in two dimensions: As mentioned in the Introduction of the chapter, arrival of Zika virus in India was reported in Ahmedabad, Gujarat, in May 2017 and it was reported in Rajasthan in September 2018, and in Madhya Pradesh in November 2018. The authors used the initial conditions as given by Hu et al. [59], to the model (5.39), to fnd the possible effect of such initial conditions on the spread of Zika virus in India. The authors [95] solved the model system numerically in the 2D spatial domain [0, 200] with time and space steps taken as t = 0.001 and h = 0.25 respectively using MATLAB R 2017a. The initial conditions are taken as follows [59]: Sh ( x, y , 0 ) = Sh* , Eh ( x, y , 0 ) = Eh* , 2 ˇ * 2 if ( x − 100 ) + ( y − 100 ) < 400,  Ih , I h ( x, y , 0 ) = ˘ otherwise,  0.0001,

ˇ * if  Rm , Rm ( x , y , 0 ) = ˘  0.0001, ˇ *  Sm , Sm ( x , y , 0 ) = ˘  0.0001,

if

2

< 400,

otherwise,

( x − 100 )2 + ( y − 100 )

2

< 400,

otherwise.

ˇ * if  Em , Em ( x , y , 0 ) = ˘  0.0001, ˇ *  Im , Im ( x, y , 0) = ˘  0.0001,

( x − 100 )2 + ( y − 100 )

if

( x − 100)2 + ( y − 100)

2

< 400,

otherwise,

( x − 100)2 + ( y − 100)

2

< 400,

otherwise,

which indicates a uniform distribution of susceptible and exposed classes throughout the domain and availability of infective and recovered populations at one place and very less at other locations. For simulations, the diffusion coeffcients were chosen as D1 = 15, D2 = 0.5, D3 = 10−5 , D4 = 5, D5 = 10, D6 = 5, D7 = 5. The EE point is obtained as Q* = ( 2.5936, 0.0637, 0.0198,6.3613, 2.3097, 0.0478, 0.1423 ) . In the numerical simulations, different types of patterns were observed and are attributed to a large variety of symmetry properties realized by different values of diffusion coeffcients. Perturbation of initial conditions leads to the formation of circular patterns. The following was concluded as a

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possible scenario of the spread of the virus: initially, at t = 10 days (approximately at the end of May 2017), the infective and exposed populations are concentrated at one main location (say, Ahmedabad). It was observed that at the time level t = 150 days, the maximum density of susceptible and recovered populations is moving to the corners of the square domain. The density of infective and exposed populations increased with time. At the time level t = 500 days, the susceptible population is now observed to be shifted toward the right upper corner of the domain and the infective and exposed population densities decreased with respect to the initial state at t = 10 days (see Figure 5.9). (The authors [95] have plotted the snapshots at times t = 10, 200, and 600.) The simulations suggest that once the disease is introduced in an area with a suitable climate like that of India, then eliminating Aedes mosquitoes completely will be a diffcult task. To achieve the goal as quickly as possible, efforts should be made for case fnding, case management, community-level mosquito surveillance and control measures, and personal protection (avoidance of mosquito bites). The ZIKV epidemic is characterized by slow growth and high spatial and seasonal heterogeneity, attributable to the dynamics of the mosquito vector and to the characteristics and mobility of the human populations. Zhang et al. [115] analyzed the spatial and

FIGURE 5.9 Snapshots of pattern formations for the time evolution of different classes of population at time t = 150 days. (Continued)

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FIGURE 5.9 (CONTINUED) Snapshots of pattern formations for the time evolution of different classes of population at time t = 500 days.

temporal dynamics of the Zika virus epidemic in the Americas with a microsimulation approach like high-defnition demographic mobility and epidemic data. The model provides probability distributions for the time and place of introduction of Zika virus in Brazil [57], the estimate of the attack rate, timing of the epidemic in the affected countries, and the projected number of newborns from women infected by the virus. These results are potentially relevant in the preparation and analysis of contingency plans aimed at Zika virus control. O’Reilly et al. [90] developed a deterministic meta-population model for ZIKV transmission between major cities (considered 90 locations consisting of large cities and islands) in Latin America and the Caribbean region where city-level population interacts according to several scenarios based on (i) a simplifed gravity model with one estimated parameter; (ii) a gravity model where three exponential terms were estimated; (iii) a radiation model; (iv) a data-driven approach based on fight data; and (v) a model of local radiation and fight movements. Gravity models assume that movement between cities is highest when located near each other and when both cities are large. Their model was calibrated using epidemiological time-series data to estimate transmission parameters. Li  et al. [66] developed a model of ZIKV transmission in Colombia on complex networks, which considers both sexual transmission among humans and the transmission by an infective vector in the process of propagation. The authors found that the transmission

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between mosquitoes and humans has a greater infuence than that of sexual contacts on ZIKV spread, which may provide new insights for ZIKV control in Colombia.

Exercise 5 5.1 Obtain the conditions on the coeffcients of the characteristic equation so that the disease-free equilibrium point E0 = ( ˙ H /µ H , 0, 0, 0, ˙V /µV , 0, 0 ) , of the model system (5.14) is locally asymptotically stable. 5.2 Show that the endemic equilibrium point E1 (see equation 5.15) of the model system (5.14) is asymptotically stable for 0 > 1. 5.3 Derive the characteristic equation of J (E0 ) given in (5.17) and study the properties of its roots. 5.4 Obtain the disease-free equilibrium of the Model (5.21). 5.5 Calculate the basic reproduction number 0 for the model system (5.21). 5.6 Investigate the local stability of N 0 for the model system (5.21) using the nextgeneration operator method. 5.7 Show that the region ˜ 1 is positively invariant for the model system (5.29). 5.8 Calculate the basic reproduction number 0 for the model system (5.29). 5.9 Discuss the global stability of the disease-free equilibrium E0 for the model system (5.29). 5.10 Prove that the disease-free equilibrium E0 of the temporal model system (5.30) is locally asymptotically stable for 0 < 1 and unstable for 0 > 1. 5.11 Discuss the global stability of the DFE E0 for the temporal model system (5.30). 5.12 Prove that the endemic equilibrium E* for the temporal model system (5.30) is locally asymptotically stable for 0 > 1 and unstable for 0 < 1. 5.13 Calculate the basic reproduction number 0 and show that no endemic equilibrium exists if 0 < 1, and a unique endemic equilibrium exists if 0 > 1. 5.14 Calculate the PRCC sensitivity indices of the basic reproduction number 0 to the system parameters given in Table 5.1.

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95. Roy, P., Upadhyay, R. K., Caur, J. 2020. Modeling the Zika transmission dynamics: prevention and control. J. Biol. Sys. 28(3), 719-749. 96. Sasmal, S. K., Ghosh, I., Huppert, A., Chattopadhyay, J. 2018. Modeling the spread of Zika virus in a stage-structured population: Effect of sexual transmission. Bull. Math. Biol. 80(11), 3038–3067. 97. Shi, W., Zhang, Z., Ling, C., Carr, M. J., Tong, Y., Gao, G. F. 2016. Increasing genetic diversity of Zika virus in the Latin American outbreak. Emerging Microbes Infect. 5(7), 68. 98. Simpson, D. I. H. 1964. Zika virus infection in man. Trans. R. Soc. Trop. Med. Hygiene. 58(4), 335–338. 99. Smith, D. L., Battle, K. E., Hay, S. I., Barker, C. M., Scott, T. W., McKenzie, F. E. 2012. Ross, Macdonald, and a theory for the dynamics and control of mosquito-transmitted pathogens. PLoS Path. 8(4), 1002588. 100. Smith, H. L., Wang, L., Li, M. Y. 2001. Global dynamics of an SEIR epidemic model with vertical transmission. SIAM J. Appl. Math. 62(1), 58–69. 101. Tang, B., Xiao, Y., Wu, J. 2016. Implication of vaccination against dengue for Zika outbreak. Sci. Rep. 6, 35623. 102. Tireito, K. F., Lawi, G. O., Okaka, C. A. 2018. HIV/AIDS treatment model with the incoporation of diffusion equations. Appl. Math. Sci. 12, 603–615. 103. Tognarelli, J., Ulloa, S., Villagra, E., Lagos, J., Aguayo, C., Fasce, R., Parra, B., Mora, J., Becerra, N., Lagos, N., Vera, L. 2016. A report on the outbreak of Zika virus on Easter Island, South Pacifc, 2014. Arch. Virol. 161(3), 665–668. 104. Van den Driessche, P., Watmough, J. 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48. 105. Vest, K. G. 2016. Zika virus: A basic overview of an emerging arboviral infection in the Western Hemisphere. Disas. Med. Pub. Heal. Preparedness 10(5), 707–712. 106. Vorou, R. 2016. Letter to the editor: Diagnostic challenges to be considered regarding Zika virus in the context of the presence of the vector Aedes albopictus in Europe. Eur. Surv. 21(10), 30161. 107. Wagner, J. G. 1973. Properties of the Michaelis-Menten equation and its integrated form which are useful in pharmacokinetics. J. Pharma. Biopharma. 1(2), 103–121. 108. Wang, W. 2012. Modeling adaptive behavior in infuenza transmission. Math. Model. Nat. Phenom. 7(3), 253–262. 109. Wang, W., Zhao, X. Q. 2012. Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J. Appl. Dyn. Sys. 11(4), 1652–1673. 110. World Health Organization (WHO), 2017. Zika situation report. Available at http://www.who. int/ emergencies/ zika-virus/ situation-report/ 10-march-2017/ en/ (accessed on 9 May 2017). 111. World Health Organization, 2016. WHO and experts prioritize vaccines, diagnostics and innovative vector control tools for Zika R&D. Saudi Medic. J. 37(4), 471–472. 112. http://www.who.int/mediacentre/news/notes/2016/research-development-zika/en. 113. World Health Organization, 2016. WHO Zika Virus (ZIKV) Vaccine Target Product Profle (TPP): Vaccine to Protect Against Congenital Zika Virus Syndrome for Use During an Emergency. Geneva: WHO/UNICEF. 114. World Health Organization, 2018. WHO Zika virus and complications: 2016 Public Health Emergency of International Concern. http://www.who.int/emergencies/zika-virus/en/. (accessed January 2018.) 115. Zanluca, C., Melo, V. C. A. D., Mosimann, A. L. P., Santos, G. I. V. D., Santos, C. N. D. D., Luz, K. 2015. First report of autochthonous transmission of Zika virus in Brazil. Memórias do Instituto Oswaldo Cruz 110(4), 569–572. 116. Zhang, Q., Suna, K., Chinazzia, M., Pastore, Y. P. A., Dean, N. E., Rojas, D. P., Merler, S., Mistry, D., Polettie, P., Rossi, L., Bray, M., Halloran, M. E., Longini Jr., I. M., Vespignani, A. 2017. Spread of Zika virus in the Americas. Proc. Natl. Acad. Sci. U. S. A. 114(22), E4334–E4343.

6 Brain Dynamics: Neural Systems in Space and Time

6.1 Introduction Brain is the most complicated self-organizing system that does not rest. It consists of cells that continuously receive information and takes decisions. It is composed of a large number of different cells. Neurons are elementary signal processing units. Neurons, or nerve cells, are electrically excitable cells that process and transmit information. There are almost 100 billion neurons in the human brain and nearly about 1011 cell bodies. About 40  types of neurons have been identifed in the cortex region of the brain [85]. The brain is also composed of glial cells. These cells are divided into two categories: (i) neuroglial cells in the brain, which are further subdivided into astrocytes and oligodendrocytes, and (ii) Schwann cells, or neurolemmocytes, in the periphery. They play the following major roles in the functions of a brain: (i) provides nutrition and energy, (ii) maintains homeostasis, (iii) forms myelin which is an electrically insulating dielectric phospholipid layer that surrounds the axons of some neurons, (iv) participates in signal transmission, (v) ensures structural stabilization of brain tissues, and (vi) destroys pathogens and removes dead neurons [85]. Recent studies have shown that astrocyte glial cells interact with neurons and infuence the information processing. Newman [113] has shown that activated glial cells, excited by focal injections of certain chemical substances, can inhibit neurons by releasing ATP. The cortex region in the brain which is the superfcial part of the encephalon is mainly composed of gray matter formed by neurons and unmyelinated fbers. The white matter below the gray matter of the cortex is formed mostly by the myelinated axons which interconnect various parts of the central nervous system. The gray matter has a horizontal organization in layers composed of various cells. Most of the cortex region is composed of six layers of neurons. Layer I is at the surface of cortex, and layer VI lies close to the white matter. In human brain, the thickness of a layer ranges from 3 to 6 mm. The detailed information about cortical structure and its functional activities can be found in the works of Peters and Jones [117], Grimbert [59], Touboul [137]. Signal processing in the brain is mainly accomplished through the neurons and synapses which are interconnected. Neurons work together and play the main role in control and cognition. Action potential plays a major role in the communication between the neurons. The propagation and transmission of signal processing in the neuronal system are described in the works of Shepherd [129], Johnston and Wu [83]. The structure of a neuron has four defned regions or parts. These are dendrites, cell body, axon, and presynaptic terminals. The roles played by these parts in the generation/communication in signal processing are the following: (i) Dendrites have tree-like branches. The input signal reaches the neuron via the dendrites. Sometimes, the signals may outfow the dendritic branches (see Figure 6.1,  [71]). 331

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FIGURE 6.1 Anatomical overview of a typical neuron. (Reproduced with permission from Dr. Mariana Ruiz Villarreal.)

(ii) The cell body or soma represents the metabolic center of the neuron. It holds the nucleus which contains the genetic information of the cell as well as the endoplasmic reticulum. The cell’s protein synthesis also takes place here. The diameter of the nucleus ranges from 3 to 18 mm. The cell body is connected to other neurons through the dendrites. (iii) Axon transmits the electrical signals along the distances ranging from 0.1 mm to meters. The neurons have one axon or may have axon with branching to communicate with other neurons. The region of an axon where it emerges from the cell body is known as axon hillock. The axon hillock is an important part of the neuron where the greatest densities of voltage-dependent sodium channels exist. It is the most excited part of the neuron. (iv) Axons divide in the branches at the end of a neuron which is referred to as presynaptic terminal. It consists of synapses where neurotransmitters are released to communicate with other neurons. The signal propagated along the neuron is emitted from the presynaptic cell and reaches the postsynaptic cell using the activity of synapse. The presynaptic terminal transmits the signal from axon hillock. The space between two communicating neurons is referred as synaptic cleft. A cubic micron area of cytoplasm may contain 1010 water molecules, 107 molecules like amino acids and nucleotides, 108 numbers of ions, and 105 number of protein molecules. Many of these small molecules carry electrical charges (positive or negative). It is found that the concentration of negative charges is very high inside the neurons. The neuronal membrane is mainly composed of some lipid bilayers which have 3 to 4 nm thickness. The membrane is impermeable to most of the charged molecules. The bilayer of the membrane is spanned by specialized protein molecules known

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as ion  channels. The ion channels select the ions and transfer them across the membrane. The ion channels can be open or closed corresponding to the response of the mechanical, electrical, or chemical signals. Mostly, different ion channels allow only one type of ions to pass through a particular channel. Many channels are gated or regulated. The channels open and close in response to different stimuli such as (i) changes in the voltages (for voltage-gated channels), (ii) the chemical transmitter, and (iii) stretch or pressure (mechanically gated channels). Nongated channels are called the resting channels [85]. Neuroscience is aimed to investigate and understand the nervous system, the brain, and spinal cord functions that rule everything that we do. The nervous system processes sensory inputs and recognizes it. The earliest study of the brain dates back to Edwin Smith who used surgical papyrus to investigate the symptoms and diagnosis of two patients wounded in the head. Invention of microscope speeded the investigations, and Golgi’s staining procedure helped to understand the brain functions. Pioneering studies were made by the Spanish physician and histologist Santiago Ramon y Cajal, who introduced the neuron doctrine – the hypothesis that the functional unit of brain is neuron. He described that neurons are discrete and provided a suitable structure for them [21]. His studies are the basis of the modern neuroscience theory. In the second half of the twentieth century, imaging techniques developed at a very fast pace, which gave access to a large amount of data and provided a better understanding of the brain functions and anatomy. Now, neuroscience has become an interdisciplinary science where biologists, physicists, technologists, and mathematicians jointly investigate. Many authors modeled the above systems and analyzed them. The authors concentrated on the following: i. To develop and examine mathematical neuron models for the evolution of the functional mechanism of the brain (single neuron models and network level). This may help in the studying neurological disorders such as epilepsy. ii. To study temporal evolution of gating variables by analyzing the ion channel’s activity as well as the generation of action potential and its spiking patterns in the cell body and along the axon and dendrite. iii. To study the qualitative behavior of temporal activity, such as periodic fring, bursting, mixed-mode oscillations, and chaotic fring, with the conductance-based mathematical models. Some examples of such models are Hodgkin–Huxley (HH), Morris–Lecar (M-L), and Hindmarsh–Rose (H-R) models. iv. To study the patterns of spiking activity which are signifcant and responsible for neural signal processing and coding for both single cells and network [8,9,52,74,80]. v. To study the working of the phase response functions and behavior of a network – how it works as a single unit when synchronized in case of coupled interacting oscillatory systems. Ermentrout and Terman [44], Amari [5], Brzychczy and Poznanski [20], Gabbiani and Cox  [49] studied the complex dynamical behavior of the neural/neuronal systems using mathematical tools. Computational neuroscience helps us (i) in studying the organizational principles and emergent dynamics of complex behavior in neural systems [35,36]; (ii) to numerically explore the characteristics of the biophysical models [31]; and (iii) to consider the neural system as a dynamical system (deterministic or stochastic) [28,29]. One of the earlier works is that of Wilfrid Rall [121], who formulated the cable model for the dendritic tree (see also Segev et al., [128]). Dendritic tree is the largest component in both

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surface area and volume of the brain region. It is used to characterize the neurons into different classes such as Purkinje, pyramidal, stellate, and amacrine cells. Cable equations were proposed using coupled PDEs to study how membrane voltage propagates along the dendritic branches in response to synaptic inputs. This “equivalent cylinder” model for dendritic trees has a simple analytical solution. Some of the authors who developed feld equations for the neural systems are Jack Cowan, Shun-ichi Amari, Hugh Wilson and Hermann Haken, etc. [30]. The analysis of the feld equations allows us to understand the mechanism of memory, EEG rhythms, motion perception, etc. [18]. Some of the mathematical tools that are required for understanding the working of the neural systems are given in the works of Kuznetsov [89], Gutkin and Ermentrout [61], Brzychczy and Poznanski [20], Gabbiani and Cox [49]. The techniques from nonlinear dynamics and chaos theory (Perko [116]; Strogatz [133,134]; Upadhyay and Iyengar [142]), bifurcation theory, perturbation theory, etc., are also useful in studying the neural systems. Sophisticated mathematical tools such as geometric singular perturbation theory, stochastic differential equations are used to investigate the neuronal noise (biophysical behavior) of the systems [10,91,96]. Yu et al. [159] investigated the spike time reliability using the Morris–Lecar model (1981) by considering type I and type II excitabilities and allowing noise in the system. Mitry et al. [105] studied the fring threshold manifolds and the roles of anesthetic and propofol on the rebound spiking. The response of single neuron models injected by the pulsatile stimuli was considered by Castejón et al. [22]. Their studies captured the information about the amplitude of response. The dynamical behavior of a neuronal population was studied by Merrison-Hort et al. [104]. They focused on a new biophysical model of the GPe-STN circuit which plays a major role in Parkinsonian conditions. The spatial behavior of single neurons and their dendrodendritic gap junctional interactions were studied by Timofeeva et  al. [136]. Using a fast spiking model, Fontolan et al. [48] described the role of theta– gamma rhythms in the brain at the network level. The works of Koch and Segev [86], Kandel et al. [85], Gerstner and Kistler [51], and Izhikevich [80] have shown that collaboration between the theoreticians and experimentalists has been very fruitful.

6.2 Properties of Neurons Neurons can be classifed using different criteria like polarity, length of synaptic projections, functional mechanism, and electrophysiological activities. However, most of the neurons can be classifed by their polarity. They are mainly of three types: (i) unipolar or pseudounipolar, (ii) bipolar, and (iii) multipolar [138]. Neurons can also be classifed by their functional mechanism as (i) afferent or sensory neurons (it transfers the information from tissues and organs to central nervous system), (ii) efferent or motor neurons (it  transmits the signals from central nervous system to effector neurons), and (iii)  interneurons (the cells which connect the neurons with the central nervous system). To study the activities of any neuron, it is important to know the particular action of this neuron on other cells. The synapses and neurotransmitters respond to these activities. The excitatory neurons depolarize the target neurons, and inhibitory neurons hyperpolarize the target cells. The activity of the presynaptic neuron on the postsynaptic neuron depends not only on the neurotransmitters but also on the receptors of the postsynaptic neuron. The modulatory neurons often release dopamine, serotonin, and acetylcholine. This type of neurons has more complex effects which are known as neuromodulation.

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6.2.1 Electrophysiological Properties of Neurons The difference between the electrical potentials (or voltages) of extracellular and intracellular potentials is a crucial measurement of the neuronal cell’s activity (membrane potential or voltage) (see [66,67]). The neurons have passive electrical properties which do not infuence sensitively on the activity of neurons. It affects the membrane capacitance and resting membrane resistance. To compare membrane properties of neurons of different sizes, the resistances of unit area of the membranes are often used (specifc membrane resistance). It depends on the densities of the resting ion channels and their conductance. The cell membrane behaves like a capacitor. The polar head directs to the intracellular cytoplasm and extracellular region, separating the external and internal conducting solution by an insulated layer of 35-50 Å. From physics, we know that if a thin insulator is charged, it acts like a capacitor. The voltage V across the capacitor is proportional to the charge Q which is stored inside it, Q = CV. In membrane dynamics, the capacitance is referred as the specifc membrane capacitance Cm, whose unit is in microfarad per square centimeter of membrane area. When the voltage changes across the capacitance, current fows and the current equation is given by I = Cm ˙˝ dVm ( t )/dt ˇˆ. The value of capacitance depends on the dielectric property of a medium and the conductance on either side. If we consider a capacitor as consisting of two parallel plates of area A, which is separated by thickness d and an insulated dielectric constant e, then the capacitance is given by C = ˜ 0 A/d , where ˜ 0 is the electric constant (˜ 0 ≈ 8.854187 × 10−12 F/m). This measures the polarization of free space universal constant. Polarization plays a major role in signal transmission [26,85]. A section of the neuronal membrane with two ion channels embedded in it as given by Hille [64] is reproduced in Figure 6.2a. A schematic diagram of voltage-gated ion channels as given by Izhikevich [80] is reproduced in Figure 6.2b. 6.2.2 Ionic Conductance The active properties of the neuron are its excitability, the specifc properties of generating action potential and transmission of signals [25]. The cell membrane acts as a capacitor,

FIGURE 6.2 (a) A section of the neuronal membrane with two ion channels embedded in it. (Reproduced with permission from Hille, B. Ion Channels of Excitable Membranes, 3rd Edition, Sunderland, MA: Sinauer Associates [64]; and from Oxford Publishing Limited, Copyright 2001.) (b) Schematic diagram of voltage gated ion channels. (Reproduced with permission from Izhikevich, E. M. 2007. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Cambridge, MA, London, England: MIT Press [80], Copyright 2007.)

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and it can conduct the electrical signal through ion transmission. The ionic conductance across the cell membrane increases during the emission of action potential. Action potential arises due to the effects of ionic fuxes through the ion channels (Figure 6.2). Alan Hodgkin and Bernard Katz [69] observed that the amplitude of action potential reduced when the concentration of external Na+ ions becomes low. Thus, the fow of sodium ions inside is responsible for the rising of action potential. The authors’ experiment suggested that the phases of action potential fall due to the effect of increase in K + permeability. To test this proposition, the authors varied the membrane voltage systematically of a squid giant axon and measured the changes in Na+ and K + conductances across the membrane (see also Hodgkin and Huxley, [68]). Their experiments showed that four types of ionic currents, sodium ions Na+ , potassium ions (K + ), calcium ions (Ca+2 ), and chloride ions (Cl− ), are responsible for the electrical activity of a neuron. The concentrations of these ions are different on the outside and inside of the cell membrane. The extracellular medium is highly concentrated with sodium and chloride ions and it has also a high concentration of calcium ions. The intracellular medium is highly concentrated with potassium ions and other different negatively charged molecules which are confned in the intracellular medium. The motions of calcium and sodium ions have no signifcant effects at resting condition. However, the movements of potassium and chloride ions have signifcant effects. There are two types of ionic conductances across the membrane. They are the following:

(

)

a. The impermeable anions attract more potassium ions inside the cell and repel more chloride ions outside the cell. It creates the concentration gradients. b. The Na+ -K + ion channels pump three sodium ions outside the cell for every two potassium ions which is inside the cell, and it creates the concentration gradients. The electric potential and concentration gradients drive the ions across the membrane channel. Potassium ions diffuse outside the cell as the external concentration of the ions is lower than the internal concentration. If we excite the cell, potassium ions carry a positive charge and fow outside the cell and leave a negative charge inside the cell. It generates outside current fow across the membrane. The positive and negative charges separate on both sides of the cell membrane which produce an electrical potential difference across the cell membrane. It is called the membrane voltage or transmembrane potential. The potential slows down the diffusion process of potassium ions as the ions are attracted to the negatively charged molecules inside the cell and they are repelled from the positively charged ions outside the cell. The concentration and electrical potential gradients apply equal and opposite forces which balance the two forces, and equilibrium is formed where the cross-membrane current becomes zero. It is denoted by the Nernst equation (see [64]): ˝ RT ˇ  [ ion ]outside  ln E=ˆ , ˙ Fz ˘  [ ion ]inside    where E is the equilibrium potential (Nernst potential) for a given ion, R is the universal gas constant, T is the temperature (in degree Kelvin), and z is the valency of the ionic species and has no units. For example, z is 1 for Na+ and K + ; −1 for Cl − ; and +2 for Ca2+ . F is Faraday’s constant and is equal to 96485 C/mol (Coulombs per mole), and [ ion ]outside and [ ion ]inside are the ion concentrations outside and inside of the cell [54]. Membrane conductance provides a measure of how the ions fow through the membrane channels. The  permeability of the membrane for a specifed ion is denoted by P (cm/sec). A Nernst

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potential develops across a membrane if the following two criteria are met: (i) a concentration gradient exists across the membrane for a given ion, and (ii) selective permeation pathways (i.e., selective ion channels) exist that allow transmembrane movement of the ion of interest. For some ion channels, where the selectivity flter strongly favors the permeation of one ion over other ions, the Nernst potential also predicts the reversal potential (Vrev) of the current–voltage (I–V) relationship. David Goldman [54] proposed a formula which relates the equilibrium potential, permeabilities of ions, and extracellular and intracellular ionic concentrations. The formula is known as Goldman–Hodgkin–Katz (GHK) equation [54]. Hodgkin and Katz [69] (see [64]) provided a method to compute the reversal potential. For the membrane separating potassium ions, sodium-positive ions, and chloride negative ions, the GHK equation is given by E=

+ + − ˝˙ + PK °K ˝˙  RT ˇ PNa °Na ˛ ˛ ˝˙out + PCl °Cl ˛ out in In  , + + − ˘ PNa ˛°Na ˝˙ + PK ˛°K ˝˙ + PCl ˛°Cl ˝˙  F in

in

out

where PNa, PK , and PCl are the permeabilities of the ionic species. The equilibrium potential is reached when the ionic currents fow across the cell membrane and balance it. As a result, the net current fow is zero across the membrane. For neurons, the value of the equilibrium potential ranges from −70 to −30 mV [36]. 6.2.3 Generation of Action Potential, Its Activity, and Signal Propagation The signals produced and propagated along the neurons are called action potential or spikes. It is the elementary unit of signal transmission. The electrical impulses transmit rapidly through the neurons. Axon potential is transmitted at a speed of around 1–100 m/s. The amplitude of the impulse throughout the axon remains almost a constant. Axon potential is regenerated at the region known as Ranvier’s nodes which is a thick insulating myelin sheath in axon. The action potential has the following properties [80]: a. An axon potential is initiated by a suffciently strong depolarization of the membrane potential in the axon hillock region. The depolarization occurs due to the injection of sodium ions inside the cell, and it comes from various sources such as synapses and sensory neurons. The membrane permeability is low for potassium ions. The depolarization triggers the voltage-gated potassium and sodium ion channels to open so that the ions can fow across the membrane. If depolarization becomes small, the potassium current dominates the sodium current, whose fow is toward the inside of the cell and the membrane starts repolarizing to its normal state of resting potential (around −70 mV). When the depolarization is strong enough, the fow of sodium current inside the cell increases and becomes more than the fow of potassium current. The voltage causes more sodium channels to open, which stimulates the membrane voltage toward the reversal potential of sodium channels. This process continues until sodium channels are completely open and the membrane voltage V converges to the sodium channel reversal potential. b. When the phase falls, the voltage which opened the sodium channels slowly decreases and the sodium channels close. During this process, voltage-dependent potassium channels open and it increases the membrane’s potassium permeability. These changes simultaneously repolarize the membrane and action potential falls down.

