Mathematical Modelling and Analysis of Infectious Diseases [1st ed.] 9783030498955, 9783030498962

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Studies in Systems, Decision and Control 302

Khalid Hattaf Hemen Dutta   Editors

Mathematical Modelling and Analysis of Infectious Diseases

Studies in Systems, Decision and Control Volume 302

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink.

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Khalid Hattaf Hemen Dutta •

Editors

Mathematical Modelling and Analysis of Infectious Diseases

123

Editors Khalid Hattaf Centre Régional des Métiers de l’Education et de la Formation (CRMEF) Casablanca, Morocco

Hemen Dutta Department of Mathematics Gauhati University Guwahati, India

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-49895-5 ISBN 978-3-030-49896-2 (eBook) https://doi.org/10.1007/978-3-030-49896-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book aims to include different topics related to mathematical modelling and analysis of infectious diseases. The emergence and re-emergence of infectious diseases are creating new health issues and causing socio-economic problem worldwide. This book is expected to be a valuable resource for researchers, students, educators, scientists, professionals and practitioners associated with diverse aspects of diseases and related issues. The general readers should also find this book interesting to understand the dynamics of various diseases, their control strategies and related several other issues. This book consists of twelve chapters, and they are organized as follows. Chapter “Pathogen evolution when transmission and virulence are stochastic” presents an analytic approach for modelling pathogen evolution by treating the vital parameters such as transmission and virulence as random variables. Starting with a general stochastic model of evolution, it derives specific equations for the evolution of transmission and virulence, and then applies these to a particular special case, the SIR model of pathogen dynamics. It shows that adding stochasticity introduces new directional components to pathogen evolution. In particular, two kinds of covariation between traits emerge as important: covariance across the population and covariance between random variables within an individual. It also shows that these different kinds of trait covariation can be of opposite sign and contribute to evolution in very different ways. It further shows that stochasticity can influence pathogen evolution through directional stochastic effects, which results from the inevitable covariance between individual fitness and mean population fitness. Chapter “On the relationship between the basic reproduction number and the shape of the spatial domain” studies a spatially diffusive SIR epidemic model with constant parameters in a bounded spatial domain and investigates the relationship between the basic reproduction number R0 and the shape of the spatial domain. Under the homogeneous Neumann boundary conditions, R0 is the same as that for the classical non-diffusive SIR epidemic model, and thus, it does not depend on the shape of the spatial domain. On the other hand, under the homogeneous Dirichlet boundary conditions, the next generation operator does not have a v

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Preface

constant eigenvector, and R0 depends on the shape of the spatial domain. By numerical simulation for the two-dimensional rectangular domain X = (0, p) x (0, 1/p), p > 0 with constant area | X | = 1, it shows that such R0 attains its maximum for p = 1 and decreases as the shape of the domain becomes long and narrow. Moreover, it observes a similar relationship between R0 and the shape of the spatial domain in a random two-dimensional lattice model. Chapter “Cause and control strategy for infectious diseases with nonlinear incidence and treatment rate” deals with cause and control strategy for infectious diseases with nonlinear incidence and treatment rate. Control strategies regarding infectious diseases can be developed with the help of mathematical modelling by including the cause of the spread of such diseases. Different diseases have different spread patterns, and a major reason for the spread of diseases can be found out with the help of incidence rates. Also, treatment therapies vary with the severity and type of diseases. The factors like the availability of vaccines for a particular disease and the number of infected people are crucial to consider for an effective treatment rate. So, nonlinear treatment rates can vary from disease to disease. Thus, it concludes that the nonlinear incidence and treatment rate can play a vital role in suggesting effective therapies to health agencies to control the spread of disease. Chapter “Global stability of a delay virus dynamics model with mitotic transmission and cure rate” studies the global properties of a basic model for viral infection with mitotic transmission, “cure” of infected cells, saturation infection rate and a discrete intracellular delay. In connection with the proposed model, it derives some threshold parameters and establishes a set of conditions which are sufficient to determine the global dynamics of the models. It uses suitable Lyapunov functionals and Lyapunov–LaSalle-type theorem for delay systems to prove the global asymptotic stability of all equilibria of the model. It also establishes the occurrence of a Hopf bifurcation and determines conditions for the permanence of model and the length of delay to preserve stability. Finally, it incorporates numerical simulations to illustrate the analytical results. Chapter “Dynamics of a fractional-order hepatitis B epidemic model and its solutions by nonstandard numerical schemes” aims to propose and analyse a fractional-order hepatitis B epidemic model. It studies dynamical properties of the proposed fractional-order model as well as its numerical solutions. It first establishes positivity and boundedness of the proposed model, and then asymptotic stability of the model is investigated by the Lyapunov stability theorem for fractional dynamical systems and numerical simulations. Finally, it constructs positivity-preserving nonstandard finite difference (NSFD) schemes for the fractional-order model. Numerical simulations have been performed to confirm the validity of the theoretical results and show advantages and superiority of NSFD schemes over the standard one. Chapter “On SICA models for HIV transmission” aims to revisit the Susceptible-Infectious-Chronic-AIDS (SICA) mathematical model for transmission dynamics of the human immunodeficiency virus (HIV) with varying population size in a homogeneously mixing population. It considers SICA models given by systems of ordinary differential equations and some generalizations given by systems with

Preface

vii

fractional and stochastic differential operators. Local and global stability results have been proved for deterministic, fractional and stochastic-type SICA models. Two case studies, in Cape Verde and Morocco, have been investigated. Chapter “Analytical and numerical solutions of a TB-HIV/AIDS co-infection model via fractional derivatives without singular kernel” aims to analyse a TB-HIV/AIDS co-infection model. The model has been extended to the Caputo– Fabrizio fractional derivative obtained using the exponential function. Then, it investigates for the uniqueness solutions with the help of a fixed-point theorem. Thereafter, the uniqueness solution of the model has been obtained by assuming certain parameters and its stability analyses have also been carried out. Finally, numerical solutions of the mathematical model have been obtained and also performed numerical simulations. Chapter “Developing a multiparametric risk index for dengue transmission” aims to discuss the framework of developing a multiparametric index to determine the transmission risk of dengue in urban zones in Sri Lanka and predict it by considering the variation of appropriate factors. It uses literature review to identify risk factors for dengue transmission and risk levels. Fuzzy analytic hierarchy process (AHP) has been used to weight the risk factors. The constructed Haddon matrices have been used to identify the risk strata of dengue transmission. The obtained results have been compared with the other records to check the validity of the model. Sensitivity analysis has been carried out to identify impacts and variation about the contribution of the risk factors towards dengue transmission. Chapter “The effect of delay and diffusion on the dynamics of wild Aedes aegypti mosquitoes” presents a study on the effect of time delay and diffusion on the dynamics of Aedes aegypti mosquitoes’ invasion with quiescent female phase. The model proposed in this chapter is given by three delay differential equations and its corresponding reaction–diffusion equations, which describe the interaction between three sub-populations, viz. eggs, pupae and female. It focuses on studying the effect of quiescent female phase represented by time delay. The existence of periodic oscillations around the persistent positive equilibrium when time delay crosses some critical value, no occurrence of Turing bifurcation and the sensitivity analysis of parameters have been established. The stability of the bifurcating branch of periodic oscillations has been shown by using normal form and centre manifold theory. Finally, numerical simulations have been carried out to support the theoretical results. Chapter “Modeling the dynamics of hepatitis B virus infection in presence of capsids and immunity” proposes three generalized systems of differential equations to model the dynamics of hepatitis B virus (HBV) infection in the presence of HBV DNA-containing capsids and immunity mediated by cytotoxic T lymphocyte (CTL) cells. The global properties of three proposed models have been investigated. Moreover, many previous studies existing in the literature have also been claimed to improve and generalize. Chapter “A class of Ebola virus disease models with post-death transmission and environmental contamination” proposes two mathematical Ebola virus disease (EVD) models that incorporate the three modes of transmission. The modes

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Preface

have been modelled by three general incidence functions that cover many types of incidence rates existing in the literature. The first model is formulated by ordinary differential equations, and the second one is governed by partial differential equations in order to describe the evolution of EVD in time and space. The qualitative analysis of both models has been investigated in detail. Further, an application has been given and numerical simulations have also been performed to support the analytical results. Chapter “A survey on sufficient optimality conditions for delayed optimal control problems” presents a survey on recent sufficient optimality conditions for optimal control problems with time delays in both state and control variables. The results have been obtained by transforming delayed optimal control problems into equivalent non-delayed problems. It is claimed that such an approach allows using of standard theorems that ensure sufficient optimality conditions for non-delayed optimal control problems. Examples have been incorporated to illustrate the results. We sincerely acknowledge the cooperation and patience of contributors during the entire process of editing this book. Reviewers deserve the most sincere thanks for their valuable contribution in a timely manner. We are thankful to numerous colleagues and friends for their continuous encouragements to develop such books for the benefit of several kinds of readers. We also thankfully acknowledge the cooperation and support of editorial staff at Springer. Morocco India April 2020

Khalid Hattaf Hemen Dutta

Contents

Pathogen Evolution When Transmission and Virulence are Stochastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pooya Aavani and Sean H. Rice

1

On the Relationship Between the Basic Reproduction Number and the Shape of the Spatial Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . Toshikazu Kuniya

37

Cause and Control Strategy for Infectious Diseases with Nonlinear Incidence and Treatment Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nilam

61

Global Stability of a Delay Virus Dynamics Model with Mitotic Transmission and Cure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eric Avila-Vales, Abraham Canul-Pech, Gerardo E. García-Almeida, and Ángel G. C. Pérez

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Dynamics of a Fractional-Order Hepatitis B Epidemic Model and Its Solutions by Nonstandard Numerical Schemes . . . . . . . . . . . . . . 127 Manh Tuan Hoang and Oluwaseun Francis Egbelowo On SICA Models for HIV Transmission . . . . . . . . . . . . . . . . . . . . . . . . 155 Cristiana J. Silva and Delfim F. M. Torres Analytical and Numerical Solutions of a TB-HIV/AIDS Co-infection Model via Fractional Derivatives Without Singular Kernel . . . . . . . . . . 181 Mustafa Ali Dokuyucu and Hemen Dutta Developing a Multiparametric Risk Index for Dengue Transmission . . . 213 I. T. S. Piyatilake and S. S. N. Perera The Effect of Delay and Diffusion on the Dynamics of Wild Aedes Aegypti Mosquitoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 R. Yafia and M. A. Aziz Alaoui

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Contents

Modeling the Dynamics of Hepatitis B Virus Infection in Presence of Capsids and Immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Khalid Hattaf and Kalyan Manna A Class of Ebola Virus Disease Models with Post-death Transmission and Environmental Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Zineb El Rhoubari, Khalid Hattaf, and Noura Yousfi A Survey on Sufficient Optimality Conditions for Delayed Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Ana P. Lemos-Paião, Cristiana J. Silva, and Delfim F. M. Torres

About the Editors

Hemen Dutta is a regular faculty member in the Department of Mathematics at Gauhati University, India. He did his Master of Science in Mathematics, Post Graduate Diploma in Computer Application, M.Phil in Mathematics and Ph.D. in Mathematics, respectively. His topics of research interest are in the areas of mathematical analysis and mathematical modelling. He has to his credit over 130 research works as journal articles, chapters and conference proceedings papers. He has also published 17 books so far as textbooks, reference books, edited volumes and conference proceedings. He has organized many academic events and associated with several others in different capacities. Khalid Hattaf is currently working as Associate Professor at Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Casablanca, Morocco. He completed his Ph.D. in Mathematics and Computer Science from Hassan II University of Casablanca, Morocco. His research interest lies in mathematical modelling in virology, epidemiology, ecology and economy. He is also interested in mathematics education including probability teaching, ICT and teaching of mathematics, modelling and simulation in teaching and learning of mathematics. He has published 8 chapters in books and more than 60 papers in well-reputed international journals, and many more are waiting for publication.

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Pathogen Evolution When Transmission and Virulence are Stochastic Pooya Aavani and Sean H. Rice

Abstract Evolutionary processes are inherently stochastic, since we can never know with certainty exactly how many descendants an individual will leave, or what the phenotypes of those descendants will be. Despite this, models of pathogen evolution have nearly all been deterministic, treating values such as transmission and virulence as parameters that can be known ahead of time. We present a broadly applicable analytic approach for modeling pathogen evolution in which vital parameters such as transmission and virulence are treated as random variables, rather than as fixed values. Starting from a general stochastic model of evolution, we derive specific equations for the evolution of transmission and virulence, and then apply these to a particular special case; the SIR model of pathogen dynamics. We show that adding stochasticity introduces new directional components to pathogen evolution. In particular, two kinds of covariation between traits emerge as important: covariance across the population (what is usually measured), and covariance between random variables within an individual. We show that these different kinds of trait covariation can be of opposite sign and contribute to evolution in very different ways. In particular, probability covariation between random variables within an individual is sometimes a better way to capture evolutionarily important tradeoffs than is covariation across a population. We further show that stochasticity can influence pathogen evolution through directional stochastic effects, which results from the inevitable covariance between individual fitness and mean population fitness. Keywords Pathogen evolution · Stochastic fitness · Virulence · Transmission · Directional stochastic effects

P. Aavani (B) · S. H. Rice Department of Biological Sciences, Texas Tech University, 2901 Main St., Lubbock, TX 79409, USA e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_1

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P. Aavani and S. H. Rice

1 Introduction All organisms live in environments that are, to some degree, unpredictable, and this unpredictability influences both an individual’s reproductive success and the phenotype of its offspring (through environmental effects on development). This uncertainty is likely to be pronounced for pathogens, which could be subject to both stochastic uncertainty within a host (such as variation in the host’s immune response), and in the host’s environment. Stochastic fitness has been shown to impact evolutionary dynamics in ways that are not obvious from deterministic models [1–3]. One well studied process that will create stochasticity in transmission rate is variation in interactions between susceptible and infected hosts. In many infectious disease models, the population is assumed to be large and evenly mixed, meaning that every individual contacts others at a fixed rate determined by their frequency [4]. In reality, however, infected individuals often have limited and variable numbers of contacts with susceptible individuals [5]. For example, some epidemiological studies have shown that during disease spreading, the population is heterogeneous, with a few particular individuals responsible for the majority of transmission events [6–10]. In particular Woolhouse et. al identified an empirical relationship that in many disease systems, the most infectious 20% of individuals are responsible for 80% of the total infections [11] . Stochastic variation in transmission thus appears to be a common feature of pathogen systems. Another important component of pathogen fitness, virulence, is also likely to be stochastic, in part because of variation in the internal environment of the host— particularly the host’s immune system. For example, Brodin and Davis document striking variation of individual immune response in humans [12]. This translates into variation in virulence rates for a particular pathogen, with virulence being relatively low in some hosts and high in others. Even within a host, variation in virulence could be a result of uncertainty in the ability of pathogens to evade the immune system and induce pathogenic effects. Variation in transmission and virulence has also been seen in other studies, with the further observation that the pattern of variation may itself be influenced by external environmental variables that the hosts experience. For example, in an experimental study of Daphnia magna infected by the bacterial parasite Pasturia ramosa, it was shown that changing environmental factors, such as nutrient availability or temperature, can alter the variance in both virulence and transmission, as well as the covariance between them [13–15]. Figure 1 shows the distribution of mortality rates as a function of time since infection (an estimator of ν1 ) for human immunodeficiency virus 1 (HIV1) [data from [16]]. This data suggests both a high variance in virulence, and that the distribution of virulence values is asymmetrical. In a stochastic evolutionary model, we treat both the number of offspring that each individual will produce and the phenotype of those offspring, as random variables— having distributions of possible values. Saying that something is a “random variable” does not mean that it is completely random in the sense that we can make no predictions about it—only that there is some uncertainty about the exact value that it

Fig. 1 Distribution of mortality/year (virulence) during 10 consecutive years after infection with HIV1. The red fitted third-degree polynomial shows the positive skewness of data and so the high variance in virulence. The data are taken from [16], that conducted the cohort study of patients after infection with HIV1 and free of clinical acquired immunodeficiency syndrome (AIDS)

Proportion dying/6 month interval

Pathogen Evolution When Transmission and Virulence are Stochastic

3

0.2

0.1

0

5

10

Years after infection

will ultimately take. Therefore, when we treat transmission as a random variable, it means only that when a pathogen infects a host, we can not predict with certainty the exact number of subsequent hosts that will be infected through transmission from the current host. Instead, we work with the probability distribution of possible transmission events. Similarly, saying that virulence is stochastic just means that we can not predict exactly how much harm a newly acquired pathogen will ultimately do to its host. A random variable has a distribution, and we can, in principle, predict the mean, variance and other moments of this distribution. For evolution with stochastic selection, we therefore calculate the expected (probability mean) change in the (frequency) mean phenotype. In the absence of migration, this is given by the stochastic Price equation [2]:  = [[φo ,  ]] + φ o ,  + δ (1) φ In Eq. (1), φ o denotes the mean phenotype of an individual’s offspring (or of that individual in the future), and  denotes relative fitness—individual fitness divided ]] is the covariance, across the popuby mean population fitness (see Table 1). [[φo ,  lation, between expected offspring phenotype and expected relative fitness. We refer to this as the frequency covariance, since it involves the frequencies of individuals in the population. Because we are now treating φ o and  as random variables, however, each individual has a probability distribution of possible values of offspring phenotype and relative fitness. These values can thus covary within an individual, producing a probability covariance that we denote φ o , . Figure 2 shows schematic representation of frequency and probability covariances.

4 Table 1 Symbols and notation Symbol X  or E(A) A A∗ [[2 X ]] [[ j X ]]  j A [[X, Y ]] A, B w  φ φo δ θ d c βi

νi

P. Aavani and S. H. Rice

Meaning Average value of X across its frequency distribution Expected value of random variable A Difference between random variable A and its  (i.e., A − A)  expectation A Variance in the value of X, across its frequency distribution jth central moment of the frequency distribution of X jth central moment of the probability distribution of random variable A Frequency covariance of X and Y Probability covariance of random variables A and B Absolute fitness (number of descendants after one generation) w Relative fitness ( = ), conditional on w w = 0 Phenotype of an individual Average phenotype of an individual’s offspring φo − φ Birth rate of susceptible hosts Death rate of the host in the absence of infection (Background Mortality) The rate that infected groups become immune to further infections (Recovery Rate) The rate at which susceptible hosts become infected by individuals in ith infected group (transmission rate) The induced death rate of the individuals in ith infected group by the pathogens (virulence rate)

To illustrate the difference between these two kinds of covariance: Saying that ]] is positive for a population means that individuals within the population [[φo ,  . By contrast, saying that have large values of φo also tend to have large values of  that φ o ,  is positive for an individual means that if an individual’s value of φ o ends up being larger than its expected value (φo ), then its value of  is also likely ). The distinction between probability and to be larger than its expected value ( frequency operations is critical in modeling stochastic processes in populations. The term δ captures processes, such as mutation, that change phenotype during the process of reproduction.

Pathogen Evolution When Transmission and Virulence are Stochastic B

Expected Fitness

Fitness

A

5

Expected Offspring Phenotype

Offspring phenotype

Fig. 2 Frequency covariance versus Probability covariance. Figure (A) shows the relation between ]] > 0. Figure (B) shows that each indiexpected offspring phenotype and expected fitness, [[φo ,  vidual has a probability distribution of possible values of offspring phenotype and relative fitness, φ o ,  < 0

Equation (1) applies to any phenotype. It assumes only that the population is closed, meaning that there is no immigration or emigration. For the case of pathogen evolution, we will consider two phenotypic traits, transmission (β) and virulence (ν). In the following section we start by deriving a general stochastic model of evolution and then derive specific equations for evolution of transmission and virulence. We will show that adding stochasticity does not just add noise to our results, it introduces new directional components to pathogen evolution.

2 The General Case , or relative fitness. In order to study Fitness enters into Eq. (1) in the form of  = w w selection, we need to write our equations in terms of individual fitness, w. This is easy in deterministic models—since w1 is a constant that we can factor out—but is challenging in stochastic models, in which both w and w are random variables that wi are correlated with one another. The solution is to approximate i = using a w Taylor expansion [2, 17] (see Methods and Appendix). Using this approach, we can rewrite the stochastic Price equation (1) as: 1 1 1  = 1 [[φ   − 1 [[ o , w, w]] − o , w φ ]] + φ o , w − 2 [[φ φ o , (w − w )(w − w) w , φ o , w]]   w w  2 2 w w w 1 1  o φ , w + O( 3 ) (2) − 2 w   w w

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Equation (2) is the general equation of change in the expected mean phenotype when descendant phenotype, and parent fitness are random variables. Often, the “descendants” are assumed to be offspring, but φ o can also measure an individual’s phenotype at a later time, or the phenotype of descendants after many generations. To apply this equation to transmission and virulence of pathogens, we consider a pathogen that has just infected a host, and interpret φ o as the phenotype after that pathogen strain has grown within the host to the point at which it can be transmitted or kill the host.

3 Evolution of Transmission and Virulence We assume that there are n pathogen strains, and define βi as the number of other hosts that will be directly infected by the descendants of strain i (i = 1, ..., n), and νi as the probability that strain i kills its host before transmission. Both β and ν are random variables, since the exact transmission rate or time until death of the host can not be specified exactly beforehand. Substituting β and ν for φ o in Eq. (2), we obtain the following equations for the evolution of transmission and virulence: 1 1  1 1  ≈ 1 [[β , 2 w]] − , w [[β [[βi , wi , w j ]] − β, (w − w )2  ]] + β, w − β 2 2 2   w w    nw n w i= j nw 1  1 1 − βi , (wi − w i )(w j − w  j ) − 2 [[ w , β, w]] − β, w  w 2 i= j  nw w

1 1  1 1  ≈ 1 [[ [[ ν, 2 w]] − [[ νi , wi , w j ]] − ν, (w − w )2  ν, w ]] + ν, w − ν 2 2   w w   2 nw n w i= j nw 1 1 1  νi , (wi − w i )(w j − w  j ) − 2 [[ w , ν, w]] − ν, w −  w 2 i= j  nw w

(3)

(4)

Note that absolute fitness, (w), is a function of both transmission (β) and virulence (ν) . We can write the linear approximation of w with respect to β and ν about origin as: (5) w(β, ν) = w(0, 0) + wβ (0, 0)β + wν (0, 0)ν In the above equation, wβ (0, 0) and wν (0, 0) are the partial derivatives of w with respect to β and ν, respectively. For convenience, we use the notations wβ instead of wβ (0, 0) and wν instead of wν (0, 0). Substituting Eq. (5) into Eqs. (3) and (4), yields: 

 β  ν



→ → → κ + g [2] · − κ + G I · σG + G o · σG + g I · σG + go · σG + H [2] · − κ = G [2] · − [3]

[3]

+ γI

[2]

[3]

· σ G + γo

[2]

· σG

[2]

[3]

[2]

[3]

[2]

[3]

[2]

(6)

Pathogen Evolution When Transmission and Virulence are Stochastic

7

Fig. 3 Schematic diagram of G [3] I

Fig. 4 Schematic diagram of G [3] o

Fig. 5 Schematic diagram of gi[3]

Detailed derivations are presented in Methods and Appendix 1. The terms on the righthand side of Eq. (6) are written in terms of vectors, matrices, and tensors of degree 3. Figures 3, 4, 5, 6, 7 and 8 show schematic diagrams of tensors in the Eq. (6). Below, we discuss the biological interpretations of each term of the Eqs. (3), (4), and (6).  2 ]] [[β , ν]] [[ β (7) G [2] = ]] [[2 [[ ν, β ν]]  1 wβ − → κ =  wν w

(8)

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P. Aavani and S. H. Rice

Fig. 6 Schematic diagram of go[3]

Fig. 7 Schematic diagram of γ I[3]

Fig. 8 Schematic diagram of γo[3]



g

σG[2]

[2]

2 β β, ν = ν, β 2 ν

1 = 2 nw H

[2]







2

−wβ −wβ wν  2 −wβ wν −wν

  β, β β, ν = ν, β ν, ν

(9)  (10)

(11)

Pathogen Evolution When Transmission and Virulence are Stochastic

9

4 Frequency Covariance versus Probability Covariance



 1  1 The first terms on the right-hand side of the Eq. (3) w [[ β , w  ]] and Eq. (4) [[ ν, w  ]]   w contain the frequency covariances between the expected absolute fitness of a pathogen strain and its expected transmission and virulence. This is the same as the covariance term in the original, deterministic, Price equation [18]. The first term → κ , defined on the right-hand side of the Eq. (6) contains a matrix, G [2] , and a vector, − as:   2 ]] [[β , 1 wβ ν]] [[ β − → (12) , κ = G [2] = ]] [[2  wν ν]] [[ ν, β w G [2] , the matrix of frequency covariances between our traits, is analogous to the standard G matrix in quantitative genetics [19]. It is multiplied by the fitness gradient, − → κ , which shows the direction of maximum increase in fitness [19, 20]. The direction of evolution is influenced by both the fitness gradient, κ, and the frequency covariance , matrix, G [2] , as shown in Fig. 9. If [[β ν]] = 0, the directions of evolution is given by , the selection gradient; increasing transmission and decreasing virulence. If [[β ν]] >  0, virulence may increase, despite the fact that it is selected to decrease. If [[β , ν]] < 0, selection tends to increase the degree to which transmission increase and virulence decrease because the covariation between β and ν is aligned with the selection gradient action on them. In the deterministic models such as [21, 22], the frequency covariance is the only term capturing the relationship between the fitness and phenotype. When phenotype and fitness are stochastic, however, two new kinds of terms appear, corresponding to evolutionary processes that are invisible to deterministic models. When fitness and phenotype are random variables, they can

covaryfor a single 1 individual. This is captured by the second terms of the Eq. (3), w  β, w and Eq. (4)

 1  ν, w . These terms contain the probability covariances between the absolute w fitness of the pathogen strain with its transmission and virulence, averaged over the entire population. The second term on the right-hand side of the Eq. (6) also contains the fitness → gradient, − κ , but here it is multiplied by a different matrix, g [2] , defined as:   2 β β, ν [2] g = (13) ν, β 2 ν The matrix g [2] contains the frequency means, across all pathogen strains, of these probability variances and covariances. The probability covariance between transmission and virulence can be positive or negative, depending on the biology of pathogen and host. For example, if the process of transmission requires harming the initial host, we would expect a positive probability covariance between virulence and transmission (Fig. 10A). Such is the case with amoebic dysentery, caused by the

10 Fig. 9 Covariance between traits can affect the direction of evolution. (a) Selection favors reducing virulence and since there is a zero frequency covariance between virulence and transmission, evolution points to the same direction as selection gradient. (b) There is a positive frequency covariance between virulence and transmission and selection tends to reduce the degree to which transmission can increase and virulence can decrease. (c) Frequency covariance between virulence and transmission is negative, and selection tends to increase the degree to which transmission can increase and virulence can decrease

P. Aavani and S. H. Rice

 ν]] = 0 [[β,

 ν]] > 0 [[β,

 ν]] < 0 [[β,

Pathogen Evolution When Transmission and Virulence are Stochastic (B)

Transmission

Transmission

(A)

11

Virulence

Virulence

Fig. 10 Two hypothetical cases of pathogen strains in which transmission and virulence are correlated. In Figure (A) we see a negative probability covariance between transmission and virulence that correspond to pathogen strains that are dependent on their hosts for transmission. But Figure (B) corresponds to pathogen strains with positive probability covariance between transmission and virulence. Those strains are not dependent on their hosts for transmission and generally the process of transmission includes harming hosts

unicellular protozoan Entamoeba histolytica. In this case, the pathogen gets out of the host by inducing diarrhea [23], which can cause dehydration and be lethal if not treated. Increasing transmission of amoebic dysentery is thus likely to accompany increasing virulence [24]. By contrast, we expect a negative probability covariance between transmission and virulence in cases in which transmission of pathogens relies on a reasonably healthy host (Fig. 10B). Sexually transmitted diseases such as HIV infection should thus show a negative value of β, ν.  Equations (6) and (13) show that the expected change in mean transmission, β,  contains the term wν · β, ν. Similarly, the expected change in mean virulence,  contains the term wβ · β, ν. As we mentioned earlier, based on the biology ν, of the pathogen wβ and wν have different signs. For example, those pathogens that need reasonably healthy hosts for their process of transmission have wβ > 0 and wν < 0. Thus a positive probability covariance between β and ν would inhibit the evolution of increased transmission and reduce the degree to which ν declines (possibly even causing it to increase). A negative value of β, ν would have the opposite effect; amplifying both the effects of selection for increased transmission and reduced virulence. To illustrate the difference between the frequency covariance, [[ ν, w ]], and the probability covariance, ν, w, Fig. 11 shows a hypothetical case in which they have different signs. We consider a population of hosts that vary in immune resistance; with some having a strong immune response and some a weak response. Those pathogens that infect hosts with a strong response will tend to have both low expected fitness ( w) and low expected virulence ( ν), while those infecting hosts with weak immune response will have higher values of both w  and  ν (black dots in the figure). We thus have a positive frequency covariance between expected fitness and expected virulence

12

P. Aavani and S. H. Rice

Fitness (

)

Hosts with weak immune response

Hosts with strong immune response

Virulence ( )

Fig. 11 Probability and frequency capture different biological processes. A hypothetical case of eight different pathogen strains, in eight hosts, for which [[ ν, w ]] > 0, but νi , wi  < 0 for each strain (so νi , wi  < 0). Each solid black dot represents the expected values of virulence ( ν) and fitness ( w ) for one strain. The shaded distributions surrounding each solid black dot indicate the probability distribution of virulence and fitness for that strain. Pathogen strains on the lower left corner show low expected virulence and low expected fitness since they happen to be in hosts with strong immune responses. The strains on the upper right corner of the figure are in hosts with weak immune systems, leading to high expected virulence but also higher fitness of the pathogen. Within any one host, the pathogen has a greater chance of transmission if it does minimal harm to its host; leading to the negative probability covariance between fitness and virulence for each strain

([[ ν, w ]] > 0). Considering any one pathogen within a particular host, however, the pathogen is likely to have higher fitness if it does not kill the host quickly. There is thus a negative probability covariance between fitness and virulence (w, ν < 0) for each individual pathogen. The concept of a probability covariance between transmission and virulence is relevant to one of the central issues in the study of pathogen evolution: the idea that there are “tradeoffs” between transmission and virulence that constrain the evolution of both traits. Anderson and May assumed a positive association between transmission and virulence [25]. Subsequently, other authors have considered different tradeoff scenarios corresponding to different biological properties of pathogens and

Pathogen Evolution When Transmission and Virulence are Stochastic

13

hosts [26, 27]. The kinds of tradeoffs described by these authors are more accurately captured by the probability covariance between β and ν (β, ν) than by the frequency covariance between their expected values. The frequency covariance , ([[β ν]]) is influenced by the current distribution of pathogen strains and available hosts (Fig. 11), whereas the probability covariance captures tradeoffs resulting from the basic biology of pathogen and host. Measuring the probability covariance between β and ν would, ideally, involve using replicate clones of both pathogen and host, to minimize variation due to genetic differences. We know of no experimental studies have specifically done this. However, some studies using the Daphnia magna-Pasturia ramosa system have provided data from which β, ν can be estimated [13, 14]. In one study, Vale et. al estimated the covariance between transmission potential and virulence for bacteria in clones of Daphnia magna (Fig. 12) [13]. (To increase the number of datapoints, these authors infected the clones of Daphnia magna by different strains of Pasturia ramosa, so this is not a pure estimate of probability covariance.) One interesting conclusion from these studies is that the value of β, ν can change sign as a result of environmental changes. Specifically, the covariance between β and ν was negative when hosts were given ample food resources, but became positive when hosts were severely nutrient stressed [13]. Thus, physiological stress on the host can lead to the evolution of increased virulence of the pathogen.

5 Directional Stochastic Effects The terms discussed above result from probability covariation between fitness and descendant phenotype. The second class of evolutionary phenomena that appears when we make fitness stochastic—directional stochastic effects—result from covariation between individual fitness (w) and mean population fitness (w) (written as w, w in the third term on the righthand side of Eq. (2)). Figure 13A illustrates the basic mechanism of directional stochastic effects. If all else is held equal, the magnitude of evolutionary change is inversely proportional to mean population fitness. This is a general property of evolutionary models, manifest in the fact that population genetic models for change in allele frequency, and quantitative genetic models for change in mean phenotype, always contain a w1 term (unless it is assumed that w = 1). Figure 13A shows a hypothetical case for a population of two individuals that have the same expected fitness, but different variances. In the example, there is an equal probability that mean phenotype will increase or decrease when φ increases, though, the absolute magnitude of change is greater than when it decreases ( 21 vs. − 16 ). The reason for this discrepancy is that φ increases when w is low, and decreases when w is large. Because any individual’s fitness is a component of mean population fitness, the probability covariance between w and w can be broken up into two components; one involving the variance in an individual’s fitness, and the other involving the

14

P. Aavani and S. H. Rice 0.08

Low Food

Mortality Rate

0.06

0.04

0.02

0.00 0

10

20

30

Parasite Transmission Potential × 105 0.08

High Food

Mortality Rate

0.06

0.04

0.02

0.00 0

10

20

30

Parasite Transmission Potential × 105

Fig. 12 The relation between Daphnia magna mortality rate and Pasturia ramosa transmission potential. The top figure shows that when clones of the hosts are severely nutrient stressed, the covariance sign becomes positive. But when hosts are given ample food resources, the sign changes and becomes negative. One striking conclusion from this study is that the value of β, ν can change sign not only as a result of the biology of pathogen, but also as a result of environmental changes. The blue line shows the least square regression line. Figure is based on data by [13]

Pathogen Evolution When Transmission and Virulence are Stochastic (A)

15 (B)

Fitness (w)

Fitness (w)

2

1

0 0

1

Expected Descendant Phenotype (

)

Descendant Phenotype (

)

Fig. 13 Directional Stochastic Effects. (A) A population of two sets of individuals with different offspring phenotypic values, 0 and 1, having different fitness distributions. When those with φo = 0 have two offspring each, φ declines by − 16 , when they have zero offspring each, φ increases by 21 ; for a total expected change of + 16 . Though the strength of selection (captured by [[φ o , w]]) is the same, and in opposite directions, the change in φ is different in the two cases because w is different. (B) Joint fitness and offspring phenotype distribution for an individual in which φ o , (w − w )2  is nonzero. This probability distribution of w and φ o within an individual leads to the same evolutionary effect as does the frequency distribution across individuals shown in (A)

probability covariance between that individual and others in the population. For the ith pathogen strain, we can write: wi , w =

2 wi  1  + wi , w j  n n j=i

(14)

This is the origin of the 3rd and 4th terms in Eqs. (3) and (4). , 2 w]]) and (−[[ ν, 2 w]]), are The third terms in the Eqs. (3) and (4), (−[[β negative, meaning that they act to pull the population towards phenotypes that have ν, 2 w]] > 0—meaning that expected minimum variance in fitness (2 w). Thus, if [[ virulence covaries positively with variance in fitness—then this directional stochastic effect will tend to reduce the expected change in virulence. Note that this effect is 1 ν, w ]] term. As a result, it distinct from selection proper, which is captured by the w  [[ is possible for a particular strain of the pathogen to have a lower expected fitness, but to nonetheless be expected to increase, if it also has a much lower variance in fitness than do other strains. The magnitude of directional stochastic effects is greatest when  either the number of pathogen strains (n) or the expected pathogen growth rate (w) is small. We thus expect these effects to be most important at times of low pathogen diversity—such as when a pathogen is introduced into a new host population—or when the pathogen population within a host is declining—such as when it is under attack by the host’s immune system. (Note that the directional stochastic effects are different from drift. Drift is nondirectional, in the sense that the expected frequency change of the phenotype due to the drift alone is zero).

16

P. Aavani and S. H. Rice

Intuitively, the reason that there is a directional stochastic effect acting to reduce variance in fitness is that a strain with high variance (high 2 w) contributes disproportionately to the variance in mean population fitness, 2 w. As a result, when that strain with a high value of φ and high 2 w happens to have a particularly high fitness, w also tends to be high—reducing the magnitude of increase. In contrast, when that strain has a lower value of w, then w also tends to be low—leading to a large decline in φ. This is the case for the individual with φ = 0 in Fig. 13A. [2] The third term on the right-hand side of the Eq. (6), G [3] I · σG , is the inner product [3] of a third order tensor, G I , and a second order tensor, the matrix σG[2] . These can be written as follows (where φ1o = β, φ2o = ν): [2] o o o G [3] Ilmk = [[φl , φm , φk ]] σG =

1 2 nw





−wβ

2

−wβ wν 

−wβ wν −wν

2

 l = 1..2, m = 1..2, k = 1..2

(15)

G [3] I is a three dimensional array, with element (l, m, k) defined by Eq. (15) and its [2] schematic diagram is represented in Fig. 3. The term G [3] I · σG captures directional stochastic effects, as discussed earlier. The elements of G [3] I are components of the [2] [[φo , w, w]] term from Eq. (2). Taken together, the combined elements of G [3] I · σG push the population towards phenotypes that minimize the variance in individual fitness and the covariance between the fitness values of different individuals. To demonstrate how expected virulence can be correlated with probability variance in pathogen strains ([[ν, 2 β]]), we can use the hypothetical example based on well known example of Myxoma virus [28]. Experimental data show that when Myxoma virus first introduced in 1950 to control the number of invasive rabbits in Australia, the early generations of the pathogen were highly virulent, killing about 99% of their hosts [28]. Later generations of pathogen, however, evolved to be less virulent. The possible explanation of this is to be found in the mechanism of pathogen transmission. When there is an ample number of hosts in the population, the virulent strains of the pathogen are favored by selection because they grow faster inside their host which in turn can boost their transmission. But as the number of the hosts declines dramatically, the contacts between hosts also decline, and the virulent strains no longer have an advantage since most of the hosts die before contacting with another host. Now consider a small population of rabbits such that there is a high variation in contacts between them that sometimes hosts form groups. Introduce two different strains of Myxoma virus, with high and low virulence to this population. As a consequence of growing faster inside its host, the high virulent strain can benefit a higher transmission from a current host to a new host only if the current host stays alive long enough for transmission. Therefore, a highly virulent pathogen can sometime gain high transmission and sometimes no transmission. But low virulent pathogen grows lower inside its host and therefore it has no chance of very high transmission. On the other hand, since it imposes low pathology on its host, a low virulence strain

Pathogen Evolution When Transmission and Virulence are Stochastic

17

usually has some moderate level of transmission. In this situation we have a positive correlation between virulence and probability variance of transmission. As a result, directional stochastic effects, tend to pull the population toward low virulence strains that have minimum variance in transmission. Equation (14) shows that there is another way in which an individual’s fitness can covary with mean fitness—if it covaries with the fitness of others. This is the source of the fourth terms in Eqs. (3) and (4) (− 1 2 i= j [[βi , wi , w j ]] and nw νi , wi , w j ]]). These terms capture the effects of covariance between − 1 2 i= j [[ nw the fitness of different pathogen strains on the evolution of transmission and virulence. An example of a case in which we might expect [[ νi , wi , w j=i ]] < 0 is a case in which there are multiple strains competing within the host, such that the most virulent strain is the best competitor (so that other strains tend to do poorly when it does well). This would make the fourth term on the righthand side of Eq. (4) positive; contributing to an increase in mean virulence. [2] The fourth term on the right-hand side of the Eq. (6), G [3] o σG , like the third term, o o is defined in terms of tensor notation: (where φi1 = βi , φi2 = ν,φ oj1 = β j , φ oj2 = ν): G [3] olmk =

 [[φiol , φiom , φ ojk ]]

(16)

i= j

Figure 4 shows that the components of this tensor involve the interactions between the phenotypes (β and ν) of the ith pathogen strain and every other strains. The biological interpretation is similar as what described for Eq. (14). The third and fourth terms in Eqs. (3) and (4), discussed above, appear whenever fitness is a random variable. If descendant phenotype is also a random variable, as we are assuming here, then we also encounter terms containing the probability w, β, w]] and covariance between phenotype and mean population fitness (− 12 [[ w

− 12 [[ w , ν, w]] in our example) which are the seventh terms in the Eqs. (3) and w (4). The interpretation of these is similar to that of the cases containing w, w, except here it is a probability covariance between phenotype and mean fitness that influences the magnitude of change when it is associated with the expected fitness of individuals. The eighth and ninth terms of the Eq. (6), γ I[3] · σG[2] and γo[3] · σG[2] , are in terms of β and ν and correspond to the effects of the seventh terms of the Eqs. (3) and (4) (where φ1o = β, φ2o = ν): γ I[3] = [[φlo , φmo , φko ]] lmk

γo[3] = lmk

 i= j

[[φiol , φiom , φ ojk ]]

(17)

(18)

18

P. Aavani and S. H. Rice

Figures 7 and 8 show a schematic diagram of the tensor γ I[3] and γo[3] . It is interesting to note that some terms such as [[ ν, 2 β]] that in previous terms had direct roles in the evolution of ν, now have direct influence on the evolution of β. )2  and The fifth terms of the Eqs. (3) and (4), − 1 2 β, (w − w nw

)2  involve an association between a stochastic phenotype and − 1 2 ν, (w − w nw uncertainty in fitness. (They are thus the stochastic phenotype analogues of the third )2  is nonzero. Note that the terms). Figure 13B shows a case in which φ o , (w − w 2 expected value of (w − w ) is just the probability variance in fitness. Saying that β, (w − w )2  < 0 is equivalent to saying that, for a particular pathogen strain: when that strain has higher than expected transmission (β), it also has lower uncertainty in fitness. The fifth and the sixth terms of the Eq. (6) correspond to the effect of this term. [2] The fifth of the Eq. (6), g [3] I · σG , is made up of the following terms (where φ1o = β, φ2o = ν): o o o o o g [3] Ilmk = φl , (φm − φm )(φk − φk ) l = 1..2 m = 1..2 l = k..2

(19)

and the Fig. 5 corresponds to the tensor g [3] I : The sixth term of the Eq. (6), go[3] · σo[2] , is made up of the following terms (where φ1o = β, φ2o = ν) : o o o  = φiol , (φiom − φ go[3] i m )(φ jk − φ jk ) l = 1..2 m = 1..2 l = k..2 lmk

(20)

and the Fig. 6 corresponds to the tensor go[3] :

A similar interpretation applies to the terms − 1 2 i= j βi , (wi − w i ) nw 

 (w j − w  j ) and − 1 2 i= j νi , (wi − w i )(w j − w  j ) . Here, it is an associanw tion between stochastic phenotype and the similarity in fitness between different individuals (or strains) that influences evolution. 1 1 Finally, the terms − w  β, w and − w  ν, w capture the direct correlation between stochastic phenotype and mean population fitness, averaged over the entire population. As in the examples above, these terms are negative because high values of w reduce the magnitude of evolutionary change, so the magnitude of change in a trait is reduced when that trait covaries with w. The last term of the Eq. (6) captures the direct correlation between the individual stochastic phenotype and population mean phenotype: H

[2]

  β, β β, ν = ν, β ν, ν

(21)

Pathogen Evolution When Transmission and Virulence are Stochastic

19

Equation (6) is based on a general linear relationship between transmission, virulence, and fitness (Eq. (5)). If we know, for a particular pathogen, how fitness is related to β and ν, we can substitute that function into Eqs. (3) and (4) to derive a specific model for that pathogen. In the next section, we do this for the widely studied SIR model of pathogen dynamics.

6 Special Case: The SIR Model The SIR model is a compartmental epedemiological model in which we follow the dynamics of three different categories of hosts: those that are uninfected but susceptible to the pathogen (S), those that are infected (I ), and those that have recovered from infection (R). For pathogen strain i, the continuous time S I R model is: S˙ = θ − d S − S

n 

βi Ii

i=1

I˙i = βi S Ii − d Ii − νi Ii − cIi , i = 1..n n  cIi − d R R˙ =

(22)

i=1

In Eq. (22), Ii represents the density of individuals infected by the ith strain of the pathogen, β is transmission rate, ν is virulence (the degree to which the infection increases the death rate of infected hosts), d is the background death rate (in the absence of infection), and c is the “clearance” rate (the rate at which infected hosts eliminate the infection, such as through immune response). A dot over a variable indicates a time derivative (see Table 1). Day and Gandon developed a way to apply the deterministic Price equation to the evolution of pathogens that follow S I R dynamics [21, 22]. They assume that a host can harbor, at most, a single pathogen strain at any given time. In this case, the fitness of pathogen strain i is equal to the per capita growth rate of infected hosts, Ii , that carry strain i. Using Eqs. (22), we can write the fitness of pathogen strain i as a function of transmission (β) and virulence (ν): wi =

I˙i = Sβi − νi − di − ci Ii

(23)

Note that using a different epidemiological model would give us a different equation for the absolute fitness.

20

P. Aavani and S. H. Rice

We consider a pathogen that has just infected a host, and treat descendant phenotype (φ o in the general equation) as the value of β or ν when that pathogen’s descendants are transmitted to another host. We thus essentially treat β and ν as having heritabilities of 1. By contrast, d and c are properties of the host, and are thus environmental variables from the perspective of the pathogen. We will include them because they may interact with β and ν to influence pathogen evolution, but we assume that d and c are not heritable by the pathogen. We introduce stochasticity by treating the parameters β, ν, d, and c in Eq. (23) as random variables. For simplicity, and consistent with the assumption that a host can harbor only one strain at a time, we will assume that different strains are stochastically independent of one another - meaning that the fourth and sixth terms of Eqs. (3) and (4) are zero. Substituting Eq. (23) into Eqs. (3) and (4) and rearranging the terms extensively, yields the following equation for the vector of expected changes in mean transmission, and virulence, natural mortality, and clearance rate over one generation (see Appendix 2): ⎡⎤ β ⎢⎥ ⎢ ν ⎥ [2] [2] [2] → → [2] − [2] − [3] [3] [3] ⎥ ⎢ ⎢  ⎥ = G · κ + 2g · κ + G · σG + g · σG + H · σ H ⎣ d ⎦

(24)

 c

The terms on the righthand side of Eq. (24) are written in terms of vectors, matrices, and tensors of degree 3. Below, we discuss the biological meaning of each term in turn. The first term on the right-hand side of the Eq. (24) contains a matrix, G [2] , and a → vector, − κ , defined as: ⎛

G [2]

]] [[β , ν]] [[2 β 2 ⎜[[  ν, β ]] [[  ν]] =⎜ ⎝[[d, β  ν]] ]] [[d, ]] [[ [[ c, β c, ν]]

 , d]] [[β  [[ ν, d]]  [[2 d]]  [[ c, d]]

⎞ ⎛ ⎞ , [[β c]] S ⎜−1⎟ 1 [[ ν, c]] ⎟ − → ⎟ ⎜ ⎟  c]]⎠ , κ = w  ⎝−1⎠ [[d, −1 c]] [[2

(25)

G [2] , the matrix of frequency covariances between our traits, is analogous to the standard G matrix in quantitative genetics. It is multiplied by the fitness gradient, − → κ , which shows the direction of maximum increase in fitness [19, 20]. The matrix → G [2] will appear in any model with parameters β, ν, d, and c. The specific form of − κ given in Eq. (25), however, is specific to the SIR model (Eq. 22)—a different model → would yield a different − κ.

Pathogen Evolution When Transmission and Virulence are Stochastic

21

→ For the case of the S I R model, the form of − κ shown in Eq. (25) shows that S selection always favors increasing the average transmission with a strength of w  (where S is the density of uninfected but susceptible hosts), while decreasing the 1 average virulence, background mortality, and recovery rate, with a strength of (− w  ). How this selection influences evolution is of course also influenced by G [2] . A positive covariance between β and ν will tend to reduce the degree to which transmission can increase and the degree which virulence can decrease.  contains the terms Note that the expected change in mean transmission, β,  , d]] and −[[β , −[[β c]]. This illustrates the importance of including the “environmental” factors d and c in our analysis. → κ is the only term that would appear in a In the absence of mutation, G [2] · − deterministic model, and is equivalent to the selection term in the models of [21, 22]. All of the other terms in Eq. (24) contain variances and covariances of random variables; and are thus invisible to deterministic models. The second term on the right-hand side of the Eq. (24) also contains the fitness → gradient, − κ , but here it is multiplied by a different matrix, g [2] , defined as: ⎛ ⎞ 2 β β, ν β, d β, c ⎜ ⎟ ⎜ ν, β 2 ν ν, d ν, c ⎟ (26) g [2] = ⎜ ⎟ ⎝d, β d, ν 2 d d, c⎠ c, β c, ν c, d 2 c Since β, ν, d, and c are random variables, each has distribution of possible values for any one pathogen strain; they can therefore covary for a single pathogen. The matrix g [2] contains the frequency means, across all pathogen strains, of these probability variances and covariances.  Equations (24) and (26) show that the expected change in mean transmission, β,  contains the term −β, ν. The expected change in mean virulence, ν, contains the term S · β, ν. Thus a positive probability covariance between β and ν would inhibit the evolution of increased transmission and amplify the evolution of increased virulence. A negative value of β, ν would have the opposite effect; amplifying the effects of selection for increased transmission while reducing the evolution of increased virulence. The third term on the right-hand side of the Eq. (24), G [3] · σG[2] , is the inner product of a third order tensor, G [3] , and a second order tensor, the matrix σG[2] . These can be written as follows (where φ1o = β, φ2o = ν, φ3o = d, φ4o = c):

[2] o o o G i[3] jk = [[φi , φ j , φk ]] σG

⎛ 2 −S 1 ⎜ ⎜ S = 2 ⎝ S  nw S

S −1 −1 −1

S −1 −2 −1

⎞ S −1⎟ ⎟ −1⎠ −2

(27)

G [3] is a three dimensional array, with element (i, j, k) defined by Eq. (27) (Fig. 14). The term G [3] · σG[2] captures directional stochastic effects, as discussed

22

P. Aavani and S. H. Rice c, β, ν]] [[ c, β, d]] [[ c, β, c]] [[ c, 2 β]] [[ [[ c, ν, β]]

c, ν, d]] [[ c, ν, c]] [[ c, 2 ν]] [[

[[ c, d, β]] [[ c, d, ν]]

c, d, c]] [[ c, 2 d]] [[

[[ c, c, β]] [[ c, c, ν]] [[ c, c, d]]

[[ c, 2 c]]

 2 β]] [[d,  β, ν]] [[d,  β, d]] [[d,  β, c]] [[d,  ν, β]] [[d,

[3]

G4::

 2 ν]] [[d,  ν, d]] [[d,  ν, c]] [[d,

 d, β]] [[d,  d, ν]] [[d,

 2 d]] [[d,  d, c]] [[d,

 c, β]] [[d,  c, ν]] [[d,  c, d]] [[d,

 2 c]] [[d,

ν , β, ν]] [[ ν , β, d]] [[ ν , β, c]] [[ ν , 2 β]] [[ [[ ν , ν, β]]

[[ ν , d, β]] [[ ν , d, ν]]

ν , d, c]] [[ ν , 2 d]] [[

[[ ν , c, β]] [[ ν , c, ν]] [[ ν , c, d]]

[[ ν , 2 c]]

 2 β]] [[β,  β, ν]] [[β,  β, d]] [[β,  β, c]] [[β,  ν, β]] [[β,

[3]

G2::

 2 ν]] [[β,  ν, d]] [[β,  ν, c]] [[β,

 d, β]] [[β,  d, ν]] [[β,

[3]

G3::

ν , ν, d]] [[ ν , ν, c]] [[ ν , 2 ν]] [[

 2 d]] [[β,  d, c]] [[β,

 c, β]] [[β,  c, ν]] [[β,  c, d]] [[β,

 2 c]] [[β, [3]

G1::

Fig. 14 An illustration of third order tensor G [3]

in the previous section. The elements of G [3] are components of the [[φo , w, w]] term from Eq. (2). Taken together, the combined elements of G [3] · σG[2] push the population towards phenotypes that minimize the variance in individual fitness and the covariance between the fitness values of different individuals. Note that the only positive terms in σG[2] are terms that multiply elements of G [3] that contain β, ν, β, d, or β, c. This means that either β or ν will tend to increase if it has a positive frequency covariance with β, ν, β, d, or β, c. The reason is that ν, d, and c all contribute negatively to fitness while β contributes positively. The fourth and the fifth terms of the Eq. (24), g [3] · σG[2] and H [3] · σ H[2] , are made up of the following terms: o o o gi[3] jk = φi , φ j φk 

⎛

[2] o o o Hi[3] jk = [[φi , φ j , φk ]] σ H

S2 ⎜ 1 ⎜−S = 2 ⎝−S  nw −S

(28) −S 1 1 1

−S 1 1 1

⎞ −S 1 ⎟ ⎟ 1 ⎠ 1

(29)

Though complicated, these terms all derive from the covariance between individual phenotypes and mean population fitness.

7 Discussion Uncertainty in both fitness and offspring phenotype can influence pathogen evolution in a variety of ways. This is clear from the fact that in the general equations for evolution of transmission and virulence, Eqs. (3) and (4), all of the terms after the first one, in each equation, contain probability operations and would thus not appear in a

Pathogen Evolution When Transmission and Virulence are Stochastic

23

deterministic model. Though the equations contain many stochastic terms, the stochastic components of directional evolution discussed in this chapter all fall into two general categories: Probability covariances between traits, and directional stochastic effects. Probability covariances between traits arise whenever the traits that influence fitness are random variables—meaning that they have a distribution of possible values. This is definitely the case for transmission and virulence of a pathogen since, when a pathogen infects a host, we can not say with certainty how long the host will survive or how many others it will infect. The concept of a probability covariance between transmission and virulence is relevant to one of the central issues in the study of pathogen evolution which is the idea that there are “tradeoffs” between transmission and virulence that constrain the evolution of both traits. We argue that the kinds of tradeoffs between the phenotypes of the pathogen are more accurately captured by the probability covariance between them than by the frequency covariance between their expected values. The frequency covariance is strongly influenced by the current distribution of pathogen strains and available hosts, but the probability covariance captures tradeoffs resulting from the basic biology of pathogen and host. Experimental evidence suggests that the sign of the probability covariance depends not only on the biology of the pathogen, but also on the physiological state of the host. The second way in which stochasticity can influence directional evolution is through directional stochastic effects. These result from the fact that the same strength of selection (the relation between expected fitness and phenotype, measured either as a covariance or a regression) will produce different magnitudes of evolutionary change depending on the mean population fitness (w), which is itself a random variable. The third through eighth terms in Eqs. (3) and (4) are all related to this kind of evolutionary process. The simplest directional stochastic effect favors phenotypes that have relatively low variance in fitness (corresponding to the − 1 2 [[φ, 2 w]] term in Eqs. (3) and (4); nw (note that this is not the same as having high geometric mean fitness [2]). Pathogen fitness involves both evading the host’s immune system—which is likely to exhibit stochastic variation both within and between hosts [29, 30]—and transmission from one host to another, which is likely to add further stochasticity. It is thus likely to be very unpredictable. A trait that reduces this uncertainty, even at the expense of reducing expected fitness, could thus spread—especially when the number of pathogen strains is small (small n in Eqs. (3) and (4)) or the population of pathogens  is declining (small w).

24

P. Aavani and S. H. Rice

Another way in which a trait can increase due to directional stochastic effects is through a negative probability covariance between the fitness of individuals possessing it and other individuals in the population. (Terms containing [[φi , wi , w j ]] in Eqs. (3) and (4)). For pathogens, this effect would favor a strain that deals differently with the host’s immune system than do other strains, even if it does no better on average. This effect would thus be a diversifying force with regard to how pathogens interact with their hosts. Note that this approach to introducing stochasticity is different from that used in stochastic differential equations (SDE). In SDE, one makes the main variable stochastic by assigning a variation to it using a Wiener process, while other variables are kept deterministic [31]. For example, a SDE model for the evolution of virulence might treat transmission and overall fitness as fixed values, while assigning random normally distributed noise to ν. By contrast, our approach treats virulence, transmission, and fitness as random variables, which can have any distributions of values and can covary with one another. A number of previous authors have noted that fitness should be treated as a random variable in models of evolution [1, 3]. Fitness is not the only stochastic parameter in evolution, however. Rice made offspring phenotype a random variable as well, but still treated parent phenotype as a deterministic value [2]. In this chapter, we have extended earlier work by treating all biological parameters (transmission, virulence, fitness, and mean fitness) as random variables. This reveals new terms such as β, ν, a probability covariance between different phenotypic traits, that capture real biological relationships that have been invisible to previous models. Beyond the application to pathogens that we consider here, this approach should be applicable to any case in which the values of some phenotypic traits can not be specified exactly prior to selection acting on them. Making our evolutionary models stochastic by allowing traits such as transmission and virulence to be random variables substantially increases the complexity of our results. Upon examination, however, the new terms that appear all correspond to real biological phenomena that would be missed by a purely deterministic analysis.

8 Methods The relative fitness of an individual,  = w conditional on w = 0, is a ratio of w correlated random variables. In order to evaluate its expected value or its probability covariance with other random variables, we need to expand it in such a way that i , all random variables are in the numerator. Following [17], we define wi∗ = wi − w and expand i as: wi wi wi = i = = ∗ +w  w w w

⎛ ⎞  ∞ ∗ j  w ∗ −1 (w ) w i ⎝ ⎠ (30) 1+ = (−1) j 1+  j   w w (w) j=1

Pathogen Evolution When Transmission and Virulence are Stochastic

25

 as [17]: We now find the expected value,  i = E 

w

∞  w w i  j w + wi , j w i  |w = 0 = (−1) j +  j+1  w w (w) i

(31)

j=1

For a population of size N , the jth order terms in these series tend to scale as i by setting j = 1. This yields: so we approximate  i ≈

 i wi (w − w)w −  2 w w

i ≈ 

w i wi , w − 2  w (w)

1 , N

(32)

]] from Eq. (1) as follow: Using Eq. (32), we write [[φo ,  ]] ≈ [[φo , 

1 o 1 ]] − 2 [[φo , w, w]] [[φ , w   w w

(33)

We also use Eq. (32) and obtain: φ o ,  ≈

1 o 1   φ , w − 2 φ o , (w − w)w   w w

(34)

The second term of the Eq. (34) can be manipulated as follow: 1   = 1 φ o , (w − w   + 1 φ o , w   (35) φ o , (w − w)w )(w − w) (w − w) 2 2   2 w w w Recall the formula ab = [[a, b]] + ab which bar indicates the average over all individuals in population. Therefore, the second term on the right-hand side of the Eq. (35), can be expanded as follows:  =w  = w i φ o , w φ o , w (w − w) φ o , w − w φ o , w φ o , w = [[ w , φ o , w]] + w

(36)

Therefore, we obtain: φ o ,  ≈

1 o 1 1  o   − 1 [[ φ , w − 2 φ o , (w − w )(w − w) w , φ o , w]] − 2 w φ , w  w  2  w w w

(37)

and substituting Eqs. (33) and (37) in Eq. (1) yields Eq. (2). The third and fourth terms of the Eq. (2) can be further expanded as: 1 1 1 [[φo , w, w]] = [[φo , wi , w j ]] = [[φo , 2 w]] 2 2 n   w nw j=1  1 + [[φio , wi , w j ]] 2  n w i= j n

(38)

26

P. Aavani and S. H. Rice

1 2

w

 = φ o , (w − w )(w − w) =

1 2

nw 1

2

nw

φ o ,

n 

(wi − w i )(w j − w  j )

j=1

φ o , (wi − w i )2  +

1   j=i nw 2

φ o , (wi − w i )(w j − w  j )

(39)

and substituting these equations in Eq. (2) and β and ν for offspring phenotype, results Eqs. (3) and (4). From Eq. (5) we derive the equation for fitness of pathogen strain i as: wi = wβ βi + wν νi

(40)

We present Eq. (6) with the tensor notations. For the strain i, we define the random variables φ1o = β, φ2o = ν. The covariance terms on the right-hand sides of Eqs. (6) can be grouped into 2 × 2 degree 2 and 2 × 2 × 2 degree 3 tensors (note that a tensor of degree 2 is just a matrix). The elements of each tensor can be shown as follow:

[2] o G lm = [[φlo , φ m ]] o o o G [3] Ilmk = [[φl , φm , φk ]]

o o o o o  g [3] Ilmk = φl , (φm − φm )(φk − φk )

γ I[3] = [[φlo , φmo , φko ]] lmk

[2] glm = φlo , φmo 

G [3] olmk =

 o , φ o , φ o ]] [[φ im jk il i= j

o o o  go[3] = φiol , (φiom − φ i m )(φ jk − φ jk ) lmk

γo[3] = lmk

 [[φiol , φiom , φ ojk ]] i= j

where i = 1...n,

j = 1...n, l = 1...2, m = 1...2, k = 1...2. Also we obtain:

  1 wβ κ − → κ = 1 =  wν κ2 w  [2] [2]     2 σG 11 σG 12 1 −wβ −wβ wν [2] σG = =  2 2 −wβ wν −wν σG[2]21 σG[2]22 nw

(41) (42)

Pathogen Evolution When Transmission and Virulence are Stochastic

27

The tensor inner products on the right-hand side of Eq. (6) is 2 by 1 vector and can be written as follows: → G [2] .− κ =

2 2  

[2] G lm κm

(43)

[2] glm κm

(44)

l=1 m=1

→ κ = g [2] .−

2 2   l=1 m=1

[2] G [3] I .σG =

2  2  2 

[2] G [3] Ilmk σG mk

(45)

[2] G [3] olmk σG mk

(46)

[2] g [3] Ilmk σG mk

(47)

l=1 m=1 k=1

G [3] o

·

σG[2]

=

2  2 2   l=1 m=1 k=1

[2] g [3] I · σG =

2  2  2  l=1 m=1 k=1

→ H [2] · − κ =

2 2  

[2] Hlm κm

(48)

l=1 m=1

γ I[3] · σG[2] =

2  2  2 

γ I[3] σ [2] lmk G mk

(49)

γo[3] σ [2] lmk G mk

(50)

m=1 l=1 k=1

γo[3] · σG[2] =

2 2  2   m=1 l=1 k=1

Appendix 1 We will use the following identities in manipulating frequency and probability operations: a,  b = 0 (51) [[a, b]] = 0, The following theorem is useful for manipulating covariance of products: Theorem 1. (i) For three arbitrary random variables a, b, and c we have the following identity a, bc = ab, c +  ca, b −  a b, c

(52)

(ii) If c is independent from a, b, and ab, Eq. (52) collapses to: a, bc =  ca, b

(53)

28

P. Aavani and S. H. Rice

Proof. (i) We extract the right side of the Eq. (52)  − ab  c) +   −  − ab, c +  ca, b −  a b, c = (abc c(ab a b) −  a (bc b c)  − ab  c + ab c −  + = abc a b c − a bc a b c  −  = a, bc = abc a bc (ii) If c is independent of a, b, and ab then b, c = ab, c = 0, and we have a, bc =  ca, b Note that the results of (1) as well can be applied to frequency operation substituting ∗. [[∗, ∗]] for ∗, ∗ and ∗ for  Recall that for the SIR model the absolute finesses of pathogen strains are independent from each other. With this assumption, the covariance between fitness of  j ]] = 0 and wi , w j  = 0. two strains will be zero, i.e, [[ wi , w The third term of the Eq. (2) is expanded as a sum of two other terms in Eq. (38). Because of the stochastic independency, the covariance between the fitness of two different strains becomes zero and therefore the second term of the Eq. (38) becomes zero and we obtain: 1 1 , w, w]] = , 2 w]] [[φ [[φ 2 2  w nw

(54)

The fourth term of the Eq. (2) is expanded as a sum two other terms in Eq. (39). By part (ii) of Theorem (1), the second term of the Eq. (39) become zero and the first term simplified as follows: 1 2 w

 = φ, (w − w )(w − w) = = =

1 2 nw 1 2

nw 1

 nw 1

2

2 nw

φ, (wi − w i )2  φ, w2  + φ, w2  − φ, w2  −

1  nw 2

2

φ, −2w w

 nw 2

2

w φ, w

2 w

[[ w, φ, w]] −

2  wφ, w (55)

2 w

Therefore, using the results of Eqs. (54) and (55) we simplify Eq. (2) as follows: 1 1 1 1 1  ≈ 1 [[φ , 2 w]] − , w φ, w2  + [[ w , φ, w]] + φ, w φ ]] + φ, w − 2 [[φ    w w nw 2 2  w nw nw

(56)

Now we try to obtain each term of the Eq. (6). Here, we use Eq. (3) and similar equations will be obtained using Eq. (4).