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c. Next, hyperpolarization occurs. Not all potassium channels which opened due to increase in voltage close while the membrane voltage comes back to the normal resting potential. The permeability of potassium channels becomes high, and it acts on the membrane voltage to reach the value of equilibrium potential of potassium channels. Hyperpolarization exists until the permeability of potassium channels returns to the base value. d. The switching of the potassium and sodium channels during the process of action potential generation may leave some of them in refractory period. In this state, the channels will not be able to open until these are recovered. When absolute refractory period occurs, many ion channels cannot work, and at this time, no action potential is produced. Recovery of the process requires that membrane voltage remains hyperpolarized for certain duration of time. Suffcient number of channels is recovered in the relative refractory period so that an action potential can be generated with a suffciently strong stimulus than usual. The action potential is produced in the cell body of the neurons and travels as a wave along the axon. The membrane of axon also consists of voltage-gated ion channels as soma’s membrane which allows transmission of electrical impulses. The signals are propagated by different ions carrying the charges. The ionic currents move toward the intracellular medium along the axon when the axon potential is generated, and depolarize the adjacent region of the membrane to evoke an action potential in the neighboring membrane patches. The fow of current passively travels from one Ranvier’s node to another. It increases the conduction velocity of action potential. In an unmyelinated axon, the axon potential propagates continuously along the axon. The signal is propagated from soma to the axon terminal. 6.2.3.1 Synapse and Its Functional Mechanism The junction between two neurons is called a synapse. A neuron sends a signal across a synapse. The sending and the receiving neurons are called as the presynaptic cell and the postsynaptic cell, respectively. The site where the axon of presynaptic neuron makes contact with the dendrite (or soma) of a postsynaptic cell is the synapse. There are mainly two types of synapses for communication between the neurons. These are chemical synapse and electrical synapse (see Figure 6.3, [119]). The synaptic strength can be increased or reduced. This plasticity of the neurons is important to many brain functions such as learning, memory, and other activities. The transmission through the electrical synapse is rapid compared with the chemical synapse. It is mainly used to transmit simple depolarized signals for rapid response. The gap between two neurons for an electrical synapse is very small (nearly 3.5nm). This gap is called a gap junction. Here, the channels are formed with protein molecules that are responsible for the fow of ionic currents from presynaptic to postsynaptic neuron. The electrical synapses allow a network of neurons to fre synchronously. Chemical synapses are asymmetric in structure and function. At a chemical synapse, the axon terminal comes very close to the postsynaptic neuron, leaving only a tiny gap between pre- and postsynaptic cell membrane, called the synaptic cell. The synaptic cleft between two intracellular regions ranges from 20 to 40nm. Neurotransmitters are the chemical substances which bind to specifc types of receptors of the postsynaptic neuron’s membrane. The transmission is based on the release of neurotransmitter from the presynaptic neuron. During the generation of a presynaptic action potential, calcium ions move toward the presynaptic terminal via voltage-gated calcium channels. The  high  concentration of calcium ions affects the vesicles to fuse with presynaptic membrane and hence release the neurotransmitters in the synaptic cleft (the process is also called as exocytosis). The chemical substances

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FIGURE 6.3 Electrical and chemical synapses. (Purves, D., et al. 2004. Neuroscience, 2nd Edition. Sunderland: Sinauer Associates, Inc. [119], Reproduced with permission from Oxford Publishing Limited, Copyright 2004.)

diffuse across the synaptic cleft and bind to the receptors of the postsynaptic membrane of neuron. It activates the receptors and ion channels. The resulting fux changes the membrane conductance and voltage of the postsynaptic neuron. These steps delay the process of transmitting the signal through chemical synapse. During the signal transmission process, the receptor of the postsynaptic neuron binds to the neurotransmitter molecules and they respond by opening the neighboring ion channels. It affects the voltage of the transmembrane potential. The resulting voltage changes the postsynaptic potential. The result of the signal processing can be excitatory for depolarizing currents and inhibitory for hyperpolarizing currents. The excitatory or inhibitory nature of synapses depends on the types of ion channels. When a signal is propagated toward an excitatory synapse, depolarization of the neuron can be suffciently strong so that the action potential can be transmitted in the postsynaptic cell. When depolarization induced by excitatory postsynaptic potential is not strong enough to initiate an action potential, the effect of depolarization will remain for some time period. The synaptic mechanism for transmitting signal and its integration has importance in neural networks and noise integration. Mathematical modeling of some of the above mentioned phenomena was done by Rotterdam et al. [148]; Gerstner and Kistler [51]. 6.2.4 Ionic Currents, Neuronal Activity and Neuronal Responses The ionic currents are commonly quantifed through voltage-clamp experiments and modeled according to a formalism introduced by Hodgkin and Huxley [68]. They experimented on a giant axon of squid and proposed mathematical models in terms of coupled differential equations to measure the total ionic currents for different ion channels across

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the cell membrane of a neuron. The membrane current is described as positive if positive ions depart from neuron and negative when positive ions enter in the neuron. The total membrane current is defned as the sum of currents for all ion channels in the membrane, and it includes the synaptic channels and voltage-dependent channels. An index i is used to describe different channels. The current fow through a particular channel of index i with reversal or Nernst potential Ei vanishes or converges to zero, when the membrane voltage satisfes the equation V = Ei. Approximately, the current fow increases or decreases linearly for different ion channels when membrane voltage deviates from the reversal potential value and the difference is known as driving force. If g i measures the conductance per unit area for a particular channel with index i, then the membrane current per unit area for that channel is given by g i (V − Ei ). The total membrane current for different channels is I =

˜g (V − E ). If we consider four types of ionic currents, then the resting potential of i

i

i

the neuron is Vrest = ( g CaECa + g NaENa + g K EK + g ClECl ) / ( g Ca + g Na + g K + g Cl ) . This equation is valid for small changes in the membrane voltage. Various models used the relations of ion permeability and selectivity of membranes [54,69]. Hodgkin and Huxley [68] worked on the voltage-gated ion channels to model their functional mechanism. They introduced two state variables, the probability m of an activation gate for the open state and the probability h of an inactivation gate for the open state, where m, h ˛[ 0,1]. If the channels are partially open, then m ˛( 0,1) . When the channels are completely activated, then m = 1, and if it is completely deactivated, then m = 0. The proportion of open channels of ion channels is taken as p = m° h˛ , where ˜ and ˜ indicate the number of active or inactive gates, respectively, per channel. If some channels do not contain inactivation gates, then ˜ = 0 and p = m° . Such types of channels result in persistent currents. The dynamics of the activation variable m is described by the ordinary differential equation: dm mˆ (V ) − m , = dt ˜ (V ) where m˝ (V ) and ˜ (V ) are the steady-state activation function and the activation time constant, respectively. The values of the above two functions of membrane voltage can be measured experimentally. Activation function shows a sigmoidal shape, and time constant has a unimodal shape. The inactivation variable is governed by the ordinary differential equation: dh hˆ (V ) − h , = dt ˜ 1 (V )

where h˝ (V ) and ˜ 1 (V ) are the steady-state inactivation function and the inactivation time constant, respectively. Some types of neurons integrate the applied input signals and produce a spike if the applied stimulus is suffciently strong or injected continuously fast. This type of neurons is called an integrator and acts as a coincidence detector. Some other types of neurons respond to inputs when received at certain frequency. They are described as resonators (such as MesV neuron). Izhikevich [77-79] observed such responses in invitro recordings, and Bryant and Segundo [19] observed it in invivo experiments. Subthreshold postsynaptic inputs can analyze this activity which is observed in many cortical cells (Llinás [95], Alonso and Klink [4], Jones [84], Mainen and Sejnowski [100], and Amir et al. [7]).

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341

A basic property that a spike and oscillations may occur is due to the fring thresholds of the neurons. When cell membrane potential reaches the threshold values, it produces fring. However, the fring threshold for neurons is not well defned in experiments and models. It was observed that the value of the membrane potential which separates the subthreshold depolarization and action potential depends on the prior behavior of the neuron. Flow frequency subthreshold oscillations may occur due to interactions between fast and slow conductances. Neurons have the property of spike latency or latency to frst spike. Izhikevich [79,80] investigated this latency in detail. His results on the neurocomputational properties of spiking neurons are given in Figure 6.4. Supra-threshold input stimulus can generate action potential with a signifcant delay. Such a response delay is attributed to slow charging of the dendritic tree. The delay corresponds to the voltagegated transient potassium current with activation and inactivation. Long spike latency is a neurocomputational property of integrators. Two basic reasons behind this property are as follows: (i) Neurons encode strength of applied input into the spike latency, and (ii) the responses of neurons are less sensitive to noise-like fuctuation, since prolonged inputs can

FIGURE 6.4 Neuro-computational properties of spiking neurons. (Reproduced with permission from Izhikevich, E. M. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Cambridge, MA, London, England: MIT Press [80], Copyright 2007, also see Izhikevich, E. M. 2004. Which model to use for cortical spiking neurons? IEEE Transact. Neural Netw. 15(5), 1063-1070 [79], Copyright 2004. Electronic version of the fgures and reproduction permissions are freely available at www.izhikevich.com.)

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cause spiking activity [137]. Cortical neurons produce different fring patterns like various trains of action potentials in response to certain injected stimulus. Detailed studies on the neural/neuronal responses were done by Izhikevich [75,80]. In general, ionic currents in neurons are not Ohmic, since the conductance may depend on time, membrane potential, neurotransmitters, neuromodulators, etc. It is the timedependent variation in conductance that allows a neuron to generate an action potential or spike [80]. Typically, neurons are in resting condition. However, they are excitable and produce spikes in response to certain injected stimulus. The cells can either add (integrate) the inputs or it can respond to certain types of input stimulus (resonator) to generate specifc spike patterns. Alan Hodgkin [66] studied the spiking activities of the excitable cell membranes in response to various intensities of step currents. It was observed that two types of excitability exist. Class I excitability: In this case, action potential is produced with arbitrary fow frequency and depends on the strength of the injected stimulus current. Class I excitable neurons (type I neurons) fre with a frequency which may vary in the ranges from 0 to 100 Hz or even higher. Class II excitability: In this case, action potential is produced in certain frequency ranges which are relatively insensitive to change in the intensity of the injected stimulus current. The frequency band of class II excitable neurons (type II neurons) is limited and ranges from 150 to 200 Hz. The following characteristics distinguish the spiking patterns [74,78,79]: Tonic spiking: Most of the neurons are excitable and can fre in response to certain stimulation. The neurons continue to fre a train of spikes when the injected stimulus is on. This is called tonic spiking, and it can be observed in the cortical neurons such as regular spiking neurons, excitatory neurons, and fast spiking inhibitory neurons [27,53]. Tonic spiking neurons are related to different functions of the nervous system, such as coding of sensory information, information processing, memory formation, attention, and motor control [14,58,70]. Phasic spiking: A neuron may produce a single spike for a certain input pulse and remain in quiescent state afterward. Tonic bursting: Some types of neurons such as chattering neurons of cat neocortex produce periodic bursts of spikes when injected by stimulus [57]. Phasic bursting: It is similar to the phasic spiking. Such types of neurons start fring in a burst in response to certain stimulation. Bursting in the brain is useful to overcome the synaptic transmission failure and reduce the noise-like random fuctuations [94]. It can transmit saliency of the injected input stimulus as the effect of burst on the postsynaptic cell is stronger than the single spiking. It can be useful for selective communication between the neurons [77,81,122]. All excitatory cortical neurons have the patterns as given in Figure 6.5 [78]. Regular spiking: They consist of a few spikes with small interspike interval initially and after that the interval increases. It is called spike frequency adaptation. The intensity of dc current stimulus increases the interspike frequency. Intrinsically bursting: In this case, the neurons produce a burst of spikes. The burst of spikes is followed by repetitive single spiking.

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FIGURE 6.5 Different types of fring patterns. (Reproduced with permission from Izhikevich, E. M. 2003. Simple model of spiking neurons. IEEE Transact. Neural Netw. 14(6), 1569-1572 [78], Copyright 2003. Electronic version of the fgure and reproduction permissions are freely available at www.izhikevich.com.)

Chattering: The neurons produce a fnite number of bursts, which are closely spaced spikes. The interburst frequency can be as high as 40 Hz. All inhibitory cortical neurons have the following fring patterns: i. Fast spiking: The neurons can produce periodic trains of action potentials with extremely high frequency without any adaptation. ii. Low-threshold spiking: The neurons can generate high-frequency trains of action potentials, but they show spike frequency adaptation. It has low fring thresholds. Mixed-mode oscillations: Intrinsically, bursting neurons in neocortex can produce mixed type of spiking. It fres a phasic bursting, frst in response to the stimulus and then shows tonic spiking [27]. Mixed-mode oscillation (MMO) consists of a number of spikes and small amplitude oscillatory behavior [126]. MMOs are trajectories that combine small oscillations and large oscillations of relaxation type, both recurring in an alternating manner [87]. The minimal neural systems which can produce intrinsic MMOs are 3D nonlinear conductance-based models. The generalized integrate-and-fre model can produce MMOs provided that the 2D subthreshold dynamics have an unstable focus [76,125]. The functional behavior of MMOs is still new in neuronal dynamics [146]. Spike frequency adaptation: The frequency of spikes is relatively high at the onset of injected input stimulus, and then, it adapts. The regular spiking excitatory neurons produce tonic spiking with decreasing frequency. Low-threshold spiking inhibitory neurons also have this property. The interspike frequency of such type of neurons may encode the time duration which has elapsed since the injected stimulus was applied to evoke fring. Rebound spikes and bursts: When a neuron gets an inhibitory input, it produces a postinhibitory spike. This is related to the anodal break excitability. Many neurons

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can exhibit spikes corresponding to inhibitory input stimulus. Some types of neurons such as thalamocortical cells exhibit postinhibitory bursting. It is believed that this type of bursting infuences the sleep oscillations in thalamocortical cells. Threshold variability: It was observed that the neurons contain variable threshold values which depend on the prior activity of the neurons. Bistability of resting and spiking states: Some types of neurons produce resting and tonic spiking or even bursting. An excitatory or inhibitory input pulse can alter between the two states. In this case, it is important to study the short-term memory. Accommodation: Neurons are very sensitive to brief simultaneous inputs. However, they do not exhibit spikes with strong but slowly increasing input stimulus. During the slow process, the inward current pulses get enough time to inactivate and outward current pulses to activate. So, the neurons accommodate and cannot produce spike. Inhibitory spiking and bursting: Many thalamocortical neurons are in quiescent states when there is no injected input. It exhibits fring when it is hyperpolarized by an inhibitory input pulse or an applied current. A thalamocortical neuron shows tonic bursting in response to longer hyperpolarization. It plays a major role in sleep rhythms. A number of mathematical models were formulated to model various neuronal responses like spiking and bursting dynamics. No single model can include all the neuro-computational properties. The models are biophysically plausible and have measurable parameters.

6.3 Hodgkin–Huxley (HH) Model Alan Hodgkin and Andrew Huxley [68] formulated a mathematical model for the generation of action potential (AP) on the surface membrane of a giant nerve fber using a voltage-clamp technique to maintain the membrane potential constant. This conductancebased model describes how action potential in neurons is initiated and propagated. It is a deterministic continuous time model and consists of a set of nonlinear ordinary differential equations that approximates the electrical characteristics of excitable cells. They ftted the functional parameters of their model of four ODEs such that the theoretical results agree with their experimental results. Hodgkin and Huxley won the 1963 Nobel Prize in Physiology or Medicine “for their discoveries concerning the ionic mechanisms involved in excitation and inhibition in the peripheral and central portions of the nerve cell membrane” [127]. The basic concept behind the famous work is that the cell membrane behaves like an electrical circuit. Different ionic currents in the circuit model are gated by the statedependent conductances. They discovered that ionic current mechanisms are involved in excitation and inhibition to the central and peripheral region of the nerve cell membrane. The original Hodgkin–Huxley model [68] represents a classical and widely used biophysical neuron model. They discovered that the squid axon involved three major currents: (i) voltage-gated persistent potassium (K+) current consisting of four activation gates (resulting in the term n4 , where n is the activation variable for potassium), (ii) voltagegated transient sodium (Na+) current involving three activation gates, and one inactivation

Brain Dynamics: Neural Systems in Space and Time

345

gate (resulting in the term m3 h), and (iii) one leak current passing mostly the chloride (Cl−) ions across the membrane. Electrical circuit representations of cell membrane and intracellular recording of the squid giant axon action potential are given in Figure 6.6a [68] and 6.6b respectively [127]. The relation between the membrane voltage and ionic currents is given by (Hodgkin and Huxley [68]): C

dV = I − I K − I Na − I L . dt

(6.1)

The model involves voltage-gated ion channels and assumes maximal conductance for each ion channel. The values of maximal conductances (g K = 36 ms/cm 2 , g Na = 120 ms/cm 2 , g L = 0.3 ms/cm 2) and shifted Nernst equilibrium potentials (EK = −12 mV, ENa = 120 mV, EL = 10.6 mV ) are given in the work of Izhikevich [80] and correspond to those of Hodgkin and Huxley [68] after the change of variables V = −V. In the model, membrane capacity is taken as C = 1 µF/cm 2. The scale of voltage is shifted so that resting potential V becomes zero (corresponding to the membrane potential being shifted approximately by +65 mV). The model is governed by the following ODEs [68]: CV = I − gK n4 (V − EK ) − g Na m3 h (V − ENa ) − g L (V − EL ) ,

(6.2a)

n = ˜ n (V ) ( 1 − n) − ° n (V ) n,

(6.2b)

 = ˜ m (V )( 1 − m ) − ° m (V ) m, m

(6.2c)

h = ˜ h (V ) ( 1 − h ) − ° h (V ) h.

(6.2d)

FIGURE 6.6 (a) Electrical circuit representation of cell membrane. (Hodgkin, A. L., Huxley, A. F. 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500-544, [68], Copyright 1952. Reprinted with permission from The Physiological Society & John Wiley & Sons.) (b)  Intracellular recording of the squid giant axon action potential. (Reproduced with permission from Schwiening, C. J. 2012. A brief historical perspective: Hodgkin and Huxley. J. Physiol. 590(11), 2571-2575, [127]; Physiological Society & John Wiley & Sons, Copyright 2012.)

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The meanings of the system variables and the parameters are as follows: V: displacement of the membrane voltage from its resting value, I: injected current into the space-clamped axon, C: membrane capacity per unit area (it is assumed a constant), g K , g Na , g L : ionic conductances for potassium, sodium and leak channels, EK , ENa : equilibrium potentials for potassium and sodium ions, EL : value of potential at which the leakage current is zero due to chloride and other ions, n, m, h : non-dimensional quantities having values in the interval (0, 1), which are associated with potassium channel activation, sodium channel activation, and sodium channel inactivation, ˜ i, ˜ i ( i = n, m, h ) : measure of the steady-state activation and inactivation functions, which depend on voltage but not time. The related activation or inactivation functions of the variables are given by xˇ = ˜ x /(˜ x + ° x ) , where x represents the system variables n, m, and h, respectively, and the time constant is given by ˜ x = 1/(° x + ˛ x ). The functions were assumed as

˜ n (V ) =

0.01( 10 − V )  V , ° n (V ) = 0.125 exp  −  ,  80  exp ˇ˘( 10 − V )/10  − 1

˜ m (V ) =

0.1( 25 − V )  V , ° m (V ) = 4exp  −  ,  18  exp ˇ˘( 25 − V )/10  − 1

1 ˇ V ˜ h (V ) = 0.07 exp  −  , ° h (V ) = . ˘ 80  exp ( 30 − V )/10  + 1 The conductance-based model was studied by many authors [34,44,80] and is widely accepted as a starting point in the study of excitable membranes. The model produces spikes very similar to the intracellular recordings when varying the system parameters. Hassard [62] and Labouriau [90] studied the Hopf bifurcation that plays an important role in locating regions of bistability in the HH model. Aihara and Matsumoto [3] obtained chaotic solutions to the HH equations with periodic forcing. Rinzel and Miller [123] located period doubling bifurcation in HH model and demonstrated the existence of chaotic solutions (with standard parameters used by Hodgkin and Huxley). Doi and Kumagai [39] showed the existence of a chaotic attractor in a modifed HH model that changes the time constant of one of the currents by a factor of 100. Guckenheimer and Oliva [60] conjectured that the basin boundary between the basins of attraction of the stable equilibrium and stable periodic orbit is a fractal set that contains the stable manifold of the chaotic invariant, showing the existence of chaotic solutions in the HH model with its original parameters. The chaotic solutions are highly unstable, but they are signifcant as they lie in the basin boundary that establishes the threshold of the system. The biological signifcance of chaos in the HH system is related to the character of the threshold that separates the states leading to repetitive fring from the states that lead to a stable steady state. The authors have also established the subtlety of the concept of threshold: The excitability of a neural

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membrane to fre an action potential may be more complex than a smooth hypersurface that divides subthreshold and superthreshold membrane potential. Clay [24] reexamined the electrical properties of the giant axon from the common squid Loligo pealeii and excitability of the squid giant axon. Benjamin Ambrosio et al. [6] modifed the HH model to consider a general network of reaction–diffusion equations as Vit = I − g K ni4 (Vi − EK ) − g Na mi3 hi (Vi − ENa ) − g L (Vi − EL ) + dVixx + H i (V1 , V2 ,…, VN ) ,

(6.3a)

nit = ˇ˘˜ n (Vi ) ( 1 − ni ) − ° n (Vi ) ni  ,

(6.3b)

mit = ˇ˘˜ m (Vi ) ( 1 − mi ) − ° m (Vi ) mi  ,

(6.3c)

hit = ˜ h (Vi ) ( 1 − hi ) − ° h (Vi ) hi  , i  {1,…, N } ,

(6.3d)

in a bounded domain ˛ = ( a, b ) ˙ , with the Neumann boundary conditions Vx ( a ) = Vx ( b ) = 0. Here, H i ( u) is defned as

˜

H i ( u) =

° ij ( S − ui ) ˇ ( u j ) , ˇ ( u j ) =

j{1,…,N }

1

1+ e (

−  u j −

)

,

with S = 100, ˜ = 20, ° = 60, and ˜ ij ° 0. H i ( u) represents excitatory nonlinear coupling (chemical synaptic coupling) between neurons [15,33]. The authors [6] have proved the existence and uniqueness of solutions, existence of invariant regions, and the existence of an attractor in the space of continuous functions defned on the real interval [a, b]. We present here their proof. For proving the existence of solution of system (6.3), the proof is divided into the following three steps: (i) Using the fxed-point theorem, existence and uniqueness of a mild solution are proved. (ii) Next, regularity of mild solution is proved. (iii) Then, the bounds of ni , mi, and hi are found and global existence of solution is proved. i. Let X = C ˇ˘ˆ  = C ([ a, b ]) be a space of continuous functions defned on the real interval [a, b]. The boundary value problem ut = uxx , ux ( a ) = ux ( b ) = 0, generates an analytical semigroup S ( t )t˛0 on X [97]. The general network equations for mild solutions can be written as t

˜

(

)

Vi ( t ) = S ( t ) Vi ( 0 ) + S ( t − s ) FV (Vi ( s ) , ni ( s ) , mi ( s ) , hi ( s )) + H i (V1 ,…,VN ) ds, 0

t

˜

ni ( t ) = ni ( 0 ) + Fn (Vi ( s ) , ni ( s )) ds, 0 t

˜

mi ( t ) = mi ( 0 ) + Fm (Vi ( s ) , mi ( s )) ds, 0

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Spatial Dynamics and Pattern Formation in Biological Populations

t

˜

hi ( t ) = hi ( 0 ) + Fh (Vi ( s ) , hi ( s )) ds,

(6.4a-d)

0

where FV (V , n, m, h ) = I − g K n4 (V − EK ) − g Na m3 h (V − ENa ) − g L (V − EL ) , Fn (V , n) = ˜ n (V ) ( 1 − n) − ° n (V ) n, Fm (V , n) = ˜ m (V ) ( 1 − m ) − ° m (V ) m, Fh (V , n) = ˜ h (V ) ( 1 − h ) − ° h (V ) h. For showing the local existence, use the fxed-point theorem to the function

(

)

(

˙U0 : C [ 0, T ] , X 4N ˆ C [ 0,T ] , X 4N

)

defned by the right-hand side of (6.4), in which we fx the initial condition U 0 ˜ X 4N . In the following, for the sake of simplicity, we drop the subscript U 0 of ˜ . Let U denote  the function (Vi , ni , mi , hi ) ( t ) , i ˇ{1,, N } , t ˇ[ 0,T ]. For all U ˛ B (U 0 , r ) , the ball of center U 0 and radius r in C [ 0,T ] , X 4 N , we have, for fxed T > 0,

(

)

(

)

˛ (U ) − U 0 ˙ KT, where ˜ stands for the sup-norm on C [ 0,T ] , X 4 N and where K is a constant depending on r which comes from the boundedness theorem. The authors [6] have also used the property of the linear semigroup S ( t ) : S ( t )u X ˛ u X and lim S ( t ) u0 = u0 , see [97]. It follows that, for T < ( r/k ) , ˝ (U ) ˙B (U 0 , r ). By anat˝0

log computations, it was shown that ˛ (U 1 ) − ˛ (U 2 ) ˙ K 2T U 1 − U 2 , which proves that for T < min ( 1/K1 ,1/K 2 ) , ˜ is a contraction mapping. Therefore, ˜ has a unique fxed point which provides the existence and uniqueness of the local mild solution.

(

)

ii. We know that the mild solutions belong to C [ 0,T ] , X 4 N . Without loss of generality, we drop the subscript i in H. The quantity ˙ˆ(V ( t + h ) − V ( t ))/h ˇ˘ is well defned

for t > 0. As h ˜ 0, the resulting function ( dV/dt ) is continuous on X . For t = 0, the derivative may not exist, since ˝˙( S ( h ) V0 − V0 )/h ˆˇ admits a limit as h ˜ 0 if and only if v0 ˛ C˝˝ [ a, b ]. For the functions m, n, and h, similar computations show that they belong to C1 ([ 0,T ] , X ) . Hence, the regularity of mild solutions is proved. iii. It is now to be proved that ni , mi, and hi remain in [0, 1]. Without loss of generality, we drop the subscript and consider only the n variable. It satisfes the equation: nt = ˜ (V ) ( 1 − n) − ° (V ) n.

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Brain Dynamics: Neural Systems in Space and Time

t

t

˜

˜

Let A ( t ) = ° (V ( s )) ds, B ( t ) = ˛ (V ( s )) ds, with ° (V ) ˇ 0 and ˛ (V ) ˇ 0. 0

0

t ˇ  − A t +B t A s +B s Then, n ( t ) = e { ( ) ( )}  n0 + ° (V ( s )) e{ ( ) ( )}ds  .   0 ˘ 

˜

It follows that n ( t ) ˛ 0. Also, ˘ − A t +B t n ( t ) ˆ e { ( ) ( )}  n0 +  

t



0



A ( s ) + B( s ) ˜ (° (V ) + ˛ (V )) e{ }ds 

(

)

− A t +B t A t +B t = e { ( ) ( )} ˘ n0 + e{ ( ) ( )} − 1    − A t +B t = 1 + ( n0 − 1) e { ( ) ( )} ˆ 1.

Global existence follows from the inequality: t

Vi (t) X ˝ Vi (0) X +

˜ (C

1

)

Vi ( s) X + C2 ds,

0

which comes from equation (6.3a). Gronwall’s inequality yields Vi (t)X ˜ Vi (0) X e C1t + C2 /C1 . 6.3.1 Simulation Results The authors have not performed any simulations for model system (6.2). For the sake of completeness and to have an insight, we have performed simulations of the deterministic model (6.2). The nontrivial equilibrium point is obtained as *

n =

( )

˜n V*

( )

( )

˜ n V + °n V *

*

*

,m =

( )

˜m V*

( )

( )

˜ m V + °m V *

*

,h =

(

( )

˜h V*

*

( )

( )

˜ h V + °h V * *

)

,

where V * is obtained by substituting these values in FV V * , n* , m* , h* = 0. The variational matrix of the system about the equilibrium point V * , n* , m* , h* is given by

(

° ˝ ˝ J=˝ ˝ ˝ ˝ ˛

a11 a21 a31 a41

a12 a22

a13 0

a14 0

0 0

a33 0

0 a44

)

˙ ˇ ˇ ˇ, ˇ ˇ ˇ ˆ

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Spatial Dynamics and Pattern Formation in Biological Populations

where a11 = −120m3 h − 36n4 − 0.3, a12 = −144n3 (V + 12), a13 = −360m2 h(V − 120), a14 = −120m3 (V − 120) , a21 =

a22 =

a31 =

a33 =

(

( n − 1)

)

100 e(1− 0.1V ) − 1

+

ne −0.0125V e(1− 0.1V ) ( 0.01V − 0.1)( n − 1) , + 2 640 10 e(1− 0.1V ) − 1

(

)

( 0.01V − 0.1) − 0.125e −0.0125V ,

( e(

1− 0.1V )

(

)

−1

( m − 1)

)

10 e( 2.5 − 0.1V ) − 1

( 0.1V − 2.5)

(

)

e( 2.5 − 0.1V ) − 1

+

2me −0.05556V e( 2.5 − 0.1V ) ( 0.1V − 2.5 )( m − 1) + , 2 9 10 e( 2.5 − 0.1V ) − 1

(

− 4e −0.05556V ,

a41 = 0.0035e −0.05V ( h − 1) −

a44 = −0.07e −0.05V −

)

(

1

(

he( 3 − 0.1V )

)

10 e( 3 − 0.1V ) + 1

)

e( 3 − 0.1V ) + 1

2

,

.

The characteristic equation of J is given by

˜ 4 + a1˜ 3 + a2 ˜ 2 + a3 ˜ + a4 = 0 where a1 = − ( a11 + a22 + a33 + a44 ) , a2 = − a12 a21 − a13 a31 − a14 a41 + a11a22 + a11a33 + a11a44 + a22 a33 + a22 a44 + a33 a44 , a3 = a13 a22 a31 + a12 a21a33 − a11a22 a33 + a14 a22 a41 + a14 a33 a41 + a12 a21a44 − a11a22 a44 + a13 a31a44 − a11a33 a44 − a22 a33 a44 , a4 = −a14 a22 a33 a41 − a13 a22 a31a44 − a12 a21a33 a44 + a11a22 a33 a44 . The characteristic equation has negative real roots or complex pairs if the following Routh–Hurwitz criterion is satisfed: a1 > 0, a2 > 0, a3 > 0, a4 > 0, A1 = a1a2 − a3 > 0 and A2 = a1a2 a3 − a12 a4 − a32 > 0. Consider the case when I = 5. The equilibrium point is obtained as V * , n* , m* , h* = ( 3.3354, 0.3698, 0.0778, 0.4769 ). The corresponding eigenvalues are obtained as ˜1 = −4.6354, ˜2 = −0.1297, ˜3, 4 = −0.0767 ± i0.5188. The equilibrium point is asymptotically stable and is in quiescence state (see Figure 6.7a). For I = 8.5, the equilibrium point is (4.9304, 0.3952, 0.0929, 0.4206) and the corresponding eigenvalues are ˜1 = −4.7624, ˜2 = −0.1367, ˜3, 4 = 0.0014 ± i0.5666, which show regular spiking phenomena (see Figure 6.7b). For I = 10, the equilibrium point is (5.5138, 0.4045, 0.0991, 0.4005). The corresponding eigenvalues are ˜1 = −4.8345, ˜2 = −0.1397, ˜3, 4 = 0.0333 ± i0.5797. In this case, the number of spikes increases (see Figure 6.7c).

(

)

Brain Dynamics: Neural Systems in Space and Time

351

FIGURE 6.7 Various neuronal response of the HH model for different stimulus current: (a) I = 5 (quiescence state), (b) I = 8.5, (regular spiking), and (c) I = 10.0, (number of spikes increase).

The bifurcation diagram of the HH model is drawn using MATCONT software and presented in Figure 6.8. The injected current stimulus (I) is treated as a predominant parameter. For I < 8.410532, the system has stable focus node and it exhibits oscillation death (quiescent state). For I > 8.410532, the system has unstable focus node. The system exhibits regular spiking from the quiescent state as we increase the value of current stimulus (I). Further increase in the value of the predominant parameter leads the system toward fast spiking. The stability of the stable equilibrium changes through the birth of unstable limit cycle (dashed red line) which indicates the subcritical Hopf bifurcation (SH) at I = 8.410532. The frst Lyapunov coeffcient at I = 8.410532 is positive having value ˜ = 0.0094, which also confrms the existence of subcritical Hopf bifurcation (SH) therein. The thick and dashed blue lines indicate the stable and unstable equilibrium branches, respectively. In numerical simulations, the authors [6] have demonstrated the bifurcation and propagation of bursting oscillations in one and two couple nonhomogeneous neurons. They have considered the model for a single nonhomogeneous neuron (N = 1) as

FIGURE 6.8 The bifurcation scenario of HH model (6.2).

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Spatial Dynamics and Pattern Formation in Biological Populations

Vt = I ( x ) − g K n4 (V − EK ) − g Na m3 h (V − ENa ) − g L (V − EL ) + DVxx , nt = ˜ n (V ) ( 1 − n) − ° n (V ) n, mt = ˜ m (V ) ( 1 − m ) − ° m (V ) m,

(6.5a-d)

ht = ˜ h (V ) ( 1 − h ) − ° h (V ) h. The initial conditions (V(0), n(0), m(0), h(0)) ˛ X 4 are such that for all x ˛[ a, b ] , n ( 0, x ) , m ( 0, x ) , h(0, x) ˙[ 0,1]. The bounded domain is taken as ˛ = ( a, b ) = ( 0,100 ), and the Neumann boundary conditions Vx ( a ) = Vx ( b ) = 0 were assumed. The regular nonhomogeneous discontinuous function I is assumed as I ( x ) = I 0 , if x < ( b − a )/10, and = 0, otherwise. We have studied the temporal as well as the spatiotemporal dynamics of the 1D diffusion model (6.5). Time series of the end oscillator of the HH model with 1D diffusion for various values of the diffusion coeffcient (D) are presented in Figure 6.9. For I = 5, regular spiking, irregular spiking, and quiescent state have been obtained for different values of diffusion coeffcient D = 0.0005, 0.001, and 0.005, respectively (see Figure 6.9a-c). For I = 8.5, only regular spiking is obtained for all values of D = 0.0005, 0.5, and 5 (see Figure 6.9d-f). For I = 10, only regular spiking is obtained for D = 0.0005, 0.05, and 5, respectively (see Figure 6.9g-i). The space-time plots corresponding to the Figure 6.9 are presented in Figure 6.10 showing similar dynamics. The model for two couple neuron (N = 2) is written as [6] V1t = I1 ( x ) − g K n14 (V1 − EK ) − g Na m13 h1 (V1 − ENa ) − g L (V1 − EL ) + DV1xx , n1t = ˜ n (V1 ) ( 1 − n1 ) − ° n (V1 ) n1 , m1t = ˜ m (V1 ) ( 1 − m1 ) − ° m (V1 ) m1 , h1t = ˜ h (V1 ) ( 1 − h1 ) − ° h (V1 ) h1 . V2t = I 2 ( x ) − g Kn24 (V2 − EK ) − g Na m23 h2 (V2 − ENa ) − g L (V2 − EL ) + DV2xx + ˜ 21 ( x ) ( S − V2 ) ˘ (V1 ) , n2t = ˜ n (V2 ) ( 1 − n2 ) − ° n (V2 ) n1 , m2 t = ˜ m (V2 ) ( 1 − m2 ) − ° m (V2 ) m1 , h2t = ˜ h (V2 ) ( 1 − h2 ) − ° h (V2 ) h1 ,

(6.6a–h)

with I 2 = 0, ˜ ij = 0 for ( i, j ) ˙ ( 2,1);˜ 21 ( x ) = 0 if x < ( b − a )/10, and=1 otherwise. For numerical simulation, the authors considered the case I ( x ) = 130, if x < ( b − a )/10, and = 0, otherwise. Further, they have shown that neuron 1 is not affected by the coupling, but for neuron 2, they observed propagation of bursting oscillations from neuron 1 to neuron 2. As the coupling acts only on the right side of the neuron 2, oscillations propagate from right to left and a short propagation was observed toward the right boundary. Bossy et al. [16] assumed that every neuron follows a stochastic version of HH model and that pairs of neurons interact through both electrical and chemical synapses, the global connectivity being of mean feld type. They studied the synchronization due to a strong coupling between neurons of stochastic mean-feld networks of HH neurons with noisy channels. The random transitions of ion channels between conducting and nonconducting states generate a source of internal fuctuations in a neuron, known as channel noise.

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353

FIGURE 6.9 Time series of the oscillator of HH model with 1D diffusion for various values of D. For I = 5: (a) D = 0.0005, (b) D = 0.001, (c) D = 0.005. For I = 8.5: (d) D = 0.0005, (e) D = 0.5, (f) D = 5. For I = 10: (g) D = 0.0005, (h) D = 0.05, (i) D = 5.

Since the classical HH formalism is deterministic, alternative models have been proposed to account for channel noise. These models assumed that the activity of ion channels is governed by random transitions among a number of possible channel conformations, which leads to intrinsically stochastic models of neuronal dynamics. Although a large number of models of this type have been proposed, including those that capture fractal properties of patch-clamp data [93] and history dependence in the activity of ion channels [132], the most widely used channel noise model is the Markov chain (MC) model. Goldwyn et al. [55], Goldwyn and Shea-Brown [56] studied the stochastic differential equation models for ion channel noise in Hodgkin–Huxley neurons and brought out their biological interpretation and their relevance. Two of the reduced models are FitzHugh–Nagumo models (FHN) [46,47] and Morris– Lecar model (M-L) [109]. One of the major advantages is that the phase plane analysis is computationally effcient and it can be extensively studied from mathematical and

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FIGURE 6.10 Space-time series of the HH model with 1D diffusion for various values of D. For I = 5: (a) D = 0.0005, (b) D = 0.001, (c) D = 0.005. For I = 8.5: (d) D = 0.0005, (e) D = 0.5, (f) D = 5. For I = 10: (g) D = 0.0005, (h) D = 0.05, (i) D = 5.

computational points of view. A reduction of the four-dimensional HH system to a two-variable system is desirable so that phase plane analysis can be done for effcient computation.

6.4 FitzHugh-Nagumo (FHN) Model This is a simplifed version of the HH model which shows spiking activity. FitzHugh (1961) observed that the gating variables n and h involve slow kinetics relative to the variable m. To electronically simulate an animal nerve axon, Nagumo et al. [111] made an active

Brain Dynamics: Neural Systems in Space and Time

355

pulse transmission line using tunnel diodes. Using these two observations, the fourdimensional HH model is reduced to a two-dimensional model called FitzHugh–Nagumo (FHN) model. The model can reproduce many characteristics and neurocomputational properties of electrical activities along the neurons and cardiac fbers such as the existence of excitation threshold, absolute and relative refractory periods. Rocsoreanu et al. [124] in his monograph entitled “The FitzHugh-Nagumo Model: Bifurcation and Dynamics” presented the complete theoretical and numerical investigation of the complex phase dynamics, bifurcation scenarios, and a global view of all possible qualitatively distinct responses of the FHN model. We present here a review of the spatial FHN model. Spiral breakup leading to turbulence can occur in a two-dimensional (2D) reaction–diffusion FitzHugh–Nagumo (FHN) system in which the spatial interaction is carried out only in membrane potential variable [11]. Acebron et al. [2] studied the noisy FitzHugh–Nagumo model and derived a Fokker–Planck equation for both a single element and a network of globally coupled elements. They have also introduced an effcient way to numerically solve this Fokker–Planck equation, especially for large noise levels. The effects of different network topologies on the noise-induced pattern formation in a two-dimensional model of excitable media with FitzHugh–Nagumo local dynamics were studied by Perc [114]. In  particular, the author showed that introduction of long-range couplings induces decoherence of otherwise coherent noise-induced spatial patterns that can be observed by regular connectivity of spatial units. The author has also studied the spatial coherence resonance in a two-dimensional model of excitable media with the FitzHugh–Nagumo local dynamics. In another work, Perc [115] studied the fuctuating excitability of FitzHugh– Nagumo neurons constituting a diffusively coupled excitable array that can induce phase slips, which in turn may lead to a symmetry break. The oscillation frequencies of the information-carrying signal are expressed analytically, and necessary conditions for the phenomenon have also been derived. Propagation of two-dimensional waves was studied in a parabolic channel fow of excitable medium of the FitzHugh–Nagumo type [42]. FHN model cannot exhibit certain interesting fring behaviors such as bursting. Faghih et al. [45] showed that by allowing the parameters to be varying in time, in the FHN model, one could overcome such limitations while still retaining the low-order complexity of the FHN model. Bashkirtseva et al. [12] have studied an excitability phenomenon for nonlinear systems with Canard cycles and sensitive equilibria under random disturbances. Parameter estimation in the FitzHugh–Nagumo model using noisy measurements for membrane potential was studied by Che et al. [23]. Wu et al. [155] introduced magnetic fux into the FitzHugh–Nagumo model to describe the effect of electromagnetic induction, and then, memristor is used to realize the feedback of magnetic fux on the membrane potential in cardiac tissue. Kuznetsov et al. [88] have investigated numerically the behavior of a two-component reaction–diffusion system of FitzHugh–Nagumo type before the onset of subcritical Turing bifurcation in response to local rigid perturbation. Gambino et  al. [50] constructed square- and target-wave patterns in a FHN reaction–diffusion system and investigated the process of pattern formation in the FHN reaction–diffusion model showing that the emerging pattern-forming solution can be successfully predicted by the amplitude equation formalism. Yao et al. [158] have numerically investigated the effects of delay and Sine-Wiener noise on synchronization and synchronization transition in the FitzHugh–Nagumo model, by calculating a synchronization measure and plotting spatiotemporal patterns. Zheng and Shen [161] investigated the effect of diffusion on the pattern formation in the FitzHugh–Nagumo model. Using linear stability analysis of the local equilibrium, they obtained the conditions for the Turing and Hopf bifurcations. Using the analysis of the amplitude equations, they demonstrated the occurrence of the Turing

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instability, Eckhaus instability, and zigzag instability. The authors [161] considered the following reaction–diffusion system:

˜u = ( a − u)( u − 1) u − v + D1ˇ 2 u, ˜t

(6.7a)

˜v = e ( bu − v ) + D2ˇ 2 v, ˜t

(6.7b)

where ˜ 2 is the two-dimensional Laplace operator and ( x , y ) ˛ R 2 . The meanings of the variables and parameters are different from the original FitzHugh–Nagumo model (see [154]). The meanings of the variables and parameters are the following: u: membrane potential; v: recovery variable; a: excitatory threshold; e: excitability; and b, e: parameters that can change the rest state and dynamics. In the remaining part of the section, we present the analysis as given by Zheng and Shen [161]. 6.4.1 Linear Stability Analysis and Hopf Bifurcation Assume that the steady-state solution of the system (6.7) is ( u0 ,  v0 ) . Since the model system is nonlinear, linear steady analysis is applied to the system (6.7) when ( u0 ,  v0 ) lies near the bifurcation point. For discussing the stability of this spatially uniform solution, consider a perturbation of the form: ˙ u ( t ) + u0 c (t ) = ˇ ˆˇ v ( t ) + v0

˘ ˙ 1+ a +  , where ( u0 , v0 ) = ˇ ˇˆ 

( 1 − a )2 − 4b 2

˘ , bu0  . 

The linearized system is obtained as ct = Jc + D˛ 2 c.

(6.8)

The coeffcient matrix is ˛ a11 J=˙ ˝ be

˛ D1 −1 ˆ ˘ , and D = ˙ 0 −e ˇ ˝

0 ˆ ˘, D2 ˇ

where a11 = 2a + 3b − ( a + 1) u0 .

° c1 ˙ ˘ t + ikr . Assume that c takes the form, c = ˝ ˇe ˛ c2 ˆ The characteristic equation of the system (6.8) is given by

˜ − a11 + k 2 D1

1

−be

˜ + e + k 2 D2

( )

= 0, ˜ k2 − Trk ˜ k + ° k 2 = 0,

The eigenvalues are given by

(

)

˜ k = Trk ± Trk2 − 4° /2,

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Brain Dynamics: Neural Systems in Space and Time

where Trk = a11 − e − k 2 ( D1 + D2 ) , Tr0 = a11 − e,

( )

δ k 2 = D1D2 k 4 − ( a11D2 − eD1 ) k 2 + e ( b − a11 ) , det ( J ) = δ 0 = e ( b − a11 ) .

∂ Re ( λ k ) ≠ 0, and Re ( λ k ) = 0, when k = 0, [Re ( λ k ) is the real ∂e part of the eigenvalue λ k , and e is the control parameter about which we try to determine the Hopf bifurcation], then the Hopf bifurcation occurs. Re ( λ k ) = 0 when k = 0, gives a11 = e. ∂ Re ( λ k ) 1 Im ( λ k ) ≠ 0 gives Trk2 ≠ 4δ . Third condition gives = − ≠ 0. That is, if e = a11, then ∂e 2 the Hopf bifurcation of the system (6.7) without diffusion occurs [since for Hopf bifurcation, Trace = Tr0 = a11 − e = 0]. Turing bifurcation: If the conditions Hopf bifurcation: If Im ( λ k ) ≠ 0,

(i) Tr0 = a11 − e < 0, (ii) det ( J ) = δ 0 = e ( b − a11 ) > 0, (iii) a11D2 − eD1 > 0, and (iv) a11D2 −

eD1 > 2 D1D2 ( be − a11e ) , are satisfied, then the Turing bifurcation (diffusion-driven instability) occurs in the reaction–diffusion system. The critical conditions are Im ( λ k ) = 0 and Re ( λ k ) = 0 when k ≠ 0. Condition (i) gives that a11 < e. Condition (ii) gives that a11 < b. The two results imply that a11 < min ( e , b ) . Conditions (iii) and (iv) imply that δ k 2 = 0 has positive roots. The two critical conditions Im ( λ k ) = 0, Re ( λ k ) = 0 imply that Trk ≠ 0, and

( )

( )

Trk2 ≠ 4δ . From δ k 2 = 0, we obtain the value of k 2 , and equating the discriminant to zero, we get a11D2 − eD1 = 2 D1D2 e ( b − a11 ) , or ( a11 + de ) − 4dbe = 0, where d = D1/D2 . This is the required condition for the Turing bifurcation of the system (6.7) to occur. Critical value of k 2 If δ k 2 < 0 for some k 2 ≠ 0, we may obtain diffusion-driven instability and also get the minimum value of δ k . In this case, the critical value is obtained as kc2 = ( a11D2 − eD1 )/2 D1D2 . 2

( )

6.4.2 Amplitude Equation Amplitude of the modes in the solution of equation (6.7) cannot be directly determined. We can obtain an approximation to the amplitude using the Taylor series expansion. Expand the right-hand sides in equation (6.7) about the equilibrium point ( u0 , v0 ) and truncate the expansion at third order. We use multiple-scale analysis to derive the amplitude equations when k = kc . When the controlled parameter e is greater than the critical value of the Turing point, the solutions of the system (6.7) can be expanded as N

c = c0 +

∑(Z e ) , i

iki r

i=1

with k = kc , Z j and the conjugate Z j are the amplitudes associated with the modes k j and −k j. Close to onset of e = ec , we have

∂ Zi = si Zi + Fi ( Zi , Z j ,) . ∂t

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where si is the coeffcient of the linear term of the variable Zi . Applying the center manifold theory near the Turing bifurcation point, it can be concluded that amplitude Z j satisfes the equation:

˜ Zi = Fi Zi , Zi , Z j , Z j , . ˜t

(

)

From the standard multiple-scale analysis, the spatiotemporal evolution of the amplitudes up to the third order in the perturbations can be written as

°0

˛ Zi = µZi + ˛t

˜h

lm

Zl Zm +

lm

˜g

lmn

Zl Zm Zn .

(6.9)

lm

Due to spatial translational symmetry, we have the following equation:

°0

˛ Zi iki r e = µZi e iki r + ˛t

˜h

lm

Zl Zm e i( kl + km )r +

lm

˜g

lmn

Zl ZmZne i( kl + km + kn )r .

(6.10)

lm

Comparing (6.9) with (6.10), we fnd that the two equations hold only if ki = kl +  + km. Let the system (6.7) be rewritten as

˜c = Lc + N ( c ) , ˜t ˝ − a + D 2 ˝ u ˇ 1 where c = ˆ ,L=ˆ  v ˆ ˙ ˘ be ˙

ˇ ˝ = , N  ˆ ˆ˙ − e + D2 2 ˘ Here, L is the linear term and N is the nonlinear term. When e is close to ec, expand e as

( a + 1) u2 − u3

−1

0

ˇ . ˘

ec − e = ˜ e1 + ˜ 2 e2 + , where ε is a small parameter. Expand c and N in terms of ˜ as ˛ u2 ˆ 2 ˛ u ˆ ˛ u1 ˆ c=˙ =˙ ˘ ˜ +… ˘˜ +˙ ˘ ˝ v ˇ ˝ v1 ˇ ˝ v2 ˇ  N= 

( a + 1) u12˜ 2 + {2 ( a + 1) u1u2 − u13 } ˜ 3 + o ( ˜ 4 ) 0

 . 

Linear operator L can be written as

(

)

L = Lc + ( ec − e ) M + o ( ec − e ) , ˙ − a + D ˝2 1 where Lc = ˇ ˇˆ bec

−1 − ec + D2˝

2

˘ ˙ 0 , M=ˇ  ˆ b

2

0 ˘ . −1 

(6.11)

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Brain Dynamics: Neural Systems in Space and Time

Let T0 = t , T1 = ˜ t , T2 = ˜ 2t…. Here, Ti is a dependent variable. We have

˜ ˜ ˜ ˜ = +° + °2 + ˜ t ˜ T0 ˜ T1 ˜ T2 The solutions of the system (6.8) have the following form: ˛ u ˆ c=˙ ˘= ˝ v ˇ

3

˜ i=1

˛ xi ˆ ik r ˙ ˘ e i + c.c. ˝ yi ˇ

This expression implies that the bases of the solutions have nothing to do with time and the amplitude Z is a variable that changes slowly. As a result, we may write

˜Z ˜Z ˜Z =° + °2 + ˜t ˜ T1 ˜ T2

(6.12)

From (6.11), comparing the different orders of ˜ , we obtain ˝ u1 ˇ ˝ u2 ˇ ˝ u1 ˇ ˝ ˜ ˝ u1 ˇ Lc ˆ  = 0, Lc ˆ = ˆ  −   e1 M ˆ  −ˆ ˙ v1 ˘ ˙ v2 ˘ ˜ T1 ˙ v1 ˘ ˙ v1 ˘ ˆ˙

( a + 1) u12 0

ˇ  ˘

° u3 ˙ ° u2 ˙ ° u1 ˙ ° 2(a + 1)u1 v1 − u13 ˙ ˜ ° u2 ˙ ˜ ° u1 ˙ Lc ˝ ˇ. ˇ= ˇ+ ˇ − e1 M ˝ ˇ − e2 M ˝ ˝ ˝ ˇ −˝ 0 ˛ v2 ˆ ˛ v1 ˆ ˛ ˛ v3 ˆ ˜ T1 ˛ v2 ˆ ˜ T2 ˛ v1 ˆ ˆ Consider the case of the frst order in ˜ . Since Lc is the linear operator of the system close T to the onset, ( u1 , v1 ) is the linear combination of the eigenvectors that correspond to the zero eigenvalue. Since ° u ˙ ˝ v ˇ= ˛ ˆ

3

° xi ˙ ik r ˇ e i + c.c , yi ˆ

˜ ˝˛ i=1

(

)

we obtain that xi = By i , where B = −1/ a + D1k 2 , ki = kc . If we assume y i = 1, then xi = B, and ˛ u1 ˆ ˛ B ˆ iki r ˙ ˘ =˙ ˘ˇ Wi e + c.c , ( i = 1, 2, 3 ) v 1 ˝ 1 ˝ ˇ

(

)

where Wi is the amplitude of the mode e iki r . Now, consider the second-order terms in ˜ 2 . According to the Fredholm solvability condition, the vector function of the right hand of the above equation must be orthogonal with the zero eigenvectors of operator L+c . The zero eigenvectors of adjoint operator L+c are ˜ 1 ˝ −iki r 2 ˛° A ˆ e , where A = a + D1k /bec . From the orthogonality condition, we obtain ˙

(

)

(1

˛ u2 ˆ A ) e −iki r Lc ˙ ˘ = 0. ˝ v2 ˇ

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Spatial Dynamics and Pattern Formation in Biological Populations

˛ ( u1 + Av1 ) + e1 A ( −bu1 + v1 ) − ( a + 1) u12 = 0. ˛T1

and

Putting the value of ( u1 , v1 ) in the above equation and equating the coeffcient of e ik1r , we obtain

( A + B)

˝W1 = e1 A ( Bb − 1) W1 + 2B2 ( a + 1) W2W3 . ˝T1

(6.13)

Write ˜ u2 ˝ ˜ a0 ˝ ˜ ai ˝ ik r ˜ aii ˝ 2ik r ˜ a12 ˝ i( k − k )r 1 2 i i +˛ ˆ +˛ ˆe +˛ ˆe ˆe ˛ ˆ =˛ ° v2 ˙ ° b0 ˙ ° bi ˙ ° b12 ˙ ° bii ˙ ˜ a23 ˝ i( k − k )r ˜ a31 ˝ i( k − k )r 2 3 3 1 +˛ + c.c. +˛ ˆe ˆe b b 23 31 ° ˙ ° ˙ The third-order terms in ˜ 3 are given by

(−a + D ˙ ) u 1

2

2

(

− v2 =

ˇu1 − ( a + 1) u12 , ˇT1

)

ˇv1 + e1 ( −bu1 + v1 ) . ˇT1

bec u2 + − ec + D2˙ 2 v2 = The elements are obtained as

(

b0 = bE1 Wi ai = Bbi , aii = E3Wi2 , bii =

2

), a

0

(

= E1 Wi

2

),

bec E3Wi2 be E W W , a12 = E4W1 W2 , b12 = c 4 2 1 2 4D2 k 2 + ec 3D2 k + ec

where E1 =

(

)

(

)

B2 ( a + 1) 4D2 k 2 + ec B2 ( a + 1) 3D2 k 2 + ec 2 ( a + 1) B2 . = , E3 = , E 4 a+b a + 4D1k 2 4D2 k 2 + ec + bec a + 3D1k 2 3D2 k 2 + ec + bec

(

)(

)

(

)(

)

Using the Fredholm solvability condition again, we obtain ˙ ˝W1 ˝b1 ˘ = e1 A ( Bb − 1) b1 + e2 A ( Bb − 1) W1 + 2B2 ( a + 1) W2b3 +W3b2 + ˆ ˝T2 ˝T1 

(

( A + B) ˇ

(

)

−  −2 ( a + 1)( E1 + E3 ) B + 3B3 W1

)(

)

2

+ (−2( a + 1) ( E1 + E4 ) B + 6B3 W2 + W3 2

2

) W . 1

(6.14)

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Brain Dynamics: Neural Systems in Space and Time

Substituting (6.13) and (6.14) into (6.12), we obtain

( A + B)

˝Z1 = ( ec − e ) A ( Bb − 1) Z1 + 2 B2 ( a + 1) Z2 Z3 − ( −2 ( a + 1)( E1 + E3 ) B ˝T

(

)

(

)

+ 3B3 Z1 + −2 ( a + 1)( E1 + E4 ) B + 6B3 Z2 + ( −2 ( a + 1) 2

2

(E1 + E4 ) B + 6B3 ) Z3 2 ) Z1 . The expressions for the coeffcients of ˜ 0 , h, g1 , and g 2 are given by the following:

˜0 =

g1

2 B2 ( a + 1) A+B ,h = , ec A ( Bb − 1) ec A ( Bb − 1)

( −2 ( a + 1)(E =

g2 =

1

+ E3 ) B + 3B3

),

+ E4 ) B + 6B3

).

ec A ( Bb − 1)

( −2 ( a + 1)(E

1

ec A ( Bb − 1)

The values of the parameters are to be chosen such that the above quantities are positive. We need the conditions Bb > 1, 3B3 > ( a + 1) B{max ˙ˆ 2 ( E1 + E3 ) , ( E1 + E4 ) ˇ˘ . Note that one can solve the amplitude equations for negative g1 , g 2 by considering the higher-order approximations to get a better stability condition [37]. This case is not considered here. The equations for determining the amplitude are given by the following:

(

)

(

)

(

)

˜0

° Z1 2 2 2 = µZ1 + hZ2 Z3 − g1 Z1 + g 2 Z2 + g 2 Z3 Z1 , °t

˜0

° Z2 2 2 2 = µZ2 + hZ1Z3 − g1 Z2 + g 2 Z1 + g 2 Z3 Z2 , °t

˜0

° Z3 2 2 2 = µZ3 + hZ2 Z1 − g1 Z3 + g 2 Z2 + g 2 Z1 Z3 . °t

(6.15a-c)

6.4.2.1 Linear Stability Analysis of the Amplitude Equation The dynamics of amplitude equation can be investigated by using the linear stability analysis. Assume that the amplitude in (6.15) can be expressed as Zi = ˜i e i˛i ,

(6.16)

where ˜i = Zi and ˜i is the phase. Substituting (6.16) into equation (6.15), separating the real and imaginary parts, and simplifying, we obtain

˜0

°˛ ˝ 2 ˝ 2 + ˝12 ˝32 + ˝32 ˝22 = −h 1 2 sin (˛ ) , °t ˝1 ˝ 2 ˝ 3

(6.17a)

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Spatial Dynamics and Pattern Formation in Biological Populations

˜0

° ˛1 2 2 2 = µ˛1 + h˛2 ˛3 cos (˙ ) − g1 ˛1 + g 2 ˛2 + g 2 ˛3 ˛1 , °t

(

)

(6.17b)

˜0

° ˛2 2 2 2 = µ˛2 + h˛1 ˛3 cos (˙ ) − g1 ˛2 + g 2 ˛1 + g 2 ˛3 ˛2 , °t

(

)

(6.17c)

˜0

° ˛3 2 2 2 = µ˛3 + h˛2 ˛1 cos (˙ ) − g1 ˛3 + g 2 ˛2 + g 2 ˛1 ˛3 , °t

(

)

(6.17d)

where ˜ = ˜1 + ˜2 + ˜3 . From the system (6.17), we fnd that the phase of pattern amplitude lies only in the phase ˜ = 0 and ˜ = ˛, when the system lies in the stationary state. Since ˜i ° 0, we can conclude that the solution of the equation (6.17a) is stable for ˜ = 0, when h > 0, and is stable for ˜ = ˛, when h < 0. If we consider only the stable solution of this equation, then the mode equation has the following form:

˜0

(

)

° ˛1 2 2 2 = µ˛1 + h ˛2 ˛3 − g1 ˛1 + g 2 ˛2 + g 2 ˛3 ˛1 . °t

(6.18)

The dynamical system (6.17) possesses four kinds of stationary solutions. In order to investigate the pattern formations, linear stability analysis was performed. Considering the perturbation to ( ˜1 , ˜2 , ˜2 ) as (˜°1 , ˜°2 , ˜°2 ) , substituting it in equation (6.18), we can derive the linear perturbation equation. The matrix of mode equation is given by

(

ˇ µ − 3g ° 2 − g ° 2 + ° 2 1 1 2 2 3   h °3 − 2g 2 °2 °1   h °2 − 2g 2 °3 °1 ˘

)

h °3 − 2g 2 °2 °1

h °2 − 2g 2 °3 °1

(

µ − 3g1 °22 − g 2 °32 + °12

)

h °1 − 2g 2 °2 °3

(

h °1 − 2g 2 °2 °3

µ − 3g1 °32 − g2 °22 + °12

)

     

(6.19)

i. The stationary state: ˜1 = ˜2 = ˜3 = 0. The stationary state corresponding to the ˜ ˝°i = µ˝°i. The stationary solution is stable when linear perturbation equation is ˜t µ < 0 = µ2; otherwise, it is unstable. ii. Stripe patterns: ˜1 = µ/g 1 , µ > 0, ˜2 = ˜3 = 0. Substituting ( ˜1 , 0, 0 ) in the perturbation equation (6.19), we obtain ˙ ˝°1 dˇ ˜ 0 ˇ ˝°2 dt ˇ ˆ ˝°3

˘ ˙ µ − 3g1 °12  ˇ 0  =ˇ ˇ  0  ˆˇ

0

µ − g2 °

0 h °1

2 1

h °1

µ − g 2 °12

˘˙ ˝°1 ˇ  ˇ ˝°2 ˇ  ˆ ˝°3

˘  .  

(6.20)

For ˜1 = µ/g1 , the characteristic equation of the coeffcient matrix is given by

( ˜ + 2 µ ) {µ (1 −   g * ) − ˜ }

2

− h2 ˛12  = 0, g * = g 2 /g1 . 

(

)

which has the eigenvalues ˜1 = −2 µ < 0, ˜2,3 = − µ g * − 1 ± h ˛1 . When g 2 > g1 , that h2 g1 is, g * > 1; ˜3 < 0. For µ > = µ3 , ˜2 < 0. Therefore, all the perturbations to ( g 2 − g 1 )2 the stripe patterns will disappear when the above two conditions are satisfed.

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iii. Hexagon patterns: ˜1 = ˜2 = ˜3 =

h ± h2 + 4 ( g1 + 2g 2 ) µ 2 ( g1 + 2g 2 )

= ˜.

˜ satisfes the equation ( g1 + 2g 2 ) ˜ 2 − h ˜ − µ = 0. One of the values is positive ( ˜ + ), and the other ( ˜ − ) is negative. Setting ˜ = ˜1 = ˜2 = ˜3 in the perturbation equation (6.19), we obtain ˝ ˛°1 d ˆ ˜ 0 ˆ ˛°2 dt ˆ ˙ ˛°3

ˇ ˝ p  ˆ  =ˆ q  ˆ q ˘ ˙

q p q

q ˇ ˝ ˛°1 ˆ q  ˆ ˛°2 p ˘ ˆ˙ ˛°3

ˇ  ,  ˘

(6.21)

where p = µ − ( 3g 1 + 2g 2 ) ° 2 , q = h ° − 2g 2 ° 2 . The characteristic equation is given 3 by ( p − ˜ ) − 3q 2 ( p − ˜ ) + 2q 3 = 0, whose eigenvalues are ˜1 = ˜2 = p − q , ˜3 = 2 q + p. Now,

˜3 = 2q + p = 2 h ° − 3 ° 2 ( g1 + 2g 2 ) + µ = 2 h ° − 3 ( h ° + µ ) + µ = − ( h ° + µ ) < 0, for ° = ° + .

˜1 = ˜2 = p − q = µ − 3g1 ˛ 2 − h ˛ = ( g1 + 2g 2 ) ˛ 2 − 2 h ˛ − 3g1 ˛ 2 = −2[ ˛ 2 ( g1 − g 2 ) + h ˛ ] < 0, for ˛ = ˛ + , when g1 > g 2 . Therefore, the pattern is stable for ˜ = ˜ + , when g1 > g 2 . ( 2g1 + g2 ) h2 = µ . Under this condiWhen g 2 > g1, we obtain the condition µ < 4 ( g 2 − g 1 )2 tion, the pattern is also stable for ˜ = ˜ + . It can be shown that for ˜ = ˜ − , the patterns are unstable. iv. The mixed states ˜1 =

µ − g1 °12 = µ − We obtain

g1 h2

( g 2 − g 1 )2

µ − g1 ˜12 , g 2 − g1 > 0, g1 + g 2 > 0, and g1 + g 2

h , ˜2 = ˜3 = g 2 − g1 > 0.