Pathogen Evolution When Transmission and Virulence are Stochastic

29

Using Eq. (5), we expand the first term on the right-hand side of the Eq. (3) as: wβ 2 1 w 1  + wν  ]] + ν [[β , w , wβ β , ν]] = [[β ]] = [[β [[ β ν]]     w w w w

(57)

The second term on the right-hand side of the Eq. (3) can be expanded as: wβ 1 1 w β, w = β, wβ β + wν ν = 2 β + ν β, ν     w w w w

(58)

The third term of the Eq. (3) is expanded as: 1 1 , 2 w]] = [[β [[β, wβ β + wν ν, wβ β + wν ν]] 2  2 nw nw 2wβ wν wβ 2 w 2 , 2 β]] + , β, ν]] + ν [[β , 2 ν]] (59) [[ β [[ β = 2 2 2 nw nw nw The fourth term on the right-hand side of the Eq. (3) is expanded below: −



1

 nw i = j 1  2

[[βi , wi , w j ]] = −



1

 nw i = j 2

[[βi , wβ βi + wν νi , wβ β j + wν ν j ]]

2 2 [[βi , βi , β j wβ + βi , ν j wβ wν + νi , β j wν wβ + νi , ν j wν ]] 2 nw i = j  1 

2 [[βi , βi , β j ]]wβ + [[βi , βi , ν j ]]wβ wν + [[βi , νi , β j ]]wν wβ + [[βi , νi , ν j ]]wβ wν − 2 nw

= −

(60)

i = j

The fifth term on the right-hand side of the Eq. (3) can be expanded as follows: − − =− −

1 2

nw 1

 nw 1



1 2

nw

2  − wν  β, wβ β + wν ν − wβ β ν 

) + (wν ν − wν  β, (wβ β − wβ β ν) 2

2 nw

wβ 2  nw

 2  = − β, (w − w)

2

]]

)2 w 2 + 2(β − β )(ν −  β, (β − β ν)wβ wν + (ν −  ν)2 wν 2  β

)2  − β, (β − β 2

2wβ wν

wν 2 )(ν −   β, (β − β ν)  − β, (ν −  ν)2  (61) 2 2 nw nw

30

P. Aavani and S. H. Rice

The sixth term on the right-hand side of the Eq. (3) can be expanded as follows: 1  i )(w j − w  j ) = βi , (wi − w 2 i= j nw 



1  i − wν  j − wν  − βi , wβ βi + wν νi − wβ β νi wβ β j + wν ν j − wβ β ν j  2  i= j nw



 1  i ) + (wν νi − wν  j ) + (wν ν j − wν  =− 2 βi , (wβ βi − wβ β νi ) (wβ β j − wβ β ν j )   i= j nw −

=− −

wβ i 2  2

nw 2

wβ wν   nw

i= j

wβ wν   nw

i )(β j − β j ) − β, (βi − β

j ) − βi , (νi −  νi )(β j − β

i= j

2

i )(ν j −  βi , (βi − β ν j )

i= j

wν 2  βi , (νi −  νi )(ν j −  ν j ) 2 i= j nw

(62)

We expand the seventh term on the right-hand side of the Eq. (3) in two steps:

−

n  1 1 [[ w ,  β, w ]] = − [[ w ,  β, βwβ + νwν ]] 2 2 w nw i=1

=−

−

1 [[ w, β, βwβ + νwν ]] 2 nw

 1 [[ w, βi , β j wβ + ν j wν ]] 2  nw i= j

(63)

(64)

Equation (63) can be further expanded as follows: 1 1 wβ +  [[ w , β, βwβ + νwν ]] = − 2 [[β νwν , β, βwβ + νwν ]] 2   nw nw  1

2 , 2 β]]wβ 2 + [[β , β, ν]]wβ wν + [[ ν, 2 β]]wν wβ + [[ ν, β, ν]]wν = − 2 [[β  nw −

=−

wβ 2 2

nw

, 2 β]] − [[β

wβ wν 2

nw

, β, ν]] − [[β

wν wβ 2

nw

[[ ν, 2 β]] −

wν 2 [[ ν, β, ν]] 2 nw

(65)

Pathogen Evolution When Transmission and Virulence are Stochastic

31

We expand Eq. (64) as follows:  1 1  i wβ +  [[ w , βi , β j wβ + ν j wν ]] = − 2 [[β νi wν , βi , β j wβ + ν j wν ]] 2   nw n w i= j i = j  1 

2 i , βi , β j ]]wβ 2 + [[β i , βi , ν j ]]wβ wν + [[ [[β =− 2 νi , βi , β j ]]wν wβ + [[ νi , βi , ν j ]]wν  i= j nw −

=−

w w  w w  wβ 2  i , βi , β j ]] − β ν i , βi , ν j ]] − ν β [[β [[β [[ νi , βi , β j ]] 2 2  i= j  i= j 2 i= j nw nw nw

−

wν 2  [[ νi , βi , ν j ]] 2 i= j nw

(66)

(67)

Finally, the eights term on the right-hand side of the Eq. (3) is expanded below: −

1 1 β, w = − β, βwβ + νwν    w w wβ w = − β, β − ν β, ν   w w

(68)

Appendix 2 From model (22), we derive the equation for fitness of pathogen strain i as: wi = Sβi − νi − di − ci

(69)

Note that in SIR model the pathogen strains are stochastically independent in fitness. By substituting Eq. (69) in Eqs. (3) and (4) we derive Eq. (24). We present Eq. (24) with the tensor notations. For the strain i, we define the random variables φ1o = β, φ2o = ν, φ3o = d, φ4o = c. The covariance terms on the right-hand sides of Eqs. (24) can be grouped into 4 × 4 degree 2 and 4 × 4 × 4 degree 3 tensors (note that a tensor of degree 2 is just a matrix). The elements of each tensor can be shown as follows:

o o G i[2] j = [[φ j , φ j ]]

o o gi[2] j = φi , φ j 

o o o G i[3] jk = [[φi , φ j , φk ]]

o o o Hi[3] jk = [[φi , φ j , φk ]]

32

P. Aavani and S. H. Rice

where i = 1...4,

j = 1...4, k = 1...4. Also we obtain:

⎛ ⎞ ⎛ ⎞ κ1 S ⎜κ 2 ⎟ 1 ⎜ −1⎟ − → ⎜ ⎟ ⎜ κ =⎝ ⎠= ⎝ ⎟  −1⎠ κ3 w κ4 −1 ⎛ [2] [2] [2] [2] ⎞ σG 11 σG 12 σG 13 σG 14 ⎛ 2 ⎞ −S S S S ⎜ [2] [2] [2] [2] ⎟ ⎜σG 21 σG 22 σG 23 σG 24 ⎟ ⎟ 1 ⎜ ⎟ ⎜ ⎜ S −1 −1 −1⎟ σG[2] = ⎜ [2] [2] [2] [2] ⎟ = 2 ⎝ S −1 −2 −1⎠ ⎜σG 31 σG 32 σG 33 σG 34 ⎟ n w  ⎠ ⎝ S −1 −1 −2 [2] [2] [2] [2] σG 41 σG 42 σG 43 σG 44 ⎛ [2] [2] [2] [2] ⎞ σ H11 σ H12 σ H13 σ H14 ⎛ 2 ⎞ S −S −S −S ⎜ [2] [2] [2] [2] ⎟ ⎜σ H21 σ H22 σ H23 σ H24 ⎟ ⎟ 1 ⎜ ⎟ ⎜ ⎜−S 1 1 1 ⎟ σ H[2] = ⎜ [2] [2] [2] [2] ⎟ = 2 ⎝ ⎠ −S 1 1 1 ⎜σ H31 σ H32 σ H33 σ H34 ⎟ n w  ⎠ ⎝ −S 1 1 1 σ H[2]41 σ H[2]42 σ H[2]43 σ H[2]44

(70)

(71)

(72)

The tensor inner products on the right-hand side of Eq. (24) is 4 by 1 vector and can be written as follows: → κ = G .− [2]

4  4 

G i[2] j κj

(73)

gi[2] j κj

(74)

i=1 j=1

→ κ = g [2] .−

4  4  i=1 j=1

G [3] .σG[2] =

4  4 4  

[2] G i[3] jk σG k j

(75)

[2] gi[3] jk σG k j

(76)

[2] Hi[3] jk σ Hk j

(77)

i=1 j=1 k=1

g [3] .σG[2] =

4  4  4  i=1 j=1 k=1

H [3] .σ H[2] =

4  4  4  i=1 j=1 k=1

Now we try to derive the Eq. (56) for each phenotype β, ν, d, and c where w is function of those variables and obtained from Eq. (69). Here we only show the  and others can be obtained similarly. detailed derivation of ν

Pathogen Evolution When Transmission and Virulence are Stochastic

33

Using Eq. (69), we expand the first term on the right-hand side of the Eq. (56) as: 1 1 −  [[ ν, w ]] = [[ ν, S β ν − d − c]]   w w S 1 1  − 1 [[ ]] − [[2 = [[ ν]] − [[ ν, β ν, d]] ν, c]]     w w w w

(78)

The second term on the right-hand side of the Eq. (56) can be expanded as: 1 1 S 1 1 1 ν, w = ν, Sβ − ν − c − d = ν, β − 2 ν − ν, c − ν, d (79)       w w w w w w The third term of the Eq. (56) is expanded as: 1  nw

2

1 [[ ν, Sβ − ν − (d + c), Sβ − ν − (d + c)]] 2 nw 2 S 2S 2S 2S 1 = [[ ν, 2 β]] − [[ ν, β, ν]] − [[ ν, β, d]] − [[ ν, β, c]] + [[ ν, 2 ν]] 2 2 2 2 2 nw nw nw nw nw 2 2 2 2 2 + [[ ν, ν, d]] + [[ ν, ν, c]] + [[ ν, c, d]] + [[ ν, 2 c]] + [[ ν, 2 d]] 2 2 2 2 2 nw nw nw nw nw

[[ ν, 2 w]] =

(80) (81)

The fourth term on the right-hand side of the Eq. (56) is derived below: 1 2 nw

ν, w2  = =

1 2 nw S2 2 nw

1

ν, (Sβ − ν − (d + c))2  = 2S

ν, β 2  −

2 nw

2

+

 nw

2

ν, βν +

2 nw 1

3 ν −

2 nw

2

ν, cν +

 nw

ν, S 2 β 2 − 2Sβν + ν 2 − 2Sβd + 2dν − 2Sβc + 2cν + 2cd + c2 + d 2 

2

2S 2 nw

ν, cd +

ν, βd +

1  nw

2

2 2 nw

ν, dν −

ν, c2  +

1  nw

2

2S 2 nw

(82)

ν, βc

(83)

ν, d 2 

Also, by substituting ν in φ and using Eq. (69), the fifth and sixth terms on the right-hand side of the Eq. (56) are expanded as follows: 1

= = =

 nw 1

2

 ν, w) = ([[ w, ν, w]] + w

 nw 1

2

 ν, Sβ − ν − c − d]] + −  [[S β ν − c − d,

 nw

2

 Sν, β + 2 ν + ν, c + ν, d]] + −  [[S β ν − c − d,

S2  nw

2

+

+

−

, ν, β]] − [[β 1 2 nw

1  nw

2

S  nw

[[ ν, 2 ν]] +

2

, 2 ν]] − [[β

1

2 nw

[[ c, ν, d]] −

[[ ν, ν, c]] +

S  nw

2

S  nw

2

1

2 nw

 ν, β]] + [[d,

1 ν, Sβ − ν − c − d  nw

S

, ν, c]] − [[β [[ ν, ν, d]] −

1  nw

2

 nw

S

2 nw

 2 ν]] + [[d,

1 1 1 2  ν − ν, c − ν, d    nw nw nw

S 1 2 1 1 ν, β −  ν − ν, c − ν, d     nw nw nw nw 2

[[ c, ν, β]] +

1  nw

2

S

, ν, d]] − [[β

 nw

1 2 nw

 ν, c]] + [[d,

2

1  nw

2

(84)

[[ ν, ν, β]]

[[ c, 2 ν]] +

1 2 nw

[[ c, ν, c]]

 ν, d]] + [[d,

S ν, β  nw

(85) (86) (87)

34

P. Aavani and S. H. Rice

The following equation shows the change in the average mean phenotype of the virulence over one generation. To track where each term comes from, we separate the corresponding terms by writing the equation numbers over them:

 νi =

Equation(78)



  1 S 1  − 1 [[ ]] − [[2 [[ ν, β ν, d]] ν, c]] ν]] − [[     w w w w



  1 1 1 S ν, β − 2 ν − ν, c − ν, d     w w w w



  2S 2S 2S 1 S2 2 2 [[ ν,   β ]] − [[ ν,  β, ν ]] − [[ ν,  β, d ]] − [[ ν,  β, c ]] + [[ ν,   ν ]] 2 2 2 2 2 nw nw nw nw nw



  2 2 2 2 2 [[ ν, ν, d]] + [[ ν, ν, c]] + [[ ν, c, d]] + [[ ν, 2 c]] + [[ ν, 2 d]] 2 2 2 2 2      nw nw nw nw nw



  S2 2S 1 3 2S 2 2S 2  −  ν, β  ν, βν  +   ν  −  ν, βd  +  ν, dν  −  ν, βc  2 2 2 2 2 2 nw nw nw nw nw nw



  2 2 1 1 ν, cν + ν, cd + ν, c2  + ν, d 2  2 2 2 2     nw nw nw nw

+

−

−

−

−

Equation(79)

Equation(80)

Equation(81)

Equation(82)

Equation(83)

Equation(84)

   S2 S S S S , ν, β]] + , 2 ν]] + , ν, c]] + , ν, d]] + − − 2 [[β [[ β [[ β [[ β [[ ν,  ν, β ]]  2 2 2 2 nw nw nw nw nw Equation(85)

   1 1 1 S 1 1 2 − − 2 [[ ν, 2 ν]] − [[ ν,  ν, c ]] − [[ ν,  ν, d ]] + [[ c ,  ν, β ]] − [[ c ,   ν ]] − [[ c ,  ν, c ]]  2 2 2 2 2 nw nw nw nw nw nw Equation(86)

   1 S  1  2 1  1  S c, ν, d]] + [[ d,  ν, β ]] − [[ d,   ν ]] − [[ d,  ν, c ]] − [[ d,  ν, d ]] −  ν, β  − − 2 [[  nw  2 2 2 2 nw nw nw nw nw  −

Equation(87)

  1 2 1 1  ν + ν, c + ν, d    nw nw nw

(88)

Note that c and d enter into Eq. (69) in the same way as does ν. We can thus derive  by substituting. Because β enters Eq. (69) as Sβ, the equation  and d equation c  is structured a bit differently. We can use the same approach to derive the for β equations for the change in the average mean phenotype of the transmission over one generation:

Pathogen Evolution When Transmission and Virulence are Stochastic  βi = + − − − − − − − −

35



S 2 1 1  − 1 [[β , , d]] , ]] − [[β [[ β ν]] − [[β c]]     w w w w  S 2 1 1 1  β − β, ν − β, c − β, d     w w w w  2 S 2S 1 2S 2 2S 3 β − β, βν + β, ν 2  − β, βd + β, dν − β, βc 2 2 2 2 2 2 nw nw nw nw nw nw  2 S 2S 2S 2S 1 , 2 β]] − , β, ν]] − , β, d]] − , β, c]] + , 2 ν]] [[β [[β [[β [[β [[β 2 2 2 2 2 nw nw nw nw nw  2 2 2 2 2 , ν, d]] + , ν, c]] + , c, d]] + , 2 c]] + , 2 d]] [[β [[β [[β [[β [[β 2 2 2 2 2      nw nw nw nw nw  2 1 1 2 2  + 2   β, cν  +  β, cd  +  β, c  β, d 2 2 2 2 nw nw nw nw  S2 S S S S , 2 β]] + , β, ν]] + , β, c]] + , β, d]] + − [[β [[β [[β [[β [[ ν, 2 β]] 2 2 2 2 2      nw nw nw nw nw  1 1 S 1 1 1 2 [[ ν,  β, ν ]] − [[ ν,  β, c ]] − [[ ν,  β, d ]] + [[ c ,   β ]] − [[ c, β, ν]] − [[ c, β, c]] − 2 2 2 2 2 2       nw nw nw nw nw nw  S  2 1  1  1  S 2 1 [[ c, β, d]] + [[d,  β]] − [[d, β, ν]] − [[d, β, c]] − [[d, β, d]] −  β − 2 2 2 2 2  nw      nw nw nw nw nw  1 1 1 β, ν + β, c + β, d    nw nw nw

(89)

References 1. Gillespie, J.H.: Natural selection for variances in offspring numbers: a new evolutionary principle. Am. Nat. 111, 1010–1014 (1977) 2. Rice, S.H.: A stochastic version of the price equation reveals the interplay of deterministic and stochastic processes in evolution. BMC Evol. Biol. 8, 1 (2008) 3. Proulx, S.: Sources of stochasticity in models of sex allocation in spatially structured populations. J. Evol. Biol. 17, 924–930 (2004) 4. Keeling, M.J., Eames, K.T.: Networks and epidemic models. J. Roy. Soc. Interface 2, 295–307 (2005) 5. McCallum, H., Barlow, N., Hone, J.: How should pathogen transmission be modelled? Trends Ecol. Evol. 16, 295–300 (2001) 6. Lloyd-Smith, J.O., Schreiber, S.J., Kopp, P.E., Getz, W.M.: Superspreading and the effect of individual variation on disease emergence. Nature 438, 355 (2005) 7. Ferrari, N., Cattadori, I.M., Nespereira, J., Rizzoli, A., Hudson, P.J.: The role of host sex in parasite dynamics: field experiments on the yellow-necked mouse apodemus flavicollis. Ecol. Lett. 7, 88–94 (2004) 8. Kilpatrick, A.M., Daszak, P., Jones, M.J., Marra, P.P., Kramer, L.D.: Host heterogeneity dominates West Nile virus transmission. Proc. Roy. Soc. Lond. B Bio 273, 2327–2333 (2006) 9. Paull, S.H., Song, S., McClure, K.M., Sackett, L.C., Kilpatrick, A.M., Johnson, P.T.: From superspreaders to disease hotspots: linking transmission across hosts and space. Front. Ecol. Environ. 10, 75–82 (2012) 10. Wolinska, J., King, K.C.: Environment can alter selection in host-parasite interactions. Trends Parasitol. 25, 236–244 (2009) 11. Woolhouse, M.E., Dye, C., Etard, J.-F., Smith, T., Charlwood, J., Garnett, G., Hagan, P., Hii, J., Ndhlovu, P., Quinnell, R., et al.: Heterogeneities in the transmission of infectious agents: implications for the design of control programs. Proc. Natl. Acad. Sci. U.S.A. 94, 338–342 (1997) 12. Brodin, P., Davis, M.M.: Human immune system variation. Nat. Rev. Immunol. 17, 21–29 (2017) 13. Vale, P.F., Wilson, A.J., Best, A., Boots, M., Little, T.J.: Epidemiological, evolutionary, and coevolutionary implications of context-dependent parasitism. Am. Nat. 177, 510–521 (2011)

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14. Little, T., Vale, P., Choicy, M.: Host nutrition alters the variance in parasite transmission potential. Biol. Lett. 9 (2013). https://doi.org/10.1098/rsbl.2012.1145 15. Mitchell, S.E., Rogers, E.S., Little, T.J., Read, A.F.: Host-parasite and genotype-byenvironment interactions: temperature modifies potential for selection by a sterilizing pathogen. Evolution 59, 70–80 (2005) 16. Mellors, J.W., Munoz, A., Giorgi, J.V., Margolick, J.B., Tassoni, C.J., Gupta, P., Kingsley, L.A., Todd, J.A., Saah, A.J., Detels, R., et al.: Plasma viral load and CD4+ lymphocytes as prognostic markers of HIV-1 infection. Ann. Intern. Med. 126, 946–954 (1997) 17. Rice, S.H.: The expected value of the ratio of correlated random variables, unpublished note (2009) 18. Price, G.R.: Selection and covariance. Nature 227, 520–521 (1970) 19. Lande, R., Arnold, S.J.: The measurement of selection on correlated characters. Evolution 37, 1210–1226 (1983) 20. Lande, R.: Natural selection and random genetic drift in phenotypic evolution. Evolution 30, 314–334 (1976) 21. Day, T., Gandon, S.: Insights from price’s equation into evolutionary epidemiology. In: Feng, Z., Dieckmann, U., Levin, S.A. (eds.) Disease Evolution: Models, Concepts, and Data Analysis, vol. 71, pp. 23–44 (2006) 22. Day, T., Gandon, S.: Applying population-genetic models in theoretical evolutionary epidemiology. Ecol. Lett. 10, 876–888 (2007) 23. Harries, J.: Amoebiasis: a review. J. Roy. Soc. Med. 75, 190 (1982) 24. Powell, S., MacLeod, I., Wilmot, A., Elsdon-Dew, R.: Metronidazole in amoebic dysentery and amoebic liver abscess. The Lancet 288, 1329–1331 (1966) 25. Anderson, R.M., May, R.: Coevolution of hosts and parasites. Parasitology 85, 411–426 (1982) 26. Alizon, S., Hurford, A., Mideo, N., Van Baalen, M.: Virulence evolution and the trade-off hypothesis: history, current state of affairs and the future. J. Evol. Biol. 22, 245–259 (2009) 27. Leggett, H.C., Cornwallis, C.K., Buckling, A., West, S.A.: Growth rate, transmission mode and virulence in human pathogens. Phil. Trans. R. Soc. B 372, 20160094 (2017) 28. Krebs, C.J.: Why Ecology Matters. University of Chicago Press, Chicago (2016) 29. Papalexi, E., Satija, R.: Single-cell RNA sequencing to explore immune cell heterogeneity. Nat. Rev. Immunol. 18, 35–45 (2017). https://doi.org/10.1038/nri.2017.76 30. Duneau, D., Ferdy, J.-B., Revah, J., Kondolf, H., Ortiz, G.A., Lazzaro, B.P., Buchon, N.: Stochastic variation in the initial phase of bacterial infection predicts the probability of survival in D. melanogaster. eLife 6 (2017). https://doi.org/10.7554/elife.28298 31. Allen, E.: Modeling with Itô Stochastic Differential Equations, vol. 22. Springer, Netherlands (2007)

On the Relationship Between the Basic Reproduction Number and the Shape of the Spatial Domain Toshikazu Kuniya

Abstract In this paper, we study a spatially diffusive SIR epidemic model with constant parameters in a bounded spatial domain and investigate the relationship between the basic reproduction number R0 and the shape of the spatial domain. Under the homogeneous Neumann boundary conditions, R0 is the same as that for the classical non-diffusive SIR epidemic model, and thus, it does not depend on the shape of the spatial domain. On the other hand, under the homogeneous Dirichlet boundary conditions, the next generation operator does not have a constant eigenvector, and R0 depends on the shape of the spatial domain. By numerical simulation for the 2-dimensional rectangular domain  = (0, p) × (0, 1/ p), p > 0 with constant area || = 1, we show that such R0 attains its maximum for p = 1 (that is,  is square) and decreases as the shape of the domain  becomes long and narrow. Moreover, we observe a similar relationship between R0 and the shape of the spatial domain in a random 2-dimensional lattice model. Keywords SIR epidemic model · Basic reproduction number · Diffusion · Dirichlet boundary condition · Neumann boundary condition · Lattice model

1 Introduction The SIR epidemic model is one of the most basic mathematical models for infectious disease dynamics, in which the total population is divided into three subpopulations called susceptible, infective and removed [13]. One of the most important concepts in this field is the basic reproduction number R0 , which implies the expected value of secondary cases produced by a typical infective individual during its entire period of infectiousness [7]. R0 indicates the intensity of the spread of diseases and it is mathematically defined by the spectral radius of a linear operator called the next generation operator [7]. T. Kuniya (B) Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_2

37

38

T. Kuniya

For an SIR epidemic model in the form of ordinary differential equations, the basic reproduction number R0 plays the role of the threshold value that completely determines the global behavior of the model: if R0 ≤ 1, then the trivial diseasefree equilibrium of the model is globally asymptotically stable, whereas if R0 > 1, then the positive endemic equilibrium of the model is globally asymptotically stable [10, Section 5.5.2]. However, some (space, age, etc.) structured SIR epidemic models are systems of partial differential equations and it is not obvious whether R0 for them still plays the role of the threshold value. Therefore, to prove the threshold property of R0 for such PDEs systems is an important mathematical problem. To consider the geographical spread of infectious diseases, many spatially diffusive epidemic models have been studied by many authors for long decades (see, e.g., [1, 3, 5, 6, 8, 9, 14, 19, 20]). In this paper, we focus on a spatially diffusive SIR epidemic model with constant parameters in a bounded spatial domain  ⊂ Rn with smooth boundary. In most of the previous works (e.g., [1, 3, 5, 14, 19, 20]), the homogeneous Neumann boundary conditions were assumed, which imply that there is no inflow and outflow in the boundary. Under the homogeneous Neumann boundary conditions, the disease-free equilibrium is spatially homogeneous, and hence, the basic reproduction number R0 is the same as that for the classical SIR epidemic model without diffusion (see Corollary 3.2 in Sect. 3). Thus, in such a case, R0 does not depend on the shape of the spatial domain. This seems unrealistic because the frequency of contacts of each individual, which determines the intensity of the geographical spread of diseases, would depend on the shape of the spatial domain. In this study, we focus instead on the homogeneous Dirichlet boundary conditions, which imply that there is no individual in the boundary. For such a case, the next generation operator does not have a constant eigenvector, and R0 depends on the shape of the spatial domain (see Sects. 3 and 6). In particular, in the case of the 2-dimensional rectangular domain  = (0, p) × (0, 1/ p), p > 0 with constant area || = 1, we show by numerical simulation that such R0 attains its maximum for p = 1 (that is,  is square) and decreases as p departs from 1 (see Fig. 7 in Sect. 6). This implies that R0 decreases as the shape of the spatial domain becomes long and narrow. This seems realistic because it would be hard for individuals to contact each other in such a long and narrow domain. In this study, we support this conjecture by constructing a random 2-dimensional lattice model. The organization of this paper is as follows. In Sect. 2, we formulate a spatially diffusive SIR epidemic model with constant parameters and a general form of boundary conditions, which includes both of the homogeneous Neumann and Dirichlet boundary conditions. In Sect. 3, we define the basic reproduction number R0 for our model by the spectral radius of the next generation operator. In Sect. 4, we show the global asymptotic stability of the disease-free equilibrium of our model for R0 < 1. In Sect. 5, we show the existence of the endemic equilibrium of our model for R0 > 1. In Sect. 6, by numerical simulation for the case of the 2-dimensional rectangular domain, we show the validity of the threshold property of R0 , and investigate the relationship between R0 and the shape of the spatial domain. In Sect. 7, we construct a random 2-dimensional lattice model to support the results in Sect. 6. Finally, Sect. 8 is devoted to the conclusion.

On the Relationship Between the Basic Reproduction Number ...

39

2 The Model In this section, we formulate our model. Let  ⊂ Rn , n ∈ {1, 2, . . .} be a bounded domain with smooth boundary ∂. Let S(t, x), I (t, x) and R(t, x) denote the susceptible, infective and removed populations at time t ≥ 0 in position x = (x1 , x2 , . . . , xn ) ∈ , respectively. Let b be the birth rate, let β be the disease transmission coefficient, let μ be the mortality rate, let γ be the removal rate and let d1 , d2 and d3 be the diffusion coefficients for the susceptible, infective and removed individuals, respectively. n ∂ 2 /∂ xi2 We assume that all of these parameters are positive constants. Let  := i=1 denote the Laplacian operator and let n denote the outward unit normal vector on the boundary ∂. In this paper, we study the following spatially diffusive SIR epidemic model: ⎧ ∂ S(t, x) ⎪ ⎪ = d1 S(t, x) + b − β S(t, x)I (t, x) − μS(t, x), ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂ I (t, x) = d2 I (t, x) + β S(t, x)I (t, x) − (μ + γ )I (t, x), ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ R(t, x) = d3 R(t, x) + γ I (t, x) − μR(t, x), t > 0, x ∈ , ∂t

(1)

with the initial conditions S(0, x) = φ1 (x),

I (0, x) = φ2 (x),

R(0, x) = φ3 (x), x ∈ ,

(2)

and the boundary conditions αS(t, x) + (1 − α) ∂ S(t,x) = 0, ∂n

= 0, α R(t, x) + (1 − α) ∂ R(t,x) ∂n

(t,x) α I (t, x) + (1 − α) ∂ I ∂n = 0,

t > 0, x ∈ ∂,

(3)

where α ∈ {0, 1}. That is, (3) implies the homogeneous Neumann and Dirichlet boundary conditions for α = 0 and α = 1, respectively. We now prove the well-posedness of the problem (1)–(3). Let  X :=

α = 0, C(, R), {ϕ ∈ C(, R) : ϕ(x) = 0 for all x ∈ ∂}, α = 1,

and Y := X × X × X, equipped with the norms ϕX := sup |ϕ(x)| and ψY := x∈

3 

ψi X , ϕ ∈ X, ψ = (ψ1 , ψ2 , ψ3 ) ∈ Y,

i=1

respectively. Let X+ and Y+ be the positive cones of X and Y, respectively. We immediately obtain the following lemma on the existence and uniqueness of the local classical solution (S, I, R) to problem (1)–(3). Lemma 2.1. Let (φ1 , φ2 , φ3 ) ∈ Y+ . Then, there exists a τ > 0 such that the unique solution (S(t, ·), I (t, ·), R(t, ·)) ∈ Y+ ∩ C 2 (, R3 ) satisfies (1)–(3) for all t ∈ (0, τ ).

40

T. Kuniya

Proof. For r := (r1 , r2 , r3 ) ∈ R3 , let F1 (r) := b − βr1r2 − μr1 , F2 (r) := βr1 r2 − (μ + γ )r2 , F3 (r) := γ r2 − μr3 . It is then easy to see that Fi (r) ≥ 0 for all i ∈ {1, 2, 3} and r ∈ R3+ such that ri = 0. The assertion then holds from [17, Theorem 3.1 and Corollary 3.2 in Chapter 7]. This completes the proof.  Before proving the existence of the global classical solution (that is, τ = +∞), we define the fundamental solution i , i ∈ {1, 2, 3} associated with ∂t − di . More precisely, let wi , i ∈ {1, 2, 3} be the solution to the following problem: ⎧ ∂wi (t,x) t > 0, x ∈ , ⎨ ∂t = di wi (t, x), w (0, x) = wi,0 (x), x ∈ , ⎩ i i (t,x) = 0, t > 0, x ∈ ∂. αwi (t, x) + (1 − α) ∂w∂n

(4)

By [11, Theorems 7.1 and 8.1], there exists the unique fundamental solution i (t, x, y), i ∈ {1, 2, 3}, t > 0, x, y ∈  such that wi (t, x) =



i (t, x, y)wi,0 (y)dy

satisfies (4), and i (t, x, ·) itself satisfies the first and third equations in (4) as a function of t and x. By [11, Theorems 8.3–8.5, 10.1], the fundamental solution i , i ∈ {1, 2, 3} satisfies the following properties: (P1) i (t, x, y) ≥ 0 for all t > 0 and x, y ∈ . (P2)

i (t, x, y) > 0 for all t > 0 and x, y ∈ . (P3)  i (t, x, y)dy ≤ 1 for all t > 0 and x ∈ .

(P4)  i (t, x, y)dy = 1 for all t > 0 and x ∈  if α = 0. (P5) i (t, x, y)

= i (t, y, x) for all t > 0 and x, y ∈ . (P6) limt→+0  i (t, x, y)ϕ(y)dy = ϕ(x) for all ϕ ∈ X and x ∈ .

(P7)  i (t, x, z) i (s, z, y)dz = i (t + s, x, y) for all t, s > 0 and x, y, z ∈ . For some examples of i , i ∈ {1, 2, 3}, see Sect. 6. We now prove the following theorem on the existence of the global classical solution (S, I, R) to problem (1)–(3). Theorem 2.2. Let (φ1 , φ2 , φ3 ) ∈ Y+ . Then, there exists the unique solution (S(t, ·), I (t, ·), R(t, ·)) ∈ Y+ ∩ C 2 (, R3 ) that satisfies (1)–(3) for all t ∈ (0, +∞). Proof. By Lemma 2.1, we see that there exists an upper solution S that satisfies ⎧ ∂ S(t,x) ⎪ ⎨ ∂t = d1 S(t, x) + b − μS(t, x), t ∈ (0, τ ), x ∈ , S(0, x) = φ1 (x), x ∈ , ⎪ ⎩ ∂ S(t,x) t ∈ (0, τ ), x ∈ ∂. αS(t, x) + (1 − α) ∂n = 0,

On the Relationship Between the Basic Reproduction Number ...

41

That is, S(t, x) ≤ S(t, x) for all t ∈ [0, τ ) and x ∈ . By using the fundamental solution 1 , we obtain, for t ∈ (0, τ ) and x ∈ , S(t, x) = e

−μt





t

1 (t, x, y)φ1 (y)dy + b

e

−μ(t−σ )

0



1 (t − σ, x, y)dydσ. (5)

From (P3), we have that, for t ∈ (0, τ ) and x ∈ , S(t, x) ≤ e−μt φ1 X + b



t

e−μ(t−σ ) dσ =

0

 b b + φ1 X − e−μt . μ μ

This implies that S is uniformly bounded above by K := max (φ1 X , b/μ), that is, S(t, x) ≤ K for all t ∈ [0, τ ) and x ∈ . We then see from the second equation in (1) that there exists an upper solution I that satisfies ⎧ ∂ I (t,x) ⎪ ⎨ ∂t = d2 I (t, x) + (β K − μ − γ )I (t, x), t ∈ (0, τ ), x ∈ , I (0, x) = φ2 (x), x ∈ , ⎪ ⎩ ∂ I (t,x) t ∈ (0, τ ), x ∈ ∂. α I (t, x) + (1 − α) ∂n = 0, As in the above discussion, we obtain the following upper estimate for I by using the fundamental solution 2 : for t ∈ (0, τ ) and x ∈ , (β K −μ−γ )t I (t, x) ≤ I (t, x) = e 2 (t, x, y)φ2 (y)dy ≤ e(β K −μ−γ )t φ2 X . (6) 

In a similar way, we obtain the following upper estimate for R by using the fundamental solution 3 : for t ∈ (0, τ ) and x ∈ , −μt R(t, x) ≤ e 3 (t, x, y)φ3 (y)dy  t −μ(t−σ ) e 3 (t − σ, x, y)I (σ, y)dydσ +γ 0  t −μt −μt φ3 X + γ e e(β K −γ )σ dσ φ2 X . (7) ≤e 0

Inequalities (6) and (7) imply that the solution never blows up in finite time. Thus, τ = +∞ follows from [17, Theorem 3.1 in Chap. 7]. This completes the proof.  By Theorem 2.2, we immediately obtain the following corollary on the existence of a semiflow. Corollary 2.3. Let φ = (φ1 , φ2 , φ3 ) ∈ Y+ . Then, there exists a semiflow  : R+ × Y+ → Y+ ∩ C 2 (, R3 ) such that (t, φ) = (S(t, ·), I (t, ·), R(t, ·)) for all t ≥ 0.

42

T. Kuniya

Proof. The assertion directly follows from Theorem 2.2 and [17, Theorem 3.1 (d) in Chap. 7]. This completes the proof.  By Theorem 2.2 and Corollary 2.3, the problem (1)–(3) is well-posed.

3 Basic Reproduction Number In this section, we define the basic reproduction number R0 for problem (1)–(3). Let E 0 : (S, I, R) = (S0 , 0, 0) ∈ Y+ ∩ C 2 (, R3 ) denote the disease-free equilibrium for problem (1)–(3). It then satisfies 

0 = d1 S0 (x) + b − μS0 (x), x ∈ , 0 (x) = 0, x ∈ ∂. αS0 (x) + (1 − α) ∂ S∂n

(8)

By using the fundamental solution 1 , we obtain the following expression of S0 : S0 (x) = b

+∞

e

−μt



0

1 (t, x, y)dydt, x ∈ .

(9)

In fact, we can easily check that this S0 satisfies both equations in (8) by using the fact that 1 (t, x, ·) satisfies the first and third equations in (4) as a function of t and x. By (P2) and (P3), we obtain the following inequalities:

+∞

0 < S0 (x) ≤ b 0

e−μt dt =

b , x ∈ . μ

(10)

In particular, by (P4), the equality S0 (x) ≡ b/μ holds if α = 0. That is, the susceptible population S0 in the disease-free equilibrium E 0 is spatially homogeneous under the homogeneous Neumann boundary conditions (α = 0). By linearizing the second equation in (1) around the disease-free equilibrium E 0 , we obtain ⎧ ∂ I (t,x) ⎨ ∂t = d2 I (t, x) + β S0 (x)I (t, x) − (μ + γ )I (t, x), t > 0, x ∈ , x ∈ , I (0, x) = φ2 (x), ⎩ (t,x) α I (t, x) + (1 − α) ∂ I ∂n = 0, t > 0, x ∈ ∂. By using the fundamental solution 2 , we obtain the following expression of I : 2 (t, x, y)φ2 (y)dy I (t, x) = e−(μ+γ )t  t e−(μ+γ )(t−σ ) 2 (t − σ, x, y)β S0 (y)I (σ, y)dydσ, t > 0, x ∈ . (11) + 0



On the Relationship Between the Basic Reproduction Number ...

43

Let u(t, x) := β S0 (x)I (t, x) denote the newly infected population at time t > 0 in position x ∈ . We then obtain by multiplying β S0 (x) by both sides of (11) that

t

u(t, x) = g(t, x) + β S0 (x)

e−(μ+γ )σ

0





2 (σ, x, y)u(t − σ, y)dydσ, t > 0, x ∈ ,

where g(t, x) := β S0 (x)e−(μ+γ )t



2 (t, x, y)φ2 (y)dy, t > 0, x ∈ .

Following the theory in [7], we define the next generation operator K : X → X by

+∞

Kϕ(x) := β S0 (x)

e−(μ+γ )σ



0

2 (σ, x, y)ϕ(y)dydσ, ϕ ∈ X,

(12)

and define R0 by the spectral radius of K: R0 := r (K). We now prove the following proposition on R0 . Proposition 3.1. R0 is a positive eigenvalue of operator K corresponding to an eigenvector v0 ∈ X+ that is strictly positive on . Proof. By (P2) and (10), we see that K is strongly positive, that is, Kϕ(x) > 0 for all x ∈  if ϕ ∈ X+ \ {0}. Moreover, K is compact as it has a continuous kernel +∞ k(x, y) := β S0 (x) 0 e−(μ+γ )σ 2 (σ, x, y)dσ , which is strictly positive on  × . Thus, the assertion follows from the Krein-Rutman theorem [2, Theorem 3.2]. This completes the proof.  In particular, we can obtain the explicit form of R0 under the homogeneous Neumann boundary conditions (α = 0). Corollary 3.2. If α = 0, then b 1 . μμ+γ

R0 = β

Proof. As stated above, by (P4), we have that S0 (x) ≡ b/μ for α = 0. The next generation operator K then becomes Kϕ(x) = β

b μ

0

+∞

e−(μ+γ )σ



2 (σ, x, y)ϕ(y)dydσ, ϕ ∈ X.

By (P4) again, every positive constant function v0 ∈ X+ \ {0} plays the role of the eigenvector of K corresponding to the eigenvalue β(b/μ)(μ + γ )−1 . By the KreinRutman theorem [2, Theorem 3.2], R0 is the only eigenvalue of K having a positive eigenvector, which implies that R0 = β(b/μ)(μ + γ )−1 . This completes the proof. 

44

T. Kuniya

Corollary 3.2 implies that the basic reproduction number R0 for system (1) under the homogeneous Neumann boundary conditions (α = 0) is equal to that for the classical SIR epidemic model without diffusion [10, Section 5.5.2]. Therefore, R0 in this case is spatially homogeneous and does not depend on the shape of the spatial domain. In contrast, R0 under the homogeneous Dirichlet boundary conditions (α = 1) can not be explicitly given and depends on the shape of the spatial domain (see Sect. 6).

4 Global Asymptotic Stability of the Disease-Free Equilibrium In this section, we prove the global asymptotic stability of the disease-free equilibrium E 0 for R0 < 1. Following the theory in [18], we define two linear operators B and C by  0 B ϕ(x) := d2 ϕ(x) − (μ + γ )ϕ(x), ϕ ∈ D(B 0 ), ϕ ∈ X, Cϕ(x) := β S0 (x)ϕ(x), where D(B 0 ) := {ϕ ∈ X ∩ C 2 (, R) : B 0 ϕ ∈ X}. Let B be the closure of B 0 in X. It then generates the uniformly bounded C0 -semigroup {T2 (t)}t≥0 defined by [T2 (t)ϕ] (x) := e

−(μ+γ )t



2 (t, x, y)ϕ(y)dy, t > 0, ϕ ∈ X, x ∈ .

 T2 (t)ϕ − ϕ exists . D(B) := ϕ ∈ X : lim t→+0 t

and

We can easily see that, for λ ∈ R, −1

(λ − B)

ϕ(x) =

+∞

e 0

−(μ+γ +λ)σ



2 (σ, x, y)ϕ(y)dydσ, ϕ ∈ X.

This implies that B is resolvent-positive and s(B) ≤ −(μ + γ ) < 0, where s(B) denotes the spectral bound of B. We see from (12) that R0 = r (K) = r (−CB −1 ). We now prove the following proposition on the local (in)stability of the disease-free equilibrium E 0 . Proposition 4.1. If R0 < 1, then the disease-free equilibrium E 0 for problem (1)– (3) is locally asymptotically stable. If R0 > 1, then E 0 is unstable.

On the Relationship Between the Basic Reproduction Number ...

45

Proof. By (P3), we have that, for t > 0, T2 (t) = sup T2 (t)ϕX ≤ e−(μ+γ )t , ϕX ≤1

and hence, we have ln T2 (t) ≤ −(μ + γ ) < 0, t→+∞ t

ω(T2 ) = lim

where ω(·) denotes the exponential growth bound of a semigroup. We see that C is linear, positive and bounded. As in the proof of Proposition 3.1, we see that CT2 (t) is compact for all t > 0. Let A := B + C be the positive perturbation of B. We then see that A generates a positive C0 -semigroup {T (t)}t≥0 , which is associated with the linearization of the semiflow  around the disease-free equilibrium E 0 . We then see from [18, Theorem 3.16] that if R0 < 1, then s(A) < 0 and ω(T ) < 0, which implies that E 0 is locally asymptotically stable. On the other hand, we see from [18, Theorem 3.17] that if R0 > 1, then s(A) > 0 and ω(T ) > 0, which implies that E 0 is unstable. This completes the proof.  We see from (5) and (9) that, for all x ∈ , 0 ≤ S(t, x) ≤ S(t, x) → S0 (x) (t → +∞). Hence, without loss of generality, we can restrict our attention to the initial condition and the solution in the following spaces:   D1 := (φ1 , φ2 , φ3 ) ∈ Y+ : φ1 (x) ≤ S0 (x) for all x ∈  , D2 := D1 ∩ C 2 (, R3 ). We easily see that (S(t, ·), I (t, ·), R(t, ·)) ∈ D2 for all t > 0 provided (φ1 , φ2 , φ3 ) ∈ D1 because S0 itself is one of the upper solution S to S. We now prove the following theorem on the global asymptotic stability of the disease-free equilibrium E 0 . Theorem 4.2. If R0 < 1, then the disease-free equilibrium E 0 ∈ D2 ⊂ D1 for problem (1)–(3) is globally asymptotically stable in D1 , that is, (S(t, ·), I (t, ·), R(t, ·)) → E 0 as t → +∞, provided (φ1 , φ2 , φ3 ) ∈ D1 . Proof. As stated above, S(t, x) ≤ S0 (x) for all t > 0 and x ∈ . Hence, there exists an upper solution I˜ that satisfies ⎧ ∂ I˜(t,x) ⎪ ⎨ ∂t = d2  I˜(t, x) + β S0 (x) I˜(t, x) − (μ + γ ) I˜(t, x), t > 0, x ∈ , x ∈ , I˜(0, x) = φ2 (x), ⎪ ˜(t,x) ⎩ ˜ α I (t, x) + (1 − α) ∂ I∂n = 0, t > 0, x ∈ ∂.

46

T. Kuniya

We see from the proof of Proposition 4.1 that I˜(t, x) = [T (t)φ2 ] (x) and ω(T ) < 0 for R0 < 1. Hence, we have that, for all x ∈ , 0 ≤ I (t, x) ≤ I˜(t, x) = [T (t)φ2 ] (x) → 0 (t → +∞). For any small  > 0, there then exists a large T > 0 such that I (t, x) ≤  for all t ≥ T and x ∈ . We next consider the lower solution S that satisfies ⎧ ∂ S(t,x) ⎨ ∂t = d1 S(t, x) + b − (β + μ) S(t, x), t > T, x ∈ , x ∈ , S(T, x) = S(T, x), ⎩ = 0, t > T, x ∈ ∂. αS(t, x) + (1 − α) ∂ S(t,x) ∂n We then have that, for all t > T and x ∈ , 1 (t − T, x, y)S(T, y)dy S(t, x) ≥ S(t, x) = e−(β+μ)(t−T )  t e−(β+μ)(t−σ ) 1 (t − σ, x, y)dydσ +b T  +∞ →b e−(β+μ)σ 1 (σ, x, y)dydσ =: S0, (x) (t → +∞). 0



Since  is arbitrary and S0, → S0 as  → +0, we obtain that S(t, x) → S0 (x) as t → +∞ for all x ∈ . In a similar manner, we can show that R(t, x) → 0 as  t → +∞ for all x ∈ . This completes the proof. Theorem 4.2 implies that the infectious disease will eventually be eradicated if R0 < 1. That is, to reduce R0 below 1 is a significant target for the eradication of the disease in our model.

5 Existence of the Endemic Equilibrium In this section, we investigate the existence of the endemic equilibrium for problem (1)–(3) for R0 > 1. Let E ∗ : (S, I, R) = (S ∗ , I ∗ , R ∗ ) ∈ Y+ ∩ C 2 (, R3 ) such that I ∗ = 0 denote an endemic equilibrium. It then satisfies ⎧ ⎨ 0 = d1 S ∗ (x) + b − β S ∗ (x)I ∗ (x) − μS ∗ (x), 0 = d2 I ∗ (x) + β S ∗ (x)I ∗ (x) − (μ + γ )I ∗ (x), ⎩ x ∈ , 0 = d3 R ∗ (x) + γ I ∗ (x) − μR ∗ (x),

(13)

On the Relationship Between the Basic Reproduction Number ...

47

and ∂ S ∗ (x) = 0, ∂n ∗ ∂ R (x) = 0, α R ∗ (x) + (1 − α) ∂n αS ∗ (x) + (1 − α)

∗

α I ∗ (x) + (1 − α) ∂ I∂n(x) = 0, x ∈ ∂.

We now establish the following theorem on the existence of the endemic equilibrium E ∗ for R0 > 1 under the homogeneous Neumann boundary conditions (α = 0). Theorem 5.1. Suppose that α = 0. If R0 > 1, then there exists at least one constant endemic equilibrium E ∗ for problem (1)–(3). Proof. There can exist a spatially homogeneous endemic equilibrium such that S ∗ (x) ≡ S ∗ , I ∗ (x) ≡ I ∗ , R ∗ (x) ≡ R ∗ and ⎧ ⎨ 0 = b − β S ∗ I ∗ − μS ∗ , 0 = β S ∗ I ∗ − (μ + γ )I ∗ , ⎩ 0 = γ I ∗ − μR ∗ . Since R0 = β(b/μ)(μ + γ )−1 by Corollary 3.2, we can obtain the following expression as in the classical argument (see, e.g., [10, Section 5.5.2]): S∗ =

μ+γ , β

I∗ =

μ (R0 − 1), β

R∗ =

γ (R0 − 1). β

Hence, the endemic equilibrium E ∗ such that I ∗ > 0 exists if R0 > 1. This completes the proof.  On the existence of the endemic equilibrium E ∗ under the homogeneous Dirichlet boundary conditions (α = 1), we restrict our attention to the special case where d1 = d2 = d3 =: d > 0. In this case, the total population N (t, x) := S(t, x) + I (t, x) + R(t, x) satisfies  ∂ N (t,x)

= dN (t, x) + b − μN (t, x), t > 0, x ∈ , ∂t N (t, x) = 0, t > 0, x ∈ ∂.

(14)

By (8), we see that S0 (x) is the positive equilibrium for problem (14). We have that, for all t > 0 and x ∈ , N (t, x) = e

−μt





→b 0

t

−μ(t−σ )



(t, x, y)N (0, y)dy + b e (t − σ, x, y)dydσ 0  e−μσ (σ, x, y)dydσ = S0 (x) (t → +∞) ,

 +∞



where := 1 = 2 = 3 . Hence, without loss of generality, we can assume that S(t, x) + I (t, x) + R(t, x) ≡ S0 (x) for all t ≥ 0 and x ∈ . We now prove the

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T. Kuniya

following theorem on the existence of E ∗ for R0 > 1 and d1 = d2 = d3 under the homogeneous Dirichlet boundary conditions (α = 1). Theorem 5.2. Suppose that α = 1 and d1 = d2 = d3 . If R0 > 1, then there exists at least one endemic equilibrium E ∗ ∈ Y+ ∩ C 2 (, R3 ) for problem (1)–(3). Proof. Since S ∗ (x) = S0 (x) − I ∗ (x) − R ∗ (x) for all x ∈ , the second equation in (13) can be rewritten as   0 = dI ∗ (x) + β S0 (x)I ∗ (x) − μ + γ + β I ∗ (x) + β R ∗ (x) I ∗ (x), x ∈ . (15) By using the Feynman-Kac formula for boundary value problems [15, Chapter 9], we obtain the following expression of I ∗ : I ∗ (x) = Ex



τ



σ ∗ ∗ β S0 (X σ )I ∗ (X σ )e− 0 [μ+γ +β I (X s )+β R (X s )]ds dσ , x ∈ ,

0

(16) where {X t }t≥0 denotes the Itô diffusion whose generator coincides with d on C02 (Rn ), Ex [·] denotes the expectation operator with respect to the probability law of X t starting at x, and τ denotes the first exit time defined by τ := / }. In particular, we have inf {t > 0 : X t ∈ Xt = x +

√ 2d Bt , t ≥ 0,

where {Bt }t≥0 denotes the n-dimensional Brownian motion such that B0 = 0. From the third equation in (13), we obtain R ∗ (x) =



+∞

e−μσ



0

(σ, x, y)γ I ∗ (y)dydσ, x ∈ .

(17)

Let us define linear operators ,  : X → X by, for ϕ ∈ X and x ∈ , (ϕ)(x) := 0

+∞

e−μσ



(σ, x, y)γ ϕ(y)dydσ

and 

τ

(ϕ)(x) := Ex



σ β S0 (X σ )ϕ(X σ )e− 0 [μ+γ +βϕ(X s )+β(ϕ)(X s )]ds dσ .

(18)

0

We then see from (16) and (17) that if  has a nontrivial fixed point in X \ {0}, then positive solutions I ∗ and R ∗ exist. To prove that  has a nontrivial fixed point in X \ {0}, we use the Krasnoselskii fixed point theorem in [12, Theorem 4.11]. To this end, we need to show that

On the Relationship Between the Basic Reproduction Number ...

49

(i) (X+ ) ⊂ X+ and (0) = 0. (ii)  has the strong Fréchet derivative  [0] and the strong asymptotic derivative  [∞] with respect to X+ . (iii) The spectrum of  [∞] lies in the circle centered at 0 with radius less than 1. (iv)  [0] has a positive eigenvector in X+ \ {0} corresponding to an eigenvalue λ0 > 1, and there is no eigenvector of  [0] in X+ \ {0} corresponding to eigenvalue 1. (v)  is compact. The assertion (i) obviously holds. We have, for all ϕ ∈ X+ and x ∈ ,  τ  x −(μ+γ )σ (ϕ)(x) dσ

σ  = E 0 β S0 (X σ )ϕ(X σ )e   τ {−β 0 [ϕ(X s )+(ϕ)(X s )]ds }n dσ , + Ex 0  β S0 (X σ )ϕ(X σ )e−(μ+γ )σ +∞ n=1 n! and thus,

lim

   ˜   (ϕ) − Kϕ

X

ϕX   τ    b ≤ lim sup β Ex e−(μ+γ )σ dσ eβ(1+)ϕX τ − 1 = 0, ϕX →0 x∈ μ 0

ϕX →0

(19)

where, for ϕ ∈ X and x ∈ ,  τ  x −(μ+γ )σ ˜ Kϕ(x) := E β S0 (X σ )ϕ(X σ )e dσ 0 +∞ e−(μ+γ )σ (σ, x, y)β S0 (y)ϕ(y)dydσ, =

(20)



0

and   = sup (ϕ)X = sup γ ϕX ≤1

x∈

+∞

e

−μσ





0



(σ, x, y)dydσ

< +∞.

˜ Moreover, we have, for all ϕ ∈ X+ and x ∈ , (19) implies that  [0] = K. (ϕ)(x) ≤ μb Ex



τ

βϕ(X σ )e−

σ 0

βϕ(X s )ds

 0 

τ = μb Ex 1 − e− 0 βϕ(X s )ds ≤ μb .

 dσ

(21)

Thus, the range (X+ ) is bounded, which implies that the strong asymptotic derivative  [∞] is equal to zero. Consequently, the assertions (ii) and (iii) hold.

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T. Kuniya

˜ = r (K) = R0 > 1. The By (12) and (20), we see that λ0 = r ( [0]) = r (K) assertion (iv) then holds by the Krein-Rutman theorem [2, Theorem 3.2] as in the proofs of Proposition 3.1 and Corollary 3.2. Let B := {ϕ ∈ X : ϕX ≤ 1}, let {ϕn }n∈N be an arbitrary function sequence in B and let vn := (ϕn ). By the Ascoli-Arzelà theorem, to prove (v), it suffices to show the equi-boundedness and the equi-continuity of {vn }. In fact, similar to (21), we see that there exists a K > 0 such that, for all n ∈ N and x ∈ , 

τ b x   E 1 − e− 0 βϕn (X s )ds  μ    b ≤ max 1, Ex eβτ < K , μ

|vn (x)| = |(ϕn )(x)| ≤

which implies the equi-boundedness of {vn }. Next, we consider the following equation.   0 = dψ(x) + β S0 (x)ϕn (x) − μ + γ + βϕn (x) + β(ϕn )(x) ψ(x), x ∈ . (22) Since ϕn is bounded and continuous, (22) with the Dirichlet boundary conditions has the solution, for all x ∈ ,   τ

σ ψ(x) = Ex β S0 (X σ )ϕn (X σ )e− 0 [μ+γ +βϕn (X s )+β(ϕn )(X s )]ds dσ = vn (x). 0

Moreover, regarding β S0 (x)ϕn (x) − β[ϕn (x) + (ϕn )(x)]vn (x) as the nonhomogeneous term, we obtain

+∞

vn (x) = 0

×

e−(μ+γ )τ



(σ, x, y) {β S0 (y)ϕn (y) − β [ϕn (y) + (ϕn )(y)] vn (y)} dydσ.

We then have, for all n ∈ N, x ∈  and small h ∈  such that x + h ∈ ,

+∞

|vn (x + h) − vn (x)| ≤ β 0



b ≤β + (1 + K )K μ

e−(μ+γ )τ





| (σ, x + h, y) − (σ, x, y)|

× [S0 (y)|ϕn (y)| + (|ϕn (y)| + |vn (y)|) |vn (y)|] dydσ  +∞ | (σ, x + h, y) − (σ, x, y)| dydσ, e−(μ+γ )τ 0



which implies the equi-continuity of {vn }. Hence, by the Ascoli-Arzelà theorem,  is compact. Thus, (v) holds. Consequently, from (i)–(v), we can apply the Krasnoselskii fixed point theorem [12, Theorem 4.11] to conclude that  has at least one nontrivial fixed point in X+ \ {0}. Thus, there exist positive I ∗ and R ∗ satisfying (16) and (17).

On the Relationship Between the Basic Reproduction Number ...

51

Finally, we show that 0 ≤ I ∗ (x) + R ∗ (x) ≤ S0 (x) for all x ∈ . Let U ∗ (x) := I (x) + R ∗ (x). Adding the third equation in (13) and (15), we obtain ∗

0 = dU ∗ (x) + β S0 (x)I ∗ (x) − [μ + β I ∗ (x)] U ∗ (x) =: L[U ∗ ](x), x ∈ . We then have that L[S0 ](x) = −b < 0 and L[0](x) = β S0 (x)I ∗ (x) ≥ 0 for all x ∈ . This implies that S0 and 0 are the upper and lower solutions to U ∗ , respectively.  Thus, 0 ≤ I ∗ (x) + R ∗ (x) ≤ S0 (x) for all x ∈ . This completes the proof. By Theorems 4.2, 5.1 and 5.2, we can state that the basic reproduction number R0 for our model is a threshold value that determines whether the infectious disease will be eradicated or not in a mathematical sense.

6 Numerical Simulation In this section, we perform numerical simulation for the case of 2-dimensional rectangular domain  := (0, 1 ) × (0, 2 ). In this case, the fundamental solution i (t, x, y), i ∈ {1, 2, 3}, t > 0, x = (x1 , x2 ), y = (y1 , y2 ) ∈ [0, 1 ] × [0, 2 ] is given by, for the homogeneous Neumann boundary conditions (α = 0), i (t, x, y) = + 122 +

4 1 2

+

2 1 2

n=1

cos

1 1 2

+∞ +∞

m,n=1

+∞

mπ y1 −di mπ x1 e m=1 cos 1 cos  21 n 2 nπ y2 −di 2 π t nπ x2 cos 2 e 2

cos

mπ x1 1

cos

nπ x2 2

cos

mπ y1 1

cos



m 1

2

π 2t

nπ y2 −di e 2



m 1

2  2  π 2t + n 2

,

and, for the homogeneous Dirichlet boundary conditions (α = 1), i (t, x, y) = +∞ 4  mπ x1 nπ x2 mπ y1 nπ y2 −di sin sin sin sin e 1 2 m,n=1 1 2 1 2



m 1

2  2  + n π 2t 2

,

(see [11, Section 16]). In what follows, we fix the following parameters, b = 1, μ = 1, γ = 2, d = d1 = d2 = d3 = 0.02, −

2  2    − x1 − 21 − x2 − 22

1 φ1 (x) = 1 − φ2 (x), φ2 (x) = √2πδ e 2 φ3 (x) = 0, x = (x1 , x2 ) ∈ [0, 1 ] × [0, 2 ],

2δ 2

× 10−2 , δ = 0.1,

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T. Kuniya

(a) t = 0

(b) t = 0 .2

(c) t = 20

Fig. 1 Time variation of the infective population I (t, x) for the case of the homogeneous Neumann boundary conditions (α = 0) and the square domain ( p = 1) with R0 ≈ 0.9667 < 1 (β = 2.9)

(a) t = 0

(b) t = 5

(c) t = 20

Fig. 2 Time variation of the infective population I (t, x) for the case of the homogeneous Neumann boundary conditions (α = 0) and the square domain ( p = 1) with R0 ≈ 1.1333 > 1 (β = 3.4)

and vary the parameters β > 0, α ∈ {0, 1} and p > 0, where 1 = p and 2 = 1/ p. Note that the area of the domain  is fixed to || = 1 2 = 1 and the shape of the domain  changes as p varies. We first consider the case of the homogeneous Neumann boundary conditions (α = 0). In this case, the basic reproduction number R0 is explicitly given as in Corollary 3.2. For β = 2.9, we obtain R0 ≈ 0.9667 < 1. Hence, by Theorem 4.2, we can expect that the disease-free equilibrium E 0 is globally asymptotically stable. In fact, Fig. 1 shows that the infective population I in the square domain ( p = 1) converges to zero as time evolves. On the other hand, for β = 3.4, we obtain R0 ≈ 1.1333 > 1. Hence, by Proposition 4.1 and Theorem 5.1, we can expect that the disease-free equilibrium E 0 is unstable and there exists at least one endemic equilibrium E ∗ . In fact, Fig. 2 shows that the infective population I in the square domain ( p = 1) converges to the spatially homogeneous solution I ∗ = (μ/β)(R0 − 1) ≈ 0.0392 as time evolves. By Corollary 3.2, we see that R0 in this case is independent of p, which implies that R0 does not depend on the shape of the spatial domain. In fact, Fig. 3 shows that the infective population I in the rectangular domain ( p = 2) with R0 ≈ 1.1333 > 1 (β = 3.4) converges to the spatially homogeneous solution I ∗ ≈ 0.0392, which is similar as in the previous example.

On the Relationship Between the Basic Reproduction Number ...

(a) t = 0

(b) t = 5

53

(c) t = 20

Fig. 3 Time variation of the infective population I (t, x) for the case of the homogeneous Neumann boundary conditions (α = 0) and the rectangular domain ( p = 2) with R0 ≈ 1.1333 > 1 (β = 3.4)

(a) t = 0

(b) t = 0.2

(c) t = 20

Fig. 4 Time variation of the infective population I (t, x) for the case of the homogeneous Dirichlet boundary conditions (α = 1) and the square domain ( p = 1) with R0 ≈ 0.9582 < 1 (β = 3.6)

We next consider the case of the homogeneous Dirichlet boundary conditions (α = 1). In this case, we numerically calculate the basic reproduction number R0 by applying the Fredholm discretization method [4, Section 3.1] to the eigenvalue problem R0 v0 = Kv0 . For β = 3.6 in the square domain ( p = 1), we obtain R0 ≈ 0.9582 < 1. Hence, by Theorem 4.2, we can expect that the disease-free equilibrium E 0 is globally asymptotically stable. In fact, Fig. 4 shows that the infective population I converges to zero as time evolves. On the other hand, for β = 4.6 in the square domain ( p = 1), we obtain R0 ≈ 1.2244 > 1. Hence, from Proposition 4.1 and Theorem 5.2, we can expect that the disease-free equilibrium E 0 is unstable and there exists an endemic equilibrium E ∗ . In fact, Fig. 5 shows that the infective population I converges to a positive non-uniform distribution as time evolves. In contrast to the case of the homogeneous Neumann boundary conditions (α = 0), R0 in this case can depend on p, that is, it depends on the shape of the spatial domain. In fact, if we change p from 1 to 2 in the previous example, then R0 for the rectangular domain becomes R0 ≈ 0.7984 < 1, and thus, the disease-free equilibrium E 0 is globally asymptotically stable (Fig. 6). Figure 7 exhibits that such R0 attains its maximum in the square domain ( p = 1) and decreases as p departs from 1 and the shape of the domain becomes long and narrow.

54

T. Kuniya

(a) t = 0

(b) t = 1

(c) t = 20

Fig. 5 Time variation of the infective population I (t, x) for the case of the homogeneous Dirichlet boundary conditions (α = 1) and the square domain ( p = 1) with R0 ≈ 1.2244 > 1 (β = 4.6)

(a) t = 0

(b) t = 0.5

(c) t = 20

Fig. 6 Time variation of the infective population I (t, x) for the case of the homogeneous Dirichlet boundary conditions (α = 1) and the rectangular domain ( p = 2) with R0 ≈ 0.7984 < 1 (β = 4.6)

This seems realistic because the individuals would be less likely to contact each other in a long and narrow region.

7 The Lattice Model In this section, by constructing a lattice model, we support our conjecture in the previous section that the frequency of contacts of each individual would decrease as the shape of the spatial domain becomes long and narrow. The general settings are as follows: • The total number of lattice points is fixed to N . • The total number of lattice points along the horizontal axis is N1 = P, and that along the vertical axis is N2 = N /P, where P > 0 is a positive divisor of N . • At the initial time t = 0, one infective individual is placed at the center (N1 /2, N2 /2) of the lattice and S0 susceptible individuals are randomly distributed among the other lattice points, where · denotes the ceiling function. • For each time step t ∈ {1, 2, . . .}, the following procedure is carried out:

On the Relationship Between the Basic Reproduction Number ...

55

Fig. 7 R0 versus p ∈ (0, 5] under the homogeneous Dirichlet boundary conditions (α = 1) with β = 4.6

1. Each individual choose the direction (up, down, left or right) it will move with the equal probability. If there is an adjacent lattice point with no individual along the chosen direction, then the individual moves there. Otherwise, it stays at the current lattice point (Fig. 8(a)). 2. Each susceptible individual can be infected by the infective individual(s) in the adjacent lattice point(s) with probability β per one infective individual (Fig. 8(b)). 3. Each infective individual can be removed with probability γ (Fig. 8(c)). 4. If there is no infective individual, then we stop the iteration and let S∞ be the number of susceptible individuals in that time. Otherwise, we continue to the next time step t + 1. • After the iteration, using the final size equation [10, Section 5.1.2], we define the basic reproduction number R0 by  R0 := −

log

S0 +1 (1 S0

p∞

− p∞ )

 =

S0 + 1 S0 log , S0 + 1 − S∞ S∞

(23)

where p∞ := (S0 + 1 − S∞ ) / (S0 + 1) denotes the proportion of the total number of individuals that finally contacts the disease. In what follows, we fix the following parameters, N = 400, S0 = 100, β = 0.5, γ = 0.1, and vary P to observe the relationship between the basic reproduction number R0 and the shape of the spatial domain {1, 2, . . . , P} × {1, 2, . . . , 400/P}. We first set P = 20. In this case, the spatial domain is the square lattice {1, 2, . . . , 20} × {1, 2, . . . , 20}. For example, we obtain a trial result as shown in Fig. 9.

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T. Kuniya

6 S ?

-

S I

(b) Each susceptible individual can be infected with probability β by an infective individual in the adjacent lattice.

(a) Each individual can move to either one of the four directions (up, down, left or right) with the equal probability.

I

I I

-

R

(c) Each infective individual can be removed with probability γ.

Fig. 8 The general rule in our lattice model

(a) t = 0

(b) t = 15

(c) t = 92

Fig. 9 A trial result of our lattice model for the case of the square lattice {1, 2, . . . , 20} × {1, 2, . . . , 20}. Green, red and blue squares mean susceptible, infective and removed individuals, respectively. In this case, S∞ = 25 and R0 ≈ 1.8423

In this case, the iteration stops at t = 92 and S∞ = 25. Hence, from (23), we obtain R0 ≈ 1.8423. We then repeat the trial 10000 times. Let R(m) 0 be the basic reproduction (m)

number R0 in the m-th trial and let R0 be its cumulative average number, that is, (m)

R0

:= (m)

m 1  (k) R , 1 ≤ m ≤ 10000. m k=1 0 (10000)

We see from Fig. 10 that R0 converges to some value close to R0 ≈ 1.2439 (10000) . as the number m of trials increases. In what follows, let R0 := R0 We next change P to 50. In this case, the spatial domain is the rectangular lattice {1, 2, . . . , 50} × {1, 2, . . . , 8}. We obtain a trial result as shown in Fig. 11. In this case, the iteration stops at t = 89 and S∞ = 40. Hence, R0 ≈ 1.5171. Similar to the above case, we repeat the trial 10000 times and obtain Fig. 12, which shows that (m) R0 converges to R0 ≈ 0.9781 as the number m of trials increases. We perform similar simulations for all P ∈ {2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200} and plot the relationship between R0 and P in Fig. 13. As in Fig. 7, R0 attains its maximum in the square lattice (P = 20) and decreases as P departs from 20 and the shape of the lattice becomes long and narrow.

On the Relationship Between the Basic Reproduction Number ...

57

Fig. 10 The basic (m) reproduction number R0 in the m-th trial and its cumulative average number (m) R0 for the case of the square lattice {1, 2, . . . , 20} × {1, 2, . . . , 20} (1 ≤ m ≤ 10000)

(a) t = 0

(b) t = 15

(c) t = 89 Fig. 11 A trial result of our lattice model for the case of the rectangular lattice {1, 2, . . . , 50} × {1, 2, . . . , 8}. Green, red and blue squares mean susceptible, infective and removed individuals, respectively. In this case, S∞ = 40 and R0 ≈ 1.5171 Fig. 12 The basic (m) reproduction number R0 in the m-th trial and its cumulative average number (m) R0 for the case of the rectangular lattice {1, 2, . . . , 50} × {1, 2, . . . , 8} (1 ≤ m ≤ 10000)

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T. Kuniya

Fig. 13 R0 versus P ∈ {2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200}

8 Conclusion In this paper, we have investigated the relationship between the basic reproduction number R0 and the shape of the spatial domain in some different situations. We have constructed the spatially diffusive SIR epidemic model with constant parameters and shown that R0 is defined by the spectral radius of the next generation operator K and it is the threshold value for the global asymptotic stability of the disease-free equilibrium E 0 and the existence of the endemic equilibrium E ∗ for both cases of the homogeneous Neumann and Dirichlet boundary conditions. Under the homogeneous Neumann boundary conditions, R0 is the same as that for the classical SIR epidemic model without diffusion, and thus, it does not depend on the shape of the spatial domain. On the other hand, under the homogeneous Dirichlet boundary conditions, R0 depends on the shape of the spatial domain, and we obtain a numerical example showing that R0 attains its maximum in the square domain and decreases as the shape of the domain becomes long and narrow. This result suggests us the importance of the homogeneous Dirichlet boundary conditions in the mathematical modeling of diffusive infectious diseases. In fact, by constructing the random 2-dimensional lattice model, we have supported the validity of our conjecture that the frequency of contacts of each individual would decrease as the shape of the spatial domain becomes long and narrow. As stated in [9] (see also [16]), the homogeneous Dirichlet boundary conditions would imply that the boundary of the spatial domain is hostile for the survival of population due to the extremely cold or hot temperature, the lack of supporting resources, and so on. If this situation seems too restrictive, then we can employ the mixed boundary conditions such that both of the homogeneous Neumann and Dirichlet boundary conditions are assumed in two disjoint subsets of the boundary, respectively. Because such a spatial domain still have the spatial heterogeneity, we may be able to obtain a similar result as in this study such that R0 decreases as the shape of the spatial domain becomes long and narrow. We leave this problem for an important future study.

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References 1. Allen, L.J.S., Bolker, B.M., Lou, Y., Nevai, A.L.: Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete Cont. Dyn. Syst. 21, 1–20 (2008) 2. Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976) 3. Capasso, V.: Global solution for a diffusive nonlinear deterministic epidemic model. SIAM J. Appl. Math. 35, 274–284 (1978) 4. Chatelin, F.: The spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators. SIAM Rev. 23, 495–522 (1981) 5. Chekroun, A., Kuniya, T.: An infection age-space structured SIR epidemic model with Neumann boundary condition. Appl. Anal. (2018). https://doi.org/10.1080/00036811.2018. 1551997 6. Chekroun, A., Kuniya, T.: An infection age-space-structured SIR epidemic model with Dirichlet boundary condition. Math. Model. Nat. Phenom. 14, 22 (2019) 7. Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990) 8. Hosono, Y., Ilyas, B.: Traveling waves for a simple diffusive epidemic model. Math. Model. Meth. Appl. Sci. 5, 935–966 (1995) 9. Huang, W., Han, H., Liu, K.: Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Math. Biosci. Eng. 7, 51–66 (2010) 10. Inaba, H.: Age-Structured Population Dynamics in Demography and Epidemiology. Springer, Singapore (2017) 11. Itô, S.: Diffusion Equations. American Mathematical Society, Providence (1992) 12. Krasnoselskii, M.A.: Positive Solutions of Operator Equations. P. Noordhoff Ltd., Groningen (1964) 13. Kermack, W.O., McKendric, A.G.: Contributions to the mathematical theory of epidemics - I. Proc. R. Soc. 115, 700–721 (1927) 14. Magal, P., Webb, G.F., Xu, Y.: On the basic reproduction number of reaction-diffusion epidemic models. SIAM J. Appl. Math. 79, 284–304 (2019) 15. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2003) 16. Okubo, A., Levin, S.A.: Diffusion and Ecological Problems: Modern Perspective. Springer, New York (2001) 17. Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society, Providence (1995) 18. Thieme, H.R.: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188–211 (2009) 19. Webb, G.F.: A reaction-diffusion model for a deterministic diffusive epidemic. J. Math. Anal. Appl. 84, 150–161 (1981) 20. Wang, W., Zhao, X.Q.: Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11, 1652–1673 (2012)

Cause and Control Strategy for Infectious Diseases with Nonlinear Incidence and Treatment Rate Nilam

Abstract Control strategies for any infectious disease can be suggested with the help of mathematical modeling after the inclusion of the cause of the spread of the disease. Different diseases are having different patterns of spread. A major reason for the spread of the disease can be found out with the help of the incidence rate. Also, treatment therapies vary with the severity and type of disease. There may be many factors; for example, the availability of vaccines for a particular disease; the number of infected people is very crucial to consider for an effective treatment rate. Hence, nonlinear treatment rates can vary from disease to disease. Therefore, the nonlinear incidence and treatment rate can play a vital role in suggesting effective therapies to health agencies to control the spread of disease. Keywords Infectious disease · Nonlinear incidence rate · Nonlinear treatment rate · Control therapy

1 Introduction Infections and infectious diseases are a great burden on many societies and a leading cause of death worldwide. An integrated approach is required to reduce the burden of infectious diseases from society. Mathematical modeling is an effective tool and plays an important role in efforts that focus on predicting, assessing, and controlling potential outbreaks. Mathematical models are capable of capturing both qualitative and quantitative aspects of complex systems. For the dynamics of epidemic/infectious diseases of the human population, the total human population is divided mainly by three epidemiological classes (compartments): susceptible (S), infected (I ), and recovered (R) individuals’ classes. Susceptible individuals are those who are healthy and do not have any infection. Individuals who are infected with the disease and are capable of transferring it to susceptible via contacts are called Infected individuals. As time progresses, infectious individuals lose the infectivity and move to the removed Nilam (B) Department of Applied Mathematics, Delhi Technological University, Delhi 110042, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_3

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or recovered compartment. These recovered individuals are immune to infectious microbes and thus do not acquire the disease again. The main focus of modeling is to involve biological features like appropriate transmission, treatment, immune responses, and social awareness in order to reduce the mortality and morbidity of the population. Some important objectives to develop epidemic models which are addressed in this chapter are: 1. Role of incidence rate on the SIR epidemic model. 2. Role of treatment rate on the SIR epidemic model. 3. Nonlinearity of incidence and treatment rates to understand the dynamics of infection and to provide the important modalities in controlling the disease transmission. 4. Stability analysis of the epidemic models to determine the behavior of the outbreak subjected to large perturbations. 5. To visualize the analytical findings through computer simulations. In the study of disease transmission dynamics, two major factors are the transmission of the diseases and the control of the spread of the diseases. The transmission of infectious diseases can be studied by incidence rate. In the next section, we will discuss the incidence rate in detail.

2 Incidence Rate The disease transmission process plays an important role in determining the incidence rate. The number of individuals who become infected per unit of time in epidemiology is known as the incidence rate. Kermack and Mckendrick model [1] introduced the incidence rate of the form k S I . This term is called a bilinear incident rate, and it defines the condition that the incidence rate increases with the increase in the number of susceptibles. So, the bilinear incidence rate is not reasonable for the large population and hence needed to be modified. Therefore, to mimic the real-world scenario, many epidemiological mathematical models have been studied with nonlinear incidences such as Holling type II incidence rate, Holling type III, Beddington-DeAngelis type, Crowley- Martin type and Hattaf-Yousfi type incidence rates. These form of nonlinear incidence rates are explored as follows: αS I , α, β > 0 is known as the saturated incidence Holling Type II: The term 1+β I rate and was proposed by C. S. Holling [2]. In Holling type II, for any outbreak of the disease, its incidence is first very low and then grows slowly with an increase in infection. Further, when the number of infected individuals is very large, the infection reaches its maximum due to the crowding effect. αI , α, β > 0 is known as Holling type III incidence Holling Type III: The term 1+β I2 rate [3]. It defines the condition in which the incidence of infection first grows very fast initially with an increase in infective, and then it grows slowly and finally settles down to maximum saturated value. After this, any increase in infective will not affect the infection rate. 2

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Beddington-DeAngelis Type: This functional response was introduced by DeAn. gelis [4] and Beddington [5] in 1975 independently and is given by the term 1+βαS S+γ I Here α is the transmission rate, β is a preventive measure adopted by susceptible individuals, and γ is a measure of inhibition effect adopted by infectives. Beddington DeAngelis’ incidence rate involves awareness/inhibitions behavior by susceptibles and infectives. It can be seen that the following three types of incidence rates can be derived from the incidence rate: if we set β = γ = 0 then bilinear incidence rate; if we set γ = 0 then saturated incidence rate with the susceptible individuals; if we set β = 0, saturated incidence rate with the infected individuals. Crowley-Martin Type: The Crowley-Martin type of functional response was introduced by P. H. Crowley and E. K. Martin in 1989 [6] and is denoted by the term f (S, I ) =

αS (1 + β S)(1 + γ I )

where α, β, γ are positive constants. From the expression, we observe that similar to the Beddington-DeAngelis type incidence rate; one can easily derive other forms of incidence rates. The important difference between the Beddington-DeAngelis type and the Crowley-Martin type incidence rate is that the latter considers the effect of inhibition among infectives even in case of the high density of susceptible populations while the former neglects the effect described above. This can be seen as follows: Beddington-DeAngelis type incidence rate for S → ∞, lim f (S, I ) =

S→∞

α . β

and Crowley-Martin type incidence rate for S → ∞, lim f (S, I ) =

S→∞

α . β(1 + γ I )

Hattaf-Yousfi Type: The Hattaf-Yousfi type of functional response was introduced βS by Hattaf and Yousfi in [7] and has the form α0 +α1 S+α , where β is a positive 2 I +α3 S I constant rate describing the infection process and α0 , α1 , α2 , α3 are non-negative constants. This functional response generalizes many functional responses and it was used in [8] to describe the dynamics of labour market. In population dynamics, the transmission of infection is the process in which susceptibles are getting infected via an infected population through various channels. Transmission plays an important role in studying the dynamical behavior of epidemic models. Therefore, it is essential to choose an appropriate incidence rate. To control and prevent the spread of infection, the treatment rate plays a crucial role. In the next section, we will discuss the forms of treatment rate in detail.

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3 Treatment Rate The treatment of infected individuals is an important strategy to control the spread of further infection and to eradicate the infection from the society. A suitable treatment rate may achieve control of infection and disease. Therefore, considering a suitable treatment rate to the classical epidemic model can change the dynamical behavior of the epidemic model. In the classical epidemic model, the treatment rate of infectives is considered to be either constant or proportional to the number of infectives. Wang and Ruan [9] considered a SIR epidemic model with constant removal rate (i.e., the recovery from infected subpopulation per unit time), called Holling type I treatment rate, as given below:  h(I ) =

r, 0,

I >0 I =0

(1)

The Holling type I treatment rate is a sensible estimate to the treatment truth of the infection that human has rich treatment sources and better treatment methods. But, medical facilities and subsequent therapies may require some time to be developed and implemented; therefore, choosing a suitable treatment rate is essential. Therefore, many researchers considered a nonlinear form of treatment rates such as Holling type II, Holling type III, and Holling type IV treatment rates, described below: aI , a, b > 0. Holling Type II: The Holling type II treatment rate is of the form 1+bI The Holling type II treatment rate is low, for there is short of effective treatment techniques at the beginning of the outbreak. At that point, the treatment rate will be increased with the improvement of the clinic’s treatment conditions, including compelling drugs, capable systems, and so forth. Finally, for the treatment limit of any network is constrained, the treatment will reach to its most extreme if the quantity of infective people is huge enough.

Holling Type III: Holling type III treatment rate was introduced to a known disease that has re-emerged and has available treatment modalities. It is of the form aI2 , a, b > 0, and it defines the condition in which removal rate first grows very 1+bI 2 fast initially with the increase in infectives and then it grows slowly and finally settles down to maximum saturated value. After this, any increase in infectives will not affect the removal rate. Holling Type IV: Holling type IV treatment rate is of the form

I2 a

βI , +I +b

β, a, b > 0.

In this treatment rate, the removal/treatment rate initially grows with the growth of infectives and reaches the maximum and then starts decaying. Such a situation may arise due to limitations in the availability of treatment for a large number of infected individuals. When supplies of treatment (medicine, immunization, etc.) are depleted, then despite a high number of infectives, the available treatment becomes very low. This case may arise when there are re-emergence and spread of disease in place of limited treatment facilities.

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Thus, various types of epidemiological models have been studied and analyzed by considering different types of incidence and treatment rates. The mathematical formulation of some epidemiological models has been discussed in the next section.

4 Development of Mathematical Model The compartmental SIR mass-action model of Kermack and McKendrick [1] is the basis of epidemic models. Kermack and McKendrick founded the deterministic compartmental epidemic modeling, and they addressed the mass—action (bilinear) incident in the disease transmission cycle, recommending that the probability of infection of a susceptible (virgin from disease) is analogous to the number of its contacts with infected individuals. The model is investigated mathematically under the assumption that the population is perfectly mixed and every susceptible has the same probability of becoming infected the probabilities are equated to the expected (mean) values of the corresponding variables in the population. These assumptions lead to the following ordinary differential equations: dS = −k S I, dt dI = k S I − β I, dt dR = β I. dt

(2)

where, S, I , and R denote the susceptible, infected, and recovered populations, respectively. The parameters k and 1/β denote the mean values of the disease transmission probability and length of the period for which an individual can transmit the disease before recovering. In this model, it has been shown that the disease becomes epidemic, i.e., d I /dt > 0, d S/dt < 0 if and only if S > β/k. Thus, the number of infected individuals will increase as long as S > β/k. At S = β/k, the number of infected cases reaches a maximum, and after this, it decreases to zero. The threshold (k/β)S is denoted by R0 , and this R0 is called the basic reproduction number. The basic reproduction number R0 represents the average number of secondary infections produced from a single infected individual introduced into a completely susceptible population.