˝ ˛°1 d ˆ ˜ 0 ˆ ˛°2 dt ˆ ˙ ˛°3

ˇ ˝ p1  ˆ  = ˆ q1  ˆ q1 ˘ ˙

(

q1 p2 q1

)

where p1 = µ − 3g1 °12 − g 2 °22 + °32 =

(

)

p2 = µ − 3g1 °22 − g 2 °22 + °12 = −

q1 ˇ ˝ ˛°1 ˆ q1  ˆ ˛°2 p2 ˘ ˆ˙ ˛°3

ˇ  ,  ˘

(6.22)

( 3g1 + g2 ) g1h2 , g1 − g 2 µ− g1 + g 2 ( g1 + g2 )( g1 − g2 )2

(

)

3g12 − g 22 h2 2g1 µ+ , g1 + g 2 ( g1 + g2 )( g1 − g2 )2

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Spatial Dynamics and Pattern Formation in Biological Populations

q1 = h ˜2 − 2g 2 ˜2 ˜1 = −

q2 = h ˜1 − 2g 2 ˜22 = −

g1 + g 2 ˘ g1 h2   h, 2 µ − ( g1 − g2 )  ( g1 − g2 )2 

(

)

g 22 − g12 + 2g1 g 2 h2 2g 2 . µ+ g1 + g 2 ( g1 + g2 )( g1 − g2 )2

The characteristic equation is ˆˇ ˜ 2 − ( p1 + p2 + q2 ) ˜ + p1 ( p2 + q2 ) − 2q12 ˘ ( ˜ − p2 + q2 ) = 0. The relationships between the roots are given by

˜1 = p2 − q2 , ˜2 + ˜3 = p1 + p2 + q2 , ˜2 ˜3 = p1 ( p2 + q2 ) − 2 q12 . The eigenvalues are negative, if i( ) ˜1 < 0, ( ii ) ˜2 + ˜3 < 0, and ( iii ) ˜2 ˜3 > 0.

(6.23)

From ( 6.23i ), we get

˜1 =

2 ( g 2 − g1 ) ( 4g1 + 2g2 )( g2 − g1 )   h2 = 2 ( g2 − g1 ) ˘   µ − ( 2g1 + g2 ) .h2  µ− g1 + g 2 g1 + g 2  ( g1 + g2 )( g1 − g2 )2 ( g1 − g2 )2  

If g 2 > g1 , and µ
0, µ −   µ − g1 + g 2  ( g1 − g2 )2   ( g1 − g2 )2  

˙ g1 h2 ˘ ˙ ( 2g1 + g2 ) h2 ˘ > 0, g > g .  ˇ or ˇ µ − µ − 2 1 ˇˆ ( g1 − g2 )2  ˇˆ ( g1 − g2 )2 

But the mixed solution gives g 2 − g1 > 0, and ˜2 = ˜3 =

µ−

g1 h2

( g 2 − g 1 )2

µ − g1 ˜12 > 0, that is, µ − g1 °12 = g1 + g 2

> 0, and g1 + g 2 > 0.

Hence, we get the condition µ >

( 2g1 + g2 ) h2 , which is a contradiction. Hence, the mixed ( g 1 − g 2 )2

structure Turing pattern is always unstable.

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The following is the summary of the results: 1. The stationary solution is stable when µ < 0 = µ2; otherwise it is unstable. g1 h2

2. Stripe pattern solution is stable for µ >

( g 1 − g 2 )2

= µ3 , g 2 > g1 . Otherwise, it is

unstable. 3. Hexagon pattern solution exists when µ > 0. If g1 , g 2 can take negative values,  ˇ h2 + then the condition is µ >  −  = µ1 . The solution ˜ = ˜ is stable when ˘ 4 ( g1 + 2g 2 )  ( 2 g1 + g2 ) h2 = µ . For ˜ = ˜ − , the patterns (i) g1 > g 2 ; or when (ii) g 2 > g1, and µ < 4 ( g 2 − g 1 )2 are always unstable. 4. The mixed-state solution exists when g 2 > g1 , g1 + g 2 > 0, and µ > µ3 = g1 °12 . The solution is always unstable. 6.4.3 Secondary Bifurcation of the Turing Pattern Newell Whitehead and Segel [112] equation, regarded as the normal form of symmetry breaking bifurcation leading to roll stripe pattern and allowing for modulation in both x and y directions, is derived from the amplitude equation and perturbation of the solution. Writing the Taylor series expansion of ˜ ( µ , k ) = Re ( ˛ ) about the critical value ( µc , kc ) , we obtain

˜ ( µ , k ) = ˜ ( µc , k c ) + + =

1 ˛ 2˜ 2 ˛ k2

˛˜ ˛µ +

˛˜ ˛µ

( µ − µc ) + c

( k − k c )2 + c

( µ − µc ) + c

˛ 2˜ ˛ µ˛ k

˛ 2˜ ˛ µ˛ k

1 ˛ 2˜ 2 ˛ µ2

˛˜ ˛k

( k − kc ) + c

1 ˛ 2˜ 2 ˛ µ2

( µ − µc ) 2 c

( µ − µc ) ( k − k c ) +  c

( µ − µc ) 2 + c

1 ˛ 2˜ 2 ˛ k2

( k − k c )2 c

( µ − µc ) ( k − k c ) +  c

since ˜ ( µ , k ) = 0, (˛˜ /˛ k ) = 0, at the critical value ( µc , kc ) . As the system nears the critical point, we assume (i) µ − µc ˝ ° . (ii) ˜ ˛ ε , on the basis of the Turing bifurcation defnition. 2 (iii) ( k − kc ) ˆ ˜ ˆ ° . (iv) ˜ k x2 ˛ ˜ k y4 ˛ ° , where ˜ k x , ˜ k y is the perturbation. Hence, the linear terms in ˜ give the equation

(

˜ (µ, k ) =

˛˜ ˛µ

( µ − µc ) + c

)

1 ˛ 2˜ 2 ˛ k2

( k − k c )2 , c

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Spatial Dynamics and Pattern Formation in Biological Populations

where the frst term is the growth rate of instability module and the second term is the linear part infuenced by other module. For our discussion, we analyze the NWS equation for stripe pattern. The critical wave vector and perturbation are given by kc = ( kc , 0 ) and ˜ k = ˜ k x , ˜ k y , respectively. By using the Taylor series expansion, we obtain the leading ε order equation as

(

( k − kc ) k=

where

2

ˇ ˜ k y2  =  ˜ kx + 2k  ˘

)

2

c

( kc + ˜ kx )2 + ˜ ky2 .

In the Fourier space, we have the following correspondence: 2

2

  ° ˜ k y2  ° ° i °2  2 ˜ k x ˙ −i , ˜ k y ˙ −i , and ( k − kc ) =  ˜ k x + ˙ − −  ° X 2 k ° Y 2  . °X °Y 2kc   c Hence, we can obtain the NWS equation as: 2

˜0

where µ ˇ

˘ ° °Z i °2  2 Z − g1 Z Z, = µZ + ˝ 02  − °t  ° X 2kc ° Y 2 

(6.24)

°˛ 1 ° 2˛ . ( µ − µc ) , and ˝02 = − °µ 2 ° k2 c

For studying the linear stability analysis of system (6.24), we assume c = c0 + Ae i (˛ kx + kc ) + c.c. where ˜ k is the perturbation and kc is the critical value. Substituting Z = Ae i° kx into the system (6.24), we obtain 2

 °  ° °A °2  °2  2 ˜0 = ( µ − ˝ 02˙ k 2 )A + 2 i˙ k˝ 02  − i 2  A + ˝ 02  − i 2  A − g1 A A. (6.25) °t °Y  °Y   °X  °X Adding the perturbation ˜ A = u + iv to the amplitude A and substituting it into (6.25), we obtain  ˜u  ˜2 ˜2 ˜4  ˜2  ˜ v, =  −2 µ − ˛ k 2˝ 02 + ˝ 02 2 + 2˝ 02˛ k˝ 02 2 − ˝ 02 4  u −  2˛ k˝ 02 − 2˝ 02 2  ˜t  ˜x ˜y ˜y  ˜y  ˜x 

(

)

ˆ ˜2 ˜v ˆ ˜2  ˜ ˜2 ˜4  u + ˛ 02 ˘ 2 + 2° k 2 − 4  v. = ˘ 2° k˛ 02 − 2˛ 02 2  ˜t ˇ ˜y  ˜x ˜y ˜y  ˇ ˜x The perturbations u and v have the following decompositions: u = Ue st cos ( px ) cos ( qy ) , v = Ve st sin ( px ) sin ( qy ) .

(6.26)

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Brain Dynamics: Neural Systems in Space and Time

The characteristic equation of (6.26) is given by the following: s + 2t1 + t2 t3

(

)

(

t3 s + t2

)

= 0,

(

)

where t1 = µ − ° 02 ˛ k 2 , t2 = p 2 + q 4 + 2˜ kq 2 ° 02 , t3 = p˜ 02 2° k + 2q 2 . The dispersion relation is given by s2 + 2(t1 + t2 )s + (2t1 + t2 )t2 − t32 = 0.

(6.27)

The equation has the following two real roots: s± = −[t1 + t2 ] ± t12 + t32 , s− < 0. The necessary condition for the stripe pattern instability is s+ > 0. The value s+ depends on the perturbation vector ( p , q ) . First, assume q = 0 in system (6.27), and obtain

(

)

(

)

s2 + 2 ˘ µ − ° 02 ˛ k 2 + p 2° 02  s + ˘ 2 µ − 3° 02 ˛ k 2 + ° 02 p 2  p 2° 02 = 0.

(6.28)

Since s− < 0, the condition for instability is

(

)

s− s+ =  2 µ − 3° 02 ˛ k 2 + p 2° 02  p 2° 02  0. The Eckhaus instability may occur when ˜ k > 3 µ /(3˛ 0 ). Furthermore, if p = 0 also, then s+ = 0. The system has a long-wave modulation as p ˜ 0, where the magnitude increases with time, when the stripe pattern of system lost stability. Next, assuming p = 0 in the system (6.27), we obtain s2 + 2(t1 + t2 )s + (2t1 + t2 )t2 − t32 = 0,

(

)

where t2 = q 2 q 2 + 2˜ k ° 02 , t3 = 0. The roots are given by

(

)

s+ = −t2 = − q 2 q 2 + 2˜ k ° 02 , s− = − ( 2t1 + t2 ) .

(

)

If s+ is to be positive, then q 2 + 2˜ k < 0, or ˜ k < − q 2 2 . Furthermore, if q = 0 also, then s+ = 0. We obtain zigzag instability which is also long-wave instability. The bifurcation diagram drawn by the authors [161] is presented in Figure 6.11. The bifurcation line divides the bifurcation space into four domains A, B, C, D; and E, F, G, H correspond to different points on the curves. In the domain D, the system is in a steady state. Domains A and C represent the regions of pure Turing and pure Hopf instabilities, respectively. In D, the two bifurcations interact. E is the intersection of the Turing and Hopf bifurcation curves. Izhikevich and FitzHugh [82] modifed the model (6.7) and presented a two-dimensional simplifcation of the Hodgkin–Huxley model of spike generation in squid giant axons as

(

)

 = (V − a − bW )/˜ , V = V − V 3 3 − W + I, W

(6.29a, b)

where V , W represent the membrane voltage and the recovery variable, respectively, and I is the magnitude of input current. a = −0.7, b = 0.8, and ˜ = 1/0.08 are the experimentally observed values of the dimensionless parameters. The system dynamics can be analyzed

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 6.11 Bifurcation diagram of the system (6.7). (Reproduced with permission form Zheng, Q., Shen, J. 2015. Pattern formation in the Fitzhugh-Nagumo model. Comp. Math. Appl. 70(5), 10821097 [161]; and Elsevier. Copyright 2015.)

by mapping the left and right branches of the cubic nullclines. The Jacobian matrix of the system is given by ˝ 1 − V2 J=ˆ ˙ 1/˜

−1 ˇ . −b/˜ ˘

We present here some numerical computations. The fxed point of the system V * , W * is given by the following: V * is the solution of the cubic equation bV 3 + 3(1 − b)V − 3a − 3bI = 0, and W * = V * − a b. Consider the two regimes where the sys-

(

)

(

)

tem (6.29) exhibits quiescent (I = 0.1) and regular spiking (I = 0.8). The fxed point at I = 0.1 is obtained as V * , W * = ( −1.13751, − 0.54689 ) . W * does not directly affect the eigenvalues. The corresponding eigenvalues are ˜1,2 = −0.178967 ± 0.258423i. It is a stable focus node.

(

)

(

)

Similarly, the fxed point at I = 0.8 is obtained as V * , W * = ( −0.272901, − 0.533874 ) . The corresponding eigenvalues are ˜1 = 0.836706, ˜2 = 0.0248192. It is a saddle point. The time series of FHN 2D model (6.29) are plotted in Figure 6.12, for the cases of external stimulus I = 0.1 and I = 0.8. Bifurcation analysis: Using MATCONT software, we have plotted the bifurcation diagrams taking I as the predominant parameter. For I < 0.331282, the system has a stable focus node and there exists a quiescent state. As we increase the value of the parameter I in the interval, 0.331282 < I < 1.418718, the system changes its behavior from a quiescent state to regular spiking. The quiescent state disappears through a subcritical Hopf bifurcation (HB1) due to the birth of an unstable limit cycle at I = 0.331282. The system has unstable focus node and unstable node in the interval, 0.331282 < I < 1.418718. After this interval,

Brain Dynamics: Neural Systems in Space and Time

369

FIGURE 6.12 Time series of uncoupled FHN 2D model for external stimulus: (a) I = 0.1 and (b) I = 0.8.

FIGURE 6.13 Bifurcation scenario of the FHN 2D model (6.29).

there arises a stable focus node. Again, the quiescent state appears through a subcritical Hopf bifurcation (HB2) at I = 1.41718. Bifurcation scenario of the model is plotted in Figure  6.13. In the fgure, the thick blue line describes the stable region and dashed blue line describes the unstable region of fxed points. An unstable limit cycle (dashed red line) changes its stability to a stable limit cycle shown in grey line. 6.4.3.1 Dynamics of 1D Diffusion in FHN Model Consider the 1D diffusion model as ˛ V3 ˆ 2 V − W + I D , V = V − ˙ + x 2 ˝ 3 ˘ˇ

(6.30a)

 = (V − a − bW )/˜ , W

(6.30b)

where D is the diffusion coeffcient. We have investigated the dynamics of the coupled FHN oscillator in two regimes for different values of external stimulus I. The impact of diffusion (D) has been studied by setting each neuron in a quiescent state (I = 0.1); that is, all the neurons are settled into the steady state. At higher (D = 0.1) and lower (D = 0.0001) values of the diffusion coeffcient, all the neurons remain in the quiescent state (see Figure 6.15a-d).

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 6.14 Spatial plot of FHN 2D model with 1D diffusion. For the 1st Panel I = 0.1, with (a) D = 0.0001, (b) D = 0.001, (c) D = 0.01, and (d) D = 0.1. For the 2nd Panel I = 0.8, with (e) D = 0.0001, (f) D = 0.001, (g) D = 0.01, and (h) D = 0.1.

The corresponding spatiotemporal patterns have also been plotted for various values of the diffusion coeffcient (see Figure 6.14a-d). Here, the diffusion coeffcient does not have any impact on the coupled system (6.30). At a lower value of diffusion (D = 0.0001), the system creates a nonhomogeneous irregular pattern (see Figures 6.14e and 6.15e). But, if we increase the value of the diffusion coeffcient, the irregular pattern becomes regular and the coupled system exhibits horizontal stripe-type pattern (see Figures 6.14f-h and 6.15f-h). But an interesting behavior is observed at higher value of the external stimulus (I = 0.8).

6.5 Morris–Lecar (M–L) Model Morris and Lecar model [109] is a combination of the Hodgkin–Huxley (HH) and FitzHugh–Nagumo (FHN) models, which considers a voltage-gated calcium channel and a delayed rectifer ionic potassium channel. The model describes the relationship between the membrane voltage and activation of ion channels across the cell membrane. The potential depends on the action of the ion channels (see Figure 6.16). It is a 2D model describing the oscillations in barnacle giant muscle fber. It is a biophysically plausible model and has measurable parameters. The assumptions made are the following: (i) The equations are based on an isopotential patch of membrane, (ii) calcium ions carry the depolarizing current, (iii) potassium ions carry the hyperpolarizing current, and (iv) ionic conductance for different ionic species relaxes quickly to the steady-state condition which is independent of voltage. The neuron model is described by the following equations (Morris and Lecar [109]):

Brain Dynamics: Neural Systems in Space and Time

371

FIGURE 6.15 Time series of the end oscillator of FHN 2D model with 1D diffusion. Value of bifurcation parameter is I = 0.1, for the 1st panel (a-d); and I = 0.8, for the 2nd panel (e-h). The values of the diffusion coeffcient are same as in Figure 6.14.

FIGURE 6.16 Equivalent circuit for a patch of space-clamped barnacle sarcolemma. (Reproduced with permission from Morris, C. and Lecar, H. 1981. Voltage oscillations in the barnacle giant muscle fber. Biophys. J. 35(1), 193-213, [109]; and Elsevier. Copyright 1981.)

CV ˛ = − g Ca Mss (V )(V − VCa ) − g KW (V − VK ) − g L (V − VL ) + I = f (V , W ) ,

(6.31a)

W˝ = ˜ W (V ) ˘ˇWss (V ) − W  = g (V , W ) .

(6.31b)

The meanings of the system variables and the parameters are as follows: V: membrane potential of the neuron, W: recovery variable, C: membrane capacitance,

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Spatial Dynamics and Pattern Formation in Biological Populations

I: applied current stimulus, g Ca , g K , g L : maximum or instantaneous conductance values for calcium, potassium channels and leak pathways respectively, VCa , VK ,VL : equilibrium potential conductance values for calcium, potassium channels and leak conductance respectively, Mss (V ), Wss (V): fractions of open calcium and potassium channels at steady state, ˜ W (V ): rate constant for opening of the calcium channels, ˜ 0 : maximum rate constant for opening of the calcium channels V1 : potential value at which Mss (V ) becomes half, V2 : reciprocal of slope of voltage dependence of Mss (V ), V3 : potential value at which Wss (V ) becomes half, V4 : reciprocal of slope of voltage dependence of Wss (V ). W (t) is equal to the instantaneous value of the probability that a potassium ion channel rests in its open state. The second equation (6.31b) describes the relaxation process where protein channels undergo transitions between ionic conductance and non-conductance states. The open state probability functions Mss (V )and Wss (V ) are derived from the assumptions that the open and closed states of a channel are partitioned under the Boltzmann distribution in equilibrium condition. The conductance functions are derived as Mss (V ) =

1 1 1 + tanh(t1 )] , Wss (V ) = [ 1 + tanh(t2 )] , [ 2 2

where t1 = (V − V1 )/V2 , t2 = (V − V3 )/V4 . The time constant for the potassium channel relaxation in response to voltage change is voltage-dependent and is derived as ˜ W (V ) = ° cosh [ 0.5t2 ]. 6.5.1 Stability and Bifurcation Analysis To study the characteristic description of the M-L model for different sets of current stimuli, perturb the system linearly around the fxed point V * , W * . The Jacobian matrix at the equilibrium point is given by

(

° a11 J=˝ ˛ a21 a11 =

)

a12 ˙ ˇ , where a22 ˆ

   ˆ V * − VCa  1 * sech 2 t1*  − g K W * − g L   −0.5g Ca 1 + tanh t1 + ˘  C  ˇ V2    

a12 = −

( )

( )

gK * V − VK , C

(

(

)

{

( )} − W

ˇ 0.5˜  a21 =  sinh 0.5t2*  0.5 1 + tanh t2*  ˘ V4 

(

(

)

)

(

)

(

*

(

)

( ))

 + cosh 0.5t2* sech 2 t2* , 

)

a22 = −˜ cosh 0.5t2* , t1* = V * − V1 V2 , t2* = V * − V3 V4 .

Brain Dynamics: Neural Systems in Space and Time

373

The condition for the equilibrium solution of the system to be stable is given by trace J = a11 + a22 < 0, and det J = a11a22 − a12 a21 > 0. We have tested the stability for the following set of parameter values: C = 1, g L = 0.5, VL = −0.5, g Ca = 1.2, VCa = 1, g K = 2, VK = −0.7, V1 = −0.01, V2 = 0.15, V3 = 0.1, V4 = 0.05, ˜ = 1/3. For I = 0.052, fxed points are (−0.368733, 0.7199 × 10−8), (−0.212277, 0.376 × 10−5), and (0.0891391, 0.3931). Corresponding eigenvalues are (−0.3297, −18.0954), (0.5716, −3.7921) and (−0.0734 ± 2.2319i). The frst fxed point is a stable node, second one is a saddle point, and the third one is a stable focus. For I = 0.054, fxed points are (−0.362542,0.92219 × 10−8), (−0.215906, 0.3252 × 10−5 ) and (0.0892734, 0.3943). Corresponding eigenvalues are (−0.3162, −17.0092), (0.5306, −3.9317), and (−0.0762 ± 2.2335i). The frst fxed point is a stable node, the second one is a saddle point, and the third one is a stable focus. For I = 0.2, fxed point and eigenvalues are (0.0986174, 0.4862), and (−0.2777 ± 2.3060i), respectively. The fxed point is a stable focus. Phase plane diagrams for I = 0.052, 0.054, and 0.2, are given in Figure 6.17. 6.5.1.1 Bifurcation Analysis Bifurcation analysis is performed for the 2D M-L system using MATCONT software by varying the injected current stimulus I.At higher injected current stimulus ( I > 0.1) , the model reveals a monostable quiescent state (stable focus). The unstable state becomes stable as a result of the subcritical Hopf bifurcation (SH) at lower positive values of stimulus current ( I ~ 0.001830 ). In Figure 6.18a, bifurcation diagram of the 2D M-L oscillator with respect to the stimulus current I is given. In the fgure, the upper solid blue line describes the changes of this quiescent state for different values of stimulus current. The lower thick blue line describes a stable node which collides with a saddle point at I ~ 0.069147 (SN point in Figure 6.18a) and vanishes together. We used phase space analysis to understand the behavior of the fxed points. The deterministic system has three equilibrium points that are the intersections of the nullclines of the system variables V and W, respectively. The left fxed point is asymptotically stable (SS1 – stable node), and the right fxed point is unstable (US) (see Figure 6.18b). When the current stimulus I is increased, the V nullcline moves upward and the two fxed points move closer to each other, collide, and mutually annihilate, resulting in a saddle node bifurcation (SN). After that, there exists

FIGURE 6.17 Phase plane diagram for I = 0.052, 0.054 and 0.2 respectively.

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 6.18 (a) Bifurcation diagram of the 2D M-L oscillator with respect to the stimulus current I. (b) Nullclines for the deterministic uncoupled 2D M-L model. The time series for different regimes: (c) for I = 0.052, (vertical green line in the inset of (a)); (d) for I = 0.054, (vertical magenta line in the inset of (a)); and (e) for I = 0.2 (vertical black line). (Reproduced with permission from Mondal, A., et al. 2019. Diffusion dynamics of a conductance-based neuronal population. Phys. Rev. E 99(4), 042307, [106]; and American Physical Society. Copyright 2019.)

only one fxed point (SS2) with a further increase of I. There is another interesting behavior appearing between I ~ 0.053 and I ~ 0.99. A stable limit cycle (thick cyan line) coexists with an unstable limit cycle shown by the dashed red line (also see the inset). Therefore, the system becomes tristable; that is, one stable node, one limit cycle, and one stable focus coexist together. The thick green lines and dotted red lines show stable and unstable limit cycles, respectively. This type of feature arises due to the impact of hyperbolic functions in the Morris–Lecar system. For our analysis, we considered three parameter spaces. The system produces phasic spiking at I = 0.052; the value is marked in the fgure with vertical green line in extreme left (Figure 6.18a, also in the inset). The corresponding time series for phasic spiking is shown in Figure 6.18c. A limit cycle (tonic spiking) is produced at I = 0.054; the value is marked with middle vertical magenta line in Figure 6.18a and the corresponding time series is shown in Figure 6.18d. The quiescent state at I = 0.2 is marked by the rightmost vertical solid black line in Figure 6.18a, and the corresponding time series is shown in Figure 6.18e. 6.5.2 Spatial Morris–Lecar Model Mondal et al. [106] investigated the dynamics of a 1D cable consisting of a chain of excitable neurons. In models of nerve conduction, only the membrane potential (V) spatially interacts; the recovery variable (W) and other variables interact only through the membrane

Brain Dynamics: Neural Systems in Space and Time

375

potential (Ermentrout and Lewis [43]). The 1D diffusion is a common scenario in many biophysical systems [43,141] in which one of the variables interacts with the others in a spatially distributed cable. Nearest-neighbor diffusion through the membrane potential is considered for the study. The 2D excitable M-L model with 1D diffusion is described by the reaction–diffusion equations [106]: C

˜V ˜ 2V = f (V , W ) + D 2 , ˜t ˜x ˜W = g (V , W ) . ˜t

(6.32a) (6.32b)

where f and g are as given in (6.31). The initial conditions are V ( t = 0, x ) > 0, W ( t = 0, x ) > 0 for x ˙ˆ, and zero-fux boundary conditions are

˜V ˜W = = 0, for x ˛˝, t > 0, ˜n ˜n where ˜ is a bounded interval, and n is the outward normal to the boundary ˜ °. In this case, ˜ is the length of the excitable cable (N = 10) and D is the strength of the synaptic coupling. Numerical simulations are done using a fnite difference scheme. The authors [106] used the “pdepe” method [157] for the numerical solution of the 1D reaction–diffusion system. The time and space steps are taken as ˜t = 0.001, ˜x = 0.1, and are fxed for all simulations. The value of C is taken as C = 1. The interval is taken as 0 ˜ x ˜ 10. The initial conditions are taken as V ( t = 0, x ) = V * +

rand rand . , W ( t = 0, x ) = W * + 10 10

The zero-fux boundary conditions indicate that the membranes are impermeable at the boundaries and it acts as an isolated cable. An external stimulus I is applied to all the excitable neurons. The external stimulus can change the dynamical behavior of the uncoupled model. The spatiotemporal patterns are investigated in three regimes, observed at different values of I. First, the impact of the diffusion coeffcient D was tested by setting each neuron in a phasic spiking state (I = 0.052); that is, all the neurons are settled into the phasic spiking regime (Figure 6.18c). At a lower value of diffusion (D = 0.0001), the system loses its stability and creates an inhomogeneous irregular pattern (Figure 6.19a), corresponding to a spiral-type instability. The vertical yellow and blue stripes signify small oscillations deviated from the original uncoupled steady states. For a better understanding, time series for an arbitrarily chosen node is drawn in Figure 6.20a. Here, spatial heterogeneity is observed. If we consider an arbitrary node (vertical yellow stripes) from the cable, it shows oscillations. At higher values of diffusion (D = 0.0005 and D = 0.0037), a more complex desynchronized fring pattern (Figure 6.19b and c) appears where a train of irregular spiking and bursting (Figure 6.20b and c) is generated. Finally, the spatial instability vanishes at a higher value of diffusion (D = 0.5), by stabilizing the whole chain or cable into a homogeneous fxed point (see Figures 6.19d and 6.20d), which is the stable node of an uncoupled neuron. We observe how fring patterns of a neuronal cable are changed

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Spatial Dynamics and Pattern Formation in Biological Populations

FIGURE 6.19 Spatiotemporal plots of the 2D ML cable with 1D diffusion for I = 0.052; (a) D = 0.0001, (b) D = 0.0005, (c)  D  =  0.0037, and (d) D = 0.5. (Reproduced with permission from Mondal, A., et al. 2019. Diffusion dynamics of a conductance-based neuronal population. Phys. Rev. E 99(4), 042307, [106]; and American Physical Society. Copyright 2019.)

FIGURE 6.20 Time series of the end oscillator of the 2D ML cable with 1D diffusion. The external current stimulus is I = 0.052. The values of the diffusion coefficients are the same as in Fig. 6.19. (Reproduced with permission from Mondal, A. et al., Diffusion dynamics of a conductance-based neuronal population, Phys. Rev. E 99(4), p.042307, 2019, [106]; and American Physical Society. Copyright 2019).

by the impact of the diffusion coeffcient D. With systematic changes in the value of D, the continuous medium (cable) passes from the regime of inhomogeneous instability to a uniform steady state through the formation of irregular structures at intermediate values of diffusion coeffcients. Now, consider the case of 2D diffusion in the frst variable V of the excitable 2D ML system (6.31). The system is given by [106] C

˜V = f (V , W ) + Dˆ 2V, ˜t ˜W = g (V , W ) , ˜t

(6.33a) (6.33b)

˜ 2V ˜ 2V + , with the same initial and boundary conditions as before. ˜ x2 ˜ y 2 The authors [106] solved the coupled diffusive 2D ML oscillator using a fnite difference scheme. The value of C was taken as C = 1. The domain was taken as 0 ˜ x ˜ 100; 0 ˜ y ˜ 100. Space and time were discretized by taking the system as N × N with N = 100 and where ° 2V =

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Brain Dynamics: Neural Systems in Space and Time

˜x = ˜y = ˜ = 0.25, ˜t = 0.0001. (Here, the domain size will remain 100 × 100 only. The method generates data for 400 × 400 points and plots it in the domain 100 × 100).˜ 2V is approximated by the fve-point formula: ˝ 2V =

˜ 2V ˜ 2V 1 + = (Vi−1, j + Vi+1, j + Vi, j− 1 + Vi, j+ 1 − 4Vi,  j ). ˜ x2 ˜ y 2 ˇ2

The time derivative is approximated by the explicit method n ˜ V Vin+1 , j − Vi , j = . ˝t ˜t

Figure 6.21 gives pattern formations in the 2D M-L cable with diffusion for I = 0.052, and for (a) D = 0.001, (b) D = 0.05, and (c) D = 0.09. For I = 0.052 and D = 0.001, the neurons with high-amplitude oscillations (i.e., generating action potentials) are distributed in a scattered way. In the spatial domain, the neighboring nodes try to fre together or set themselves in the steady states, although neither the synchronous fring nor synchronous steady states dominate in the spatial domain. If the diffusive coupling is increased to D  =  0.05, hexagon-like patterns (shown in red or yellow) become broader in size, suggesting that small groups of nodes are fring asynchronously, whereas inside the blue domain, the neighboring nodes stay below the subthreshold oscillations. With a further increase of diffusion strength to D = 0.09, the neurons in the spatial domain form distinct clusters (shown in red) of fring surrounded by a large subthreshold population, which fnally leads to a homogeneous state for higher diffusion or for a long-time evaluation. 6.5.3 Multiple-Scale Analysis (Amplitude Equations) To study the dynamics near a bifurcation point, the authors applied the multiple-scale analysis method near this point. The relevant patterns can be expressed in terms of three active resonant pairs of modes (k j , −k j ), such that k j = kT for j = 1, 2, 3, where kT is the critical wave number (where the instability occurs). Using the approximations, tanh ( x ) ˛ x − x 3 /3 , cosh ( x ) ˛ 1 + x 2 /2 , and setting C = 1, ˜ = 1/3, in equation (6.31), we obtain

(

)

(

)

FIGURE 6.21 Pattern formation in the 2D ML cable with diffusion for I = 0.052. (a) D = 0.001, (b) D = 0.5, (c) D = 0.09. (Reproduced with permission from Mondal, A., et al. 2019. Diffusion dynamics of a conductance-based neuronal population. Phys. Rev. E 99(4), 042307, [106]; and American Physical Society. Copyright 2019.)