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Consideration of the treatment modalities in mathematical modeling has been an important step to describe the real situations during an epidemic. To understand the effect of the treatment capacity on disease transmission, an SIR epidemic model with bilinear incidence rate and constant treatment rate is considered in [9] as given below: dS = A − d S − λS I, dt dI = λS I − (d + γ )I − h(I ), dt dR = γ I + h(I ) − d R. dt

(3)

where S, I , and R denote the susceptible, infected, and recovered populations, respectively. The parameters d and γ denote the natural death rate and recovery rate, respectively. λ is the force of infection. h(I ) is the treatment rate given by (1). Here the treatment rate is taken to be dependent on the capacity of treatment of infected individuals. The optimal capacity of treatment is determined to depend on their outcomes. The stability analysis of the above epidemic model has been studied, and various bifurcations have been proved. It is observed that the inclusion of constant treatment rate exhibits periodic oscillations in diseases while the model without the treatment is globally stable. It has been shown that the model is more realistic and useful since the ultimate behavior of the equilibria depends on the initial positions. This model is suitable for measles, AIDS, flu, etc. The quantitative analysis carried out can be adapted to an SI model, which is useful for sexually transmitted diseases or bacterial infections. For the dynamical behavior of the epidemic model, incidence rate, and treatment rate play a major role. Therefore, the choice of appropriate incidence and treatment rates are crucial in the study of transmission and control of infectious diseases. The above two mathematical epidemic models (2) and (3), were studied with the consideration of bilinear incidence rates. However, the model (3) considered the effect of the linear treatment rate. But, to achieve the dynamical behavior of the epidemic model at more realism level, nonlinearity must be considered. Therefore, many nonlinear mathematical models have been formulated and analyzed to study disease transmission and provide disease prevention techniques [10–14]. Zhonghua and Yaohong [15] modified the model of Wang and Ruan [9] to incorporate the saturated treatment rate in place of constant treatment rate along with eternal immunity. It has been argued that this treatment rate is a better alternative for new emerging diseases such as Severe Acute Respiratory Syndrome (SARS). On the onset of an outbreak, the treatment will be less, and this will increase with the improvement in the hospital’s condition, the supply of drugs, etc. and approaches to maximum capacity with limited resources available in the community. Here coexistence of disease-free equilibrium and endemic equilibrium has been shown. This suggests that not only the reduction of threshold value R0 to the values less than unity is always effective to control the spread of disease but also there is a need to eliminate such diseases, to restrict the

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initial value of each sub-population to the domain of attraction of the disease-free equilibrium. Goel and Nilam [16] incorporated the nonlinear incidence rate of Beddington DeAngelis type functional form and saturated nonlinear treatment rate of the form aI2 , which contains the form of Holling type III functional form also, into bI 2 +cI +1 their epidemic model and studied the dynamic behavior of the infectious disease. A control strategy of the infectious diseases has been provided, and it has been shown that if this behavioral response is delayed, then this simple looking system can exhibit interesting dynamics. The latent period as time delay has also been considered to capture the effect of the varying infectious period. In the present chapter, we have studied the SIR epidemic model with Beddington DeAngelis and Holling type IV treatment rates with time delay as follows: β S(t)I (t − τ ) dS = A − μS − , dt 1 + αS(t) + γ I (t − τ ) dI β S(t)I (t − τ ) aI = − (μ + d + θ )I − 2 , dt 1 + αS(t) + γ I (t − τ ) I + bI + c dR aI = θI + 2 − μR. dt I + bI + c

(4)

The analysis and simulation of the model (4) will be discussed in detail as given below: The initial conditions φ = (φ1 , φ2 , φ3 ) of (4) are defined in the Banach space 3 ) : φ1 ( ) = S( ), φ2 ( ) = I ( ), φ3 ( ) = R( )}, C+ = {φ ∈ C([−τ, 0], R+ 3 = {(S, I, R) ∈ R 3 : S ≥ 0, I ≥ 0, R ≥ 0}. Biologically, it is assumed that where R+ φi > 0 (i = 1, 2, 3). In the above model (4), we consider the total population N(t) at time t, with immigration of susceptible population with a constant rate A. The constants μ, d and θ represent the natural death rate, disease-induced death rate, and recovery rate, respectively. β is the transmission rate, α and γ are the measure of inhibitions adopted aI is the treatment by susceptibles and infectives, respectively. The term g(I ) = I 2 +bI +c rate, where a > 0 represents the treatment rate of infected individuals, b > 0 is the limitation rate in treatment availability and c > 0 is the saturation constant.

5 Analysis of the Model The basic properties of the model (4) are presented as follows: The equations of the model (4) monitor populations. From the proposition 2.1 in Hattaf et al. [17] and proposition 2.3. in Yang et al. [18], it can be shown that all state variables of the

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3 model (4) are nonnegative. That is, (S, I, R) ∈ R+ . Also, for ecological reasons, we supposed that all parameters A, μ, β, α, γ , d, θ , a, b are positive and c is nonnegative. Since N (t) = S(t) + I (t) + R(t), the governing equations of model (4) can be rewritten as,

dN = A − μN − d I dt ≤ A − μN

(5) (6)

3 Lemma 1. All solutions of the model (4) starting in R+ are bounded and eventually enter a compact attracting set 3

= {(S, I, R) ∈ R+ : S(t) + I (t) + R(t) = N (t) ≤

A }. μ

Proof. Continuity of the right hand side of the model (4) and its derivative assure the well-possess of the model for N (t) > 0. The invariant region for the existence of the solutions can be determined as given below:

0 < lim inf N (t) ≤ lim sup N (t) ≤ t→∞

t→∞

A μ

Since N (t) > 0 on [−τ, 0] by assumption, N (t) > 0 for all t ≥ 0. Therefore, with the help of Eq. (6), it can be seen that for any finite time t, N(t) cannot blow up to infinity. The model system is dissipative (solutions are bounded) and consequently, the solution exists globally for all t > 0 in the invariant and compact set 3 : S(t) + I (t) + R(t) = N (t) ≤

= {(S, I, R) ∈ R+

A }, μ

As N (t) approaches zero, S(t), I (t), and R(t) also approach zero. Thus, each of these terms tend towards zero as N (t) does. Thus, it is reasonable to interpret these terms as zero when N (t) = 0.  Remark 1. In this region , basic results such as usual local existence, uniqueness and continuation of solutions are valid for model (4). Hence, there exists a unique solution (S(t), I (t), R(t)) of model (4) starting in the interior of that exists on a maximal interval [0, ∞) if solutions remain bounded [19].

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Further the equilibrium points of the model are obtained as follows: From the model (4) it has been observed that, since R(t) does not appear in equations for ddtS and ddtI , it is sufficient to analyze the behaviors of solutions of (4) by the following system of delay differential equations: dS β S(t)I (t − τ ) = A − μS − dt 1 + αS(t) + γ I (t − τ ) dI β S(t)I (t − τ ) aI = − (μ + d + θ )I − 2 dt 1 + αS(t) + γ I (t − τ ) I + bI + c

(7)

with initial conditions φ = (φ1 , φ2 ) of (7) are defined in the Banach space 2 ) : φ1 ( ) = S( ), φ2 ( ) = I ( )}, C+ = {φ ∈ C([−τ, 0], R+ 2 = {(S, I ) ∈ R 2 : S ≥ 0, I ≥ 0},φi > 0 (i = 1, 2). where R+ Now, the equilibria of the model has been obtained as follows:   (i) E 0 (S0 , 0) = E 0 μA , 0 , disease-free equilibrium (DFE). (ii) E e (S ∗ , I ∗ ), positive or endemic equilibrium (EE), will be discussed later.   The characteristic equation of the model (7) at disease-free equilibrium E 0 μA , 0 is given by   a  β Ae−λτ − (μ + d + θ ) − −λ =0 (8) (μ + λ) μ + αA c

One of the root of the Eq. (8) is given by λ1 = −μ and and other root can be obtained from a  β Ae−λτ − (μ + d + θ ) − −λ=0 μ + αA c A −λτ The term (μ+α A) βμ+d+θ+ at τ = 0, is known as basic reproduction number. In a e ( c) epidemiological research, the basic reproduction number, denoted by R0 , is defined as the average number of secondary infections caused by a single infected agent, during his/her entire infectious period, in a completely susceptible population [20]. Thus, basic reproduction number for our model is given by

R0 =

βA  . (μ + α A) μ + d + θ + ac

The local stability of disease-free is investigated as follows: Theorem 1. The disease free equilibrium E 0 is locally asymptotically stable if R0 < 1 and unstable if R0 > 1.

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Proof. Clearly, the Eq. (8) has one negative root λ1 = −μ and other root is obtained by the equation βA a e−λτ = 0 λ+μ+d +θ + − c (μ + α A) Let f (λ) = λ + μ + d + θ +

  a a βA 1 − R0 e−λτ − e−λτ = λ + μ + d + θ + c (μ + α A) c

If R0 > 1 , it is readily seen that for real λ, f (0) = μ + d + θ +

a βA − < 0, c (μ + α A)

lim f (λ) = +∞

λ→∞

Hence, f (λ) = 0 and f  (λ) > 0, so f (λ) = 0 has a unique positive real root if R 0 > 1. If R 0 < 1, we assume that Reλ ≥ 0. We notice that Re λ =

 a βA a β Ae−(Re λ)τ cos(I mλ)τ  − μ+d +θ + < − μ+d +θ + 0 if =⇒ β − pα > 0.

c(β − pα) − αa > 0

(11)

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On adding first and second equations of the system (7) and put it to zero, we get: A − μS ∗ − (μ + d + θ )I ∗ −

I ∗2

aI∗ =0 + bI ∗ + c

(12)

Substituting the value of S ∗ from the Eq. (10) into the Eq. (12), we get the following equation in I ∗ : A0 + A1 I ∗ + A2 I ∗2 + A3 I ∗3 + A4 I ∗4 + A5 I ∗5 = 0

(13)

where A0 = −c A(c(β − αp) − αa) + μc(a + pc) A1 = ab(Aα + μ) + (a + pc)(c(β − pα) − αa + cγ μ) + 2bcμp − 2bc A(β − pα) − a 2 α A2 = pμ(b2 + 2c) + b(β − αp + γ μ)(2cp + a) + a(Aα + μ) − abαp − A(b2 + 2c)(β − αp) A3 = 2bμp + (β − αp + γ μ)(a + p(b2 + 2c)) − αpa − 2b A(β − αp) A4 = p(α A + μ + 2bβ) − Aβ − 2bp(γ μ + αp) A5 = p(β − αp + γ μ)

Under condition (11), β − pα > 0. Using Descartes’ rules of signs, Eq. (13) has a unique solution if any of the conditions is satisfied: (i) A5 > 0, A4 < 0, A3 < 0, A2 < 0, A1 < 0 and A0 < 0 (ii) A5 > 0, A4 > 0, A3 < 0, A2 < 0, A1 < 0 and A0 < 0 (iii) A5 > 0, A4 > 0, A3 > 0, A2 < 0, A1 < 0 and A0 < 0

(14)

(iv) A5 > 0, A4 > 0, A3 > 0, A2 > 0, A1 < 0 and A0 < 0 (v) A5 > 0, A4 > 0, A3 > 0, A2 > 0, A1 > 0 and A0 < 0 After determining the value of I ∗ , we can determine the value of S ∗ from Eq. (10). This implies that there exists a unique positive endemic equilibrium E e (S ∗ , I ∗ ) if one of the conditions (14) hold. We now analyze the local stability of endemic equilibrium E e as follows: The characteristic equation of the system (7) evaluated at E e is a second degree transcendental equation: λ2 + p0 λ + q0 + ( p1 λ + q1 )e−λτ = 0

(15)

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where

  a c − I ∗2 β I ∗ (γ I ∗ + 1) + d + θ + 2μ + , ∗ ∗ 2 (I (b + I ) + c) (αS ∗ + γ I ∗ + 1)2       μ(αS ∗ + γ I ∗ + 1)2 + β I ∗ (γ I ∗ + 1) a c − I ∗2 + (I ∗ (b + I ∗ ) + c)2 (d + θ + μ) q0 = , (I ∗ (b + I ∗ ) + c)2 (αS ∗ + γ I ∗ + 1)2 β S ∗ (αS ∗ + 1) , p1 = − (αS ∗ + γ I ∗ + 1)2 βμS ∗ (αS ∗ + 1) q1 = − . (αS ∗ + γ I ∗ + 1)2 p0 =

(16)

√ √ Theorem 2. At τ = 0, E e is locally asymptotically stable if − c < I ∗ < c, ∗ ∗ I > 1+αS are satisfied. S∗ 1+γ I ∗ Proof. At E e , the characteristic equation at τ = 0 is given by λ2 + p 0 λ + q 0 + ( p 1 λ + q 1 ) = 0 ∗2

∗

∗

β S (1+αS ) ac−a I ≥ (1+αS ∗ +γ I ∗ )2 is satisfied then ( I ∗2 +bI ∗ +c)2 a (c−I ∗2 ) β I ∗ (γ I ∗ +1)−β S ∗ (αS ∗ +1) > 0, p0 + p1 = (I ∗ (b+I ∗ )+c)2 + d + δ + 2μ + (αS ∗ +γ I ∗ +1)2 q0 + q1 = (y(b+y)+c)2ν(αx+γ y+1)2 > 0,

It is easy to verify that if

where      ν = a c − y 2 μ(αx + γ y + 1)2 + βy(γ y + 1) + (y(b + y) + c)2 μ d(αx + γ y + 1)2 +   β(y(γ y + 1) − x(αx + 1)) + δ(αx + γ y + 1)2 + βy(d + δ)(γ y + 1) + μ2 (αx + γ y + 1)2

Hence, by Routh-Hurwitz criterion, it is concluded that the endemic equilibrium E e of the system (7) is locally asymptotically stable when τ = 0.  Theorem 3. For τ > 0, E e is locally asymptotically stable if the conditions p02 − 2q0 − p12 > 0 and q02 − q12 > 0 are satisfied simultaneously, where p0 , p1 , q0 , q1 are given in Eq. (16). Proof. The characteristic equation for τ > 0 evaluated at E e is given by the Eq. (15) which is λ2 + p0 λ + q0 + ( p1 λ + q1 )e−λτ = 0 For τ > 0, if instability occurs for a particular value of the delay τ , a characteristic root of (15) must intersect the imaginary axis. Assume that λ = iη, η > 0 is the root of the characteristic Eq. (15).

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Substituting λ = iη in Eq. (15), we get (−η2 + q0 + p1 η sin ητ + q1 cos ητ ) + i( p0 η + p1 η cos ητ − q1 sin ητ )

(17)

On separating real and imaginary part of Eq. (17), we get p1 η sin ητ + q1 cos ητ = η2 − q0

(18)

p1 η cos ητ − q1 sin ητ = −η p0

(19)

Eliminating τ by squaring and adding Eqs. (18) and (19),we obtain a polynomial in η as η4 + ( p02 − 2q0 − p12 )η2 + (q02 − q12 ) = 0

(20)

Letting η2 = x, Eq. (20) becomes: x 2 + Ax + B = 0

(21)

where, we assume that A = p02 − 2q0 − p12 > 0,

B = q02 − q12 > 0.

By Routh-Hurwitz criterion, this contradicts to our assumption of instability i.e λ = iη . Hence it is proved that the endemic equilibrium E e of the system (7) is locally asymptotically stable for τ > 0.  Theorem 4. If B = q02 − q12 < 0, the endemic equilibrium E e of the system (7) is asymptotically stable for τ ∈ [0, τ0 ) and it undergoes Hopf bifurcation at τ = τ0 . Proof. If B = q02 − q12 < 0 then there is a unique positive η0 satisfying Eq. (21) i.e. there is single pair of purely imaginary roots ±iη0 to Eq. (15). From Eqs. (18) and (19), τk corresponding to η0 can be obtained as τk =

  2 2nπ 1 η (q1 − p0 p1 ) − q0 q1 + arccos 0 , n = 0, 1, 2 . . . 2 2 2 η0 η0 p 1 η0 + q 1

Endemic equilibrium E e is stable for τ < τ0 if transversality condition holds i.e. d = 0. λ) dt (Re λ=iη0

Differentiating Eq. (15) with respect to τ , we get  dλ  = λ( p1 λ + q1 )e−λτ 2λ + p0 + p1 e−λτ − ( p1 λ + q1 )τ e−λτ dτ

(22)

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dλ dτ

−1

2λ + p0 + p1 e−λτ − ( p1 λ + q1 )τ e−λτ λ( p1 λ + q1 )e−λτ τ 2λ + p0 p1 − = + λ( p1 λ + q1 )e−λτ λ( p1 λ + q1 ) λ 2λ + p0 p1 τ = + − 2 −λ(λ + p0 λ + q0 ) λ( p1 λ + q1 ) λ =

 −1 d dλ = Re (Re λ) λ=iη0 λ=iη0 dτ dτ   p1 2iη0 + p0 iτ + = Re + η0 −iη0 (−η02 + i p0 η0 + q0 ) − p1 η0 + iq1 η0   2 2 2 1 2η0 (η0 − q0 ) + p0 η0 p 1 η0 = − 2 2 2 η0 ( p0 η0 ) + (η0 − q0 ) ( p1 η0 )2 + q12 =

2(η02 − q0 ) + p02 p12 − 2 2 2 ( p0 η0 ) + (η0 − q0 ) ( p1 η0 )2 + q12

(23)

Now, on squaring and adding Eqs. (18) and (19), we get ( p1 η0 )2 + q12 = ( p0 η0 )2 + (η02 − q0 )2 so that Eq. (23) can be written as d 2η2 + ( p02 − 2q0 − p12 ) = 0 (Re λ) λ=iη0 dτ p12 η02 + q12

(24)

Under the condition A = p02 − 2q0 − p12 > 0, it can be seen that d >0 (Re λ) λ=iη0 dτ Therefore, the transversality condition holds and Hopf bifurcation occurs at η = η0 ,  τ = τ0 .

6 Numerical Simulation To verify the model numerically, the system (7) has been simulated with the following set of parameter values:

Cause and Control Strategy for Infectious Diseases ... Table 1 List of parameters Parameter A α β μ d γ θ a b c

75

Interpretation

Value

Recruitment rate of susceptible Measure of inhibition taken by susceptibles Transmission rate Natural death rate Disease-induced death rate Measure of inhibition taken by infectives Recovery rate Treatment rate of infected individuals/Cure rate Limitation rate in treatment availability Saturation constant

5 0.002 0.004 0.05 0.001 0.002 0.002 0.1 0.002 0.004

The computer simulations are performed for S and I for various values of τ . The trajectory of S and I with initial conditions S(0) = 80, I (0) = 5, approach to the endemic equilibrium as shown in Fig. 1. Figure 1 shows the effect of time delay on susceptible population and infective population, respectively, for time for the set of parameters given in Table 1. It is shown that as the time delay of τ increases, the number of susceptible starts decreasing, and the number of infectives starts increasing, which shows the significance of time delay in the present paper. 120

100

Population

80

60

40

20

0 0

20

40

60

80

Time t (days)

Fig. 1 Susceptible and Infected population for various values of τ .

100

120

76

Nilam 60 = 0.004 = 0.0045 = 0.005

Infected Population (I)

50

40

30

20

10

0

0

20

40

60

80

100

120

Time t (days) =1

Fig. 2 Infected population for various values of β at τ = 1 day. 70 = 0.004 = 0.0045 = 0.005

Infected Population (I)

60

50

40

30

20

10

0

0

20

40

60

80

Time t (days) =2

Fig. 3 Infected population for various values of β at τ = 2 days.

100

120

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Figures 2 and 3 show the influence of transmission rate β and variation of time delay τ on infected population for τ = 1 and τ = 2 respectively. Here, the infected population is drawn for various values of β, (i.e., β = 0.004, 0.0045, 0.005) and for two different values of time delays (i.e., τ = 1 day in Fig. 2 and τ = 2 days in Fig. 3). In these figures, we note that when the effective contact rate β is high then more people will be infected and when the effective contact rate β is low, then less people are infected. Also, the number of infectives will be more for same transmisssion rate at a increased value of delay. Thus, the transmission rate and the time delay have a substantial role in the spread of disease. Figure 4 shows the effect of Holling type IV treatment rate on the infected individuals for the time lag τ = 1 day. This figure shows that infection is increasing in both the cases and attaining its steady state, but if we consider Holling type IV treatment rate then infection will occur at a lower level i.e. Holling type IV treatment rate is reducing the infection. In Figs. 5, 6, 7 and 8, the infected population is drawn for the transmission rate β = 0.000489 and time lag τ = 5.5, 6, 7 and 10 days, respectively. These figures show the oscillatory and periodic solutions of the infected population, which confirms the occurrence of Hopf bifurcation. We see that as the value of time delay gets higher, the periodicity also increases at a higher rate. In Fig. 5, we see the oscillation in the infected population, but after a certain time, they reached their steady-state, which shows the stability of endemic equilibrium. As the value of timed delay increases, then the endemic equilibrium loses its stability, as seen in Figs. 6, 7 and 8 respectively. Some other recent models have been proposed (by the author of the chapter) with the combination of different incidence and treatment rates to get a deep insight into the causes and control strategies for different types of infectious diseases. For instance,

45 Without treatment rate With treatment rate

40

Infected Population (I)

35 30 25 20 15 10 5 0

0

20

40

60

80

100

120

Time t (days) =1

Fig. 4 Infected population with and without Holling type IV treatment rate at τ = 1 day.

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Fig. 5 Behaviour of infected population for β = 0.00489 and τ = 5.5 days.

Fig. 6 Behaviour of infected population for β = 0.00489 and τ = 6 days.

to gain a better understanding of transmission and subsequent control of the diseases which are re-emerged and for which treatment strategies are available, Kumar et al. [13] proposed and analyzed the susceptible-infected-recovered mathematical model via the combination of nonlinear saturated incidence and Holling type III treatment rates. This model is special case of (4), it suffices to take α = 0 and c = 0.

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Fig. 7 Behaviour of infected population for β = 0.00489 and τ = 7 days.

Fig. 8 Behaviour of Infected population for β = 0.00489 and τ = 10 days.

7 Conclusion In this chapter, we have studied the role of time delay, incidence rate, treatment rate on susceptible-infected-recovered (SIR) epidemic model. We have elaborated on the importance of nonlinearity of incidence and treatment rates to understand the dynamics of infection and provide the important modalities in controlling the disease transmission. For this, we have considered a delayed SIR epidemic model with Beddington- DeAngelis type incidence rate and Monod-Haldane type treatment rate. The Stability analysis of the epidemic models is performed to determine the

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behavior of the outbreak subjected to large perturbations is performed. Finally, the analytical findings are visualized through numerical simulations. We conclude that the identification of various causes/factors behind the spread of infection is essential to provide an effective control strategy to minimize the spread of the disease. These factors should be incorporated to the maximum extent into the formulation of incidence rate to represent a more realistic state. After that, to provide an effective strategy to control different types of infections requires different factors to be considered into the formulation of treatment rate A pertinent combination of incidence and treatment rate into the mathematical model can be proved vital in proposing an effective control strategy to prevent the spread of infection.

References 1. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A115, 700–721 (1927) 2. Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91(07), 385–398 (1959) 3. Dubey, P.: Stability Analysis and simulation of deterministic models in epidemiology and immunology. Doctor of Philosophy, Birla Instutute of Technology and Science, Pilani, India (2016) 4. DeAngelis, D.L., Goldstein, R., Oneill, R.: A model for tropic interaction. Ecology 56(4), 881–892 (1975) 5. Beddington, J.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44(1), 331–340 (1975) 6. Crowley, P.H., Martin, E.K.: Functional responses and interference within and between year classes of a dragonfly population. J. N. Am. Benthol. Soc. 8(3), 211–221 (1989) 7. Hattaf, K., Yousfi, N.: A class of delayed viral infection models with general incidence rate and adaptive immune response. Int. J. Dyn. Control 4(3), 254–265 (2016) 8. Riad, D., Hattaf, K., Yousfi, N.: Dynamics of capital-labour model with Hattaf-Yousfi functional response. J. Adv. Math. Comput. Sci. 18(5), 1–7 (2016). https://doi.org/10.9734/BJMCS/2016/ 28640 9. Wang, W., Ruan, S.: Bifurcation in an epidemic model with constant removal rates of the infectives. J. Math. Anal. Appl. 21, 775–793 (2004) 10. Dubey, B., Patra, A., Srivastva, P.K., Dubey, U.: Modeling and analysis of an SEIR model with different types of nonlinear treatment rates. J. Biol. Syst. 21(3), 1350023 (2013) 11. Dubey, B., Dubey, P., Dubey, U.S.: Dynamics of an SIR model with nonlinear incidence and treatment rate. Appl. Appl. Math. 10(2), 718–737 (2015) 12. Kumar, A., Nilam, : Stability of a time delayed SIR epidemic model along with nonlinear incidence rate and holling Type II treatment rate. Int. J. Comput. Methods 15(6), 1850055 (2018) 13. Kumar, A., Goel, K., Nilam: A deterministic time-delayed SIR epidemic model: mathematical modeling and analysis, in theory in biosciences (2019). https://doi.org/10.1007/s12064-01900300-7 14. Goel, K., Nilam, : Stability behavior of a nonlinear mathematical epidemic transmission model with time delay. Nonlinear Dyn. 98(2), 1501–1518 (2019) 15. Zhonghua, Z., Yaohong, S.: Qualitative analysis of a SIR epidemic model with saturated treatment rate. J. Appl. Math. Comput. 34(1–2), 177–194 (2010) 16. Goel, K., Nilam, : A mathematical and numerical study of a SIR epidemic model with time delay, nonlinear incidence and treatment rates. Theor. Biosci. 132(2), 203–213 (2019)

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17. Hattaf, K., Lashari, A.A., Louartassi, Y., Yousfi, N.: Electron. J. Qual. Theor. Differ. Equ. A delayed SIR epidemic model with a general incidence rateA delayed SIR epidemic model with a general incidence rate 3, 1–9 (2013) 18. Yang, M., Sun, F.: Global stability of SIR models with nonlinear Incidence and discontinuous treatment. Electron. J. Differ. Equ. 2015(304), 1–8 (2015) 19. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993) 20. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

Global Stability of a Delay Virus Dynamics Model with Mitotic Transmission and Cure Rate Eric Avila-Vales, Abraham Canul-Pech, Gerardo E. García-Almeida, and Ángel G. C. Pérez

Abstract In this paper, we study the global properties of a basic model for viral infection with mitotic transmission, “cure” of infected cells, saturated infection rate, and a discrete intracellular delay. For our model, we derive some threshold parameters and establish a set of conditions which are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functionals and the Lyapunov– Lasalle type theorem for delay systems, we prove the global asymptotic stability of all equilibria of our model. We also establish the occurrence of a Hopf bifurcation, determine conditions for the permanence of model, and the length of delay to preserve stability. Furthermore, we present some numerical simulations to illustrate the analytical results.

1 Introduction Mathematical modeling is a tool to study population dynamics between virus particles and target cells. Modeling processes in a viral infection are mainly: target-cell dynamics, viral transmission routes and production of viral particles. The viral transmission may occur by different routes: cell-to-free virus transmission, direct cellto-cell contacts to neighboring cells or mitotic transmission, that is to say, mitotic division of infected cells that contain viral genome. Most of the models focus on cell-free virus transmission [1–6] and few models incorporate mitotic transmission [3, 5, 7–9]. E. Avila-Vales (B) · A. Canul-Pech · G. E. García-Almeida · Á. G. C. Pérez Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte, Tablaje 13615, 97119 Mérida, Yucatán, Mexico e-mail: [email protected] A. Canul-Pech e-mail: [email protected] G. E. García-Almeida e-mail: [email protected] Á. G. C. Pérez e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_4

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The “curative” mechanisms of infected cells is another innovation in the modeling of viral infections. Technically, a term δy(t) out of the equation of infected cells and incorporated into the equation of uninfected cells, corresponding to the rate at which uninfected cells are created through “cure”. This kind of models focus on “cure” of infected cells for hepatitis B and C virus [7, 10–14]. The basic model of virus dynamics with mitotic transmission and cure of infected cells was proposed by Dahari and collaborators [7] for hepatitis B and C virus. Some related models can be found in [15–18], which also include different types of time delays. See also the model studied in [19], which includes time delay and a saturated infection rate but omits the cure of infected cells. The model given in [7] is formulated by the following system of nonlinear differential equations: d x(t) dt dy(t) dt dv(t) dt

 = s + r x(t) 1 −



− μx(t) − βx(t)v(t) + δy(t),  − αy(t) − δy(t), = βx(t)v(t) + r y(t) 1 − x(t)+y(t) xmax 

x(t)+y(t) xmax

(1)

= σ y(t) − γ v(t).

Here, x(t), y(t) and v(t) denote the concentration of uninfected cells (or target cells), infected cells and free virus, respectively. All parameters are assumed to be positive constants. Here, target cells are generated at a constant rate s and die at a rate μ per uninfected cell. These cells are infected at rate β per target cell per virion. Infected cells die at rate α per cell by cytopathic effects and are possibly “cured” by noncytolytic processes at a constant rate δ per cell. Because of the viral burden on the virus-infected cells, we assume that μ ≤ α. In other words, we assume that the average life-time of infected cells (1/α) is shorter than the average life-time of uninfected cells (1/μ) [20, 21]. The proliferation of infected and uninfected cells due to mitotic division obeys alogistic growth.  The mitotic proliferation of uninfected x(t)+y(t) cells is described by r x(t) 1 − xmax , and mitotic transmission occurs at a rate   , which is the mitotic division of infected cells. Uninfected and r y(t) 1 − x(t)+y(t) xmax infected cells grow at the same constant rate r , and xmax is the maximal number of total cell population proliferation. Infected cells produce virions at an average rate σ per infected cell, and γ is the clearance rate of virus particles. In several of models with or without delay described above, the process of cellular infection by free virus particles is typically modeled by the mass action principle, that is to say, the infection rate is assumed to occur at a rate proportional to the product of the concentration of virus particles and uninfected target cells. This principle is insufficient to describe the cellular infection process in detail, and some nonlinear infection rates have been proposed. Li and Ma [22] and Song and Neumann [23] considered a virus dynamics model with Monod functional response, bxv/(1 + αv). Regoes et al. [24] and Song and Neumann [23] considered a virus dynamics model with the nonlinear infection rates bx(v/κ) p /[1 + (v/κ) p ] and bxv q /(1 + αv p ), where p, q, k > 0 are constants, respectively. Recently, Huang et al. [25] considered a class of models of viral infections with a nonlinear infection rate and two discrete intracellular

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delays, and assumed that the infection rate is given by a general nonlinear function of uninfected target cells and free virus particles F(x, v), which satisfies certain conditions. The parameter τ accounts for the time between viral entry into a target cell and the production of new virus particles. The recruitment of virus producing cells at time t is given by the number of cells that were newly infected at time t − τ and are still alive at time t. Here, m is assumed to be a constant death rate for infected but not yet virus-producing cells. Thus, the probability of surviving the time period from t − τ to t is e−mτ . Motivated by the above comments, we propose here a new model, which is described by the following system of delay differential equations: ⎧  d x(t) ⎪ = s + r x(t) 1− ⎪ ⎨ dt −mτ

dy(t) x(t−τ )v(t−τ ) = βe 1+kv(t−τ dt ) ⎪ ⎪ ⎩ dv(t) = σ y(t) − γ v(t). dt



− μx(t) − βx(t)v(t) + δy(t), 1+kv(t)   x(t)+y(t) − (α + δ)y(t), + r y(t) 1 − xmax

x(t)+y(t) xmax

(2)

Let C([−τ, 0], R3+ ) be the Banach space of continuous functions mapping the interval [−τ, 0] into R3+ , where R3+ = {(x, y, v) ∈ R3 : x ≥ 0, y ≥ 0, v ≥ 0}.

(3)

For elements of the space C([−τ, 0], R3+ ) we use the norm φ = sup {|φ1 (θ )| , |φ2 (θ )| , |φ3 (θ )|} . −τ ≤θ≤0

The biologically reasonable history of the host for model (2) is as follows: x(θ ) = φ1 (θ ) ≥ 0, y(θ ) = φ2 (θ ) ≥ 0, v(θ ) = φ3 (θ ) ≥ 0, θ ∈ [−τ, 0].

(4)

In this paper, our primary goal is to carry out a complete mathematical analysis of system (2) and establish its global dynamics. The organization of this paper is as follows. In the next section we discuss about positive invariance and boundedness. The existence of equilibrium points and computation of the basic reproduction number are discussed in Sect. 3. In Sect. 4, we study the local stability by analyzing the corresponding characteristic equation, and we analyze the global asymptotic stability of the infection-free equilibrium of the model (2) using Lyapunov functionals. In Sect. 5, we discuss the local stability of the chronic-infection equilibrium. In Sect. 6, the Hopf bifurcation at the infected equilibrium is determined. In Sect. 7, we discuss the global stability of the chronic-infection equilibrium by means of suitable Lyapunov functionals and the Lyapunov–LaSalle type theorem for delay differential equations. Conditions for the permanence are established in Sect. 8. In Sect. 9, we perform an estimation of the length of delay to preserve stability. The numerical validation is found in Sect. 10. A brief remark is given in Sect. 11 to conclude this work.

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2 Positive Invariance, Boundedness and Basic Reproduction Number Theorem 1. All solutions of system (2) with initial conditions (4), where x(0) > 0, y(0) > 0, v(0) > 0, are positive. Proof. From the last two equations of system (2) we obtain

y(t) = y(0)e

t dφ −(α+δ)t+ 0 r 1− x(φ)+y(φ)) xmax

t dφ −(α+δ)t+ 0 r 1− x(φ+y(φ)) xmax

+e  t −mτ

)) βe x(φ − τ )v(φ − τ ) (α+δ)φ−0φ r 1− x(θx+y(θ dθ max dφ e × 1 + κv(φ − τ ) 0 and v(t) = e

−γ t

   t γφ v(0) + σ y(φ)e dφ .

(5)

(6)

0

Let t ∈ [0, τ ]. Then we have φ − τ ∈ [−τ, 0] for all φ ∈ [0, τ ]. x(t) ≥ 0, y(t) ≥ 0, v(t) ≥ 0 for t ∈ [−τ, 0] and x(0) > 0, y(0) > 0, v(0) > 0. If t ∈ [0, τ ], then the second term of (5) is non-negative, therefore y(t) > 0. Consequently, the second term of (6) is positive, implying v(t) > 0. Next, let t1 be the first value of t such that x(t1 ) = 0. If t1 ≤ τ , from the first equation of system (2) we obtain that d x(t) = s + δy(t) > 0, dt giving us a contradiction, because this implies that there exists an ε > 0 such that x(t) < 0 for t ∈ (t1 − ε, t1 ). Hence x(t) > 0 for all t ∈ [0, τ ]. Therefore, we have proved that x(t) > 0, y(t) > 0, and v(t) > 0 for all t ∈ [0, τ ]. This argument can now be repeated to deduce positivity of x(t), y(t) and v(t) on the interval [τ, 2τ ] and then on successive intervals [nτ, (n + 1)τ ] n ≥ 2, to include all positive times.   Theorem 2. Let (x(t), y(t), v(t)) be the solution of system (2) satisfying initial conditions (4). Then x(t), y(t), and v(t) are all bounded for all t ≥ 0 at which the solution exists. Proof. Let (x(t), y(t), v(t)) be any solution with nonnegative initial conditions. We define a function  t x(s)v(s) α U (t) = x(t) + y(t) + e−m(t−s) v(t) + β ds, n  1. nσ 1 + kv(s) t−τ

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The time derivative along a solution of (2) is:   dU (t) α(n − 1) αγ x(t) + y(t) = s − μx(t) − y(t) − v(t) + r (x(t) + y(t)) 1 − dt n nσ xmax  t −m(t−s) x(s)v(s) − mβ e ds 1 + kv(s) t−τ xmax 2 α(n − 1) αγ r  r xmax x(t) + y(t) − − μx(t) − y(t) − v(t) − =s+ 4 n nσ xmax 2  t −m(t−s) x(s)v(s) − mβ e ds 1 + kv(s) t−τ  t x(s)v(s) α(n − 1) αγ 4s + r xmax e−m(t−s) − μx(t) − y(t) − v(t) − mβ ds. ≤ 4 n nσ 1 + kv(s) t−τ

It follows that

dU (t) 4s + r xmax + aU (t) ≤ , dt 4

where a = min{ α(n−1) , γ , μ, m}. Thus lim supt−→∞ U (t) ≤ 4s+r4axmax . Therefore, n x(t), y(t), and v(t) are all bounded for all t ≥ 0. This completes the proof.  

3 Existence of Equilibria A constant solution that satisfies system (2) is known as an equilibrium point. System (2) always has an uninfected equilibrium given as E 0 = (x0 , 0, 0), where xmax x0 = 2r

 (r − μ) +

 (r −

μ)2

4r s + xmax

 .

(7)

To show the existence of an infected equilibrium, we define the basic reproduction number. We derive the basic reproduction number for viral infection using the next generation operator [26]. Using the notation in [26], the non-negative matrix, F, of the new infection terms, and the M-matrix, V , of the transition terms associated with the model (2), are given, respectively, by    

x0 α+δ 0 βe−mτ x0 r 1 − xmax and V := F := . −σ γ 0 0 It follows that the basic reproduction number, denoted by R0 = ρ(F V −1 ), where ρ is the spectral radius, is given by   −mτ  βe σ x0 1 x0 , R0 (τ ) = +r 1− (α + δ) γ xmax

(8)

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where x0 is the concentration of uninfected cells at the uninfected equilibrium state in R3+ , given by (7). The parameter R0 has an interesting biological meaning, it is the sum of average numbers of secondary infected cells produced by a single infected cell in a population of target cells, by cell-free virus transmission and mitotic division, respectively. Par−mτ ticularly, the term βe γ σ x0 describes the per unit time secondary infections through   x0 the direct target cell-to-virus contact, and the term r 1 − xmax those through mitotic division. We introduce the following result. Proposition 3. Suppose that R0 (τ ) > 1, then the system (2) admits an equilibrium point E 1 or infected equilibrium, where every component is strictly positive. Proof. We assume that there exists a constant solution (x1 , y1 , v1 ) for system (2), then this constant solution satisfies:   βx1 v1 x1 + y1 + δy1 , (9) 0 = s + r x1 1 − − μx1 − xmax 1 + kv1   x1 + y1 βe−mτ x1 v1 − (α + δ)y1 , 0= + r y1 1 − (10) 1 + kv1 xmax (11) 0 = σ y1 − γ v1 . Then the Eq. (11) implies v1 = σγy1 , which allows us to reduce system (9)–(11) to two equations. We express the Eq. (9) as the following quadratic equation in x1

r y1 βσ y1 x1 − (δy1 + s) = 0. x1 2 + −r + +μ+ xmax xmax γ + κσ y1 r

Note that this quadratic equation has two real roots of opposite sign that depend on y1 . We are interested in the positive one, which is clearly a function of y1 . Defining x1 = f (y1 ), we express Eqs. (9) and (10) as   f (y1 ) + y1 βσ f (y1 )y1 − μf (y1 ) − s + r f (y1 ) 1 − + δy1 = 0, xmax γ + κσ y1   βe−mτ σ f (y1 )y1 f (y1 ) + y1 − (α + δ)y1 = 0. + r y1 1 − γ + κσ y1 xmax We define F(y1 ) :=

  βe−mτ σ f (y1 ) f (y1 ) + y1 − (α + δ) for y1 > 0. +r 1− γ + κσ y1 xmax

From the Eq. (13), we consider the existence of positive roots of F(y1 ) = 0.

(12) (13)

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We now show that F(y1 ) is a strictly decreasing function.



− f  (y1 ) − 1 (γ + κσ y1 ) f  (y1 ) − f (y1 )κσ + r F  (y1 ) = βe−mτ σ (γ + κσ y1 )2 xmax

−mτ r σ βe f  (y1 ) − = (γ + κσ y1 ) xmax

r βe−mτ σ 2 κ . (14) f (y ) + − 1 (γ + κσ y1 )2 xmax Note that F  (y1 ) depends on f  (y1 ). To calculate f  (y1 ) we first rewrite the Eq. (12) as   βσ y1 f (y1 ) + y1 δy1 s −μ− +r 1− = 0. (15) + f (y1 ) xmax γ + kσ y1 f (y1 ) Using implicit differentiation we get



f (y1 ) = −

r xmax

γβσ δ + − (γ + kσ y1 )2 f (y1 )



s r δy1 + + 2 f 2 (y1 ) xmax f (y1 )

−1

.

(16) Defining a =

κσ γ

and b =

βe−mτ σ γ

, we can rewrite (14) and (16).

r b ab r f  (y1 ) − (17) − f (y ) + 1 (1 + ay1 ) xmax (1 + ay1 )2 xmax



−1 r s r bemτ δ δy1  + f (y1 ) = − + − + 2 . xmax (1 + ay1 )2 f (y1 ) f 2 (y1 ) xmax f (y1 ) (18)

F  (y1 ) =



Substituting (18) into (17), we obtain:





−1 s + δy1 r b r bemτ δ r − − − + + 1 + ay1 xmax xmax f (y1 ) xmax (1 + ay1 )2 f 2 (y1 )

ab r − f (y1 ) + xmax (1 + ay1 )2   rb b2 emτ r bemτ bδ r2 rδ = − − + + + 2 − 3 2 xmax (1 + ay1 ) (1 + ay1 ) f (y1 ) xmax f (y1 ) (1 + ay1 ) xmax (1 + ay1 ) xmax

−1

s + δy1 ab r r × − f (y1 ) + + xmax xmax f 2 (y1 ) (1 + ay1 )2  

2

−1 r s + δy1 r bemτ bδ r < + + + xmax (1 + ay1 ) f (y1 ) xmax xmax (1 + ay1 )2 f 2 (y1 )

ab r . − f (y ) + 1 xmax (1 + ay1 )2

F  (y1 ) =



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From (15), we have bδ δe−mτ = (1 + ay1 ) f (y1 ) y1



s f 2 (y1 )

+

δy1 r + xmax f 2 (y1 )

−

2r xmax y1

+

r μ−r + xmax f (y1 ) f (y1 )y1

δe−mτ .

and

s δy1 r r f (y1 ) r bemτ + 2 + = (1 + ay1 )2 xmax xmax y1 (1 + ay1 ) f 2 (y1 ) f (y1 ) xmax

2r μ−r r f (y1 ) r . + − + (1 + ay1 )xmax xmax y1 xmax f (y1 ) f (y1 )y1 Then F  (y1 )
+ + xmax f (y1 ) xmax x0 x0 xmax x0

.

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91

Hence F(y1 ) is a strictly decreasing function with respect to y1 . On the other hand,   x0 βe−mτ σ x0 − (α + δ) +r 1− lim F(y1 ) = y1 →+0 γ xmax −mτ 

 βe r x0 σ x0 = (α + δ) −1 + 1− (α + δ)γ α+δ xmax = (α + δ) (R0 (τ ) − 1) > 0. We will now calculate the limit of F(y1 ) as y1 → +∞. We first note that lim

y1 →+∞

f (y1 ) ≥ 0,

because x1 = f (y1 ) is positive by definition. We also have lim

y1 →+∞

βe−mτ σ r − γ + kσ y1 xmax

so lim

y1 →+∞

f (y1 )

=−

r < 0, xmax

βe−mτ σ r − γ + kσ y1 xmax

≤ 0.

Hence, we have 

   βe−mτ σ f (y1 ) f (y1 ) + y1 − (α + δ) +r 1− lim F(y1 ) = lim y1 →+∞ y1 →+∞ γ + κσ y1 xmax 

 r βe−mτ σ r f (y1 ) − = lim − y1 + r − (α + δ) y1 →+∞ γ + kσ y1 xmax xmax   r y1 + r − (α + δ) ≤ lim − y1 →+∞ xmax = −∞, which implies lim y1 →+∞ F(y1 ) = −∞. Combining this with lim y1 →+0 F(y1 ) > 0, we can see that there exists a unique positive root of F(y1 ), which is the second component of the infected equilibrium. Then the first and third components are   given by the quadratic equation in x1 and v1 = σγy1 , respectively.

4 Asymptotic Stability of the Uninfected Equilibrium State The objective of this section is to study the local and global stability of the infectionfree equilibrium.

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4.1 Local Analysis of the Viral Free Equilibrium E0 The following theorem gives us conditions for system (2) to be locally asymptotically stable. Theorem 4. If R0 (τ ) < 1, then the infection-free steady state E 0 is locally asymptotically stable. If R0 (τ ) > 1, then it is unstable. Proof. The characteristic equation associated with the Jacobian matrix at the infection-free equilibrium, E 0 = (x0 , 0, 0), is given by the following determinant   λ − r − μ −   0    0  

2x0 r xmax



   

 x0 −βe−(m+λ)τ x0  = 0. λ − r − (α + δ) − xrmax   −σ λ+γ   r x0 xmax

−δ

βx0

(19)

Calculating the determinant (19), then the characteristic equation is as follows: 



 

 2x0 r r x0 λ− r −μ− λ − r − (α + δ) − (λ + γ ) − βσ x0 e−(m+λ)τ = 0. xmax xmax

(20)

x0 = μ − xs0 , and using We also consider that E 0 satisfies system (2), so r 1 − xmax the previous fact we can rewrite the factors of the characteristic Eq. (20) as λ=r −μ−

s r x0 2x0 r , =− + xmax x0 xmax

which have a negative eigenvalue, and the other two eigenvalues satisfy the following transcendental polynomial λ2 + a1 λ + a0 + b0 (τ )e−λτ = 0.

(21)

where a1 = γ +

s γs + α − μ + δ, a0 = γ ((α + δ) − μ) + , b0 (τ ) = −βσ x0 e−mτ . x0 x0

When τ = 0, Eq. (21) becomes λ2 + a1 λ + a0 + b0 (0) = 0.

(22)

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Note that a1 = γ +

s x0

93

+ (α − μ) + δ > 0 and

γs a0 + b0 (0) = γ ((α + δ) − μ) + − βσ x0 x0   s βσ x0 +μ− = −γ − (α + δ) γ x0

  x0 βσ x0 e−mτ − (α + δ) +r 1− = −γ γ xmax 

 1 βσ x0 x0 +r 1− = −γ (α + δ) −1 (α + δ) γ xmax = −γ (α + δ) [R0 (0) − 1] . If R0 (0) < 1, we have a0 + b0 (0) > 0. It follows that for τ = 0 the polynomial (22) is Hurwitz stable. Now, let us consider the distribution of the roots of (21) when τ > 0. Suppose λ = ωi (ω > 0) is a solution of (21). Substituting λ = ωi (ω > 0) into (21), then separating in real and imaginary parts, we obtain a0 − ω2 = −b0 cos(ωτ ), a1 ω = b0 sin(ωτ ). Squaring and adding the last two equations and after simplifications we get ω4 + (a12 − 2a0 )ω2 + (a02 − b02 (τ )) = 0

(23)

with s + α − μ + δ, x0 γs a0 = γ ((α + δ) − μ) + , x0 b0 (τ ) = −βσ x0 e−mτ . a1 = γ +

Let ω2 = z, A = a12 − 2a0 and B(τ ) = a02 − b02 (τ ). Then we have F(z) = z 2 + Az + B(τ ) = 0.

(24)

Note that

s 2 > 0, A = − 2a0 = γ + α + δ − μ + x0

s βσ x0 e−mτ 2 2 2 B(τ ) = a0 − b0 = γ (α + δ) δ + (α − μ) + (1 − R0 (τ )). + x0 γ

a12

2

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If R0 (τ ) < 1 then B(τ ) > 0. Now as A, B(τ ) > 0 and ω > 0, then F(z) > 0 for any z > 0, which contradicts (24). This shows that characteristic Eq. (20) does not have pure imaginary roots when R0 (τ ) < 1. The fact that (20) is stable for τ = 0 and the continuity of the roots with respect to the delay imply that (20) has all its roots with real negative part when R0 (τ ) < 1. Therefore, if R0 (τ ) < 1, the infection-free steady state E 0 of system (2) is locally asymptotically stable. If R0 (τ ) > 1, let f (λ) = λ2 + a1 λ + a0 + b0 e−λτ .

(25)

Note that f (0) = a0 + b0 = γ (α + δ) [1 − R0 (τ )] < 0 since R0 (τ ) > 1 and limλ→∞ f (λ) = ∞. It follows from the continuity of the function f (λ) on (−∞, ∞) that the equation f (λ) = 0 has at least one positive root. Hence, the characteristic Eq. (25) has at least one positive root. Accordingly, the infection-free steady state E 0 is unstable.  

4.2 Global Analysis of the Viral Free Equilibrium E0 For biological models and in particular virological models, it makes sense to study the stability of positive equilibrium points, it is necessary for all cell populations to persist and it is also necessary for all cell populations to be present initially, so a natural concept of global stability for positive equilibria in biological models is that every model solution that starts in the positive octant R3+ must remain in the positive octant for all finite values of t and converge to the equilibrium point when t → ∞. In this section, applying Lyapunov functionals as in Vargas-De-León [27], we consider the global stability of the uninfected equilibrium E 0 . Theorem 5. Assume that the condition δ ∈ [ r (exmax−1) x0 , r (exmax+3) x0 ] holds for m ≥ 0. If R0 (τ ) ≤ 1, then the uninfected equilibrium state E 0 of (2) is globally asymptotically stable in R3+ . mτ

mτ

Proof. Define the Lyapunov functional  U (t) =

x(t) x0

σ − x0 βx0 dσ + emτ y(t) + v(t) + β σ γ



τ 0

x(t − ω)v(t − ω) dω. 1 + kv(t − ω)

U is defined and is continuous for any positive solution (x(t), y(t), v(t)) of system (2) and U = 0 at E 0 = (x0 , 0, 0). Calculating the derivative of U (t) along a positive solution of (2), it follows that

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 τ (x − x0 ) dU (t) βx0 d x(t − ω)v(t − ω) = x(t) ˙ + emτ y˙ (t) + v(t) ˙ −β dω dt x γ dω 1 + kv(t − ω) 0



βxv (x − x0 ) x+y βx(t − τ )v(t − τ ) − = s − μx + r x 1 − + δy + x xmax 1 + kv 1 + kv(t − τ )

x + y βx 0 − emτ (α + δ)y + + emτ r y 1 − (σ y − γ v) xmax γ βx(t − τ )v(t − τ ) βxv − + . 1 + kv(t − τ ) 1 + kv

Using r − μ = ing, we get

r x0 xmax

−

s , x0

0) δ(x − x0 ) xy = −δy (x−x + x x0 2

δ (x x0

− x0 )y and simplify-

y βxv dU (t) s(x − x0 ) r ry βv = (x − x0 ) − +δ + − (x − x0 ) − − dt x x0 xmax xmax 1 + kv x 1 + kv

ry r y2 x0 βx0 σ + emτ r y 1 − y − βx0 v − emτ (x − x0 ) − emτ − emτ (α + δ)y + xmax xmax xmax γ

(s + δy)(x − x0 )2 r r r emτ δ r y(x − x0 ) − =− − (x − x0 )2 + + − y2 x x0 xmax xmax xmax x0 xmax

βx0 v x0 βx0 σ r (emτ − 1) 2 mτ mτ − e (α + δ)y + + e ry 1 − y − βx0 v y + − xmax 1 + kv xmax γ (s + δy)(x − x0 )2 x x0

r 2r r emτ r δ r y(x − x0 ) − − (x − x0 )2 + + − − y2 xmax xmax xmax xmax x0 xmax

r (emτ − 1) 2 βx0 v x βx 0 0σ − emτ (α + δ)y + + emτ r y 1 − y − βx0 v − y + xmax 1 + kv xmax γ

⎛ ⎞2 2r r emτ r δ xmax + xmax − xmax − x0 r ⎝ (s + δy)(x − x0 )2 − y⎠ =− (x − x0 ) + 2r x x0 xmax xmax

mτ

mτ r (e −1) r (e +3) − xδ − xδ xmax xmax r (emτ − 1) 2 βx0 kv 2 0 0 + y2 − y − r 4 xmax xmax 1 + kv

−mτ σ e βx x 1 0 0 + emτ (α + δ)y +r 1− −1 α+δ γ xmax

⎞2 ⎛ 2r r emτ r δ xmax + xmax − xmax − x0 (s + δy)(x − x0 )2 r ⎝ =− − y⎠ (x − x0 ) + 2r x x0 xmax xmax

mτ

mτ r (e −1) r (e +3) − xδ − xδ xmax xmax r (emτ − 1) 2 βx0 kv 2 0 0 + y2 − y − r 4 xmax xmax 1 + kv =−

+ emτ (α + δ)y (R0 (τ ) − 1) .

Note that emτ − 1 ≥ 0 for m ≥ 0, then r (exmax−1) y 2 ≥ 0. If δ <

mτ

mτ r (e −1) − xδ0 < 0 and r (exmax+3) − xδ0 > 0. Therefore, xmax mτ

r (emτ +3) x0 , xmax

then

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r (emτ −1) xmax

−

δ x0



r (emτ +3) xmax

−

δ x0



r 4 xmax

y 2 < 0.

Thus, R0 (τ ) ≤ 1. Then dUdt(t) ≤ 0 and E 0 is globally asymptotically stable. Note that dU = 0 if and only if x(t) = x0 , y(t) = 0, v(t) = 0. Therefore, the largest dt = 0} when R0 (τ ) ≤ 1 is E 0 (x0 , 0, 0). By the invariant set in {(x(t), y(t), v(t)) : dU dt Lyapunov–LaSalle type theorem for delay systems (Theorem 2.5.3 in [28]), E 0 is globally asymptotically stable.  

5 Local Asymptotic Stability of the Infected Equilibrium State We now study the local stability behavior of the infected equilibrium state E 1 when R0 (τ ) > 1 for system (2). Thus, linearizing system (2) at the infected equilibrium state E 1 = (x1 , y1 , v1 ), we obtain that the associated transcendental characteristic equation is given by   r (2x1 + y1 ) − λ − r − μ − xmax   r y1 βe−(m+λ)τ v1  xmax − 1+kv1   0

βv1 1+kv1



  βx1 r x1  2 xmax − δ 1) 

(1+kv −(m+λ)τ βe x1  = 0, r λ − r − (α + δ) − xmax (x1 + 2y1 ) − (1+kv ))2  1   −σ λ+γ

which can be rewritten as  λ + s + r x1 + δy1  x1 max  r y1 x1 βex−(m+λ)τ  x − 1+kv v1 λ + 1  max  0

r x1 −δ xmax y1 βe−mτ x1 v1 + xrmax (1+kv1 )y1

−σ



βx1  (1+kv1 )2  βe−(m+λ)τ x1  − (1+kv1 )2  

λ+γ



= 0.

When we take into account the identities

s βv1 x1 + y1 δy1 r −μ = − +r + − , x1 xmax 1 + kv1 x1 βe−mτ x1 v1 r (x1 + y1 ) r −α =− + + δ, (1 + kv1 )y1 xmax the characteristic equation reduces to λ3 + a2 (τ )λ2 + a1 (τ )λ + a0 (τ ) + (b1 (τ )λ + b0 (τ ))e−λτ = 0,

(26)

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97

where s r x1 r y1 δy1 βe−mτ x1 v1 + + + + +γ x1 xmax xmax x1 (1 + kv1 )y1

−mτ



−mτ βe x1 v1 x1 v1 s r x1 δy1 r y1 βe r y1 γ a1 (τ ) = + + + +γ + + x1 xmax x1 (1 + kv1 )y1 xmax (1 + kv1 )y1 xmax

r x1 r y1 −δ − xmax xmax

−mτ

−mτ βe r y1 x1 v1 x1 v1 r x1 δy1 s δy1 βe r y1 s γ + + +γ + + + + = x1 xmax x1 (1 + kv1 )y1 x1 x1 xmax (1 + kv1 )y1 xmax r y1 + δ xmax

−mτ



βe r y1 r x1 x1 v1 βx1 v1 s r x1 δy1 r y1 r y1 γ− + + + −δ γ − γ a0 (τ ) = x1 xmax x1 (1 + kv1 )y1 xmax xmax xmax xmax (1 + kv1 )2 y1

r y1 βx1 v1 r x1 δy1 βe−mτ x1 v1 s δy1 r y1 δ r y1 s + + γ+ + γ+ γ− γ = x1 xmax x1 (1 + kv1 )y1 x1 x1 xmax xmax xmax (1 + kv1 )2 y1

−mτ −mτ v1 βx1 σ e r x1 βe + −δ b1 (τ ) = − (1 + kv1 )2 xmax 1 + kv1 a2 (τ ) =

βx1 v1 e−mτ r x1 βe−mτ v1 βe−mτ v1 γ+ −δ (1 + kv1 )2 y1 xmax 1 + kv1 1 + kv1



s r x1 βx1 σ e−mτ r x1 δy1 βe−mτ v1 βσ x1 βe−mτ v1 + b0 (τ ) = − + + −δ γ + . 2 (1 + kv1 ) x1 xmax x1 xmax 1 + kv1 (1 + kv1 )2 1 + kv1 =−

Define P(λ, τ ) = λ3 + a2 (τ )λ2 (τ ) + a1 (τ )λ + a0 (τ ), Q(λ, τ ) = b1 (τ )λ + b0 (τ ). When τ = 0, we have from (26) that P(λ, 0) + Q(λ, 0) = λ3 + a2 (0)λ2 + (a1 (0) + b1 (0))λ + a0 (0) + b0 (0) = 0. (27) By the Routh-Hurwitz criterion the conditions for Re λ < 0 are a2 (0) > 0, a0 (0) + b0 (0) > 0, Q = a2 (0)(a1 (0) + b1 (0)) − (a0 (0) + b0 (0)) > 0, in our case a2 (0) > 0 and a1 (0) + b1 (0) =

s x1

+



r y1 βx1 v1 + +γ (1 + kv1 )y1 xmax

+

r (x1 + y1 ) βx1 v1 γ+ γ kv1 xmax (1 + kv1 )y1

δr y12 rβx12 v1 rβx1 v1 δy1 δr y1 + γ+ + > 0, + xmax (1 + kv1 ) xmax (1 + kv1 )y1 x1 x1 xmax xmax

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r x1 δy1 δy1 r y1 r y1 δ βx1 v1 s s + + γ kv + + γ+ γ 1 x1 xmax x1 (1 + kv1 )2 y1 x1 x1 xmax xmax

r x1 βv1 + −δ γ kv1 xmax (1 + kv1 )2

βγ v1 βx1 v1 + − δ (1 + kv1 )2 (1 + kv1 )y1

r x1 δy1 r y1 r y1 δ βx1 v1 s s + γ kv + + γ+ γ = 1 x1 xmax (1 + kv1 )2 y1 x1 x1 xmax xmax r x1 βv1 + γ kv1 xmax (1 + kv1 )2

βx1 v1 βγ v1 + − δ . (1 + kv1 )2 (1 + kv1 )y1

a0 (0) + b0 (0) =

Using βx1 v1 r x1 r y1 − = α +δ −r + , (1 + kv1 )y1 xmax xmax we obtain



s s βx1 v1 r x1 δy1 r y1 r y1 δ a0 (0) + b0 (0) = + γ kv1 + + γ+ γ x1 xmax (1 + kv1 )2 y1 x1 x1 xmax xmax βv1 r x1 γ kv1 + xmax (1 + kv1 )2

x1 + y1 βγ v1 α −r 1− . + (1 + kv1 )2 xmax

−mτ x1 +y1 > 0, then a0 + b0 > 0. If α − r 1 − e xmax If τ = 0, by the Routh-Hurwitz criterion, we have the following theorem:

+y1 > 0, then the Theorem 6. Assume R0 (0) > 1, if Q > 0 and α − r 1 − xx1max infected equilibrium E 1 = (x1 , y1 , v1 ) is locally asymptotically stable as τ = 0.

6 Hopf Bifurcation from the Endemic Equilibrium In this section, we shall regard τ as a parameter to study the existence of Hopf bifurcation from the endemic equilibrium E 1 . The characteristic equation of the linearization of system (2) near the endemic equilibrium E 1 is given by (26), which can be rewritten as P(λ, τ ) + Q(λ, τ )e−λτ = 0,

Global Stability of a Delay Virus Dynamics Model ...

where

P(λ, τ ) = λ3 + a2 (τ )λ2 (τ ) + a1 (τ )λ + a0 (τ ), Q(λ, τ ) = b1 (τ )λ + b0 (τ ).

99

(28)

When τ = 0, from Theorem 6 we know that the positive E 1 is locally

equilibrium x1 +y1 asymptotically stable if R0 (0) > 1, Q > 0, and α − r 1 − xmax > 0. In the following, we investigate the existence of purely imaginary roots λ = iω (ω > 0) to Eq. (26), which takes the form of a third-degree exponential polynomial in λ, where all the coefficients of P and Q depend on τ . Beretta and Kuang [29] established a geometrical criterion which gives the existence of purely imaginary roots of a characteristic equation with delay dependent coefficients. In order to apply the criterion due to Beretta and Kuang [29], we need to verify the following properties for all τ ∈ [0, τmax ), where τmax is the maximum value at which E 1 exists. (a) P(0, τ ) + Q(0, τ ) = 0; (b) P(iω, τ ) + Q(iω,  τ ) = 0;  Q(λ,τ )  (c) lim sup{ P(λ,τ )  : |λ| → ∞, Re λ ≥ 0} < 1; (d) F(ω, τ ) = |P(iω, τ )|2 − |Q(iω, τ )|2 has a finite number of zeros; (e) each positive root ω(τ ) of F(ω, τ ) = 0 is continuous and differentiable in τ whenever it exists. Here, P(λ, τ ) and Q(λ, τ ) are defined as in (28). Let τ ∈ [0, τmax ). It easy to see that P(0, τ ) + Q(0, τ ) = a0 (τ ) + b0 (τ )

r x1 δy1 r y1 r y1 δ βe−mτ x1 v1 s s = + γ kv + + γ+ γ 1 x1 xmax (1 + kv1 )2 y1 x1 x1 xmax xmax r x1 βe−mτ v1 γ kv1 xmax (1 + kv1 )2



βγ v1 e−mτ x1 + y1 −mτ + ) α − r 1 − + δ(1 − e (1 + kv1 )2 xmax +

= 0.

This implies that (a) is satisfied, and (b) is obviously true since P(iω, τ ) + Q(iω, τ ) = −iω3 − a2 (τ )ω2 + a1 (τ )ω + a0 (τ ) + b1 (τ )iω + b0 (τ ) = a0 (τ ) + b0 (τ ) − a2 (τ )ω2 + iω(a1 (τ ) + b1 (τ ) − ω2 ) = 0. From (28), the degree of P(λ, τ ) is greater than the degree of Q(λ, τ ); hence, by infinite limit process, we know that    Q(λ, τ )   = 0. lim  |λ|→+∞ P(λ, τ ) 

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Therefore, (c) follows. Let F be defined as in (d), from |P(λ, τ )|2 = ω6 + (a22 (τ ) − 2a1 (τ ))ω4 + (a12 (τ ) − 2a0 (τ )a2 (τ ))ω2 + a02 (τ ), |Q(λ, τ )|2 = b02 (τ ) + b12 (τ )ω2 ,

we have F(λ, τ ) = ω6 + c1 (τ )ω4 + c2 (τ )ω2 + c3 , where c1 (τ ) = a22 (τ ) − 2a1 (τ ), c2 (τ ) = a12 (τ ) − 2a0 (τ )a2 (τ ) − b12 (τ ), c3 (τ ) = a02 (τ ) − b02 (τ ). It is obvious that property (d) is satisfied, and by the Implicit Function Theorem, (e) is also satisfied. Now let λ = iω (ω > 0) be a root of (26). Substituting λ = iω in (26) and separating in real and imaginary parts, we obtain a0 (τ ) − a2 (τ )ω2 = −b0 (τ ) cos(ωτ ) − b1 (τ )ω sin(ωτ ),

(29)

a1 (τ )ω − ω3 = −b1 (τ )ω cos(ωτ ) + b0 (τ ) sin(ωτ ). From (29), it follows that sin(ωτ ) = cos(ωτ ) =

  [a1 (τ )b1 (τ ) − b0 (τ )] ω3 + a1 (τ )b0 (τ ) − a3(τ ) b1 (τ ) ω b02 (τ ) + b12 (τ )ω2

,

b0 (τ )ω4 + [b1 (τ )b0 (τ ) − a2 (τ )b1 (τ )] ω2 + a3 (τ )b0 (τ )ω . b02 (τ ) + b12 (τ )ω2

(30) (31)

By the definitions of P(λ, τ ), Q(λ, τ ) as in (28), and applying the property (a), (30) and (31) can be written as Q(λ, τ ) , P(λ, τ ) Q(λ, τ ) cos(ωτ ) = − Re , P(λ, τ ) sin(ωτ ) = Im

if ω satisfies (30) and (31), then ω(τ ) must satisfy that |P(iω, τ )|2 = |Q(iω, τ )|2 .

(32) (33)

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101

i.e, ω(τ ) must be a positive root of F(ω, τ ) := |P(iω, τ )|2 − |Q(iω, τ )|2 .

(34)

/ I, Assume that I ⊆ R+0 is the set where ω(τ ) is a positive root of (34) and for τ ∈ ω(τ ) is not defined. Then for all τ in I , ω(τ ) satisfies F(ω, τ ) = 0. By (32), we can define the angle θ ∈ [0, 2π ] as the solution to Q(λ, τ ) , P(λ, τ ) Q(λ, τ ) , cos(θ τ ) = − Re P(λ, τ ) sin(θ τ ) = Im

(35) (36)

and the relation between ωτ in (32) for τ > 0 and the argument θ in (35) must be ωτ = θ + 2π n, n = 0, 1, 2, ... Hence, we can define the maps τn : I → R+0 given by τn (τ ) :=

θ (τ ) + 2π n , τ ∈ I, n = 0, 1, 2, . . . ω(τ )

where ω(τ ) is a positive root of (26). Let us introduce the functions I → R Sn (τ ) := τ − τn (τ ), τ ∈ I, n = 0, 1, 2, . . . that are continuous and differentiable in τ . Thus, we give the following theorem which is due to Beretta and Kuang [29]. Theorem 7. Assume that ω(τ ) is a positive root of (26) defined for τ ∈ I, I ⊆ R+0 , and at some τ ∗ ∈ I , Sn (τ ∗ ) = 0 for some n ∈ N0 . Then a pair of simple conjugate pure imaginary roots λ = ±iω exists at τ = τ ∗ which crosses the imaginary axis from left to right if δ(τ ∗ ) > 0 and crosses the imaginary axis from right to left if δ(τ ∗ ) < 0. Where δ(τ ∗ ) = sign{ F˙ω (ωτ ∗ , τ ∗ )} sign



 d Sn (τ ) |τ =τ ∗ . dτ

Applying Theorem 7 and the Hopf bifurcation theorem for functional differential equations [30], we can conjecture the existence of a Hopf bifurcation as stated in Theorem 8.

+y1 Theorem 8. Assume that R0 (0) > 1, Q > 0 and α − r 1 − xx1max > 0. For system (2), then there exists τ ∗ ∈ I , such that the equilibrium E 1 is asymptotically stable for 0 ≤ τ < τ ∗ , and becomes unstable for τ staying in some right neighborhood of τ ∗ , with a Hopf bifurcation occurring when τ = τ ∗ .

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7 Global Asymptotic Stability of the Infected Equilibrium State In this section, we study the global stability of the infected equilibrium of system (2). The global stability of the infected equilibrium state is proved by constructing a global Lyapunov function. The Lyapunov functionals used here are similar in nature to those used in [19, 27] in which the global dynamics are resolved for virus models with time delay, respectively. Theorem 9. Assume that δ ∈ [ r (exmax−1) x1 , r (exmax+3) x1 ] for m ≥ 0. If R0 (τ ) > 1 and

+y1 ≥ δyx11 , then the unique infected equilibrium E 1 = (x1 , y1 , v1 ) of μ − r 1 − xx1max system (2) is globally asymptotically stable for any τ ≥ 0. mτ

mτ

Proof. Define a Lyapunov functional for E 1 ˜ + βx1 v1 L + , L(t) = L(t) 1 + kv1 where L˜ =



x x1

(σ − x1 ) dσ + emτ σ



y y1

(σ − y1 ) βx1 v1 dσ + σ σ y1 (1 + kv1 )



v v1



v1 (1 + kσ ) 1− dσ σ (1 + kv1 )

and  L+ =

0

τ



x(t − ω)v(t − ω)(1 + kv1 ) x(t − ω)v(t − ω)(1 + kv1 ) − 1 − ln dω. x1 v1 (1 + kv(t − ω)) x1 v1 (1 + kv(t − ω))

At the infected equilibrium, we have βv1 r y1 s + + (x1 + y1 ) − δ , x1 1 + kv1 xmax x1 r βx1 v1 + emτ emτ r − emτ (α + δ) = − (x1 + y1 ), y1 (1 + kv1 ) xmax y1 γ =σ . v1 r −μ=−

(37) (38) (39)

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The derivative of L˜ with respect to t along the solutions of system (2) is

(x − x1 ) βx1 v1 v1 (1 + kv) (y − y1 ) d L˜ = x˙ + emτ y˙ + 1− v˙ dt x y σ y1 (1 + kv1 ) v(1 + kv1 )

s βv r y = (x − x1 ) (x + y) − − +r −μ+δ x xmax 1 + kv x −mτ

r x(t − τ )v(t − τ ) βe − + emτ (y − y1 ) (x + y) + r − (α + δ) y(1 + kv(t − τ )) xmax

v1 (1 + kv) βx1 v1 1− (σ y − γ v). + σ y1 (1 + kv1 ) v(1 + kv1 ) Using (37)–(39) and y1 y (x − x1 )2 δ − = −y + (y − y1 ), x x1 x x1 x1 we get

d L˜ (x − x1 ) r v v1 − [(x − x1 ) + (y − y1 )] − β = (x − x1 ) −s − dt x x1 xmax 1 + kv 1 + kv1

δy(x − x1 ) δ + (x − x1 ) − + (y − y1 ) x x1 x1

x(t − τ )v(t − τ ) x1 v1 r [(x − x1 ) + (y − y1 )] − − emτ = (y − y1 ) β y(1 + kv(t − τ )) y1 (1 + kv1 ) xmax



v1 (1 + kv) βx1 v1 v 1− σ y − σ y1 . = σ y1 (1 + kv1 ) v(1 + kv1 ) v1

Canceling identical terms with opposite signs and collecting terms yields 2  2r r emτ r δ d L˜ r (x − x1 )2 x max + x max − x max − x 1 − (y − y1 ) (x − x1 ) + = − (s + δy) 2r dt x x1 xmax x max

mτ

mτ r (e −1) r (e +3) − xδ1 − xδ1 x max x max r (emτ − 1) + (y − y1 )2 − (y − y1 )2 4r x max x max

βx1 v1 xv(1 + kv1 ) x(t − τ )v(t − τ )(1 + kv1 ) + − + 1 + kv1 x1 v1 (1 + kv) x1 v1 (1 + kv(t − τ ))

βx1 v1 x(t − τ )y1 v(t − τ )(1 + kv1 ) yv1 (1 + kv) + − − 1 + kv1 y1 v(1 + kv1 ) x1 yv1 (1 + kv(t − τ )

v(1 + kv1 ) (1 + kv) βx1 v1 v x + + + − . 1 + kv1 x1 v1 (1 + kv) v1 1 + kv1

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We can rewrite

d L˜ dt

as

2  2r r emτ r δ (x − x1 )2 d L˜ r x max + x max − x max − x 1 = − (s + δy) − (y − y1 ) (x − x1 ) + 2r dt x x1 xmax x max

mτ

mτ r (e −1) r (e +3) − xδ1 − xδ1 x max x max r (emτ − 1) + (y − y1 )2 − (y − y1 )2 4r x max x max

βx1 v1 xv(1 + kv1 ) x(t − τ )v(t − τ )(1 + kv1 ) + − + 1 + kv1 x1 v1 (1 + kv) x1 v1 (1 + kv(t − τ ))

x1 βx1 v1 yv1 (1 + kv) x(t − τ )y1 v(t − τ )(1 + kv1 ) 3− + − − 1 + kv1 x y1 v(1 + kv1 ) x1 yv1 (1 + kv(t − τ ))

(1 + kv) βx1 v1 v(1 + kv1 ) x1 βx1 v1 v x + −2 . + − −1 + + 1 + kv1 v1 (1 + kv) v1 (1 + kv1 ) 1 + kv1 x x1

Replacing the term

x1 x

+

x x1

− 2 by

(x−x1 )2 , x x1

we get

 2 mτ 2r r + rxemax − xmax − xδ1 d L˜ r (x − x1 )2 x (x − x1 ) + max − (y − y ) = −(s + δy) 1 2r dt x x1 xmax x max

mτ

mτ r (e −1) r (e +3) − xδ1 − xδ1 x max x max r (emτ − 1) + (y − y1 )2 − (y − y1 )2 4r xmax x max

βx1 v1 xv(1 + kv1 ) x(t − τ )v(t − τ )(1 + kv1 ) + − + 1 + kv1 x1 v1 (1 + kv) x1 v1 (1 + kv(t − τ ))

βx1 v1 x1 yv1 (1 + kv) x(t − τ )y1 v(t − τ )(1 + kv1 ) + 3− − − 1 + kv1 x y1 v(1 + kv1 ) x1 yv1 (1 + kv(t − τ ))

v1 (1 + kv) v(1 + kv1 ) βx1 v1 v 1− . + − 1 + kv1 v(1 + kv1 ) v1 (1 + kv) v1

Using s −

βx1 v1 1+kv1

= (μ − r )x1 +

r x1 xmax

(x1 + y1 ) − δy1 , we have

d L˜ y1 y (x − x1 )2 r (x1 + y1 ) − δ + δ = − (μ − r ) + dt xmax x1 x1 x  2 2r r emτ r δ r x max + x max − x max − x 1 (x − x1 ) + − (y − y1 ) 2r xmax x max

mτ

mτ r (e −1) r (e +3) − xδ1 − xδ1 x max x max r (emτ − 1) + (y − y1 )2 − (y − y1 )2 4r xmax x max

xv(1 + kv1 ) βx1 v1 x(t − τ )v(t − τ )(1 + kv1 ) − + + 1 + kv1 x1 v1 (1 + kv) x1 v1 (1 + kv(t − τ ))

βx1 v1 x1 yv1 (1 + kv) x(t − τ )y1 v(t − τ )(1 + kv1 ) + 3− − − 1 + kv1 x y1 v(1 + kv1 ) x1 yv1 (1 + kv(t − τ )) − kβx1

(v − v1 )2 . (1 + kv)(1 + kv1 )2

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It is easy to see that

 τ d L+ x(t − ω)v(t − ω)(1 + kv1 ) d x(t − ω)v(t − ω)(1 + kv1 ) dω = − 1 − ln dt dt 0 x1 v1 (1 + kv(t − ω)) x1 v1 (1 + kv(t − ω))

 τ d x(t − ω)v(t − ω)(1 + kv1 ) x(t − ω)v(t − ω)(1 + kv1 ) = − 1 − ln dω x1 v1 (1 + kv(t − ω)) x1 v1 (1 + kv(t − ω)) 0 dt

 τ d x(t − ω)v(t − ω)(1 + kv1 ) x(t − ω)v(t − ω)(1 + kv1 ) dω =− − 1 − ln x1 v1 (1 + kv(t − ω)) x1 v1 (1 + kv(t − ω)) 0 dω   x(t − ω)v(t − ω)(1 + kv1 ) x(t − ω)v(t − ω)(1 + kv1 ) τ =− − 1 − ln x1 v1 (1 + kv(t − ω)) x1 v1 (1 + kv(t − ω)) ω=0 x(t − τ )v(t − τ )(1 + kv1 ) xv(1 + kv1 ) x(t − τ )v(t − τ )(1 + kv1 ) =− + + ln x1 v1 (1 + kv(t − τ )) x1 v1 (1 + kv) x1 v1 (1 + kv(t − τ )) xv(1 + kv1 ) + ln x1 v1 (1 + kv) x(t − τ )v(t − τ )(1 + kv1 ) xv(1 + kv1 ) x(t − τ )y1 v(t − τ )(1 + kv1 ) =− + + ln x1 v1 (1 + kv(t − τ )) x1 v1 (1 + kv) x1 yv1 (1 + kv(t − τ )) x1 yv1 (1 + kv) = ln + ln . x y2 v(1 + kv1 )

Since

d L˜ βx1 v1 d L + dL = + , dt dt 1 + kv1 dt

we obtain

y1 y (x − x1 )2 dL r (x1 + y1 ) + δ + δ = − (μ − r ) + dt xmax x1 x1 x 2  mτ 2r r + rxemax − xmax − xδ1 r x − (y − y ) (x − x1 ) + max 1 2r xmax x max

mτ

mτ r (e −1) r (e +3) − xδ1 − xδ1 x max x max r (emτ − 1) + (y − y1 )2 − (y − y1 )2 4r xmax x max βx1 v1 x1 x1

− − 1 − ln 1 + kv1 x x

βx1 v1 yv1 (1 + kv)) yv1 (1 + kv)) − − 1 − ln 1 + kv1 y1 v(1 + kv1 ) y1 v(1 + kv1 )

βx1 v1 x(t − τ )y1 v(t − τ )(1 + kv1 )) x(t − τ )y1 v(t − τ )(1 + kv1 )) − − 1 − ln 1 + kv1 x1 yv1 (1 + kv(t − τ )) x1 yv1 (1 + kv(t − τ )) − kβx1

From δ
0. Therefore,

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+y1 ≥ δyx11 , which implies ddtL ≤ 0. Furthermore, ddtL = 0 if and Thus, μ − r 1 − xx1max only if x(t) = x(t − τ ) = x1 , v(t) = v(t − τ ) = v1 and y(t) = y1 . Therefore, the largest compact invariant set M is the singleton {E 1 }, where E 1 is the infected equilibrium. This shows that limt−→∞ (x, y, v) = (x1 , y1 , v1 ). By the Lyapunov– LaSalle type theorem for delay systems (Theorem 2.5.3 in [28]), this implies that E 1 is globally asymptotically stable in the interior of R3+ .  