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Spatial Dynamics and Pattern Formation in Biological Populations

ˇ t3  V ˛ = I − g L (V − VL ) − 0.5g Ca (V − VCa )  1 + t1 − 1  − g KW (V − VK ) , 3 ˘ W˜ =

(6.34a)

ˇ 0.5 ˝ t2 ˇ ˝ t3 1 + 2  ˆ 1 + t2 − 2 − 2W  , ˆ 3 ˙ 8 ˘˙ 3 ˘

(6.34b)

where t1 = (V − V1 )/V2 , t2 = (V − V3 )/V4 . Simplifying and retaining terms up to order three, we obtain V˜ = I + a6 + a1V + a2W + a3V 2 + a4VW + a5V 3 ,

(6.35a)

W˜ = b7 + b1V + b2W + b3V 2 + b4VW + b5V 3 + b6V 2W,

(6.35b)

ˆ  0.5g Ca t3 V V a1 = − g L − 0.5g Ca ˘1 − t3 + 3 − Ca 1 − t32  , t3 = 1 , a2 = g KVK , a3 = 3 V V V2 2 2 ˇ  ˝ −0.5g Ca ˝ VCa ˇ t33 ˇ ˝V t ˇ t + a = V g + 0.5 V g 1 − t + , , × ˆ Ca 3 + t32 − 1 , a4 = − g K , a5 = 3 6 L L Ca Ca 3 ˆ˙ V22 ˆ˙ 3V2 ˘ 3 ˘ ˙ V2 ˘ 0.5 ˛ 1˝ 1 ˇ 0.5 ˛ 1 5 5 t t 5 5 b1 = 1 − 4 − t42 − t44 ˆ˘ , b2 = − ˆ 1 + t42  , b3 = + t4 + t43 ˆ˘ , b4 = 4 , 3V4 ˙˝ 3˙ 8 ˘ 3V42 ˙˝ 8 8 12 ˇ 4 8 24 ˇ 12V4 −2.5 −1 0.5 ˝ 1 5 1 V , b7 = b5 = 1 + 2t42 , b6 = 1 − t4 + t42 + t43 + t45 ˇ , t4 = 3 . 72V43 24V42 3 ˆ˙ 8 24 24 ˘ V4  + W * , about the equilibrium point V * , W * . Consider a perturbation V = V + V * and W = W Substituting in (6.35), simplifying, and retaining terms up to third order, we obtain

(

where

(

)

)

(

(

)

 + a3 + 3a5V * V 2 + a4VW  = a11V + a12W   + a5V 3 , V˛

(

)

 ˛ = a21V + a22W  + t5V 2 + b4 + 2b6V * VW .   + b5V 3 + b6V 2W W

( ) , a =a +a V , a + b (V ) , t = ( b + 3b V + b W ) .

where a11 = a1 + 2a3V * + a4W * + 3 a5 V * *

*

2b6V W , a22 = b2 + b4V

*

6

* 2

5

2

12

3

2

*

5

*

4

21

)

(6.36a) (6.36b)

( )

= b1 + b4W * + 2b3V * + 3b5 V *

2

+

*

6

The corresponding diffusive system for (6.36) is

˜ V  + a3 + 3 a5V * V 2 + a4V W  + a5V 3 + Dˆ 2V,  = a11V + a12W ˜t

(

)

 ˜W  + t5V 2 + b4 + 2b6V * V W  + b5V 3 + b6V 2W.  = a21V + a22W ˜t

(

)

These equations can be written in the vector form as

˜X = LX + H, ˜t

(6.37)

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Brain Dynamics: Neural Systems in Space and Time

° V X=˝  ˝˛ W

where

(

)

˙ ˇ, ˇˆ

˝ a + D˛ 2 L = ˆ 11 ˆ˙ a21

˙  + a5V 3 a3 + 3a5V * V 2 + a4V W H=ˇ ˇ t V 2 + b + 2b V * V W  + b5V 3 + b6V 2W  ˇˆ 5 4 6

(

)

ˇ , ˘

a12 a22

˘ .  

and

( )

Expand the bifurcation parameter I as I − IT = ˜ I1 + ˜ 2 I 2 + ˜ 3 I 3 + o ˜ 4 , where ˜  1, and IT is the Hopf bifurcation point. Similarly, expand the variable X and the nonlinear term H, as the following: ˙ V X=ˇ  ˆˇ W

˘ ˙ p1  =ˇ  ˆˇ q1

˘ ˙ p2 ˜ +ˇ  ˆˇ q2

˘ ˙ p3 2 ˜ +ˇ  ˇˆ q3

˘ 3 4 ˜ +o ˜ , 

( )

(6.38)

and

( )

H = ˜ 2 h2 + ˜ 3 h3 + o ˜ 4 , ˙ h2 = ˇ ˇ ˇˆ

(

(a

)

(6.39)

+ 3a5V * p12 + a4 p1q1 ˘ , t5 p12 + b4 + 2b6V * p1q1   3

(

)

)

˙ 2 a3 + 3a5V * p1 p2 + a4 ( p1q2 + p2 q1 ) + a5 p13 ˇ h3 = ˇ 2t p p + b + 2b V * p q + p q + b p 3 + b p 2 q ( 1 2 2 1) 5 1 6 1 1 ˇˆ 5 1 2 4 6

(

)

˘ .  

The linear operator L can be written as

(

)

L = LT + ( I − IT ) L1 + ( I − IT ) L2 + o ( I − IT ) , 2

(

3

)

L = LT + ( I − IT ) M + o ( I − IT ) , 2

(6.40) (6.41)

T ˙ aT + D˝ 2 ˘ ˙ m11 a12 m12 ˘ 1 ˜iL 11 T T T T , L = ˇ , M=ˇ T  , and a11, a12 , a21 , a22 are i T T m m ˇˆ i! ˜ I 21 22 a22  a21 ˆ  the critical values of the elements of the Jacobian matrix at the instability point or Turing point. We have

where Li =

L1 = M =

where m11 =

˜ L ˜ L ˜ V ˜ L ˜ W ˝ m11 = + =ˆ ˜ I ˜ V ˜ I ˜ W ˜ I ˙ m21

˜ a11 ˜ V ˜ a11 ˜ W + , etc. ˜V ˜ I ˜W ˜ I

m12 ˇ  m22 ˘

380

Spatial Dynamics and Pattern Formation in Biological Populations

Denote t1* =

m11 =

V1 − V * * VCa − V * * V3 − V * , t2 = , t3 = . We obtain V2 V4 V2  ˆ 1  −0.5gCa ˘ˇ *  ˘ V2 cosh t1 

( )

2

( )

tanh t1*

−

2

−1

V2

 g Ca  * *  g L + g KV − 0.5g Ca tanh t1 − 1 + 

(

( ) )

(

+

( )(

)

2 sinh t1* V * − 1 ˘   3  V22 cosh t1* ˘ 

( ) (tanh (t ) − 1)(V * 2 1

Ca

2V2

(

( ) )

( )

m12 = ( − g K ) ˆ g L + g KV * − 0.5g Ca tanh t1* − 1 + 0.5g Ca tanh t1* ˇ˘

(

)(

 m21 =  0.25 ˜ sinh 0.5t3* tanh t3* 

(

(

(

− 0.25 ˜ sinh 0.5t3*

( )

))

)) (

2

( ) + (˜cosh ( 0.5t ) sinh (t )) 2

(

(

(

m22 = 0.5 ˜ sinh 0.5t3*

)

) (V g (V 4

K

* 3

* 3

(

( ) )

*

)(

)

− 1 t2*  , 

)(

(

( )

V42  g L + g KV * − 0.5g Ca tanh t1* − 1 + 0.5g Ca tanh t1* 

+ 0.5 ˜ sinh 0.5t3*

2

1 , V − VK *

( )

− 1 − 0.25 ˜ cosh 0.5t3* W * + 0.5tanh t3* − 0.5

cosh t3*

{

)

− V*    

−

))

( )

3 cosh t3*   2

)

− 1 t2*  

}

))

− VK ,

{

(

(

( ) )

( )

V4  g L + g KV * − 0.5g Ca tanh t1* − 1 + 0.5g Ca tanh t1* 

2

)

}

− 1 t2*  . 

Now, split the typical time scale in the time derivative as

˜ ˜ ˜ =° + °2 + , ˜t ˜ T1 ˜ T2

(6.42)

where T1 = ˜ t and T2 = ˜ 2t. We have, from equation (6.37),

˜  ˇ p1 °  ˜ t  ˘ q1 

 ˇ p2 2  +°   ˘ q2

 ˘ °   ˘ p1 2 °  ˜ ° T + ˜ ° T +  ˜  q 1 1 2    ° p1  ˜ ˝  ˝˛ q1

 ˇ p3 3  +°   ˘ q3  ˘ p2 2  +˜    q2

˙ ° p2 2 ˇ +˜ ˝ ˇˆ ˝˛ q2

  3 2  = {LT + ( I − IT ) M } X + ° h2 + ° h3 .    ˘ p3 3  +˜    q3

˙ ° p3 3 ˇ +˜ ˝ ˇˆ ˝˛ q3

  2 3  = LT + ˜ I1 + ˜ I 2 + ˜ I 3 M  

{

(

˙  2 3 ˇ  + ˜ h2 + ˜ h3 . ˇˆ  

) }

381

Brain Dynamics: Neural Systems in Space and Time

Comparing the orders of ˜ , and ˜ 3 , we obtain ˜ p1 LT ˛ ˛° q1

° p3 LT ˝ ˝˛ q3

˝ ˆ = 0, ˆ˙

(6.43)

° p2 LT ˝ ˝˛ q2

˙ ˜ ° p1 ˇ= ˝ ˇˆ ˜ T1 ˝˛ q1

˙ ° p1 ˇ − I1 M ˝ ˇˆ ˝˛ q1

˙ ˇ − h2 , ˇˆ

˙ ˜ ° p2 ˇ= ˝ ˇˆ ˜ T1 ˝˛ q2

˙ ˜ ° p1 ˇ+ ˝ ˇˆ ˜ T2 ˝˛ q1

˙ ° p2 ˇ − I1 M ˝ ˇˆ ˝˛ q2

˙ ° p1 ˇ − I2 M ˝ ˇˆ ˝˛ q1

(6.44) ˙ ˇ − h3 . ˇˆ

(6.45)

T a12 = f and q1 = 1 (Here, we fnd the eigenvecT Dk − a11 tor ( f, 1) corresponding to zero eigenvalue, which is also equivalent to the condition of existence of solution of the homogenous system. Corresponding to an eigenvalue there may exist more than one eigenvector). The authors have taken ( f, 1) as one of the possible combinations. We can write

Solving equation (6.43), we get p1 =

° p1 ˝ ˝˛ q1

˙ ° ˇ =˝ ˇˆ ˛

2

f ˙° ˇ˝ 1 ˆ ˝˛

3

˜W e j

ik j r

j=1

˙ + c.c ˇ , ˇˆ

(6.46)

that is, ( p1 , q1 ) is a linear combination of the eigenvectors that correspond to the zero ik r eigenvalue of the linear operation LT , where Wj is the amplitude of the mode e j , and c.c represents complex conjugate. Now, to get the nontrivial solution of equation (6.44), the Fredholm solvability criterion [160] was used, which requires that the zero eigenvectors of operator L+T (the adjoint operator of LT ) must be orthogonal to the right-hand side of equation (6.44). Note that the zero eigenvectors of the operator L+T are described by (1 g )T e −ik j r + c.c , j = 1, 2, 3, where g = a12T /a22T . Now, from equation (6.44), we can write T

(

)

˝ ˜ Fp ˆ ˛ ˆ˙ ˛° Fq

˜ p2 LT ˛ ˛° q2 Let, Fpj and Fqj represent the coeffcients of e

(

nality condition, ( 1 g ) F F j p

)

j T q

ik j r

˝ ˆ. ˆ˙

in Fp and Fq , respectively. Using the orthogo-

= 0, we obtain

˜ W1 = I1 ˆˇ fm11 + m12 + g ( fm21 + m22 ) ˘ W1 + 2 ( l1 + gl2 ) W2W3 , ˜ T1 ˜ W2 ( f + g) = I1 ˆˇ fm11 + m12 + g ( fm21 + m22 ) ˘ W2 + 2 ( l1 + gl2 ) W1W3 , ˜ T1 ˜ W3 = I1 ˆˇ fm11 + m12 + g ( fm21 + m22 ) ˘ W3 + 2 ( l1 + gl2 ) W1W2 , ( f + g) ˜ T1 ( f + g)

(

)

(

) (

)

where l1 = f 2 a3 + 3a5V * + fa4 , and l2 = f 2 b3 + 3b5V * + b6W * + f b4 + 2b6V * .

382

Spatial Dynamics and Pattern Formation in Biological Populations

Solving equation (6.44), we obtain ° p2 ˝ ˝˛ q2

˙ ° P0 ˙ ˇ =˝ ˇ+ ˇˆ ˛ Q0 ˆ

° Pj ˝ ˝ Qj j=1 ˛

˙ ik r ˇe j + ˇˆ

3

˜

°

3

Pjj

˜ ˝˝˛

Q jj

j=1

˙ 2ik r ˇe j ˇˆ

° P12 ˙ i( k − k )r ° P23 ˙ i( k − k )r ° P31 ˙ i( k − k )r 1 2 2 3 3 1 + c.c. +˝ +˝ +˝ ˇe ˇe ˇe Q Q Q 12 ˆ 23 ˆ 31 ˆ ˛ ˛ ˛

(6.47)

The coeffcients in equation (6.47) are given by the following: ˛ P0 ˆ ˛ Zp0 ˙ ˘ =˙ ˝ Q0 ˇ ˙˝ Zq0 ˝ ˜ Zp 1 ˆ =˛ ˙ˆ ˛° Zq 1

˜ Pjj ˛ ˛° Q jj

ˆ 2 2 2 ˘ W1 + W2 + W3 , Pj = fQ j , ˘ˇ

(

)

˝ ˜ Pjk ˆ Wj2 , ˛ ˆ˙ ˛° Q jk

˜ Zp0 ˛ ˛° Zq0

˝ −2 ˜ aT l − aT l 22 1 12 2 ˆ= T˛ T T ˆ˙  0 °˛ a11l2 − a21 l1

˛ Zp1 ˙ ˙˝ Zq1

˛ ˆ −1 ˙ = ˘ T T T T ˘ˇ − 4DkT2T a22 − a12 a11 a21 ˙ ˝

˛ Zp2 ˙ ˙˝ Zq2

˛ ˆ −2 ˙ = ˘ T T T T a11 − 3DkT2T a22 − a12 a21 ˙ ˇ˘ ˝

(

˝ ˜ Zp 2 ˆ =˛ ˆ˙ ˛° Zq 2

˝ T T T T T a21 , kT2T = 0T , a22 − a12 ˆ ,  T0 = a11 ˆ˙ Da22 T T a22 l1 − a12 l2

(a

)

(

˝ ˆ WjWk , ˆ˙

T 11

T T a22 l1 − a12 l2

(a

)

)

T l1 − 4DkT2T l2 − a21

T 11

)

T l1 − 3DkT2T l2 − a21

ˆ ˘, ˘ ˇ ˆ ˘. ˘ ˇ

Using the approach described above and following the Fredholm solvability criterion, the authors have shown that (from equation (6.45))

˜ Q1 ˘ 1 =  fm11 + m12 + g ( fm21 + m22 )  ( I1Q1 + I 2W1 ) + ( f + g ) ˙ˇˆ ˜˜W T ˜ T   2

1

(

)

(

+ H Q2 W3 + Q3 W2 − G1 W1 + G2 W2 + W3  2

2

2

) W ,

where H = 2 ( l1 + gl2 ) ,

(

)

(

)

−G1 = ( 2˜ 1 f + °1 ) zp0 + zp1 + °1 f zq0 + zq1 + 3˛ 1 f 3 

(

)

(

)

+ g ( 2˜ 2 f + ° 2 ) zp0 + zp1 + ° 2 f zq0 + zq1 + 3˝ 2 f 2 + 3˛ 2 f 3  ,

(

)

(

)

−G2 = ( 2˜ 1 f + °1 ) zp0 + zp2 + °1 f zq0 + zq2 + 6˛ 1 f 3 

(

)

(

)

+ g ( 2˜ 2 f + ° 2 ) zp0 + zp2 + ° 2 f zq0 + zq2 + 6˝ 2 f 2 + 6˛ 2 f 3  ,

1

(6.48)

383

Brain Dynamics: Neural Systems in Space and Time

(

)

(

)

˜ 1 = a3 + 3a5V * , °1 = a4 , ˛ 1 = a5 , ˝ 2 = b6 , ˜ 2 = b3 + 3b5V * + b6W * ,

(

)

˜ 2 = b4 + 2b6V * , ° 2 = b5 . The remaining two equations are obtained through the transformation of the subscripts of W and Q. Here, A j and its conjugate A j ( j = 1, 2, 3 ) are the amplitudes of the modes k j and −k j, respectively. The amplitude A j is expanded as A j = ˜ Wj + ˜ 2Q j + o ˜ 3 . Using the expression for A j and equation (6.42), the amplitude equation corresponding to A1 was obtained as

( )

˜0

(

)

° A1 2 2 2 = µ A1 + hA2 A3 − g1 A1 + g 2 A2 + g2 A3 A1 , °t

(6.49)

where µ = ( I − IT )/IT is a normalized distance to the onset, and g1 and g 2 explore the type of instability [37], where g1 =

G2 G1 , g2 = , IT ˆ˙ fm11 + m12 + g ( fm21 + m22 ) ˇ˘ IT ˆ˙ fm11 + m12 + g ( fm21 + m22 ) ˇ˘

˜0 =

f +g H ,h= . IT ˇˆ fm11 + m12 + g ( fm21 + m22 ) ˘ IT ˇˆ fm11 + m12 + g ( fm21 + m22 ) ˘

In the same way, the remaining two equations are derived for evaluating A2 and A3 . 6.5.3.1 Amplitude Stability The amplitude equations (equation (6.49) for A2 and A3 ) are transformed from rectangular i˛ to polar coordinates by setting the complex amplitude as A j = ˜ j e j , where ˜ j = A j and ˜ j represents the phase angle in the system. The authors obtained a set of coupled equations with a constraint ˜ = ˜ 1 + ˜ 2 + ˜ 3 , as

˜0

°˛ ˝ 2 ˝ 2 + ˝12 ˝32 + ˝32 ˝22 = −h 1 2 sin˛ , °t ˝1 ˝ 2 ˝ 3

˜0

° ˛1 = µ˛1 + h˛2 ˛3 cos˙ − g1 ˛13 − g 2 ˛22 + ˛32 ˛1 , °t

˜0

° ˛2 = µ˛2 + h˛1 ˛3 cos˙ − g1 ˛23 − g 2 ˛12 + ˛32 ˛2 , °t

˜0

° ˛3 = µ˛3 + h˛2 ˛1 cos˙ − g1 ˛33 − g 2 ˛12 + ˛22 ˛3 . °t

( (

(

) )

)

(6.50)

Depending on the parameters, µ , g1 , g 2 , and h, the 2D cable can reveal structurally different patterns including stationary, striped, or hexagons. i. The stationary state is given by˜1 = ˜2 = ˜3 = 0, and is stable when µ < µ2 = 0 and unstable for µ > µ2 .

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ii. The striped pattern is given by ˜1 = µ/g1 ˝ 0, ˜2 = ˜3 = 0. The striped pattern is g1 h2 stable when µ > µ3 = and unstable for µ < µ3 . ( g 2 − g 1 )2 iii. The hexagonal patterns exist when

˜1 = ˜ 2 = ˜ 3 =

h ± h2 + 4 ( g1 + 2g 2 ) µ

with ˜ = 0 or ˜ and when µ > µ1 = −

2 ( g1 + 2g 2 )

.

h2 . 4 ( g1 + 2g 2 )

The hexagonal pattern H ˜ (when ϕ = π ) is stable when µ < µ 4 = H 0 (when ˜ = 0) is always unstable.

2g1 + g 2

( g 2 − g 1 )2

.h2 .

The authors have plotted the characterization of patterns of the coupled diffusive 2D M-L model, which are given in Figure 6.22. The boundaries of the emergence of various structures (hexagons and stripes) for different values of I are given in the following: (a) I = 0.052, (b) I = 0.054, and (c) I = 0.2. The dashed magenta line indicates the values of µ for (a) I = 0.052, (µ * = 27.4153), (b) I = 0.054, (μ* = 28.508), and (c) I = 0.2, (μ* = 108.2896). The thick black line indicates the condition for the existence of hexagons, whereas the thick blue and green lines indicate the boundary of the stability of stripes and H ˜ hexagons, respectively. Figure 6.22d gives the time integration of equation (6.50) for I = 0.2. 6.5.4 Spiking and Bursting in Single M-L Neuron Model Two-variable M-L neural model [109] described by equations (6.31a) and (6.31b) does not support bursting. The applied current can be taken as the third variable, which can be assumed to be slowly varying so that it is the slowest among the three system variables. The fast–slow system governing the variables (V , W ) and Z is given by the system of ordinary differential equations [74]: dV = Z − g Ca m˙ (V )(V − VCa ) − g KW (V − VK ) − g L (V − VL ) , dt

(6.51a)

dW wˆ (V ) − W = , dt ˜ w (V )

(6.51b)

dZ = − µ (V0 + V ) , dt

(6.51c)

where m (V ) = 0.5 ˆ˙1 + tanh ( t1 ) ˘ˇ , w (V ) = 0.5 ˙ˆ1 + tanh ( t2 ) ˇ˘ , −1

and ˜ w (V ) = ˆˇ° cosh ( 0.5t2 ) ˘ , t1 = (V − V1 ) V2 , t2 = (V − V3 ) V4 , ° = 1/3, V0 = 0.2. The model exhibits bursting. In the absence of diffusion, model system (6.51) presents a square-wave bursting pattern for the set of parameter values given below. The model consists of the voltage-gated Ca+2 current, voltage-gated delayed rectifer K + current, and

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FIGURE 6.22 Characterization of patterns of the diffusively coupled 2D M-L model. (Reproduced with permission from Mondal, A., et al. 2019. Diffusion dynamics of a conductance-based neuronal population. Phys. Rev. E 99(4), 042307, [106]; and American Physical Society. Copyright 2019.)

the leakage current. The variable V represents the membrane potential of the neuronal cell, W is the activation variable for the K + ion channels, and Z is the external input current which decays exponentially with a rate that is decided by both the current value of the membrane potential and factors external to the neural cell under study. The parameters g Ca , g K , and g L represent the maximum conductances corresponding to Ca+2 , K + , and leak currents respectively. VK and VL represent the reversal potential corresponding to the above ionic currents. The parameter ˜ represents the temperature scaling factor for the K + channel opening. The parameters V1 , V2 , V3 , and V4 are appropriately written in terms of the hyperbolic functions so that they can attain their equilibrium points instantaneously. µ lies in the interval (0, 1), [74,150]. Upadhyay and Mondal [143] studied the synchronization of bursting neurons with slowly varying dc current using identical as well as nonidentical coupled bursting Morris–Lecar neurons. Mondal et al. [106] considered the spatiotemporal form of the above system (6.51). The authors take the system with 1D diffusion as C

˜u ˜2u = I − g Ca mˇ ( u)( u − VCa ) − g K v ( u − VK ) − g L ( u − VL ) + D1 2 , ˜t ˜x

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Spatial Dynamics and Pattern Formation in Biological Populations

˜v = ° ( u) ˘V ( u) − v , ° ( u) = ˛ cosh ( 0.5t2 ) , ˜t V ( u) = 0.5 ˙ˆ1 + tanh ( t2 ) ˇ˘ , t2 = ( u − V3 )/V4 . For simulations, the authors assumed zero-fux boundary conditions. The parameter values are taken as C = 1, g Ca = 1.2, V1 = −0.01, V2 = 0.15, V3 = 0.1, V4 = 0.05, V0 = 0.2, g K = 2, g L = 0.5, VK = −0.7, VL = −0.5, µ = 0.005, ° = 1/ 3. For these values of the parameters, the nonzero equilibrium point is not locally asymptotically stable. Finite length of excitable cable was considered. The authors used the pdepe tools of MATLAB to solve the above system. The space and time step lengths are taken as 0.1 and 0.01, respectively. The system shows an irregular spike at low diffusion D1 = 0.026 (see Figure 6.23a), which eventually leads to a nonhomogeneous irregular pattern (Figure  6.23d). The twisted red lines in Figure 6.23d show the high amplitude of spikes. At a higher value of diffusion D1 = 0.4, the system shows irregular bursting and a more complex pattern is obtained (see Figure 6.23b). The red horizontal stripes show the weakly synchronized oscillations in the systems (see Figure 6.23e). For D1 = 0.7, the

FIGURE 6.23 Time series of the end oscillator and spatial plot of the improved 3D M-L cable with 1D diffusion. For fgures (a)  and (d): D1 = 0.026; for (b) and (e): D1 = 0.4; for (c) and (f): D1 = 0.07. (Reproduced with permission from Mondal,  A., et al. 2019. Diffusion dynamics of a conductance-based neuronal population. Phys. Rev. E 99(4), 042307, [106]; and American Physical Society. Copyright 2019.)

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system exhibits bursting (Figure 6.23c) and a wavelike spatiotemporal pattern (the nodes appear correlated to each other) is generated as plotted in Figure 6.23e. The authors have also considered the model with 2D diffusion by taking the term D1 (˜ 2 u/˜ x 2 + ˜ 2 u/˜ y 2 ) in the frst equation. Upadhyay et al. [145] investigated the dynamical behavior of two types of planner burster using fractional exponents. The integer and corresponding fractional-order M-L systems were analyzed using the dissipative condition and LEs. A fractional-order coupled 3D M-L system shows the variations in the fring patterns from resting state → oscillatory pattern → bursting and the synchronous behavior by designing a bidirectional coupling mechanism. Larter et al. [92] developed a coupled ODE lattice model for the CA3 region of the hippocampus for the simulation of epileptic seizures. The model consists of a hexagonal lattice of nodes, each describing a subnetwork consisting of a group of prototypical excitatory pyramidal cells with membrane potential V and a group of prototypical inhibitory interneurons with membrane potential Z interconnected via on/off excitatory and inhibitory synapses and both fed by current from the excitatory pathway (see Figure 1 of Larter et al., [92]). These two types of neurons are included in the subnetwork which forms the basis of the model. The growth of mean membrane potential of excitatory principal cells is controlled by that of the inhibitory interneurons. The dynamical behavior of this subnetwork was described by a system of three differential equations based on a two-variable reduction of the H-H model. To model the behavior of the entire subnetwork, Larter et al. [92] have added an equation to the original M-L model system to simulate the effect of a population of inhibitory interneurons synapsing on pyramidal cells. The model consists of an interconnected population of excitatory and inhibitory elements, both of which are fed by an external current. The system of equations governing the subnetwork model is given as dV = I − g Ca mˇ (V ) (V − 1) − g KW V − V K − g L V − V L + I − ˜ inh ( Z ) Z, dt

(6.52a)

w (V ) − W dZ =˜ ˇ , ° w (V ) dt

(6.52b)

dW = bcI + b˜ exc (V ) V. dt

(6.52c)

(

)

(

)

Rai et al. [120] reported the existence of phase coupled oscillations in electrical activity of the neuronal cells that carry together amplitude, phase, and time information for cellular signaling in the model developed by Larter et al. [92]. Nadar and Rai [110] numerically simulated this model and commented that chaos is vital for functioning of a healthy brain and synchronization of the neural system occurs when all its regions are in transient periodicity represented by chaotic saddles in state space. This is how the intermittent pathology of epileptic seizure is created. Tateno and Pakdaman [135] have studied the Morris–Lecar neural model driven by a white Gaussian noise current. Depending on parameter selections, the deterministic Morris–Lecar model can be considered as a canonical prototype for widely encountered classes of neuronal membranes, referred to as class I and class II membranes. This work examined how random perturbations affect class I and class II membranes bifurcation scenarios. Tusmoto et al. [140] have studied bifurcations in Morris–Lecar neuron model in a fve-dimensional parameter space and identifed

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an essential parameter of the half-activated potential of the potassium activation curve that contributes to the alternation of the membrane properties of the M-L neuron. Wang et al. [151] have investigated the bifurcations associated with the bursting oscillations in the modifed M-L neuron and synchronization transitions of two coupled identical bursters. Ditlevsen and Greenwood [38] showed that the Morris–Lecar neuron model embeds a leaky integrate-and-fre (LIF) model. A stochastic LIF model constructed with a radial Ornstein–Uhlenbeck (OU) process and fring mechanism of either logistic or exponential type has been shown to mimic the ISI statistics of a M-L neuron model. It captures subthreshold dynamics, not of the membrane potential alone, but of a combination of the membrane potential and ion channels. Shi and Wang [130] have studied the fractionalorder M-L neuron model with fast–slow variables. They showed that the fractional-order derivative can activate the slow potassium ion channel faster and play an important role to modulate the fring activity of the new model. Wu and Ma [156] have investigated a regular network of M-L neuron with the ion channels, and the potential mechanism of the formation of a spiral wave was studied in detail. Meier et al. [102] have presented a physically motived modifcation to excitable models based on the charge depletion which occurs in brain neurons. By incorporating this modifcation into the conventional Morris– Lecar reaction–diffusion model, they have explored several examples of the emergent complex behavior which have particular relevance to synchronization of neurons in the brain and the relevance of synchronization to diseases such as epilepsy. An electronic implementation for the Morris–Lecar neuron model was done by Hu et al. [72]. The circuit implementation of the Morris–Lecar neuron model presents a new way to study dynamics of the neuron in real time; namely, the designed circuit can be used as a simulation tool to investigate the fring patterns when the neuron has different parameters. Upadhyay and Mondal [146] have studied the mixed-mode oscillations and synchronous activity in noiseinduced modifed Morris–Lecar neural system. The Lyapunov spectrum was computed to present the nature of the system dynamics. Varying the noise intensity while keeping the predominant parameters of the model fxed, they observed the changes in the dynamical behavior of the system. Brandibur and Kaslik [17] have obtained necessary and suffcient conditions for the asymptotic stability of a two-dimensional incommensurate order linear autonomous system with one fractional-order derivative and one frst-order derivative. These theoretical results were successfully applied to investigate the equilibrium states of a fractional-order Morris–Lecar neuronal model. Upadhyay et al. [147] have studied parameter estimation technique of a 3D M-L model. The method depends on an algorithm where the least square method is used to estimate the biophysical parameters. The presence of noise was considered as an additive white noise. Song et al. [131] investigated the autapse-induced fring pattern transitions in the three classes of M-L neurons with excitatory or inhibitory autapses. Excitatory autapses enhance spiking, whereas inhibitory autapses suppress it.

6.6 Hindmarsh–Rose (H-R) Model The 3D Hindmarsh–Rose [65] neural model for a single neuron is a simplifed version of the Hodgkin–Huxley neural model and a modifed form of the FitzHugh–Nagumo neural model. It was originally proposed to model the synchronization of fring of two snail

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389

neurons in which the complicated current–voltage relationships of the conductance-based models are replaced by polynomials in the dynamical variables [118]. When the nonlinear time series analysis was applied to isolate neurons from the stomatogastric ganglion of the California spiny lobster Panulirus interruptus, it was observed that the number of degrees of freedom in their membrane potential oscillation ranges typically from 3 to 5 [1]. Based on this observation, models of the action potential activity in biological systems were developed using the H-R framework [65]. The action potential fring and its nature (cyclic or chaotic) depend on some key parameters of the model like the applied current I, which leads to the appearance of the neuronal activity via the instability of the equilibrium points [40]. Neuronal activity is a mixture of two phases: (i) rest phase when neuron does not emit any action potential and (ii) spiking or fast dynamic phase which is characterized by a repeated emission of action potential. These two phases lead to the bursting phase, the fast-repeated emission of action potential, separated by slow periods [40]. Models were proposed to describe the spiking and bursting behavior for an excitatory applied current and the fast–slow processing system. The models are of low dimension and are capable of producing many qualitative phenomena of spiking neurons. They can be decomposed into slow–fast systems and show regular bursting or chaotic bursting for certain values of system parameters [99,152]. Hindmarsh and Rose [65] proposed the following model to describe the neuronal bursting: x = y + ax 2 − x 3 − z + I,

(6.53a)

y = 1 − bx 2 − y,

(6.53b)

z = r ˙ˆ s ( x − x0 ) − z ˇ˘ .

(6.53c)

where the meanings of the variables and parameters are the following: x: membrane voltage, y: rate of transport of sodium and potassium ions passing through fast ion channels, z: rate of transport of other ions across slow ion channels, I: applied current stimulus which is also a control parameter of the system, a: governs bursting and spiking behavior, r: represents difference scale between fast and slow variables and controls the speed of variation of the slow variable z (Effciency of slow channels in exchanging ions. It controls the number of spikes per burst), s: measures the adaptation. The transport of potassium and sodium ions occurs through the fast ion channels, and  it  measures the variable y. The transport of the other ions is passed through the slow  channels, and it is measured by the variable z, and responsible for the bursting behavior of the system. It is a fast–slow system. This model shows the spiking and bursting  behavior  of the membrane voltage. The model is a natural extension of the FitzHugh model of nerve impulse (FitzHugh, 1961) and can be regarded in the same way as a qualitative representation of a further set of neural properties in phase

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Spatial Dynamics and Pattern Formation in Biological Populations

plane. Using stability analysis, Hindmarsh and Rose [65] showed that the model has three equilibrium points. One of them is a saddle point from which two separatrices form and divide the whole phase plane into two regions. In one region, all phase paths approach a limit cycle, and in the other, all the phase paths approach a stable equilibrium point. Consequently, a short depolarizing current pulse will change an initially silent model neuron into one that fres repetitively. The third equation of the model limits this fring to either an isolated burst or a depolarizing after potential. When steady depolarizing current was applied to this model, it resulted in periodic bursting. Djeundam et al. [40] set the value of the control parameter a = 3 and studied the deterministic and stochastic bifurcation structures of 3D H-R model with and without a random component and concluded that large noise always leads to the suppression of the neuronal activity (bursting or spiking). Duarte et al. [41] investigated the applicability and effectiveness of the step homotopy analysis method (SHAM) for fnding the accurate analytical solutions to the 3D H-R model. Bashkirtseva et al. [13] studied the order and chaos in a 2D stochastic H-R model of the neuron bursting and showed that noise-induced transitions, forming the bursting regime in H-R model, are accompanied by transformation between chaos and order. The authors have also investigated the phenomenon of noiseinduced bursting in a 2D stochastic H-R model and proposed the method of analysis of such phenomena based on the stochastic sensitivity technique and confdence domain method. Upadhyay et al. [144] applied the reconstruction method in the 3D stochastic H-R model to estimate the unknown time-varying parameters with multiplicative noise from the time series. Tsaneva-Atanasova et al. [139], and Mondal and Upadhyay [107] studied a modifed H-R model. Its structure is based on the 3D H-R model. The model consists of the following four coupled ODEs with a decaying injected current: du = − s − a1u3 + u2 − v − b1w + z = f1 ( u, v , w , z ) , dt

(6.54a)

dv = ˜ u 2 − v = f 2 ( u, v , w , z ) , dt

(6.54b)

dw = ˜ ( sa2 u + b2 − kw ) = f3 ( u, v , w , z ) , dt

(6.54c)

dz = − µ u = f 4 ( u, v , w , z ) , dt

(6.54d)

(

)

(

)

where the right-hand sides f1 , f2 , f3, f 4 are suffciently smooth functions. The variables u ( t ) , v ( t ), and w ( t ) respectively, are the membrane potential of the cell, the gating dynamics of the (K +) ion channels, and the dynamics of cytosolic (Ca+2 ) channels. The variable z ( t ) is the decaying injected current function. The meanings of the terms in equation (6.54a) are the following: f1 ( u, v , w , z ) is a cubic polynomial function which is an N-shaped u-nullcline [139] which measures the contribution of the Ca+2 ionic inward current; ( −v ) is the contribution of the outward voltage-sensitive ionic (K+) currents; and (−b1w) represents the contribution of the outward calcium-sensitive potassium current. The function f2 ( u, v , w , z ) represents the delayed rectifer activation kinetics. It depends on the membrane potential value. The meanings of the terms in equation (6.54c) are the following: f3 ( u, v , w , z ) is linear and represents the dynamics of Ca+2 ionic channel; ( sa2 u + b2 ) exchanges the source of

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391

calcium ion across the voltage-gated Ca+2 ion channels where b2 determines the location of the fxed point of the complete system and the parameter s < 0 controls the location and type of bifurcation in the system; and (−kw) is responsible for decay dynamics of cytosolic Ca+2 channels. In equation (6.54d), the parameter µ measures the rate constants of the changes in the applied current stimulus and it takes small values (0 < µ  1).s, ˜ , and b2 are the predominant parameters which vary in certain ranges. The model presents a fast– slow dynamical system [74,80,139]. The frst two equations represent the fast subsystem, and the remaining two equations represent the slow subsystem. In a biophysical neural system, the parameters ˜ and ˜ represent the rate constants that control the separation of time scale. Both the parameters ˜ and ˜ are such that the variables u and v stay on the faster time scale than w and z. These two parameters measure the time scale of changes in the ion channel concentrations across the cell membrane. Therefore, it was assumed that ˜ = 1 in the polynomial model and ˜ has a small positive value. The model system (6.54) has the equilibrium point u* , v * , w * , z* = ( 0, 0, b2 /k , b1b2 /k ) , which is a function of b2 (on which the nature of stability depends). The characteristic equation at the equilibrium point u* , v * , w * , z* is given by

(

(

)

)

P ( ˜ ) = c0 ˜ 4 + c1˜ 3 + c2 ˜ 2 + c3 ˜ + c 4 µ = 0, where c0 = 1 > 0, c1 = 1 + ˜ k > 0, c2 = µ + ˜ k + ˜ sa2b1 > 0, c3 = µ ( 1 + ° k ) + ° sa2b1 > 0, c 4 = ° kµ > 0. Since ˜ > 0, k > 0,b1 > 0, a2 < 0, s < 0, and 0 < µ  1. The coeffcients are positive and do not contain b2 . They depend on the predominant parameters s and ˜ . Suppose that ˜ is fxed and s is varied. Then, c1c2 − c0 c3 = k˜ ( 1 + k˜ + ˜ sa2b1 ) > 0, for all a2 < 0, s < 0. Also, c1c2 c3 − c12 c 4 − c0 c32 = k˜ 2 sa2b1 {1 + µ ( 1 + k˜ ) + ˜ ( k + sa2b1 } > 0, for all a2 < 0, s < 0. The Routh–Hurwitz criterion is satisfed for all a2 < 0, s < 0. The equilibrium solution is stable for all a2 < 0, s < 0. Tsaneva-Atanasova et al. [139] have chosen the parameter values as k = 0.2, a2 = −0.1, b1 = 1, s = −2.6. For the parameter set a1 = 0.5, a2 = −0.1, b1 = 1, µ = 0.00005, k = 0.2, s = −2.6, the system exhibits chaotic dynamics for ˜ = 0.66, b2 = −0.21 and a limit cycle for ˜ = 0.07, b2 = −0.01, for long time range. Mondal and Upadhyay [107] computed the corresponding Lyapunov exponents as

˜1 = 0.000461, ˜2 = −0.000265, ˜3 = −0.315384, and ˜ 4 = −0.972526; ˜1 = −0.000224, ˜2 = −0.002911, ˜3 = −0.131186, ˜ 4 = −1.130690. The chaotic and limit cycle attractors are presented in Figure 6.24 for long time range. Many authors studied the Hindmarsh–Rose model theoretically as well as experimentally. The studies of the H-R oscillator revealed various interesting dynamical features. To understand the wave propagation in coupled neuronal oscillators, Mondal et al. [108]

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FIGURE 6.24 Chaotic and Limit cycle attractors of the model system (6.54). (Reproduced with permission from Mondal, A., Upadhyay, R. K. 2017. Dynamics of a modifed Hindmarsh–Rose neural model with random perturbations: Moment analysis and fring activities. Phys. A Statist. Mech. Appl. 486, 144-160, [107]; and Elsevier. Copyright 2017.)

considered the reaction–diffusion system of a modifed H-R oscillator which consists zero as an equilibrium point. The authors have studied the spatial dynamics analytically to explore the wave profles of traveling wave solutions that typically arise in an excitable system. The waves were analyzed using the transformation ξ = x − ct, where c is the wave speed. The solitary traveling wave solution is stationary in the phase space with oscillatory and other patterns. 6.6.1 Formulation of a Modified H-R System Mondal et al. [108] examined the spatiotemporal dynamics of an excitable system by considering the applied current as a system variable which is varying at a slow rate. The cells interact with its medium and environment. The interaction of the H-R system [32,98,139] with an additional state variable is represented by the following set of nonlinear coupled ODEs [108]: du = v + ˜ ( u ) − w + z = f 1 ( u, v , w , z ) , dx

(6.55a)

dv = ˜ ( u) − v = f2 ( u, v , w , z ) , dx

(6.55b)

dw = r ˙ˆ s ( u − x0 ) − wˇ˘ = f3 ( u, v , w , z ) , dx

(6.55c)

dZ = − µ u = f 4 ( u, v , w , z ) . dx

(6.55d)

The meanings of the variables and parameters are the following: u: membrane voltage, v: transportation rate of sodium and potassium ions, w: transportation rate of other ions,

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z: applied current which decays at a slow rate, ˜ ( u) : assumed as ˜ ( u) = −u3 + au2, ˜ ( u) : assumed as ˜ ( u) = bu − du2, a: describes the bursting and spiking behavior, v: function of the parameters b and d, r: r (0 < r  1) is a small positive quantity. It controls the speed of variation of the variable w (the effciency in exchanging ions across slow channels). This parameter measures the number of spikes per burst. s: describes adaptation, µ: describes the time scale relationship between spiking and modulation activity and assumes a small value (0 < µ  1). x0 : the equilibrium of the two-dimensional H-R system that corresponds to the threshold potential to trigger bursts. The equilibrium point is given by u* = 0, v * = 0, w * = z* = − sx0 . The Jacobian of the model system evaluated at the u* , v * , w * , z* is given by

(

)

˝ 0 ˆ b J* = ˆ ˆ rs ˆ −µ ˙

1 −1 0 0

−1 0 −r 0

1 0 0 0

ˇ  .   ˘

The characteristic equation of J * is given by ˜ 4 + a1˜ 3 + a2 ˜ 2 + a3 ˜ + a4 = 0, where a1 = ( r + 1) > 0, a2 = ( r − b + rs + µ ) , a3 = ( rs − br + µr + µ ), and a4 = µr > 0. The system becomes degenerate at µ = 0, as the Jacobian becomes singular. The Routh–Hurwitz criterion for stability of the fxed point is a1 > 0, a2 > 0, a3 > 0, a4 > 0, A1 ( b ) = a1a2 − a3 > 0 and A2 ( b ) = a1a2 a3 − a12 a4 − a32 > 0. The control parameters are taken as b > 0, and r > 0. Now, a2 > 0, gives µ > b − r ( 1 + s ) ; a3 > 0, gives µ >

r (b − s) . r+1

(6.56a)

 r (b − s)  Hence, we obtain that µ > max  ˇ˘b − r ( 1 + s )  ,  > 0. r+1  

(6.56b)

A1 ( b ) = a1a2 − a3 = r 2 ( 1 + s ) + r − b > 0, gives b < r ˘ r ( 1 + s ) + 1. 

(6.56c)

A2 ( b ) = a1a2 a3 − a12 a4 − a32

(

)

(

(6.56d)

)

= µ ( r + 1) r 2 s − b + r ( s − b ) ˇ˘ r ( r + 1) + r 2 s − b  > 0.

The chosen values of the parameters should satisfy the conditions (6.56b)–(6.56d) so that the equilibrium point is stable. The authors tested the stability criterion using the particular set of parameter values: a = 3, d = 5, s = 4, x0 = 0, 0 < µ  1. The coeffcients a2 , a3 , and the minor A1 ( b ) fulfll the stability criteria for µ > r ( b − 4 )/( r + 1) = µ * , and b < r ( 1 + 5r ). From A2 ( b ) > 0, the stability criterion was obtained as µ > µ * 5r 2 + r − b / 4r 2 − b .

(

)(

)

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i. For fxed r = 0.008001, and μ = 0.00005 and 0.005, the equilibrium point is locally asymptotically stable for the ranges b < 0.008308 and b < 0.0072, respectively. It shows tonic spiking in the ranges 0.0083 ≤ b ≤ 2.2 and 0.0072 ≤ b ≤ 3.7, and periodic bursting for b > 2.2 and b > 3.7, respectively. (These results are obtained from simulation experiments done using MATCONT software, see also Figure 6.25). For (i) μ = 0.00005, and b > 2.2; and (ii) μ = 0.005, b > 3.7, the dynamics shows periodic bursting. ii. For fxed r = 0.001, and μ = 0.00005 and 0.005, the stability criterion is satisfed in the ranges b < 0.001, and b < 0.00045, respectively. It shows tonic spiking in the ranges 0.001 ≤ b ≤ 2.23, and 0.00045 ≤ b ≤ 3.41 and periodic bursting for b > 2.23, and b > 3.41, respectively. 6.6.2 Bifurcation Analysis The authors consider b as a predominant parameter with fxed r = 0.008001, and μ = 0.00005 with the other parameters taking values as in (i). The authors used the MATCONT software to plot the bifurcation scenarios. For b < 0.008308, the system has a stable focus node and a quiescent state exists. For b > 0.008308, the system frst shows tonic spiking up to b  =  2.2, and then, it shows periodic bursting with the increase in the value of the parameter b. The system has a saddle focus up to b = 0.406, and after that, the system has a saddle node. The quiescent state disappears through a supercritical Hopf bifurcation (HB1) at b = 0.008308. The authors used the following values of the parameters for numerical computations: a = 3, d = 5, s = 4, µ = 0.00005. In Figure 6.25a, the bifurcation diagram with respect to the parameter b, at r = 0.008001, is given. In Figure 6.25b, the bifurcation diagram with respect to the parameter r at b = 0.1 is given. In Figure 6.25a, the thick blue line describes the quiescent region and the dashed blue line describes the unstable region. A stable limit cycle (in thick green line) changes its stability to an unstable limit cycle shown in a dashed red line. Now, r is considered as the predominant parameter with fxed b = 0.1 and μ = 0.00005. The system has a saddle focus for r < 0.073213, where it shows tonic

FIGURE 6.25 Bifurcation diagram of the modifed H-R system. (Reproduced with permission from Mondal, A., et al. 2018. Dynamics of a modifed excitable neuron model: Diffusive instabilities and traveling wave solutions. Chaos Interdisc. J. Nonlinear Sci. 28(11), 113104, [108]; and American Institute of Physics. Copyright 2018.)

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spiking (hence fast spiking), and in this regime, the system is unstable. For higher values of r > 0.073213, the system has a stable focus node and there exists a quiescent state. The quiescent state appears through a supercritical Hopf bifurcation (HB1) for r = 0.073213. In Figure 6.25b, the thick blue line describes the quiescent state region and the dashed blue line describes the unstable region. A stable limit cycle (in thick green line) changes to an unstable limit cycle shown in a dashed red line. 6.6.3 Modified Reaction–Diffusion H-R System The propagation of traveling waves and its pattern formation in an excitable H-R system can be effectively explored by studying the reaction–diffusion equations. Mondal et al., [108] considered the following system:

˜u ˜2u = f1 ( u, v , w , z ) + Du 2 , ˜t ˜x

(6.57a)

˜v ˜2v = f2 ( u, v , w , z ) + Dv 2 , ˜t ˜x

(6.57b)

˜w ˜2w = f3 ( u, v , w , z ) + Dw , ˜t ˜ x2

(6.57c)

˜z ˜2z = f 4 ( u, v , w , z ) + Dz 2 . ˜t ˜x

(6.57d)

with initial conditions u ( x , 0 ) > 0, v ( x , 0 ) > 0, w ( x , 0 ) > 0, z ( x , 0 ) > 0, and Du , Dv , Dw and Dz are positive constant diffusion coeffcients. To investigate the stability of the uniform steady-state solution ( u0 , v0 , w0 , z0 ) , consider the perturbations as u ( x , t ) = u0 + ˙u ( x , t ) , v ( x , t ) = v0 + ˙v ( x , t ) , w ( x , t ) = w0 + ˙w ( x , t ) , z ( x , t ) = z0 + ˙z ( x , t ) . The perturbations ˛u ( x , t ) , ˛v ( x , t ) , ˛w ( x , t ) and ˛z ( x , t ) depend on the spatial domain, and are written as ˛u ( x , t ) = n1e ( ˙ t + ikx ) , ˛v ( x , t ) = n2 e ( ˙ t + ikx ) , ˛w ( x , t ) = n3 e ( ˙ t + ikx ) , ˛z ( x , t ) = n4 e ( ˙ t + ikx ) , where n1 , n2 , n3 , and n4 are suffciently small constants, ˜ is an eigenvalue, and k is the wave number. The linearized spatial system about the equilibrium point is obtained as ˙ 0 ˇ b J=ˇ ˇ rs ˇ −µ ˆ

1 −1 0 0

−1 0 −r 0

1 0 0 0

˙ D k2 + ° ˘ u ˇ  ˇ 0  −ˇ  ˇ 0  ˇ  0 ˆ

0

0

2

Dv k + °

0

0

Dw k 2 + °

0

0

˘  0   0  2 Dz k + °  0

(6.58)

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The characteristic equation is of the form

˜ 4 + m1˜ 3 + m2 ˜ 2 + m3 ˜ + m4 = 0.

(6.59)

where m1 = ( Du + Dv + Dw + Dz ) k 2 + r + 1, m2 = ( DuDz + Dw Dz + Dv Dz + DuDw + DuDv + Dv Dw ) k 4 + ˆˇ( Dz ( r + 1) + Du ( r + 1) + ,Dv r + Dw ) ˘ k 2 + r − b + rs + µ m3 = ( DuDw Dz + DuDv Dz + Dv Dw Dz + DuDv Dw ) k 6 + ˆˇ DuDz ( r + 1) + Dv Dz r + Dw Dz + DuDv r + DuDw ˘ k 4 + ˆˇ Dz r + Dur − b ( Dz + Dw ) + rs ( Dv + Dz ) + µ ( Dv +   Dw ) ˘ k 2 + r ( s + µ − b) + µ, m4  = DuDv Dw Dz k 8 + ( DuDv Dz r + DuDw Dz ) k 6 + ( DuDz r − Dw Dzb + Dv Dz rs + Dv Dw µ ) k 4 + [ Dz r( s − b) + Dv rµ + Dw µ ] k 2 + rµ.

(

)

The nonspatial system k 2 = 0 is stable if all the roots of the characteristic equation have

negative real parts, Re ( λ ) < 0. The steady-state solution becomes unstable in the presence of diffusion if at least one eigenvalue has Re ( λ ) > 0, with positive diffusion coefficients. The diffusion-driven instability (k 2 ≠ 0) occurs when λ = 0 and dλ/dk = 0. From these two equations,  the critical values of wave number and the predominant parameters can be obtained. The critical value of the wave number can also be obtained from the coefficient m4 as it is the product of the eigenvalues. The critical wave number can be derived from the equation dm4 /dη = 0, where k 2 = η. The wave numbers are the positive roots of dm4 /dη = 0. The derivative of m4 is given by f (˜ ) = 4A0˜ 3 + 3A1˜ 2 + 2A2˜ + A3 = 0,

(6.60)

where A0 = DuDv Dw Dz , A1 = ( DuDv Dz r + DuDw Dz ) , A2  = ( DuDz r − Dw Dz   b + Dv Dz   rs + Dv Dw µ ) , A3 = ( Dz r ( s − b ) + Dv rµ + Dw µ ) . Note that A0 > 0, and A1 > 0. The discriminant of the above cubic equation is

(

)

˛ 1 = 432 A0 A1 A2 A3 − A02 A32 − 108A13 A3 + 36A12 A22 − 128A0 A23 . If the discriminant ˜ 1 ° 0, then the cubic has only real roots. However, using the Descartes rule of signs of the coeffcients of (6.60), the authors found that the above cubic equation

Brain Dynamics: Neural Systems in Space and Time

397

does not have three positive real roots (for the above parameter values) when the nonspatial system is asymptotically stable. Now, f (˜ ) can be written as f (˜ ) = ˜ 3 + 3p1˜ 2 + 3p2˜ + p3 = 0,

(6.61)

where p1 = A1 4A0 , p2 = A2 6A0 , p3 = A3 4A0 , and A0 > 0. Note that p1 > 0. If p3 < 0, that is, A3 = ˇ˘ Dz r ( s − b ) + Dv rµ + Dw µ  < 0 or ( Dz rs + Dw µ + Dv rµ ) < rbDz , the cubic equation has one positive real root. When p3 ˜ 0, the cubic equation has one positive real root, if the polynomial f (˜ ) has a local minimum value for ˜ > 0. The authors verifed the occurrence of diffusive instability numerically and presented its graphical presentation (see Figure 6.26) for two sets of parameter values, where the equilibrium solution of the nonspatial system is stable. The parameter values chosen are a = 3, d = 5, s = 4,b = 0.005, and r = 0.008001. For µ = 0.00005, the occurrence of diffusive instabilities of the reaction–diffusion neural system (6.57) with the diffusion coeffcients (a) Du = 0.001, Dv = 0.05, Dw = 8 and Dz = 1, 2 and 5 (red, blue, and green curves) is given in Figure 6.26a; and for µ = 0.005, the occurrence of diffusive instabilities of the system with (b) Du = 0.001, Dv = 0.05, Dw = 8, and Dz = 7,10 and 11 (red, blue, and green curves) is given in Figure 6.26b. The diffusion coeffcients of the slow subsystem differ signifcantly from the fast subsystems [103,149]. When the spatial system exhibits diffusive instability, the authors obtained the critical values of the wave number k corresponding to the parameter sets r = 0.008001, b = 0.005, µ = 0.00005, and µ = 0.005 with the diffusion coeffcients Du = 0.001, Dv = 0.05, Dw = 8, and  Dz = 1, 2, 5 as kT = 0.7097, 1.5034, and kT = 0.7051, 1.5057, respectively. For Du = 0.001, Dv = 0.05, Dw = 8, and Dz = 7,10 and 11, the values of wave numbers were obtained as kT = 0.7879, 1.4616; kT = 0.7622, 1.4759, and kT = 0.7569, 1.4788, respectively. Next, the authors considered the case when the eigenvalues of the characteristic equation  (6.59) are complex. The nature of the roots mainly depends on the sign of the discriminant. The diffusive wave instability occurs when the real part of an eigenvalue becomes zero with k ˜ 0. Now, assume that the eigenvalues are complex with k ˜ 0.

FIGURE 6.26 The occurrence of diffusive instabilities of the reaction-diffusion neural system. (Reproduced with permission from Mondal, A. et al., 2018. Dynamics of a modifed excitable neuron model: Diffusive instabilities and traveling wave solutions. Chaos Interdisc. J. Nonlinear Sci. 28(11), 113104, [108]; and American Institute of Physics. Copyright 2018.)

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The  condition that equation (6.59) has at least one purely imaginary root with k ˜ 0 was derived as m32 + m12 m4 = m1m2 m3 , and ( m3 /m1 ) > 0.

(6.62)

Since m1 > 0, the second condition becomes m3 > 0. Equation (6.62) can be written as S ( k ) = B0 k 12 + B1k 10 + B2 k 8 + B3 k 6 + B4 k 4 + B5 k 2 + B6 = 0.

(6.63)

where m1 = a0 k 2 + a1 , m2 = b0 k 4 + b1k 2 + b2 , m3 = c0 k 6 + c1k 4 + c2 k 2 + c3 , m4 = d0 k 8 + d1k 6 + d2 k 4 + d3 k 2 + d4 . B0 = c02 + a02 d0 − a0b0 c0 , B1 = a02 d1 + 2c0 c1 + 2a0 a1d0 − a0b0 c1 − a0b1c0 − a1b0 c0 , B2 = c12 + 2c0 c2 + a02 d2 + a12 d0 + 2a0 a1d1 − a0b0 c2 − a0b1c1 − a0b2 c0 − a1b0 c1 − a1b0 c1 − a1b1c0 , B3 = 2c1c2 + 2c0 c3 + a02 d3 + a12 d1 + 2a0 a1d2 − a0b0 c3 − a0b1c2 − a0b2 c1 − a1b0 c2 − a1b1c1 − a1b2 c0 , B4 = c22 + 2c1c3 + a02 d4 + a12 d2 + 2a0 a1d3 − a0b1 c 3 − a0b2 c2 − a1b0 c3 − a1b1c2 − a1b2 c1 , B5 = 2c2 c3 + a12 d3 + 2a0 a1d4 − a0b2 c3 − a1b1c3 − a1b2 c2 , B6 = c32 + a12 d4 − a1b2 c3 . The authors [108] examined the instabilities numerically for nonzero values of the wave number. The critical values for the occurrence of wave instability are the roots of (6.63) with the condition m3 > 0 satisfed simultaneously. For numerical computations, the authors have taken the values of the parameters as were used in plotting Figure 6.26. In Figure 6.27a and b, the diffusive wave instabilities of the system (6.57) are presented for the values of the diffusive coeffcients Du = 0.001, Dv = 0.05, Dw = 0.7 and Dz = 11, 13, 14 (red, blue, and green curves) at µ = 0.00005. In Figure 6.27c and d, the diffusive wave instabilities are presented for the values of the diffusive coeffcients Du = 5, Dv = 1, Dw = 0.1, and Dz = 10 (blue curves) at µ = 0.005.(The colors are visible in e-book). 6.6.4 Construction of Traveling Front Solution The authors [108] showed that traveling pulses are obtained both when the system is in stable state and when the diffusive instability occurs. The authors used the following steps for constructing the traveling wave solutions: Step 1: Consider a general form of nonlinear equation as F °˛ p , pt , px , pxx ,…˝˙ = 0,

(6.64)

where the dependent variable p contains the components ui or ( u, v , w , z,) and the independent variable x contains the components ( x , y , t ,…) .

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FIGURE 6.27 Diffusive wave instabilities of the reaction-diffusive neural system (6.57). (Reproduced with permission from Mondal, A., et al. 2018. Dynamics of a modifed excitable neuron model: Diffusive instabilities and traveling wave solutions. Chaos Interdisc. J. Nonlinear Sci. 28(11), 113104, [108]; and American Institute of Physics. Copyright 2018.)

Step 2: To fnd a traveling or stationary wave solution, introduce the variable, ˜ = ( x − ct ) , where c measures the wave speed. The system of (6.57) reduces to a system of ODEs, since (˜ /˜ t ) = −c ( d/d° ) , (˜ /˜ x ) = ( d/d° ) , and ˜ 2 /˜ x 2 = d 2 /d° 2 and so on for other derivatives. Using these transformations, the PDE (6.64) changes to an ODE

(

F ˙ˆ p ( x ) , p˛ ( x ) , p˛˛ ( x ) ,…, p( n) ( x ) ˇ˘ = 0.