8 Permanence Lemma 10. For any solution (x(t), y(t), v(t)) of system (2), we have xmax lim sup x(t) ≤ x0 = 2r t→+∞

 (r − μ) +

 4r s (r − μ)2 + xmax

 .

Then, for all  > 0, there is a t1 > 0 such that x(t) < x0 +  for all t > t1 . Let M y = lim sup y(t) and Mv = lim sup v(t), t→+∞

t→+∞

which we know are finite quantities due to Theorem 2. Define  = {(x, y, v) : 0 < x ≤ x0 , 0 ≤ y ≤ M y , 0 ≤ v ≤ Mv }. System (2) satisfies, for some t1 > 0,

x + My βx − , x ≥ s − μx + r x 1 − xmax k 

which implies that ⎤ ⎡ 

r My r My 2 β xmax ⎣ 4r s ⎦ β lim inf x(t) ≥ r −μ− − . + + r −μ− − x→+∞ 2r k xmax k xmax xmax Now we shall prove that the instability of E 0 implies that system (1) is permanent. Definition 11. System (2) is said to be uniformly persistent if there is an η > 0 (independent of the initial data) such that every solution (x(t), y(t), v(t)) with initial condition of system (2) satisfies lim inf x(t) ≥ η lim inf y(t) ≥ η lim inf v(t) ≥ η t−→∞

t−→∞

t−→∞

For dissipative systems, uniform persistence is equivalent to permanence.

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We present the persistence theory for infinite dimensional systems from Hale [31]. Let X be a complete space metric. Suppose that X 0 ⊂ X , X 0 ⊂ X , X 0 ∩ X 0 = ∅, X = X 0 ∪ X 0 . Assume that Y (t) is a C 0 -semigroup on X satisfying 

Y (t) : X 0 → X 0 , Y (t) : X 0 → X 0 .

(40)

Let Yb (t) = Y (t)| X 0 and let Ab be the global attractor for Yb (t). Lemma 12. Suppose that Y (t) satisfies (40) and we have the following: there is a t0 ≥ 0 such that Y (t) is compact for t > t0 , Y (t) is a point dissipative in X , A¯ b = ∪x∈Ab ω(x) is isolated and has an acyclic covering M, where M = {M1 , M2 , . . . , Mn }, 4. W s (Mi ) ∩ X 0 = ∅, for i = 1, 2, . . . , n.

1. 2. 3.

Then X 0 is a uniform repeller with respect to X 0 , i.e., there is an  > 0 such that for x ∈ X 0 lim inf d(Y (t)x, X 0 ) ≥ , t→∞

where d is the distance of Y (t)x from X 0 . Theorem 13. System (2) is permanent provided R0 (τ ) > 1. Proof. We begin by showing that the boundary planes of R3+ repel the positive solutions of system (2) uniformly. Let us define C0 = {(ψ, φ1 , φ2 ) ∈ C([−τ, 0], R3+ ) : ψ(θ ) = 0, φ1 (θ ) = φ2 (θ ) = 0, (θ ∈ [−τ, 0])}.

If C 0 = int C([−τ, 0], R3+ ), it suffices to show that there exists an 0 such that for any solution u t of system (2) initiating from C 0 , lim inf t→+∞ d(u t , C0 ) ≥ 0 . To this end, we verify below that the conditions of Lemma 12 are satisfied. It is easy to see that C 0 and C0 are positively invariant. Morever, conditions 1 and 2 of Lemma 12 are satisfied. Thus, we only need to verify conditions 3 and 4. There is a constant solution E 1 in C0 , to x(t) = x0 , y(t) = v(t) = 0. If(x(t), y(t), v(t)) is a solution of system (2) initiating from C0 , then x(t) → x0 , y(t) → 0, v(t) → 0, as t → +∞. It is obvious that E 0 is an isolated invariant. Now, we show that W s (E 1 ) ∩ C 0 = ∅. Assuming the contrary, then there exists a positive solution (x(t), ˜ y˜ (t), v(t)) ˜ of system (2) such that ((x(t), ˜ y˜ (t), v(t))) ˜ → (x0 , 0, 0) as t → ∞. Let us choose  > 0 small enough and t0 > 0 sufficiently large such that ˜ < x0 + , 0 < y˜ (t) < , 0 < v(t) ˜ t0 − τ . Then we have for t > t0 ,

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˙ x0 + 2 − (α + δ) y˜ , ˜ − τ ) + r y˜ 1 − y˜ (t) ≥ βe−mτ (x0 − )v(t xmax ˙˜ = σ y˜ − γ v. v(t) ˜ Let us consider the matrix defined by ⎛

⎞ x0 + 2 −mτ r 1 − − (α + δ) βe (x − ) 0 ⎠. A = ⎝ xmax σ −γ Since A admits positive off-diagonal elements, the Perron–Frobenius theorem implies that there is a positive eigenvector Vˆ for the maximum eigenvalue λ1 of A .

x0 +2 Morever, since R0 (τ ) > 1, then γ (α + δ) − r γ 1 − xmax − σβe−mτ (x0 − ) < 0 for  small enough, by a simple computation we see that λ1 is positive. Let us consider

 x0 + 2 y˙ (t) = βe−mτ (x0 − )v(t − (α + δ) y˜ , ˜ − τ ) + r y˜ 1 − (41) xmax ˙˜ = σ y˜ − γ v. v(t) ˜ Let v = (v1 , v2 ) and l > 0 be small enough such that lv1 < y˜ (t0 + θ ), lv2 < v(t ˜ 0 + θ ), for θ ∈ [−τ, 0] if (y(t), v(t)) is a solution of system (41) satisfying y(t) = lv1 , v(t) = lv2 for t0 − τ ≤ t ≤ t0 . Since the semiflow of system (41) is monotone and A v > 0, it follows that y(t) and v(t) are strictly increasing and y(t) → ∞, v(t) → ˜ → ∞ as t → ∞. Note that y˜ ≥ y(t), v(t) ˜ ≥ v(t) for t > t0 . We have y˜ (t) → ∞, v(t) ∞ as t → ∞. At this time, we are able to conclude from Theorem 2 that C0 repels the positive solutions of system (2) uniformly. Incorporating this into Lemma 12 and Theorem 2, we know that the system (2) is permanent.  

9 Estimation of the Length of Delay to Preserve Stability In this section, we consider system (2) when m = 0. We can use the Nyquist criterion to obtain an estimate for the maximum value of the delay τ for which the infected equilibrium is locally asymptotically stable. We omit the calculations and present the results only since the analysis is similar to that performed in [19] and is not affected by the cure term δy(t). Let

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r x1 βv1 βx1 v1 r y1 a2 = p + + + + + γ, xmax 1 + kv1 (1 + kv1 )y1 xmax





βx1 v1 r x1 βv1 r y1 βx1 v1 r y1 γ + + +γ + + a1 = p + xmax 1 + kv1 (1 + kv1 )y1 xmax (1 + kv1 )y1 xmax



r y1 r x1 − −δ , xmax xmax







βx1 v1 r y1 r x1 r x1 βv1 r y1 a0 = p + γ− + + −δ γ xmax 1 + kv1 (1 + kv1 )y1 xmax xmax xmax



r y1 βσ x1 − , xmax (1 + kv1 )2

βx1 σ r x1 βv1 b1 = − + − δ , xmax 1 + kv1 (1 + kv1 )2



βv1 r x1 r x1 βx1 σ βv1 βv1 βσ x1 p + + b0 = − + − δ γ + . xmax 1 + kv1 xmax 1 + kv1 (1 + kv1 )2 (1 + kv1 )2 1 + kv1

Define v+ = K1 =

|b2 − a1 b1 | + 2 (v ) , 2

|b1 | +

" b12 + 4a2 (|a0 | + |b0 |) 2a2

K 2 = |b1 |(v + )2 + |a2 b2 |(v + ),

,

K 3 = a2 a1 + a1 b1 − a0 − b2 .

Then a positive root of K 1 τ 2 + K 2 τ = K 3 is given by τ+ =

1 2K 1



" −K 2 + K 22 + 4K 1 K 3 .

For 0 ≤ τ ≤ τ+ , the Nyquist criterion holds. The value of τ+ gives an estimate for the length of delay for which stability is preserved. Hence, the estimate for the delay is totally dependent on system parameters for which the equilibrium E 1 is locally asymptotically stable. By a similar analysis to that used in [19], we obtain the following result. Theorem 14. If there exists a parameter 0 ≤ τ ≤ τ+ such that K 1 τ 2 + K 2 τ < K 3 , then τ+ is the maximum value (length of delay) of τ for which E 1 is asymptotically stable.

10 Numerical Simulations In the following, we present some numerical simulations of examples which validate our theoretical results. We used dde23 [32, 33], which is based on the Runge– Kutta methods, and obtained some figures (see Figs. 1–8). To explore system (2) and illustrate the stability of equilibrium solutions, we considered parameter values similar to those in [19, 34–36]. In Figs. 1, 3, 5, and 7, constant initial conditions a, b,

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and c were given; in Figs. 2, 4, 6, and 8, variable initial conditions of the form aeθ , beθ , ceθ were provided. With the set of parameter values τ = 5, r = 0.05, s = 20, μ = 0.02, α = 0.021, xmax = 1200, β = 9.2419 × 10−7 , σ = 0.21, γ = 0.02, k = 0.001, δ = 0.01, m = 0.021,

(42)

we have R0 (τ ) = 0.4011 < 1 and E 0 = (1141, 0, 0), so Theorem 4 is satisfied, 4r x0 = 0.1901, and when m = 0, moreover when m = 0, we have δ = 0.01 < xmax r (emτ −1) r (emτ +3) x = 0.0053 < δ = 0.01 < x = 0.1954, by Theorem 5 the unin0 0 xmax xmax fected equilibrium is stable (see Figs. 1 and 2). Considering the set of parameter values τ = 2, r = 0.03, s = 10, μ = 0.02, α = 0.26, xmax = 1500, (43) β = 0.0027, σ = 5.9, γ = 2.4, k = 0.001, δ = 0.01, m = 0.26,

−mτ x1 +y1 = we find that R0 (τ ) = 10.9893 > 1, Q = 5.3371 > 0, α − r 1 − e xmax 0.2315 > 0, so the conditions of Theorem 6 are satisfied. Then the trajectories of system (43) converge to the infected equilibrium E 1 = (89.02, 23.3, 57.23), as shown in Figs. 3 and 4. The infected steady state is locally asymptotically stable when τ = 0. With the set of parameter values τ = 10, r = 0.001, s = 5, μ = 0.02, α = 0.05, xmax = 1200, β = 0.0027, σ = 0.5, γ = 2.1, k = 0.001, δ = 0.0001, m = 0.05, (44) we have E 1 = (127.2, 31.65, 7.536) and R0 (τ ) = 2.0406 > 1, when m = 0, mτ r (emτ −1) x = 6.88 × 10−5 < δ = 10−4 < r (exmax+3) x1 = 4.93 × 10−4 , and μ − r

xmax 1 +y1 1 − xx1max − δ xy11 = 0.0191 > 0. By Theorem 9, the infected steady state is globally asymptotically stable, as shown in Figs. 5 and 6. For the case m = 0, we used the set of parameter values τ = 10, r = 0.02, s = 5, μ = 0.02, α = 0.302, xmax = 1200, β = 0.0027, σ = 3.02, γ = 10, k = 0.001, δ = 0.01, m = 0;

(45)

+y1 we find R0 (τ ) = 1.4663 > 1, δ = 0.01 < x4r x = 0.0245, and μ − r 1 − xx1max − 1 max

δ xy11 = 0.0060 > 0. Therefore, by Theorem 9 with m = 0, the infected steady state E 1 = (366.8, 9.366, 2.829) is globally asymptotically stable when m = 0 (see Figs. 7 and 8).

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1200 500

1000 400

800 v(t)

300

600

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400

100

200

0 80 60

0 0

200

400

600

1200 40

800

1000 20

t

y(t)

800 0

600

x(t)

(b)

(a)

Fig. 1 Dynamical behavior of system (2) with the set of parameter values (42). We have R0 = 0.4011 < 1 and E 0 = (1141, 0, 0) is globally asymptotically stable. The graph in a shows the time series of the solutions with constant initial conditions. The graph in b shows the trajectory in the phase diagram of system (42), which illustrates the stability of infection-free equilibrium E 0 with the history functions φ1 (θ) = 800, φ2 (θ) = 80, φ3 (θ) = 300 (first trajectory); φ1 (θ) = 750, φ2 (θ) = 75, φ3 (θ) = 250 (second trajectory); φ1 (θ) = 700, φ2 (θ) = 70, φ3 (θ) = 200 (third trajectory) Phase space

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200

0 80 60

0 0

200

400

t

(a)

600

800

1200 40

1000 20

y(t)

800 0

600

x(t)

(b)

Fig. 2 Dynamical behavior of system (2) with the set of parameter values (42). We have R0 = 0.4011 < 1 and E 0 = (1141, 0, 0) is globally asymptotically stable. The graph in a shows the time series of the solutions with variable initial conditions. The graph in b shows the trajectory in the phase diagram of system (42), which illustrates the stability of infection-free equilibrium E 0 with the history functions φ1 (θ) = 800eθ , φ2 (θ) = 80eθ , φ3 (θ) = 300eθ (first trajectory); φ1 (θ) = 750eθ , φ2 (θ) = 75eθ , φ3 (θ) = 250eθ (second trajectory); φ1 (θ) = 700eθ , φ2 (θ) = 70eθ , φ3 (θ) = 200eθ (third trajectory)

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x(t)

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We illustrate the stability of E 1 according to Theorem 8. For that, we choose a set of parameters as follows: r = 0.84, s = 1, μ = 0.02, α = 0.9, xmax = 1500, β = 0.0027, σ = 9, γ = 2.4, k = 0.0001, δ = 0.01, m = 1.2.

(46)

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One can see that there are two critical values of the delays, denoted by τ ∗ and τ ∗∗ , τ ∗ = 0.6894 and τ ∗∗ = 1.9458. Simple examination shows that the infected equilibrium is locally asymptotically stable for τ ∈ [0, τ ∗ ). In this case, we select τ = 0.66 < τ ∗ = 0.6894, see Fig. 9. The positive equilibrium of the system is unstable for τ ∈ (τ ∗ , τ ∗∗ ). In this case, we select τ = 0.75 > τ ∗ and τ = 1.9 < τ ∗∗ = 1.9458, see Figs. 10 and 11. When τ = 1.96 > τ ∗∗ , the positive equilibrium is asymptotically stable again, see Fig. 12. At τ ∗ and τ ∗∗ , Hopf bifurcation occurs. Note that in Figs. 1–8, there is very little change between the figures with constant initial conditions and the corresponding figures with variable initial conditions. This also happens when in Figs. 9–12 we replace the constant initial conditions with similar variable conditions. Therefore, we only provide the figures with constant initial conditions in Figs. 9–12.

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To illustrate Theorem 13, we used the set of parameter values τ = 5, r = 0.02567, s = 5, μ = 0.02, α = 0.05, xmax = 1200, β = 0.0027, σ = 0.5, γ = 2.1, k = 0.001, δ = 0.0007, m = 0.05. (47) We showed that uniform persistence occurs if the basic reproduction number is greater than unity. We note that the condition for the global stability of endemic equilibrium is not satisfied, as it also happens in another similar system with R0 > 1 (see Fig. 10).

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Finally, we present some bifurcation diagrams obtained when the time delay in system (2) is varied. First, we take a set of parameters as follows: r = 0.8, s = 1, μ = 0.02, α = 0.9, xmax = 1500, β = 0.0027, σ = 9, γ = 2.4, k = 0.0001, δ = 0.01, m = 1.2.

(48)

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System (2) has a process from stability to oscillation and then to stability again when τ changes from 0 to 3, as we can see in Fig. 14. At the left end of the τ range, up to τ ∗ = 0.6894, the infected steady state is asymptotically stable, but the virus decreases. As τ increases, the time delay destabilizes the infected steady state, leading to periodic oscillations. At the right of τ ∗∗ = 1.9458, the infected steady state is again asymptotically stable and the virus level v(t) decreases. These results imply that the time delay has an effect on the dynamics of the model.

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Next, we take the set of parameters r = 0.8, s = 1, μ = 0.02, α = 0.5, xmax = 1500, β = 0.0027, σ = 9, γ = 2.4, k = 0.0001, δ = 0.01, m = 1.2,

(49)

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Our results not only improve our understanding of viral pathogens, but also can aid to develop appropriate therapeutic strategies to control viral infections. The information given by the bifurcation diagrams shows those intervals for the delay that prevent oscillations of the solution, hence giving valuable insight for therapeutic strategies aimed to keep the delay within desirable values in the aforementioned intervals.

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Fig. 13 Plot of the basic reproduction number R0 as a function of the delay τ (with δ = 0.01) and as a function the cure rate δ (τ = 1.9). We consider the values for parameters as r = 0.8, s = 1, μ = 1.2, α = 0.9, xmax = 1500, β = 0.0027, σ = 9, γ = 2.4, k = 0.0001, and m = 0.02

10.1 Comparison with Other Models We will now make some comparisons of the viral dynamics for our model with other related models studied in the literature.

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Effect of Cure Rate

Consider the case when m = 0 and δ = 0, we can find that system (2) becomes: ⎧   x(t)+y(t) d x(t) ⎪ − μx(t) − βx(t)v(t) = s + r x(t) 1 − , ⎪ xmax  ⎨ dt  1+kv(t) dy(t) x(t)+y(t) βx(t−τ )v(t−τ ) − αy(t), = 1+kv(t−τ ) + r y(t) 1 − xmax dt ⎪ ⎪ ⎩ dv(t) = σ y(t) − γ v(t), dt

(50)

which is precisely the model analyzed in [19]. Let R0 and R 0 be the basic reproduction numbers of model (2) and model (50), respectively. It is easy to see that R0 ≤ R 0 . We observe that the virus level of the endemic equilibrium state for system (2) is less than the corresponding level for system (50) (see Fig. 16), which shows that the infection speed of the virus in our model is slower than for system (50). Hence, the disease can easily be controlled if we improve the cure rate.

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We also observe in our model the effect of the nonlinear saturated rate. If we consider τ = 0 and k = 0, we can find that system (2) becomes: ⎧   x(t)+y(t) d x(t) ⎪ − μx(t) − βx(t)v(t) + δy(t), = s + r x(t) 1 − ⎪ xmax ⎨ dt   dy(t) x(t)+y(t) − (α + δ)y(t), = βx(t)v(t) + r y(t) 1 − xmax ⎪ dt ⎪ ⎩ dv(t) = σ y(t) − γ v(t). dt

(51)

System (51) has been studied in [34, 35]. When τ = 0, comparing system (2) with (51), the main difference is that the infection rate is represented by a nonlinear saturated infection rate. Varying the parameter k of the nonlinear saturated rate in system (2), we observe that when k grows both infected cells and viruses decrease, while the concentration of uninfected cells increases (see Figs. 17, 18, and 19). The results show that the saturation and cure rate can control the disease progression by reducing the concentration of free virus particles and infected cells.

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11 Conclusion We have studied a viral infection model with mitotic transmission, cure rate, the effect of saturated infection function and intracellular time delay between infection of target cells and production of new particles. The steady state of this model depends on the delay τ through a factor e−mτ , where m is the death rate of infective cells. One would expect that the characteristic equation has coefficients depending on the steady state coordinates and hence on this exponential factor e−mτ . The resulting characteristic equation linearized around the steady state also contains this factor. To carry out the linear stability analysis here is a harder task than in the case where the model is independent of delay τ . From our results of model (2), we can see that, as the time delay τ increases, oscillatory dynamics may appear, and further increase of τ will return the oscillatory dynamics to the stable state form. At last, when the time delay τ is too long, the positive equilibrium disappears and the virus cells v(t) die out, see Fig. 13. Studying the reversion rate of infected cells into the uninfected state as a control parameter of a viral infection is very important. We plot R0 versus δ, and vary the values of δ between (0, 10). In Fig. 13, we note that if δ = 0, then R0 = 16.3078 and if we increase the value of cure rate of the infected cells, then the basic reproductive number decreases below one. A relevant result for model (2) is that the viral infection gets eradicated if δ is larger. For a cytopathic virus (α > μ), we established conditions to ensure the local stability of the infected equilibrium state. Our findings also improve our understanding of viral pathogenesis and may help to develop novel therapeutic strategies in order to control viral infections. Future work includes finding the direction of the Hopf bifurcation. Another improvement to the current model is to use a more general incidence rate as a function with certain desired properties, as considered in [37]. There is also room for improvement if we consider distributed delays in the equations for the infected cells and the virus particles, as is discussed in [38]. These last two possible improvements to the current model pose more challenging problems to be analyzed. Acknowledgements This work was supported by Universidad Autónoma de Yucatán and Mexican CONACYT under SNI grant numbers 15284 and 33365.

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Dynamics of a Fractional-Order Hepatitis B Epidemic Model and Its Solutions by Nonstandard Numerical Schemes Manh Tuan Hoang and Oluwaseun Francis Egbelowo

Abstract In this chapter, we propose and analyze a fractional-order hepatitis B epidemic model, which is an extension of a recognized ordinary differential model. Our main objective is to study dynamical properties of the proposed fractional-order model and its numerical solutions. Firstly, we establish positivity and boundedness of the proposed model. Secondly, the asymptotic stability of the model is investigated rigorously by the Lyapunov stability theorem for fractional dynamical systems and numerical simulations. Finally, we construct positivity-preserving nonstandard finite difference (NSFD) schemes for the fractional-order model. It should be emphasized that the constructed NSFD schemes preserve the positivity of solutions and the stability of the continuous model for all finite step sizes and therefore, they reflect exactly dynamics of the continuous model. Meanwhile, the standard Grunwald-Letnikov scheme fails to preserve the correct behavior of the continuous model for a given step size. Actually, the standard scheme provides numerical approximations that are completely different from the exact solution of the fractional-order model. Some numerical simulations are performed to confirm the validity of the theoretical results and to show advantages and superiority of NSFD schemes over the standard ones. The numerical experiments indicate that there is a good agreement between numerical simulations and the theoretical results. Keywords Hepatitis B epidemic model · Global asymptotic stability · Lyapunov stability theorem · Nonstandard finite difference schemes · Numerical simulations

M. T. Hoang (B) Department of Mathematics, FPT University, Hoa Lac Hi-Tech Park, Km29 Thang Long Blvd, Hanoi, Vietnam e-mail: [email protected]; [email protected] O. F. Egbelowo Division of Clinical Pharmacology, Department of Medicine, University of Cape Town, Cape Town, South Africa e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_5

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1 Introduction Hepatitis B virus (HBV) infection is one of the most devastating disease in the world, it is a viral infection that attacks the liver and can cause both acute and chronic diseases. Nowadays, HBV has become a major health problem worldwide. Therefore, effective approaches to the predict and eradicate the HVB are needed. For this purpose, many systems of ordinary differential equations (ODEs) that mathematically modeled the transmission of HBV have been constructed and analyzed (see [7, 33, 54, 55, 59] and references therein). The study of these systems is one of the most effective approaches to prevent and control the transmission of the HVB. Consequently, this could lead to eradication of the disease. We now consider a hepatitis B epidemic model with saturated incidence rate which was proposed in a recent work [33]. The model assumes the human population is divided into the following categories: susceptible individuals (S), those who are infected with the HVB (I ), and those who recover of Hepatitis B infection (R). The susceptible individuals are recruited into the population at per capita rate  and there is a per capita natural mortality rate μ0 . Individuals with the HVB suffer disease induced death at rate μ1 or recover at rate β while the vaccination and saturation rate are ν and γ and the transmission rate is denoted α. The model is given by the following system of ODEs: d S(t) αS(t)I (t) =− − (μ0 + ν)S(t), dt 1 + γ I (t) αS(t)I (t) d I (t) = − (μ0 + μ1 + β)I (t), dt 1 + γ I (t) d R(t) = β I (t) + ν S(t) − μ0 R(t). dt

(1)

More details of the model (1) can be found in [33]. In [33], the essential qualitative properties of the the model (1) including the positivity of the solutions and the asymptotic stability of the equilibrium points were established. In this chapter, we generalize the model (1) by considering its fractional-order version. More precisely, the fractional-order version under consideration is given by α q S(t)I (t) q − (μ0 + ν q )S(t), 1 + γ q I (t) α q S(t)I (t) q q C q − (μ0 + μ1 + β q )I (t), 0 Dt I (t) = 1 + γ q I (t) q C q q q 0 Dt R(t) = β I (t) + ν S(t) − μ0 R(t), C q 0 Dt S(t)

q

= q −

(2)

where q ∈ (0, 1) and C0 Dt f (t) is the Caputo derivative of a given function f (t) (see [8, 19, 52, 53]). When q → 1 then the model (2) reduces to the model (1) and therefore, the model (2) is a generalization of the model (1).

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It is well-known that fractional calculus is a generalization of classical calculus which studies derivatives and integrals of fractional order. It has become a major tool in almost every branch of science and engineering [19, 34, 35, 38, 41, 52, 60] due to its superior accuracy over integer-order calculus. In particular, fractional differential equations (FDEs) have become an important and powerful tool in mathematical modelling and analysis nowadays [19, 38, 41, 52]. These equations play an important role in both theory and practice. Many important phenomena and processes arising in the applied sciences were described mathematically and analyzed based on FDEs (see [2, 4, 9, 10, 32, 53, 56] and references therein). The advantage of fractional-order over the integer-order is that fractional-order are related to systems with memory which is possessed by most biological systems [47]. Therefore, the dynamics of HBV transmission will be better understand with the concept of FDEs. To the best of our knowledge, there have been many differential models proposed to predict the dynamics of the HVB transmission but the study of the model (1) with fractionalorder derivatives has not been conducted. This motivates us to do this research. Our main objective is to study dynamical properties of the model (2) and to construct effective numerical schemes to solve it. Firstly, positivity and boundedness of solutions of the model (2) are established. Next, the asymptotic stability of equilibrium points of the model (2) are investigated by the Lyapunov stability theorem for fractional dynamical systems and numerical simulations. More clearly, by using a suitable Lyapunov function we show that the disease free equilibrium (DFE) point is uniformly asymptotically stable if the reproduction number is less than one. On the other hand, when the reproduction number is greater than one, the disease endemic equilibrium (DEE) point is locally asymptotically stable and especially, numerical simulations suggest that it is not only locally asymptotically stable but uniformly asymptotically stable. Since it is impossible to find the exact solution to the fractional model (2), our next objective is to propose effective numerical schemes to solve the model (2). For this purpose, we use nonstandard finite difference (NSFD) schemes as an effective and powerful techniques for numerical simulations of the model (2). It is well-known that the NSFD schemes for solving differential equations were first introduced by Mickens [43–46] to compensate for the weaknesses of the standard finite difference (SFD) schemes. For many years, NSFD methods have been widely used for solving differential models arising in applied sciences and engineering (see [1, 5, 20–23, 28, 37, 43–46, 57, 58] and references therein). The NSFD schemes overcome drawbacks of the SFD schemes by providing reliable numerical solutions for differential models. Meanwhile, the SFD schemes can not exactly preserve the essential dynamical properties of the differential models and consequently, they can generate numerical solutions which are different from the solutions of the differential models. Recently, some results on NSFD schemes for ODE models have been proposed [11–18, 24– 27, 30, 31]. Besides, the NSFD schemes have been constructed to solve systems of FDEs (see [6, 29, 50, 51]). All the results agree that the NSFD schemes are very effective for both ordinary differential and fractional-order models. However, to the best of our knowledge, NSFD schemes for the fractional-order systems are few in general and NSFD schemes for the model (2) has not been considered in particular.

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This motivates us to propose NSFD schemes for the fractional-order model (2). It is worth noting that the constructed NSFD schemes preserve the positivity of solutions and the stability of the continuous model for all finite step sizes and hence, essential properties of the continuous model are reflected exactly. Meanwhile, the standard Grunwald-Letnikov scheme fails to preserve the correct behavior of the fractional-order model for a given step size. Actually, the standard scheme provides numerical approximations which are completely different from the exact solution of the continuous model. The chapter is organized as follows. Some preliminaries are provided in Sect. 2. The positivity, boundedness and stability of the model (2) are investigated in Sect. 3. In Sect. 4, NSFD schemes for the model (2) are constructed and analyzed. In Sect. 5, some numerical simulations are reported. Some conclusions and open problems are presented in the last section.

2 Preliminaries 2.1 Fractional Derivatives In this subsection, we present some preliminaries that will be used in the next sections. Definition (See [8]). Suppose that α > 0, t > 0, α, a, t ∈ R. The Caputo fractional derivative is given by C α a Dt

f (t) =

1 (n − α)

 a

t

f (n) (ξ ) dξ, (t − ξ )α+1−n

n − 1 < α < n ∈ N.

Remark 1. The Caputo fractional derivative of order 0 < α < 1 for a smooth function f = f (t) is given by C α a Dt

f (t) =

1 (1 − α)

 a

t

1 d f (ξ ) dξ. α (t − ξ ) dξ

Property 1. (Linearity property [19]). Let f (t), g(t) : [a, b] → R be such that f (t) and Ca Dtα g(t) exist everywhere and let c1 , c2 ∈ R. Then, Ca Dtα (c1 f (t) + c2 g(t)) exists everywhere, hence

C α a Dt

C α a Dt (c1

f (t) + c2 g(t)) = c1 Ca Dtα f (t) + c2 Ca Dtα g(t).

Property 2. (Caputo derivative of a constant [52]). The fractional derivative for a constant function f (t) = c is zero.

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Lemma 1. ([48]). Suppose that w ∈ C[a, b] and Ca Dtα w(t) ∈ C[a, b] for 0 < α ≤ 1, then we have 1 C α w(t) = w(a) + D w(ξ )(t − ξ )α , (α) a t with a ≤ ξ ≤ b, for all t ∈ (a, b]. Remark 2. ([36, Remark 1]). Assume that w ∈ C[a, b] and Ca Dtα w(t) ∈ C[a, b] for 0 < α ≤ 1. Then, Lemma 1 implies that if Ca Dtα w(t) ≥ 0, for all t ∈ (a, b), then w(t) is non-decreasing on [a, b], and if Ca Dtα w(t) ≤ 0, for all t ∈ (a, b), then w(t) is non-increasing on [a, b].

2.2 Fractional Dynamical Systems Consider the following general type of equations involving the Caputo derivative: C α t0 Dt x(t)

= f (t, x),

α ∈ (0, 1),

(3)

with the initial condition x0 = x(t0 ). Lemma 2. (A fractional comparison principle [39, Lemma 10]). Let x(0) = y(0) β β and Ct0 Dt x(t) ≥ Ct0 Dt y(t), where β ∈ (0, 1). Then x(t) ≥ y(t). Definition (See [39, 40]). The constant x ∗ is an equilibrium point of Caputo fractional dynamical system (3), if and only if f (t, x ∗ ) = 0. Theorem 1. (See [42]). Consider the following commensurate fractional-order system C α x(0) = x0 ∈ Rn , t0 Dt x(t) = f (x), with 0 < α ≤ 1. Let x ∗ be an equilibrium point of the system, that is, f (x ∗ ) = 0. If ∂f ∗ απ (x ) satisfy | arg(λi )| > , all the eigenvalues λi of the Jacobian matrix J = ∂x 2 then the equilibrium point is locally asymptotically stable. The following criterion called the Schur-Hurwitz criterion [3]. Theorem 2. Consider the quadratic polynomial of the form P(λ) = λ2 + a1 λ + a2 . Then, all roots of P(λ) have the negative parts if a1 > 0 and a2 > 0. We now present the Lyapunov direct method for fractional-order nonlinear systems [10, 53].

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Theorem 3. Let x ∗ be an equilibrium point for the non autonomous fractional-order system (3) and ⊂ Rn be a domain containing x ∗ . Let L(t, x(t)) : [0, ∞] × → R be a continuously differentiable function such that W1 (x) ≤ L(t, x(t)) ≤ W2 (x), C α t0 Dt L(t, x(t))

≤ −W3 (x),

∀α ∈ (0, 1), ∀x ∈ , where W1 (x), W2 (x) and W3 (x) are continuous positive definite functions on . Then the equilibrium point of system (3) is uniformly asymptotically stable. Lemma 3. (See [2]). Let x(t) ∈ R be a continuous and derivable function. Then, for any time instant t ≥ t0 1C α 2 D x (t) ≤ x(t)Ct0 Dtα x(t), ∀α ∈ (0, 1). 2 t0 t

2.3 The Grunwald-Letnikov Approximation We recall from [6, 49] some results related to the Grunwald-Letnikov approximation. Assume that the function Dtα y(τ ) satisfies some smoothness conditions in every finite interval (0, t), t ≤ T . Choosing the grid 0 = τ0 < τ1 < . . . < τn+1 = t = (n + 1)h, τn+1 − τn = h, and using the classical notation of finite differences   n+1  1 α 1 α y(τ  y(t) = ) − c y(τ ) , n+1 n+1−ν ν hα h hα ν where cνα = (−1)ν−1

  α . ν

Then, the Grunwald-Letnikov definition reads [52] Dtα y(t) = lim

t→0

1 α  y(t). hα h

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By applying the Grunwald-Letnikov definition to the FDE of the form C α 0 Dt y(t)

= f (y(t)),

y(t0 ) = y0 ,

we obtain the explicit Grunwald-Letnikov method for an equidistant grid as follows (see [49]) n+1  α cνα yn+1−ν − rn+1 y0 = h α f (yn ), (4) yn+1 − ν=1

where

α α rn+1 = h α r0α (τn+1 ) = γ0,−1 (n + 1)−α ,

and the coefficient α γμ,k =

(μa + 1) , (kα + 1)

μ, k ∈ N0 ∪ {−1}.

For some important properties of the coefficients cνα and rνα , we refer the readers to [49].

3 Dynamical Properties of Model (2) 3.1 Positivity and Boundedness In this subsection, we investigate the positivity and boundedness of the model (2).  Theorem 4 (Positivity of solutions). The set R3+ := (S, I, R)|S ≥ 0, I ≥ 0, R ≥  0 isa positively invariant set for the model (2), i.e., if S(0), I (0), R(0) ∈ R3+ , 3 then S(t), I (t), R(t) ∈ R+ for all t > 0. Proof. Firstly, it is easy to show the existence and uniqueness of the solution of the model (2) by using Theorem 3.1 and Remark 3.1 in [41]. On the other hand, we have that



q

q

C q

= q > 0, C0 Dt I

= 0, C0 Dt R

= β q I + ν q S ≥ 0. (5) 0 Dt S

S=0

I =0

R=0

 Let S(0), I (0), R(0)) be any initial data with S(0), I (0), R(0) ≥ 0. Then, by using  Remark 2 and (5), we conclude that the solution S(t), I (t), R(t) cannot escape from the hyperplanes of S = 0, I = 0 and R = 0, and on each hyperplane the vector field is tangent to that hyperplane or points toward the interior of the set R3+ . Therefore,  the solution will remain in the set R3+ . This completes the proof.

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Let us define    q  q q m 1 := min μ0 + ν q , μ0 + μ1 + β q , m 2 := max β q , ν q .

(6)

 T Theorem 5 (Boundedness of solutions). Let S(0), I (0), R(0) be any initial data  T for the model (2) with S(0), I (0), R(0) ≥ 0 and S(t), I (t), R(t) be the solution of the model (2) corresponding to the initial data. Then, we have lim sup S(t) ≤ t→∞

q μ0

 q q m 2 q , lim sup , lim sup R(t) ≤ S(t) + I (t) ≤ q . m1 + νq m 1 μ0 t→∞ t→∞ (7)

Proof. From the first equation of the model (2), we have C q 0 Dt S(t)

q

≤ q − (μ0 + ν q )S(t).

(8)

Consider an auxiliary equation given by C q 0 Dt z(t)

q

= q − (μ0 + ν q )z(t),

z(0) ≥ S(0).

(9)

By using (8) and Lemma 2, we get S(t) ≤ z(t) for all t ≥ 0. On the other hand, q the equation (9) has a unique equilibrium point given by z ∗ = q /(μ0 + ν q ). We ∗ will prove that z is uniformly asymptotically stable. Indeed, consider the Lyapunov function of the form 1 V (z) = (z − z ∗ )2 . 2 From Lemma 3, we obtain C q 0 Dt V

≤ (z − z ∗ )C0 Dt z = −(μ0 + ν q )(z − z ∗ )2 . q

q

By Theorem 3, we conclude that z ∗ is uniformly asymptotically stable. This implies that limt→∞ z(t) = z ∗ . q . Hence, the Note that S(t) ≤ z(t). Therefore, lim supt→∞ S(t) ≤ z ∗ = q μ0 + ν q first estimate of (7) is proved. The second estimate of (7) is proved similarly. Indeed, from the first and second equations of the model (2), we get  C q 0 Dt S

q q q + I = q − (μ0 + ν q )S − (μ0 + μ1 + β q )I ≤ q − m 1 (S + I ),

which implies that lim sup(S + I ) ≤ t→∞

q . m1

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Finally, by using the third equation of the model (2), we get C q 0 Dt R

  q q q q ≤ max β q , ν q (S + I ) − μ0 R ≤ m 2 (S + I ) − μ0 R ≤ m 2 − μ0 R, m1

for t large enough. Consequently, lim sup R(t) ≤ t→∞

m 2 q q. m 1 μ0 

The proof is complete. From Theorems 4 and 5, we have the following corollary. Corollary 1. The compact set



= (S, I, R) ∈ R3+ 0 ≤ S ≤

q q m 2 q , 0 ≤ S + I ≤ , 0 ≤ R ≤ q q m1 μ0 + ν q m 1 μ0



can be considered as a positively invariant set of the model (2).

3.2 The Reproduction Number and Equilibria Based on the reproduction number of the model (1), we can compute the reproduction number of the model (2) as follows R0 :=

q (μ0

α q q . q q + ν q )(μ0 + μ1 + β q )

(10)

It is important to note that the basic reproduction number is a threshold quantity defined as the expected number of secondary infectious produced by an index case, that is, the average number of secondary infections during its entire infection period arising from a single individual introduced into the susceptible population (see [33, Section 3]). Hence, the reproduction number of the model (2) can be understood in the same way. Importantly, as will be seen later, the asymptotic stability of the equilibria of the model (2) depends on the value of the reproduction number. As a consequence of this, the dynamics of the model (2) is also dependent on R0 . In order to determine equilibrium points of the model (2), we consider the following system αq S I q − (μ0 + ν q )S = 0, 1 + γqI αq S I q q − (μ0 + μ1 + β q )I = 0, 1 + γqI q β q I + ν q S − μ0 R = 0.

q −

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After some algebraic manipulations, the solutions to this system are given by S0 =

q μ0

q , + νq

I0 = 0,

R0 =

ν q q , + νq )

(11)

q q μ0 (μ0

and  q  q q q α q − μ0 + ν q μ0 + μ1 + β q 1 , S ∗ = q (μq0 + μq1 + β q )(1 + γ q I ∗ ),  I∗ =  q q α μ0 + μq + β q α q + γ q (μ0 + ν q )

R∗ =

β q I ∗ + νq S∗ . q μ0

(12)

Note that I ∗ > 0 if and only if R0 > 1.

Proposition 1 (Equilibrium points). Model (2) always possesses a disease free equilibrium (DFE) point E 0 = (S0 , I0 , R0 ) for all the values of the parameters, whereas, only if R0 > 1, there is a (unique) disease endemic equilibrium (DEE) point E ∗ = (S ∗ , I ∗ , R ∗ ) in the interior of R3+ .

3.3 Local Asymptotic Stability We now analyze the local asymptotic stability of the model (2) with the help of Theorem 1. The Jacobian matrix of the system (2) is ⎞

⎛ −α q I αq S q − (μ0 + ν q ) − q ⎜1 + γ I (1 + γ q I )2 ⎜ ⎜ ⎜ αq S αq I J (S, I, R) = ⎜ q q − (μ0 + μ1 + β q ) ⎜ 1 + γqI (1 + γ q I )2 ⎜ ⎝ νq

0

⎟ ⎟ ⎟ ⎟ . 0 ⎟ ⎟ ⎟ ⎠ q

βq

−μ0

Consequently, we have ⎛ ⎞ q −(μ0 + ν q ) −α q S 0 0 ⎜ ⎟ ⎜ ⎟ q q q 0 q ⎟. S − (μ + μ + β ) 0 0 α J (E 0 ) = ⎜ 0 1 ⎜ ⎟ ⎝ ⎠ q βq −μ0 νq Three eigenvalues of the matrix J (E 0 ) are q

q

q

q

λ1 = −(μ0 + ν q ), λ2 = −μ0 , λ3 = α q S 0 − (μ0 + μ1 + β q ). Clearly, λ1 < 0 and λ2 < 0. Moreover, it is easy to verify that λ3 < 0 if and only if R0 < 1.

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Proposition 2. The DFE point E 0 of the model (2) is locally asymptotically stable if R0 < 1. Similarly, we have ⎛ −α q I ∗ αq S∗ q q − (μ + ν ) − 0 ⎜1 + γ q I∗ (1 + γ q I ∗ )2 ⎜ ⎜ ⎜ αq S∗ αq I ∗ J (E ∗ ) = ⎜ q q − (μ0 + μ1 + β q ) ⎜ q ∗ ⎜ 1+γ I (1 + γ q I ∗ )2 ⎝ νq

⎞ 0

⎟ ⎟ ⎟ ⎟ . 0 ⎟ ⎟ ⎟ ⎠ q

βq

−μ0

Thus, one of three eigenvalues of the matrix J (E ∗ ) is λ1 = −μ0 < 0 and two remaining eigenvalues are the eigenvalues of the following sub-matrix q

⎞ ⎛ −α q I ∗ αq S∗ q q − (μ + ν ) − 0 ⎟ ⎜1 + γ q I∗ (1 + γ q I ∗ )2 ⎟ ⎜ ∗  J (E ) = ⎜ ⎟. q ∗ q ∗ ⎠ ⎝ α I α S q q q − (μ + μ + β ) 0 1 1 + γq I∗ (1 + γ q I ∗ )2 The characteristic polynomial of J is P2 (x) = x 2 − trace( J)x + det( J), where −α q I ∗ αq S∗ q q q − (μ0 + ν q ) + − (μ0 + μ1 + β q ), 1 + γq I∗ (1 + γ q I ∗ )2    αq I ∗ −α q I ∗ αq S∗ αq S∗ q q q q) q) + det( J) = − (μ + ν − (μ + μ + β 0 0 1 1 + γq I∗ (1 + γ q I ∗ )2 (1 + γ q I ∗ )2 1 + γ q I ∗

trace( J) =

We prove that trace( J) < 0 and det( J) > 0 if R0 > 1. To do this we need only prove that αq S∗ q q < (μ0 + μ1 + β q ). R0 > 1 =⇒ (1 + γ q I ∗ )2 Indeed, note that if R0 > 1, then I ∗ > 0. From the second equation of the model (2), we have α q S ∗ = (μ0 + μ1 + β q )(1 + γ q I ∗ ) < (μ0 + μ1 + β q )(1 + γ q I ∗ )2 , q

q

q

q

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which is equivalent to αq S∗ q q < (μ0 + μ1 + β q ). (1 + γ q I ∗ )2 Hence, we conclude that if R0 > 1, then trace( J) < 0 and det( J) > 0. Form Theorems 1 and 2, we have the following result. Proposition 3. The DEE point E ∗ of the model (2) is locally asymptotically stable if R0 > 1. From Propositions 2 and 3, we obtain the following theorem. Theorem 6 (Local asymptotic stability). (i) If R0 < 1, then the DEF point E 0 of the model (2) is locally asymptotically stable. (ii) If R0 > 1, The DEE point E ∗ of the model (2) is locally asymptotically stable.

3.4 Uniform Asymptotic Stability We now study the uniform asymptotic stability of the model (2). Theorem 7. If R0 < 1, then the DFE point E 0 of the fractional-order model (2) is uniformly asymptotically stable. Proof. Since the first and second equations of the model (2) only consist of the components S and I , it is sufficient to consider the following sub-system α q S(t)I (t) q − (μ0 + ν q )S(t), 1 + γ q I (t) α q S(t)I (t) q q C q − (μ0 + μ1 + β q )I (t). D I (t) = 0 t 1 + γ q I (t)

C q 0 Dt S(t)

= q −

(13)

By Corollary 1, the set 

 := (S, I ) 0 ≤ S ≤ S0 ,

I ≥0



can be considered as a positively invariant set of the model (13).  Model (13) always has a boundary equilibrium point Z 0 =

(14)

 q ,0 = q μ0 + ν q (S0 , 0) for all values of the parameters. To investigate the uniform asymptotic stability  → R+ defined by of Z 0 , we use a Lyapunov function V : V (S, I ) =

1 a(S − S0 )2 + bI, 2

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where a and b are positive constants that will be selected later. Clearly, the function V is continuous. Moreover, V (Z 0 ) = 0 and V (Z ) > 0 for all Z = Z 0 . The derivative of the function V along with solutions of the model (13) satisfies C q 0 Dt V

q

q

C ≤ a(C 0 Dt S)(S − S0 ) + b(0 Dt I )   q   q α SI α SI q q q = a q − − (μ0 + ν q )S (S − S0 ) + b − (μ0 + μ1 + β q )I . q q 1+γ I 1+γ I

Note that

q

q

q − (μ0 + ν q )S = −(μ0 + ν q )(S − S0 ), αq S I . ≤ α q S I ≤ α q S0 I, ∀(S, I ) ∈ 1 + γqI

By these estimates, we get   q 2 q αq S q q C D q V ≤ −a(μq + ν q )(S − S )2 − a α S I + a α S0 S I + b q) I − (μ + μ + β 0 0 t 0 0 1 1 + γqI 1 + γqI 1 + γqI 

q

≤ −a(μ0 + ν q )(S − S0 )2 + aα q S02 I + b

α q q



q q − (μ0 + μ1 + β q ) I. q μ0 + ν q

It is important to note that if R0 < 1, then α q q q q − (μ0 + μ1 + β q ) < 0. q μ0 + ν q Therefore, there always exist positive constants a and b satisfying  aα q S02 + b

 α q q q q q − (μ + μ + β ) < 0. q 0 1 μ0 + ν q

Consequently, the function V satisfies the conditions of Theorem 3. In other words, the equilibrium point Z 0 is uniformly asymptotically stable of the model (13). The proof is complete.  Because it is not easy to study the uniform asymptotic stability of the DEE point E ∗ theoretically, we will investigate this problem by numerical simulations. As will be seen in Sect. 5, numerical simulations suggest that the DEE point E ∗ is not only locally asymptotically stable but also uniformly asymptotically stable when R0 > 1. Conjecture 1. If R0 > 1, then the DEE point E ∗ of the fractional-order model (2) is not only locally asymptotically stable but also globally asymptotically stable.

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4 Positivity-Preserving NSFD Schemes for Model (2) Our main objective in this section is to design NSFD schemes preserving the positivity of the model (2). Let [0, T ] be a finite interval and N be a positive integer. We define h = t = T /N as the time step size of the discretization t0 = 0 < t1 < t2 < . . . < t N −1 < t N = T = N h, and let tn = nh with n = 0, 1, 2 . . . N . Let Sn , In and Rn be approximations of  T S(tn ), I (tn ) and R(tn ), respectively, where S(t), I (t), R(t) is the exact solution to the model (2). We use the following non-local approximations for the right-hand side function of the model (2) α q Sn+1 In α q S(t)I (t) α q Sn In q q q − (μ0 + ν q )S(t) → q − τ1 q − − τ2 − τ3 (μ0 + ν q )Sn+1 − τ4 (μ0 + ν q )Sn , 1 + γ q I (t) 1 + γ q In 1 + γ q In α q Sn+1 In α q S(t)I (t) α q Sn In q q q q q q − (μ0 + μ1 + β q )I (t) → τ5 + τ6 − τ7 (μ0 + μ1 + β q )In+1 − τ8 (μ0 + μ1 + β q )In , 1 + γ q I (t) 1 + γ q In 1 + γ q In q q q β q I (t) + ν q S(t) − μ0 R(t) → τ9 β q In+1 + τ10 β q In + τ11 ν q Sn+1 + τ12 ν q Sn − τ13 μ0 Rn+1 − τ14 μ0 Rn ,

where

(15)

  τi + τi+1 = 1, i ∈ 1, 3, 5, 7, 9, 11, 13 .

Using the Grunwald–Letnikov discretization and (15), we obtain the following NSFD scheme for the model (2) Sn+1 −

n+1 q q q q q i=1 ci Sn+1−i − rn+1 S0 = q − τ α Sn+1 In − τ α Sn In − τ (μq + ν q )S q 1 1 + γq I 2 1 + γq I 3 0 n+1 − τ4 (μ0 + ν )Sn , [ϕ(h)]q n n

n+1 q q q q q q i=1 ci In+1−i − rn+1 I0 = τ α Sn+1 In + τ α Sn In − τ (μq + μq + β q )I q 6 1 + γq I 7 0 n+1 − τ8 (μ0 + μ1 + β )In , 5 1 + γq I 1 [ϕ(h)]q n n n+1 q q Rn+1 − i=1 ci Rn+1−i − rn+1 R0 q q = τ9 β q In+1 + τ10 β q In + τ11 ν q Sn+1 + τ12 ν q Sn − τ13 μ0 Rn+1 − τ14 μ0 Rn , [ϕ(h)]q

In+1 −

(16) where, the denominator function ϕ(h) satisfies ϕ(h) = h + O(h 2 ) as h → 0 (see [43–46]).

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We now investigate the positivity of the NSFD scheme (16). For this purpose, we introduce the following condition τ1 ≥ 0, τ3 ≥ 0, τ5 ≥ 0,

τ2 ≤ 0, τ4 ≤ 0, τ6 ≥ 0,

τ1 + τ2 = 1, τ3 + τ4 = 1, τ5 + τ6 = 1,

τ7 ≥ 0, τ9 ≥ 0,

τ8 ≤ 0, τ10 ≥ 0,

τ7 + τ8 = 1, τ9 + τ10 = 1,

τ11 ≥ 0, τ13 ≥ 0,

τ12 ≥ 0, τ14 ≤ 0,

(17)

τ11 + τ12 = 1, τ13 + τ14 = 1.

 T Theorem 8. Let S0, I0 , R0 be any initial data for the initial problem (2) with T   be the approximations generated by the S0 , I0 , R0 ≥ 0 and Sn , In , Rn n≥1

NSFD scheme (16). If the condition (17) is satisfied, then Sn , In , Rn ≥ 0 for all n ≥ 1. In other words, the NSFD scheme (16) preserves the positivity of the model (2) under the condition (17). Proof. The theorem is proved by the mathematical induction. Indeed, we rewrite the scheme (16) in the form   α q Sn In q q q ci Sn+1−i + rn+1 S0 + ϕ q q − τ2 − τ4 (μ0 + ν q )Sn 1 + γ q In Sn+1 = ,   α q In q q) 1 + ϕ q τ1 + τ (μ + ν 3 0 1 + γ q In  αq S  n+1 q α q Sn In n+1 In q q q q + τ6 − τ8 (μ0 + μ1 + β q )In i=1 ci In+1−i + rn+1 I0 + ϕ τ5 1 + γ q In 1 + γ q In , In+1 = q q 1 + ϕ q τ7 (μ0 + μ1 + β q )   n+1 q q q q q q q i=1 ci Rn+1−i + rn+1 R0 + τ9 β In+1 + τ10 β In + τ11 ν Sn+1 + τ12 ν Sn − τ14 μ0 Rn . Rn+1 = q 1 + ϕ q τ13 μ0 n+1 i=1

(18)

From (18), we see that if the condition (17) holds and S0 , I0 , R0 ≥ 0, then  Sn , In , Rn ≥ 0 for all n ≥ 1. The proof is complete.

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Obviously, there are many ways to choose the parameters τi (i = 1, 2, . . . , 14) satisfying the condition (17). In particular, Theorem 8 leads to the following positive NSFD scheme n+1 q q q q i=1 ci Sn+1−i + rn+1 S0 + ϕ   αq I  , n q q) 1 + ϕq + (μ + ν 0 1 + γ q In   q q n+1 q q q 1 α Sn+1 In + 1 α Sn In c I + r I + ϕ 0 n+1−i n+1 i=1 i 2 1 + γ q In 2 1 + γ q In In+1 = , q q 1 + ϕ q (μ0 + μ1 + β q )  n+1 q 1 q 1 q 1 q  q q 1 q i=1 ci Rn+1−i + rn+1 R0 + ϕ 2 β In+1 + 2 β In + 2 ν Sn+1 + 2 ν Sn . Rn+1 = q 1 + ϕ q μ0

Sn+1 =

(19) The NSFD scheme (19) will be used to solve the model (2) in numerical simulations.

5 Numerical Simulations In this section, we report some numerical simulations to validate the theoretical results and to show advantages of the NSFD schemes (18) over the standard Grunwald-Letnikov scheme. For this purpose, we will consider three examples with the parameters given in Table 1. Also, in Examples 2 and 3, we consider the model (2) with the following initial conditions  (i) S(0), I (0), R(0) = (20, 10, 5);  (ii) S(0), I (0), R(0) = (30, 20, 15);  (iii) S(0), I (0), R(0) = (25, 15, 10);  (iv) S(0), I (0), R(0) = (15, 10, 5);  (v) S(0), I (0), R(0) = (15, 10, 10);

Table 1 The parameters in numerical simulations Example 

α

γ

μ0

β

q

0.7

0.75

0.6

0.2

μ1 0.25

ν

1

0.3

0.4

0.85

R0 0.9222

2

0.5

0.55

0.6

0.1

0.2

0.25

0.45

0.75

0.6963

3

0.85

0.75

0.5

0.01

0.025

0.35

0.15

0.7

3.6930

Equilibria E 0 = (1.2027, 0, 1.6977) E 0 = (1.1190, 0, 2.2247) E ∗ = (0.8175, 1.2298, 18.0349)

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143

60

S-component 40

20

S

0

-20

-40

-60

-80

-100

0

10

20

30

40

50

60

70

80

90

100

t Fig. 1 The S-component generated by the standard scheme with h = 0.8

Example 1 (Comparision of the NSFD scheme and the SFD scheme). Consider model (2) with the parameters given in Example 1 in Table 1. In this case, R0 < 1 and hence, the equilibrium point E 0 is uniformly asymptotically stable. We use the standard Grunwal-Letnikov scheme (4) and NSFD scheme (19) to simulate the model. The obtained numerical solutions are depicted in Figs. 1, 2, 3, 4, 5, 6, 7, 8 and 9. From these figures, we see that the standard scheme does not preserve the positivity and the stability of the model (2). Actually, the standard scheme provides the numerical solutions which oscillate rapidly around the equilibrium position and are completely different from the exact solution. Conversely, the NSFD scheme generates the numerical solutions which reflect exactly the dynamics of the continuous model. The NSFD scheme not only preserves the positivity and stability of the model (2) but also operates well for all finite step sizes. Hence, the NSFD scheme is very effective when studying the continuous model in a very long time.

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I-component 80 60 40

I

20 0 -20 -40 -60 -80

0

10

20

30

40

50

60

70

80

90

100

t Fig. 2 The I-component generated by the standard scheme with h = 0.8

Example 2 (Dynamics of NSFD schemes when R0 ≤ 1). Consider model (2) with the parameters given by Example 2 in Table 1. In this case, R0 < 1 and hence, the equilibrium point E 0 is uniformly asymptotically stable. To confirm this result, we use the NSFD scheme with a small step size, namely, h = 0.05 and ϕ(h) = h. The obtained numerical simulation is shown in Fig. 10. Clearly, E 0 is uniformly asymptotically stable. Consequently, the validity of Theorem 7 is confirmed.

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14

R-component 12 10 8

R

6 4 2 0 -2 -4 -6

0

10

20

30

40

50

60

70

80

90

100

t Fig. 3 The R-component generated by the standard scheme with h = 0.8

Example 3 (Dynamics of NSFD schemes when R0 > 1). Consider model (2) with the parameters given by Example 3 in Table 1. In this case, R0 > 1 and hence, the equilibrium point E ∗ is asymptotically stable. Similarly to Example 2, we use NSFD scheme (12) with a small step size to confirm the local asymptotic stability and to investigate the uniform asymptotic stability of E ∗ . We select h = 0.05, ϕ(h) = h and t ∈ [0, 500]. The obtained numerical solutions are sketched in Fig. 11. Clearly, E ∗ is not only locally asymptotically stable but also uniformly asymptotically stable. This confirms the validity of the Theorem 6 and supports Conjecture 1.

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S-component

80 60 40

S

20 0 -20 -40 -60 -80

0

10

20

30

40

50

60

70

80

90

100

t Fig. 4 The S-component generated by the standard scheme with h = 1 150

I-component

100

I

50

0

-50

-100

-150

0

10

20

30

40

50

60

70

t Fig. 5 The I-component generated by the standard scheme with h = 1

80

90

100

Dynamics of a Fractional-Order Hepatitis B Epidemic Model ...

147

20

R-component

15

10

R

5

0

-5

-10

-15

-20 0

10

20

30

40

50

60

70

80

90

100

t Fig. 6 The R-component generated by the standard scheme with h = 1 10

S-component

9 8 7

S

6 5 4 3 2 1 0 0

10

20

30

40

50

60

70

t

Fig. 7 The S-component generated by the NSFD scheme with h = 2

80

90

100

148

M. T. Hoang and O. F. Egbelowo 6

I-component

5

I

4

3

2

1

0 0

10

20

30

40

50

60

70

80

90

100

t Fig. 8 The I-component generated by the NSFD scheme with h = 2 10

R-component 9 8 7

R

6 5 4 3 2 1

0

10

20

30

40

50

60

70

t Fig. 9 The R-component generated by the NSFD scheme with h = 2

80

90

100

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30 25

R

20 15 10 5 0 20

(i) (ii) (iii) (iv) (v) 30

15

25 10

I

20

E0

5

15 10 5

0

S

0

Fig. 10 The solutions obtained by the NSFD scheme with ϕ(h) = h, h = 0.05 and t ∈ [0, 500] in Example 2

(i) (ii) (iii) (iv) (v)

40 35 30

R

25 20 15 10 5 25

30 20 20

10

15

10

I

5

0

S

0

Fig. 11 The solutions of the model (2) generated by NSFD scheme with ϕ(h) = h, h = 0.05 and t ∈ [0, 500] in Example 3. The red circle indicates the equilibrium point E ∗

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6 Conclusions In this chapter, we have proposed and analyzed a fractional-order hepatitis B epidemic model to predict the dynamics of the HVB transmission. The dynamical properties of the proposed model including the positivity, the boundedness and the stability are studied. By a suitable Lyapunov function, we have proved that the DFE point is uniformly asymptotically stable if the reproduction number is less than one. On the other hand, when the reproduction number is greater than one the DFE point is locally asymptotically stable. It should be emphasized that the numerical simulations indicate that the DFE point is not only locally asymptotically stable but uniformly asymptotically stable when the reproduction number is greater than one. Besides, NSFD schemes are designed to solve the proposed fractional-order model. The result is that we obtain NSFD schemes which are dynamically consistent with the continuous model. Meanwhile, the standard Grunwald-Letnikov scheme produces numerical solutions that are dynamically inconsistency and could not replicate the dynamics of the continuous model. The advantages and superiority of the proposed NSFD schemes over the standard Grunwald-Letnikov one are confirmed by a set of numerical simulations. In the near future, we will extend the constructed results in the work to other models described by ODEs and FDEs. Especially, the study of effective strategies to eliminate the HVB transmission in the community will be considered. Conflict of Interest Authors have no conflict of interest. Acknowledgements The authors thank the editor and the anonymous referees for useful comments that led to a great improvement of the paper.

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On SICA Models for HIV Transmission Cristiana J. Silva and Delfim F. M. Torres

Abstract We revisit the SICA (Susceptible–Infectious–Chronic–AIDS) mathematical model for transmission dynamics of the human immunodeficiency virus (HIV) with varying population size in a homogeneously mixing population. We consider SICA models given by systems of ordinary differential equations and some generalizations given by systems with fractional and stochastic differential operators. Local and global stability results are proved for deterministic, fractional, and stochastictype SICA models. Two case studies, in Cape Verde and Morocco, are investigated. Keywords HIV/AIDS · SICA compartmental models · Deterministic models · Fractional models · Stochastic models · Stability analysis · Lyapunov functions · Cape Verde and Morocco case studies

1 Introduction In this work, we make an overview on SICA compartmental models for HIV/AIDS transmission dynamics with varying population size in a homogeneously mixing population, given by a system of four equations. The SICA model divides the total human population into four mutually-exclusive compartments: susceptible individuals (S); HIV-infected individuals with no clinical symptoms of AIDS (the virus is living or developing in the individuals but without producing symptoms or only mild ones) but able to transmit HIV to other individuals (I ); HIV-infected individuals under antiretroviral therapy (ART), the so called chronic stage with a viral load remaining low (C); and HIV-infected individuals with AIDS clinical symptoms (A). The total population at time t, denoted by N (t), is given by N (t) = S(t) + I (t) + C(t) + A(t). C. J. Silva (B) · D. F. M. Torres Department of Mathematics, Center for Research & Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193 Aveiro, Portugal e-mail: [email protected] D. F. M. Torres e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_6

155

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The deterministic SICA model was firstly proposed as a sub-model of a TB– HIV/AIDS co-infection model and published in 2015, see [55]. After, it was generalized to fractional [58] and stochastic systems of differential equations [20] and calibrated to the HIV/AIDS epidemic situation in Cape Verde [56, 57] and Morocco [40]. One of the main goals of SICA models is to show that a simple mathematical model can help to clarify some of the essential relations between epidemiological factors and the overall pattern of the AIDS epidemic [46, 56]. The assumptions of the SICA models are now described. The susceptible population is increased by the recruitment of individuals into the population, assumed susceptible, at a rate . All individuals suffer from natural death, at a constant rate μ. Susceptible individuals S acquire HIV infection, following effective contact with people infected with HIV, at a rate λ, given by λ(t) =

β (I (t) + ηC C(t) + η A A(t)) , N (t)

(1)

where β is the effective contact rate for HIV transmission. The modification parameter η A ≥ 1 accounts for the relative infectiousness of individuals with AIDS symptoms, in comparison to those infected with HIV with no AIDS symptoms. Individuals with AIDS symptoms are more infectious than HIV-infected individuals (pre-AIDS) because they have a higher viral load and there is a positive correlation between viral load and infectiousness [63]. On the other hand, ηC ≤ 1 translates the partial restoration of immune function of individuals with HIV infection that use correctly ART [16]. HIV-infected individuals, with and without AIDS symptoms, have access to ART treatment. HIV-infected individuals with no AIDS symptoms, I , progress to the class of individuals with HIV infection under ART treatment C at a rate φ, and HIV-infected individuals with AIDS symptoms are treated for HIV at rate γ . An HIV-infected individual with AIDS symptoms, A, that starts treatment, moves to the class of HIV-infected individuals, I , and will move to the chronic class, C, only if the treatment is maintained. HIV-infected individuals with no AIDS symptoms, I , that do not take ART treatment, progress to the AIDS class, A, at rate ρ. Only HIVinfected individuals with AIDS symptoms A suffer from an AIDS induced death, at a rate d. These assumptions are translated into the following mathematical model, given by a system of four ordinary differential equations: ⎧ ˙ =  − βλ(t)S(t) − μS(t), S(t) ⎪ ⎪ ⎪ ⎨ I˙(t) = βλ(t)S(t) − (ρ + φ + μ)I (t) + γ A(t) + ωC(t), ˙ ⎪ C(t) = φ I (t) − (ω + μ)C(t), ⎪ ⎪ ⎩˙ A(t) = ρ I (t) − (γ + μ + d)A(t). The region

  4

= (S, I, C, A) ∈ R+ : N ≤ /μ

(2)

(3)

On SICA Models for HIV Transmission

157

is positively invariant and attracting [55]. Thus, the dynamics of the HIV model evolves in . Model (2) has a disease-free equilibrium, given by 

0 = S , I , C , A 0

0

0

0



=

  , 0, 0, 0 . μ

(4)

Following the approach from [60], the basic reproduction number R0 for model (2), which represents the expected average number of new HIV infections produced by a single HIV-infected individual when in contact with a completely susceptible population, is given by R0 =

N β (ξ2 (ξ1 + ρ η A ) + ηC φ ξ1 ) = , μ (ξ2 (ρ + ξ1 ) + φ ξ1 + ρ d) + ρ ω d D

(5)

where along all manuscript ξ1 = γ + μ + d, ξ2 = ω + μ, and ξ3 = ρ + φ + μ, see [57]. To find conditions for the existence of an equilibrium for which HIV is endemic in the population (i.e., at least one of I ∗ , C ∗ or A∗ is nonzero), denoted by + = (S ∗ , I ∗ , C ∗ , A∗ ), the equations in (2) are solved in terms of the force of infection at the steady-state λ∗ , given by λ∗ =

β (I ∗ + ηC C ∗ + η A A∗ ) . N∗

(6)

Setting the right hand side of the equations of the model to zero, and noting that λ = λ∗ at equilibrium gives S∗ =

λ∗

 , +μ

I∗ = −

λ∗ ξ1 ξ2 φλ∗ ξ1 , C∗ = − , D D

A∗ = −

ρ1 λ∗ ξ2 D

(7)

with D = − (λ∗ + μ) (μ (ξ2 (ρ + ξ1 ) + ξ1 φ + ρd) + ρωd), we use (7) in the expression for λ∗ in (6) to show that the nonzero (endemic) equilibrium of the model satisfies λ∗ = −μ(1 − R0 ). The force of infection at the steady-state λ∗ is positive only if R0 > 1. Thus, the existence and uniqueness of the endemic equilibrium follows. Remark 1.1. The explicit expression of the endemic equilibrium + of model (2) is given by (ρdξ2 − D) ξ1 ξ2 (D − N ) , I∗ = , S∗ = μ(ρdξ2 − N ) D(ρdξ2 − N ) (8) φξ1 (D − N ) ρξ2 (D − N ) ∗ ∗ C = , A = . D(ρdξ2 − N ) D(ρdξ2 − N )

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Adding the equations of model (2), with d = 0, gives N˙ =  − μN , so that as t → ∞. Thus,  is an upper bound of N (t) provided that N (0) ≤  . N→ μ μ μ   Further, if N (0) > μ , then N (t) decreases to this level. Using N = μ in the force of

infection λ = Nβ (I + ηC C + η A A) gives a limiting (mass action) system (see, e.g., [1]). Then, the force of infection becomes λ = β1 (I + ηC C + η A A) , where β1 =

βμ . 

Therefore, we have the following model: ⎧ ˙ =  − β1 (I (t) + ηC C(t) + η A A(t)) S(t) − μS(t), S(t) ⎪ ⎪ ⎪ ⎨ I˙(t) = β (I (t) + η C(t) + η A(t)) S(t) − ξ I (t) + γ A(t) + ωC(t), 1 C A 3 ˙ ⎪ C(t), C(t) = φ I (t) − ξ 2 ⎪ ⎪ ⎩˙ A(t) = ρ I (t) − ξ1 A(t),

(9)

where ξ1 = γ + μ. Here, different mathematical models based on (2) and (9) are considered. The paper is organized as follows. In Sect. 2, a general fractional SICA model is proposed and the uniform stability of the equilibrium points is given. In Sect. 3, a stochastic environmental noise is introduced into the SICA model (9). Existence and uniqueness of a positive global solution is proved and conditions for the extinction and persistence in mean of the disease are provided. In Sect. 4, the deterministic model is analyzed, proving the uniform persistence of the total population and local and global stability of the equilibrium points, through Lyapunov’s direct method and LaSalle’s invariance principle. Lyapunov functions are provided. Then, with Sect. 5, two case studies are analyzed, being shown that models (2) and (9), after an adequate calibration of the parameters, describe well the HIV/AIDS situation in Cape Verde and Morocco from 1987 to 2014 and 1986 to 2015, respectively. The paper ends with Sect. 6 of conclusion.

2 Fractional SICA Model Fractional differential equations (FDEs), also known in the literature as extraordinary differential equations, are a generalization of differential equations through the

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application of fractional calculus, that is, the branch of mathematical analysis that studies different possibilities of defining differentiation operators of noninteger order [3, 22, 58]. FDEs are naturally related to systems with memory, which explains their usefulness in most biological systems [49]. Indeed, FDEs have been considered in many epidemiological models [58]. In this section, we analyze the general fractional SICA model and its uniform stability, proved in [58]. We first recall some important definitions and results that are used in the proofs of the uniform asymptotic stability of the equilibrium points.

2.1 Preliminaries: Fractional Calculus and Uniform Asymptotic Stability We begin by recalling the definition of Caputo fractional derivative. Definition 2.1 (See [8]). Let a > 0, t > a, and α, a, t ∈ R. The Caputo fractional derivative of order α of a function f ∈ C n is given by C α a Dt

f (t) =

1 (n − α)



t

a

f (n) (ξ ) dξ, (t − ξ )α+1−n

n − 1 < α < n ∈ N.