) (

)

(6.65)

Step 3: The solution has the form ui ( x ) = U i (T ), where T = tanh ( c1x + c2 y + c3 z + c4t ) = tanh ˜ . It can be derived that cos h 2˜ − sin h 2˜ = 1, (tanh ˜ )˝ = 1 − tan h 2˜ , and T˜ = 1 − T 2. Step 4: The localized wave solutions with wave speed c and wave number k are represented by the associated system variables u ( x , t ) = U (˜ ) , v ( x , t ) = V (˜ ) , w ( x , t ) = W (˜ ) , and z ( x , t ) = Z (˜ ) . The ODEs system can be simplifed by introducing the change of variable, T = tanh ( k˜ ). The terms in the ODE system are simplifed as we can write

(

)

(d/d˜ ) = k 1 − T 2 (d/dT ),

( d /d˜ ) = k (1 − T ) ˇ˘ −2T(d/dT ) + (1 − T )( d /dT ) . 2

2

2

2

2

2

2

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Spatial Dynamics and Pattern Formation in Biological Populations

It illustrates that a traveling wave has characteristic width k −1 . Generally, the wave number k has arbitrary values. However, it assumes appropriate fixed values in some solutions [73,153]. The solution of the system is assumed as a series of the form:  u( x, t )   v ( x, t )   w ( x, t )  z ( x, t ) 



  u ( kξ )     v ( kξ )  =   w ( kξ )     z ( kξ )

  U (T )     V (T )  =   W (T )     Z (T )

  ∑ a Tn 1n     ∑ a2 nT n  = n   ∑ a3 nT   ∑ a Tn 4n  

    , (6.66)   

where the summation is taken as n = 0,1,..., N , and N is a positive integer. The largest value of N can be determined by balancing the highest degree coefficients of T. From the works of earlier authors [63,101,153], the authors assumed N = 2 and obtained the analytic solution of the system with excitability in a closed form. Using Step 3, the system (6.57) is transformed to a system of ODEs as    −c      

′ U (ξ )   0   V (ξ )  b =  W (ξ )  rs   Z(ξ )   − µ

1 −1 0 0

−1 0 −r 0

1 0 0 0

      

U V W Z

  aU 2 − U 3   2  +  − dU 0     0 

     +      

′′ DuU (ξ )   DvV (ξ )  . (6.67) DwW (ξ )   Dz Z(ξ ) 

Using Step 4 [63,153], the system (6.67) is transformed to the system   2  − ck(1 − T )      

′ U (T )   0   V (T )  b =  rs W (T )   Z(T )   − µ

1 −1 0 0

−1 0 −r 0

  − 2T (1 − T 2 )k 2   

1 0 0 0

      

DuU (T ) DvV (T ) DwW (T ) Dz Z(T )

U V W Z

  aU 2 − U 3   2  +  − dU 0     0 

′     + k 2 (1 − T 2 )2       

     

DuU (T ) DvV (T ) DwW (T ) Dz Z(T )

′′    . (6.68)   

The boundary conditions are taken as X (ξ ) → 0, as ξ → +∞ or −∞ where X ≡ X (U , V , W , Z ) . This implies that S(T ) ≡ (U , V , W , Z )  → 0 as T → 1 or −1, [33,34]. The expressions for the wave  speed were derived using the above boundary conditions, and substituting the a­ symptotic values, X (ξ ) ≈ exp ( −2ξ ) as ξ → +∞, where X ≡ X (U , V , W , Z ) into system (6.67). The wave speed equations are of the parabolic type. The speed of the wave corresponding to the four components (u, v, w, z) was obtained as 2 c = 1 + 4Du k 2 , 2 c = b − 1 + 4Dv k 2 , 2 c = r ( s − 1) + 4Dw k 2 , and 2 c = − µ + 4Dz k 2 , which becomes constant in the absence of diffusion. Substituting the series solutions as given in (6.66), in the transformed system (6.68), the following system is obtained:

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Brain Dynamics: Neural Systems in Space and Time

(

)

(

)

− ck 1 − T 2 ( a11 + 2a12T ) = a20 + a21T + a22T 2 − (a10 + a11T + a12T 2 )3

(

+ a( a10 + a11T + a12T 2 )2 − a30 + a31T + a32T 2

(

+ a40 + a41T + a42T 2

(

)

)

)

(

)

+ 2Du k 2 1 − T 2 ˆˇ −T ( a11 + 2a12T ) + a12 1 − T 2 ˘ ,

(

)

(

) (

(

)

− ck 1 − T 2 ( a21 + 2a22T ) = b a10 + a11T + a12T 2 − d a10 + a11T + a12T 2

(

)

2

)

(

)

− a20 + a21T + a22T 2 + 2Dv k 2 1 − T 2 ˆˇ −T ( a21 + 2a22T ) + a22 1 − T 2 ˘ ,

(

)

(

) (

−ck 1 − T 2 ( a31 + 2a32T ) = rs a10 + a11T + a12T 2 − r a30 + a31T + a32T 2

(

)

)

(

)

(

)

+ 2Dw k 2 1 − T 2 ˆˇ −T ( a31 + 2a32T ) + a32 1 − T 2 ˘ ,

(

)

(

−ck 1 − T 2 ( a41 + 2a42T ) = − µ a10 + a11T + a12T 2

(

)

)

+ 2Dz k 2 1 − T 2 ˇ˘ −T ( a41 + 2a42T ) + a42 1 − T 2  . Equating the powers of T on both sides of the above system, the following systems are obtained: Power of T 0 : 3 2 −cka11 = a20 − a10 + aa10 − a30 + a40 + 2Du k 2 a12 , 2 −cka21 = ba10 − da10 − a20 + 2Dv k 2 a22 ,

−cka31 = rsa10 − ra30 + 2Dw k 2 a32 , −cka41 = − µ a10 + 2Dz k 2 a42 . Power of T 1 : 2 −2cka12 = a21 − 3a10 a11 + 2 aa10 a11 − a31 + a41 − 2 Du k 2 a11 ,

−2cka22 = ba11 − 2da10 a11 − a21 − 2Dv k 2 a21 , −2cka32 = rsa11 − ra31 − 2Dw k 2 a31 , −2cka42 = − µ a11 − 2Dz k 2 a41 . Power of T 2 :

(

) (

)

2 2 2 cka11 = a22 − 3a10 a11 + 3a10 a12 + a a11 + 2a10 a12 − a32 + a42 − 8Du k 2 a12 ,

(

)

2 cka21 = ba12 − d a11 + 2a10 a12 − a22 − 8Dv k 2 a22 ,

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Spatial Dynamics and Pattern Formation in Biological Populations

cka31 = rsa12 − ra32 − 8Dw k 2 a32 , cka41 = − µ a12 − 8Dz k 2 a42 . Power of T 3 : 2cka12 = − ( a11 + 6a10 a11a12 ) + 2 aa11a12 + 2Duc12 a11 , 2cka22 = −2da11a12 + 2Dv k 2 a21 , 2cka32 = 2Dw k 2 a31 , 2cka42 = 2Dz k 2 a41 . From the above equations, the values of a31 , a41 are obtained in terms of a32 , a42 which are assumed as free variables. All the remaining coeffcients are also obtained in terms of a32 and a42. The required values are the following: a31 = ( c/Dw k ) a32 , a41 = ( c/Dz k ) a42 , a11 = ( c/Dw ks ) a32 , a12 = ( 1/rs ) ˙ˆ c 2 /Dw + r + 8Dw k 2 ˇ˘ a32 ,

(

)

2 2 2 a11 = ( 1/µ ) ˆ2c ˇ k − 2Dz k ( c/Dz k ) ˘ a42 , a12 = − ( 1/µ ) ˆˇ c /Dz + 8Dz k ˘ a42 .

The following equations give the values of a10 and a22 in terms of the coeffcients a11 and a12 .

(

)

a10 = ( 1/6 a11a22 ) −a11 + 2aa11a12 + 2Du k 2 a11 − 2cka12 ,

{

(

}

)

(

)

2 + 2a10 a12 − ( cd/Dv k ) a11a12  . a22 = 1/ ˇ˘ c 2 /Dv + 1 + 8Dv k 2  ˇ˘ba12 − d a11

Now, a21 is given by

(

)

a21 = ˙ˆ1/ 1 + 2 Dv k 2 ˇ˘ ( ba11 − 2da10 a11 + 2cka22 ) . Substituting the values of a10 , a11 , a12 , and a22 from previous expressions, a21 is expressed in terms of a32 and a42. The remaining coeffcients are derived from the values of a32 and a42 . The expressions for the remaining coeffcients are the following: 2 a20 = ba10 − da10 + 2Dv k 2 a22 + ( c/Dv k )( cka22 + da11a22 ) ,

(

(

) )

a30 = ( 1/r ) rsa10 + 2Dw k 2 a32 + c 2 /Dw a32 ,

(

)

3 2 a40 = −a20 + a10 − aa10 − cka11 + a30 − 2Du k 2 a12 .

Substituting in the series solution, the solution of the spatial variables for traveling wave profles are obtained as the following:

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Brain Dynamics: Neural Systems in Space and Time

(

u ( x,t ) = ( 1/6 a11a22 ) − a11 + 2aa11a12 + 2Du k 2 a11 − 2cka12

(

)

)

+ ( c/Dw ks ) a32T + ( 1/rs ) ˆˇ c 2 /Dw + r + 8Dw k 2 ˘ a32T 2 , 2 + 2Dv k 2 a22 + ( c/Dv k )( cka22 + da11a12 ) v ( x , t ) = ba10 − da10

(

)

+ ˘1/ 1 + 2 Dv k 2  ( ba11 − 2da10 a11 + 2cka22 ) T

{

(

)

}

(

)

2 + 1/ ˘ c 2 /Dv + 1 + 8Dv k 2  ˘ba12 − d a11 + 2a10 a12 − ( cd/Dv k ) a11a12  T 2 ,

(

)

w ( x,t ) = ( 1/r ) ˙ˆ rsa10 + 2Dw k 2 a32 + c 2 /Dw a32 ˇ˘ + ( c/Dw k ) a32T + a32T 2 ,

(

)

3 2 z ( x , t ) = −a20 + a10 − aa10 − cka11 + a30 − 2Du k 2 a12 + ( c/Dz k ) a42T + a42T 2 .

6.6.4.1 Numerical Results The coeffcients of the traveling wave solutions (6.66) are expressed in terms of a32 and a42, as these coeffcients are considered as free values. Small appropriate values of the free coeffcients are to be chosen for numerical simulation. The wave numbers also take small positive values. For numerical computations, the authors have taken the values of the parameters as were used in plotting Figure 6.26. The remaining parameter set was taken as r = 0.008001, µ = 0.005, b = 0.005 and 0.008. The speed diagrams are derived when diffusive instability occurs in the spatial system for suitable values of diffusion coeffcients. The speed diagrams are plotted in Figure 6.28a and b with respect to the wave numbers. For a chosen set of values of the diffusion coeffcients, when the diffusive Turing instability sets in for a fxed parameter set, the value of the wave number k increases with respect to the excitation variable b. The speeds of the waves increase with the wave numbers. However, the rate of change is different and it depends on the values of the diffusion coeffcients. The speed diagrams have the same type of features for the same diffusion coeffcients. The plots of wave speed ( c ) vs wave number ( k ) , when the values of diffusion coeffcients are taken as (a) Du = 0.001, Dv = 0.05, Dw = 8, Dz = 1 and (b) Du = 5, Dv = 1, Dw = 0.1, Dz = 10, are given in Figure 6.28a and b respectively. At these values, diffusive instability occurs for the system (6.57). To observe the exact solitary traveling wave solution analytically [16,42], the diffusion coeffcients are all set as 1. Assuming Du = 1, Dv = 1, Dw = 1, and Dz = 1, r = 0.001, µ = 0.00005 and b = 0.0008 with the remaining parameters being the same as given in Figure 6.25, traveling wave profles are plotted in Figure 6.29. It represents the wave profles of the system variables u, v , w and z with the transformed functional variable ˜ . The values of a32 , a42 were assumed as a32 = a42 = 0.1 with a = 3, d = 5, s = 4, µ = 0.00005, r = 0.001 and b = 0.0008, to observe the solitary wavy fronts of all the spatial variables. At these parameter sets, the deterministic system goes to the oscillatory region (square wave bursting) from the quiescent stable (stable focus node). As the deterministic system generates oscillatory behavior from the excitable regime at these parameter sets, its behavior was studied using larger/increasing values of b. The wave shapes remain invariant in time with only changes in their amplitudes. It was observed that the modifed H-R single neuron model exhibits spiking bursting activity. It was assumed that diffusion is responsible for the spatial spreading behavior of

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FIGURE 6.28 Wave speed c vs wave number k. (Reproduced with permission from Mondal, A., et al. 2018. Dynamics of a modifed excitable neuron model: Diffusive instabilities and traveling wave solutions. Chaos Interdisc. J. Nonlinear Sci. 28(11), 113104, [108]; and American Institute of Physics. Copyright 2018.)

FIGURE 6.29 Traveling wave solutions. (Reproduced with permission from Mondal, A., et al. 2018. Dynamics of a modifed excitable neuron model: Diffusive instabilities and traveling wave solutions. Chaos Interdisc. J. Nonlinear Sci. 28(11), 113104, [108]; and American Institute of Physics. Copyright 2018.)

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the membrane voltage due to the semiconductor behavior of the cell membrane which is caused by the ionic-conductance and various ionic-channels of the cell membrane. The diffusion driven instabilities and propagation of wave profiles in an excitable medium have important applications.

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Solutions to Odd-Numbered Problems

Exercise 3 3.1

Derive the conditions for cross-diffusion-induced Turing patterns to emerge in the spatial epidemic model (3.9). Solution: The endemic equilibrium point of the model is S* ,  I * , 1/ p 1/ p 1/ p  d˝˜ˇ  ˝˜ˇ ˛˜ˆ where S* = K 1 − ˆ   , I * = ˆ  S* , with r > d ˙ ˘ . ˙ d˘ ˝ dˇ r ˙ d ˘  

(

(

)

)

The elements of the Jacobian matrix J at S* ,  I * are given by 2S ˇ ˝ Sˇ ˝ fS = r ˆ 1 −  − ˜ p ˆ  ˙ ˙ I˘ K˘

p−1

p

˝ Sˇ ˝ Sˇ ,  f I = − ˜ q ˆ  ,  gS = ˜ p ˆ  ˙ I˘ ˙ I˘

p−1

p

˝ Sˇ ,  g I = ˜ q ˆ  − d. ˙ I˘

Using the eigenvalue analysis, the Jacobian of the system is obtained as ˙ J (k) = ˇ ˇ ˆ

fS − DS k 2

f I − Dk 2

gS

g I − DI k 2

˘ .  

The characteristic equation is given by ˜ 2 − ˜T + ˝ = 0, where T = trace ( J ) = fS + g I − k 2 ( DS + DI ) , ˛   = det ( J ) = fS g I − f I g s − k 2 ( fSDI + g I DS − gSD ) + k 4 ( DSDI ) . The system is stable without diffusion, implying that 2S ˘ ˙ S˘ ˙ fS + g I = r ˇ 1 −  − d + ˜ ˇ  ˆ I ˆ  K

p−1

 ˙ S˘   q ˇˆ I  − p  < 0.  

Hence, T < 0. Irrespective of the sign of T, one of the eigenvalues is positive when ˜ < 0. Therefore, Turing instability occurs when ˜ < 0. Now, y = ˝ k 2 is a parabola which attains its minimum value for

( )

˝ fSDI + g I DS − gSD ˆˇ kc2 = ˙ = 2DSDI

fS g I − f I g s − g S D . DSDI

413

414

Solutions to Odd-Numbered Problems

This determines the critical wave number (the wave number of the modes that frst becomes unstable). For k 2 = kc2 , we have

( )

˛ kc2 = fS g I − f I g s −

( fSDI + gI DS − gSD)2 . 4 ( DSDI )

( )

˛ kc2 < 0 gives the condition ( fSDI + g I DS − gSD ) > 4DSDI ( fS g I − f I g s ). 2

Together with the condition that k is real, this inequality determines the Turing space. Hence, the conditions for yielding cross-diffusion-induced Turing patterns are given by (i) fS + g I < 0, (ii) ( fS g I − f I g s ) > 0, (iii) ( fSDI + g I DS − gSD ) > 0, (iv) ( fSDI + g I DS − gSD) > 4DSDI ( fS g I − f I g s ). 2

(

)

3.3 Determine the conditions such that the endemic equilibrium point S* ,  I * of the model (3.36) is asymptotically stable. Solution: The equilibrium point S* , I * is locally asymptotically stable if the zeros of the characteristic equation ˜ 2 − trace ( J ) ˜ + det( J ) = 0 have negative real parts. The suffcient conditions are trace ( J ) < 0 and det ( J ) > 0. The following results are obtained:

(

)

p = 1: trace ( J ) = a11 + a22 = − AA1q + a − b , det ( J ) = a11a22 − a21a12 = − ( −A1 + a + ˜ ) ( AA1q ) . If AA1q + b > a, then trace ( J ) < 0; and if ˜ + b > a, then det ( J ) > 0. When the two conditions are satisfed, the unique endemic equilibrium point S* ,  I * is asymptotically stable.

(

)

p > 1: trace ( J ) = a11 + a22 = ( −AA1q + a − b ) + ( p − 1) A1 = A1 ( p − 1 − Aq ) + a − b. det ( J ) = a11a22 − a21a12 = ( − AA1q + a − b )( p − 1) A1 − AA1q ( − pA1 + a + ˜ ) = A1 ˘ Aq (° + b − a ) + ( p − 1) ( a − b )  . If trace ( J ) < 0, and det ( J ) > 0, then the characteristic equation has negative real parts and the equilibrium point S* ,  I * is asymptotically stable. If det ( J ) < 0, then the equilibrium point S* ,  I * is unstable, irrespective of the sign of trace ( J ) . 3.5 Derive the conditions under which Turing instabilities occur in the system (3.70a)–(3.70b). Solution: To study the conditions under which Turing instabilities occur in the system, one studies how perturbation of a homogeneous steady-state solution behaves in the long term. Using the eigenvalue analysis, the Jacobian of the system (3.70a) and (3.70b) is obtained as

(

)

(

)

415

Solutions to Odd-Numbered Problems

˛ J(k ) = ˙ ˙ ˝

fS − DS k 2

fI

gS

g I − DI k 2

ˆ ˘, ˘ ˇ

where fS = − d − ˜ I 2 , f I = − ˜ S2 , gS = ˜ I 2 , g I = − ( d + ˜ ) + ° S2 , and k 2 = k x2 + k y2 are wave numbers. The characteristic equation is given by µ 2 − Tµ + ˝ = 0, where T = trace ( J ) = fS + g I − k 2 ( DS + DI ) , ˛   = det ( J ) = fS g I − f I g s − k 2 ( fSDI + g I DS ) + k 4 ( DSDI ) . The conditions for yielding Turing patterns are given by (see equation 3.8) (i) fS + g I < 0, (ii) ( fS g I − f I g s ) > 0, (iii) ( fSDI + g I DS ) > 0, (iv) ( fSDI + g I DS ) > 4DSDI ( fS g I − f I g s ) . 2

3.7 Find the basic reproduction number of the model (3.90). Solution: Defne  = The rate at which secondary infections increase (newly infected terms) ˙ ˜ E° N + I° N  = ˇ 0 ˇˆ

˘ . 

V = The rate at which disease progresses (death and recovery decrease; i.e., the total outfow from infected compartments E and I) ˇ ˜E = ˜ − E − °I) ( ˘

F|DFE

˙ ˜ 1  ˜      1 ˇ ˜ ˜I E =ˇ ˇ ˜ 2  ˜ 2      ˇ ˜I ˆ ˜E

V

−1

ˆ 1 ˜       0 =˘ ˘ˇ 1 °      1 °

  

˘  ˙ °˛ N       ˛ N ˘ ˙ ˜  =ˇ  ;  V|DFE = ˇ  0         0  ˆ −˜ ˆˇ  DFE ˆ ˛N  ˝ 1  ˘ ˛ N  +          −1  , K L = FV = ˘ ˜ °  °  ˘ 0                     0 ˇ

0 ˘ . ° 

  .  

Since the rank of K L is one, it has only one nonzero eigenvalue. The basic repro˘ ˛ 1 duction number is 0 = ˜( K L ) = ° N  +  . ˝ ˙ 

(

)

3.9 Show the existence of an endemic equilibrium point P* S* ,  E* ,  I * , R* for the model (3.113).

416

Solutions to Odd-Numbered Problems

Solution: Denote t1 = ˜ + ° + r, t2 = ˛ + ˝ + r. From the third equation, we obtain (˜° + ˛ t2 )  I * − ˛˝ I *2 . ˜ E* = ˙ˆt2 I * − ° I *2 ˇ˘ . From the fourth equation, we obtain R* = ˜ ( r + ˙ ) − ˝ I *  From the second equation, we obtain S* =

(t

1

)

− ˜ I * E*

°I

*

. Substituting in the frst

equation and simplifying, we get

{

(

)(

)

( r + ˜ ) − °   I *  (° − ˛ )  I * − r  t1 − °   I * t2   I * − °   I *2 + ˝˛   I * r

}

+ ˜˛   I *2 (˝˙ + ˆ t2 ) − ˆ°   I *  = 0. This equation is of ffth-degree in I * . The equation has at least one real positive root. Substituting in the earlier equations, we get E* ,  R * and S* .

Exercise 4 4.1

Show that the endemic-free state EFE (N, 0, 0, 0, 0) of the model (4.11) is always unstable as long as ˜ is positive. Solution: The Jacobian matrix of the system (4.11) at (N, 0, 0, 0, 0) is given by ° ˝ ˝ J=˝ ˝ ˝ ˝ ˛

0 0 0 0 0

a12 a22 a32 0 a52

0 0 0 0 0

a13 a23 a33 a43 a53

a15 a25 0 0 0

˙ ˇ ˇ ˇ, ˇ ˇ ˇ ˆ

where a12 = − pˆ L + ( p − 1) ˜ , a13 = − p˛ I , a15 = −˜ , a22 = pˇ L − ( p − 1) ˜ − ° − ˛ , a23 = p° I , a25 = ˜ , a32 = ˜ , a33 = − (˜ + ° ) , a43 = ˜ , a52 = ˜ ,  a53 = ° . Two eigenvalues of J are 0, 0 and the other three eigenvalues are the roots of the equation, ˜ 3 + c1˜ 2 + c2 ˜ − c3 = 0, where c1 =  − p L + ˜ ( p − 1) + ° + ˛ + ˝ + ˙ , c2 =  − p L + ˜ ( p − 1)  (° + ˛ ) + ˝ (° + ˛ − p I ) + ˙ (° + ˛ − ˆ ) , and c3 = ˜ (°˛ + ˝˙ + °˙ ) > 0. For ˜ > 0,  the equation has at least one positive root showing that the EFE is always unstable.

(

)

4.3 Obtain the endemic equilibrium point E1 S* ,  E* ,  I * ,  R* ,  H * ,  D* , of the temporal model of the system (4.18). Solution: From the third equation, we get E* = ( t1 ˜ ) I * , t1 = ° + ° h + µ.

417

Solutions to Odd-Numbered Problems

˜h . d1 + r1 1 From the sixth equation, we get D* = t3 I * , t3 = ( d1t2 + f° ) . ˜ 1 From the fourth equation, we get R* = t4 I * , t4 = (1 − f )° + r1t2 . µ From the ffth equation, we get H * = t2 I * , t2 =

(

)

˜ i I * + ˜ h H * + ˜ d D* = t5 I * , t5 = ˜ i + ˜ ht2 + ˜ dt3 . N = S + t6 I * , t6 = 1 + ( t1 ˜ ) + t2 + t3 + t4 . Substituting in the second equation, canceling   I *, and simplifying, we get S* = t7 I * , t7 =

(˜ + µ ) t1t6 . ˜ t5 − (˜ + µ ) t1

Substituting in the frst equation, canceling I * , and simplifying, we get I* =

A ( t6 + t7 ) . µt7 ( t6 + t7 ) + t5t7

This gives the endemic equilibrium point. 4.5 Establish the linear stability of the endemic equilibrium point for the model system (4.19). Solution: Endemic equilibrium point E = S , E , I a , I s , O, R ,  D is given by

(

)

O = ˜ 1 Is , D = ˜ 2 Is , R = ˜ 6 Is , Ia = ˜ 3˜ 5 Is , E = (Q2 P° ) I a = ˜ 5 Is , Is = ˇ ( 0 − 1) ˜ * , S = ˇ° 2 ˜ * , ˜ * = Q1˜ 5 ( 0 − 1) + µ° 2 .

˜ 1 and other terms are defned in the text. For studying the local asymptotic stability, the authors have used a method based on the central manifold theory (Castillo-Chavez and Song [23], see Theorem 4.1). The bifurcation parameter is taken as ˜ = ° c. From (4.20), we obtain for 0 = 1, ˜ = ˜c =

°Q1Q2Q3Q4 . ˛ t1t2

The Jacobian matrix J ac of the system (4.19), linearized about ( E0 , ˜ c ) , is given by  −µ  0   0 J ac =  0   0  0   0

0 −Q1 p˝

0 0 −Q2

−° c °c 0

0 0 0

0 0 0

−˛° c ˛° c 0

0 0 0

˙ 0 ˇ1 0

−Q3 ˆ ˇ2 ˘1

0 − Q4 ˇ3 ˘2

0 0 −µ 0

0 0 0 −

( 1 − p )˝





.





418

Solutions to Odd-Numbered Problems

Zero is a simple eigenvalue of J ac . The corresponding right and the left eigenvecT tors are w = ( w1 , w2 ,  w3 ,  w4 ,  w5 , w6 , w7 ) and v = (ν 1 , ν 2 , ν 3 , ν 4 , ν 5 , ν 6 ,  v7 ) , satisfying v ⋅ w = 1. The components of the eigenvector w are given by w1 = −

pQ3Q4 Q Q1Q2Q3Q4 QQQ , w2 = 2 3 4 , w3 = , w4 = 4 , γµω t1 γω t1 ω t1 ω

w5 = 1,  w6 =

1  pQ3Q4δ 1 Q σ + ωσ 2  . + Q4δ 2 + ωδ 3  , w7 = 4 1  µω  ρω t1

The components of the eigenvector v are given by

ν 1 = 0, ν 2 = ν 5 =  

γρω t1 θρω Q1 ρω Q1Q2 , ν 3 = , ν 4 = , ξ ξ ξ ξ + + ( 5 6) ( 5 6) (ξ 5 + ξ6 )

ηρω Q1Q2Q3Q4t1 ηρω Q1Q2Q3t1σ 2 , ν 6 = 0, ν 7 = , ρQ2 (ξ 3 + ξ 4 ) ρQ2 (ξ 3 + ξ 4 )

(

)

where ξ 3 = t1Q3Q4 Q4 ( ρ + ησ 1 ) + ηωσ 2 = t1t2Q3Q4 , pθ Q3   ξ 4 = Q1 (ψ 7 + t1ψ 8 ) , ξ 5 = ρQ1Q4  Q2 + ,  t1 

ξ6 =

( (

)

Q2Q3  Q4 Q4 ρ 2 + η ( ρ + Q1 )σ 1 + ηρωσ 2 ( 1 + Q1 ) + ηρω Q1σ 2  .  t2 

)

v and w satisfy the equalities v ⋅ J 0 = 0 and J 0 ⋅ w = 0. For the above system, the coefficients a and b are defined as 7

7

∑

∑

∂2 f k ∂2 f k a= v k wi w j (E0 , ϕ c ) ,  b = vk wi (E0 , ϕ c ) , ∂ xi ∂ x j ∂ xi ∂ β k ,  i ,  j= 1 k ,  i = 1 which are explicitly computed as a=

−2Q1Q2ψ 7ψ 9 γ t1t2 > 0, < 0, b = pθρωγπ (  ξ 3 ρQ2 + Q1Q2t1ψ 8 + ψ 7 ) ξ ( 5 + ξ6 )

where

ψ 7 = pθ Q3Q4 (Q4 ρ + η (σ 1 + ωσ 2 )) ρ , ψ 8 = Q42 ( ρ 2 + η ( ρ + Q3 )σ 1 + ηωσ 2 ( ρ (Q3 + Q4 ) + Q3Q4 ) , 

(

(

)

ψ 9 = Q2 Q4γ ( ρ (δ 2 + µ ) + µσ 1 ) pθ + ( 1 − p ) µρQ3 + pγ Q4 ρQ3 ( µ + δ 1 )

(

)(

))

+ ωγ ρ (δ 3 + µ ) + µσ 2 pθ + ( 1 − p ) Q4  . Since a < 0 and b > 0, at ϕ = ϕ c , the endemic equilibrium is locally asymptotically stable for 0 > 1.

419

Solutions to Odd-Numbered Problems

4.7 Discuss the asymptotic stability of the disease-free equilibrium point E0 of system (4.33). Find the conditions under which the endemic equilibrium is asymptotically stable. Also, plot the time series for the following parameter set ˙ = 600, c = 0.05, d = 0.5,  µ = 0.00047, ° = 0.23, ˛ = 0.5, ˝ = 5.0. Solution: At the DFE point E0 ( ˝ µ , 0, 0, 0, 0 ), the Jacobian matrix JE0 is given by ˘ −µ   0  JE0 =  0  0   0 

0

−c

0

−d

−µ

c − ce− µ e− 

0

d − de− µ e− 

0 0

−t1 + ce− µ e−  °

0 −µ

de− µ e−  0

0

µ +˛

0

−˝

    .    

The characteristic equation is obtained as ˆ ( ˜ , ° ) = (˜ + µ )3 h ( ˜ , ° ) = 0, where h ( ˜ , ° ) = ( ˜ + t1 ) ( ˜ + ˛ ) − ( c˜ + t2 ) e − ( ˜° + µ° ) . ˜ = − µ is a negative root. Consider the equation h ( ˜ , ° ) = 0. Now, h ( 0, ˜ ) = ° t1 − t2 e − µ˜ = ° t1 ( 1 − 0 ) . When 0 > 1, h ( 0, ˜ ) < 0. Since h ( ˜ , ° ) is continuous with respect to ˜ and lim h ( ˜ , ° ) = +ˇ , there exists ˜ > 0 such that h ˜ , ° = 0. Therefore,

(

˜ ˘ˇ

)

E0 ( ˝ µ , 0, 0, 0, 0 ) is unstable for 0 > 1. Now, when 0 < 1, we need to show that all the eigenvalues of h ( ˜ , ° ) = 0 have negative real parts. From

(

)

h ( ˜ , ° ) = ( ˜ + t1 ) ( ˜ + ˛ ) − c˜ e − µ° + t2 e − µ° e − ˜° = 0,

   ce − µ° ˘ ˜  ˘˜ ˘˜ we obtain h ( ˜ , ° ) =  + 1  + 1 −  + 0 e − ˜° = 0,    t1 ˛   t1  ˛  

 ce − µ ˝ ˜ ˇ  − ˜  ˇ  ce − µ ˝ ˜ ˇ ˝˜ ˇ˝˜ e =  or ˆ + 1 ˆ + 1 =   + 1 e − ˜ . + 0 0   ˆ  ˆ  ˘˙° ˙ t1 ˘  t1 ˙ ° ˘   0t1 ˙ ° ˘  Assume that there exists a root of h ( ˜ , ° ) = 0, with Re ( ˜ ) ˝ 0. Then,

and e − °˛ ˜ 1. Now, 0 = Hence,

− µ e − µ t2 ˘ c + ( d ˜ ) ( µ + ˛ )  e c c − µ = e < 1. > e − µ , or 0t1 ˜ t1 t1 t1

˜ ce − µ ˙ ˜ ˘ +1 > + 1 . Therefore, when 0 < 1, we obtain ° 0t1 ˇˆ °   ce − µ ˝ ˜ ˇ  ˇ ˇ˝ ˜ ˝˜ + > + 1 e − ˜ , + 1 1  0  ˘ ˆ˙ ° ˆ˙ t ˘  ˆ 1  0t1 ˙ ° ˘ 

˜ + 1 ˛ 1, t1

420

Solutions to Odd-Numbered Problems

which is a contradiction. Hence, the eigenvalues of h ( ˜ , ° ) = 0 have negative real parts and E0 is locally asymptotically stable. The authors have also shown that if 0 < 1, the DFE E0 is also globally asymptotically stable. Now, consider the EE point E1 . The linearized system of (4.33) about E1 is dS ( t ) = a11S ( t ) + a12E ( t ) + a13 I ( t ) + a12 R ( t ) + a15 D ( t ) , dt dE ( t ) = − ( a11 + µ ) S ( t ) − ( a12 + µ ) E ( t ) − a13 I ( t ) − a12 R ( t ) − a15 D ( t ) dt + e − µ° ( a11 + µ ) S ( t − ° ) + e − µ° a12E ( t − ° ) + e − µ° a13 I ( t − ° ) + e − µ° a12 R ( t − ° ) + e − µ° a15 D ( t − ° ) ,  

dI ( t ) = −t1I ( t ) − e − µ° ( a11 + µ ) S ( t − ° ) − e − µ° a12E ( t − ° ) dt − e − µ° a13 I ( t − ° ) − e − µ° a12 R ( t − ° ) − e − µ° a15 D ( t − ° ) , dR ( t ) = ˜ I ( t ) − µR ( t ) , dt dD ( t ) = ( µ + ° ) I (t ) − ˛ D (t ). dt

˙ ( 1 − 0 ) µ ˙ − µS* ˙ − µS* c ˙ − µS* d − , a = , a = − , a = − , 13 12 15 * * * * 0S 0 0S 0S 0 0S 0 and t1 = ( ˜ + µ + ˛ ) . The characteristic equation of the Jacobian of the system is given by where a11 =

( ˜ + µ )2 ( P ( ˜ , ˛ ) + Q ( ˜ , ˛ ) e − ˜˛ ) = 0,  where P ( ˜ , ° ) = ˜ 3 + A1˜ 2 + A2 ˜ + A3 , Q ( ˜ , ° ) = B1˜ 2 + B2 ˜ + B3 , ˜t ˆ ˆ ˆ ˜ ˆ ˇ with A1 = ( t1 + ˜ ) + * ,  A2 = t1  ˜ + *  + * ,  A3 =    1* , ˘ S S S  S B1 =

µe − µ −ce − µ ,  B2 = 0 0

ˇ e − µ B3 = t5  ˘ 0

− µ  ˆ˙  ˆ cµe + ° t1  , ˘ˇ * − µ  − ˘ S  ˇ 0

ˇ    ˘ S* − µ  − °µt1 , t5 = µ ( µ + ˛ ) − ˛° .

˜ = − µ is a double root. Now, consider the roots of P ( ˜ , ° ) + Q ( ˜ , ° ) e − ˜° = 0.  e − µ˜   ˘  > 0. We fnd P ( 0, ˜ ) + Q ( 0, ˜ ) = A3 + B3 =  * − µ   ˛ t1 + t5  S   0  

421

Solutions to Odd-Numbered Problems

Thus, ˜ = 0 is not a root of the equation. When ˜ = 0, P ( ˜ , 0 ) + Q ( ˜ , 0 ) = ˜ 3 + (A1 + B1 )˜ 2 + (A2 + B2 )˜ + A3 + B3 = 0. Routh-Hurwitz stability criteria can be applied to show that the roots are negative or have negative real parts. Since 0 > c t1 , and 0 > 1, we have A1 + B1 ˝ ˜ +

˙ µ  ˘˙ ˘ > 0, and A3 + B3 > 0. > 0,  A2 + B2 ˝  * − µ   t1 + S  S* 0 

   (A1 + B1 ) ( A2 + B2 ) ˘  * − µ   ° + *  ( ˛ + µ + ˝ ) , S  S 

µ (µ + ˝ )   ˘  * − µ  ° ( ˛ + µ + ˝ ) + , S  0  µ ( µ + ˝ ) − ˝°    ˘  * − µ   ° t1 +  = A3 + B3 . S  0 Routh-Hurwitz stability criteria are satisfed and therefore, E1 is locally asymptotically stable when ˜ = 0. When ˜ > 0,  assume that ˜ = i° (° > 0) is a root. Substituting in the equation, separating the real and imaginary parts, setting sin 2˜° + cos 2˜° = 1, and simplifying, one obtains

(

)

(

)

(

)

˜ 6 + A12 − 2A2 − B12 ˜ 4 + A22 − 2A1 A3 − B22 + 2B1B3 ˜ 2 + A32 − B32 = 0. 2

ˆ ˙ Now, A12 − 2A2 − B12 > ˜ 2 + ˘ *  > 0, ˇS  A32 − B32 = ( A3 + B3 )( A3 − B3 ) > ˜µt1 ( A3 + B3 ) + A22 − 2 A1 A3 − B22 + 2 B1B3 >

2e − µ 0

˛˜ e − µ    (˜ − µ ) t1 = ˝1 ,  * − µ  + S S* 0

 cµe − µ t µ + ° ˛ − µ + 1 + ° ( ) ( ) ( ) 1  0

     S* − µ 

  2    2 + t12   *  − µ 2  +  *  ˛ 2 − µ 2 = ˝ 2 .  S   S 

(

)

Using Descartes’ rule of signs, it can be seen that the cubic equation in ˜ 2 has no positive roots if all the coeffcients are positive. That is, if ˜ 1 > 0, and ˜ 2 > 0, then roots of the form ˜ = i° do not exist. Hence, all the eigenvalues have negative real parts when ˜ > 0. This implies that E1 is locally asymptotically stable and the variation of time delay cannot destroy this stability. Therefore, when 0 > 1, the endemic equilibrium point E1 is locally asymptotically stable if ˜ 1 > 0 and ˜ 2 > 0. An obvious local stability condition is ˜ ˛ µ. For the parameter set ˝ = 600, c = 0.05, d = 0.5,  µ = 0.00047, ° = 0.23, ˛ = 0.5, ˜ = 5.0, time series are plotted in the Figures A1 and  A2.

422

Solutions to Odd-Numbered Problems

FIGURE A1 Time series for 0 < 1.

FIGURE A2 Time series for 0 > 1.

For ˜ = 0.01, 0.04934, and 0.09, the values of 0 are obtained as 0 > 1,  = 1, and < 1 respectively. The initial conditions are taken as 10 ) . For ˜ = 0.09, 0 = 0.873129 < 1 (S( 0) , E ( 0) , I ( 0) ,  R ( 0) , D ( 0)) = ( 500, 30, 10, 10, and the DFE E0 = 1.2766 × 106 , 0, 0, 0, 0 is globally asymptotically stable, which is shown in Figure A1. When ˜ = 0.01, 0 ˛ 1.16 > 1 and E1 S* , E* , I * , R * , D* = ( 283404, 2331.26, 1936.64, 41205.2, 892.676 ). From the time series plotted in Figure A2, the long-term dynamical behavior displaying the persistence of the infection and the locally asymptotic stability of the endemic equilibrium point can be observed.

(

(

)

)

Exercise 5 5.1

Obtain the conditions on the coeffcients of the characteristic equation so that the disease-free equilibrium point E0 = ( ˙ H µ H , 0, 0, 0, ˙V µV , 0, 0 ) , of the model system (5.14) is locally asymptotically stable.

423

Solutions to Odd-Numbered Problems

Solution: The associated Jacobian matrix of the system (5.14), evaluated at E0 = ( ˙ H µ H , 0, 0, 0, ˙V µV , 0, 0 ) , is given by ˘ − µH  0   0  0 J 0 (E0 ) =   0   0  0 

0

− °t1

0

0

0

−t1

− k1

°t1 − k2 ˝ −t2 t2 0

0

0

0

t1

˛H 0 0 0 0

0

0

0

0

− µH

0

0

0

0

− µV

0

0

0 0

0 0

− k3 ˙v

0 − µV

     .     

Three of the eigenvalues of J 0 are − µ H , − µ H , and − µV . The remaining four eigenvalues are the roots of the equation G ( ˜ ) = ˜ 4 + a1˜ 3 + a2 ˜ 2 + a3 ˜ + a4 = 0, where a1 = k1 + k2 + k3 + µV, a2 = ( k1 + k2 + k3 ) µV + ( k1 + k2 ) k3 + k1k2 ( 1 − 21 ) , a3 = k1k2 ( 1 − 21 ) ( k3 + µV ) + k3 µV ( k1 + k2 ) , a4 = k1k2 k3 µV  ( 1 − 0 ) ( 1 + 0 − 2 1 ) . The equation G ( ˜ ) = 0 has four negative roots, if the Routh Hurwitz criterion, ai > 0, i = 1,, 4, and a1a2 a3 > a12 a4 + a32 are satisfed. The suffcient condition for the coeffcients, ai, i = 1,  , 4, to be positive is 0 < 1, (if 0 < 1, then obviously 21 < 1). Thus, at the disease-free equilibrium E0 , the system is locally asymptotically stable if 0 < 1, and a1a2 a3 > a12 a4 + a32 [0  and 2 are defned on pp. 284 in Section 5.4, Model 2]. 5.3 Derive the characteristic equation of J (E0 ) given in (5.17) and study the properties of its roots. Solution: Jacobian matrix (5.17), evaluated at E0 = ( ˙ H µ H , 0, 0, 0, ˙V µV , 0, 0 ), is given by  − µH  0   0  0  J 0 ( E0 ) =   0    0    0

0

− °t1

0

0

0

−t1

−k1 ˛H

°t1 −k2 ˝

0

0

0

t1

0 0 0 0

µv k3 ( °t1 ˛ H − k2 k1 ) t1 ˛ H˙ v −

µ v k3 ( °t1 ˛ H − k2 k1 ) t1 ˛ H˙ v 0

0

0

0

0

− µH

0

0

0

0

− µv

0

0

0

0

−k3

0

0

0

˙v

− µv

      ,       

424

Solutions to Odd-Numbered Problems

Three of the eigenvalues of J (E0 ) are − µ H , − µ H , and − µ v. The remaining four eigenvalues are the roots of the equation G ( ˜ ) = ˜ 4 + a1˜ 3 + a2 ˜ 2 + a3 ˜ + a4 = 0, where a1 = k1 + k2 + k3 + µ v, a2 = ( k1 + k2 + k3 ) µ v + ( k1 + k2 ) k3 + k1k2 ( 1 − 21 ) , a3 = k1k2 ( 1 − 21 ) ( k3 + µ v ) + k3 µ v ( k1 + k2 ) , and a4 = k1k2 k3 µ v ( 1 − 0 )( 1 + 0 − 21 ) . The equation G ( ˜ ) = 0 has four negative roots if the Routh Hurwitz criteria, ai > 0, i = 1,, 4, and a1a2 a3 > a12 a4 + a32 are satisfed. Suffcient condition for the coeffcients, ai, i = 1,  , 4, to be positive is 0 < 1, (if 0 < 1, then obviously 21 < 1). Thus, at the disease free equilibrium E0 , the system is locally asymptotically stable if 0 < 1 and a1a2 a3 > a12 a4 + a32 . 5.5 Calculate the basic reproduction number 0 for the model system (5.21). Solution: Let x = ( Eh , I h , ,Ev   I v ). Then, from (5.21) we get ˇ Chv I vSh  .  ˇ − ( pE + qI ) B +   K E  h h h 1 h Nh       dx −˜ Eh +   K 2 I h 0 = F − V , where F =   , V =  C I S dt hv h v  − ( rEv + sI v ) Bv +   K 3Ev  .     Nh −° Ev +   K 4 I v ˘   0 ˘  ˛ ˙ and F1 =  Jacobian of F at DFE = ˙ ˙ ˙ ˝

0 0 0 0

0 0 t1 0

ˆ − pBh + K1 ˘ −˜ ˘ V1 =  Jacobian of V at DFE = ˘ 0 ˘ ˘ˇ 0

0 0 0 0

Chv 0 0 0

ˆ ˘ ˘ , t1 = Chv v µ h . ˘ h µv ˘ ˇ

− qBh

0

0

K2

0

0

0 0

− rBv + K 3 −°

−sBv K4

The next-generation matrix for the epidemic model is given by ˇ  0    0 −1 K = F1V1 =   ° t1  m 1  ˘ 0

0 0 ( K1 − pBh ) t1 m1 0

   .   

˜ Chv m2 0

( K3 − rBv ) Chv

0

0

0

0

m2 0

m1 = ( K1 − pBh ) K 2 − q˜ Bh , m2 = K 4 ( K 3 − rBv ) − s° Bv .

    ,    

   .  

425

Solutions to Odd-Numbered Problems

Chvt1˜° m1m2 The above expression can be re-written as

0 is given by c .= 02 =

c   =

2 Chv  v µh°˛ . µv h    K 2 ( µh − pBh )  +  ° (˝ + ˙ I   + µh − qBh )  K  4 ( µ v − r   Bv )   +  ˛ (˙ v   +   µ v   −   sBv ) 

5.7 Show that the region ˜ 1 is positively invariant for the model system (5.29). Solution: By adding the components of the model (5.29), we obtain dN H ( t ) = ˇ B − µB N B ( t ) − µW NW ( t ) = ˇ B − µH N H ( t ) , dt dNV ( t ) = ˇ V − µV NV ( t ) , where µH = min { µB , .µW } dt A standard comparison theorem can be used to show that N H ( t ) ˝ N H ( 0 ) e − µH t +

ˆB ˆ ˆ 1 − e − µH t , in particular, N H ( t ) ˝ B , if N H ( 0 ) ˝ B . µH µH µH

(

)

ˆV ˆ ˆ 1 − e − µV t , in particular, NV ( t ) ˝ V , if NV ( 0 ) ˝ V . µV µH µH Thus, the region ˜ 1 is positively invariant. Thus, every solution of the model system (5.29) with an initial condition in ˜ 1 remains in ˜ 1 for all t > 0. Therefore, the ˜-limit sets of the model system (5.29) are contained in ˜ 1. 5.9 Discuss the global stability of the disease-free equilibrium E0 for the model system (5.29). Solution: Following the approach given in [4], denote Also, NV ( t ) ˝ NV ( 0 ) e − µV t +

(

)

X = ( SB ,  RB ,  SW ,  RW ,  SV ) ,  Z = ( EB ,  AB ,  I B ,  I BM , EW ,  AW ,  IW ,  IWM , EV ,  IV ) and group the system (5.29) as dX dZ = F ( X , 0 ) , = ,G ( X Z ) , dt dt where F ( X , 0 ) denotes the right-hand sides of the equations governing SB ,  RB ,  SW ,  RW ,  SV with EB =   AB = I B =   I BM =   EW =   AW =   IW =   IWM =   EV =   IV =  0; and G ( X , Z ) denotes the right-hand sides of the equations governing EB ,  AB ,  I B ,  I BM , EW , AW ,  IW ,  IWM , EV , and IV . Consider the reduced system as dX = ,F ( X  0 ) , which is given by dt SB˙ ( t ) = ˇ B − qR ˇ B RW ( t ) − (˜ + µB ) SB , RB˙ ( t ) = ( 1 − r ) qR ˘ B RW ( t ) − (˜ + µB ) RB ( t ) , SW ˙ ( t ) = ˜ SB ( t ) − µW SW ( t ) ,

426

Solutions to Odd-Numbered Problems

RW ˙ ( t ) = ˜ RB ( t ) − µW RW ( t ) , SV˝ ( t ) = ˆ V − µV SV ( t ) . ˘ ˆB ˜ˆ B ˆ  * * Let X * = SB* ,  RB* ,  SW ,  RW ,  SV* =  , 0,  , 0,  V  be an equilibrium µW (˜ + µB ) µV   ˜ + µB point of the reduced system. We can show that X * is globally stable in ˜ 1. Integrating the frst and second equations, we obtain

(

)

ˇt  SB ( t ) =  e ( + µB ) u˙ B ˘ˇ1 − qR RW ( u)  du + SB ( 0 )  e − ( + µB )t ,   ˘0 

˜

 ˘t RB ( t ) =  ( 1 − r ) qR ˆ B RW ( u) e ( + µB ) udu + RB ( 0 )  .e − ( + µB )t    0

˜

Simplifying, we get SB ( t ) =

ˇ B 1 − e − (° + µB )t 

t  −  e (° + µB ) uˇ B qR RW ( u) du − SB ( 0 )  e − (° + µB )t .   0 

˜

(° + µB )

Taking the limit as t ˜ °, we obtain lim SB ( t ) = t˘

ˆB

(˜ + µB )

, and lim RB ( t ) = 0. t˘

Solving the third and fourth equations in (5.21), we obtain  ˇt SW ( t ) =  ° SB ( z ) e µw z dz + SW ( 0 )  e − µw t    ˘0

˜

 ˇt and RW ( t ) =  ° RB ( z ) e µw z dz + RW ( 0 )  e − µw t .    ˘0 Substituting the expressions for SB ( t ) and RB ( t ) , we obtain

˜

(

)

 t  − k1 z  z    πB 1 − e  SW ( t ) =  α −  e k1uπ B qR RW ( u) du − SB ( 0 ) e − k1 z  e k4 z dz + SW ( 0 )  e − k4 t ,   k1    0   0 

∫

 t   RW ( t ) =  α   0 

∫

    − k1 z  k 4 z e dz + RW ( 0 )  e − k4 t , ( 1 − r ) qR π BRW ( u) e du + RB ( 0)   e    0  z

∫ ∫

k1 u

where k1 = α + µB ,   k 4 = µW . Taking the limit as t → ∞, we obtain

427

Solutions to Odd-Numbered Problems

απ B and lim RB ( t ) = 0. t→∞ µW (α + µB   )

limSW ( t ) = t →∞

Similarly, solving for SV ( t ), we obtain SV ( t ) =

 πV π  + e − µV t   SV ( 0 ) − V    , µV µV  

which converges to π V µV , as t → ∞. The asymptotic dynamics is independent of the initial conditions in Γ. Hence, the convergence of solutions is global in Γ 1. Following the works of Castillo-Chavez et al. [26], G ( X , Z ) is to satisfy the two conditions

(

)

G ( X , 0 ) = 0 and G ( X , Z ) = DZG X * , 0 Z − Gˆ  ( X , Z ) ,   Gˆ ( X , Z ) ≥ 0, 



( X , 0) =  α π+ µ , 0, µ (ααπ+ µ ) , 0, πµ , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 and D G ( X , 0 )  is the Jacobian of G ( X , Z ) taken with respect to ( E ,  A , I ,  I , E ,  A , I ,  I , E ,  I ) and evaluated at ( X , 0). The Jacobian is an M-matrix. D G ( X , 0 ) is obtained as *

where

B

B

B

W

B

V

*

Z

B

V

B

B

BM

W

Z

W

W

WM

V

V

*

*

 −k2 0 0 0 0 0 0 0 0 ˜ B    0 0 0 0 0 − k3 q˛ W 0 0   (1 − p)° B  p° B 0 0 0 0 − k3 q˛ W 0 0 0    0 0 0 0 0 0  0 0 0 − k1   − k7 0 0 0 0 0 0 0 0  W  DZ G(X * , 0) =  ,  −k6 0 0 0 0 (1 − p)° W 0 0 0 0    0 0  p° W 0 0 0 0 0 −k6 0   0  0 0 0 ˝ 0 0 0 − k4 0   0 0 0 − k5 0  ˜˙B  V ˜ V 0 ˙W  V  V   0 0 0 0 0 0 0 0 ° V − µV   * * NW , ° V = ˜V bV SV* N H* and where ° B = ˜ B bV SB* N B* , ° W = ˜W bV SW T Gˆ ( X , Z ) = ( ˝ B IV , 0, 0, 0, ˝W IV , 0, 0, 0,  t1 , 0 ) ,

t1 = ˜°B ˝V AW + ˜˝V IW + °W ˝V AB + ˝V I B , °B =

˜ BbV SB* ˙ ˜ b S* S N* ˘ 1 − B *B  , °W = W V* W * ˇ NB ˆ N B SB  NW

* ˙ ˘ ˜V bV SV* SW NW ˇˆ 1 − N S*  , and °V = N * W W H

˙ SV N H* ˘ ˇˆ 1 − N S*  . H V

428

Solutions to Odd-Numbered Problems

˜ˇ B ˇB ˛ * , SB* = , N H* = SW + ˛SB* , and SV* = V , we have in ˜ 1 µW (˜ + µB ) ˜ + µB µV * that SW ˜ SW , SB ˜ SB* , and SV ˜ SV* . Therefore, it follows that N H ˜ N H* . Hence, if the human population is at equilibrium level, we have * Since SW =

* ° ° ° N B* SB ˙ NW SW ˙ SV N H* ˙ ˝˛ 1 − S* N ˇˆ > 0,   ˝˛ 1 − S* N ˇˆ > 0, and ˝ 1 − N * ˇ > 0. W W B B ˛ HSV ˆ

Thus, Gˆ ( X , Z ) ˛ 0. Now, ˙ ˇ ˇ ˇ ˇ ˇ G( X , Z ) = ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇˆ

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 ˜°B ˝V AW

0 0 0 0 0 0 0 0 ˜˝V IW

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 °W ˝V AB

0 0 0 0 0 0 0 0 ˝V I B

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

˝ B IV 0 0 0 ˝W I V 0 0 0 0

0

0

0

0

0

0

0

0

0

0

˘      .       

5.11 Discuss the global stability of the DFE E0 for the temporal model system (5.30). Solution: Castillo-Chavez’s theorem [26] can be applied to study the global stability of the disease-free equilibrium. We rewrite the model with D1 = D2 = D3 = D4 = 0 in the form dX dZ = H (X;Z); = G ( X ; Z ) ; G ( X ;0 ) = 0, dt dt where X = ,( Sh Rh ,  Sv ) ˝   3 denotes the uninfected Z   =  ( I h , I v ) ˝   2 denotes the infected compartments and ˘ ˆ h − t2 I v − µ h Sh  ° I h − µ h Rh H (X; Z) =   ˆ − t I − µ S   v 3 h v v 

(

) (

   , G(X ; Z) =  

compartments

˘ t I −  ( µ h + ° ) I h    2 v  t3 I h −   µ v I v   

and

 .  

)

At the DFE E0 , E 0 = X * , Z * = X * , 0 ,  X * =  ( ˙ h µ h , 0, ˙ v µv ) . The conditions G ( X ; Z ) = BZ   −  Gˆ ( X ; Z ) , Gˆ ( X ; Z ) ˙  0,

(

)

where B  =   DZG X * , 0 is an M-matrix (B is irreducibly diagonal dominant and has non-positive off-diagonal elements), must be satisfed to guarantee global asymptotic stability.

429

Solutions to Odd-Numbered Problems

(

)

The fxed point E 0   =   X * , 0 is globally asymptotically stable when 0 < 1 and the above conditions are satisfed; otherwise, it is unstable, [29]. ˇ ˙ h − µ hSh  We have H ( X , 0 ) =  0  ˙ v − µ vSv ˘ ˘ − ( µh + ° ) B=  t3*  t2* =

t2* − µv

  ˆ  and G ( X , Z ) = BZ − G ( X , Z ) , where    ˆ  G1 ( X , Z )  , Gˆ ( X , 0 ) =   Gˆ 2 ( X , Z )   

  0 

= ,

 0  

˜ Sh* ˜ h ˝ Sv* ˝ v * = , . = = t 3 * * ( µh° +  h ) ° + Sh ˙ + Sv ( µv˙ +  v )

(

)

(

)

Now, ( −B ) is an M-matrix. Diagonal dominance gives some conditions on the parameters. Since Gˆ ( X ,  Z ) ˛ 0 and the above conditions are satisfed, E 0 is globally asymptotically stable for 0 < 1. The result implies that given a large perturbation of the DFE, the solutions of the model (5.30) will eventually converge to the DFE whenever 0 < 1. Epidemiologically, it implies that if a large number of infectious individuals are introduced into a fully susceptible population, the disease would die out if 0 < 1 is maintained. Otherwise, the disease would spread. 5.13 Calculate the basic reproduction number 0 and show that no endemic equilibrium of the model system (5.38) exists if 0 < 1, and a unique endemic equilibrium exists if 0 > 1. Solution: Let x = (E, I). Then, from the model system (5.38), we have x = ( Eh ,  I h ,  Em ,  I m ) , and  ˜ h I mSh   1 + hI m dx = F − V , where F =  0 dt  ˜ IS m h m   0

      ,V =      

(° h + µh ) Eh (˝ h + µh ) I h − ° hEh (° m + µm ) Em µm I m − ° mEm

   .   

Denote F1 = Jacobian of F at DFE and V1 = Jacobian of V at DFE. Then, ˆ ˘ ˘ ˘ F1 = ˘ ˘ ˘ ˘ ˇ

0

0

0

0

0 M˜ m µm 0

0

A˜ h µh 0

0

0

0

0

0 0

  ˆ ˛ h + µh  ˘ −˛ h  ˘ , and V = 1  ˘ 0  ˘ ˘ˇ  0  

0 ˝ h + µh

0 0

0 0

0 0

˛ m + µm −˛ m

0 µm

  .   

430

Solutions to Odd-Numbered Problems

The next-generation matrix for the epidemic model is given by K = F1V1−1. We obtain     K=    

A˜ h   µ h µm   0 .  0   0 

A˜ h° m µh µm (° m + µm )

0

0

0 M˜ m° h µm (˝ h + µh ) (° h + µh )

0 M˜ m µm (˝ h + µ h )

0

0

0

0

0

The basic reproduction number 0 is defned as the spectral radius of K. We obtain 02 =

AM˜ h ˜ m° h° m t = 1. 2 µh (˝ h + µh ) µm (° h + µ h )(° m + µm ) t2

In terms of 02 , we obtain Eh* =

I h* =

t2 t1 − t2 = 02 − 1 , ˜ m° h (° h + µh ) t3 ˜ m° h (° h + µ h ) t3

(

t2

˜m

 (° + µ ) (˝ + µ ) t ( h

h

Em* =

2 0

h

h

3

)

− 1 , Rh* =

)

(1 − f )° ht2

˜ m µh (° h + µh ) (˝ h + µh ) t3

(

2 0

)

−1 ,

t2 t2 02 − 1 , I m* = 02 − 1 . t5 (˜ m + µm ) µmt5 (˜ m + µm )

(

)

(

)

Hence, the model (5.38) has no endemic equilibrium if 0 < 1, and a unique endemic equilibrium exists if 0 > 1.

Index action potential 7, 35, 331, 337 activator-depleted substrate model 62 adjoint equations 282 Adomian decomposition method 54, 158 Adomian polynomials 55 advection 2, 3 advection rate 152 Allee effect 64, 114 amplitude equation 25, 31, 355, 357, 361 axon 331, 332 axon potential 337 backward bifurcation 162, 302 basic reproduction number 128, 143, 148, 152 basic reproductive number/ratio 112 Belousov-Zhabotinsky reaction 72, 75 Belousov-Zhabotinsky reaction-diffusion systems 72 bifurcating periodic solutions 22, 25 bifurcation analysis 64, 85, 167, 229, 257, 286, 293, 319, 368, 373, 394 bilinear incidence rate 115, 161 biological dispersal 3 Bogdanov-Takens bifurcation 166, 170 Brownian motion 2, 3, 19, 50 Brusselator model 75 bursting phase 389 bursting neurons 385 cable equations 334 cable model 333 Cattaneo equation 45 cell body 331, 332, 333 central manifold theorem 24 central manifold theory 228, 257, 319, 417 Chapman equation 17 chattering 343 Class K function 20 concentration gradient 2, 4, 15, 62, 336 conservation law 10 contact number 113 contact rate of infection 113 convection 3 cortex region 331 coupled-patch models 41 Criss-cross Interactions Model 308 cross-diffusion 5

cross-diffusion-driven instability 63 Crowley-Martin functional response 66 Darcy’s law 9 dendrites 331 diffusion 2 diffusion equation 9 diffusion-driven instability 60, 62, 66, 357 diffusive infuenza epidemic model 116 disease-free equilibrium 117, 197, 199 dispersal 1, 3, 169 dispersion relation 61 distributed contact models 116 double-zero bifurcation 76 Ebola Epidemic Models 217 Ebola Epidemic SEIORD Model 243 Ebola Epidemic SEIR Model 226 Ebola Epidemic SEIRD Model 248 Ebola Epidemic SEIRHD Model 238 Ebola virus 215 ecosystem function 2 ecosystem theory 2 Einstein’s relation 3 endemic disease 112 endemic equilibrium point 118, 164 excitability 335, 342 exposed population 111 Fickian equation 17 Fick’s frst law 4, 7, 61 Fick’s second law 8 FKPP reaction-diffusion equation 53 Fisher’s equation 50, 51, 54, 55 Fisher’s model 1 FitzHugh-Nagumo Model 355 Fokker-Planck equation 64, 355 Fourier's law 9 Gibbs equation 3 Gibbs free energy 3 Gierer-Meinhardt model 64 glial cells 331 Gray-Scott model 85 Hastings-Powell model 88 heteroclinic solutions 47 hexagon patterns 32, 363

431

432

Hindmarsh-Rose Model 35, 391 Hodgkin-Huxley equation 6, 56 Hodgkin-Huxley Model 344, 367 homoclinic solutions 47 Hopf bifurcation 22, 25, 61, 66, 99 Hopf bifurcation analysis 22, 84 human and vector mobility 303 hyperbolic reaction-diffusion equations 44 incidence decay with exponential adjustment (IDEA) model 220 incidence rate 113, 119, 144, 163 incubation period 112 infected population 111 infuenza epidemic models 126 inhibitory autapses 388 inhibitory input 343 inhibitory neurons 334, 342, 343 inhibitory spiking 344 interneurons 334, 387 intrinsically bursting 342 ion channels 333 ionic conductance 335 Ivlev type predator-prey model 71 KISS model 49, 50 Krasnoselskii sublinearity trick 295 Lengyel-Epstein model 80 Leslie-Gower scheme 91, 96 Levins model 1, 41 Local Auto-activation-Lateral Inhibition (LALI) 61 Lotka-Volterra system 21 Lyapunov direct method 18 Lyapunov-A and Lyapunov-B functions 20 mass action law 1, 113 maximum principle 35, 147, 273, 280 Maxwell-Stefan diffusion 4 mean square stable 20 mean-feld equations 43 membrane voltage 334, 336, 345, 370 meta-population models 42, 324 mixed mode oscillations 343 modifed Upadhyay-Rai model 91 Morris-Lecar (M-L) Model 370 motor neurons 334 multiple-scale analysis 25, 357, 358, 377 multispecies reaction-diffusion models 88 Nagumo reaction-diffusion equation 58 Nagumo telegraph equation 57 Nernst equation 336

Index

Nerve Axon 56, 354 neuronal membrane 332, 335 neurons 6, 331, 334, 335 next-generation matrix 117, 227 next generation operator 117 nonhyperbolic equilibrium point 228 nonlinear Fisher equation 50 optimal control analysis 280 Oregonator model 73 Painleve property 53 phasic bursting 342 phasic spiking 342 Pontryagin's maximum principle 273, 280 quarantine population 111 radial diffusion 12 reaction-Cattaneo (RC) system 45 reaction-diffusion equation 42, 43 reaction-Telegraph (RT) equation 45 rebound spikes 343 recovered population 112 regular spiking 342 replacement number 113 Rosenzweig-MacArthur reaction diffusion model 64 Schauder’s fxed point theorem 75, 167 Schnakenberg model 79 second Law of Thermodynamics 3 self-diffusion 4, 5, 93 Self-Enhancement and Lateral Inhibition 61 Sel’kov model 82 sensitivity analysis 230 SIRS endemic model 171 SIS reaction-diffusion model 146 soma 332, 338 spatial heterogeneity 114, 149, 310 spatial SI epidemic model 127 spatiotemporal patterns 66, 90, 100, 225, 235, 242 spike frequency adaptation 343 standard incidence 113, 162, 226 stochastic discrete-time SEIR model 218 stochastic Fisher-Kolmogorov equation 21 stochastic model 22, 41, 158, 222 stochastic reaction-diffusion systems 18 stochastic SIRS epidemic model 127, 178 stochastically asymptotically stable 19 stochastically globally asymptotically stable 20 stochastically stable 19 stripe patterns 31, 362

433

Index

subcritical Hopf bifurcation 351, 368 susceptible population 111 Susceptible-Exposed-Infected-Recovered (SEIR) model 179 Susceptible-Infected (SI) Model 118 Susceptible-Infected-Removed (SIR) model 153 Susceptible-Infected-Removed-Susceptible (SIRS) model 170 Susceptible-Infected-Susceptible (SIS) model 142 supercritical Hopf bifurcation 394 synaptic cleft 332, 338 Takens-Bogdanov bifurcation 76 threshold variability 344 tonic bursting 342 tonic spiking 342 transcritical bifurcation 119 transmission dynamics 111 transversality condition 24, 84

travelling wave solutions 306 Turing instability 60 Turing patterns 43, 66, 67 vertical incidence 113 vertical transmission 113 Volterra scheme 91 Wang’s model 139 Weber’s disk problem 15 Wereide equation 17 Zika epidemic model 270 Zika virus 267 Zika virus SEIR horizontal and vertical transmission model 290 Zika virus SEIR transmission model 314 Zika virus SIR transmission model 273 Zika virus transmission 268, 303

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