Let us consider the following general fractional differential equation involving the Caputo derivative: C α a Dt x(t)

= f (t, x(t)),

α ∈ (0, 1),

(10)

subject to a given initial condition x0 = x(t0 ). Definition 2.2 (See, e.g., [37]). The constant x ∗ is an equilibrium point of the Caputo fractional dynamic system (10) if, and only if, f (t, x ∗ ) = 0. We recall an extension of the celebrated Lyapunov direct method for Caputo type fractional order nonlinear systems [18]. Theorem 2.3 (Uniform Asymptotic Stability[18]). Let x ∗ be an equilibrium point for the nonautonomous fractional order system (10) and ⊂ Rn be a domain containing x ∗ . Let L : [0, ∞) × → R be a continuously differentiable function such that W1 (x) ≤ L(t, x(t)) ≤ W2 (x) and

C α a Dt L(t, x(t))

≤ −W3 (x)

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for all α ∈ (0, 1) and all x ∈ , where W1 (·), W2 (·) and W3 (·) are continuous positive definite functions on . Then the equilibrium point x ∗ of system (10) is uniformly asymptotically stable. A useful lemma is proved in [61], where a Volterra-type Lyapunov function is obtained for fractional-order epidemic systems. Lemma 2.4 (See [61]). Let x(·) be a continuous and differentiable function with x(t) ∈ R+ . Then, for any time instant t ≥ t0 , one has C α t0 D t

x(t) − x ∗ − x ∗ ln



 x(t) x∗ ≤ 1 − x∗ x(t)

C α t0 D t x(t),

x ∗ ∈ R+ ,

∀α ∈ (0, 1).

2.2 Fractional SICA Model: Local and Uniform Stability Analysis Let us consider the Caputo fractional order SICA epidemiological model for HIV/AIDS transmission with constant recruitment rate, mass action incidence, and variable population size, firstly proposed in [58]: ⎧C α ⎪ ⎪t0 Dt S(t) =  − β (I (t) + ηC C(t) + η A A(t)) S(t) − μS(t), ⎪ ⎨C D α I (t) = β (I (t) + η C(t) + η A(t)) S(t) − ξ I (t) + ωC(t) + γ A(t), C A 3 t0 t ⎪Ct0 Dtα C(t) = φ I (t) − ξ2 C(t), ⎪ ⎪ ⎩C α t0 Dt A(t) = ρ I (t) − ξ1 A(t). (11) The local asymptotic stability of the disease free equilibrium 0 (4), comes straightforward from [45] and [2]. Stronger stability results are stated next. Theorem 2.5 (See [58]). Let α ∈ (0, 1). The disease free equilibrium 0 (4), of the fractional order system (11), is uniformly asymptotically stable in (3), whenever (5) satisfies R0 < 1. The uniform asymptotic stability of the disease free equilibrium 0 (4) and endemic equilibrium + (8) of the fractional order system (11) are based on Theorem 2.3 and Lemma 2.4. Theorem 2.6 (See [58]). Let α ∈ (0, 1) and (5) be such that R0 > 1. Then the unique endemic equilibrium + (8) of the fractional order system (11) is uniformly asymptotically stable in the interior of (3). For the numerical implementation of the fractional derivatives, the implementation of the Adams–Bashforth–Moulton scheme is carried out in [58], which is based in the Matlab code fde12 by Garrappa [21]. This code implements a predictor-corrector PECE method of Adams–Bashforth–Moulton type, as described in [19].

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3 Stochastic SICA Model Here, we begin with the deterministic SICA epidemic model for HIV transmission (9). Then, following [23–25, 43, 59, 67], a stochastic environmental noise is introduced making the model biologically more realistic. Precisely, we consider the model ⎧ d S(t) = [ − β (I (t) + ηC C(t) + η A A(t)) S(t) − μS(t)] dt, ⎪ ⎪ ⎪ ⎨d I (t) = β (I (t) + η C(t) + η A(t)) S(t) − ξ I (t) + γ A(t) + ωC(t) dt, C A 3 ⎪ dC(t) = [φ I (t) − ξ2 C(t)] dt, ⎪ ⎪ ⎩ d A(t) = [ρ I (t) − ξ1 A(t)] dt. Next, the fluctuations in the environment are assumed to manifest themselves in ˙ the transmission coefficient rate β, so that β → β + σ B(t), where B(t) is a standard Brownian motion with intensity σ 2 > 0. The stochastic model takes then the following form (see [20]): ⎧ d S(t) = [ − β (I (t) + ηC C(t) + η A A(t)) S(t) − μS(t)] dt ⎪ ⎪ ⎪ ⎪ ⎪ − σ (I (t) + ηC C(t) + η A A(t)) S(t)d B(t), ⎪ ⎪ ⎪ ⎨d I (t) = β (I (t) + η C(t) + η A(t)) S(t) − ξ I (t) + γ A(t) + ωC(t) dt C A 3 ⎪ C(t) + η A(t)) S(t)d B(t), + σ (t) + η (I C A ⎪ ⎪ ⎪ ⎪ ⎪dC(t) = [φ I (t) − ξ2 C(t)] dt, ⎪ ⎪ ⎩ d A(t) = [ρ I (t) − ξ1 A(t)] dt. (12) Let ( , F, {F}t≥0 , P) be a complete probability space with filtration {F}t≥0 , which is right continuous and such that F contains all P-null sets. The scalar Brownian motion B(t) of (12) is defined on the   given probability space. Moreover, denote R4+ = (x1 , x2 , x3 , x4 )|xi > 0, i = 1, 4 . The existence and uniqueness of a positive global solution of model (12) is easily proved using similar arguments as the ones used in [35]. Theorem 3.1 (See [20]). For any t ≥ 0 and any initial value (S(0), I (0), C(0), A(0)) ∈ R4+ , there is a unique solution (S(t), I (t), C(t), A(t)) to the SDE (12) and the solution remains in R4+ with probability one. Moreover, N (t) →

 as t → ∞, μ

where N (t) = S(t) + I (t) + C(t) + A(t). The next theorem provides a condition for the extinction of the disease. Theorem 3.2 (See [20]). Let Y (t) = (S(t), I (t), C(t), A(t)) be the solution of system (12) with positive initial value. Assume that σ 2 > 2ξβ3 . Then,

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I (t), C(t), A(t) → 0 a.s. and S(t) →

 a.s., μ

as t → +∞. Let us now recall the notion of persistence in mean. 1 t→∞ t



Definition 3.3. System (12) is said to be persistent in mean if lim ds > 0 a.s.

t

I (s)

0

In what follows, we use the notation 1 [I (t)] := t



t

I (s)ds.

0

Theorem 3.4. Let β K1 = μ



γρ ωφ − ξ3 + ξ1 ξ2

 +

μ(ξ1 ξ2 − μ) . (1 + ηC + η A )

(13)

For any initial value (S(0), I (0), C(0), A(0)) ∈ R4+ such that S(t) + I (t) + C(t) + A(t) = N (t) →  if K 1 = 0, K11 β − ξ1 ξ2 − μ C(t), A(t)) satisfies

σ 2 2 2μ2



lim inf [I (t)] ≥ t→∞

 as t → ∞, μ

> 0 and ξ1 , ξ2 > 1, then the solution (S(t), I (t),

1 K1



β σ 2 2 − ξ1 ξ2 − μ 2μ2

 .

Proofs of Theorems 3.2 and 3.4 follow the large number theorem for martingales [24] and L’Hôpital’s rule, after applying Itô’s formula in an appropriate way. For numerical simulations that illustrate Theorems 3.2 and 3.4, see [20].

4 Deterministic SICA Model Let us first consider model (2), that we recall here: ⎧ β ˙ ⎪ ⎪ S(t) =  − βλ(t) N (t) (I (t) + ηC C(t) + η A A(t)) S(t) − μS(t), ⎪ ⎨ I˙(t) = βλ(t) β (I (t) + η C(t) + η A(t)) S(t) − ξ I (t) + γ A(t) + ωC(t), C A 3 N (t) ˙ ⎪C(t) C(t), = φ I (t) − ξ ⎪ 2 ⎪ ⎩˙ A(t) = ρ I (t) − ξ1 A(t).

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Theorem 4.1 (See [55]). The population N (t) is uniformly persistent, that is, lim inf N (t) ≥ ε t→∞

with ε > 0 not depending on the initial data. The local and global stability analysis of the disease free equilibrium 0 given by (4) and endemic equilibrium point + given by (8) is derived in [55]. The local asymptotic stability of the disease-free equilibrium, 0 , follows from Theorem 2 of [60], and holds whenever R0 < 1. Lemma 4.2 (See [55]). The disease free equilibrium 0 is locally asymptotically stable if R0 < 1, and unstable if R0 > 1. The global asymptotic stability of the disease free equilibrium 0 is proved in [55], following [10]. Theorem 4.3 (See [55]). The disease free equilibrium 0 is globally asymptotically stable for R0 < 1. For the endemic equilibrium point, existence and local asymptotic stability holds whenever R0 > 1. Lemma 4.4 (See [55]). The model (2) has a unique endemic equilibrium whenever R0 > 1. The local asymptotic stability of the endemic equilibrium + , can be proved using the center manifold theory [9], as described in [11, Theorem 4.1]. Theorem 4.5 (See [55]). The endemic equilibrium + is locally asymptotically stable for R0  1. Now, assume that the AIDS-induced death rate can be neglected, i.e., d = 0, and consider the deterministic SICA model given by (9). The model (9) has a unique ˜ + = + |d=0 , whenever R˜ 0 = R0 |d=0 > 1. Defining endemic equilibrium given by 

0 = {(S, I, C, A) ∈ : I = C = A = 0} , and considering the Lyapunov function  ω γ   C − C ∗ ln(C) + A − A∗ ln(A) , V = S − S ∗ ln(S) + I − I ∗ ln(I ) + ξ2 ξ1 (14) it follows from LaSalle’s invariance principle [36] that the endemic equilibrium + is globally asymptotically stable. ˜ + of model (9) is globally Theorem 4.6 (See [57]). The endemic equilibrium  asymptotically stable in \ 0 whenever R˜ 0 > 1.

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4.1 General Incidence Function for ηC = η A = 0 In this subsection, we consider a SICA deterministic model with a general incidence function f , and assume that η A = ηC = 0. The assumption η A = ηC = 0 is justified by the following arguments: – although individuals in the chronic stage, with a low viral load and under ART treatment, can still transmit HIV infection, as ART greatly reduces the risk of transmission and individuals that take ART treatment correctly are aware of their health status, one can assume that individuals in the class C do not have risky behaviours for HIV transmission and do not transmit HIV virus, i.e., ηC = 0; – individuals with AIDS clinical symptoms, A, are responsible and do not have any behaviour that can transmit HIV infection or, in other cases, are too sick to have a risky behavior, i.e., η A = 0. Mathematically, the assumption η A = ηC = 0 translates to the fact that the incidence function f depends only on S and I . Accordingly, we consider the SICA deterministic model with a general incidence function f given by (see [40]) ⎧ ˙ =  − μS(t) − f (S(t), I (t)) I (t), S(t) ⎪ ⎪ ⎪ ⎨ I˙(t) = f (S(t), I (t)) I (t) − ξ I (t) + γ A(t) + ωC(t), 3 ˙ ⎪ C(t), C(t) = φ I (t) − ξ 2 ⎪ ⎪ ⎩˙ A(t) = ρ I (t) − ξ1 A(t),

(15)

with initial conditions S(0) = S0 ≥ 0, I (0) = I0 ≥ 0, C(0) = C0 ≥ 0, A(0) = A0 ≥ 0.

(16)

As in [26, 30–32, 42], the incidence function f (S, I ) is assumed to be non-negative and continuously differentiable in the interior of R2+ . Moreover, we assume the following hypotheses ([40]): f (0, I ) = 0, for all I ≥ 0, ∂f (S, I ) > 0, for all S > 0 and I ≥ 0, ∂S ∂f (S, I ) ≤ 0, for all S ≥ 0 and I ≥ 0. ∂I

(H1 ) (H2 ) (H3 )

The reason for adopting hypothesis (H3) is the fact that susceptible individuals take measures to reduce contagion if the epidemics breaks out. This idea has first been explored in [7].

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165

Theorem 4.7 (See [40]). All solutions of (15) starting from non-negative initial conditions (16) exist for all t > 0 and remain bounded and non-negative. Moreover, N (t) ≤ N (0) +

 . μ

The basic reproduction number R0 is given by R0 =

f (/μ, 0)ξ2 ξ1 . D

(17)

Theorem 4.8 (See [40]) (i) If R0 ≤ 1, then system (15) has a unique disease-free equilibrium 0 given by (4). (ii) If R0 > 1, then the disease-free equilibrium 0 given by (4) is still present and system (15) has a unique endemic equilibrium of the form E ∗ = (S ∗ , I ∗ , C ∗ , A∗ ),   , I ∗ > 0, C ∗ > 0, and A∗ > 0. with S ∗ ∈ 0, μ Theorem 4.9 (See [40]). The disease-free equilibrium 0 given by (4) is globally asymptotically stable if R0 ≤ 1. The proof of Theorem 4.9 comes from LaSalle’s invariance principle [36], choosing the Lyapunov functional at 0 as follows: V1 (S, I, C, A) = S − S0 −

S S0

ω f (S0 , 0) γ d X + I + C + A, f (X, 0) ξ2 ξ1

 . μ Assume now that function f also satisfies the following condition:

where S0 =

 

f (S, I ) f (S, I ∗ ) I 1− ≤ 0, for all S, I > 0. − f (S, I ∗ ) f (S, I ) I∗

(H4 )

Theorem 4.10 (See [40]) (i) If R0 > 1, then the disease-free equilibrium E f is unstable. (ii) If R0 > 1 and (H4) holds, then the endemic equilibrium E ∗ is globally asymptotically stable.

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 For any arbitrary equilibrium E = S, I , C, A , the characteristic equation is given by   ∂f ∂f   I −λ − I − f (S, I ) 0 0  −μ −    ∂S ∂I   ∂f ∂f     = 0. I I + f (S, I ) − ξ3 − λ ω γ   ∂ S ∂ I   0 φ −(ξ2 + λ) 0    0 ρ 0 −(ξ1 + λ)  Evaluating the characteristic equation at 0 , we have λ3 + a1 λ2 + a2 λ + a3 = 0, where a1 = ξ1 + ξ2 + ξ3 − f (S0 , 0), a2 = ξ1 ξ3 + ξ3 ξ2 + ξ2 ξ1 − (ξ2 + ξ1 ) f (S0 , 0) − φω − ργ , a3 = (1 − R0 )D. It is clear that a3 < 0 when R0 > 1. Then, the disease-free equilibrium 0 is unstable. The global stability of the endemic equilibrium E ∗ comes from LaSalle’s invariance principle [36], choosing a Lyapunov functional V2 as follows: V2 (S, I, C, A) = S − S ∗ −

 S f (S ∗ , I ∗ ) I ∗ − I ∗ ln d X + I − I ∗ I∗ S ∗ f (X, I )







γ C A ω ∗ − A∗ ln C − C ∗ − C ∗ ln + A − A . + ξ2 C∗ ξ1 A∗

Remark 4.11 (See [40]). The incidence function f can take many forms. Table 1 collects the most popular of such forms that one can find in the existing literature. For f (S, I ) ∂ f (S, I ) = , any form of f (S, I ) given in Table 1, it is easy to verify that ∂β β which is important for examining the robustness of model (15) to β. One way to determine the robustness of model (15) to some specific parameter values, e.g. β, consists to examine the sensitivity of the basic reproduction number R0 with respect to such parameter by the so called sensitivity index. Definition 4.12 (See [12, 53]). The normalized forward sensitivity index of a variable u, that depends differentially on a parameter p, is defined as ϒ pu :=

p ∂u × . ∂p u

(18)

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Table 1 Some special incidence functions f (S, I ), where αi ≥ 0, i = 0, . . . , 3 (see [40]) Incidence functions f (S, I ) References Bilinear

βS

[33, 34, 62]

Saturated

βS βS or 1 + α1 S 1 + α2 I

[38, 67]

Beddington–DeAngelis

βS 1 + α1 S + α2 I

[4, 6, 15]

Crowley–Martin

βS 1 + α1 S + α2 I + α1 α2 S I

[14, 39, 68]

Specific nonlinear

βS 1 + α1 S + α2 I + α3 S I

[27, 28, 31, 41, 47]

Hattaf–Yousfi

βS α0 + α1 S + α2 I + α3 S I

[29, 44]

From (17) and Definition 4.12, we derive the normalized forward sensitivity index of R0 with respect to β, using any form for the incidence function, as the ones found in Table 1, and we get the following proposition. Proposition 4.13 (See [40]). The normalized forward sensitivity index of R0 with respect to β is given by ϒβR0 =

β ∂ f (S0 , 0) × . ∂β f (S0 , 0)

Remark 4.14 (See [40, 56]). The sensitivity index of R0 (17) of the model with respect μφξ3 ργ (μ + d)ξ2 , ϒγR0 = to φ, ρ, γ and ω are given, respectively, by ϒφR0 = − , D Dξ3 ρ(μ + d)ξ2 μωφξ3 and ϒωR0 = ϒρR0 = − . D Dξ2 Remark 4.15 (See [40]). For all incidence functions in Table 1, β is always the most sensitive parameter and has a high impact on R0 . Indeed, ϒβR0 is independent of any parameter of system (15) with ϒβR0 = +1.

5 Two HIV/AIDS Case Studies: Cape Verde and Morocco Two case studies are now given, showing that models (2) and (9) describe well the HIV/AIDS situations in Cape Verde and Morocco, from 1987 to 2014 and 1986 to 2015, respectively.

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5.1 Cape Verde (1987–2014) In 2014, 409 new HIV cases were reported in Cape Verde, accumulating a total of 4,946 cases. Of this total, 1,766 developed AIDS and 1,066 have died. The municipality with more cases was Praia, followed by Santa Catarina (Santiago island) and São Vicente. Cape Verde has developed a Strategic National Plan to fight against AIDS, which includes ART treatment, monitoring of patients, prevention actions, and HIV testing. From the first diagnosis of AIDS in 1986, Cape Verde got significant progress in the fight, prevention and treatment of HIV/AIDS [52, 56]. Model (2) was calibrated to the cumulative cases of infection by HIV and AIDS in Cape Verde from 1987 to 2014. Following [57], we show that model (2) predicts well this reality. In Table 2, the cumulative cases of infection by HIV and AIDS in Cape Verde are depicted for the years 1987–2014 [52]. The values of the initial conditions (19) are based on [52, 64] (see [57]): S0 = S(0) = 323911 ,

I0 = I (0) = 61 , C0 = C(0) = 0 ,

A0 = A(0) = 0. (19) The values of the parameters ρ = 0.1 and γ = 0.33 are taken from [54] and [5], respectively. It is assumed that after one year, the HIV infected individuals I that are under ART treatment have a low viral load [50] and, therefore, are transferred to class C. In agreement, it is taken φ = 1. It is well known that taking ART therapy is a long-term commitment. Following [56], it is assumed that the default treatment rate for C individuals is approximately 11 years (1/ω years, to be precise). The AIDS induced death rate is assumed to be d = 1 based on [69]. Following the World Bank data [64, 65], the natural death rate is assumed to take the value μ = 1/69.54. The recruitment rate  = 13045 was estimated in order to approximate the values of the total population of Cape Verde given in Table 2. See Fig. 1, were it is observable that

Table 2 Cumulative cases of infection by HIV/AIDS and total population in Cape Verde in the period 1987–2014 [52, 65] Year 1987 1988 1989 1990 1991 1992 1993 HIV/AIDS Population Year HIV/AIDS Population Year HIV/AIDS Population Year HIV/AIDS Population

61 323972 1994 358 378763 2001 913 447357 2008 2610 483824

107 328861 1995 395 389156 2002 1064 455396 2009 2929 486673

160 334473 1996 432 399508 2003 1233 462675 2010 3340 490379

211 341256 1997 471 409805 2004 1493 468985 2011 3739 495159

244 349326 1998 560 419884 2005 1716 474224 2012 4090 500870

303 358473 1999 660 429576 2006 2015 478265 2013 4537 507258

337 368423 2000 779 438737 2007 2334 481278 2014 4946 513906

On SICA Models for HIV Transmission 5.2

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×10 5 real data model

5 4.8

Total population

4.6 4.4 4.2 4 3.8 3.6 3.4 3.2

0

5

10

15

20

25

30

time (years)

Fig. 1 Model (2) fitting the total population of Cape Verde between 1987 and 2014 [52, 65]. The l2 norm of the difference between the real total population of Cape Verde and our prediction gives an error of 1.9% of individuals per year with respect to the total population of Cape Verde in 2014 (see [57])

model (2) fits well the total population of Cape Verde. The AIDS induced death rate is assume to be d = 1 based on [69]. Two cases are considered: ηC = 0.04, based on a research study known as HPTN 052, where it is found that the risk of HIV transmission among heterosexual serodiscordant couples is 96% lower when the HIV-positive partner is on treatment [13]; and ηC = 0.015, which means that HIVinfected individuals under ART treatment have a very low probability of transmitting HIV, based on [17]. For the modification parameter η A ≥ 1 that accounts for the relative infectiousness of individuals with AIDS symptoms, in comparison to those infected with HIV with no AIDS symptoms, we assume η A = 1.3 and η A = 1.35, based in [63]. We estimated the value of the HIV transmission rate β for (ηC , η A ) = (0.04, 1.35) equal to 0.695 and for (ηC , η A ) = (0.015, 1.3) equal to 0.752, and show that model (2) predicts well the reality of Cape Verde for these parameter values: see Fig. 2. All the considered parameter values are resumed in Table 3. For the triplets (β, ηC , η A ) = (0.752, 0.015, 1.3) and (β, ηC , η A ) = (0.695, 0.04, 1.35), and the other parameter values from Table 3, we have that the basic reproduction number is given by R0 = 4.0983 and R0 = 4.5304, respectively.

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5.2 Particular Case η A = ηC = 0, Case Study in Morocco (1986–2015) In [56] and [40], it is assumed that η A = ηC = 0, based on the assumptions made in Subsect. 4.1. Based on these two assumptions, susceptible individuals acquire HIV infection by following effective contact with individuals in the class I at a rate λ = β NI . Taking into account the data from the Ministry of Health in Morocco [48], in [40] the value of the HIV transmission rate is estimated to be β = 0.755. Moreover, the following initial conditions are considered based on Moroccan data: S0 = (N0 − (2 + 9))/N0 ,

I0 = 2/N0 ,

C0 = 0,

A0 = 9/N0 ,

(20)

with the initial total population N0 = 23023935 [66]. The values of the parameters ρ = 0.1, φ = 1, ω = 0.09 and d = 1 are the same as the ones used in Subsect. 5.1. Following [51], the natural death and recruitment rates are assumed to take the values μ = 1/74.02 and  = 2.19μ. All the considered parameter values are summarized in Table 3. 5500

5500 real data model

4500 4000 3500 3000 2500 2000 1500

4500 4000 3500 3000 2500 2000 1500

1000

1000

500

500

0

0

5

10

15

20

time (years)

(a)(β,η C ,η A )=(0.752, 0.015, 1.3)

25

real data model

5000

cumulative HIV and AIDS cases

cumulative HIV and AIDS cases

5000

0

0

5

10

15

20

25

time (years)

(b)(β,η C ,η A )=(0.695, 0.04, 1.35)

Fig. 2 Model (2) fitting the data of cumulative cases of HIV and AIDS infection in Cape Verde between 1987 and 2014 [52]. The l2 norm of the difference between the real data and the cumulative cases of infection by HIV/AIDS given by model (2) gives, in both cases, an error of 0.03% of individuals per year with respect to the total population of Cape Verde in 2014 (see [57])

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Table 3 Parameter values of HIV/AIDS models (2) and (15) for Cape Verde [57] and Morocco [40] Symbol Description Cape Verde Morocco References N (0)  μ β ηC ηA φ ρ γ ω

d

Initial population Recruitment rate Natural death rate HIV transmission rate Modification parameter Modification parameter HIV treatment rate for I individuals Default treatment rate for I individuals AIDS treatment rate Default treatment rate for C individuals AIDS induced death rate

323972 13045 1/69.54 0.752

23023935 2.19 μ 1/74.02 0.755

[64, [51, [51, [40,

66] 64] 64] 57]

0.015, 0.04

0

[57]

1.3, 1.35

0

[57]

1

1

[50, 55]

0.1

0.1

[54, 55]

0.33

0.33

[5, 55]

0.09

0.09

[55]

1

1

[69]

In Fig. 3, we observe that model (15) fits the real data reported in [48]. The HIV cases described by model (15) are given by I (t) + C(t) + μ (I (t) + C(t)) for t ∈ [0, 29], which corresponds to the interval of time between the years of 1986 (t = 0) and 2015 (t = 29). βS [38], see Fig. 4. For the saturated incidence function 1 + α1 S βS For the saturated incidence function [67], see Fig. 5. 1 + α2 I βS For the Beddington–DeAngelis incidence function [4, 6, 15], see 1 + α1 S + α2 I Fig. 6. βS [27, 28, For the specific non-linear incidence function 1 + α1 S + α2 I + α3 S I 31, 41, 47], see Fig. 7. In Tables 4 and 5, the basic reproduction number R0 is computed for each incidence function proposed in Table 1 with the parameter values from Table 3, see [40]. In Table 6, we present the sensitivity index of parameters β, φ, ρ, γ and ω, computed for the parameter values given in Table 3.

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3.5 ×10

real data Bilinear incidence

3

HIV cases

2.5 2 1.5 1 0.5 0

0

5

10

15

20

25

30

time (years) Fig. 3 Model (15) fitting the data of HIV cases in Morocco between 1986 (t = 0) and 2015 (t = 29)

3.5

×10 -5 real data Saturated

3

Saturated Saturated Saturated

HIV cases

2.5

1 1 1 1

=0 = 0.01 = 0.1 = 0.2

2 1.5 1 0.5 0 0

5

10

15

20

25

30

time (years) Fig. 4 Saturated incidence function to the year 1986

βS with α1 ∈ {0, 0.01, 0.1, 0.2}. Time t = 0 corresponds 1 + α1 S

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×10 -5 real data Saturated α2 = 0

3

Saturated α2 = 10000 Saturated α2 = 100000 Saturated α2 = 1000000

HIV cases

2.5

2

1.5

1

0.5

0 0

5

10

15

20

25

30

time (years)

Fig. 5 Saturated incidence function to the year 1986 3.5

βS with α2 ∈ {0, 104 , 105 , 106 }. Time t = 0 corresponds 1 + α2 I

×10-5 real data α1 = 0, α2 = 0 α1 = 0.1, α2 = 10 4

3

α1 = 0.01, α2 = 10 5 α1 = 0.1, α2 = 10 6

HIV cases

2.5

2

1.5

1

0.5

0 0

5

10

15

20

25

30

time (years)

βS with α1 ∈ {0, 0.01, 0.1} and 1 + α1 S + α2 I 4 5 6 α2 ∈ {0, 10 , 10 , 10 }. Time t = 0 corresponds to the year 1986, see [40]

Fig. 6 Beddington–DeAngelis incidence function

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3.5

×10

real data α1 = α2 = α3 = 0,

3

α1 = 0.1, α2 = 10 4, α3 = 0.0000001 α1 = 0.1, α2 = 10 5, α3 = 10 α1 = 0.1, α2 = 10 6, α3 = 10 2

HIV cases

2.5

2

1.5

1

0.5

0 0

5

10

15

20

25

30

time (years)

βS with α1 ∈ {0, 0.1}, α2 ∈ 1 + α1 S + α2 I + α3 S I {0, 104 , 105 , 106 } and α3 ∈ {0, 10−6 , 10, 102 }. Time t = 0 corresponds to the year 1986, see [40]

Fig. 7 Specific non-linear incidence function

Table 4 Basic reproduction number for some special incidence functions, see [40] Incidence functions f (S, I ) R0 Bilinear

βS

7.5340

Saturated I

βS 1 + α1 S

0.0031 0.0009 α1 + 0.0004

Saturated II

βS 1 + α2 I

7.5340

Beddington–DeAngelis

βS 1 + α1 S + α2 I

0.0031 0.0009 α1 + 0.0004

Crowley–Martin

βS 1 + α1 S + α2 I + α1 α2 S I

0.0031 0.0009 α1 + 0.0004

Specific non-linear

βS 1 + α1 S + α2 I + α3 S I

0.0031 0.0009 α1 + 0.0004

Hattaf–Yousfi

βS α0 + α1 S + α2 I + α3 S I

0.0031 0.0004 α0 + 0.0009 α1

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Table 5 Basic reproduction number for different values of α1 for the incidence function Saturated I, Beddington–DeAngelis, Crowley–Martin and Specific non-linear, see [40] α1 0.01 0.1 0.2 R0 7.3725 6.1804 5.2392

Table 6 Sensitivity index of R0 for parameter values given in Table 3 for the bilinear incidence function f (S, I ) = β S, see [40]

3.5

Parameter Sensitivity index β φ ρ γ ω

×10 -5 real data fractional α = 0.7 fractional α = 0.8 fractional α = 0.9 deterministic f(S,I)=β S stochastic

3

2.5

HIV cases

+1 −0.5947 −0.3437 +0.0844 +0.5170

2

1.5

1

0.5

0 0

5

10

15

20

25

30

time (years)

Fig. 8 Comparison of integer-order, fractional, deterministic and stochastic models, (11) and (12), with η A = ηC = 0

6 Conclusion We have treated different SICA models for HIV/AIDS transmission: deterministic and stochastic, with integer- and fractional-order derivatives. The superiority or better usefulness of one model over the others depends always on the concrete situation one is studying. For example, let us consider the case study of HIV/AIDS infection in Morocco from 1986 to 2015, and compare the deterministic model (11), both

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for integer-order (α = 1) and fractional-order cases (α ∈ (0, 1)), with the stochastic model (12). In this case, the results are shown in Fig. 8. We see that, in this example, the best modeling is obtained using the deterministic model (11) with integer-order derivatives (α = 1). All our simulations have been done using the numerical computing environment Matlab, release R2016a, on an Apple MacBook Air Core i5 1.3 GHz with 4 Gb RAM. The solutions of the models were found in “real time”. For example, the total computing time needed to generate and plot the 5 solutions in Fig. 8, obtained by solving deterministic, stochastic, integer-order and fractional models, was as small as 3.03 s. Acknowledgements The authors were supported by CIDMA through the Portuguese Foundation for Science and Technology (FCT), within project UIDB/04106/2020. Silva was also supported by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in numbers 4, 5 and 6 of art. 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. They are sincerely grateful to two anonymous reviewers for their useful comments.

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15. DeAngelis, D.L., Goldstein, R.A., O’Neill, R.V.: A model for tropic interaction. Ecology 56(4), 881–892 (1975) 16. Deeks, S.G., Lewin, S.R., Havlir, D.V.: The end of AIDS: HIV infection as a chronic disease. Lancet 382(9903), 1525–1533 (2013) 17. Del Romero, J., et al.: Natural conception in HIV-serodiscordant couples with the infected partner in suppressive antiretroviral therapy: a prospective cohort study. Medicine (Baltimore) 95(30), e4398 (2016) 18. Delavari, H., Baleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67(4), 2433–2439 (2012) 19. Diethelm, K., Freed, A.D.: The FracPECE Subroutine for the Numerical Solution of Differential Equations of Fractional Order. In: Forschung und Wissenschaftliches Rechnen 1998, Gessellschaft fur Wissenschaftliche Datenverarbeitung, Heinzel, S. and Plesser, T., pp. 57–71 (1999) 20. Djordjevic, J., Silva, C.J., Torres, D.F.M.: A stochastic SICA epidemic model for HIV transmission. Appl. Math. Lett. 84, 168–175 (2018) 21. Garrappa, R.: Predictor-corrector PECE method for fractional differential equations, MATLAB Central File Exchange (2011). File ID: 32918 22. George, A.J., Chakrabarti, A.: The Adomian method applied to some extraordinary differential equations. Appl. Math. Lett. 8(3), 91–97 (1995) 23. Grafton, R.Q., Kompas, T., Lindenmayer, D.: Marine reserves with ecological uncertainty. Bull. Math. Biol. 67(5), 957–971 (2005) 24. Gray, A., Greenhalgh, D., Hu, L., Mao, A., Pan, J.: A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 71(3), 876–902 (2011) 25. Greenhalgh, D., Liang, Y., Mao, X.: Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching. Phys. A 462, 684–704 (2016) 26. Hattaf, K., Lashari, A.A., Louartassi, Y., Yousfi, N.: A delayed SIR epidemic model with general incidence rate. Electron. J. Qual. Theory Differ. Equ. 2013(3), 9 (2013) 27. Hattaf, K., Mahrouf, M., Adnani, J., Yousfi, N.: Qualitative analysis of a stochastic epidemic model with specific functional response and temporary immunity. Phys. A 490, 591–600 (2018) 28. Hattaf, K., Yousfi, N.: Global dynamics of a delay reaction-diffusion model for viral infection with specific functional response. Comput. Appl. Math. 34(3), 807–818 (2015) 29. Hattaf, K., Yousfi, N.: A class of delayed viral infection models with general incidence rate and adaptive immune response. Int. J. Dyn. Control 4(3), 254–265 (2016) 30. Hattaf, K., Yousfi, N., Tridane, A.: Mathematical analysis of a virus dynamics model with general incidence rate and cure rate. Nonlinear Anal. Real World Appl. 13(4), 1866–1872 (2012) 31. Hattaf, K., Yousfi, N., Tridane, A.: Stability analysis of a virus dynamics model with general incidence rate and two delays. Appl. Math. Comput. 221, 514–521 (2013) 32. Hattaf, K., Yousfi, N., Tridane, A.: A delay virus dynamics model with general incidence rate. Differ. Equ. Dyn. Syst. 22(2), 181–190 (2014) 33. Ji, C., Jiang, D.: Threshold behaviour of a stochastic SIR model. Appl. Math. Model. 38(21–22), 5067–5079 (2014) 34. Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics, part I. Proc. Roy. Soc. Edinburgh A 115, 700–721 (1927) 35. Lahrouz, A., Omari, L.: Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence. Stat. Probab. Lett. 83(4), 960–968 (2013) 36. LaSalle, J.P.: The Stability of Dynamical Systems. SIAM, Philadelphia (1976) 37. Li, Y., Chen, Y., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Autom. J. IFAC 45(8), 1965–1969 (2009) 38. Liu, X., Yang, L.: Stability analysis of an SEIQV epidemic model with saturated incidence rate. Nonlinear Anal. Real World Appl. 13(6), 2671–2679 (2012) 39. Liu, X.-Q., Zhong, S.-M., Tian, B.-D., Zheng, F.-X.: Asymptotic properties of a stochastic predator-prey model with Crowley-Martin functional response. J. Appl. Math. Comput. 43(1– 2), 479–490 (2013)

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Analytical and Numerical Solutions of a TB-HIV/AIDS Co-infection Model via Fractional Derivatives Without Singular Kernel Mustafa Ali Dokuyucu and Hemen Dutta

Abstract Human beings, who have encountered a lot of diseases and viruses in the last century, conduct a lot of studies to defeat such diseases, particularly HIV, known as the human immunodeficiency virus, and AIDS disease known as advanced immunodeficiency syndrome. The said diseases can be quite dangerous for countries and even for the whole world as the mode of transmission and transmission rate of such diseases increase. The literature needs mathematical modeling and subsequent analysis of such diseases. In the light of this information, TB-HIV/AIDS coinfection model was analyzed. To this end, the model was firstly extended to the CaputoFabirizo fractional derivative obtained using the exponential function. Then, uniqueness of solution was investigated by the help of the fixed-point theorem. Thereafter, the uniqueness of solution of the model was made by assuming certain parameters, and its stability analyzes were examined. Finally, numerical solutions of the mathematical model were made and its numerical simulations were shown. Keywords Co-infection · Treatment · Caputo-Fabrizio fractional derivative · Numerical approximation

1 Introduction In epidemiology, epidemic refers to a disease, which seems to be new cases, but has more than expected effects compared to previous experiences in a certain human population within a certain period. Epidemics have greatly affected cities, countries, continents and sometimes the entire world throughout human history. Epidemics have occurred in almost all ages. The examples include the bubonic plague known M. A. Dokuyucu (B) Department of Mathematics, A˘grı ˙Ibrahim Çeçen University, A˘grı, Turkey e-mail: [email protected] H. Dutta Department of Mathematics, Gauhati University, Guwahati 781014, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_7

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as the black plague that occurred in the middle ages and the Spanish flu that emerged at the end of the First World War. Viruses are the greatest factor in occurrence and spread of epidemics. For most of infectious diseases, the easiest way to catch the disease is direct contact with an infected person or animal. Types of direct contract can be listed as follows: • From person to person: Infectious diseases usually spread by direct transfer of bacteria, viruses or other germs from one person to another. Bacteria or viruses can be transported if a person carrying bacteria or viruses sneezes, coughs, touches or hugs a healthy person. • From animal to human: Being bitten or scratched by an infected animal, including pets, can cause disease. Some infectious diseases, such as rabies or tetanus, can be fatal if left untreated. • To unborn babies: A pregnant woman can pass on germs that cause infectious diseases to her unborn baby. Some microbes can be transmitted through the placenta or breast milk. The new corona virus identified on 13/01/2020 and named COVID-19, which is thought to have emerged from Wuhan city of China, has an impact on the whole world. Although many countries try to take precautions, the virus, which is thought to have been transmitted from animal to human, is also passed on from person to person very simply and quickly. According to two-month studies, its mortality rate is around 3%. Although its transmission rate is quite high, the mortality is lower compared to many other epidemics. Nowadays, many scientists are making great efforts to find an antivirus. Unfortunately, there is no definitive treatment for each epidemic, particularly the HIV-TB coinfection diseasese. Tuberculosis (TB) is an infectious disease that can spread form person to person. TB is caused by bacteria called Mycobacterium tuberculosis. TB bacteria spread into the air and usually affect the lungs. However, the bacteria that cause TB can attack any part of the body, including the kidneys, spine, or brain. If left untreated, TB can cause death. HIV stands for human immunodeficiency virus, which is the virus that leads to HIV infection. AIDS stands for Acquired Immune Deficiency Syndrome. AIDS is the most advanced stage of HIV infection. HIV attacks and destroys the immune system’s infection-fighting CD4 cells. Loss of CD4 cells makes it difficult for the body to fight infections and some cancers. Without treatment, HIV can gradually destroy the immune system and progress to AIDS. ART is recommended for everyone with HIV. ART cannot cure HIV, but HIV drugs may help people with HIV live longer and healthier. ART also reduces the risk of HIV transmission. Infection with both HIV and TB is called HIV/TB coinfection. Latent TB is likely to progress to TB disease in people with HIV. TB disease may cause HIV to worsen. Treatment with HIV drugs is called antiretroviral therapy (ART). ART protects the immune system and prevents HIV infection from progressing to AIDS. In people with HIV/TB coinfection, ART reduces the chance of latent TB to switch to TB. TB disease is one of the leading causes of death among people with HIV worldwide. In the United States, where HIV drugs are widely used, people with HIV receive less TB compared to many other countries. However, TB affects many people with HIV in the United States, especially those who were born outside the United States.

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The literature includes very serious and important studies in this respect. Some of those studies are as follows: Sharma et al. [1] suggest that the treatment of HIV-TB coinfection requires strong commitment and a focused approach. They also concluded that proper use of highly active antiretroviral therapy (HAART) to maintain immunity and treat HIV infection requires a high level of containment and adjustment to prevent TB. In their analysis, Gao et al. [2] showed that the prevalence of HIV/TB coinfection in China deserves special attention, and more attention should be paid to TB screening among HIV/AIDS populations for treatment of both diseases. Pawlowski et al. [3] showed that, in light of the information on the interaction mechanisms of the two pathogens, there are many gaps to be filled in order to develop preventive measures against the both diseases. McShane [4] has examined HAART’s effects on TB in many aspects. Furthermore, an increased risk of TB in HIV-positive patients and differences in the clinical picture in the HIV-positive population suggest that a high degree of clinical suspicion should remain. Naresh [5] analyzed a nonlinear mathematical model for transmission dynamics of HIV and a curable TB pathogen in a variable-size population. It was shown that the positive coinfection balance is always locally stable, but it may be globally stable, indicating that the disease becomes endemic due to the continuous migration of the population into the habitat under certain circumstances. Datiko et al. [6] showed that HIV infection rate in TB patients and pregnant women was higher among study participants in urban areas, and found that the HIV infection rate in TB patients is associated with the prevalence of HIV infection in pregnant women participating in attending antenatal care (ANC). Moreover, some studies were conducted on the mathematical modeling and analysis of HIV-TB coinfection disease. Some of those studies are as follows: Agusto and Adekunle [7] showed that the most cost-effective control strategy is to practice combination strategy involving the prevention of treatment failure in drug-sensitive TB-infected individuals and treat individuals with drug-resistant TB. Okosun and Makinde [8] created a mathematical model to investigate synergistic relationships in the presence of treatments for malaria-cholera coinfection. They also analyzed the steady state of single infection. The literature includes following studies conducted by extending the HIV-TB coinfection disease model to fractional derivative: Carvalho and Pinto [9] studied a delayed fractional mathematical model for malaria and human immunodeficiency virus infection, taking into account personal protection and vaccination against malaria. The reproduction number of the model was calculated to examine its balance stability. Zafar et al. [10] studied a fractional order nonlinear mathematical model to analyze and control the spread of HIV/AIDS. Both disease-free equilibrium E0 and endemic equilibrium E* were found and stability results were obtained using the stability theorem of fractional order differential equations. Kheiri and Jafari [11] presented a general formulation for a fractional optimal control problem (FOCP) in which state and co-state equations are given in terms of left fraction derivatives. Then, the forward/backward sweeping method (FBSM) was developed using Adams-type predictive-correction method to solve FOCP. The numerical calculations of Mallela et al. [12] model solution showed that the results of treatment programs aimed at reducing the total load of this co-infection largely depend on both the strength and the start time of antiretroviral therapy (ART). Khan et al. [13] discussed an analysis

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of the HIV-TB-infected model in Atangana—Baleanu fractional differential form, maintaining the importance of the HIV model. Besides, the model was also examined for existence and uniqueness of solution, Hyers–Ulam (HU) stability and numerical simulations assuming certain parameters. In the literature, many mathematical modeling and their analysis related to disease models have been made. For instance, Dokuyucu and Dutta [14] extended the Ebola virus model to the Caputo-Fabrizio fractional derivative. Then they examined the existence and uniqueness solutions. Finally the results are compared with numerical simulations. Dokuyucu et al. [15] investigated cancer treatment model via new Caputo fractional derivative. They proved the existence and uniqueness solution for the mathematical model. Singh et al. [16] explore a fractional smoking model of a new non-singular fractional derivative. The existence of the solution was investigated with the fixed-point theorem. Then the uniqueness of the solution is shown. Basavarajaiah and Murthy [17] have published a book on the statistical modeling of HIV transmission. In this book, they organized the information available in the literature in a frame. They examined the relationship between the social and biological mechanisms that affect AIDS and its spread. There are many fractional operators in the literature. Some of these are GrünwaldLetnikov, Riemann-Liouville, Caputo, Caputo-Fabrizio, Atangana-Baleanu. In addition to the advantages of operators, they also have some disadvantages. For example, the Caputo fractional derivative operator has a singularity problem. Due to this problem, the Caputo-Fabrizio fractional derivative has been identified [18]. This operator eliminated the problem of singularity using the exponential kernel. In addition, solutions made with the exponential kernel give results very close to the exact solution. The motivation of using this kernel in the analysis of our model is the reasons we have listed above. This study will analyze the HIV-TB coinfection disease model. The model will first be expanded to the Caputo-Fabrizio fractional derivative containing an exponential core. Then, the existence and uniqueness solutions of the extended model will be made. Finally, numerical results will be obtained and numerical simulations will be done.

2 Preliminaries This section will give necessary definitions, theorems and lemmas with the new Caputo fractional derivative and integral operator. Definition 2.1. The Caputo’s fractional derivative are defined as follows [19], C ρ a Dt φ(t))

1 = (n − ρ)

 a

t

φ (n) (ω) dω, n − 1 < ρ < n ∈ N . (t − ω)ρ+1−n

(1)

Definition 2.2. Let f ∈ H 1 (a, b), b > a, ρ ∈ (0, 1) then, the definition of the new Caputo fractional derivative is [18],

Analytical and Numerical Solutions of a TB-HIV/AIDS Co-infection Model ... CF ρ a Dt (φ(t))

=

ρ M(ρ) 1−ρ

 a

t

  t −ω dφ(ω) ex p − ρ dω, dω 1−ρ

185

(2)

where M(ρ) is a normalization function. Furthermore, M(0) = M(1) = 1. The definition is also written if the function does not belong to H 1 (a, b),     t −ω ρ M(ρ) t  CF ρ φ(t) − φ(ω) ex p − ρ dω. (3) a Dt (φ(t)) = 1−ρ a 1−ρ 1 ∈ (0, ∞), ρ = 1+η ∈ [0, 1], in this case the following Remark 2.1. If η = 1−ρ ρ equation can be written according to the above equation.

η

Dt (φ(t)) =

N (η) η



t

a

  t −ω dφ(ω) ex p − dω, dω η

N (0) = N (∞) = 1.

(4)

In addition,   t −ω 1 ex p − = δ(ω − t). ρ→0 ρ ρ lim

Thus, Nieto and Losada proposed that the new Caputo derivative of order 0 < ρ < 1 can be reformulated as below, Definition 2.3. The Caputo-Fabrizio fractional derivative of order ρ is as follows [20],    t 1 t −ω CF ρ  dω. (5) D (φ(t)) = f (ω)ex p − ρ 1−ρ 0 1−ρ Remark 2.2. The Laplace transform of the new Caputo fractional derivative with s variable ρ

L T [Dt (φ(t))] =

   ∞  t (s L T (φ(t) − f (0))) dφ(x) 1 t−x dωdt = ex p(−st) ex p − ρ . 1−ν 0 dω 1−ρ s + ν(1 − s) 0

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Theorem 2.1. For NFDt , if the function φ(t) is such that [18], φ (s) (a) = 0, s = 1, 2, . . . , n,

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 ρ   (ρ)  Dt φ(t) = Dt Dnt φ(t) . D(n) t

(8)

then, we have

(ρ+1)

Proof 2.1. We begin considering n = 1, then Dt

φ(t), we obtain

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Dt

 M(ρ) D(1) t φ(t) = 1−ρ

 a

t

  ρ(t − ω) dω. f  (ω)ex p − 1−ρ

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Hence, after an integration by parts and assuming f  (a) = 0, we have,  (1)  Dt φ(t) D(ρ) t

   M(ρ) t d  ρ(t − ω) dω = ( φ (ω))ex p − 1 − ρ a dω 1−ρ  t   d M(ρ) ρ(t − ω)  (φ (ω))ex p − dω = 1 − ρ a dω 1−ρ (10)     t ρ ρ(t − ω)  dω − φ (ω)ex p − 1−ρ a 1−ρ      t M(ρ)  ρ ρ(t − ω)  φ (t) − dω . = φ (ω)ex p − 1−ρ 1−ρ a 1−ρ

Otherwise,       (ρ)  d M(ρ) t  ρ(t − ω) dω D φ(t) = φ (ω))ex p − D(1) t t dt 1 − ρ a 1−ρ    (11)   t M(ρ)  ρ ρ(t − ω) φ (t) − dω . = φ  (ω)ex p − 1−ρ 1−ρ a 1−ρ It is easy to generalize the proof for any n > 1 [18]. Definition 2.4. Let 0 < ρ < 1. The fractional integral of order ρ of a function f is defined by [18], CF ρ

I φ(t) =

2ρ 2(1 − ρ) u(t) + (2 − ρ)M(ρ) (2 − ρ)M(ρ)



t

u(s)ds,

t ≥ 0.

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0

Definition 2.5. The Sobolev space of order 1 in (a, b) is defined as [21]: H 1 (a, b) = {u ∈ L 2 (a, b) : u  ∈ L 2 (a, b)}.

3 Materials and Methods 3.1 Classical Model Our model consisting of 11 differential equations (13) is given in the system of equations. The population is divided into certain groups as A(t), BT (t), C T (t), E(t), C H (t), F(t), G H (t), BT H (t), C T H (t), E H (t) and FT (t), namely susceptible individuals A(t),

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individuals with no TB disease symptom and non-infected individuals BT (t), TBinfected individuals with active TB disease and infectious TB C T (t), individuals with recurrent TB E(t), individuals who do not show AIDS symptoms but who are infected with HIV C H (t), individuals who are being treated for HIV infections F(t), HIVinfected individuals with AIDS symptoms G H (t), HIV-infected TB-latent individuals (pre-AIDS) BT H (t), HIV-infected individuals with active TB disease (pre-AIDS) C T H (t), HIV-infected individuals who are recovered without TB and AIDS symptoms E H (t), individuals infected with HIV and also infected with active TB, showing AIDS symptoms FT (t), respectively. Total population denoted by N (t) at time T , N (t) = A(t) + BT (t) + C T (t) + E(t) + C H (t) + F(t) + G H (t) + BT H (t) + C T H (t) + E H (t) + FT (t).

The following system differential equations can be taken as the TB-HIV/AIDS model [22]: d A(t) dt d BT (t) dt dC T (t) dt d E(t) dt dC H (t) dt d F(t) dt dG H (t) dt d BT H (t) dt dC T H (t) dt d E H (t) dt d FT (t) dt

= − μT (t)A(t) − μ H (t)A(t) − ξ A(t), = μT (t)A(t) + α1 μT (t)E(t) − (m 1 + χ1 + ξ )BT (t),   = m 1 BT (t) − χ2 + r T + ξ + θμ H (t) C T (t), = χ1 BT (t) + χ2 C T (t) − (α1 μT (t) + μ H (t) + ξ )E(t), = μ H (t)A(t) − (ϑ1 + ψ + νμT (t) + ξ )C H (t) + β1 F(t) + μ H (t)E(t) + υ1 C H (t), = ϑ1 C H (t) + υ2 E H (t) − β1 F(t) − (ξ + r A )F(t),

(13)

= ψC H (t) + qϑ2 C T H (t) + eχ3 BT H (t) − (υ1 + ξ )G H (t), = α2 μT (t)E H (t) − (m 2 + χ3 + ξ )BT H (t), = θμ H (t)C T (t) + νμT (t)C H (t) + β2 FT (t) + m 2 BT H (t) − (ϑ2 + ξ + r T )C T H (t), = pϑ2 C T H (t) + (1 − e)χ3 BT H (t) − (α2 μT (t) + υ2 + ξ )E H (t), = (1 − (q + p))ϑ2 C T H (t) − (β2 + ξ + r T A )FT (t).

Approximately 10% of people infected with mycobacterium tuberculosis develop active TB disease. Therefore, it remains hidden in about 90% of infected people. Latent infected TB patients are asymptomatic and do not transmit TB [23]. The transmission rate in treatment of individuals in the latent TB category is χ1 and the recovery rate is m 1 . Recovery rate of active individuals infected with TB is χ2 . Supposing that individuals with recurrent TB have become partially immune and the transmission rate of this class is limited to α1 ≤ 1. Individuals with active TB disease are exposed to evoked death at the rate of r T . We assume that individuals in class E are susceptible to HIV infection at the rate of μ H . On the other hand, TB-active infected individuals are susceptible to HIV infection at the rate of C T , θ μ H (t), whereas the modification parameter θ ≤ 1 explains that individuals in the class C T are more likely to be HIV positive. HIV-infected individuals in the class

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F (without AIDS symptoms) have an AIDS ratio of ϑ1 and the recovery rate of the HIV-infected individuals progresses to the rate of ψ. Individuals in the class G H differ from individuals in the class C H by υ1 . HIV-infected individuals with AIDS symptoms have an HIV recovery rate of β1 and they experience induced death at the rate of r A . Individuals in the class C H are sensitive to TB infection at the rate of νμt [24]. HIV-positive individuals infected with individuals with TB disease (pre-AIDS) leave the class C T H at the rate of ϑ2 . C T H Individuals in the class G H progress to the class at the rate of qϑ2 and to the class E H at the rate of qϑ2 . Individuals in the class who do not receive any treatment for TB or HIV progress to the class at the rate of (1 − (q + p))ϑ2 Individuals leave the class BT H at the rate of χ3 . Individuals in the class BT H are more likely to progress to active TB disease compared to individuals infected with latent TB only. The progress rate in the model is expressed in m 2 . Similarly, HIV-infected individuals become more susceptible to TB infection compared to the patients who are not HIV positive. For individuals in the class E H with recurrent TB, the modification parameter associated with the rate of infection is expressed in α2 and α2 ≤ 1. Individuals in this class progress to the class F at the rate of υ2 . Individuals with both HIV and TB (with AIDS symptoms) are treated at the rate of β2 for HIV. The individuals in the class FT have a mortality rate of r T A in AIDS-TB co-infection.

3.2 Existence of Solution to Fractional Model In this section, existence solution will be obtained by using fixed point technique of fractional TB-HIV/AIDS mathematical model. Dokuyucu and Dutta [14] have proven the Ebola virus mathematical model with a fixed point theorem. Similarly, when the system (13) is extended to the Caputo-Fabrizio fractional derivative, we have the following system: ⎧C F ρ ⎪ t D0 A(t) ⎪ ⎪C ⎪ F ρ ⎪ ⎪ t D0 BT (t) ⎪ ⎪C ⎪ F ρ ⎪ ⎪ ⎪t D0 C T (t) ⎪ ⎪ C F Dρ E(t) ⎪ ⎪ 0 ⎪t ⎪ ⎪ F ρ ⎪ ⎪C t D0 C H (t) ⎨ C F Dρ F(t) t 0 ⎪ ⎪ C F Dρ G (t) ⎪ ⎪ t ⎪ 0 H ⎪ ⎪ CF ρ ⎪ ⎪ ⎪t D0 BT H (t) ⎪ ⎪ ⎪ CF ρ ⎪ ⎪t D0 C T H (t) ⎪ ⎪ ⎪ F ρ ⎪C ⎪ t D E H (t) ⎪ ⎩C F 0ρ t D0 FT (t)

= − μT (t)A(t) − μ H (t)A(t) − ξ A(t), = μT (t)A(t) + α1 μT (t)E(t) − (m 1 + χ1 + ξ )BT (t),   = m 1 BT (t) − χ2 + r T + ξ + θμ H (t) C T (t), = χ1 BT (t) + χ2 C T (t) − (α1 μT (t) + μ H (t) + ξ )E(t), = μ H (t)A(t) − (ϑ1 + ψ + νμT (t) + ξ )C H (t) + β1 F(t) + μ H (t)E(t) + υ1 C H (t), = ϑ1 C H (t) + υ2 E H (t) − β1 F(t) − (ξ + r A )F(t), = ψC H (t) + qϑ2 C T H (t) + eχ3 BT H (t) − (υ1 + ξ )G H (t), = α2 μT (t)E H (t) − (m 2 + χ3 + ξ )BT H (t), = θμ H (t)C T (t) + νμT (t)C H (t) + β2 FT (t) + m 2 BT H (t) − (ϑ2 + ξ + r T )C T H (t), = pϑ2 C T H (t) + (1 − e)χ3 BT H (t) − (α2 μT (t) + υ2 + ξ )E H (t), = (1 − (q + p))ϑ2 C T H (t) − (β2 + ξ + r T A )FT (t).

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In addition, the initial values are as follows, A0 (t) = A(0), BT (t) = B(0), C T (t) = C(0), E 0 (t) = E(0), C H (t) = C H (0), F0 (t) = F(0) 0 0 0 G H (t) = G H (0), BT H (t) = BT H (0), C T H (t) = C T H (0), E H (t) = E H (0), FT (t) = FT (0). 0 0 0 0 0

Using the Definition (2.4), the system of equations above can be written as follows:   2(1 − ρ)

− μT (t)A(t) − μ H (t)A(t) − ξ A(t) 2M(ρ) − ρ M(ρ)  t   2ρ

− μT (y)A(y) − μ H (y)A(y) − ξ A(y) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) μ (t)A(t) + α1 μT (t)E(t) − (m 1 + χ1 + ξ )BT (t) BT (t) − BT (0) = 2M(ρ) − ρ M(ρ) T  t   2ρ μT (y)A(y) + α1 μT (y)E(y) − (m 1 + χ1 + ξ )BT (y) dy, + 2M(ρ) − ρ M(ρ) 0     2(1 − ρ) m B (t) − χ2 + r T + ξ + θμ H (t) C T (t) C T (t) − C T (0) = 2M(ρ) − ρ M(ρ) 1 T  t     2ρ m 1 BT (y) − χ2 + r T + ξ + θμ H (y) C T (y) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) χ B (t) + χ2 C T (t) − (α1 μT (t) + μ H (t) + ξ )E(t) E(t) − E(0) = 2M(ρ) − ρ M(ρ) 1 T  t   2ρ + χ1 BT (y) + χ2 C T (y) − (α1 μT (y) + μ H (y) + ξ )E(y) dy, 2M(ρ) − ρ M(ρ) 0 A(t) − A(0) =

  2(1 − ρ) μ (t)A(t) − (ϑ1 + ψ + νμT (t) + ξ )C H (t) + β1 F(t) + μ H (t)E(t) + υ1 C H (t) 2M(ρ) − ρ M(ρ) H  t  2ρ μ H (y)A(y) − (ϑ1 + ψ + νμT (y) + ξ )C H (y) + β1 F(y) + μ H (y)E(y) + 2M(ρ) − ρ M(ρ) 0  + υ1 C H (y) dy,   2(1 − ρ) ϑ C (t) + υ2 E H (t) − β1 F(t) − (ξ + r A )F(t) F(t) − F(0) = 2M(ρ) − ρ M(ρ) 1 H  t   2ρ ϑ1 C H (y) + υ2 E H (y) − β1 F(y) − (ξ + r A )F(y) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) ψC H (t) + qϑ2 C T H (t) + eχ3 BT H (t) − (υ1 + ξ )G H (t) G H (t) − G H (0) = 2M(ρ) − ρ M(ρ)  t   2ρ ψC H (y) + qϑ2 C T H (y) + eχ3 BT H (y) − (υ1 + ξ )G H (y) dy, + 2M(ρ) − ρ M(ρ) 0    2(1 − ρ) BT H (t) − BT H (0) = α μ (t)E H (t) − (m 2 + χ3 + ξ )BT H (t) 2M(ρ) − ρ M(ρ) 2 T  t    2ρ α2 μT (y)E H (y) − (m 2 + χ3 + ξ )BT H (y) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) θμ H (t)C T (t) + νμT (t)C H (t) + β2 FT (t) + m 2 BT H (t) − (ϑ2 + ξ + r T )C T H (t) C T H (t) − C T H (0) = 2M(ρ) − ρ M(ρ)  t  2ρ + θμ H (y)C T (y) + νμT (y)C H (y) + β2 FT (y) + m 2 BT H (y) 2M(ρ) − ρ M(ρ) 0  − (ϑ2 + ξ + r T )C T H (y) dy,   2(1 − ρ) pϑ2 C T H (t) + (1 − e)χ3 BT H (t) − (α2 μT (t) + υ2 + ξ )E H (t) E H (t) − E H (0) = 2M(ρ) − ρ M(ρ)  t   2ρ pϑ2 C T H (y) + (1 − e)χ3 BT H (y) − (α2 μT (y) + υ2 + ξ )E H (y) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) (1 − (q + p))ϑ2 C T H (t) − (β2 + ξ + r T A )FT (t) FT (t) − FT (0) = 2M(ρ) − ρ M(ρ)  t   2ρ (1 − (q + p))ϑ2 C T H (y) − (β2 + ξ + r T A )FT (y) dy. + 2M(ρ) − ρ M(ρ) 0 C H (t) − C H (0) =

(15)

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To simplify, the kernels can be written as follows, S1 (t, A) = − μT (t)A(t) − μ H (t)A(t) − ξ A(t), S2 (t, BT ) = μT (t)A(t) + α1 μT (t)E(t) − (m 1 + χ1 + ξ )BT (t),   S3 (t, C T ) = m 1 BT (t) − χ2 + r T + ξ + θ μ H (t) C T (t), S4 (t, E) = χ1 BT (t) + χ2 C T (t) − (α1 μT (t) + μ H (t) + ξ )E(t), S5 (t, C H ) = μ H (t)A(t) − (ϑ1 + ψ + νμT (t) + ξ )C H (t) + β1 F(t) + μ H (t)E(t) + υ1 C H (t), S6 (t, F) = ϑ1 C H (t) + υ2 E H (t) − β1 F(t) − (ξ + r A )F(t), S7 (t, G H ) = ψC H (t) + qϑ2 C T H (t) + eχ3 BT H (t) − (υ1 + ξ )G H (t), S8 (t, BT H ) = α2 μT (t)E H (t) − (m 2 + χ3 + ξ )BT H (t), S9 (t, C T H ) = θ μ H (t)C T (t) + νμT (t)C H (t) + β2 FT (t) + m 2 BT H (t) − (ϑ2 + ξ + r T )C T H (t), S10 (t, E H ) = pϑ2 C T H (t) + (1 − e)χ3 BT H (t) − (α2 μT (t) + υ2 + ξ )E H (t), S11 (t, FT ) = (1 − (q + p))ϑ2 C T H (t) − (β2 + ξ + r T A )FT (t).

(16) and 1 = μT − μ H − ξ 2 = m 1 + χ1 + ξ 3 = χ2 + r T + ξ + θ μ H 4 = α1 μ+ μ H + ξ 5 = ϑ1 + ψ + νμT + ξ 6 = β1 + ξ + r A 7 = υ1 + ξ 8 = m 2 + χ3 + ξ 9 = ϑ2 + ξ + r T 10 = α2 μT + υ2 + ξ 11 = β2 + ξ + r T A It can be assumed that the following assumption H : A(t), BT (t), C T (t), E(t), C H (t), F(t), G H (t), BT H (t), C T H (t), E H (t), FT (t), A1 (t), BT1 (t), C T1 (t), E 1 (t), C H1 (t), F1 (t), G H1 (t), BT H1 (t), C T H1 (t), E H1 (t), FT1 (t) ∈ L[0, 1] are continuous functions, so that ||A(t)|| ≤ L1 , ||B(t)|| ≤ L2 , ||C T (t)|| ≤ L3 , ||E(t)|| ≤ L4 , ||C H (t)|| ≤ L5 , ||F(t)|| ≤ L6 , ||G H (t)|| ≤ L7 , ||BT H (t)|| ≤ L8 , ||C T H (t)|| ≤ L9 , ||E H (t)|| ≤ L10 , ||FT (t)|| ≤ L11 . Theorem 3.1. The kernels Si , i = 1, . . . 11 are satisfying the Lipschitz condition if the assumption H is true and are contractions provided that i < 1 for ∀ ∈ i = 1 . . . 11. Proof 3.1. First of all, we will begin to proof that S1 (t, A) satisfies Lipschitz condition. Let A(t) and A1 (t) are two functions, then

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  ||S1 (t, A) − S1 (t, A1 )|| = || − μT A − μ H A − ξ A   − − μT A1 − μ H A1 − ξ A1 ||   ≤ μT − μ H − ξ ||A − A1 || = 1 ||A − A1 ||. Next, we will proof that S2 (t, BT ) satisfies Lipschitz condition. Let BT (t) and BT1 (t) are two functions, then   ||S2 (t, BT ) − S2 (t, BT1 )|| = || μT A + α1 μT E − (m 1 + χ1 + ξ )BT   − μT A + α1 μT E − (m 1 + χ1 + ξ )BT1 ||   ≤ m 1 + χ1 + ξ ||BT − BT1 || = 2 ||BT − BT1 ||. Next, we will proof that S3 (t, C T ) satisfies Lipschitz condition. Let C T (t) and C T1 (t) are two functions, then     ||S3 (t, C T ) − S3 (t, C T1 )|| = || m 1 BT − χ2 + r T + ξ + θ μ H C T     − m 1 BT − χ2 + r T + ξ + θ μ H C T1 ||   ≤ χ2 + r T + ξ + θ μ H ||C T − C T1 || = 3 ||C T − C T1 ||. Next, we will proof that S4 (t, E) satisfies Lipschitz condition. Let E(t) and E 1 (t) are two functions, then   ||S4 (t, E) − S4 (t, E 1 )|| = || χ1 BT + χ2 C T − (α1 μ+ μ H + ξ )E   − χ1 BT + χ2 C T − (α1 μ+ μ H + ξ )E 1 ||   ≤ α1 μ+ μ H + ξ ||E − E 1 || = 4 ||E − E 1 ||. Next, we will proof that S5 (t, C H ) satisfies Lipschitz condition. Let C H (t) and C H1 (t) are two functions, then   ||S5 (t, C H ) − S5 (t, C H1 )|| = || μ H A − (ϑ1 + ψ + νμT + ξ )C H + β1 F + μ H E + υ1 C H   − μ H A − (ϑ1 + ψ + νμT + ξ )C H1 + β1 F + μ H E + υ1 C H ||   ≤ ϑ1 + ψ + νμT + ξ ||C H − C H1 || = 5 ||C H − C H1 ||.

Next, we will proof that S6 (t, F) satisfies Lipschitz condition. Let F(t) and F1 (t) are two functions, then

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  ||S6 (t, F) − S6 (t, F1 )|| = || ϑ1 C H + υ2 E H − β1 F − (ξ + r A )F   − ϑ1 C H + υ2 E H − β1 F1 − (ξ + r A )F1 ||   ≤ β1 + ξ + r A ||F − F1 || = 6 ||F − F1 ||. Next, we will proof that S7 (t, G H ) satisfies Lipschitz condition. Let G H (t) and G H1 (t) are two functions, then   ||S7 (t, G H ) − S7 (t, G H1 )|| = || ψC H + qϑ2 C T H + eχ3 BT H − (υ1 + ξ )G H   − ψC H + qϑ2 C T H + eχ3 BT H − (υ1 + ξ )G H1 ||   ≤ υ1 + ξ ||G H − G H1 || = 7 ||G H − G H1 ||. Next, we will proof that S8 (t, BT H ) satisfies Lipschitz condition. Let BT H (t) and BT H 1 (t) are two functions, then   ||S8 (t, BT H ) − S8 (t, BT H 1 )|| = || α2 μT E H − (m 2 + χ3 + ξ )BT H   − α2 μT E H − (m 2 + χ3 + ξ )BT H1 ||   ≤ m 2 + χ3 + ξ ||BT H − BT H1 || = 8 ||BT H − BT H1 ||. Next, we will proof that S9 (t, C T H ) satisfies Lipschitz condition. Let C T H (t) and C T H 1 (t) are two functions, then   ||S9 (t, C T H ) − S9 (t, C T H 1 )|| = || θμ H C T + νμT C H + β2 FT + m 2 BT H − (ϑ2 + ξ + r T )C T H   − θμ H C T + νμT C H + β2 FT + m 2 BT H − (ϑ2 + ξ + r T )C T H 1 ||   ≤ ϑ2 + ξ + r T ||C T H − C T H1 || = 9 ||C T H − C T H1 ||.

Next, we will proof that S10 (t, E H ) satisfies Lipschitz condition. Let E H (t) and E H1 (t) are two functions, then   ||S10 (t, E H ) − S10 (t, E H1 )|| = || ϑ2 C T H + (1 − e)χ3 BT H − (α2 μT + υ2 + ξ )E H   − ϑ2 C T H + (1 − e)χ3 BT H − (α2 μT + υ2 + ξ )E H1 ||    ≤ α2 μT + υ2 + ξ ||E H − E H1 || = 10 ||E H − E H1 ||.

Finally, we will proof that S11 (t, FT ) satisfies Lipschitz condition. Let FT (t) and FT1 (t) are two functions, then

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  ||S11 (t, FT ) − S11 (t, FT1 )|| = || (1 − (q + p))ϑ2 C T H − (β2 + ξ + r T A )FT   − (1 − (q + p))ϑ2 C T H − (β2 + ξ + r T A )FT1 ||   ≤ β2 + ξ + r T A ||FT − FT1 || = 11 ||FT − FT1 ||. All kernels which Si , i = 1 . . . 11 are satisfying conditions, so that they are contractions with i < 1, i ∈ 1 . . . 11. As a result, this completes the proof. Let all the initial values be zero. If the system (15) is then rewritten,

A(t) = BT (t) = C T (t) = E(t) = C H (t) = F(t) = G H (t) = BT H (t) = C T H (t) = E H (t) = FT (t) =

 t 2(1 − ρ) 2ρ S1 (t, A(t)) + S1 (y, A(y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S2 (t, BT (t)) + S2 (y, BT (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S3 (t, C T (t)) + S3 (y, C T (y))dy, 2M(ρ) − ρ M(ρ)0 2M(ρ) − ρ M(ρ) 0  t 2ρ 2(1 − ρ) S4 (t, E(t)) + S4 (y, E(y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S5 (t, C H (t)) + S5 (y, C H (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2ρ 2(1 − ρ) S (t, F(t)) + S (y, F(y))dy, 2M(ρ) − ρ M(ρ) 6 2M(ρ) − ρ M(ρ) 0 6  t 2(1 − ρ) 2ρ S7 (t, G H (t)) + S7 (y, G H (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S (t, BT H (t)) + S (y, BT H (y))dy, 2M(ρ) − ρ M(ρ) 8 2M(ρ) − ρ M(ρ) 0 8  t 2(1 − ρ) 2ρ S9 (t, C T H (t)) + S9 (y, C T H (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S10 (t, E H (t)) + S10 (y, E H (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S11 (t, FT (t)) + S11 (y, FT (y))dy. 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0

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Then the following system of equations can be defined with the help of a recursive formula.

194

M. A. Dokuyucu and H. Dutta An (t) = BTn (t) = C Tn (t) = E n (t) = C Hn (t) = Fn (t) = G Hn (t) =

BT Hn (t) =

 t 2(1 − ρ) 2ρ S1 (t, An−1 (t)) + S1 (y, An−1 (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S2 (t, BTn−1 (t)) + S2 (y, BTn−1 (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S3 (y, C Tn−1 (y))dy, S3 (t, C Tn−1 (t)) + 2M(ρ) − ρ M(ρ)0 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S4 (t, E n−1 (t)) + S4 (y, E n−1 (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S5 (t, C Hn−1 (t)) + S5 (y, C Hn−1 (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S (t, Fn−1 (t)) + S (y, Fn−1 (y))dy, 2M(ρ) − ρ M(ρ) 6 2M(ρ) − ρ M(ρ) 0 6  t 2(1 − ρ) 2ρ S7 (t, G Hn−1 (t)) + S7 (y, G Hn−1 (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S8 (t, BT Hn−1 (t)) + S (y, BT Hn−1 (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0 8

 t 2(1 − ρ) 2ρ S9 (t, C T Hn−1 (t)) + S9 (y, C T Hn−1 (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S10 (t, E Hn−1 (t)) + S10 (y, E Hn−1 (y))dy, E Hn (t) = 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S11 (t, FTn−1 (t)) + S11 (y, FTn−1 (y))dy. FTn (t) = 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0

C T Hn (t) =

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Also, the differences of each equation can be written as follows:   2(1 − ρ) S1 (t, An (t)) − S1 (t, An−1 (t)) 2M(ρ) − ρ M(ρ)  t   2ρ S1 (y, An (y)) − S1 (y, An−1 (y)) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) S2 (t, BTn (t)) − S2 (t, BTn−1 (t)) (BTn+1 − BTn )(t) = 2M(ρ) − ρ M(ρ)  t   2ρ S2 (t, BTn (y)) − S2 (t, BTn−1 (y)) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) S3 (t, C Tn (t)) − S3 (t, C Tn−1 (t)) (C Tn+1 − C Tn )(t) = 2M(ρ) − ρ M(ρ)  t   2ρ S3 (t, C Tn (y)) − S3 (t, C Tn−1 (y)) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) S4 (t, E n (t)) − S4 (t, E n−1 (t)) (E n+1 − E n )(t) = 2M(ρ) − ρ M(ρ)  t   2ρ S4 (y, E n (y)) − S4 (y, E n−1 (y)) dy, + 2M(ρ) − ρ M(ρ) 0

(An+1 − An )(t) =

Analytical and Numerical Solutions of a TB-HIV/AIDS Co-infection Model ...

195

  2(1 − ρ) S5 (t, C Hn (t)) − S5 (t, C Hn−1 (t)) 2M(ρ) − ρ M(ρ)  t   2ρ S5 (t, C Hn (y)) − S5 (t, C Hn−1 (y)) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) S6 (t, Fn (t)) − S6 (t, Fn−1 (t)) (Fn+1 − Fn )(t) = 2M(ρ) − ρ M(ρ)  t   2ρ S6 (y, Fn (y)) − S6 (y, Fn−1 (y)) dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) S7 (t, G Hn (t)) − S7 (t, G Hn−1 (t)) (G Hn+1 − G Hn )(t) = 2M(ρ) − ρ M(ρ)  t   2ρ S7 (t, G Hn (y)) − S7 (t, G Hn−1 (y)) dy, + 2M(ρ) − ρ M(ρ) 0 (C Hn+1 − C Hn )(t) =

  2(1 − ρ) S8 (t, BT Hn (t)) − S8 (t, BT Hn−1 (t)) 2M(ρ) − ρ M(ρ)  t   2ρ S8 (t, BT Hn (y)) − S8 (t, BT Hn−1 (y)) dy, + 2M(ρ) − ρ M(ρ) 0

(BT Hn+1 − BT Hn )(t) =

  2(1 − ρ) S9 (t, C T Hn (t)) − S9 (t, C T Hn−1 (t)) 2M(ρ) − ρ M(ρ)  t   2ρ + S9 (t, C T Hn (y)) − S9 (t, C T Hn−1 (y)) dy, 2M(ρ) − ρ M(ρ) 0

(C T Hn+1 − C T Hn )(t) =

  2(1 − ρ) S10 (t, E Hn (t)) − S10 (t, E Hn−1 (t)) 2M(ρ) − ρ M(ρ)  t   2ρ S10 (t, E G Hn (y)) − S10 (t, E Hn−1 (y)) dy, + 2M(ρ) − ρ M(ρ) 0

(E Hn+1 − G Hn )(t) =

  2(1 − ρ) S11 (t, FTn (t)) − S11 (t, FTn−1 (t)) 2M(ρ) − ρ M(ρ)  t   2ρ S11 (t, FTn (y)) − S11 (t, FTn−1 (y)) dy. + 2M(ρ) − ρ M(ρ) 0

(FTn+1 − BTn )(t) =

When we take the norm of both sides of the above equations,   2(1 − ρ) || S1 (t, An (t)) − S1 (t, An−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ || S1 (y, An (y)) − S1 (y, An−1 (y)) ||dy, + 2M(ρ) − ρ M(ρ) 0

||(An+1 − An )(t)|| =

  2(1 − ρ) || S2 (t, BTn (t)) − S2 (t, BTn−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ + || S2 (t, BTn (y)) − S2 (t, BTn−1 (y)) ||dy, 2M(ρ) − ρ M(ρ) 0

||(BTn+1 − BTn )(t)|| =

196

M. A. Dokuyucu and H. Dutta   2(1 − ρ) || S3 (t, C Tn (t)) − S3 (t, C Tn−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ || S3 (t, C Tn (y)) − S3 (t, C Tn−1 (y)) ||dy, + 2M(ρ) − ρ M(ρ) 0

||(C Tn+1 − C Tn )(t)|| =

  2(1 − ρ) || S4 (t, E n (t)) − S4 (t, E n−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ + || S4 (y, E n (y)) − S4 (y, E n−1 (y)) ||dy, 2M(ρ) − ρ M(ρ) 0

||(E n+1 − E n )(t)|| =

  2(1 − ρ) || S5 (t, C Hn (t)) − S5 (t, C Hn−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ || S5 (t, C Hn (y)) − S5 (t, C Hn−1 (y)) ||dy, + 2M(ρ) − ρ M(ρ) 0

||(C Hn+1 − C Hn )(t)|| =

  2(1 − ρ) || S6 (t, Fn (t)) − S6 (t, Fn−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ || S6 (y, Fn (y)) − S6 (y, Fn−1 (y)) ||dy, + 2M(ρ) − ρ M(ρ) 0   2(1 − ρ) − G Hn )(t)|| = || S7 (t, G Hn (t)) − S7 (t, G Hn−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ || S7 (t, G Hn (y)) − S7 (t, G Hn−1 (y)) ||dy, + 2M(ρ) − ρ M(ρ) 0

||(Fn+1 − Fn )(t)|| =

||(G Hn+1

  2(1 − ρ) || S8 (t, BT Hn (t)) − S8 (t, BT Hn−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ || S8 (t, BT Hn (y)) − S8 (t, BT Hn−1 (y)) ||dy, + 2M(ρ) − ρ M(ρ) 0

||(BT Hn+1 − BT Hn )(t)|| =

  2(1 − ρ) || S9 (t, C T Hn (t)) − S9 (t, C T Hn−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ + || S9 (t, C T Hn (y)) − S9 (t, C T Hn−1 (y)) ||dy, 2M(ρ) − ρ M(ρ) 0

||(C T Hn+1 − C T Hn )(t)|| =

  2(1 − ρ) || S10 (t, E Hn (t)) − S10 (t, E Hn−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ + || S10 (t, E G Hn (y)) − S10 (t, E Hn−1 (y)) ||dy, 2M(ρ) − ρ M(ρ) 0

||(E Hn+1 − G Hn )(t)|| =

  2(1 − ρ) || S11 (t, FTn (t)) − S11 (t, FTn−1 (t)) || 2M(ρ) − ρ M(ρ)  t   2ρ || S11 (t, FTn (y)) − S11 (t, FTn−1 (y)) ||dy. + 2M(ρ) − ρ M(ρ) 0

||(FTn+1 − BTn )(t)|| =

Theorem 3.2. The TB-HIV/AIDS coinfection model has a solution if the following inequality is achieved:

Analytical and Numerical Solutions of a TB-HIV/AIDS Co-infection Model ...

ϒ = max{i } < 1, i = 1, 2, . . . 11.

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Proof 3.2. Let us consider the following equations, K1n (t) = An+1 (t) − An (t), K2n (t) = BT (t) − BTn (t), K3n (t) = C T (t) − C Tn (t), K4n (t) = E n+1 (t) − E n (t), n+1 n+1

K5n (t) = C Hn+1 (t) − C Hn (t), K6n (t) = Fn+1 (t) − Fn (t), K7n (t) = G Hn+1 (t) − G Hn (t)), K8n (t) = BT Hn+1 (t) − BT Hn (t), K9n (t) = C T Hn+1 (t) − C T Hn (t), K10n (t) = E Hn+1 (t) − G Hn (t), K11n (t) = FTn+1 (t) − FTn (t).

Firstly, we will start with K1n (t), 2(1 − ρ) ||S1 (t, An (t)) − S1 (t, A(t)|| 2M(ρ) − ρ M(ρ)  t 2ρ ||S1 (y, An (y)) − S1 (y, A(y))||dy + 2M(ρ) − ρ M(ρ) 0   2ρ 2(1 − ρ) + 1 ||An − A|| ≤ 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ)  n 2(1 − ρ) 2ρ + ≤ ϒ n ||A − A1 ||. 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ)

||K1n (t)|| ≤

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Similarly,  ||K2n (t)|| ≤  ||K3n (t)|| ≤  ||K4n (t)|| ≤  ||K5n (t)|| ≤  ||K6n (t)|| ≤  ||K7n (t)|| ≤  ||K8n (t)|| ≤  ||K9n (t)|| ≤  ||K10n (t)|| ≤  ||K11n (t)|| ≤

2(1 − ρ) 2ρ + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 2(1 − ρ) 2ρ + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 2(1 − ρ) 2ρ + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 2(1 − ρ) 2ρ + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 2(1 − ρ) 2ρ + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 2(1 − ρ) 2ρ + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 2(1 − ρ) 2ρ + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 2(1 − ρ) 2ρ + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 2(1 − ρ) 2ρ + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 2ρ 2(1 − ρ) + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ)

n ϒ n ||BT − BT1 ||, n ϒ n ||C T − C T1 ||, n ϒ n ||E − E 1 ||, n ϒ n ||C H − C H1 ||, n ϒ n ||F − F1 ||, n

(21) ϒ n ||G H − G H1 ||,

n ϒ n ||BT H − BT H1 ||, n ϒ n ||C T H − C T H1 ||, n ϒ n ||E H − E H1 ||, n ϒ n ||FT − FT1 ||.

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So that, it can be said that, we can find Kin (t) → 0, i = 1, 2, . . . 11, as n → ∞. Thus, the proof is complete.

3.3 Uniqueness of Solution to Fractional Model In this section we will show you the unique solution of TB-HIV/AIDS mathematical model as [14]. Theorem 3.3. The TB-HIV/AIDS mathematical model shown in system (14) will have a unique solution if the following inequality hold true: 

 2ρ 2(1 − ρ) + i ≤ 1, i = 1, 2, . . . 11. 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ)

Proof 3.3. Let us assume that the system (16) has solutions A(t), BT (t), C T (t), E(t), C H (t), F(t), G H (t), BT H (t), C T H (t), ˜ ˜ ˜ B˜ T (t), C˜ T (t), E(t), C˜ H (t), F(t), G˜ H (t), B˜ T H (t), E H (t), FT (t), as well as A(t), C˜ T H (t), E˜ H (t), F˜T (t). So that, the following system can be written, ˜ = A(t) B˜ T (t) = C˜ T (t) = ˜ E(t) = C˜ H (t) =

 t 2(1 − ρ) 2ρ ˜ ˜ + S1 (y, A(y))dy, S1 (t, A(t)) 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S2 (t, B˜ T (t)) + S2 (y, B˜ T (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S3 (t, C˜ T (t)) + S3 (y, C˜ T (y))dy, 2M(ρ) − ρ M(ρ)0 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ ˜ ˜ S4 (t, E(t)) + S4 (y, E(y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S5 (t, C˜ H (t)) + S5 (y, C˜ H (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0

˜ F(t) = G˜ H (t) = B˜ T H (t) = C˜ T H (t) = E˜ H (t) = F˜T (t) =

 t 2(1 − ρ) 2ρ ˜ ˜ + S6 (y, F(y))dy, S6 (t, F(t)) 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S7 (y, G˜ H (y))dy, S7 (t, G˜ H (t)) + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S (t, B˜ T H (t)) + S8 (y, B˜ T H (y))dy, 2M(ρ) − ρ M(ρ) 8 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S9 (y, C˜ T H (y))dy, S9 (t, C˜ T H (t)) + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S10 (y, E˜ H (y))dy, S10 (t, E˜ H (t)) + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S11 (y, F˜T (y))dy. S11 (t, F˜T (t)) + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0

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Analytical and Numerical Solutions of a TB-HIV/AIDS Co-infection Model ...

199

When the norm is taken from both sides of the system of equations above, firstly 2(1 − ρ) ˜ ||S1 (t, A(t)) − S1 (t, A(t))|| 2M(ρ) − ρ M(ρ)  t 2ρ ˜ + ||S1 (y, A(y)) − S1 (y, A(y))||dy 2M(ρ) − ρ M(ρ) 0 2(1 − ρ) 2ρ1 ˜ ˜ + ≤ 1 ||A − A|| ||A − A||. 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) (23) The following inequality can be written,   2(1 − ρ) 2ρ1 ˜ ≥ 0. 1 + − 1 ||A − A|| 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) ˜ ||A(t) − A(t)|| ≤

˜ = 0. This implies A(t) = A(t). ˜ Thus ||A − A|| When the same method is applied ˜ ˜ that BT (t) = B˜ T (t), C T (t) = C˜ T (t), E(t) = E(t), C H (t) = C˜ H (t), F(t) = F(t), and G H (t) = G˜ H (t), BT H (t) = B˜ T H (t), C T H (t) = C˜ T H (t), E H (t) = E˜ H (t) FT (t) = F˜T (t). According to these results, the model has a unique solution.

4 Stability Analysis In this section, we will examine the stability of the TB-HIV/AIDS model. First of all, the following definition should be given. Definition 4.1. The system (18) Hyers-Ulam stable [25] if exists constants ϒi > 0, i = 1, 2, . . . 11 satisfying for every ςi > 0, i = 1, 2, . . . 11,  t 2(1 − ρ) 2ρ S1 (t, A(t)) + S1 (y, A(y))dy| ≤ ς1 , 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S2 (t, BT (t)) + S2 (y, BT (y))dy| ≤ ς2 , |BT (t) − 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S3 (t, C T (t)) + S3 (y, C T (y))dy| ≤ ς3 , |C T (t) − 2M(ρ) + ρ M(ρ)0 2M(ρ) − ρ M(ρ) 0

|A(t) −

200

M. A. Dokuyucu and H. Dutta  t 2(1 − ρ) 2ρ S4 (t, E(t)) + 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ) 0 2(1 − ρ) 2ρ |C H (t) − S5 (t, C H (t)) + 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ)  t 2ρ 2(1 − ρ) S6 (t, F(t)) + |F(t) − 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ) 0 |E(t) −

S4 (y, E(y))dy ≤ ς4 ,  t S5 (y, C H (y))dy ≤ ς5 , 0

S6 (y, F(y))dy ≤ ς6 ,  t S7 (y, G H (y))dy ≤ ς7 ,

2(1 − ρ) 2ρ S7 (t, G H (t)) + 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S8 (t, BT H (t)) + |BT H (t) − S8 (y, BT H (y))dy ≤ ς8 , 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S9 (y, C T H (y))dy ≤ ς9 , S9 (t, C T H (t)) + |C T H (t) − 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ |E H (t) − S10 (y, E H (y))dy ≤ ς10 , S10 (t, E H (t)) + 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S11 (y, FT (y))dy ≤ ς11 . S11 (t, FT (t)) + |FT (t) − 2M(ρ) + ρ M(ρ) 2M(ρ) − ρ M(ρ) 0 |G H (t) −

(24)

There exist, A(t), B T (t), C T (t), E(t), C H (t), F(t), G H (t), B T H (t), C T H (t), E H (t), F T (t) are satisfying  t 2(1 − ρ) 2ρ S1 (t, A(t)) + S1 (y, A(y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S2 (t, B T (t)) + B T (t) = S2 (y, B T (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S3 (t, C T (t)) + C T (t) = S3 (y, C T (y))dy, 2M(ρ) − ρ M(ρ)0 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S4 (t, E(t)) + E(t) = S4 (y, E(y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ C H (t) = S5 (y, C H (y))dy, S5 (t, C H (t)) + 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ F(t) = S (y, F(y))dy, S (t, F(t)) + 2M(ρ) − ρ M(ρ) 6 2M(ρ) − ρ M(ρ) 0 6  t 2(1 − ρ) 2ρ S7 (t, G H (t)) + G H (t) = S7 (y, G H (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S (t, B T H (t)) + B T H (t) = S (y, B T H (y))dy, 2M(ρ) − ρ M(ρ) 8 2M(ρ) − ρ M(ρ) 0 8  t 2(1 − ρ) 2ρ S9 (t, C T H (t)) + C T H (t) = S9 (y, C T H (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S10 (t, E H (t)) + E H (t) = S10 (y, E H (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0  t 2(1 − ρ) 2ρ S11 (t, F T (t)) + F T (t) = S11 (y, F T (y))dy, 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) 0 A(t) =

(25)

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201

such that |A(t) − A(t)| ≤ σ1 ς1 , |BT (t) − BT (t)| ≤ σ2 ς2 , |C T (t) − C T (t)| ≤ σ3 ς3 , |E(t) − E(t)| ≤ σ4 ς4 , |C H (t) − C H (t)| ≤ σ5 ς5 , |F(t) − F(t)| ≤ σ6 ς6 , |G H (t) − G H (t)| ≤ σ7 ς7 , |BT H (t) − BT H (t)| ≤ σ8 ς8 , |C T H (t) − C T H (t)| ≤ σ9 ς9 , |E H (t) − E H (t)| ≤ σ10 ς10 , |FT (t) − FT (t)| ≤ σ11 ς11 .

Theorem 4.1. The fractional system (18) is Hyers-Ulam stable with assumption H. Proof 4.1. In Theorem (3.3), A(t), BT (t), C T (t), E(t), C H (t), F(t), G H (t), BT H (t), C T H (t), E H (t) and FT (t) were shown to have a unique solution. Let A(t), B T (t), C T (t), E(t), C H (t), F(t), G H (t), B T H (t), C T H (t), E H (t) and F T (t) be an approximate solution of system (13) satisfying system (18). After, we can say that 2(1 − ρ) ||S1 (t, A(t)) − S1 (t, A(t))|| 2M(ρ) − ρ M(ρ)  t 2ρ ||S1 (y, A(y)) − S1 (y, A(y))||dy + 2M(ρ) − ρ M(ρ) 0   2ρ 2(1 − ρ) + 1 ||A − A|| ≤ 2M(ρ) − ρ M(ρ) 2M(ρ) − ρ M(ρ) (26) 2(1−ρ) 2ρ + , we have when we take ς1 = 1 , ϒ1 = 2M(ρ)−ρ M(ρ) 2M(ρ)−ρ M(ρ) ||A(t) − A(t)|| ≤

||A(t) − A(t)|| ≤ ς1 ϒ1 . In this way, the following inequalities can be easily written. ||BT (t) − BT (t)|| ≤ ς2 ϒ2 ||C T (t) − C T (t)|| ≤ ς3 ϒ3 ||E(t) − E(t)|| ≤ ς4 ϒ4 ||C H (t) − C H (t)|| ≤ ς5 ϒ5 ||F(t) − F(t)|| ≤ ς6 ϒ6 ||G H (t) − G H (t)|| ≤ ς7 ϒ7

(27)

||BT H (t) − BT H (t)|| ≤ ς8 ϒ8 ||C T H (t) − C T H (t)|| ≤ ς9 ϒ9 ||E H (t) − E H (t)|| ≤ ς10 ϒ10 ||FT (t) − FT (t)|| ≤ ς11 ϒ11 With the help of Eqs. (26) and (27), the system (18) Hyers-Ulam is stable. Thus, the theorem is proved.

202

M. A. Dokuyucu and H. Dutta

5 Numerical Simulations Atangana and Owolabi [26] have found a new numerical approach using the new Caputo fractional derivative for the discretization of fractional differential equations. The authors consider the following fractional differential equation first. CF ρ 0 Dt x(t)

or ( f (t, x(t))) =

M(ρ) 1−ρ



t 0

= ( f (t, x(t))),

 x  (τ )ex p −

(28)

 ρ (t − τ ) dτ. 1−ρ

(29)

When they edit the above equation using the fundamental theorem of analysis, they get,  t ρ 1−ρ f (t, x(t)) + x(t) − x(0) = f (τ, x(τ ))dτ. (30) M(ρ) M(ρ) 0 Therefore, x(tn+1 ) − x(0) =

1−ρ ρ f (tn , x(tn )) + M(ρ) M(ρ)



tn+1

f (t, x(t))dt,

(31)

f (t, x(t))dt.

(32)

0

and

x(tn+1 ) − x(0) =

1−ρ ρ f (tn−1 , x(tn−1 )) + M(ρ) M(ρ)



tn

0

When they removing (32) from (31), the following equation system is obtained.

x(tn+1 ) − x(tn ) =

1−ρ ρ { f (tn , x(tn )) − f (tn−1 , xn−1 )} + M(ρ) M(ρ)

 0

tn

f (t, x(t))dt, (33)

where 

tn+1 tn

 f (tn , xn ) f (tn−1 , xn−1 ) (t − tn−1 − )(t − tn ) dt h h tn h 3h f (tn , xn ) − f (tn−1 , xn−1 ). = 2 2 (34) 

f (t, x(t))dt =

tn+1

Analytical and Numerical Solutions of a TB-HIV/AIDS Co-infection Model ...

203

Thus,  1−ρ 3ρh f (tn , xn ) − f (tn−1 , xn−1 ) + f (tn , xn ) M(ρ) 2M(ρ) ρh f (tn−1 , xn−1 ), − 2M(ρ)

x(tn+1 ) − x(tn ) =

(35)

which implies that  x(tn+1 ) − x(tn ) =

1−ρ 3ρh + M(ρ) 2M(ρ)



 f (tn , xn ) +

1−ρ 3ρh + M(ρ) 2M(ρ)

 f (tn−1 , xn−1 ).

(36) Hence,  xn+1 = xn +

3ρh 1−ρ + M(ρ) 2M(ρ)



 f (tn , xn ) +

3ρh 1−ρ + M(ρ) 2M(ρ)

 f (tn−1 , xn−1 ),

(37) which is the corresponding two-step Adams-Bashforth method for the CaputoFabrizio fractional derivative. Theorem 5.1. Let x(t) be a solution of Ca F Dρt (x(t)) = f (t, x(t)) where f is a continuous function bounded for the Caputo-Fabrizio fractional derivative [26],  xn+1 = xn +

3ρh 1−ρ + M(ρ) 2M(ρ)



 f (tn , xn ) +

ρh 1−ρ + M(ρ) 2M(ρ)

 f (tn−1 , xn−1 ) + Rρn

(38) where ||Rρn || ≤ M.

5.1 Numerical Simulations for the Model The expanded TB-HIV/AIIDS model for the Caputo-Fabrizio fractional derivative was introduced in system (15). When the system (15) is rearranged by the fundamental theorem of analysis, the next system of equations is obtained for Si , i = 1, 2, . . . 11 kernels as follows:

204

M. A. Dokuyucu and H. Dutta  t 1−ρ ρ S (t, A(t)) + S1 (ω, A(ω))dω, M(ρ) 1 M(ρ) 0  t 1−ρ ρ BT (t) − BT (0) = S2 (t, BT (t)) + S2 (ω, BT (ω))dω, M(ρ) M(ρ) 0  t 1−ρ ρ S3 (ω, C T (ω))dω, S (t, C T (t)) + C T (t) − C T (0) = M(ρ) 3 M(ρ) 0  t 1−ρ ρ E(t) − E(0) = S4 (ω, E(ω))dω, S (t, E(t)) + M(ρ) 4 M(ρ) 0  t 1−ρ ρ S5 (t, C H (t)) + C H (t) − C H (0) = S5 (ω, C H (ω))dω, M(ρ) M(ρ) 0  t ρ 1−ρ S6 (ω, F(ω))dω, S (t, F(t)) + F(t) − F(0) = M(ρ) 6 M(ρ) 0  t 1−ρ ρ G H (t) − G H (0) = S7 (ω, G H (ω))dω, S7 (t, G H (t)) + M(ρ) M(ρ) 0  t 1−ρ ρ S8 (ω, BT H (ω))dω, S (t, BT H (t)) + BT H (t) − BT H (0) = M(ρ) 8 M(ρ) 0  t 1−ρ ρ C T H (t) − C T H (0) = S9 (ω, C T H (ω))dω, S (t, C T H (t)) + M(ρ) 9 M(ρ) 0  t 1−ρ ρ S10 (ω, E H (ω))dω, S (t, E H (t)) + E H (t) − E H (0) = M(ρ) 10 M(ρ) 0  t 1−ρ ρ FT (t) − FT (0) = S11 (ω, FT (ω))dω. S (t, FT (t)) + M(ρ) 11 M(ρ) 0

(39)

 t n+1 1−ρ ρ S1 (t, A(t))dt, S1 (tn , A(tn )) + M(ρ) M(ρ) 0  t n+1 1−ρ ρ BTn+1 − BT (0) = S2 (t, BT (t))dt, S (tn , BT (tn )) + M(ρ) 2 M(ρ) 0  t n+1 1−ρ ρ S3 (t, C T (t))dt, S (tn , C T (tn )) + C Tn+1 − C T (0) = M(ρ) 3 M(ρ) 0  t n+1 1−ρ ρ E n+1 − E(0) = S4 (t, E(t))dt, S (tn , E(tn )) + M(ρ) 4 M(ρ) 0  t n+1 1−ρ ρ S5 (t, C H (t))dt, S (tn , C H (tn )) + C Hn+1 − C H (0) = M(ρ) 5 M(ρ) 0  t n+1 1−ρ ρ Fn+1 − F(0) = S6 (t, F(t))dt, S (tn , F(tn )) + M(ρ) 6 M(ρ) 0  t n+1 1−ρ ρ G Hn+1 − G H (0) = S7 (t, G H (t))dt, S (tn , G H (tn )) + M(ρ) 7 M(ρ) 0  t n+1 1−ρ ρ BT Hn+1 − BT H (0) = S (tn , BT H (tn )) + S8 (t, BT H (t))dt, M(ρ) 8 M(ρ) 0  t n+1 1−ρ ρ C T Hn+1 − C T H (0) = S9 (t, C T H (t))dt, S (tn , C T H (tn )) + M(ρ) 9 M(ρ) 0

(40)

A(t) − A(0) =

Thus,

An+1 − A(0) =

 tn+1 1−ρ ρ S10 (tn , E H (tn )) + S10 (t, E H (t))dt, M(ρ) M(ρ) 0  tn+1 1−ρ ρ S11 (tn , FT (tn )) + − FT (0) = S11 (t, FT (t))dt, M(ρ) M(ρ) 0

E Hn+1 − E H (0) = FTn+1 and

Analytical and Numerical Solutions of a TB-HIV/AIDS Co-infection Model ...  tn 1−ρ ρ S (t , A(tn−1 )) + S1 (t, A(t))dt, M(ρ) 1 n−1 M(ρ) 0  tn 1−ρ ρ BTn − BT (0) = S (t , B (t )) + S2 (t, BT (t))dt, M(ρ) 2 n−1 T n−1 M(ρ) 0  tn 1−ρ ρ , C (t )) + S3 (t, C T (t))dt, S (t C Tn − C T (0) = M(ρ) 3 n−1 T n−1 M(ρ) 0  tn 1−ρ ρ E n − E(0) = , E(tn−1 )) + S4 (t, E(t))dt, S (t M(ρ) 4 n−1 M(ρ) 0  tn 1−ρ ρ S5 (tn−1 , C H (tn−1 )) + C Hn − C H (0) = S5 (t, C H (t))dt, M(ρ) M(ρ) 0  tn 1−ρ ρ , F(tn−1 )) + S6 (t, F(t))dt, S (t Fn − F(0) = M(ρ) 6 n−1 M(ρ) 0  tn 1−ρ ρ G Hn − G H (0) = S7 (t, G H (t))dt, S7 (tn−1 , G H (tn−1 )) + M(ρ) M(ρ) 0  tn 1−ρ ρ ,B (t )) + S8 (t, BT H (t))dt, S (t BT H n − BT H (0) = M(ρ) 8 n−1 T H n−1 M(ρ) 0  tn 1−ρ ρ C T H n − C T H (0) = ,C (t )) + S9 (t, C T H (t))dt, S (t M(ρ) 9 n−1 T H n−1 M(ρ) 0  tn 1−ρ ρ , E (t )) + S1 0(t, E H (t))dt, S (t E Hn − E H (0) = M(ρ) 10 n−1 H n−1 M(ρ) 0  tn 1−ρ ρ FTn − FT (0) = , F (t )) + S11 (t, FT (t))dt. S (t M(ρ) 11 n−1 T n−1 M(ρ) 0

205

An − A(0) =

(41)

When we removing (41) from (40), the following equation system is obtained.

  t n+1 1−ρ ρ S1 (tn , A(tn )) − S1 (tn−1 , A(tn−1 )) + S1 (t, A(t))dt, M(ρ) M(ρ) tn

  t n+1 1−ρ ρ S2 (tn , BT (tn )) − S2 (tn−1 , BT (tn−1 )) + S2 (t, BT (t))dt, BTn+1 − BT (0) = M(ρ) M(ρ) tn

  t n+1 1−ρ ρ S3 (tn , C T (tn )) − S3 (tn−1 , C T (tn−1 )) + S3 (t, C T (t))dt, C Tn+1 − C T (0) = M(ρ) M(ρ) tn 

 t n+1 1−ρ ρ S4 (t, E(t))dt, S4 (tn , E(tn )) − S4 (tn−1 , E(tn−1 )) + E n+1 − E(0) = M(ρ) M(ρ) tn 

 t n+1 1−ρ ρ S5 (t, C H (t))dt, C Hn+1 − C H (0) = S5 (tn , C H (tn )) − S5 (tn−1 , C H (tn−1 )) + M(ρ) M(ρ) tn 

 t n+1 1−ρ ρ S6 (t, F(t))dt, S6 (tn , F(tn )) − S6 (tn−1 , F(tn−1 )) + Fn+1 − F(0) = M(ρ) M(ρ) tn 

 t n+1 1−ρ ρ S7 (t, G H (t))dt, S7 (tn , G H (tn )) − S7 (tn−1 , G H (tn−1 )) + G Hn+1 − G H (0) = M(ρ) M(ρ) tn An+1 − A(0) =

(42)



 t n+1 1−ρ ρ S8 (t, BT H (t))dt, S8 (tn , BT H (tn )) − S8 (tn−1 , BT H (tn−1 )) + M(ρ) M(ρ) tn 

 t n+1 1−ρ ρ S9 (t, C T H (t))dt, S9 (tn , C T H (tn )) − S9 (tn−1 , C T H (tn−1 )) + C T H n+1 − C T H (0) = M(ρ) M(ρ) tn 

 t n+1 1−ρ ρ S10 (t, E H (t))dt, S10 (tn , E H (tn )) − S10 (tn−1 , E H (tn−1 )) + E Hn+1 − E H (0) = M(ρ) M(ρ) tn 

 t n+1 1−ρ ρ S11 (t, FT (t))dt, S11 (tn , FT (tn )) − S11 (tn−1 , FT (tn−1 )) + FTn+1 − FT (0) = M(ρ) M(ρ) tn BT H n+1 − BT H (0) =

206

M. A. Dokuyucu and H. Dutta

where 

tn+1

 S1 (t, A(t))dt =

tn

tn+1

tn

S1 (tn , An ) S1 (tn−1 , An−1 ) (t − tn−1 ) − (t − tn ) h h



h 3h S2 (tn , BTn ) − S2 (tn−1 , BTn−1 ), = 2 2   tn+1  tn+1 S2 (tn−1 , BTn−1 ) S2 (tn , BTn ) S2 (t, BT (t))dt = (t − tn−1 ) − (t − tn ) h h tn tn h 3h S2 (tn , BTn ) − S2 (tn−1 , BTn−1 ), 2 2   tn+1  tn+1 S3 (tn−1 , C Tn−1 ) S3 (tn , C Tn ) S3 (t, C T (t))dt = (t − tn−1 ) − (t − tn ) h h tn tn =

h 3h S3 (tn , C Tn ) − S3 (tn−1 , C Tn−1 ), 2 2   tn+1 S4 (tn , E n ) S4 (tn−1 , E n−1 ) (t − tn−1 ) − (t − tn ) S4 (t, E(t))dt = h h tn =



tn+1 tn

h 3h S4 (tn , E Tn ) − S4 (tn−1 , E Tn−1 ), 2 2   tn+1 S5 (tn−1 , C Hn−1 ) S5 (tn , C Hn ) (t − tn−1 ) − (t − tn ) S5 (t, C H (t))dt = h h tn =



tn+1 tn

h 3h S5 (tn , C Hn ) − S5 (tn−1 , C Hn−1 ), 2 2   tn+1  tn+1 S6 (tn , Fn ) S6 (tn−1 , Fn−1 ) S6 (t, F(t))dt = (t − tn−1 ) − (t − tn ) h h tn tn

(43)

=

h 3h S6 (tn , FTn ) − S6 (tn−1 , FTn−1 ), 2 2

  tn+1 tn+1 S (t , G ) S7 (tn−1 , G Hn−1 ) 7 n Hn S7 (t, G H (t))dt = (t − tn−1 ) − (t − tn ) h h tn tn =



h 3h S7 (tn , G Hn ) − S7 (tn−1 , G Hn−1 ), 2 2   tn+1  tn+1 S8 (tn−1 , BT H n−1 ) S8 (tn , BT H n ) S8 (t, BT H (t))dt = (t − tn−1 ) − (t − tn ) h h tn tn =

h 3h S8 (tn , BT H n ) − S8 (tn−1 , BT H n−1 ), 2 2   tn+1 S9 (tn−1 , C T H n−1 ) S9 (tn , C T H n ) (t − tn−1 ) − (t − tn ) S9 (t, C T H (t))dt = h h tn =



tn+1

tn

=

 t n+1 tn

S10 (t, E H (t))dt =

h 3h S9 (tn , C T H n ) − S9 (tn−1 , C T H n−1 ), 2 2

  t S10 (tn−1 , E Hn−1 ) n+1 S10 (tn , E H ) n (t − tn−1 ) − (t − tn ) h h tn

h 3h S10 (tn , E Hn ) − S10 (tn−1 , E Hn−1 ), 2 2

  t  t S11 (tn−1 , FTn−1 ) n+1 n+1 S11 (tn , FT ) n (t − tn−1 ) − (t − tn ) S11 (t, FT (t))dt = h h tn tn =

=

h 3h S11 (tn , FTn ) − S11 (tn−1 , FTn−1 ). 2 2

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207

Therefore, 

1−ρ 3ρh ρh S1 (tn , An ) − S1 (tn−1 , An−1 ) + S1 (tn , An ) − S1 (tn−1 , An−1 ), M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh S2 (tn , BTn ) − S2 (tn−1 , BTn−1 ), BTn+1 − BTn = S2 (tn , BTn ) − S2 (tn−1 , BTn−1 ) + M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh S3 (tn , C Tn ) − S3 (tn−1 , C Tn−1 ), C Tn+1 − C Tn = S3 (tn , C Tn ) − S3 (tn−1 , C Tn−1 ) + M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh E n+1 − E n = S4 (tn , E n ) − S4 (tn−1 , E n−1 ) + S4 (tn , E n ) − S4 (tn−1 , E n−1 ), M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh C Hn+1 − C Hn = S5 (tn , C Hn ) − S5 (tn−1 , C Hn−1 ) + S5 (tn , C Hn ) − S5 (tn−1 , C Hn−1 ), M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh Fn+1 − Fn = S6 (tn , Fn ) − S6 (tn−1 , Fn−1 ) + S6 (tn , Fn ) − S6 (tn−1 , Fn−1 ), M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh G Hn+1 − G Hn = S7 (tn , G Hn ) − S7 (tn−1 , G Hn−1 ) + S7 (tn , G Hn ) − S7 (tn−1 , G Hn−1 ), M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh S8 (tn , BT H n ) − S8 (tn−1 , BT H n−1 ), BT H n+1 − BT H n = S8 (tn , BT H n ) − S8 (tn−1 , BT H n−1 ) + M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh C T H n+1 − C T H n = S9 (tn , C T H n ) − S9 (tn−1 , C T H n−1 ) + S9 (tn , C T H n ) − S9 (tn−1 , C T H n−1 ), M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh E Hn+1 − E Hn = S10 (tn , E Hn ) − S10 (tn−1 , E Hn−1 ) + S10 (tn , E Hn ) − S10 (tn−1 , E Hn−1 ), M(ρ) 2M(ρ) 2M(ρ) 

1−ρ 3ρh ρh FTn+1 − FTn = S11 (tn , FTn ) − S11 (tn−1 , FTn−1 ) + S11 (tn , FTn ) − S11 (tn−1 , FTn−1 ), M(ρ) 2M(ρ) 2M(ρ) An+1 − An =

(44)

which implies that, 

   1−ρ 1−ρ 3ρh ρh + S1 (tn , An ) + + S1 (tn−1 , An−1 ), M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh S2 (tn , BTn ) + S2 (tn−1 , BTn−1 ), BTn+1 = BTn + + + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh C Tn+1 = C Tn + + S3 (tn , C Tn ) + + S3 (tn−1 , C Tn−1 ), M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh E n+1 = E n + + S4 (tn , E n ) + + S4 (tn−1 , E n−1 ), M(ρ) 2M(ρ) M(ρ) 2M(ρ) An+1 = An +

    3ρh ρh 1−ρ 1−ρ + S5 (tn , C Hn ) + + S5 (tn−1 , C Hn−1 ), C Hn+1 = C Hn + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh S6 (tn , Fn ) + S6 (tn−1 , Fn−1 ), Fn+1 = Fn + + + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     3ρh ρh 1−ρ 1−ρ + S7 (tn , G Hn ) + + S7 (tn−1 , G Hn−1 ), G Hn+1 = G Hn + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh BT H n+1 = BT H n + + + S8 (tn , BT H n ) + S8 (tn−1 , BT H n−1 ), M(ρ) 2M(ρ) M(ρ) 2M(ρ)     3ρh ρh 1−ρ 1−ρ C T H n+1 = C T H n + + S9 (tn , C T H n ) + + S9 (tn−1 , C T H n−1 ), M(ρ) 2M(ρ) M(ρ) 2M(ρ)     3ρh ρh 1−ρ 1−ρ E Hn+1 = E Hn + + S10 (tn , E Hn ) + + S10 (tn−1 , E Hn−1 ), M(ρ) 2M(ρ) M(ρ) 2M(ρ)     3ρh ρh 1−ρ 1−ρ FTn+1 = FTn + + S11 (tn , FTn ) + + S11 (tn−1 , FTn−1 ), M(ρ) 2M(ρ) M(ρ) 2M(ρ)

(45)

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According to Theorem (4.1), we get, 

   1−ρ 1−ρ 3ρh ρh + S1 (tn , An ) + + S1 (tn−1 , An−1 ) + 1 Rρn , M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh S2 (tn , BTn ) + S2 (tn−1 , BTn−1 ) + 2 Rρn , BTn+1 = BTn + + + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh C Tn+1 = C Tn + + S3 (tn , C Tn ) + + S3 (tn−1 , C Tn−1 ) + 3 Rρn , M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh + S4 (tn , E n ) + + S4 (tn−1 , E n−1 ) + 4 Rρn , E n+1 = E n + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh S5 (tn , C Hn ) + S5 (tn−1 , C Hn−1 ) + 5 Rρn , C Hn+1 = C Hn + + + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh Fn+1 = Fn + + S6 (tn , Fn ) + + S6 (tn−1 , Fn−1 ) + 6 Rρn , M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh + S7 (tn , G Hn ) + + S7 (tn−1 , G Hn−1 ) + 7 Rρn , G Hn+1 = G Hn + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh S8 (tn , BT H n ) + S8 (tn−1 , BT H n−1 ) + 8 Rρn , BT H n+1 = BT H n + + + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh C T H n+1 = C T H n + + S9 (tn , C T H n ) + + S9 (tn−1 , C T H n−1 ) + 9 Rρn , M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh + S10 (tn , E Hn ) + + S10 (tn−1 , E Hn−1 ) + 10 Rρn , E Hn+1 = E Hn + M(ρ) 2M(ρ) M(ρ) 2M(ρ)     1−ρ 1−ρ 3ρh ρh FTn+1 = FTn + + S11 tn , FTn ) + + S11 (tn−1 , FTn−1 ) + 11 Rρn , M(ρ) 2M(ρ) M(ρ) 2M(ρ) An+1 = An +

(46) where ||i Rρn ||∞ 1, the persistent positive equilibrium E 1 is asymptotically stable. iii) If R = 1, E 1 = E 0 . Remark 2.1. Note that for τ = 0 in Saulo et al. [12], Theorem 2.1 states that the equilibrium point E 0 is asymptotically stable if ϕ(u) < 1. Then, the quantity ϕ(0) corresponds to R in our situation and we have the same results of stability.

3 Delay Model Without Diffusion 3.1 Local Stability of Mosquito-Free Equilibrium The linearized system of system (1) around the mosquito-free equilibrium point E 0 = (0, 0, 0) is given by: dX = L 0 X (t) + L τ X (t − τ ) dt ⎛ ⎛ ⎞ ⎞ 0 0 α −μ F 0 0 ⎠ 0 where L 0 = ⎝0 −(γ + μ E ) and L τ = ⎝ βσ 0 0⎠ 0 γ −(α + μ P ) 0 00 and the associated characteristic equation is as follows P(λ) + Q(λ)e−λτ = 0 where P(λ) = λ3 + (α + μ P + γ + μ E )λ2 + (α + μ P )(γ + μ E )λ, Q(λ) = μ F λ2 + μ F (α + μ P + γ + μ E )λ + μ F (α + μ P )(γ + μ E ) − βγ ασ. Define F by F(y) = |P(i y)|2 − |Q(i y)|2 , = Y 3 + a1 Y 2 + a2 Y + a3 where Y = y 2 and a1 = (α + μ P )2 + (γ + μ E )2 − μ2F ,  a2 = (α + μ P )2 (γ + μ E )2 − μ2F (α + μ P )2 + (γ + μ E )2 − 2μ F βγ ασ,  2 a3 = − μ F (α + μ P )(γ + μ E ) − βγ ασ . Hence F is continue and F(0) = a3 < 0 and lim F(Y ) = +∞, then F crosses xY →+∞

axis in some positive value. Then, E 0 is asymptotically stable for time delay smaller

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than a critical value and unstable otherwise (see [3, 5, 14]). As we are interested to the coexistence of all sub-populations of mosquitoes we will study the qualitative behaviour and bifurcation at the persistent equilibrium.

3.2 Stability of Persistent Positive Equilibrium and Hopf Bifurcation Linearizing system (1) around the persistent positive equilibrium E 1 = (F ∗ , E ∗ , P ∗ ), we obtain the following system dX = J0 X (t) + Jτ X (t − τ ) dt where ⎛

0

⎜ J0 = ⎝0 − 0



0

βσ F ∗ k



⎞

∗ ⎟ − βσkF ⎠ + γ + μE γ −(α + μ P )

⎛

and

α

 −μ F ∗ ∗ 0 Jτ = ⎝βσ 1 − E +P 0 k 0 0

⎞ 0 0⎠ . 0

(4)

(5)

The associated characteristic equation is given by (λ) = λ3 + α1 λ2 + α2 λ + (μ F λ2 + β1 λ + β2 )e−λτ = 0

(6)

where βσ F ∗ + γ + μE + α + μP , k βσ F ∗ (α + μ P + γ ) + (γ + μ E )(α + μ P ), α2 = k  βσ F ∗ + γ + μE + α + μP , β1 = μ F k   βγ σ F ∗ βσ F ∗ E∗ + μF − αγ (γ + μ E ) ∗ . β2 = μ F (α + μ P ) γ + μ E + k k F α1 =

Since the persistent positive equilibrium E 1 is asymptotically stable for τ = 0 (see Proposition 2.1) and by the continuity property, it’s still asymptotically stable for small τ > 0 (see [3, 5, 14]).

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To obtain the switch of stability, one needs to find a purely imaginary root for some critical value of τ . Let iω (ω > 0) be a root of Eq. (6), then we have − iω3 − α1 ω2 + iα2 ω + (−μ F ω2 + iβ1 ω + β2 )(cos ωτ − i sin ωτ ) = 0.

(7)

Separating the real and imaginary parts, we find 

α1 ω2 = −μ F ω2 cos ωτ + β1 ω sin ωτ + β2 cos ωτ, ω3 − α2 ω = μ F ω2 sin ωτ + β1 ω cos ωτ − β2 sin ωτ.

(8)

By computation, we obtain ω6 + (α12 − 2α2 − μ2F )ω4 + (α22 + 2μ F β2 + β12 )ω2 − β22 = 0.

(9)

Let z = ω2 , Eq. (9) becomes h(z) = z 3 + (α12 − 2α2 − μ2F )z 2 + (α22 + 2μ F β2 + β12 )z − β22 = 0.

(10)

As h(0) = −β22 < 0 and lim h(z) = +∞, Eq. (10) has at least one positive root, z→∞

denoted by z 0 . Consequently, Eq. (9) has at least one positive root, denoted by ω0 . This implies that the characteristic equation (6) has a pair of purely imaginary roots ±iω0 at the critical values of time delay τ = τ j , where   2 jπ 1 β1 ω04 − α1 μ F ω03 + (α1 β2 − β1 α2 )ω02 + arccos ; τj = 4 2 2 2 ω0 ω0 μ F ω0 + (β1 − 2μ F β2 )ω0 + β2

j = 0, 1, 2, .... (11)

Lemma 3.1 [13]. Consider the transcendental equation: Q(λ) = λn + p10 λn−1 + 0 1 λ + pn0 + e−λτ1 ( p11 λn−1 + ... + pn−1 λ + pn1 ) + ... + e−λτs ( p1s λn−1 + ... + ... + pn−1 s k s pn−1 λ + pn ), where τk > 0 and p j , k = 1, ..., s; j = 1, ..., n are constants. As (τk )k=1,...,s vary, the sum of the order of the zeros of the polynomial Q on the right half plane can change only if a zero appears on or crosses the imaginary axis. Using Lemma 3.1, we obtain the following lemma. Lemma 3.2 [14]. As −β22 < 0, all roots with positive real parts of Eq. (6) has the same sum to those of the same equation without delay for τ ∈ [0, τ0 ). From Lemma 3.2, we have the following result. Corollary 3.1. i) The mosquito-free equilibrium point E 0 is unstable for all τ > 0. ii) If R > 1, the persistent positive equilibrium E 1 is asymptotically stable for τ < τ0 .

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Let λ(τ ) = η(τ ) + iω(τ ) be the eigenvalue of Eq. (6) such that η(τ0 ) = 0 and ω(τ0 ) = ω0 . Then, one needs to verify the transversality condition d d Reλ(τ )|τ =τ0 = η(τ )|τ =τ0 = 0. dτ dτ Lemma 3.3. Let z 0 = ω02 and β2 = μ F ω02 and h (z 0 ) = 0, where h is given by (10). Then     d(Reλ) d(Reλ) = 0 and sign = sign(h (z 0 )) dτ dτ |τ =τ j |τ =τ j From Lemma 3.3 and Corollary 3.1, we deduce the following theorem. Theorem 3.4. Suppose R > 1 and h (z 0 ) = 0, then, a Hopf bifurcation occurs at the persistent positive equilibrium E 1 when τ = τ j , j = 0, 1, 2, .....

4 Delay Model with Diffusion 4.1 Stability of Mosquito-Free Equilibrium Note that the operator − with the homogeneous Neumann boundary condition on

has the eigenvalues 0 = μ0 < μ1 < μ2 < ... < μn < ... and lim μn = ∞. n→+∞

Let S(μn ) be the eigenspace corresponding to μn with multiplicity m n ≥ 1. Let n j (1 ≤ j ≤ m n ) be the normalized eigenfunctions corresponding to μn . Then, the set { ji : i ≥ 0, 1 ≤ j ≤ m n } forms a complete orthonormal basis [6, 11]. The linearized system of (2) at the mosquito-free equilibrium E 0 can be expressed by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ F F F Ft ⎝ E t ⎠ = L ⎝ E ⎠ = D ⎝E ⎠ + J0 ⎝ E ⎠ , (12) P P P Pt ⎛ ⎞ d00 D = ⎝ 0 0 0⎠ , 000

⎛

⎞ −μ F 0 α ⎠. 0 J0 = ⎝ βσ −(γ + μ E ) 0 γ −(α + μ P )

Let X n j = {c.n j : c ∈ R3 } where {n j : 1 ≤ j ≤ Dim[S(μn )]} is an orthonormal basis of S(μn ).  Dim[S(μn )]  X n j is invariFor n ≥ 0, it can be observed that X = ∞ n=1 X n , X n = j=1 ant under the operator L and λ is an eigenvalue of L if and only if λ is an eigenvalue of the matrix Jn = −μn D + J0 for some n ≥ 0.

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Therefore, the stability is translated into the distribution of roots of the following characteristic equation : λ3 + a2n λ2 + a1n λ + a0n = 0,

(13)

where a2n = γ + μ E + α + μ P + μn d + μ F , a1n = (μn d + μ F )(α + μ P + γ + μ E ) + (γ + μ E )(α + μ P ), a0n = μn d(γ + μ E )(α + μ P ) + μ F (γ + μ E )(α + μ P )(1 − R). It is clear that all coefficients a2n , a1n , a0n are positives when R ≤ 1. Thus, λ = 0 is not a root of (13). Moreover, the characteristic equation (13) does not change sign and from Descartes method we deduce that all roots of Eq. (13) have negative real parts for any n ∈ N with R ≤ 1 and we obtain the following lemma. Lemma 4.1. If R ≤ 1, then the mosquito-free equilibrium E 0 is locally asymptotically stable without time delay.

4.2 Stability of the Persistent Equilibrium 4.2.1

Without Delay

In this section, we study the locally asymptotical stability of the persistent positive equilibrium E 1 of system (2). Linearizing system (2) at E 1 , we obtain the following linearized system ⎛

⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Ft F F F ⎝ E t ⎠ = L ⎝ E ⎠ = D ⎝E ⎠ + J1 ⎝ E ⎠ , P P P Pt

(14)

⎛ ⎞ d00 D = ⎝ 0 0 0⎠ , 000

where

⎛

−μ F

∗ ∗ ⎜  J1 = ⎝βσ 1 − E +P − k 0



0

βσ F ∗ k



α

⎞

∗ ⎟ − βσkF ⎠ . + γ + μE γ −(α + μ P )

The corresponding characteristic equation is given by λ3 + b2n λ2 + b1n λ + b0n = 0,

(15)

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where βσ F ∗ + γ + μ E + α + μ P + μn d + μ F , k   βσ F ∗ + γ + μE = (α + μ P ) k   βσ γ F ∗ βσ F ∗ + γ + μE + α + μP + , + (μn d + μ F ) k k βσ F ∗ + μn d(α + μ P )(γ + μ E ). = (μn d + μ F )(α + μ P + 1) k

b2n = b1n

b0n

By the same way as in the previous paragraph, all coefficients b2n , b1n , b0n are positives, we deduce the following lemma. Lemma 4.2. Let R > 1 and τ = 0, – the persistent positive equilibrium E 1 is locally asymptotically stable, – the mosquito-free equilibrium E 0 is unstable.

4.2.2

With Delay

The linearization of system (2) at the trivial steady state E 0 can be expressed by: ⎛

⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Ft F F F F(t − τ, x, y) ⎝ E t ⎠ = L ⎝ E ⎠ = D ⎝E ⎠ + J11 ⎝ E ⎠ + J22 ⎝ E(t − τ, x, y)⎠ , P P P P(t − τ, x, y) Pt

(16)

with ⎛ ⎞ d00 D = ⎝ 0 0 0⎠ , 000 ⎛

J11

⎞ 0 0 α ⎠, 0 = ⎝0 −(γ + μ E ) 0 γ −(α + μ P )

J22

⎛ ⎞ −μ F e−λτ 0 0 = ⎝ βσ e−λτ 0 0⎠ . 0 00

Note that λ is an eigenvalue of L if and only if λ is an eigenvalue of the matrix Jn = −μn D + J11 + J22 for some n ≥ 0. Therefore, the stability is translated into the distribution of roots of the following characteristic equation:

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R. Yafia and M. A. Aziz Alaoui

λ3 + α2n λ2 + α1n λ + α0n + [β2 λ2 + β1 λ + β0 ]e−λτ = 0,

(17)

where α2n = γ + μ E + α + μ P + μn d, α1n = (γ + μ E )(α + μ P ) + μn d(γ + μ E + α + μ P ), α0n = μn d(γ + μ E )(α + μ P ), β2 = μ F , β1 = μ F (γ + μ E + α + μ P ), β0 = μ F (γ + μ E )(α + μ P )(1 − R). It is known that the mosquito-free equilibrium is locally asymptotically stable if and only if all roots of Eq. (17) have negative real parts for every n ∈ N (see [5] and [3]). Conversely, the mosquito-free equilibrium is unstable if there exists a n ∈ N, such that the Eq. (17) has at least one root with a positive real part. Assume that λ = iω(ω > 0) is a root of (17), then we have − iω3 − α2n ω2 + iα1n ω + α0n + (−β2 ω2 + iβ1 ω + β0 )(cos ωτ − i sin ωτ ) = 0. (18) Separating the real and imaginary parts, we have 

−α2n ω2 + α0n = (β2 ω2 − β0 ) cos ωτ − β1 ω sin ωτ, −ω3 + α1n ω = (β0 − β2 ω2 ) sin ωτ − β1 ω cos ωτ.

(19)

Adding up the squares of both the equations, we obtain ω6 + A2n ω4 + A1n ω2 + A0n = 0.

(20)

Let z = ω2 , Eq. (20) becomes h(z) = z 3 + A2n z 2 + A1n z + A0n = 0,

(21)

where 2 − 2α1n − β22 , A2n = α2n 2 A1n = α1n − 2α2n α0n + 2β2 β0 − β12 , 2 A0n = α0n − β02 .

For n = 0, we have A00 = −β02 < 0. Hence h is continue and lim h(z) = +∞, z→+∞

then h crosses x-axis in some positive value. As E 0 is asymptotically stable for τ = 0, then it becomes unstable for time delay greater than some critical value (see [5] and [3]).

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5 Occurrence of the Hopf Bifurcation In this section, we study the local asymptotic stability of the persistent positive equilibrium E 1 of system (2) and the occurrence of Hopf bifurcation by considering time delay τ as a bifurcation parameter. By linearizing system (2) at E 1 , we get the following corresponding characteristic equation (22) λ3 + An λ2 + Bn λ + Cn + (Dλ2 + Eλ + F)e−λτ = 0, where An = Bn = Cn = D= E1 = F1 =

βσ F ∗ + γ + μ E + α + μ P + μn d, k ∗ βσ F βσ F ∗ βσ γ F ∗ ( + γ + μ E )(α + μ P ) + μn d( + γ + μE + α + μP ) + , k k k ∗ βσ F μn d (α + μ P + γ ) + μn d(γ + μ E )(α + μ P ), k μF , βσ F ∗ μF ( + γ + μ E + α + μ P ), k ∗ βσ F μF (α + μ P + γ ). k

If λ = iω(ω > 0) is a root of (22), we have − iω3 − An ω2 + i Bn ω + Cn + (−Dω2 + i E 1 ω + F1 )(cos ωτ − i sin ωτ ) = 0. (23) Separating the real and imaginary parts, we have 

−An ω2 + Cn = (Dω2 − F1 ) cos ωτ − E 1 ω sin ωτ, −ω3 + Bn ω = (F1 − Dω2 ) sin ωτ − E 1 ω cos ωτ.

(24)

Adding up the squares of both the equations, we obtain ω6 + η2n ω4 + η1n ω2 + η0n = 0.

(25)

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Let z = ω2 , Eq. (25) becomes g(z) = z 3 + η2n z 2 + η1n z + η0n = 0,

(26)

where η2n = A2n − 2Bn − D 2 , η1n = Bn2 − 2 An Cn + 2D F1 − E 12 , η0n = Cn2 − F12 . To attest the instability of E 1 , we assume n = 0, we have η00 = −F 2 < 0 and g(z) −→ +∞ when z −→ +∞. We conclude that there exists N1 ∈ N0 such that all roots the Eq. (26) are positives and real (noted z n ) for all n ≤ N1 and no positive real roots for n ≥ N1 + 1. According to the above analysis, we deduce the following result. Lemma 5.1. Assume that R > 1 holds, then (22) has a pair of purely imaginary roots √ ±iωn with ωn = z n for each n ≤ N1 and from (24), we obtain τn j = τn0 +

2 jπ ; ωn

j = 0, 1, 2, ....

(27)

and τn0

1 = ωn

  (E 1 − D An )ωn4 + (An F1 − Bn E 1 + DCn )ωn2 − F1 Cn arccos . (28) D 2 ωn4 + (E 12 − 2D F1 )ωn2 + F12

Let λ(τ ) = η(τ ) + iω(τ ) be the root of Eq. (22) near τn j such that η(τn j ) = 0 and ω(τn j ) = ωn . Then, one needs to verify the transversality condition d d Reλ(τ )|τ =τn j = η(τ )|τ =τn j = 0. dτ dτ Lemma 5.2 [21]. Let z n = ωn2 and F = Dωn2 and g (z n ) = 0 where g is given by (26). Then 

d Reλ dτ

 |τ =τn j

 = 0 and sign

d Reλ dτ

 |τ =τn j

 = sign g (z n ) .

Let τ ∗ = min {τn j }. Then from Lemma 5.2 and the above analysis, we deduce the 0≤n≤N1

following theorem.

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Theorem 5.3. Suppose R > 1 and g (z n ) = 0, then – the persistent positive equilibrium E 1 is asymptotically stable when τ < τ ∗ and unstable when τ > τ ∗ , – a Hopf bifurcation occurs near E 1 at τ = τn j for 0 ≤ n ≤ N1 and j ∈ {0, 1, 2....}.

6 Stability of Bifurcating Branch In this section, we prove the stability of the bifurcating branch of periodic solutions. The methods used here follows the normal form theory and center manifold theorem (see [9, 18]). Without loss of generality, we denote any one of the critical value τn j , 0 ≤ n ≤ N1 and j ∈ {0, 1, 2....} by τ0 and ωn by ω0 . Let w1 (t) = F(t) − F ∗ (t), w2 (t) = E(t) − E ∗ (t), w3 (t) = P(t) − P ∗ (t), μ = τ − τ0 ; then μ = 0 is the Hopf bifurcation value of (2). Rescale the time by t −→ t/τ to normalize the delay. Define C = C([−1, 0], R3 ); then system (2) can be transformed into a functional differential equation as: (29) w˙ t = Lμ wt + f (μ, wt ) where w(t) = (w1 (t), w2 (t), w3 (t))T ∈ R3 and Lμ : C → R3 and f : R × C → R3 are respectively represented by: ⎞⎛ ⎞ ⎛ 0 α φ1 (0) −dμn   ∗ ∗ ⎟⎜ ⎟ ⎜ Lμ φ = τ ⎝ 0 − βσkF + γ + μ E − βσkF ⎠ ⎝φ2 (0)⎠ φ3 (0) 0 γ −(α + μ P ) ⎛

⎞⎛ ⎞ φ1 (−1)  −μ FE ∗ +P ∗ 0 0 +τ ⎝βσ 1 − k 0 0⎠ ⎝φ2 (−1)⎠ , φ3 (−1) 0 00 and

⎞   0 ⎜ φ2 (0) + φ3 (0) ⎟ ⎟, f (μ, ) = τ ⎜ ⎠ ⎝βσ φ1 (−1) k 0 ⎛

where φ = (φ1 , φ2 , φ3 )T ∈ C. By the Riesz representation Theorem, there exists a 3 × 3 matrix η(θ, μ), whose elements are of bounded variation functions, such that  Lμ φ =

0

−1

[dη(θ, μ)]φ(θ ),

f or φ ∈ C.

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Choosing ⎛ ⎞ −dμn  0 α  ∗ ∗ ⎜ ⎟ η(θ, μ) = τ ⎝ 0 − βσkF + γ + μ E − βσkF ⎠ δ(θ ) 0 γ −(α + μ P ) ⎛

 −μ F ∗ ∗ 0 +τ ⎝βσ 1 − E +P 0 k 0 0

⎞ 0 0⎠ δ(θ + 1) 0

where δ is a Dirac delta function. For φ ∈ C([−1, 0], R3 ), define 

, −1 dη(s, μ)φ(s) = Lμ φ,

f or θ ∈ [−1, 0), f or θ = 0,

dφ(θ)

A(μ)φ =  dθ 0



and R(μ)φ =

f or θ ∈ [−1, 0), f or θ = 0.

0, f (μ, φ),

System (29) is equivalent to w˙ t = A(μ)wt + R(μ)wt ,

(30)

where wt (θ ) = w(t + θ ) for θ ∈ [−1, 0]. For ψ ∈ C([−1, 0], (R3 )∗ ), the adjoint operator A∗ of A = A(0) is defined as 

− dψ(s) A ψ(s) =  0 ds T −1 dη (t, 0)ψ(−t)

s ∈ (0, 1], s = 0,

∗

and a bilinear product defined as ¯ < ψ(s), φ(θ ) >= ψ(0).φ(0) −



0 θ=−1



θ ξ =0

¯ − θ )dη(θ )φ(ξ )dξ, ψ(ξ

where η(θ ) = η(θ, 0). From the discussion in Sect. 5, we know that ±iω0 τ0 are eigenvalues of A(0), and so they are also eigenvalues of A∗ . We assume that q(θ ) = (q1 , q2 , q3 )T eiω0 τ0 θ be the eigenvector of A corresponding to eigenvalue iω0 τ0 . Then, we have Aq(θ ) = iω0 τ0 q(θ ).

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Based on the definition of A, we can get the following linear algebraic equations: ⎛

⎞⎛ ⎞ iω0 + dμn + μ F eiω0 τ0 0 −α q1    iω τ ⎜ ⎟⎜ ⎟ βσ F ∗ βσ F ∗ E ∗ +P ∗ 0 0 e iω0 + + γ + μE ⎝−βσ 1 − k ⎠ ⎝q2 ⎠ = 0 k k 0 −γ iω0 + (α + μ P ) q3 (31) Solving the system (31) and choosing q1 = 1 we get q2 =

iω0 + (α + μ P ) q3 ; γ

q3 =

iω0 + dμn + μ F eiω0 τ0 . α

Analogously, if q ∗ (s) = M(q1∗ , q2∗ , q3∗ )T eiω0 τ0 s be the eigenvector of A∗ corresponding to eigenvalue −iω0 τ0 , we have   ∗ ∗ eiω0 τ0 −iω0 + dμn + μ F eiω0 τ0 −βσ 1 − E +P k   ⎜ ∗ ⎜ 0 −iω0 + βσkF + γ + μ E ⎝ ⎛

βσ F ∗ k

−α

⎞ ⎛ ∗⎞ q1 ⎟ ⎜ ∗⎟ ⎟ ⎜q ⎟ = 0, −γ ⎠⎝ 2⎠ q3∗ −iω0 + (α + μ P ) 0

where q1∗ = 1,

−iω0 + dμn + μ F eiω0 τ0 q2∗ = ,  ∗ ∗ βσ 1 − E +P eiω0 τ0 k

q3∗ =

−iω0 +



βσ F ∗ k

+ γ + μE

γ

 q2∗ .

From < q ∗ (s), q(θ) > = q¯ ∗ (0)q(0) −



0



θ

θ =−1 ξ =0

q¯∗ (ξ − θ)dη(θ)q(ξ )dξ,

¯ q¯∗ , q¯∗ , q¯∗ )(q1 , q2 , q3 )T − = M( 1 2 3



0



θ

θ =−1 ξ =0

q¯∗ (ξ − θ)dη(θ)q(ξ )dξ,

    βσ F ∗ E∗ + P∗ = M¯ q1 q¯1∗ + q2 q¯2∗ + q3 q¯3∗ − q1 q¯2∗ τ0 βσ 1 − e−iω0 τ0 + q3 q¯2∗ τ0 . k k

We can determine M by < q ∗ (s), q(θ ) > = 1. Thus, we can obtain     βσ F ∗ −1 E∗ + P∗ M = q¯1 q1∗ + q¯2 q2∗ + q¯3 q3∗ − q¯1 q2∗ τ0 βσ 1 − . eiω0 τ0 + q¯3 q2∗ τ0 k k

We compute the coordinates to describe the center manifold C0 at μ = 0 (see [9]). Let wt be solution of Eq. (6) when μ = 0. Define z(t) = < q ∗ , wt >,

(32)

W (t, θ ) = wt (θ ) − 2Re{z(t)q(θ )},

(33)

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On the center manifold C0 , we have W (t, θ ) = W (z(t), z¯ (t), θ ) = W20 (θ )

z2 z¯ 2 + W11 (θ )z z¯ + W02 (θ ) + ..., 2 2

(34)

z and z¯ are local coordinates for centre manifold C0 in direction of q ∗ and q¯∗ . Noting that W is also real if wt is real. We are only concerned with the real solutions. For solution wt ∈ C0 of Eq. (6), we get: z˙ (t) = < q ∗ , w˙ t >, = iω0 τ0 z + q¯∗ (0). f (0, W (z, z¯ , 0) + 2Re{zq(0)}), = iω0 τ0 z + q¯∗ (0) f 0 (z, z¯ ). The above equation can be rewritten as z˙ (t) = iω0 τ0 z + g(z, z¯ ), where g(z, z¯ ) = q¯∗ (0). f 0 (z, z¯ ) = g20

z2 z¯ 2 z 2 z¯ + g11 z z¯ + g02 + g21 . 2 2 2

Then, we can obtain wt (θ ) = W (z, z¯ , θ ) + 2Re{zq(θ )}, z2 z¯ 2 ¯ = W20 (θ ) + W11 (θ )z z¯ + W02 (θ ) + zq + z q, 2 2 z2 z¯ 2 = W20 (θ ) + W11 (θ )z z¯ + W02 (θ ) + z(1, q2 , q3 )T eiω0 τ0 θ 2 2 T −iω0 τ0 θ +¯z (1, q¯2 , q¯3 ) e + ....

Substituting the values of f and wt (θ ) into (35), we have: g(z, z¯ ) = q¯∗ (0) f 0 (z, z¯ ) = q¯∗ (0) f 0 (z, wt ), ⎞ ⎛ 0   ⎜ 3 (0) ⎟ ¯ = τ0 M(1, q¯2∗ , q¯2∗ ). ⎝βσ φ1 (−1) φ2 (0)+φ ⎠. k 0

(35)

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Simplifying the above equation and comparing with Eq. (35) we get ⎧ ⎪ g20 ⎪ ⎪ ⎪ ⎪ g ⎪ ⎪ ⎨ 11 g02 ⎪ ⎪ ⎪g21 ⎪ ⎪ ⎪ ⎪ ⎩

∗

q q βσ = 4τ0 M¯ 2 k2 e−iω0 τ0 , q ∗ βσ = 4τ0 M¯ 2 k (q¯2 e−iω0 τ0 + q2 eiω0 τ0 ), q ∗ βσ = 4τ0 M¯ 2 k qe ¯ iω0 τ0 , ∗ q βσ (1) (2) (3) 2q¯2 W20 = τ0 M¯ 2 k (−1) + (W20 (0) + W20 (0))eiω0 τ0  (1) (2) (3) +8q2 W11 (−1) + 2(W11 (0) + W11 (0))eiω0 τ0

(36)

Since g21 depends on W20 (θ ) and W11 (θ ), we should also compute the values of W20 (θ ) and W11 (θ ). Following the procedures in [16, 19], we have W20 (θ ) =

ig20 i g¯ 02 −iω0 τ0 θ q(0)eiω0 τ0 θ + q(0)e ¯ + E 1 e2iω0 τ0 θ ω0 τ0 3ω0 τ0

and W11 (θ ) =

ig11 i g¯ 11 −iω0 τ0 θ q(0)eiω0 τ0 θ + q(0)e ¯ + E2 , ω0 τ0 ω0 τ0

where E 1 and E 2 are the solutions of the following linear algebraic equations, respectively: ⎛

⎞ 2iω0 + dμn μ F e−iω0 τ0 −α    −iω τ ∗ ⎜ ⎟ βσ F ∗ E ∗ +P ∗ e 0 0 2iω0 + βσkF + γ + μ E ⎝−βσ 1 − k ⎠ E1 k 0 −γ 2iω0 + (α + μ P ) ⎞ ⎛ 0 3 ⎠, = ⎝βσ q2 +q k 0 and

⎛



⎜ ⎝−βσ 1 −

0 E ∗ +P k

0

 ∗

e−iω0 τ0

βσ F ∗ k

μF

−α

⎞

⎛

∗ ⎟ ⎜ + γ + μ E βσkF ⎠ E 2 = ⎝βσ −γ (α + μ P )



⎞  Re{q2 }+Re{q3 } ⎟ ⎠. k 0 0

Referring to [9], the Hopf bifurcating periodic solutions of the system (2) at τ0 on the center manifold are determined by the following formulas   g21 i g02 2 2 g11 g20 − 2 g11 − + , c1 (0) = 2ω0 τ0 3 2 Re{c1 (0)} μ2 = − , Re{λ (τ0 )}

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β2 = 2Re{c1 (0)}, I m{c1 (0)} + μ2 I m{λ (τ0 )} T2 = − . ω0 τ0 Thus, we have the results on the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Proposition 6.1. The direction of the Hopf bifurcation is determined by μ2 : if μ2 > 0 (μ2 < 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ > τ0 (τ < τ0 ). The stability of the bifurcating periodic solutions is determined by β2 : if β2 < 0 (β2 > 0), then the bifurcating periodic solutions are stable (unstable). The period of the bifurcating periodic solutions is determined by T2 : if T2 > 0 (T2 < 0), then the period increases (decreases).

7 Sensitivity Analysis The normalized forward sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter, it is used to discover parameters that have a high impact on R. If the variable is a differentiable function of the parameter, the sensitivity index is then defined using partial derivatives. Definition 7.1 [10]. The normalized forward sensitivity index of a variable f that depends differentiability on a parameter p is defined as: ϒ pf =

∂f p ∂p f

Table 1 The sensitivity indices of the population reproduction number R Parameter Sensitivity index Index at parameters value γ α σ β μF μE μP

μE γ +μ E μP α+μ P

+1 +1 −1 μE − γ +μ E

μP − γ +μ P

+0.1428 +0.366 +1 +1 −1 −0.1428 −0.336

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Surface of R=R( γ , α ) with σ = 4, β = 0.4, μ F = 0.16, μ E =0.15, μ P = 0.01

10 8

R

6 4 2 0 1 0.5

α

0

0

0.2

0.4

0.6

0.8

1

γ

Fig. 1 Surfaces representing the effect of parameters (left γ and α) and (right σ and α) on the variations of the population reproduction number R

To reduce the rate of the reproduction of mosquitoes, we need to know the importance of different factors involved in its production. So we investigate the sensitivity indices of the reproduction number R relative to the parameters involved. The sensitivity indices of the reproduction number R is given in Table 2. These indices allow us to measure the relative change in R with the change in a parameter. Using these indices, we find the parameters that highly effect R, and need to be targeted by intervention strategies (Table 1 and Fig. 1).

8 Numerical Simulations In this section we give some numerical simulation in order to illustrate our theoretical results. The parameters values are estimated in Table 2 (Figs. 2, 3, 4, 5, 6, 7, 8, 9 and 10). Table 2 Parameters estimation Parameter Value γ α σ β μF μE μP k

0.90 0.6 4 0.4 0.05–2 0.15 0.01 500

Reference Assumed Assumed [12] [12] Assumed Assumed Assumed Assumed

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R. Yafia and M. A. Aziz Alaoui Surface of R=R( μ F , α ) σ = 4, γ = 0.90, β = 0.4, μ E =0.15, μ P = 0.01

80

80

60

60

40

40

R

R

Surface of R=R( β , α ) σ = 4, γ = 0.90, μ F = 0.16, μ E =0.15, μ P = 0.01

20

20

0 1

0 1 3 0.5

0.5

2 1

α

0

α

β

0

0

0

0.2

0.4

0.6

0.8

1

μF

Fig. 2 Surfaces representing the effect of parameters (left β and α) and (right μ F and α) on the variations of the population reproduction number R Surface of R=R( μ E , α ) σ = 4, γ = 0.90, β = 0.4, μ F =0.16, μ P = 0.01

Surface of R=R( μ P , α ) σ = 4, γ = 0.90, β = 0.4, μ F =0.16, μ E = 0.15

10

10

8

8

6

R

R

6

4

4

2

2

0 1

0 1 0.5

α

0

0

0.2

0.4

0.6

0.8

1 0.5

α

μE

0

0

0.2

0.4

0.6

0.8

1

μP

Fig. 3 Surfaces representing the effect of parameters (left μ E and α) and (right μ P and α) on the variations of the population reproduction number R Surface of R=R( μ P , β ) σ = 4, γ = 0.90, μ F = 0.4, α =0.60.16, μ E = 0.15

Surface of R=R( μ P , μ F ) σ = 4, γ = 0.90, β = 0.4, α =0.60.16, μ E = 0.15

25

80

20

60

R

R

15 40

10 20

5 0 1

0 1 0.5

μF

0

0

0.2

0.4

μP

0.6

0.8

1 0.5

β

0

0

0.2

0.4

0.6

0.8

1

μP

Fig. 4 Surfaces representing the effect of parameters (left μ P and μ F ) and (right μ P and β) on the variations of the population reproduction number R

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Surface of R=R( μ P , γ ) σ = 4, β = 0.4, μ F = 0.4, α =0.60.16, μ E = 0.15

Fig. 5 Surface representing the effect of parameters μ P and γ on the variations of the population reproduction number R

10 8

R

6 4 2 0 1 0.5

γ

0

0.2

0

0.6

0.4

0.8

1

μP

1800 F E P

1600

280

1400 260 240

1000

P

F,E,P

1200

220 200

800

180

600

160 200

400

2000 200 0

1500

150

1000 500

0

50

100

150

200

250

300

350

400

450

E

500

100

0

F

time t

Fig. 6 Periodic solution bifurcated from the steady state E 1 for τ = τ0 = 10.35 in (t, F E P) plane (left) and in (F, E, P) space (right) 1800 F E P

1600

300

1200

250

1000

200

P

F,E,P

1400

800

150

600 100 200

400

2000

150

200

1500 1000

100 0

0

100

200

300

400

500

600

700

800

900

1000

E

500 50

0

F

time t

Fig. 7 Periodic solution bifurcated from the steady state E 1 for τ = τ0 +  = 10.85 with  = 0.5 in (t, F E P) plane (left) and in (F, E, P) space (right)

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1800 F E P

1600 1400

300 250

1200

200 P

F,E,P

1000

150

800 100 600

50

400

0 200

200

100

0

0 -200

0

100

200

300

400

500

600

700

800

900

1000

E

-100

-500

0

500

1000

1500

2000

F

time t

Fig. 8 Chaotic solution bifurcated from the steady state E 1 for τ = τ0 +  = 11.15 with  = 0.8 in (t, F E P) plane (left) and in (F, E, P) space (right)

Fig. 9 Periodic solutions for τ = τ0 = 60, μ F = 0.05, d = 0.05, and R = 26.97 > 1 with respect to varying time space variables (above) and varying time and fixed space variable (below)

Fig. 10 Chaotic solutions for τ = 70.5 > τ0 , μ F = 0.05, d = 0.05, and R = 26.97 > 1 with respect to varying time space variables (above) and varying time and fixed space variable (below)

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References 1. Aghriche, A., Yafia, R., Aziz Alaoui, M.A., Tridane, A.: Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes. Int. J. Bifurcat. Chaos 29(13), 1950189 (2019) 2. Aghriche, A., Yafia, R., Aziz Alaoui, M.A., Tridane, A., Rihan, A.F.: Oscillations induced by quiescent adult female in a reaction diffusion model of wild aedes aegypti mosquitoes. DCDSS J. (to appear). https://doi.org/10.3934/dcdss.2020194 3. Boese, F.G.: Stability with respect to the delay: on a paper of K. L. Cooke and P. van den Driessche. J. Math. Anal. Appl. 228(2), 293–321 (1998) 4. Cauchemez, S., Ledrans, M., Poletto, C., Quenel, P., De Valk, H., Colizza, V., Boelle, P.Y.: Local and regional spread of chikungunya fever in the Americas. Euro Surveill Biomet. 19, 20854 (2014) 5. Cooke, K.L., van den Driessche, P.: On the zeroes of some transcendental equations. Funkcial. Ekvac. 29, 77–90 (1986) 6. Faria, T.: Stability and bifurcation for a delayed predator-prey model and the effect of diffusion. J. Math. Anal. Appl. 254(2), 433–463 (2001) 7. Fauci, A.S., Morens, D.M.: Zika virus in the Americas - yet another arbovirus threat. New Eng. J. Med. 374, 601–604 (2016) 8. Gao, D., Lou, Y., He, D., Porco, T.C., Kuang, Y., Chowell, G., Ruan, S.: Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis. Sci. Rep. 6, 28070 (2016). http://dx.doi.org/10.103 8/srep 28070 9. Hassard, B., Kazarinoff, D., Wan, Y.: Theory and Applications of Hopf Bifurcation, 1st edn. Cambridge University Press, Cambridge (1981) 10. Ngoteya, F.N., Gyekye, Y.N.: Sensitivity analysis of parameters in a competition model. Appl. Comput. Math. 4(5), 363–368 (2015) 11. Ouyang, Q.: Pattern Formation in Reaction Diffusion Systems. Shanghai Scientific and Technological Education Publishing House (2000) 12. Saulo, A.C.T., Bermudez, A.E., Loaiza, A.M.: Controlling aedes aegypti mosquitoes by using ovitraps: a mathematical model. Appl. Math. Sci. 11(23), 1123–1131 (2017) 13. Ruan, S., Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Continuous Discrete Impuls Syst. Ser. A 10, 863–74 (2003) 14. Song, Y., Yuan, S.: Bifurcation analysis in a predator-prey system with time delay. Nonlinear Anal. 7, 265–84 (2006) 15. Takahashi, L.T., Maidana, N.A., Ferreira Jr., W.C., Pulino, P., Yang, H.M.: Mathematical models for the Aedes aegypti dispersal dynamics: travelling waves by wing and wind. Bull. Math. Biol. 67, 509–528 (2005) 16. Tian, C.R., Zhang, L.: Hopf bifurcation analysis in a diffusive food-chain model with time delay. Comput. Math. Appl. 66, 2139–2153 (2013) 17. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 1–2 (2002) 18. Wu, J.H.: Theory and Applications of Partial Functional Differential Equations, 1st edn. Springer, New York (1996) 19. Zhang, Q.Y., Tian, C.R.: Pattern dynamics in a diffusive Rossler model. Nonlinear Dyn. 78, 1489–1501 (2014) 20. Zhang, M., Lin, Z.: A reaction-diffusion-advection model for Aedes aegypti mosquitoes in a time-periodic environment. Nonlinear Anal.: Real World Appl. 46, 219–237 (2019) 21. Zhou, X., Wu, Y., Li, Y., Yao, X.: Stability and Hopf bifurcation analysis on a two-neuron network with discrete and distributed delays. Chaos Solit. Fract. 40, 1493–1505 (2009)

Modeling the Dynamics of Hepatitis B Virus Infection in Presence of Capsids and Immunity Khalid Hattaf and Kalyan Manna

Abstract In this chapter, we propose three generalized systems of differential equations to model the dynamics of hepatitis B virus (HBV) infection in presence of HBV DNA-containing capsids and immunity mediated by cytotoxic T lymphocyte (CTL) cells. The global properties of three proposed models are rigorously investigated. Moreover, many previous studies existing in the literature are improved and generalized. Keywords HBV dynamics · Capsids · Mathematical modeling · Stability · Immunity

1 Introduction Hepatitis B virus (HBV) infection is a hepatological condition leading to critical global health concern. The pathogenesis of HBV infection is typically two types in nature: (a) acute illness which lasts for several weeks before eventually getting resolved in majority of cases in presence of dominant immune responses; and (b) chronic illness which can potentially give rise to a range of severe long-term implications such as acute necrotizing vasculitis, liver cirrhosis, membranous glomerulonephritis and hepatocellular carcinoma (HCC) [1–3]. Due to these spectrum of severe long-term complications, about 780 thousand individuals die annually with roughly 240 million chronically infected individuals [4]. Transmission of HBV generally occurs through two different routes: (a) vertical transmission where the virus K. Hattaf Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Derb Ghalef, 20340 Casablanca, Morocco e-mail: [email protected] K. Manna (B) Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_10

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carries to infant from mother at the time of birth; and (b) horizontal transmission where the virus passes through bites, sanitary habits and lesions in case of infants, and through sexual contacts, drug uses and medical procedures etc. in case of adults [4–6]. The pretty complex replication procedure of HBV inside liver has been elucidated in [1, 7]. Being a prototype to the Hepadnaviridae family of viruses, HBV represents an enveloped virus comprising a small (approximately 3.2 kilobase) partially doublestranded (ds) open circular DNA genome which gets transformed into covalently closed circular DNA (cccDNA) during the infection process of hepatocytes [1, 7]. Potentially several copies of cccDNA, which are situated in the nucleus of the infected hepatocyte, abet the pre-genomic and sub-genomic mRNAs production [1, 7]. Then pre-genomic mRNA passes through reverse transcription into HBV DNA by the viral polymerase which eventually leads to the formation of new virions [1, 7]. However, immune responses can play a significant part in controlling the viral load and in determining the nature of the infection (whether acute or chronic) [3, 8]. The resistance against HBV infection progression within a host comprises of the combined effect of innate and adaptive immunity of the host [9]. Generally very early in the infection, innate immunity acts to restrict the infection spread and initiates the activation of an adaptive immune response [9]. The adaptive immune response comprises of a complex web of effector cells such as cytotoxic T lymphocyte (CTL) cells and B cells. CTL cells play a crucial role in clearance of infected hepatocytes through cytolytic and non-cytolytic mechanisms, while B cells prevent re-infection by neutralizing free virus particles [9]. Traditionally several mathematical models have been introduced in order to gain insights into the pathogenesis of HBV infection by using ordinary differential equations (ODEs) and delay differential equations (DDEs). Nowak et al. [10] first introduced a basic ODE-driven model for HBV infection comprising uninfected hepatocytes, infected hepatocytes and free virus particles, and an extension of this basic viral infection model by including CTL immune responses was presented in [11]. Min et al. [12] presented an improved HBV infection model by incorporating standard incidence function for the infection process and they pointed out that basic model leads to unrealistic relationship between susceptibility to infection and number of hepatocytes due to mass action term for the infection process. A delayed version of HBV infection model with standard incidence function was proposed and analyzed in [13] by taking into account the time required for production of the matured virions from the exposed cells. Eikenberry et al. [14] showed the existence of sustained oscillations apart from the other two well-known dynamical behaviors such as convergence to an infection-free steady state and to a chronic steady state for a delayed HBV infection model with logistic hepatocyte growth. Wang et al. [15] performed the global stability analysis for an HBV infection model with standard incidence function and cytokine-mediated cure for infected hepatocytes based on empirical evidences. Hews et al. [16] proposed an improved HBV infection model by incorporating both logistic hepatocyte growth and standard incidence function. They identified a Hopf and a homoclinic bifurcation point, and presented their dependence on the intrinsic growth rate of hepatocytes and the basic reproduction number.

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A similar type of model for HBV infection with logistic hepatocyte growth and mass action term was presented and analyzed in [17]. Yousfi et al. presented an HBV infection model with adaptive immune response and discussed about the role on adaptive immunity in HBV infection in [18]. A viral dynamics model with cure rate and general incidence function was presented in [19]. Xu investigated the dynamics of a viral infection model with Crowley-Martin type functional response in [20]. The effect of the export of precursor CTL cells from thymus and the role of their cytolytic and non-cytolytic mechanisms was studied in [21]. Wang and Tian [22] investigated the global stability properties for an HBV infection model with time delay and CTL immune response. Manna and Chakrabarty [23] modeled HBV infection by incorporating both uninfected hepatocytes and HBV DNA-containing capsids for the first time, and the corresponding delayed version of the model was analyzed in [24]. An extended version of this model by taking into account CTL immune responses was presented in [25]. In recent years, many researchers have been focusing on the spatiotemporal dynamics of the viral infections (such as HBV, HIV etc.). Generally, the mathematical modeling of virus infections assumes that the concerned cells and viruses are well-mixed within their habitat and ignores the individual spatial movements [26], whereas in reality these biological motions can play a significant role in several biological processes [27]. By taking into account the dispersal of viruses through Fickian diffusion and neglecting the same for liver cells, Wang and Wang introduced a spatiotemporal HBV infection model and showed the existence of traveling wave solutions by using the geometric singular perturbation method in [28]. By incorporating the intracellular time delay between infection of a cell and production of new virus particles, Wang et al. presented a delayed version of this model and investigated the effects of delay and diffusion together on the dynamics via numerical simulations in [29]. Gan et al. [30] derived the existence of traveling wave solutions for this delayed spatiotemporal HBV infection model by using cross-iteration method, Schauder’s fixed point theorem and by constructing a pair of upper-lower solutions. The dynamics of similar types of delayed diffusive HBV infection models with saturation response rate and standard incidence rate for infection process have been investigated respectively in [31, 32]. Zhang and Xu [33] studied the global stability of spatially homogeneous steady states and existence of traveling wave solutions in terms of the basic reproduction number for a diffusive HBV infection model with time delay and Beddington-DeAngelis type functional response. Hattaf and Yousfi [34] presented a general procedure for the construction of Lyapunov functions in order to prove global stability of steady states for a class of reaction-diffusion systems in epidemiology and employed the technique for several examples in viral dynamics. They further carried out the global dynamical analysis for a delayed spatiotemporal viral infection model with specific functional response in [35]. In [36], the authors studied the stability property of a diffusive HBV infection model with two delays and general incidence rate. Global dynamics of a spatiotemporal model of HBV infection with nonlinear incidence function and CTL immune response have been explored in [37]. Manna and Chakrabarty [38] studied a diffusive HBV infection model with HBV DNA-containing capsids in terms of global stability and

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non-standard finite difference (NSFD) scheme and the corresponding delayed version of the model has been studied in terms of global dynamics in [39]. Some other related works on NSFD scheme for diffusive viral infection models can be found in [40–43]. Considering both single-strain and multi-strain viral infection models, Wang et al. [44] explored the role of density-dependent diffusion in infection process. A delayed spatiotemporal viral infection model with Crowley-Martin type functional response and CTL immune response has been proposed and investigated for the global stability properties in [45]. Hattaf [46] extended this delayed model by considering two general nonlinear incidence functions that model the infection routes and include the Crowley-Martin functional response. Further, Hattaf and Yousfi [47] explored the global dynamical behaviors of a five-dimensional diffusive viral infection model with general incidence function and adaptive immune response. Also, the global stability results for a delayed reaction-diffusion system of HBV infection with general incidence function and HBV DNA-containing capsids has been presented in [48]. But the authors did not consider the role of immune responses in the infection and also ignored the dispersal of capsids. Recently, we investigated the detailed global spatiotemporal dynamics of a generalized HBV infection model with capsids, adaptive immune response and three discrete delays in [49]. However, we concentrated only on the virus-to-cell mode of transmission, and accordingly, considered general functional form for this mode of transmission in [49]. The main purpose of this chapter is to provide detailed analytical results on global dynamics of the generalized models for within host HBV infection in DDE and reaction-diffusion settings with CTL immune response. The considered models in this study form a system of nested models, and hence, the well ordered exploration about their global dynamics can be helpful for comparison purpose. Also, the consideration of general forms for both the modes of transmission along with biologically feasible restrictions gives rise to a class of HBV infection models which will directly follow our theoretically obtained results. The rest of this chapter is organized in the following manner. In the next section, we propose an ODE-driven generalized model for HBV infection incorporating both the virus-to-cell and cell-to-cell modes of transmission with capsids and CTL immune response. Then the model is extended by incorporating distributed delays and we examine theoretically the well-posedness, feasibility conditions for the existence of different steady states and global dynamics of this model in Sect. 3. In Sect. 4, we further extend the model by considering the diffusivity of capsids, virions and CTL cells, and present the well-posedness and global stability results. Thereafter, we describe the applicability of our theoretically obtained results by considering specific forms for both the modes of transmission in Sect. 5. Finally, we end this chapter with a brief conclusion in Sect. 6.

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2 Dynamics of HBV Infection with ODEs In this section, we propose and analyze a mathematical model for HBV infection with capsids, CTL cells and two modes of transmission, one by virus-to-cell infection through the extracellular space and the other by cell-to-cell transfer involving direct cell-to-cell contact [50, 51]. Other viruses can spread via the last mode such as human immunodeficiency virus (HIV) and hepatitis C virus (HCV) [52–56]. The proposed model is governed by the following nonlinear system of ODEs: ⎧ dU ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ dI ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ dt dD ⎪ dt ⎪ ⎪ dV ⎪ ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎩ dZ dt

    = s − μU (t) − f U (t), I (t), V (t) V (t) − g U (t), I (t) I (t),     = f U (x, t), I (t), V (t) V (t) + g U (t), I (t) I (t) − δ I (t) − p I (t)Z (t), = a I (t) − (β + δ)D(t), = β D(t) − cV (t), = q I (t)Z (t) − σ Z (t),

(1) where U (t), I (t), D(t), V (t) and Z (t) are the densities of the uninfected hepatocytes, infected hepatocytes, capsids, virions and CTL cells at time t, respectively. Uninfected cells are produced at rate s, die at rate μU and become infected by contact with virions at rate f (U, I, V )V and by contact with infected cells at rate g(U, I )I . The parameter δ is the death rate of infected cells and capsids. The parameters a, β and c are, respectively, the production rate of capsids from infected cells, the rate at which the capsids are transmitted to blood which gets converted to virions, and the clearance rate of virions. The parameter p denotes the rate at which infected cells are killed by CTL cells, q is the CTL responsiveness rate, and σ is the death rate of CTL cells. As in [46, 57, 58], the incidence functions f (U, I, V ) and g(U, I ) for both modes of infection are continuously differentiable and satisfy the following hypotheses:  ∂g (U, I ) ≥ 0 or g(U, I ) is a strictly monotone (H0 ) g(0, I) = 0, for all I ≥ 0; ∂U  ∂g increasing function with respect to U when f ≡ 0 and (U, I ) ≤ 0, for all ∂I U ≥ 0 and I ≥ 0. (H1 ) f (0, I, V ) = 0, for all I ≥ 0 and V ≥ 0, (H2 ) f (U, I, V ) is a strictly monotone increasing function with respect to  ∂f (U, I, V ) ≥ 0 when g(U, I ) is a strictly monotone increasing funcU or ∂U  tion with respect to U , for any fixed I ≥ 0 and V ≥ 0, (H3 ) f (U, I, V ) is a monotone decreasing function with respect to I and V . From a biological point of view, these four hypotheses on the incidence functions for the two modes are reasonable and consistent with reality. For more details on the biological meanings of these four hypotheses, we refer the reader to the works

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Fig. 1 Schematic representation of ODE model (1)

[57, 59, 60]. Besides, the schematic representation of ODE model (1) is illustrated in Fig. 1. It is very important to note that the ODE model presented in [25] is a special case of model (1), it suffices to take f (U, I, V ) = kU and g(U, I ) = 0, where k is a positive constant rate describing the infection process by a free virus. In absence of capsids and cell-to-cell transmission, our ODE model can reduced to the generalized virus dynamics model with the immune response studied in [61].

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3 Dynamics of HBV Infection with DDEs To model the delay in the production of productively infected cells and the delays in the production of matured capsids and virons, we propose the following model of DDEs with three distributed delays: ⎧ dU ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ dI ⎪ ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎨ dD ⎪ ⎪ ⎪ dt ⎪ ⎪ dV ⎪ ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎩ dZ dt

    = s − μU (t) − f U (t), I (t), V (t) V (t) − g U (t), I (t) I (t), ∞   = 0 f 1 (τ )e−α1 τ [ f U (t − τ ), I (t − τ ), V (t − τ ) V (t − τ )   +g U (t − τ ), I (t − τ ) I (t − τ )]dτ − δ I (t) − p I (t)Z (t), ∞ = a 0 f 2 (τ )e−α2 τ I (t − τ )dτ − (β + δ)D(t), ∞ = β 0 f 3 (τ )e−α3 τ D(t − τ )dτ − cV (t),

(2)

= q I (t)Z (t) − σ Z (t).

Here, we suppose that a susceptible cell attacked by a virion or by a infected cell infected at time t − τ and it becomes infected at time t, where τ is a random variable with a probability distribution f 1 (τ ). The factor e−α1 τ represents the probability of surviving from time t − τ to time t, where α1 is the death rate for infected but not yet virus-producing cells. Further, we suppose that the time spent in the production of matured capsids is a random variable with a probability distribution f 2 (τ ). The probability of survival of immature capsids is given by e−α2 τ and the average life time of an immature capsid is given by α12 . Similarly, we suppose that the time necessary for the newly produced capsids to become virions is a random variable with a probability distribution f 3 (τ ). The factor e−α3 τ denotes the probability of surviving from time t − τ to time t, where α13 is the average life time of an immature virion. The three probability distribution  ∞ functions f 1 (τ ), f 2 (τ ) and f 3 (τ ) are supposed to be nonnegative and satisfy 0 f i (τ )dτ = 1 for all i ∈ {1, 2, 3}. On the other hand, the delayed HBV infection model with HBV DNA-containing capsids and CTL immune response introduced in [25] is a particular case of (2) when f (U, I, V ) = kU , g(U, I ) = 0, f 1 (τ ) = δ(τ − τ1 ), f 2 (τ − τ2 ), f 3 (τ ) = δ(τ ) and α1 = α2 = α3 = 0, where δ(.) is the Dirac delta function. In reality, the state variables and parameters of model (2) are nonnegative and bounded. Hence, we assume that the initial conditions of (2) satisfy: U (θ ) = χ1 (θ ) ≥ 0, V (θ ) = χ4 (θ ) ≥ 0,

I (θ ) = χ2 (θ ) ≥ 0, Z (θ ) = χ5 (θ ) ≥ 0,

D(θ ) = χ3 (θ ) ≥ 0, θ ∈ (−∞, 0].

(3)

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For the existence of solution, we consider the following Banach space: Eα = ϕ ∈ C((−∞, 0], IR5+ ) : ϕ(θ )eαθ is uniformly continuous

αθ on (−∞, 0] and ϕ = sup |ϕ(θ )|e < ∞ , θ≤0

where α is a positive constant. Similarly to [58, 62], it is not hard to show the following theorem. Theorem 3.1. For any given initial condition χ ∈ Eα satisfying (3), there exists a unique solution of model (2) defined on [0, +∞). Moreover, this solution remains nonnegative and bounded for all t ≥ 0. Clearly, DDE model (2) has always a unique infection-free equilibrium E 0 (U0 , 0, s 0, 0, 0), where U0 = , which represents the patient’s state without infection or μ disease. Then the basic reproduction number of (2) can be defined as follows R0 =

aβη1 η2 η3 f ( μs , 0, 0) + c(β + δ)η1 g( μs , 0) δc(β + δ) 

where

∞

ηi =

,

f i (τ )e−αi τ dτ, i = 1, 2, 3.

(4)

(5)

0

This basic reproduction number is the sum of two basic reproduction numbers due to virus-to-cell and cell-to-cell transmissions. Also, 0 < ηi ≤ 1. The other steady states of (2) satisfy the following equations:

η1



s − μU − f (U, I, V )V − g(U, I )I = 0,  f (U, I, V )V + g(U, I )I − δ I − p I Z = 0,

(6) (7)

aη2 I − (β + δ)D = 0, βη3 D − cV = 0,

(8) (9)

q I Z − σ Z = 0.

(10)

From (10), we have Z = 0 or Z = σq . Case (i). If Z = 0, then I = and

η1 (s − μU ) aβη1 η2 η3 (s − μU ) aη1 η2 (s − μU ) ,V = ,D = δ δ(β + δ) cδ(β + δ)

 η1 (s − μU ) aη1 η2 η3 β(s − μU )  η1 (s − μU ) + c(β + δ)η1 g(U, aβη1 η2 η3 f U, , ) = cδ(β + δ). δ cδ(β + δ) δ

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s η1 (s − μU ) ≥ 0, we have U ≤ . Then there is no biological steady δ μ s state if U > . So, we define a function ψ1 on the closed interval [0, μs ] by μ Since I =

 η1 (s − μU ) aη1 η2 η3 β(s − μU )  , ψ1 (U ) = aβη1 η2 η3 f U, δ cδ(β + δ) η1 (s − μU ) ) − cδ(β + δ). + c(β + δ)η1 g(U, δ s We have, ψ1 (0) = −cδ(β + δ) < 0, ψ1 ( ) = cδ(β + δ)(R0 − 1) and μ ψ1 (U ) = aβη1 η2 η3



∂f μη1 ∂ f aβμη1 η2 η3 ∂ f − − ∂U δ ∂I cδ(β + δ) ∂ V



+ c(β + δ)η1

∂g μη1 ∂g − ∂U δ ∂I

> 0.

s If R0 > 1, then ψ1 ( ) > 0. Since ψ1 is a continuous and strictly increasing μ function on the interval [0, μs ], we deduce that there exists a unique U1 ∈ (0, μs ) such as ψ1 (U1 ) = 0. Hence, model (2) has a unique infection steady state withη1 (s − μU1 ) out cellular immunity E 1 (U1 , I1 , D1 , V1 , 0), where I1 = > 0, D1 = δ aη1 η2 (s − μU1 ) aβη1 η2 η3 (s − μU1 ) > 0 and V1 = > 0. δ(β + δ) cδ(β + δ) η2 η3 aσ η2 Case (ii). If Z = 0, then I = σq , D = q(β+δ) , V = aβσ and cq(β+δ)  σ aβσ η2 η3  σ cq(β + δ)(s − μU ) + c(β + δ)g(U, ) = . aβη2 η3 f U, , q cq(β + δ) q σ )−δσ δσ From (7), we have Z = qη1 (s−μU ≥ 0 which implies that U ≤ μs − qμη . So, pσ 1 s δσ s δσ there is no positive equilibrium when U > μ − qμη1 or μ − qμη1 ≤ 0. Then we δσ ] as follows define a function ψ2 on the interval [0, μs − qμη 1

 σ aβσ η2 η3  σ cq(β + δ)(s − μU ) + c(β + δ)g(U, ) − . ψ2 (U ) = aβη2 η3 f U, , q cq(β + δ) q σ It is obvious that ψ2 (0) = − cq(β+δ)s < 0 and σ ψ2 (U ) = aβη2 η3

∂f ∂g cqμ(β + δ) + c(β + δ) + > 0. ∂U ∂U σ

When the cellular immunity has not been established, we have q I1 − σ ≤ 0. So, we define the reproduction number for cellular immunity as follows

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R1Z =

q I1 , σ

(11)

1 is the average life span of CTL cells and I1 is the quantity of infected cells σ at the steady state E 1 . Thus, the threshold parameter R1Z denotes the average number of CTL cells activated by infected cells. δσ σ s and If R1Z < 1, then I1 < , U1 > − q μ qμη1

which

ψ2 (

δσ δσ s 1 s 1 − ) = ψ1 ( − ) < ψ1 (U1 ) = 0, μ qμη1 η1 μ qμη1 η1

which leads that there is no positive steady state if R1Z < 1. δσ δσ σ s s If R1Z > 1, then I1 > , U1 < − and ψ2 ( − ) > 0. Consequently, q μ qμη1 μ qμη1 model (2) has a unique infection steady state with cellular immunity E 2 (U2 , I2 , δσ s σ aσ η2 aβσ η2 η3 , V2 = ), I2 = , D2 = D2 , V2 , Z 2 ), where U2 ∈ (0, − μ qμη1 q q(β + δ) cq(β + δ) qη1 (s − μU2 ) − δσ . and Z 2 = pσ According to the above discussions, we get the following result. Theorem 3.2. (i) If R0 ≤ 1, then model (2) has a unique infection-free steady state E 0 (U0 , 0, 0, s 0, 0), where U0 = . μ (ii) If R0 > 1, then model (2) has a unique infection steady state without cellular η1 (s − μU1 ) , D1 = immunity E 1 (U1 , I1 , D1 , V1 , 0) besides E 0 , where I1 = δ aη1 η2 (s − μU1 ) aβη1 η2 η3 (s − μU1 ) and V1 = . δ(β + δ) cδ(β + δ) (iii) If R1Z > 1, then model (2) has a unique infection steady state with cellus lar immunity E 2 (U2 , I2 , D2 , V2 , Z 2 ) besides E 0 and E 1 , where U2 ∈ (0, − μ δσ σ aσ η2 aβσ η2 η3 qη (s − μU2 ) − δσ , V2 = and Z 2 = 1 ), I2 = , D2 = . pσ qμη1 q q(β + δ) cq(β + δ) For the asymptotic stability of the steady state E 0 , we have the following result. Theorem 3.3. The infection-free steady state E 0 of model (2) is globally asymptotically stable if R0 ≤ 1 and unstable if R0 > 1.

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Proof. Consider a Lyapunov functional as follows L0 =

f (U0 , 0, 0) p 1 βη3 f (U0 , 0, 0) D(t) + V (t) + I (t) + Z (t) η1 c(β + δ) c qη1  ∞  t       1 f 1 (τ )e−α1 τ f U (θ), I (θ), V (θ) V (θ) + g U (θ), I (θ) I (θ) dθdτ + η1 0 t−τ  t  aβη3 f (U0 , 0, 0)) ∞ + f 2 (τ )e−α2 τ I (θ)dθdτ c(β + δ) 0 t−τ  t  β f (U0 , 0, 0)) ∞ + f 3 (τ )e−α3 τ D(θ)dθ dτ. c 0 t−τ

For convenience, we let ϕ = ϕ(t) and ϕτ = ϕ(t − τ ) for any ϕ ∈ {H, I, D, V, Z }. Clearly, L 0 is positive and zero at the equilibrium E 0 . By taking the time derivative of L 0 along the solution of model (2), we obtain d L0 = dt



pσ f (U, I, V ) − f (U0 , 0, 0) V − Z qη1

aβη1 η2 η3 f (U0 , 0, 0) + c(β + δ)η1 g(U, I ) δ + I −1 . η1 δc(β + δ)

From the first equation of model (2), we get lim sup U (t) ≤ U0 . t→∞

This implies that all omega limit points satisfy U (t) ≤ U0 . Hence, it is sufficient to consider solutions for which U (t) ≤ U0 . It follows from (4) and (H0 )–(H3 ) that d L0 ≤ dt ≤



f (U, 0, 0) − f (U0 , 0, 0) V +

 δ pσ R0 − 1 I − Z. η1 qη1

 δ pσ R0 − 1 I − Z η1 qη1

d L0 d L0 We can see that ≤ 0 if R0 ≤ 1. In addition, we have = 0 if and only if dt dt    f (U, 0, 0) − f (U0 , 0, 0) V = 0, (R0 − 1 I and Z = 0. We will make discussions according to the following two cases: If U = U0 . From the first equation of (2), we get f (U0 , I, V )V = 0 and g(U0 , I )I = 0. Since U0 > 0, we have f (U0 , I, V ) > 0 and g(U0 , I ) > 0. Hence, V = 0 and I = 0. By the fourth equation of (2), we have D = 0. If U = U0 , then V = 0. By the third and fourth equations of (2), we have D = 0 dU = s − μU . This shows that and I = 0. By the first equation of (2), we obtain dt s U → = U0 as t → ∞. μ

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d L0 = 0}. Thus, the singleton {E 0 } is the largest invariant subset of {(U, I, D, V, Z )| dt It follows from LaSalle’s invariance principle [63] that the infection-free steady state E 0 is globally asymptotically stable when R0 ≤ 1. It remains to prove the instability of E 0 when R0 > 1. To do this, we need to find the characteristic equation about the steady state. Let E e (Ue , Ie , De , Ve , Z e ) be an arbitrary steady state of (2). Then the characteristic equation about this point is given by     −μ − Ce,1 − λ −Ce,2 0 −Ce,3 0     η1 (λ)Ce,1 η1 (λ)Ce,2 − δ − p Z e − λ 0 η (λ)C − p I e e,3 1    = 0,  0 aη (λ) −β − δ − λ 0 0 2     (λ) −c − λ 0 0 0 βη 3    0 0 q Ie − σ − λ  0 q Ze

(12) ∂f ∂g (Ue , Ie , Ve ) + Ie ∂U (Ue , Ie ), Ce,2 = Ve ∂∂ fI (Ue , Ie , Ve ) + Ie ∂∂gI where Ce,1 = Ve ∂U (Ue , Ie ) + g(Ue , Ie ), Ce,3 = Ve ∂∂Vf (Ue , Ie , Ve ) + f (Ue , Ie , Ve ) and  ηi (λ) =

∞

f i (τ )e−(αi +λ)τ dτ, i = 1, 2, 3.

(13)

0

At E 0 (U0 , 0, 0, 0, 0), (12) reduces to (μ + λ)(σ + λ)F0 (λ) = 0,

(14)

where   F0 (λ) = λ3 + c + β + 2δ − η1 (λ)g(U0 , 0) λ2 + [c(β + 2δ − η1 (λ)g(U0 , 0)) + (β + δ)(δ − η1 (λ)g(U0 , 0))]λ + cδ(β + δ)(1 − R0 ). We have lim F0 (λ) = +∞ and F0 (0) = cδ(β + δ)(1 − R0 ) < 0 if R0 > 1. Hence, λ→+∞

the characteristic equation (14) has a positive root if R0 > 1. This implies that E 0 becomes unstable when R0 > 1.  Now, we investigate the asymptotic stability of the two infection steady states E 1 and E 2 of model (2) by assuming that R0 > 1 and the incidence functions f and g satisfy for all U, I, V > 0 the following further hypothesis

f (U, I, V ) f (U, Ii , Vi ) V 1− ≤ 0, − f (U, Ii , Vi ) f (U, I, V ) Vi

f (Ui , Ii , Vi )g(U, I ) f (U, Ii , Vi )g(Ui , Ii ) I 1− ≤ 0, − f (U, Ii , Vi )g(Ui , Ii ) f (Ui , Ii , Vi )g(U, I ) Ii

(H4 )

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where Ui , Ii and Vi are the uninfected cell, infected cell and virus components of the infection steady state E i for i = 1, 2. Theorem 3.4. Suppose that R0 > 1 and (H4 ) holds for E 1 . Then the infection steady state without cellular immunity E 1 is globally asymptotically stable if R1Z ≤ 1 and it becomes unstable if R1Z > 1. Proof. Consider a Lyapunov functional  U p 1 f (U1 , I1 , V1 ) I + dS + I1  Z η1 I1 qη1 U1 f (S, I1 , V1 )

(β + δ) f (U1 , I1 , V1 )V1 D V f (U1 , I1 , V1 )V1 + D1  V1  aη2 I1 D1 aβη2 η3 I1 V1 

  ∞  t f U (θ ), I (θ ), V (θ ) V (θ ) 1 dθ dτ f (U1 , I1 , V1 )V1 f 1 (τ )e−α1 τ  η1 f (U1 , I1 , V1 )V1 0 t−τ  

 ∞  t g U (θ ), I (θ ) I (θ ) 1 dθ dτ g(U1 , I1 )I1 f 1 (τ )e−α1 τ  η1 g(U1 , I1 )I1 0 t−τ

 ∞  t 1 I (θ ) dθ dτ f (U1 , I1 , V1 )V1 f 2 (τ )e−α2 τ  η2 I1 0 t−τ

 ∞  t 1 D(θ ) dθ dτ, f (U1 , I1 , V1 )V1 f 3 (τ )e−α3 τ  η3 D1 0 t−τ

L 1 = U − U1 − + + + + +

where (u) = u − 1 − ln u, u > 0. It is clear that (u) ≥ 0 for all u > 0, and (u) = 0 if and only if u = 1. So, the functional L 1 is nonnegative. By a simple computation, we find



pσ U f (U1 , I1 , V1 ) d L1 + 1− (R Z − 1)Z = μU1 1 − dt U1 f (U, I1 , V1 ) qη1 1

f (U, I, V )V V f (U, I1 , V1 ) + f (U1 , I1 , V1 )V1 − 1 − + + V1 f (U, I, V ) f (U, I1 , V1 )V1

I f (U1 , I1 , V1 )g(U, I )I f (U, I1 , V1 )g(U1 , I1 ) + + + g(U1 , I1 )I1 − 1 − I1 f (U1 , I1 , V1 )g(U, I ) f (U, I1 , V1 )g(U1 , I1 )I1 



 ∞ , I , V f (U f (Uτ , Iτ , Vτ )Vτ I1 1 1 1 1) −α τ + f (U1 , I1 , V1 )V1 f 1 (τ )e 1  − η1 f (U, I1 , V1 ) f (U1 , I1 , V1 )V1 I 0 



 ∞ f (U, I1 , V1 ) f (U1 , I1 , V1 ) 1 −α τ 1 +  dτ − g(U1 , I1 )I1 f 1 (τ )e f (U, I, V ) η1 f (U, I1 , V1 ) 0



 f (U, I1 , V1 )g(U1 , I1 ) g(Uτ , Iτ )Iτ + dτ + g(U1 , I1 )I f (U1 , I1 , V1 )g(U, I )

 ∞ 1 D1 I τ − dτ f (U1 , I1 , V1 )V1 f 2 (τ )e−α2 τ  η2 D I1 0

 ∞ V1 Dτ 1 f (U1 , I1 , V1 )V1 f 3 (τ )e−α3 τ  − dτ. η3 V D1 0

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Since the incidence function f (U, I, V ) is strictly monotonically increasing with respect to U , we have for all i ∈ {1, 2} that



U f (Ui , Ii , Vi ) 1− 1− ≤ 0. Ui f (U, Ii , Vi )

(15)

From (H4 ), we deduce for all i ∈ {1, 2} that −1 −

V Vi

+

f (U,Ii ,Vi ) f (U,I,V )

+

V f (U,I,V ) Vi f (H,Ii ,Vi )

= 1−

f (U,I,V ) f (U,Ii ,Vi )



f (U,Ii ,Vi ) f (U,I,V )

−

V Vi

≤ 0, (16)

and

(Ui ,Ii ,Vi )g(U,I )I i ,Vi )g(Ui ,Ii ) −1 − IIi + ff (H,I + ff(U,I (Ui ,Ii ,Vi )g(U,I ) i ,Vi )g(Ui ,Ii )Ii

f (Ui ,Ii ,Vi )g(U,I ) f (U,Ii ,Vi )g(Ui ,Ii ) = 1 − f (U,Ii ,Vi )g(Ui ,Ii ) − IIi ≤ 0. f (Ui ,Ii ,Vi )g(U,I )

(17)

d L1 ≤ 0 if R1Z ≤ 1. It is obvious that the singleton {E 1 } is the dt d L1 = 0}. Thus, LaSalle’s largest compact invariant subset of {(U, I, D, V, Z )| dt invariance principle ensures the global asymptotic stability of E 1 when R1Z ≤ 1. Now, it remains to show the instability of E 1 when R1Z > 1. At this steady state, Eq. (12) becomes (18) (q I1 − σ − λ)F1 (λ) = 0, This implies that

where    −μ − C1,1 − λ −C1,2 0 −C1,3    η1 (λ)C1,1 η1 (λ)C1,2 − δ − λ 0 η1 (λ)C1,3  F1 (λ) =  . 0 aη2 (λ) −β − δ − λ 0    −c − λ  0 0 βη3 (λ) Hence, λ1 = q I1 − σ is a root of the characteristic equation (18). Since R1Z = qσI1 > 1, we get λ1 > 1 which implies that the characteristic equation (18) has a real positive root when R1Z > 1. Therefore, E 1 becomes unstable when R1Z > 1. This completes the proof.  Theorem 3.5. Suppose that R0 > 1 and (H4 ) holds for E 2 . Then the infection steady state with cellular immunity E 2 is globally asymptotically stable if R1Z > 1.

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Proof. Define a Lyapunov functional as follows  U 1 p f (U2 , I2 , V2 ) I Z dS + + I2  Z2 η1 I2 qη1 Z2 U2 f (S, I2 , V2 )

f (U2 , I2 , V2 )V2 (β + δ) f (U2 , I2 , V2 )V2 D V + D2  V2  aη2 I2 D2 aβη2 η3 I2 V2 

  ∞  t f U (θ ), I (θ ), V (θ ) V (θ ) 1 dθ dτ f (U2 , I2 , V2 )V2 f 1 (τ )e−α1 τ  η1 f (U2 , I2 , V2 )V2 0 t−τ  

 ∞  t g U (θ ), I (θ ) I (θ ) 1 dθ dτ g(U2 , I2 )I2 f 1 (τ )e−α1 τ  η1 g(U2 , I2 )I2 0 t−τ

 ∞  t 1 I (θ ) dθ dτ f (U2 , I2 , V2 )V2 f 2 (τ )e−α2 τ  η2 I2 0 t−τ

 ∞  t 1 D(θ ) dθ dτ, f (U2 , I2 , V2 )V2 f 3 (τ )e−α3 τ  η3 D2 0 t−τ

L 2 = U − U2 − + + + + +

δ p By using s = μU2 + f (U2 , I2 , V2 )V2 + g(U2 , I2 )I2 = μU2 + I2 + I2 Z 2 , aη2 I2 = η1 η1 σ (β + δ)D2 , I2 = and βη3 D2 = cV2 , we obtain q



U f (U2 , I2 , V2 ) d L2 = μU2 1 − 1− dt U2 f (U, I2 , V2 )

V f (U, I2 , V2 ) f (U, I, V )V + + f (U2 , I2 , V2 )V2 − 1 − + V2 f (U, I, V ) f (U, I2 , V2 )V2

f (U2 , I2 , V2 )g(U, I )I I f (U, I2 , V2 )g(U2 , I2 ) + + + g(U2 , I2 )I2 − 1 − I2 f (U2 , I2 , V2 )g(U, I ) f (U, I2 , V2 )g(U2 , I2 )I2 



 ∞ , I , V f (U f (Uτ , Iτ , Vτ )Vτ I1 1 2 2 2) −α τ + f (U2 , I2 , V2 )V2 f 1 (τ )e 1  − η1 f (U, I2 , V2 ) f (U2 , I2 , V2 )V2 I 0



  ∞ 1 f (U f (U, I2 , V2 ) 2 , I2 , V2 ) dτ − g(U2 , I2 )I2 f 1 (τ )e−α1 τ  + f (U, I, V ) η1 f (U, I2 , V2 ) 0



 g(Uτ , Iτ )Iτ f (U, I2 , V2 )g(U2 , I2 ) + + dτ g(U2 , I2 )I f (U2 , I2 , V2 )g(U, I )

 ∞ 1 D2 I τ − dτ f (U2 , I2 , V2 )V2 f 2 (τ )e−α2 τ  η2 D I2 0

 ∞ 1 V2 Dτ − dτ. f (U2 , I2 , V2 )V2 f 3 (τ )e−α3 τ  η3 V D2 0

d L2 d L2 ≤ 0 if R1Z > 1. It is easy to show that =0 From (15)–(17), we deduce that dt dt if and only if U = U2 , I = I2 , D = D2 , V = V2 and Z = Z 2 . Therefore, {E 2 } is the d L2 = 0}. It follows from LaSalle’s largest invariant subset of {(U, I, D, V, Z )| dt invariance principle that the steady state E 2 is globally asymptotically stable when  R1Z > 1.

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4 Dynamics of HBV Infection with PDEs In this section, we extend model (2) by taking into account the mobility of hepatocytes, capsids, virions and CTL cells. Thus, model (2) becomes ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

  ∂U = dU U (x, t) + s − μU (x, t) − f U (x, t), I (x, t), V (x, t) V (x, t) ∂t   − g U (x, t), I (x, t) I (x, t),    ∂I = d I I (x, t) + 0∞ f 1 (τ )e−α1 τ [ f U (x, t − τ ), I (x, t − τ ), V (x, t − τ ) V (x, t − τ ) ∂t   + g U (x, t − τ ), I (x, t − τ ) I (x, t − τ )]dτ − δ I (x, t) − p I (x, t)Z (x, t),  ∂D = d D D(x, t) + a 0∞ f 2 (τ )e−α2 τ I (x, t − τ )dτ − (β + δ)D(x, t), ∂t  ∂V = dV V (x, t) + β 0∞ f 3 (τ )e−α3 τ D(x, t − τ )dτ − cV (x, t), ∂t ∂Z = d Z Z (x, t) + q I (x, t)Z (x, t) − σ Z (x, t), ∂t

(19) where U (x, t), I (x, t), D(x, t), V (x, t) and Z (x, t) denote the densities of the uninfected cells, infected cells, capsids, virions and CTL cells at position x ∈  and time t, respectively. Here,  is a bounded domain in IRn with smooth boundary ∂ and  is the Laplacian operator. Further, dU , d I , d D , dV are d Z represent the diffusion coefficients of uninfected cells, infected cells, capsids, virions and CTL cells, respectively. The importance of our model (19) resides in the generalization and extension of the recent mathematical models with partial differential equations (PDEs) proposed and analyzed in [39, 43, 48, 64]. On the other hand, the hepatocytes, capsids, virions and CTL cells are biological quantities which should be non-negative and they also not cross the boundary ∂. So, we consider model (19) with initial conditions as follows U (x, θ ) = φ1 (x, θ ) ≥ 0, I (x, θ ) = φ2 (x, θ ) ≥ 0, D(x, θ ) = φ3 (x, θ ) ≥ 0, ¯ × (−∞, 0], V (x, θ ) = φ4 (x, θ ) ≥ 0, Z (x, θ ) = φ5 (x, θ ) ≥ 0, (x, θ ) ∈  (20) and Neumann boundary conditions ∂I ∂D ∂V ∂Z ∂U = = = = = 0, on ∂ × (0, +∞), ∂ν ∂ν ∂ν ∂ν ∂ν

(21)

∂ is the outward normal derivative on the boundary ∂. where ∂ν We first show the existence, uniqueness, non-negativity and boundedness of solu¯ IR5 ) be the Banach space of contintions of problem (19)–(21). So, we let X = C(, ¯ to IR5 . Then we define Cα = Cα ((−∞, 0], X) to be the Banach uous functions from  space of continuous functions ϕ from (−∞, 0] to X, where ϕ(θ )eαθ is uniformly continuous on (−∞, 0] and ϕ = supϕ(θ )X eαθ < ∞ with α is a positive constant. θ≤0

¯ × (−∞, 0] to IR5 defined Also, we identify an element ϕ ∈ Cα as a function from  by ϕ(x, θ ) = ϕ(θ )(x). Moreover, for any continuous function u(.) : (−∞, b) → X we define u t ∈ Cα by u t (θ ) = u(t + θ ), for b > 0 and θ ∈ (−∞, 0].

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Theorem 4.1. For any given initial condition φ ∈ Cα satisfying (20), there exists a unique nonnegative solution of the problem (19)–(21). If the hepatocytes and CTL cells have the same diffusion coefficient (dU = d I = d Z ), then this solution ¯ × [0, +∞). is bounded and defined on  ¯ we define G = (G 1 , G 2 , Proof. For any ϕ = (ϕ1 , ϕ2 , ϕ3 , ϕ4 , ϕ5 )T ∈ Cα and x ∈ , G 3 , G 4 , G 5 ) : Cα → X by   G 1 (ϕ)(x) = s − μϕ1 (x, 0) − f ϕ  1 (x, 0), ϕ2 (x, 0), ϕ4 (x, 0) ϕ4 (x, 0) − g ϕ1 (x, 0), ϕ2 (x, 0) ϕ2 (x, 0),  G 2 (ϕ)(x) = 0∞ f 1 (τ )e−α1 τ [ f ϕ1 (x,  −τ ), ϕ2 (x, −τ ), ϕ4 (x, −τ ) ϕ4 (x, −τ ) + g ϕ1 (x, −τ ), ϕ2 (x, −τ ) ϕ2 (x, −τ )]dτ − δϕ2 (x, 0) − pϕ2 (x, 0)ϕ5 (x, 0), G 3 (ϕ)(x) = a 0∞ f 2 (τ )e−α2 τ ϕ2 (x, −τ )dτ − (β + δ)ϕ3 (x, 0), G 4 (ϕ)(x) = β 0∞ f 3 (τ )e−α3 τ ϕ3 (x, −τ )dτ − cϕ4 (x, 0), G 5 (ϕ)(x) = qϕ2 (x, 0)ϕ5 (x, 0) − σ ϕ5 (x, 0).

Then we can rewrite problem (19)–(21) as the following abstract functional differential equation: u  (t) = Au + G(u t ), t > 0, (22) u(0) = φ ∈ Cα , where u = (U, I, D, V, Z )T , Au = (dU U, d I I, d D D, dV V, d Z Z )T and φ = (φ1 , φ2 , φ3 , φ4 , φ5 )T . Clearly, G is locally Lipschitz in Cα . Thus, we deduce that system (22) has a unique local solution on [0, tmax ), where [0, tmax ) is the maximal time interval on which the solutions are guaranteed to exist [65–69]. Since 0 = (0, 0, 0, 0, 0) is a lower-solution of (19)–(21), we have U (x, t) ≥ 0, I (x, t) ≥ 0, D(x, t) ≥ 0, V (x, t) ≥ 0 and Z (x, t) ≥ 0. Next, we show that the solution of (19)–(21) is bounded. From the first equation of (19), we have ⎧ ∂U ⎪ ⎪ ⎪ ⎨ ∂t − dU U ≤ s − μU (x, t), ∂U = 0, ⎪ ⎪ ⎪ ⎩ ∂ν U (x, 0) = φ1 (x, 0) ≥ 0. By applying Lemma 1 in [46], we get

s ¯ × [0, tmax ). , maxφ1 (x, 0) , ∀(x, t) ∈  U (x, t) ≤ max μ x∈¯

Define  N (x, t) = I (x, t) + 0

∞

f 1 (τ )e−α1 τ U (x, t − τ )dτ +

p Z (x, t). q

(23)

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Hence,  ∞ p ∂N = d I I (x, t) + d Z Z (x, t) + dU f 1 (τ )e−α1 τ U (x, t − τ )dτ ∂t q 0  ∞ pσ −α1 τ Z (x, t) + sη1 − μ f 1 (τ )e U (x, t − τ )dτ − δ I (x, t) − q 0  ∞ p ≤ d I I (x, t) + d Z Z (x, t) + dU f 1 (τ )e−α1 τ U (x, t − τ )dτ q 0 + sη1 − 1 N (x, t), where 1 = min{μ, δ, σ }. Since dU = d I = d Z = d, we have ⎧ ∂N ⎪ ⎪ − dN ≤ sη1 − 1 N (x, t), ⎪ ⎨ ∂t ∂N = 0, ⎪ ⎪ ∂ν ⎪ ⎩ N (x, 0) = φ (x, 0) + p φ (x, 0) +  ∞ f (τ )e−α1 τ φ (x, −τ )dτ ≥ 0. 2 1 1 0 q 5 From Lemma 1 in [46], we obtain

sη1 ¯ × [0, tmax ). , max N (x, 0) , ∀(x, t) ∈  N (x, t) ≤ max 1 x∈¯

(24)

By (24) and (19)–(21), we find that D satisfies the following system ⎧ ∂D ⎪ ⎪ ⎪ ⎨ ∂t − d D D ≤ a2 η2 − (β + δ)D, ∂D = 0, ⎪ ⎪ ⎪ ⎩ ∂ν D(x, 0) = φ3 (x, 0) ≥ 0,

sη1 where 2 = max , max N (x, 0) . By applying Lemma 1 in [46], we get 1 x∈¯ D(x, t) ≤ max

a2 η2 ¯ × [0, tmax ). , maxφ3 (x, 0) , ∀(x, t) ∈  β + δ x∈¯

(25)

Similarly to above, we can get

β3 η3 ¯ × [0, Tmax ), , maxφ4 (x, 0) , ∀(x, t) ∈  V (x, t) ≤ max ¯ c x∈



a2 η2 , maxφ3 (x, 0) . where 3 = max β + δ x∈¯

(26)

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From (23)–(26), we deduce that U (x, t), I (x, t), D(x, t), V (x, t) and Z (x, t) are ¯ × [0, tmax ). By the standard theory for semilinear parabolic equations bounded on   [70], we conclude that tmax = +∞. On the other hand, E 0 (U0 , 0, 0, 0, 0) is also a steady state of model (19). Then we have the following result. Theorem 4.2. The infection-free steady state E 0 of model (19) is globally asymptotically stable if R0 ≤ 1 and unstable if R0 > 1. Proof. By applying the Hattaf-Yousfi method presented in [34], we construct a Lyapunov functional for PDE model (19) at E 0 as follows  1 f (U0 , 0, 0) p βη3 f (U0 , 0, 0) D(x, t) + V (x, t) + L0 = I (x, t) + Z (x, t) η c(β + δ) c qη 1 1   ∞  t    1 + f 1 (τ )e−α1 τ f U (x, θ ), I (x, θ ), V (x, θ ) V (x, θ ) η1 0 t−τ    + g U (x, θ ), I (x, θ ) I (x, θ ) dθ dτ  t  aβη3 f (U0 , 0, 0)) ∞ f 2 (τ )e−α2 τ I (x, θ )dθ dτ + c(β + δ) 0 t−τ

  t β f (U0 , 0, 0)) ∞ f 3 (τ )e−α3 τ D(x, θ )dθ dτ d x. + c 0 t−τ For simplicity, we denote  = (x, t) and τ = (x, t − τ ) for any  ∈ {U, I, D, V, Z }. Then 

pσ f (U, I, V ) − f (U0 , 0, 0) V − Z qη 1 

aβη1 η2 η3 f (U0 , 0, 0) + c(β + δ)η1 g(U, I ) δ − 1 dx + I η1 δc(β + δ)

   δ pσ R0 − 1 I − ≤ Z d x. qη1  η1

dL0 = dt

dL0 ≤ 0 with equality if and only if U = U0 , I = 0, D = 0, V = 0 dt and Z = 0. This indicates that the largest compact invariant set in {(U, I, D, V, Z )| dL0 = 0} is the singleton {E 0 }. Therefore, it follows from LaSalle’s invariance dt principle [63] that E 0 is globally asymptotically stable whenever R0 ≤ 1. Now, it remains to establish the stability property of E 0 when R0 > 1. Let 0 = λ1 < λ2 < ... < λn < ... be the eigenvalues of the operator − on  with the no-flux boundary conditions, and E(λi ) be the eigenfunction space corresponding to λi in C 1 (). Let {ei j : j = 1, 2, ..., dim E(λi )} be an orthonormal basis of E(λi ), Y = [C 1 ()]5 , and Yi j = {αei j : α ∈ IR5 }. Then If R0 ≤ 1, then

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Y=

∞ 

Yi and Yi =

i=1

dim E(λi ) 

Yi j ,

j=1

 where represents the direct sum of the subspaces. Linearizing the PDE model (19) about an arbitrary steady state, we obtain the following system    ∂u 1 ∂f ∂g  u 1 (x, t) − Ve ∂∂ fI + Ie ∂∂gI + g(Ue , Ie ) u 2 (x, t) + Ie ∂U = dU u 1 − μ + Ve ∂U ∂t   − Ve ∂∂Vf + f (Ue , Ie , Ve ) u 4 (x, t),  ∂f ∂u 2 ∂g   ∞ −α1 τ u (x, t − τ )dτ = d I u 2 + Ve ∂U + Ie ∂U 1 0 f 1 (τ )e ∂t ∞  ∂f ∂g + Ve ∂ I + Ie ∂ I + g(Ue , Ie ) 0 f 1 (τ )e−α1 τ u 2 (x, t − τ )dτ   + Ve ∂∂Vf + f (Ue , Ie , Ve ) 0∞ f 1 (τ )e−α1 τ u 4 (x, t − τ ) − ( p Z e + δ)u 2 (x, t) − p Ie u 5 (x, t),  ∂u 3 = d D u 3 + a 0∞ f 2 (τ )e−α2 τ u 2 (x, t − τ )dτ − (β + δ)u 3 (x, t), ∂t  ∂u 4 = dV u 4 + β 0∞ f 3 (τ )e−α3 τ u 3 (x, t − τ )dτ − cu 4 (x, t), ∂t ∂u 5 = d Z u 5 + q Z e u 2 (x, t) + (q Ie − σ )u 5 (x, t). ∂t

This system can be rewritten as follows  ∂u = Du + A0 u(x, t) + ∂t i=1 3



∞

f i (τ )e−αi τ Ai u(x, t − τ )dτ,

0

where u = (u 1 , u 2 , u 3 , u 4 , u 5 )T , D = diag(dU , d I , d D , dV , d Z ), ⎡

⎤ −μ − Ce,1 −Ce,2 0 −Ce,3 0 ⎢ 0 0 − p Ie ⎥ 0 −δ − p Z e ⎢ ⎥ ⎢ 0 0 −β − δ 0 0 ⎥ A0 = ⎢ ⎥, ⎣ 0 0 0 −c 0 ⎦ 0 0 q Ie − σ 0 q Ze ⎡

0 ⎢Ce,1 ⎢ A1 = ⎢ ⎢ 0 ⎣ 0 0 and

0 Ce,2 0 0 0

0 0 0 0 0

0 Ce,3 0 0 0 ⎡

0 ⎢0 ⎢ A3 = ⎢ ⎢0 ⎣0 0

⎡ ⎤ 0 00 ⎢0 0 0⎥ ⎢ ⎥ ⎢ 0⎥ ⎥ , A2 = ⎢0 a ⎣0 0 ⎦ 0 00 0 0 0 0 0 0

0 0 0 β 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

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3  ∞ −αi τ Define Lu = Du + A0 u(x, t) + i=1 Ai u(x, t − τ )dτ . For each 0 f i (τ )e i ≥ 1, Yi is invariant under the operator L. Further, ξ is an eigenvalue of L if and only if it is a root of the following characteristic equation det(A0 + η1 (ξ )A1 + η2 (ξ )A2 + η3 (ξ )A3 − ξ I − λi D) = 0,

(27)

for some i ≥ 1, for which there exists an eigenvector in Yi . Here, I is the identity matrix. The characteristic equation at E 0 is given by    ξ + μ + λi dU ξ + σ + λi d Z i (ξ ) = 0,

(28)

where   i (ξ ) = ξ 3 + c + β + 2δ + λi d D + λi dV − η1 (ξ )g(U0 , 0) ξ 2 + [(β + 2δ + λi d I + λi d D − η1 (ξ )g(U0 , 0))(c + λi dV ) + (β + δ + λi d D )(δ + λi d I − η1 (ξ )g(U0 , 0))]ξ + δ(β + δ + λi d D )(c + λi dV ) − aβη1 (ξ )η2 (ξ )η3 (ξ ) f (U0 , 0, 0) − (β + δ + λi d D )(c + λi dV )η1 (ξ )g(U0 , 0). We have lim i (ξ ) = +∞ and 1 (0) = cδ(β + δ)(1 − R0 ) < 0 if R0 > 1. Thus, ξ →+∞

the characteristic equation (28) has a positive root if R0 > 1. Consequently, E 0 becomes unstable when R0 > 1.  When R0 > 1, the equilibria E 1 and E 2 are also the steady states of model (19). At the same way, by evaluating the characteristic equation (27) at E 1 and based on the Lyapunov functionals L 1 and L 2 for DDE model (2), it is not difficult to prove the following theorem. Theorem 4.3. Suppose that R0 > 1 and (H4 ) holds. (i) If R1Z ≤ 1, then the infection steady state without cellular immunity E 1 of model (19) is globally asymptotically stable. (ii) If R1Z > 1, then E 1 becomes unstable and the infection steady state with cellular immunity E 2 of model (19) is globally asymptotically stable.

5 Application In this section, we present an example with specific forms for both the modes of transmission and apply our theoretical results obtained in the preceding section. Accordingly, we consider the following particular reaction-diffusion system for HBV infection:

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⎧ ∂U ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ∂I ⎪ ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂D ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ∂V ⎪ ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎩ ∂Z ∂t

U (x,t)V (x,t) U (x,t)I (x,t) = dU U (x, t) + s − μU (x, t) − k11+b − k21+b , 1 V (x,t) 2 I (x,t) ∞ U (x,t)V (x,t) U (x,t)I (x,t) + k21+b ]dτ − δ I (x, t) = d I I (x, t) + 0 f 1 (τ )e−α1 τ [ k11+b 1 V (x,t) 2 I (x,t)

− p I (x, t)Z (x, t), ∞ = d D D + a 0 f 2 (τ )e−α2 τ I (x, t − τ )dτ − (β + δ)D(x, t), ∞ = dV V + β 0 f 3 (τ )e−α3 τ D(x, t − τ )dτ − cV (x, t), = d Z Z + q I (x, t)Z (x, t) − σ Z (x, t),

(29) subjected to the non-negative initial conditions (20) and homogeneous Neumann boundary conditions (21). The above model (29) is a particular case of the reactionk2 U k1 U and g(U, I ) = . Here, diffusion system (19) with f (U, I, V ) = 1 + b1 V 1 + b2 I the virus-to-cell and cell-to-cell transmission rates are denoted by k1 and k2 , respectively. The non-negative parameters b1 and b2 measure the corresponding saturation effects respectively. The state variables and other parameters have the same biological meanings as in the system (19). s The model (29) always admits a unique infection-free steady state E 0 ( , 0, 0, μ 0, 0) and the basic reproduction number in this case is given by R0 =

aβsk1 η1 η2 η3 + cs(β + δ)k2 η1 . cδμ(β + δ)

(30)

For the case R0 > 1, the model (29) admits a unique infection steady state without cellular immunity E 1 (U1 , I1 , D1 , V1 , 0) along with the steady state E 0 . Also, one can easily compute the reproduction number for cellular immunity R1Z numerically for this model and there exists a unique infection steady state with cellular immunity E 2 (U2 , I2 , D2 , V2 , Z 2 ) when R1Z > 1. From the considered specific forms for both the modes of transmission, it is clear that the hypotheses (H0 )–(H3 ) are satisfied. On the other hand, we have

f (U, I, V ) −b1 (V − Vi )2 f (U, Ii , Vi ) V 1− = − ≤ 0, f (U, Ii , Vi ) f (U, I, V ) Vi Vi (1 + b1 Vi )(1 + b1 V )

f (Ui , Ii , Vi )g(U, I ) −b2 (I − Ii )2 f (U, Ii , Vi )g(Ui , Ii ) I 1− = − ≤ 0, f (U, Ii , Vi )g(Ui , Ii ) f (Ui , Ii , Vi )g(U, I ) Ii Ii (1 + b2 Ii )(1 + b2 I )

with i = 1, 2. Hence, the last hypothesis (H4 ) is also satisfied. Therefore, from the Theorems 4.2 and 4.3, we can easily deduce the following result. Corollary 5.1. (i) The infection-free steady state E 0 of the model (29) is globally asymptotically stable whenever R0 ≤ 1 and is unstable if R0 > 1.

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(ii) If R0 > 1 and R1Z ≤ 1, then the infection steady state without cellular immunity E 1 of the model (29) is globally asymptotically stable. On the other hand, E 1 is unstable whenever R0 > 1 and R1Z > 1. (iii) If R0 > 1 and R1Z > 1, then the infection steady state with cellular immunity E 2 of the model (29) is globally asymptotically stable.

6 Conclusions In this chapter, we have investigated the role of diffusion on global dynamics possessed by a delayed generalized HBV infection model with capsids and CTL immune response. First, we have considered a five-dimensional non-diffusive delayed HBV infection model with both the virus-to-cell and cell-to-cell modes of transmission. Both these modes of transmission have been incorporated in the modeling approach through two general functional forms with some associated biologically feasible assumptions. Also, three distributed delays have been included in this model to capture the time needed for the biological processes such as latently infected hepatocytes to become infectious, capsids and virions to become mature. Investigating for the biologically feasible steady states of this model, we have obtained two threshold parameters such as basic reproduction number R0 and reproduction number for CTL immune response R1Z which completely determine the dynamics of the model. We have provided the result regarding the well-posedness of the model in terms of nonnegativity, existence, uniqueness and boundedness of the solutions. By constructing appropriate Lyapunov functionals, the global properties of the feasible steady states have been established. Our analysis suggests that the infection-free steady state E 0 is globally asymptotically stable whenever R0 ≤ 1. This indicates the complete eradication of the infection in this parameter regime, and otherwise, the infection persists. For the case R0 > 1, the following two situations emerge depending on the value of R1Z . The infection steady state without cellular immunity E 1 is globally asymptotically stable whenever R1Z ≤ 1, and otherwise, the infection steady state with cellular immunity E 2 becomes globally asymptotically stable. These results indicate that the value of R1Z needs to be greater than unity in order to trigger the CTL immune response, and otherwise, it can not be triggered. Also, our study suggests that the activation of CTL immune response is unable to eradicate the infection completely. Further, we have extended the delayed generalized HBV infection model by incorporating dispersion of each component, and investigated for the well-posedness and global properties of the extended model. Our analysis predicted the similar dynamics as that of the model without diffusion. This indicates that the incorporation of diffusion does not have any drastic effect on the dynamical behaviors of the solutions. As we have dealt with generalized models, the results obtained in this chapter are valid for a wide variety of models with specific forms for modes of transmission, and accordingly, we have provided an appropriate example in the preceding section. Overall, our study suggests that complete cure is not possible for chronic HBV infected patients with only cellular immunity. Therefore, a well-designed

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treatment protocol should be administered on the patient with chronic infection keeping in mind the goal of diminishing the value of R0 below unity.

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49. Manna, K., Hattaf, K.: Spatiotemporal dynamics of a generalized HBV infection model with capsids and adaptive immunity. Int. J. Appl. Comput. Math. 5, 65 (2019) 50. Karayiannis, P.: Hepatitis B virus: virology, molecular biology, life cycle and intrahepatic spread. Hep. Intl. 11(6), 500–508 (2017) 51. Goyal, A., Murray, J.M.: Modelling the impact of cell-to-cell transmission in hepatitis B virus. PLoS One 11(8), e0161978 (2016) 52. Marsh, M., Helenius, A.: Virus entry: open sesame. Cell 124, 729–740 (2006) 53. Sattentau, Q.: Avoiding the void: cell-to-cell spread of human viruses. Nat. Rev. Microbiol. 6, 815–826 (2008) 54. Martin, N., Sattentau, Q.: Cell-to-cell HIV-1 spread and its implications for immune evasion. Curr. Opinion HIV AIDS 4, 143–149 (2009) 55. Mothes, W., Sherer, N.M., Jin, J., Zhong, P.: Virus cell-to-cell transmission. J. Virol. 84, 8360– 8368 (2010) 56. Timpe, J.M., Stamataki, Z., Jennings, A., Hu, K., Farquhar, M.J., Harris, H.J., Schwarz, A., Desombere, I., Roels, G.L., Balfe, P., McKeating, J.A.: Hepatitis C virus cell-cell transmission in hepatoma cells in the presence of neutralising antibodies. Hepatology 47(1), 17–24 (2008) 57. Hattaf, K., Yousfi, N.: A generalized virus dynamics model with cell-to-cell transmission and cure rate. Adv. Differ. Equ. 2016, 1–11 (2016) 58. Hattaf, K., Yousfi, N.: Qualitative analysis of a generalized virus dynamics model with both modes of transmission and distributed delays. Int. J. Differ. Equ. 2018, 1–7 (2018) 59. Hattaf, K., Yousfi, N.: A numerical method for a delayed viral infection model with general incidence rate. J. King Saud Univ. Sci. 28, 368–374 (2016) 60. Wang, X.-Y., Hattaf, K., Huo, H.-F., Xiang, H.: Stability analysis of a delayed social epidemics model with general contact rate and its optimal control. J. Ind. Manag. Optim. 12, 1267–1285 (2016) 61. Hattaf, K., Yousfi, N., Tridane, A.: Global stability analysis of a generalized virus dynamics model with the immune response. Can. Appl. Math. Q. 20, 499–518 (2012) 62. Hattaf, K.: Global stability and Hopf bifurcation of a generalized viral infection model with multi-delays and humoral immunity. Physica A Statist. Mech. Appl., 1–14 (2019) 63. Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) 64. Hattaf, K., Yousfi, N.: Global properties of a diffusive HBV infection model with cell-to-cell transmission and three distributed delays. In: Disease Prevention and Health Promotion in Developing Countries, pp. 117–131. Springer, Cham (2020) 65. Travis, C.C., Webb, G.F.: Existence and stability for partial functional differential equations. Trans. Am. Math. Soc. 200, 395–418 (1974) 66. Fitzgibbon, W.E.: Semilinear functional differential equations in Banach space. J. Differ. Equ. 29, 1–14 (1978) 67. Martin, R.H., Smith, H.L.: Abstract functional differential equations and reaction-diffusion systems. Trans. Am. Math. Soc. 321, 1–44 (1990) 68. Martin, R.H., Smith, H.L.: Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence. J. für die reine und angewandte Mathematik 413, 1–35 (1991) 69. Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996) 70. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)

A Class of Ebola Virus Disease Models with Post-death Transmission and Environmental Contamination Zineb El Rhoubari, Khalid Hattaf, and Noura Yousfi

Abstract Ebola virus disease (EVD) is an infectious disease caused by Ebola virus which is a member of Filoviridae family that can provoke severe hemorrhagic fever in humans and other primates. It is mainly transmitted by an infected human who is still alive, from dead to the living during funerals or by a contaminated environment. In this chapter, we propose two mathematical EVD models that incorporate the three above modes of transmission. These modes are modeled by three general incidence functions that cover many types of incidence rates existing in the literature. Also, the first model is formulated by ordinary differential equations (ODEs) and the second is governed by partial differential equations (PDEs) in order to describe the evolution of EVD in time and space. The qualitative analysis of both models is rigorously investigated. Our analytical results are illustrated by an application and numerical simulations. Keywords Ebola virus disease · General incidence rate · Contaminated environment · Global stability · Spatial diffusion

1 Introduction The Ebola virus disease (EVD) was discovered in 1976 simultaneously in Anzara in South Sudan and in a village in Democratic Republic of Congo (DRC) near a river called Ebola from which the disease derived its name. Since then, dozens of Ebola outbreaks have affected the West Africa countries. The last outbreak was declared in August 2018 in DRC with 3421 cases and 2242 deaths, up to the end of 28 JanZ. El Rhoubari (B) · K. Hattaf · N. Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, Sidi Othman, P.O Box 7955, Casablanca, Morocco e-mail: [email protected] K. Hattaf e-mail: [email protected] K. Hattaf Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Derb Ghalef, 20340 Casablanca, Morocco © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_11

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uary 2020 [1]. It is now classified as the second biggest Ebola epidemic ever recorded behind the 2014–2016 one. It has been shown that fruit bats are natural Ebola virus (EV) hosts [2]. The virus is transmitted to the other animals through close contact with saliva, blood or other bodily fluids. The human infection occurs by an unprotected contact with bodily fluids of infected bats or animals such as chimpanzees, gorillas, monkeys, forest antelopes or porcupines found ill or dead in the rainforest. Then the virus spreads in the human population through contact of broken skin or mucous membranes with 1. Bodily fluids of a person who is sick with Ebola. 2. Infectious corpses during burial ceremonies. In [3], the study has been shown that EV persists for up to 7 days on bodies surfaces of deceased individuals which favorises the post-death transmission. 3. EV contaminated environment. People can catch Ebola through contact with EV contaminated objects. In [4], the authors have detected viable EV in human blood for up to 14 days using simulated West Africa countries environmental conditions and they found that EV survives in water from 3 to 6 days. Unfortunately, there is no licensed drug to treat the EVD. The standard treatment settle for a supportive therapy providing hydration, oxygen and medication to support blood pressure and to manage the possible symptoms such as fever and diarrhea. A protective vaccine against Ebola is under development and it is not yet commercially licensed. The natural reservoir of EV and other human viral infections including measles and mumps are bats [5–7]. These mammals can live more than 40 years with highly synchronous birthing [8–10]. Moreover, they carry the EV without being ill. In [11], the authors have shown that EV replicates in bats without any reported infection. Another study [12] has revealed that a seropositive bat for EV lived healthy over 13 months after sampling. In order to understand the transmission and the spread of EV in bat population, many mathematical models have been proposed and analyzed. For instance, Buceta and Kaylynn studied in [13] a SIR compartmental model including the bats mobility and the spatiotemporal climate variability. They are used to understand the migration patterns of bats and to predict the hot spots of Ebola outbreaks in space and time. Also, Berge et al. [14] presented an epizootic model for Ebola. This model shows the transmission dynamics of EVD in bat population using bilinear incidence functions that follow the mass action principle. Moreover, they assumed that bats do not recover from EVD which was declined by many studies [13, 15, 16]. With this in mind, Rhoubari et al. [17] improved the model of [14] and extended it to more general incidence functions. Further, they investigated the global stability of the two equilibria (disease-free and endemic) theoretically and numerically. The frequent emergence and the large damage of EVD boosted the mathematical modeling of EVD in the human population. Wang and Zhong [18] estimated the basic reproduction number, the inflection point, the final outbreak size and the peak time by dividing the population into two classes, susceptible and the infective. In [19], Rachah and Torres modeled the 2014 Ebola outbreak in Liberia using the classical SIR model, then they applied the optimal control theory in order to show the impact of vaccina-

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tion on the spread of the EV. In a follow-up work [20], they added the exposed class into the SIR model. Using the SEIR model, they proposed three strategies to control the spread of Ebola: the first by vaccination of susceptible, the second by minimizing exposed and infected, and the third by vaccination and education. Later in [21], they compared the results of the modeling of EVD by the SIR and the SEIR models. The comparison was based on numerical simulations and on the study of the optimal control strategies to control the spread of EV. Using a SEIR model, Althaus [22] estimated in September 2014, the evolution of the reproduction number of EV during the 2014 Ebola outbreaks in Guinea, Sierra Leone and Liberia. More precisely, he predicted no decline of this number in Liberia by the end of August 2014 and estimated that it might dropped in the two other countries by the end of May and July 2014. Also, Ghowell et al. [23] estimated the basic reproduction number in the two Ebola outbreaks of Congo in 1995 and Uganda in 2000 using a SEIR model. Moreover, they quantified the sensitivity of this number to some disease-related parameters. Grigorieva and Khailov [24] proposed two intervention control strategies by protecting susceptible individuals from infected and exposed individuals. A fractional SEIR model was proposed and analyzed in [25] to provide a prediction of the outbreak in Liberia, Guinea, and Sierra Leone. All the above models assumed that Ebola can be transmitted only from an alive infected individual. Motivated by the statistics in [26] and the work in [27], Rhoubari et al. [28] proposed a generalized model for EVD by considering the second mode of transmission from a dead to a living person. They modeled the mechanism of Ebola transmission by two general incidence functions and established a threshold parameter and conditions that determine if the disease is still in the population or is eradicated. The first purpose of this study is to propose a mathematical model with ODEs to describe the dynamics of EVD by taking into account the three modes of transmission by adding the environmental route of Ebola infection. This model is given by the following system ⎧ dS = A − μS − f (S, I )I − g(S, D)D − h(S, P)P, ⎪ ⎪ ⎪ ddtI ⎪ ⎨ dt = f (S, I )I + g(S, D)D + h(S, P)P − (μ + d + r )I, dR = r I − μR, dt ⎪ dD ⎪ = (μ + d)I − bD, ⎪ ⎪ ⎩ ddtP = σ I + γ D − η P, dt

(1)

where S, I , R, D and P are susceptible, infectious, recovered, died and environmental classes, respectively. A susceptible individual can be infected either by an infectious individual at rate f (S, I )I , or by a deceased individual at rate g(S, D)D, or by a contaminated environment at rate h(S, P)P. Susceptible individuals increase at the recruitment rate A and die naturally at rate μ. Infectious individuals increase at the total Ebola infection rate f (S, I )I + g(S, D)D + h(S, P)P and decrease either at the natural death rate μ, or at the death rate due to EVD d. Recovered individuals convert from the infectious class at the recovery rate r and die only at the rate μ. In addition, the deceased individuals who carry EV are assumed to be buried directly during

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Fig. 1 Schematic diagram of model (1)

funerals at rate b. On the other hand, Ebola viruses are generated in the environment from infectious and deceased human individuals at rates σ and γ respectively and decay at rate η. A schematic diagram of model (1) is provided Fig. 1. Since the third equation of system (1) does not depend to the others, we can reduce (1) to the following system ⎧ dS = A − μS − f (S, I )I − g(S, D)D − h(S, P)P, ⎪ ⎪ ⎨ ddtI = f (S, I )I + g(S, D)D + h(S, P)P − (μ + d + r )I, dt dD = (μ + d)I − bD, ⎪ ⎪ ⎩ ddtP = σ I + γ D − η P. dt

(2)

As in [17, 28, 29], we assume that the general incidences f , g and h are continuously differentiable in the interior of IR2+ and satisfy the following hypotheses: ∂f ∂f (S, I ) > 0, (S, I ) ≤ 0 for all S, I ≥ 0. ∂S ∂I ∂g ∂g (H2 ) g(0, D) = 0, (S, D) > 0, (S, D) ≤ 0 for all S, D ≥ 0. ∂S ∂D ∂h ∂h (H3 ) h(0, P) = 0, (S, P) > 0, (S, P) ≤ 0 for all S, P ≥ 0. ∂S ∂P It is very important to note that the simple epidemic model for Ebola in humans introduced by Berge et al. [14] is a special case of our model (2) when taking f (S, I ) = β1 S, g(S, D) = β2 S and h(S, P) = β3 S. (H1 ) f (0, I ) = 0,

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On the other hand, it has been shown that West Africa population mobility is seven times higher than elsewhere in the world [30]. This high mobility contributes to losing control of the spread of EVD. Therefore, the spatial analysis should not be neglected. For this reason, the second purpose of this study is to incorporate the evolution of EVD in space. So, we assume that populations follow the Fickian diffusion, which means that the fluxes of these populations are proportional to their concentration gradient and move from regions of high concentration to regions of low concentration. Hence, system (2) becomes, ⎧ ∂S ⎪ ⎪ = d S S + A − μS(x, t) − f (S(x, t), I (x, t))I (x, t) ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ − g(S(x, t), D(x, t))D(x, t) − h(S(x, t), P(x, t))P(x, t), ⎪ ⎪ ⎪ ⎪ ∂ I ⎪ ⎪ ⎨ = d I I + f (S(x, t), I (x, t))I (x, t) + g(S(x, t), D(x, t))D(x, t) ∂t ⎪ + h(S(x, t), P(x, t))P(x, t) − (μ + d + r )I (x, t), ⎪ ⎪ ⎪ ⎪ ∂ D ⎪ ⎪ ⎪ = d D D + (μ + d)I (x, t) − bD(x, t), ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎩ ∂ P = d P P + σ I (x, t) + γ D(x, t) − η P(x, t), ∂t

(3)

where S(x, t), I (x, t), D(x, t) and P(x, t) denote the concentrations of susceptible, infectious, deceased individuals and EV in the environment, respectively at location x ∈  and time t. d S , d I , d D and d P are the diffusion coefficients for these four classes and  is the Laplacien operator. The domain  is bounded in IRn with smooth boundary ∂. Further, we assume that the four populations do not move across ∂. For this reason, we consider system (3) with the homogeneous Neumann boundary conditions ∂I ∂D ∂P ∂S = = = = 0, on ∂ × (0, +∞), ∂ν ∂ν ∂ν ∂ν

(4)

and initial conditions S(x, 0) = S0 (x) ≥ 0, I (x, 0) = I0 (x) ≥ 0, D(x, 0) = D0 (x) ≥ 0, ¯ P(x, 0) = P0 (x) ≥ 0, x ∈ .

(5)

∂ is the outward normal derivative on ∂. Here, ∂ν The rest of the chapter is organized as follows. The mathematical analysis of the ODE and the PDE models is given in Sects. 2 and 3, respectively. An application and numerical simulations are presented in Sect. 4 in order to illustrate our theoretical results. We give a conclusion in the last section.

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2 Mathematical Analysis of the ODE Model In this section, we first prove the well-posedness of ODE model (2) and determine the basic reproduction number and equilibria. After, we investigate the global stability of the corresponding equilibria.

2.1 Well-Posedness and Equilibria Before discussing the existence of equilibria, we first show that our ODE model (2) is well-posed. Let IR4+ = {(S, I, D, P) ∈ IR4 : S ≥ 0, I ≥ 0, D ≥ 0, P ≥ 0}. We have the following result. Theorem 2.1. The first quadrant IR4+ is positively invariant with respect (2). Moreover, all solutions of (2) are uniformly bounded in the compact subset   A A(μ + d) σ A γ A(μ + d)

= (S, I, D, P) ∈ IR4+ : S + I ≤ , D ≤ , P≤ + . μ μb ημ bημ

Proof. Since d S  dt S=0 d I  dt I =0 d D  dt D=0 d P  dt P=0

= A > 0, = g(S, D)D + h(S, P)P ≥ 0, for all S, D, P ≥ 0, = (μ + d)I ≥ 0, for all I ≥ 0, = σ I + γ D ≥ 0, for all I, D ≥ 0,

we deduce that IR4+ is positively invariant with respect (2). Adding the first two equations of system (2), we obtain S˙ + I˙ = A − μS − (μ + d + r )I ≤ A − μ(S + I ). Then



A lim sup S(t) + I (t) ≤ . μ t→∞

Similarly, from the two last equations of system (2), we get lim sup D(t) ≤ t→∞

A(μ + d) , bμ

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and lim sup P(t) ≤ t→∞

301

σ A γ A(μ + d) + . ημ bημ

Therefore, all solutions of model (2) which start in IR4+ are eventually confined in the region . This completes the proof.  Now, we calculate the basic reproduction number and discuss the existence of equilibria. Obviously, model (2) has always one disease-free equilibrium E f ( μA , 0, 0, 0). We introduce the following number, which in the sequel (Theorem 2.3) we shall prove to be the basic reproduction number of (2), R0 =

ηb f

A μ







, 0 + η(μ + d)g uA , 0 + σ b + γ (μ + d) h μA , 0 η(μ + d + r )b

.

(6)

This number represents biologically the average number of secondary infections produced by one infectious individual when all individuals in the population under study are uninfected. Also, R0 can be rewritten as R01 +R02 +R03 , where R01 =

f

A μ

,0



μ+d +r

, R02 =

(μ + d)g

A μ

,0

(μ + d + r )b

and R03



σ b + γ (μ + d) h uA , 0 . = η(μ + d + r )b

R01 , R02 and R03 represent respectively the contribution of infectious living individuals, infectious corpses and contaminated environment on Ebola infection. The other equilibrium of (2) satisfies the following equations A − μS − f (S, I )I − g(S, D)D − h(S, P)P = 0,

(7)

f (S, I )I + g(S, D)D + h(S, P)P − (μ + d + r )I = 0, (μ + d)I − bD = 0, σ I + γ D − η P = 0.

(8) (9) (10)

By (7) to (10), we have  (μ + d) (μ + d)(A − μS) A − μS + g S, f S, μ+d +r b b(μ + d + r )  σ σ γ (μ + d)

γ (μ + d) A − μS

+ h S, + = (μ + d + r ). + η ηb η ηb μ+d +r 

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Since I =

Z. El Rhoubari et al. A−μS μ+d+r

≥ 0, we have S ≤

librium when S >

A . Therefore, there is no epidemiological equiμ

A . Hence, we define a function ψ on the interval [0, μA ] by μ

  A − μS (μ + d) (μ + d)(A − μS) ψ(S) = f S, + g S, μ+d +r b b(μ + d + r )  σ σ γ (μ + d)

γ (μ + d) A − μS

+ h S, + − (μ + d + r ). + η ηb η ηb μ+d +r A We have ψ(0) = −(μ + d + r ) < 0, ψ( ) = (μ + d + r )(R0 − 1) and μ ψ  (S) =

μ ∂f ∂f − ∂S μ + d + r ∂I  μ + d ∂g μ(μ + d) ∂g + − b ∂S b(μ + d + r ) ∂ D 

∂h σ γ (μ + d) ∂h σ γ (μ + d) μ + − + > 0. + η ηb ∂S η ηb μ + d +r ∂P



Thus, for R0 > 1, there exists a unique endemic equilibrium E ∗ (S ∗ , I ∗ , D ∗ , P ∗ ) with A S ∗ ∈ (0, ), I ∗ > 0, D ∗ > 0 and P ∗ > 0. μ The above discussions can be summarized in the following theorem. Theorem 2.2. Let R0 defined by Eq. (6). (i) If R0 ≤ 1, then model (2) has a unique disease-free equilibrium of the form E f (S 0 , 0, 0, 0), where S 0 = μA . (ii) If R0 > 1, the disease-free equilibrium is still present and model (2) has a unique endemic equilibrium of the form E ∗ (S ∗ , I ∗ , D ∗ , P ∗ ) with S ∗ ∈ (0, μA ), I ∗ > 0, D ∗ > 0 and P ∗ > 0.

2.2 Stability Analysis The aim of this subsection is to investigate the stability of the disease-free equilibrium E f and the endemic equilibrium E ∗ . First, we have the following result. Theorem 2.3. The disease-free equilibrium E f of model (2) is globally asymptotically stable when R0 ≤ 1 and it becomes unstable if R0 > 1.

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Proof. Consider the following functional V , which we shall prove to be Lyapunov at Ef

A A A V (S, I, D, P) = ηbI + ηg( , 0) + γ h( , 0) D + bh( , 0)P. (11) μ μ μ The derivative of the functional V with respect to time t and along the positive solutions of (2) satisfies dV = dt





A A A ηb f (S, I ) + (μ + d) ηg( , 0) + γ h( , 0) + bσ h( , 0) − ηb(μ + d + r ) I μ μ μ 

A A + ηbg(S, D) − b ηg( , 0) + γ h( , 0) + γ bh(S, P) D μ μ  A + bηh(S, P) − bηh( , 0) P. μ

A , which implies that all omega limit points satisfy S(t) ≤ μ t→∞ A . Hence, it suffices to consider solutions for which S(t) ≤ μA . From the expression μ (6) of R0 and (H1 )–(H3 ), we get

We have lim sup S(t) ≤



dV ≤ η(μ + d + r ) R0 − 1 I. dt Consequently, ddtV ≤ 0 for R0 ≤ 1. Further, it is easy to show that the largest compact invariant set in {(S, I, D, P)| ddtV = 0} is the singleton {E f }. By LaSalle invariance principle [31], we conclude that E f is globally asymptotically stable if R0 ≤ 1. In order to prove the remaining part of the theorem, we determine the characteristic equation at the disease-free equilibrium E f . This characteristic equation is given by (μ + λ) (λ) = 0, where

A

(λ) = λ3 + λ2 η + b + μ + d + r − f ( , 0) μ

A A A + λ (b + η)(μ + d + r ) − (b + η) f ( , 0) − (μ + d)g( , 0) − σ h( , 0) μ μ μ + bη(μ + d + r )(1 − R0 ).

We have lim (λ) = +∞ and (0) = bη(μ + d + r )(1 − R0 ). If R0 > 1, then λ→+∞

(0) < 0. Thus, there exists a λ0 ∈ (0, +∞) such that (λ0 ) = 0, which implies that the characteristic equation at E f has a positive root when R0 > 1. Therefore, E f is  unstable when R0 > 1.

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Next, we establish the global stability of the endemic equilibrium E ∗ by assuming that R0 > 1 and the functions f , g and h satisfy, for all S, I, D, P > 0, the following hypothesis  1−  1−  1−

 f (S, I ) f (S, I ∗ ) I − ∗ ≤ 0, f (S, I ∗ ) f (S, I ) I  ∗ ∗ f (S , I )g(S, D) f (S, I ∗ )g(S ∗ , D ∗ ) − f (S, I ∗ )g(S ∗ , D ∗ ) f (S ∗ , I ∗ )g(S, D)  f (S ∗ , I ∗ )h(S, P) f (S, I ∗ )h(S ∗ , P ∗ ) − f (S, I ∗ )h(S ∗ , P ∗ ) f (S ∗ , I ∗ )h(S, P)

D ≤0 D∗ P ≤ 0. P∗

(H4 )

Theorem 2.4. Assume that R0 > 1 and (H4 ) holds. The endemic equilibrium E ∗ is globally asymptotically stable if σ = 0 or γ = 0. Proof. We define the following functional W , which we shall prove to be Lyapunov at E ∗ 

S f (S ∗ , I ∗ ) I ∗ W (S, I, D, P) = S − S ∗ − H d X + I ∗ I∗ S ∗ f (X, I )   ∗ ∗ γ g(S , D ) D (12) + h(S ∗ , P ∗ ) D ∗ H + b ηb D∗  h(S ∗ , P ∗ ) ∗ P , P H + η P∗ where H (X ) = X − 1 − ln X , X > 0. Obviously, H : (0, +∞) → [0, +∞) attains its strict global minimum at X = 1 and H (1) = 0. Then H (X ) ≥ 0 for all X > 0. Since f (S, I ∗ ) is an increasing function with respect S, we deduce that the function S ∗ ∗ ,I ) S → S − S ∗ − S ∗ ff (S d X is nonnegative. Thus, the functional W is also (X,I ∗ ) nonnegative. The time derivative of W along the positive solution of system (2) is given by   f (S ∗ , I ∗ ) d S I∗ dI dW = 1− + 1 − dt f (S, I ∗ ) dt I dt   γ g(S ∗ , D ∗ ) D∗ d D + h(S ∗ , P ∗ ) 1 − + b ηb D dt  P∗ d P h(S ∗ , P ∗ ) 1− . + η P dt

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Using A = μS ∗ + f (S ∗ , I ∗ )I ∗ + g(S ∗ , D ∗ )D ∗ + h(S ∗ , P ∗ )P ∗ = μS ∗ + (μ + d + r )I ∗ , (μ + d)I ∗ = bD ∗ and η P ∗ = σ I ∗ + γ D ∗ , we obtain   S f (S ∗ , I ∗ ) dW ∗ 1− = μS 1 − ∗ dt S f (S, I ∗ )  f (S ∗ , I ∗ ) f (S, I )I I ∗ ∗ ∗ + + f (S , I )I 2 − − ∗ f (S, I ∗ ) f (S, I ∗ )I ∗ I  ∗ ∗ ∗ ∗ f (S , I ) f (S , I )g(S, D)D D + + g(S ∗ , D ∗ )D ∗ 3 − − ∗ f (S, I ∗ ) f (S, I ∗ )g(S ∗ , D ∗ )D ∗ D D∗ I g(S, D)D I ∗ − + g(S ∗ , D ∗ )D ∗ I DI∗  f (S ∗ , I ∗ )h(S, P)P P f (S ∗ , I ∗ ) + + h(S ∗ , P ∗ )P ∗ 3 − − ∗ ∗ f (S, I ) f (S, I ∗ )h(S ∗ , P ∗ )P ∗ P P∗ I h(S, P)P I ∗ − + h(S ∗ , P ∗ )P ∗ I PI∗  γ D∗ I P∗ D P∗ I + h(S ∗ , P ∗ )D ∗ 1 + − − . η PI∗ DI∗ P D∗ Thus,   S f (S ∗ , I ∗ ) dW 1− = μS ∗ 1 − ∗ dt S f (S, I ∗ )  f (S, I )I I f (S, I ∗ ) + + f (S ∗ , I ∗ )I ∗ − 1 − ∗ + I f (S, I ) f (S, I ∗ )I ∗  ∗ f (S ∗ , I ∗ )g(S, D)D D f (S, I )g(S ∗ , D ∗ ) + + g(S ∗ , D ∗ )D ∗ − 1 − ∗ + D f (S ∗ , I ∗ )g(S, D) f (S, I ∗ )g(S ∗ , D ∗ )D ∗  ∗ ∗ ∗ f (S ∗ , I ∗ )h(S, P)P P f (S, I )h(S , P ) ∗ ∗ ∗ + + h(S , P )P − 1 − ∗ + P f (S ∗ , I ∗ )h(S, P) f (S, I ∗ )h(S ∗ , P ∗ )P ∗  ∗) ∗, I ∗) f (S f (S, I ) f (S, I − − + f (S ∗ , I ∗ )I ∗ 3 − f (S, I ) f (S, I ∗ ) f (S ∗ , I ∗ )  ∗ ∗ ∗ f (S, I )g(S ∗ , D ∗ ) g(S, D)D I ∗ I D∗ f (S , I ) − − − + g(S ∗ , D ∗ )D ∗ 4 − f (S, I ∗ ) f (S ∗ , I ∗ )g(S, D) g(S ∗ , D ∗ )D ∗ I I∗D  ∗ ∗ ∗ ∗ ∗ ∗ f (S, I )h(S , P ) h(S, P)P I I P∗ f (S , I ) − − − + h(S ∗ , P ∗ )P ∗ 4 − f (S, I ∗ ) f (S ∗ , I ∗ )h(S, P) h(S ∗ , P ∗ )P ∗ I I∗P  ∗ ∗ ∗ P I γ D I P D . + h(S ∗ , P ∗ )D ∗ 1 + − − η PI∗ DI∗ P D∗

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Since the arithmetic mean is greater than or equal to the geometric mean, we have 3−

f (S, I ∗ ) f (S ∗ , I ∗ ) f (S, I ) − − ≤ 0, f (S, I ) f (S, I ∗ ) f (S ∗ , I ∗ )

4−

f (S ∗ , I ∗ ) f (S, I ∗ )g(S ∗ , D ∗ ) g(S, D)D I ∗ I D∗ − − − ≤ 0, f (S, I ∗ ) f (S ∗ , I ∗ )g(S, D) g(S ∗ , D ∗ )D ∗ I I∗D

4−

f (S, I ∗ )h(S ∗ , P ∗ ) h(S, P)P I ∗ I P∗ f (S ∗ , I ∗ ) − − − ≤ 0. f (S, I ∗ ) f (S ∗ , I ∗ )h(S, P) h(S ∗ , P ∗ )P ∗ I I∗P

and

Moreover, with the help of the hypothesis (H1 ), we have  1−

S S∗

 f (S ∗ , I ∗ ) 1− ≤ 0. f (S, I ∗ )

According to (H4 ), we obtain −1 −

  f (S, I ∗ ) f (S, I ∗ ) f (S, I ) I f (S, I )I I ≤ 0, + = 1 − + − I∗ f (S, I ) f (S, I ∗ )I ∗ f (S, I ∗ ) f (S, I ) I∗ D f (S ∗ , I ∗ )g(S, D)D f (S, I ∗ )g(S ∗ , D ∗ ) −1 − ∗ + + D f (S ∗ , I ∗ )g(S, D) f (S, I ∗ )g(S ∗ , D ∗ )D ∗   ∗ ∗ ∗ f (S, I )g(S ∗ , D ∗ ) D f (S , I )g(S, D) − ∗ ≤ 0, = 1− f (S, I ∗ )g(S ∗ , D ∗ ) f (S ∗ , I ∗ )g(S, D) D

and f (S ∗ , I ∗ )h(S, P)P P f (S, I ∗ )h(S ∗ , P ∗ ) + −1 − ∗ + P f (S ∗ , I ∗ )h(S, P) f (S, I ∗ )h(S ∗ , P ∗ )P ∗   ∗ ∗ ∗ f (S, I )h(S ∗ , P ∗ ) P f (S , I )h(S, P) ≤ 0. − = 1− f (S, I ∗ )h(S ∗ , P ∗ ) f (S ∗ , I ∗ )h(S, P) P∗ If γ = 0, then

dW dt

≤ 0.

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If σ = 0, then η P ∗ = γ D ∗ , which implies that P ∗ =

γ η

307

D ∗ . In this case, we have

  dW S f (S ∗ , I ∗ ) 1− = μS ∗ 1 − ∗ ∗ dt S f (S, I )  f (S, I )I f (S, I ∗ ) I ∗ ∗ ∗ + f (S , I )I + −1− ∗ + I f (S, I ) f (S, I ∗ )I ∗  ∗ D f (S, I )g(S ∗ , D ∗ ) f (S ∗ , I ∗ )g(S, D)D + g(S ∗ , D ∗ )D ∗ − 1 − ∗ + + D f (S ∗ , I ∗ )g(S, D) f (S, I ∗ )g(S ∗ , D ∗ )D ∗  ∗ )h(S ∗ , P ∗ ) f (S, I P f (S ∗ , I ∗ )h(S, P)P + h(S ∗ , P ∗ )P ∗ − 1 − ∗ + + P f (S ∗ , I ∗ )h(S, P) f (S, I ∗ )h(S ∗ , P ∗ )P ∗  ∗) ∗, I ∗) f (S, I f (S f (S, I ) − − + f (S ∗ , I ∗ )I ∗ 3 − ∗ ∗ ∗ f (S, I ) f (S, I ) f (S , I )  ∗, I ∗) ∗ )g(S ∗ , D ∗ ) f (S, I g(S, D)D I ∗ I D∗ f (S + g(S ∗ , D ∗ )D ∗ 4 − − − − ∗ ∗ ∗ ∗ ∗ ∗ ∗ f (S, I ) f (S , I )g(S, D) g(S , D )D I I D  f (S ∗ , I ∗ ) f (S, I ∗ )h(S ∗ , P ∗ ) h(S, P)P I ∗ D∗ I P∗ D ∗ ∗ ∗ + h(S , P )P 5 − − − − − f (S, I ∗ ) f (S ∗ , I ∗ )h(S, P) h(S ∗ , P ∗ )P ∗ I DI∗ P D∗

Since 5−

f (S, I ∗ )h(S ∗ , P ∗ ) h(S, P)P I ∗ D∗ I P∗ D f (S ∗ , I ∗ ) − − − − ≤ 0, f (S, I ∗ ) f (S ∗ , I ∗ )h(S, P) h(S ∗ , P ∗ )P ∗ I DI∗ P D∗

≤ 0. we get dW dt = 0 if and only if S = S ∗ , I = I ∗ , On the other hand, it is not hard to show that dW dt ∗ ∗ D = D and P = P . It follows from LaSalle invariance principle that E ∗ is globally asymptotically stable. 

3 Mathematical Analysis of the PDE Model In this section, we study the well-posedness, equilibria and stability analysis of PDE model (3).

3.1 Well-Posedness and Equilibria Firstly, we prove the well-posedness of the PDE model (3) by establishing the existence, uniqueness, nonnegativity and boundedness of solutions. ¯ 4 satTheorem 3.1. For any initial condition u 0 = (S0 , I0 , D0 , P0 )T ∈ C = [C()] isfying (5), there exists a unique solution of problem (3)–(5) defined on [0, +∞) and this solution remains nonnegative and bounded for all t ≥ 0.

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Proof. System (3)–(5) can be written abstractly in the Banach space C as follows u  (t) = Au(t) + F(u(t)), u(0) = u 0 ∈ C,

t > 0,

(13)

where u = (S, I, D, P)T , Au = (d S S, d I I, d D D, d P P)T and ⎛

⎞ A − μS − f (S, I )I − g(S, D)D − h(S, P)P ⎜ f (S, I )I + g(S, D)D + h(S, P)P − (μ + d + r )I ⎟ ⎟. F(u) = ⎜ ⎝ ⎠ (μ + d)I − bD σ I + γ D − ηP It is clear that F is locally Lipschitz in C. From [32], we deduce that system (13) admits a unique local solution on [0, Tmax ), where Tmax is the maximal existence time for solution of system (13). In addition, all solutions are nonnegative since 0 is a lower solution for each equation of model (3). Now, we show the boundedness of the solution. From (3)–(5), we have ⎧ ∂S ⎪ ⎪ − d S S ≤ A − μS, ⎪ ⎪ ⎨ ∂t ∂S (14) = 0, ⎪ ⎪ ∂ν ⎪ ⎪ ⎩ S(x, 0) = S0 (x) ≤ S0 ∞ = max S0 (x), x∈

By applying Lemma 1 given by Hattaf in [33], we get A ¯ × [0, Tmax ). S(x, t) ≤ max{ , S0 ∞ }, ∀(x, t) ∈  μ From Theorem 3.1 given by Alikakos in [34], to establish the L ∞ uniform boundedness of I (x, t), it is sufficient to show the L 1 uniform boundedness of I (x, t). Since ∂I ∂S = = 0 and ∂ν ∂ν





∂ S + I − dS S + d I I ≤ A − μ S + I , ∂t we get

Hence,

∂ ∂t



(S + I )d x ≤ mes()A − μ



(S + I )d x .

A (S + I )d x ≤ mes() max{ , S0 + I0 ∞ }, μ 

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A I (x, t)d x ≤ K := mes() max{ , S0 + I0 ∞ }. μ Using Theorem 3.1 in [34], we deduce that there exists a positive constant K ∗ that depends on K and on S0 + I0 ∞ such that

which implies that, supt≥0



sup I (., t)∞ ≤ K ∗ . t≥0

¯ × From the above, we have proved that S(x, t) and I (x, t) are L ∞ bounded on  [0, Tmax ). Now, by the boundedness of I and the system (3)–(5), D satisfies the following system ⎧ ∂D ⎪ ⎪ − d D D ≤ (μ + d)M1 − bD, ⎪ ⎪ ⎨ ∂t ∂D = 0, ⎪ ⎪ ∂ν ⎪ ⎪ ⎩ D(x, 0) = D0 (x) ≤ D0 ∞ = max D0 (x),

(15)

x∈

where M1 = max{I (x, t), (x, t) ∈  × [0, Tmax )}. Therefore, we get D(x, t) ≤ max{

(μ + d)M1 ¯ × [0, Tmax ). , D0 ∞ }, ∀(x, t) ∈  b

Similarly, using the boundedness of I and D and the system (3)–(5), P satisfies the following system ⎧ ∂P ⎪ ⎪ − d P P ≤ σ M1 + γ M2 − η P, ⎪ ⎪ ⎨ ∂t ∂P = 0, ⎪ ⎪ ∂ν ⎪ ⎪ ⎩ P(x, 0) = P0 (x) ≤ P0 ∞ = max P0 (x),

(16)

x∈

(μ + d)M1 , D0 ∞ }. where M2 = max{ b Then σ M1 + γ M2 ¯ × [0, Tmax ). , P0 ∞ }, ∀(x, t) ∈  P(x, t) ≤ max{ η ¯ × [0, Tmax ). It follows Therefore, all compartments of model (2) are bounded on  from the standard theory for semilinear parabolic systems [35] that Tmax = +∞. This completes the proof.  It is obvious that the equilibrium points E f (S 0 , 0, 0, 0) and E ∗ (S ∗ , I ∗ , D ∗ , P ∗ ) of ODE model (2) are also the steady states of our PDE model (3). It remains to study the stability analysis of these equilibria.

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3.2 Stability Analysis For the stability of the disease-free equilibrium E f , we have the following result. Theorem 3.2. The disease-free equilibrium E f of model (3) is globally asymptotically stable when R0 ≤ 1 and it becomes unstable if R0 > 1. Proof. Based on the Hattaf-Yousfi method described in [36], we construct the Lyapunov functional for the PDE model (3) at E f with the help of (11) as follows

V=



V (S(x, t), I (x, t), D(x, t), P(x, t))d x.

Calculating the time derivative of L along the solution of model (3), we have dV = dt

 d S S + d I I + d D D + d P P + ηb f (S(x, t), I (x, t)) 



A A A + (μ + d) ηg( , 0) + γ h( , 0) + bσ h( , 0) − ηb(μ + d + r ) I (x, t) μ μ μ 

A A + ηbg(S(x, t), D(x, t)) − b ηg( , 0) + γ h( , 0) + γ bh(S(x, t), P(x, t)) D(x, t) μ μ   A + bηh(S(x, t), P(x, t)) − bηh( , 0) P(x, t) d x. μ

From (14) and by applying Lemma 1 of [33], we obtain

lim sup S(x, t) ≤ S 0 . t→∞

This implies that all omega limit points satisfy S(x, t) ≤ S 0 . So, It sufficient to consider solutions for which S(x, t) ≤ S 0 . By the explicit formula of R0 in (6) and the hypotheses (H1 )–(H3 ), we get dV ≤ dt



 η(μ + d + r )(R0 − 1)I d x. 

dV ≤ 0 for R0 ≤ 1. In addition, it is clear that the singleton {E f } is the dt dV = 0}. By LaSalle invariance principle, we largest invariant set in {(S, I, D, P)| dt deduce that E f is globally asymptotically stable when R0 ≤ 1. Now, we establish the instability property of the disease-free equilibrium E f when R0 > 1. Let 0 = λ1 < λ2 < ... < λn < ... be the eigenvalues of the operator − on  under the homogeneous boundary conditions and F(λi ) be the eigenfunction space corresponding to the eigenvalue λi in C 1 (). Therefore,

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  Let ϕi j : j = 1, 2, ..., dim F(λi ) be an orthonormal basis of F(λi ), X = [C 1 ()]4   and Xi j = dϕi j : d ∈ IR4 . Then we have X=

∞  i=1

Xi

and

Xi =

dimF(λ  i)

Xi j

j=1

  Let E ∗ = S∗ , I∗ , D∗ , P∗ be an arbitrary spatially homogeneous equilibrium of the system (3)–(5). Introduce the following change of variables: U1 (x, t) = S(x, t) − S∗ , U2 (x, t) = I (x, t) − I∗ , U3 (x, t) = D(x, t) − D∗ and U4 (x, t) = P(x, t) − P∗ . Then the linearization of system (3) at E ∗ is given by ∂U = D U + J U (x, t), ∂t where

(17)

  J = JS , J I , J D , J P ,

with  JS =

 JI =

− I∗

∂f ∂g ∂h ∂f ∂g ∂h − D∗ − P∗ − μ, I∗ + D∗ + P∗ , 0, 0 ∂S ∂S ∂P ∂S ∂S ∂S

T

∂f ∂f + f (S∗ , I∗ ) − (μ + d + r ), μ + d, σ − f (S∗ , I∗ ) − I∗ , I∗ ∂I ∂I 

JD =

− D∗ 

JP =

∂g ∂g − g(S∗ , D∗ ), D∗ + g(S∗ , D∗ ), −b, γ ∂D ∂D

∂h ∂h − h(S∗ , P∗ ), P∗ + h(S∗ , P∗ ), 0, −η − P∗ ∂P ∂P

,

T ,

T ,

T ,

D = diag(d S , d I , d D , d P ) and U = (U1 , U2 , U3 , U4 )T . Let LU = D U + J U (x, t). For each i ≥ 1, Xi is invariant under the operator L. In addition, ξ is an eigenvalue of L if and only if it is a root of the equation det(−λi D + J − ξ I4 ) = 0,

(18)

for some i ≥ 1, for which there exists an eigenvector in Xi . On the other hand, the characteristic equation at the disease-free equilibrium E f is given by (μ + λi d S + ξ )i (ξ ) = 0,

(19)

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where  A i (ξ ) = ξ 3 + λi (d I + d D + d P ) + μ + r + η + b + d − f ( , 0) ξ 2 μ   2 + λi (d D d P + d I d P + d D d I ) + λi b(d P + d I ) + d(d P + d P ) A A + (μ + d + r − f ( , 0))(d D + d P ) + b(μ + d + r ) + ηb − (η + b) f ( , 0) μ μ  A A − (μ + d)g( , 0) − σ h( , 0) + η(μ + d + r ) ξ μ μ  A + λi3 (d I d P d D ) + λi2 d D d P (μ + d + r ) − f ( , 0) + bd I d P + ηd D d I ) μ  A A + λi (ηd D + bd P )(μ + d + r ) + ηbd I − d P (b f ( , 0) + (μ + d)g( , 0)) μ μ A A − d D (η f ( , 0) + σ h( , 0)) + ηb(μ + d + r )(1 − R0 ) μ μ

It is obvious that ξ = −μ − λi d S is a negative root of the Eq. (19). The remaining roots are the solutions of the equation i (ξ ) = 0. Since λ1 = 0 and R0 > 1, we have 1 (0) = ηb(μ + d + r )(1 − R0 ) < 0. In addition, we have lim i (ξ ) = ξ →+∞

+∞. Thus, there exists a real positive root ξ ∗ of the equation 1 (ξ ) = 0. This implies that the disease-free equilibrium E f is unstable if R0 > 1. This completes the proof  Next, we investigate the global stability of the endemic equilibrium E ∗ . We have the following result Theorem 3.3. Assume that R0 > 1 and (H4 ) holds. The endemic equilibrium E ∗ of model (3) is globally asymptotically stable if γ = 0 or σ = 0. Proof. Similarly to above and based on the Lyapunov functional (12) of ODE model, we build the Lyapunov functional of PDE model for the endemic equilibrium E ∗ as

W=



W (S(x, t), I (x, t), D(x, t), P(x, t))d x.

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Calculating the time derivative of P along the solution of model (3), we have 

 dW f (S ∗ , I ∗ ) d S I∗ dI 1− = + 1− ∗ dt f (S, I ) dt I dt    g(S ∗ , D ∗ ) γ D∗ d D ∗ ∗ + h(S , P ) 1 − + b ηb D dt   P∗ d P h(S ∗ , P ∗ ) 1− d x. + η P dt  

 S f (S ∗ , I ∗ ) = μS ∗ 1 − ∗ 1− S f (S, I ∗ )   I f (S, I )I f (S, I ∗ ) + + f (S ∗ , I ∗ )I ∗ − 1 − ∗ + ∗ ∗ I f (S, I ) f (S, I )I  ∗ )g(S ∗ , D ∗ ) f (S ∗ , I ∗ )g(S, D)D D f (S, I + g(S ∗ , D ∗ )D ∗ − 1 − ∗ + + ∗ ∗ ∗ ∗ ∗ ∗ D f (S , I )g(S, D) f (S, I )g(S , D )D  P f (S ∗ , I ∗ )h(S, P)P f (S, I ∗ )h(S ∗ , P ∗ ) ∗ ∗ ∗ + h(S , P )P − 1 − ∗ + + P f (S ∗ , I ∗ )h(S, P) f (S, I ∗ )h(S ∗ , P ∗ )P ∗  ∗ ∗ ∗ f (S , I ) f (S, I ) f (S, I ) + f (S ∗ , I ∗ )I ∗ 3 − − − f (S, I ) f (S, I ∗ ) f (S ∗ , I ∗ )  ∗, I ∗) ∗ )g(S ∗ , D ∗ ) f (S, I g(S, D)D I ∗ I D∗ f (S + g(S ∗ , D ∗ )D ∗ 4 − − − − f (S, I ∗ ) f (S ∗ , I ∗ )g(S, D) g(S ∗ , D ∗ )D ∗ I I∗D  ∗, I ∗) ∗ )h(S ∗ , P ∗ ) ∗ f (S, I h(S, P)P I I P∗ f (S − − − ∗ + h(S ∗ , P ∗ )P ∗ 4 − ∗ ∗ ∗ ∗ ∗ ∗ f (S, I ) f (S , I )h(S, P) h(S , P )P I I P  ∗I ∗I ∗D  ∗, I ∗) γ P f (S D P + h(S ∗ , P ∗ )D ∗ 1 + + 1− d S S − − η PI∗ DI∗ P D∗ f (S, I ∗ )    I∗ γ g(S ∗ , D ∗ ) D∗ + 1− d I I + + h(S ∗ , P ∗ ) 1 − d D D I b ηb D   P∗ h(S ∗ , P ∗ ) 1− d P P d x. + η P

Using the zero-flux boundary (4) and the divergence theorem, we get



d S Sd x = d S

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 P∗ ∇ P2 1− d P Pd x = −d P P ∗ d x. P P2  

This implies that dW = dt

 

 S f (S ∗ , I ∗ ) μS ∗ 1 − ∗ 1− S f (S, I ∗ )   f (S, I ∗ ) f (S, I )I I + f (S ∗ , I ∗ )I ∗ − 1 − ∗ + + I f (S, I ) f (S, I ∗ )I ∗  ∗ )g(S ∗ , D ∗ ) D f (S, I f (S ∗ , I ∗ )g(S, D)D + g(S ∗ , D ∗ )D ∗ − 1 − ∗ + + D f (S ∗ , I ∗ )g(S, D) f (S, I ∗ )g(S ∗ , D ∗ )D ∗  ∗ )h(S ∗ , P ∗ ) P f (S, I f (S ∗ , I ∗ )h(S, P)P + h(S ∗ , P ∗ )P ∗ − 1 − ∗ + + P f (S ∗ , I ∗ )h(S, P) f (S, I ∗ )h(S ∗ , P ∗ )P ∗  ∗) ∗, I ∗) f (S, I f (S f (S, I ) + f (S ∗ , I ∗ )I ∗ 3 − − − f (S, I ) f (S, I ∗ ) f (S ∗ , I ∗ )  ∗ ∗ ∗ f (S , I ) f (S, I )g(S ∗ , D ∗ ) g(S, D)D I ∗ I D∗ + g(S ∗ , D ∗ )D ∗ 4 − − − − f (S, I ∗ ) f (S ∗ , I ∗ )g(S, D) g(S ∗ , D ∗ )D ∗ I I∗D  ∗ ∗ ∗ ∗ ∗ ∗ f (S , I ) f (S, I )h(S , P ) h(S, P)P I I P∗ + h(S ∗ , P ∗ )P ∗ 4 − − − − f (S, I ∗ ) f (S ∗ , I ∗ )h(S, P) h(S ∗ , P ∗ )P ∗ I I∗P   ∗ ∗ ∗ D I P D γ P I − − + h(S ∗ , P ∗ )D ∗ 1 + dx η PI∗ DI∗ P D∗

∇ S2 ∂f ∇ I 2 d x − dI I ∗ dx − f (S ∗ , I ∗ )d S (S, I ∗ ) ∗ 2  f (S, I ) I2  ∂S  

∇ D2 g(S ∗ , D ∗ ) γ − dx + h(S ∗ , P ∗ ) d D D ∗ b ηb D2 

∇ P2 h(S ∗ , P ∗ ) − d x. dP P∗ η P2 

∂f dW (S, I ∗ ) > 0. Then ≤ 0 if γ = 0. According to (H1 ), we have ∂S dt When σ = 0, we obtain  

 S f (S ∗ , I ∗ ) dW = μS ∗ 1 − ∗ 1− dt S f (S, I ∗ )   f (S, I )I I f (S, I ∗ ) ∗ ∗ ∗ + −1− ∗ + + f (S , I )I I f (S, I ) f (S, I ∗ )I ∗  f (S ∗ , I ∗ )g(S, D)D D f (S, I ∗ )g(S ∗ , D ∗ ) + + g(S ∗ , D ∗ )D ∗ − 1 − ∗ + D f (S ∗ , I ∗ )g(S, D) f (S, I ∗ )g(S ∗ , D ∗ )D ∗  ∗ ∗ ∗ f (S ∗ , I ∗ )h(S, P)P P f (S, I )h(S , P ) + + h(S ∗ , P ∗ )P ∗ − 1 − ∗ + P f (S ∗ , I ∗ )h(S, P) f (S, I ∗ )h(S ∗ , P ∗ )P ∗

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 f (S, I ∗ ) f (S ∗ , I ∗ ) f (S, I ) + f (S ∗ , I ∗ )I ∗ 3 − − − f (S, I ) f (S, I ∗ ) f (S ∗ , I ∗ )  ∗ ∗ ∗ f (S , I ) f (S, I )g(S ∗ , D ∗ ) g(S, D)D I ∗ I D∗ + g(S ∗ , D ∗ )D ∗ 4 − − − − f (S, I ∗ ) f (S ∗ , I ∗ )g(S, D) g(S ∗ , D ∗ )D ∗ I I∗D  ∗ ∗ ∗ ∗ ∗ ∗ f (S , I ) f (S, I )h(S , P ) h(S, P)P I D∗ I + h(S ∗ , P ∗ )P ∗ 5 − − − − f (S, I ∗ ) f (S ∗ , I ∗ )h(S, P) h(S ∗ , P ∗ )P ∗ I DI∗ 

∗ 2 ∂f ∇ I 2 P D ∇ S ∗ d x − f (S ∗ , I ∗ )d S − d x − d I dx (S, I ∗ ) I ∗ PD  f (S, I ∗ )2 I2  ∂S  

γ ∇ D2 g(S ∗ , D ∗ ) + h(S ∗ , P ∗ ) d D D ∗ − dx b ηb D2 

∇ P2 h(S ∗ , P ∗ ) dP P∗ d x. − η P2 

dW ≤ 0. Moreover, it can be easily shown that the largest compact invaridt dW = 0} is the singleton {E ∗ }. By LaSalle’s invariance prinant set in {(S, I, D, P)| dt ciple, we deduce that the endemic equilibrium E ∗ is globally asymptotically stable  when R0 > 1. This ends the proof.

Therefore,

4 Application and Numerical Simulations In this section, we apply the main above results to the following examples. Example 1: Consider the following ODE model ⎧ β2 S D β3 S P β1 S I dS ⎪ ⎪ ⎪ dt = A − μS − 1 + α I − 1 + α D − 1 + α P , ⎪ 1 2 3 ⎪ ⎨ β2 S D β3 S P β1 S I dI + + − (μ + d + r )I, = dt 1 + α1 I 1 + α2 D 1 + α3 P ⎪ ⎪ dD ⎪ ⎪ = (μ + d)I − bD, ⎪ ⎩ ddtP = σ I + γ D − η P, dt

(20)

where β1 , β2 and β3 are the infection rates caused by infectious individuals, deceased individuals and contaminated environment, respectively. The nonnegative constants αi with i ∈ {1, 2, 3} measure the saturation effect. System (20) is a particular case of model (2) with f (S, I ) =

β1 S , g(S, D) = 1 + α1 I

β2 S β3 S and h(S, P) = . In this case, the basic reproduction number R0 is 1 + α2 D 1 + α3 P given by

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R0 =

 μ+d σ b + γ (μ + d) A β1 + β2 + β3 . μ(μ + d + r ) b ηb

(21)

Clearly, the incidence functions f , g and h satisfy the hypotheses (H1 )−(H4 ). By applying Theorems 2.3 and 2.4, we have the following result. Corollary 4.1. (i) If R0 ≤ 1, then the disease-free equilibrium E f of model (20) is globally asymptotically stable. (ii) If R0 > 1, then E f becomes unstable and the endemic equilibrium E ∗ of model (20) is globally asymptotically stable when γ = 0 or σ = 0.

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globally asymptotically stable, which means that the disease disappears. This result is illustrated by Fig. 2. Now, we simulate the second case when R0 is greater than one. To this end, we change only the value of β1 to 0.008 and γ to 0. By a simple computation, we have R0 = 1.4283 > 1. Therefore, model (20) has a unique endemic equilibrium E ∗ (80.7455, 15.7823, 10.8522, 0.3946) and it is globally asymptotically stable. This result is shown in Fig. 3 and reflects that the disease is still present in the population. Example 2: Consider the following PDE model ⎧ ∂S ⎪ ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂I ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂D ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ∂P ⎪ ⎩ ∂t

= d S S + A − μS(x, t) −

β1 S(x, t)I (x, t) β2 S(x, t)D(x, t) − 1 + α1 I (x, t) 1 + α2 D(x, t)

β3 S(x, t)P(x, t) , 1 + α3 P(x, t) β1 S(x, t)I (x, t) β2 S(x, t)D(x, t) β3 S(x, t)P(x, t) = d I I + + + 1 + α1 I (x, t) 1 + α2 D(x, t) 1 + α3 P(x, t) − (μ + d + r )I (x, t), −

= d D D + (μ + d)I (x, t) − bD(x, t), = d P P + σ I (x, t) + γ D(x, t) − η P(x, t), (22)

with Neumann boundary conditions ∂I ∂D ∂P ∂S = = = = 0, on ∂ × (0, +∞), ∂ν ∂ν ∂ν ∂ν

(23)

and initial conditions S(x, 0) = S0 (x) ≥ 0, I (x, 0) = I0 (x) ≥ 0, D(x, 0) = D0 (x) ≥ 0, ¯ P(x, 0) = P0 (x) ≥ 0, x ∈ .

(24)

By applying Theorems 3.2 and 3.3, we can easily get the following corollary. Corollary 4.2. (i) If R0 ≤ 1, then the disease-free equilibrium E f of model (22) is globally asymptotically stable. (ii) If R0 > 1, then E f becomes unstable and the endemic equilibrium E ∗ of model (22) is globally asymptotically stable when γ = 0 or σ = 0. For numerical simulations of model (22), we choose d S = 0.1, d I = 0.5, d D = 0.01 and d P = 0.001. Also, we keep the same other parameters values as for model (20). The results presented in the last corollary are illustrated by Fig. 4 for R0 ≤ 1 and by Fig. 5 for R0 > 1.

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Fig. 4 Demonstration of the global stability of the disease-free equilibrium E f of model (22) for R0 = 0.9379 ≤ 1

Fig. 5 Demonstration of the global stability of the endemic equilibrium E ∗ of model (22) for R0 = 1.4283 > 1

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5 Conclusion People can catch Ebola by contact with bodily fluids of an alive infected individual, an infectious corpse during burial ceremonies or by contact with a contaminated environment. In this study, we have proposed two mathematical EVD models that take into account the three above modes of transmission. Further, the first model was formulated by ODEs and the second was governed by PDEs in order to consider the mobility of populations. For both models, the modes of transmission were modeled by three general incidence functions that include many incidence rates existing in the literature such as the saturated incidence, the classical bilinear incidence, the BeddingtonDeAngelis functional response, the Hattaf-Yousfi functional response and the Crowley-Martin functional response. Moreover, the qualitative properties such as positivity, boundedness and global behaviors of solutions have been rigorously established. On the other hand, the mathematical models and results presented in recent works [14, 28] have been improved and extended by considering general incidence functions for the three modes of transmission and by taking into account the mobility of individuals and EV contaminated environment. When a disease spreads within a community, individuals acquire knowledge about this disease. It is more reasonable to extend our models presented in this chapter by using fractional order derivative as in [37–40] in order to model the memory effect on the spreading of Ebola disease. We will study this in our future work.

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A Survey on Sufficient Optimality Conditions for Delayed Optimal Control Problems Ana P. Lemos-Paião, Cristiana J. Silva, and Delfim F. M. Torres

Abstract The aim of this work is to make a survey on recent sufficient optimality conditions for optimal control problems with time delays in both state and control variables. The results are obtained by transforming delayed optimal control problems into equivalent non-delayed problems. Such approach allows to use standard theorems that ensure sufficient optimality conditions for non-delayed optimal control problems. Examples are given with the purpose to illustrate the results. Keywords Delayed optimal control problems · Constant time delays in state and control variables · Sufficient optimality conditions · Equivalent and augmented non-delayed optimal control problems Mathematics Subject Classification (2010) Primary 49K15 · Secondary 34H99 · 49L99

1 Introduction The study of delayed systems, which can be optimized and controlled by a certain control function, has a long history and has been developed by many researchers (see, e.g., This work is part of first author’s Ph.D. project, which was carried out at the University of Aveiro under the support of the Portuguese Foundation for Science and Technology (FCT), fellowship PD/BD/114184/2016. It was also supported by FCT within project UIDB/04106/2020 (CIDMA). Silva was also supported by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in numbers 4, 5 and 6 of art. 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. A. P. Lemos-Paião · C. J. Silva · D. F. M. Torres (B) Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal e-mail: [email protected] A. P. Lemos-Paião e-mail: [email protected] C. J. Silva e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 K. Hattaf and H. Dutta (eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control 302, https://doi.org/10.1007/978-3-030-49896-2_12

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[2, 3, 6, 9, 17, 22, 26, 29, 54, 69] and references cited therein). Such systems can be called retarded, time-lag, or hereditary processes/optimal control problems. There are many applications of such systems, in diverse fields as Biology, Chemistry, Mechanics, Economy and Engineering (see, e.g., [3, 17, 27, 37, 41, 64, 69–71]). Dynamical systems with time delays, in both state and control variables, play an important role in the modelling of real-life phenomena, in various fields of applications (see [26, 27]). For instance, in [60] the incubation and pharmacological delays are modelled through the introduction of time delays in both state and control variables. In [67], Silva et al. introduce time delays in the state and control variables for tuberculosis modelling. They represent the time delay on the diagnosis and commencement of treatment of individuals with active tuberculosis infection and the delays on the treatment of persistent latent individuals, due to clinical and patient reasons. Delayed linear differential systems have also been investigated, their importance being recognized both from a theoretical and practical points of view. For instance, in [22], Friedman considers linear hereditary processes and apply to them Pontryagin’s method, deriving necessary optimality conditions as well as existence and uniqueness results. Analogously, in [54], delayed linear differential equations and optimal control problems involving this kind of systems are studied. Since these first works, many researchers have devoted their attention to linear quadratic optimal control problems with time delays (see, e.g., [9, 16, 18, 40, 55]). It turns out that for delayed linear quadratic optimal control problems it is possible to provide an explicit formula for the optimal controls (see [9, 40, 55]). Delayed optimal control problems with differential systems, which are linear both in state and control variables, have been studied in [9, 13, 16, 18, 40, 42, 43, 46, 53, 55]. In [16, 42, 55], the system is delayed with respect to state and control variables. In [13, 53], the system only considers delays in the state variable. Chyung and Lee derive necessary and sufficient optimality conditions in [13], while O˘guztöreli only proves necessary conditions in [53]. Certain necessary conditions analysed by Chyung and Lee in [13] have been already derived in [30, 58, 59]. However, the system considered in [13] is different from the previously studied hereditary systems, which do not require a initial function of state. In [18], Eller et al. derive a sufficient condition for a control to be optimal for certain problems with time delay. The problems studied by Eller et al. and Khellat in [18] and [40], respectively, consider only one constant lag in the state. The research done by Lee in [46] is different from that of the current work (more specifically from that of Sect. 3), because in [46] the aim is to minimize a cost functional, which does not consider delays, subject to a linear differential system (with respect to state and control variables) and to another constraint. In their differential system, the state variable depends on a constant and a fixed delay, while the control variable depends on a constant lag, which is not specified a priori. Note that the differential system of the problem considered in [43] is similar to the one of [46]. Although Banks has studied delayed non-linear problems without lags in the control, he has also analysed problems that are linear and delayed with respect to control (see [2]). Later, in 2010, Carlier and Tahraoui investigated optimal control problems with a unique delay in the state (see [10]). In 2012 and 2013, Frederico and Torres devoted their attention to optimal control problems that only contain delays in the state variables and the dependence on the control is linear

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(see [20, 21]). Recently, Cacace et al. studied optimal control problems that involve linear differential systems with variable delays only in the control (see [9]). The problems analysed in the current work are different from those considered in the mentioned papers. In Sect. 3, the optimal control problems involve differential systems that are linear with respect to state, but not with respect to the control. In Sect. 4, we study optimal control problems with non-linear differential systems. Furthermore, in both Sects. 3 and 4, we consider a constant time delay in the state and another one in the control. These two delays are, in general, not equal. In [35], Hughes firstly consider variational problems with only one constant lag and derive various necessary and a sufficient optimality conditions for them. The variational problems in [35] can easily be transformed into control problems with only one constant delay (see, e.g., [49, p. 53–54]). Hughes also investigates an optimality condition for a control problem with a constant delay, which is the same for state and control. The problems analysed by Chan and Yung in [11], and by Sabbagh in [62], are similar to the first problems studied by Hughes in [35]. Therefore, the problems investigated in [11, 35, 62] are different from the problems studied by us here, because in the present work the state delay is not necessarily equal to the control delay. The problems considered in [35, 62] are also considered in [56] by Palm and Schmitendorf. For such problems, they derive two conjugate-point conditions, which are not equivalent. Note that their conditions are only necessary and do not give a set of sufficient conditions (see [56]). Recent results include Noether type theorems for problems of the calculus of variations with time delays (see [19, 52, 65]), necessary optimality conditions for quantum (see [21]) and Herglotz variational problems with time delays (see [63, 64]), as well as delayed optimal control problems with integer (see [5, 8, 20]) and non-integer (fractional order) dynamics (see [14, 15]). Applications of such theoretical results are found in Biology and other Natural Sciences, e.g., in tuberculosis (see [67]) and HIV (see [60, 61]). In [39], Jacobs and Kao investigate delayed problems that consist to minimize a cost functional without delays subject to a differential system defined by a nonlinear function with a delay in state and another one in the control. Similarly to our problems, these delays do not have to be equal. In contrast, all types of cost functionals considered in our work also contain time delays. Therefore, we study here problems that are more general than the one considered in [39]. Jacobs and Kao transform the problem using a Lagrange-multiplier technique and prove a regularity result in the form of a controllability condition, as well as some necessary optimality conditions. Then, in some special restricted cases, they prove existence, uniqueness, and sufficient conditions. Such restricted problems consider a differential system that is linear in state and in control variables. Thus, the sufficient conditions of [39] are derived for problems that are less general than ours. As it is well-known, and as Hwang and Bien write in [36], many researchers have directed their efforts to seek sufficient optimality conditions for control problems with delays (see, e.g., [13, 18, 35, 39, 48, 66]). Therefore, it is not a surprise that there are authors that already proved some sufficient optimality conditions for delayed optimal control problems similar but, nevertheless, different from ours. In what respects to research done in [13, 18, 35, 39], we have already seen why they are different.

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The delayed optimal control problems analysed by Schmitendorf in [66] have a cost functional and a differential system that are more general than ours. However, in [66] the control takes its values in all Rm , while in the present work the control values belong to a set U ⊆ Rm , m ∈ N. In [48], Lee and Yung study a problem that is similar to the one considered in [66], where the control belongs to a subset of Rm , as we consider here. First and second-order sufficient conditions are shown in [48]. Nevertheless, the conditions of [48] are not constructive and not practical for the computation of the optimal solution. Indeed, as hypothesis, it is assumed the existence of a symmetric matrix under some conditions, for which is not given a method to calculate its expression. Another similar problem to ours is studied by Bokov in [8], in order to arise a necessary optimality condition in an explicit form. Moreover, a solution to the problem with infinite time horizon is given in [8]. In contrast, in the present work we are interested to derive sufficient optimality conditions. In [36], Hwang and Bien prove a sufficient condition for problems involving a differential affine time delay system with the same time delay for the state and the control. The differential systems considered in the present work are more general. In 1996, Lee and Yung, considering functions that do not have to be convex, derived various first and second-order sufficient conditions for non-linear optimal control problems with only a constant delay in the state (see [45]). Their class of problems is obviously different from our. In particular, we consider delays for both state and control variables. As in [11, 48], second-order sufficient conditions are shown to be related to the existence of solutions of a Riccati-type matrix differential inequality. Optimal control problems with multiple delays have also been investigated. In [29], Halanay derive necessary conditions for some optimal control problems with various time lags in state and control variables, using the abstract multiplier rule of Hestenes (see [34]). In [29], all delays related to state are equal to each other and the same happens with the delays associated to the control. Note that the results of [22, 30] are obtained as particular cases of problems considered in [29]. Later, in 1973, a necessary condition is derived for an optimal control problem that involves multiple constant lags only in the control. This delayed dependence occurs both in the cost functional and in the differential system, which is defined by a non-linear function (see [68]). In [32], Haratišvili and Tadumadze prove the existence of an optimal solution and a necessary condition for optimal control systems with multiple variable time lags in the state and multiple variable commensurable time delays in the control. Later, an optimal control problem where the state variable is solution of an integral equation with multiple delays, both on state and control variables, is studied by Bakke in [1]. Furthermore, necessary conditions and Hamilton–Jacobi equations are derived. In 2006, Basin and Rodriguez-Gonzalez proved a necessary and a sufficient optimality condition for a problem that consists to minimize a quadratic cost functional subject to a linear system with multiple time delays in the control variable (see [4]). In their work, they begin by deriving a necessary condition through Pontryagin’s Maximum Principle. Afterwards, sufficiency is proved by verifying if the candidate found, through the Maximum Principle, satisfies the Hamilton–Jacobi– Bellman equation. Although Basin and Rodriguez-Gonzalez consider multiple time delays, the dependence of the state and control in the differential system is linear. In our current work, the dependence of the control, in the differential systems, is,

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in general, non-linear. In 2013, Boccia et al. derived necessary conditions for a free end-time optimal control problem subject to a non-linear differential system with multiple delays in the state (see [6]). The control variable is not influenced by time lags in [6]. Recently, in 2017, Boccia and Vinter obtained necessary conditions for a fixed end-time problem with a constant and unique delay for all variables, as well as free end-time problems without control delays (see [7]). As Guinn wrote in [28], the classical methods of obtaining necessary conditions for retarded optimal control problems (used, for instance, by Halanay in [29], Haratišvili in [31] and Oˇguztöreli in [54]) require complicated and extensive proofs (see, e.g., [2, 22, 29, 31, 54]). In 1976, Guinn proposed a method whereby we can reduce some specific time-lag optimal control problems to equivalent and augmented optimal control problems without delays (see [28]). By reducing delayed optimal control problems into non-delayed ones, we can then use well-known theorems, applicable for optimal control problems without delays, to derive desired optimality conditions for delayed problems (see [28]). In [28], Guinn study specific optimal control problems with a constant delay in state and control variables. These two delays are equal. Later, in 2009, Göllmann et al. studied optimal control problems with a constant delay in state and control variables subject to mixed control-state inequality constraints (see [26]). In that research, the delays do not have to be equal. For technical reasons, the authors need to assume that the ratio between these two time delays is a rational number (see [26]). In [26], the method used by Guinn in [28] is generalized and, consequently, a non-delayed optimal control problem is again obtained. Pontryagin’s Minimum Principle, for non-delayed control problems with mixed state-control constraints, is used and first-order necessary optimality conditions are derived for retarded problems (see [26]). Furthermore, Göllmann et al. discuss the Euler discretization of the retarded problem and some analytical examples versus correspondent numerical solutions are given. For more on numerical methods, for solving applied optimal control problems of systems governed by delay differential equations, see [12, 33, 38]. Later, in 2014, Göllmann and Maurer generalized the research mentioned before, by studying optimal control problems with multiple and constant time delays in state and control, involving mixed state-control inequality constraints (see [27]). Again, necessary optimality conditions are derived (see [27]). Note that the works [26–29] consider delayed non-linear differential systems. In Sect. 3, we consider optimal control problems that consist to minimize a delayed non-linear cost functional subject to a delayed differential system that is linear with respect to state, but not with respect to control. Note that the cost functional does not have to be quadratic, but it satisfies some continuity and convexity assumptions. In Sect. 4, we consider optimal control problems that consist to minimize a delayed non-linear cost functional subject to a delayed non-linear differential system. In both Sects. 3 and 4, the delay in the state is the same for the cost functional and for the differential system. The same happens with the time lag of the control variable. Analogously to Göllmann et al. in [26], we ensure the Commensurability Assumption between the, possibly different, delays of state and control variables. The proofs of our sufficient optimality conditions consider the technique proposed by Guinn in [28] and used by Göllmann et al. in [26, 27] (see [50, 51]). As we have already mentioned before, the technique consists to transform a delayed optimal control problem into an

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equivalent non-delayed optimal control problem. After doing such transformation, one can apply well-known results for non-delayed optimal control problems and then return to the initial delayed problem. Here we restrict ourselves to delayed problems with deterministic controls. For the stochastic case, we refer the reader to [19, 23, 25, 37, 44]. This work is organised as follows. We begin by recalling the Commensurability Assumption, introduced by Göllmann et al. in [26], and by defining some needed notations, in Sect. 2. In Sect. 3.1, we define a state-linear optimal control problem with constant time delays in state and control variables. Then, in Sect. 3.2, we present a sufficient optimality condition associated with the problem stated in Sect. 3.1. A concrete example is solved in detail in Sect. 3.3, with the purpose to illustrate Theorem 3.3 of Sect. 3.2. In Sect. 4.2, we present a sufficient optimality condition associated with the non-linear optimal control problem with time lags both in state and control variables, defined in Sect. 4.1. An example that illustrates the obtained theoretical result— Theorem 4.3 of Sect. 4.2—is given. We end with some conclusions, in Sect. 5.

2 Commensurability Assumption and Notations In this section, we recall the Commensurability Assumption introduced by Göllmann et al. in [26]. Assumption 2.1 (See Assumption 4.1 of [26]). We consider r, s ≥ 0, not simultaneously equal to zero, and commensurable, that is, (r, s) = (0, 0) and

s r ∈ Q for s > 0 or ∈ Q for r > 0. s r

Actually, Commensurability Assumption 2.1 holds for any couple of rational numbers (r, s) for which at least one number is non-zero (see [26]). With the purpose to simplify the writing, we introduce some notations. Notation 2.2. We define tτ , t τ and tττ12 as follows: tτ = t − τ, t τ = t + τ and tττ12 = t − τ1 + τ2 for time delays τ, τ1 , τ2 ∈ {r, s} and for all t ∈ [a, b].   we Notation 2.3. Let xa = x(a) = ϕ(a) and xr (t) = x(t),x(t − r ) . Moreover,   define the operators [·, ·]r and ·, ·r by [x, ζ ]r (t) := t, xr (t), ζ t, xr (t) and    x, ζ r (t) := t, xr (t), ζ t, x(t) , respectively. While Notation 2.2 is used in Sects. 3 and 4, Notation 2.3 is only considered in Sect. 4.

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3 Delayed State-Linear Optimal Control Problem This section is devoted to state-linear optimal control problems with constant time delays in state and control variables. We make a survey on a sufficient optimality condition for this type of problems. Its proof, and more details associated with the contents of the current section, can be found in [50]. To finish this section, an illustrative example is given.

3.1 Statement of the Optimal Control Problem We start by defining a delayed state-linear optimal control problem. Definition 3.1. Consider that r ≥ 0 and s ≥ 0 are constant time delays associated with the state and control variables, respectively. We assume that (r, s) = (0, 0). A non-autonomous state-linear optimal control problem (OCPLD ) with time delays and with a fixed initial state, on a fixed finite time interval [a, b], consists in   min C L D x(·), u(·) =

 a

b

    f x0 t, x(t), x(t − r ) + f u0 t, u(t), u(t − s) dt

subject to the delayed differential system     x(t) ˙ = A(t)x(t) + A D (t)x(t − r ) + g t, u(t) + g D t, u(t − s)

(3.1)

with the following initial conditions: x(t) = ϕ(t), t ∈ [a − r, a], u(t) = ψ(t), t ∈ [a − s, a[,

(3.2)

where i. the state trajectory is x(t) ∈ Rn for each t ∈ [a − r, b]; ii. the control is u(t) ∈ U ⊆ Rm for each t ∈ [a − s, b]; iii. A(t) and A D (t) are real n × n matrices for each t ∈ [a, b]. Next we define admissible pair for (OCPLD ).   Definition 3.2. We say that x(·), u(·) is an admissible pair for (OCPLD ) if it respects the following conditions:   i. x(·), u(·) ∈ W 1,∞ ([a − r, b], Rn ) × L ∞ ([a − s, b], Rm ), where W 1,∞ is the space of Lipschitz functions;   ii. x(·), u(·)  satisfies the conditions (3.1) and (3.2); iii. x(t), u(t) ∈ Rn × U for all t ∈ [a, b].

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3.2 Main Result In what follows, we consider that the time delays r and s respect the Commensurability Assumption 2.1 and we use Notation 2.2. The following theorem supplies a sufficient optimality condition associated with (OCPLD ) (see Definition 3.1). Such result generalizes Theorem 5 in Chapter 5.2 of [47]. Theorem 3.3. Consider (OCPLD ) and assume that i. functions f x0 , ∂2 f x0 , ∂3 f x0 , f u0 , g, g D , A and A D are continuous for all their arguments; ii. f x0 (t, x, xr ) is a convex function in (x, xr ) ∈ R2n for each t ∈ [a, b]; iii. for almost all t ∈ [a, b], u ∗ is a control with response x ∗ that satisfies the maximality condition    max H D1 t, x ∗ (t), x ∗ (tr ), u, u ∗ (ts ), η(t) u∈U    + H D0 t s , x ∗ (t s ), x ∗ (trs ), u ∗ (t s ), u, η(t s ) χ[a,b−s] (t)   = H D1 t, x ∗ (t), x ∗ (tr ), u ∗ (t), u ∗ (ts ), η(t)   + H D0 t s , x ∗ (t s ), x ∗ (trs ), u ∗ (t s ), u ∗ (t), η(t s ) χ[a,b−s] (t), where

p H D (t, x, y, u, v, η) = − f x0 (t, x, y) + f u0 (t, u, v)

+ η A(t)x + A D (t)y + pg(t, u) + (1 − p)g D (t, v) for p ∈ {0, 1} and η(t) is any non-trivial solution of the adjoint system     η(t) ˙ = ∂2 f x0 t, x ∗ (t), x ∗ (tr ) + ∂3 f x0 t r , x ∗ (t r ), x ∗ (t) χ[a,b−r ] (t) − η(t)A(t) − η(t r )A D (t r )χ[a,b−r ] (t) that satisfies the transversality condition η(b) = [0 · · · 0]1×n .   Then, x ∗ (·), u ∗ (·) is an optimal solution of (OCPLD ) that leads to the minimal cost  C L D x ∗ (·), u ∗ (·) . The detailed proof of Theorem 3.3 can be found in [50].

3.3 An Illustrative Example In this section we provide an illustrative example associated with Theorem 3.3.

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Let us consider the delayed state-linear optimal control problem given by 





min C L D x(·), u(·) =

4

x(t) + 100u 2 (t)dt

0

s.t. x(t) ˙ = x(t) + x(t − 2) − 10u(t − 1),

(3.3)

x(t) = 1, t ∈ [−2, 0], u(t) = 0, t ∈ [−1, 0[, where u(t) ∈ U = R for  each t ∈ [−1, 4].  Thus, we have  that n = m =1, a = 0, b = 4, r = 2, s = 1, f x0 t, x(t), x(t − 2) = x(t), f u0 t,u(t), u(t − 1) = 100u 2 (t), A(t) = A D (t) = 1, g t, u(t) = 0 and g D t, u(t − 1) = −10u(t − 1). Note that our functions respect hypothesis i and ii of Theorem 3.3. Let u¯ be an admissible control of problem (3.3) and let us maximize function   − f u0 t, u, u(t ¯ − 1) + η(t)g(t, u)    + − f u0 t + 1, u(t ¯ + 1), u + η(t + 1)g D (t + 1, u) χ[0,3] (t)  = − 100u 2 + − 100u¯ 2 (t + 1) − 10η(t + 1)u χ[0,3] (t)

−100u 2 − 10η(t + 1)u − 100u¯ 2 (t + 1), t ∈ [0, 3] = t ∈ ]3, 4] −100u 2 , with respect to u ∈ R. We obtain u(t) = −

η(t + 1) 20

for t ∈ [0, 3] and u(t) = 0 for t ∈ ]3, 4]. Furthermore, we know that η(t) is any non-trivial solution of     η(t) ˙ = ∂2 f 0 t, x(t), x(t − 2) + ∂3 f 0 t + 2, x(t + 2), x(t) χ[0,2] (t) − η(t)A(t) − η(t + 2)A D (t + 2)χ[0,2] (t)

1 − η(t) − η(t + 2), t ∈ [0, 2] ⇔ η(t) ˙ = 1 − η(t) − η(t + 2)χ[0,2] (t) = 1 − η(t), t ∈ ]2, 4] that satisfies the transversality condition η(4) = 0. The adjoint system is given by

⎧ ⎪ 1 − η(t) − η(t + 2), t ∈ [0, 2] ⎨η(t) ˙ = 1 − η(t), t ∈ ]2, 4] ⎪ ⎩ η(4) = 0.

(3.4)

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For t ∈ ]2, 4], the solution of differential equation

η(t) ˙ = 1 − η(t) η(4) = 0

is given by η(t) = 1 − e4−t . Knowing η(t), t ∈ ]2, 4], and attending to the continuity of function η for all t ∈ [0, 4], we can determine η(t) for t ∈ [0, 2] solving the differential equation

η(t) ˙ = 1 − η(t) − η(t + 2) η(2) = 1 − e4−2 = 1 − e2   for t ∈ [0, 2]. Therefore, we have that η(t) = e2−t t − e2 − 1 for t ∈ [0, 2]. Consequently, the solution of the adjoint system (3.4) is given by

η(t) =

  e2−t t − e2 − 1 , t ∈ [0, 2] t ∈ ]2, 4]. 1 − e4−t ,

So, the control is given by ⎧ ⎪ ⎪0, ⎪ 1 ⎨e3−t − e1−t t, u(t) = ⎪e3−t − 1, 20 ⎪ ⎪ ⎩ 0,

t t t t

∈ [−1, 0[ ∈ [0, 1[ ∈ [1, 3] ∈ ]3, 4].

(3.5)

Knowing the control, we can determine the state by solving the differential equation

x(t) ˙ = x(t) + x(t − 2) − 10u(t − 1) x(t) = 1, t ∈ [−2, 0].

The state solution is given by ⎧ 1, ⎪ ⎪ ⎪ ⎪ t ⎪ −1 ⎪ ⎪  2 + 2e 4,    ⎪ ⎪ e + 2e − 2e2 t e−t − 8 + 17 − 2e2 et ⎪ ⎨ , 8 x(t) =   4−t + 4 + −47e−2 + 17 − 2e2 + 16e−2 t et ⎪ 2e ⎪ ⎪ ⎪ , ⎪ ⎪ 8   ⎪ 6 ⎪ 4 t e−t + 4 + −51e−2 + 24 − 2e2 + 17e−2 t − 2t et ⎪ + e −e ⎪ ⎩ , 8

t ∈ [−2, 0] t ∈ ]0, 1] t ∈ ]1, 2]

(3.6)

t ∈ ]2, 3] t ∈ ]3, 4].

Such analytical expressions can be obtained with the help of a modern computer algebra system. We have used Mathematica. In Fig. 1, we observe that the numerical solutions for control and state, obtained using AMPL [24] and IPOPT [57], are in agreement with their analytical solutions, given by (3.5) and (3.6), respectively.

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Fig. 1 Optimal solution of problem (3.3): red line—initial data; dark green line—analytical solution; dashed light green line—numerical solution

The numerical solutions were obtained using Euler’s forward difference method in AMPL and IPOPT, dividing the interval of time [0, 4] into 2000 subintervals. The minimal cost is 23 + e2 + 34e4 − 2e6  67.491786. 16

4 Delayed Non-linear Optimal Control Problem This section is devoted to non-linear optimal control problems with constant time delays in state and control variables. We make a survey on a sufficient optimality condition for this type of problems. Its proof, and more details associated with the contents of the current section, can be found in [51]. We finish this section with an illustrative example.

4.1 Statement of the Optimal Control Problem We start by defining the delayed non-linear optimal control problem. Definition 4.1. Consider that r ≥ 0 and s ≥ 0 are constant time delays associated with the state and control variables, respectively. A non-autonomous optimal control problem with constant time delays and with a fixed initial state, on a fixed finite time interval [a, b], is denoted by (OCPD ) and consists in

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min C D x(·), u(·) = g x(b) + 0

b

  f 0 t, x(t), x(t − r ), u(t), u(t − s) dt

a

subject to the delayed differential system   x(t) ˙ = f t, x(t), x(t − r ), u(t), u(t − s) for a.a. t ∈ [a, b]

(4.1)

with initial and final conditions x(t) = ϕ(t), t ∈ [a − r − s, a] ⊂ R, u(t) = ψ(t), t ∈ [a − s, a[, x(b) ∈ ⊆ Rn ;

(4.2)

where i. the state trajectory is x(t) ∈ Rn for all t ∈ [a − r − s, b]; ii. the control is u(t) ∈ U ⊆ Rm for all t ∈ [a − s, b];  T iii. f = f 1 · · · f n . Next we define admissible pair for (OCPD ).   Definition 4.2. We say that x(·), u(·) is an admissible pair for (OCPD ) if it respects the following conditions:       i. x(·), u(·) ∈ W 1,∞ [a − r − s, b], Rn × L ∞ [a − s, b], Rm ; ii. x(·), u(·)  satisfies conditions (4.1) and (4.2); iii. x(t), u(t) ∈ Rn × U for all t ∈ [a, b].

4.2 Main Result In what follows we also consider that the time delays r and s respect Commensurability Assumption 2.1. Moreover, here we use Notations 2.2 and 2.3. The following theorem provides a sufficient optimality condition associated with (OCPD ) (see Definition 4.1). Such result generalizes Theorem 7 in Chapter 5.2 of [47].

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Theorem 4.3. Consider (OCPD ). Let the interval [a, b] be divided into N ∈ N > 0 and suppose that the functions g 0 , f 0 subintervals of amplitude h = b−a N 1 and f are of class C with respectto all their arguments. Assume there exists a  1+3n m    1 ∗ C R , R feedback control u t, xr (t), η t, xr (t) = u ∗ [x, η]r (t) such that     max H t, xr (t), u, u ∗ [x, η]r (ts ), η t, xr (t) u∈U     + H t s , xr (t s ), u ∗ [x, η]r (t s ), u, η t s , xr (t s ) χ[a,b−s] (t)    = H t, xr (t), u ∗ [x, η]r (t), u ∗ [x, η]r (ts ), η t, xr (t)    + H t s , xr (t s ), u ∗ [x, η]r (t s ), u ∗ [x, η]r (t), η t s , xr (t s ) χ[a,b−s] (t) =: H 0 [x, η]r (t) + H 0 [x, η]r (t s )χ[a,b−s] (t) for all t ∈ [a, b], where H (t, x, y, u, v, η) = − f 0 (t, x, y, u, v) + η f (t, x, y, u, v). Furthermore,  h(i + 1)], i = 0, . . . , N − 1, and suppose that  let Ii = [a+ hi, a + function S t, x(t) ∈ C 2 R1+n , R , t ∈ [a, b], is a solution of equation N −1       ∂1 S t, x(t) + − f 0 t, xr (t), u ∗ x, ∂2 Sr (t), u ∗ x, ∂2 Sr (ts ) i=0

   + ∂2 S t, x(t) f t, xr (t), u ∗ x, ∂2 Sr (t), u ∗ x, ∂2 Sr (ts ) χ Ii (t) = 0 

(4.3)

    with S b, x(b) = −g 0 x(b) , x(b) ∈ . Finally, consider that the control law    u ∗ t, xr (t), ∂2 S t, x(t) = u ∗ x, ∂2 Sr (t), t ∈ [a, b], determines a response x(t) ˜ steering (a, xa ) to (b, ). Then,   ˜ x(t ˜ − r ), ∂2 S(t, x(t) ˜ u(t) ˜ = u ∗ t, x(t), is an optimal control of (OCPD ) that leads to the minimal cost   ˜ u(·) ˜ = −S(a, xa ). C D x(·), The detailed proof of Theorem 4.3 can be found in [51].

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4.3 An Illustrative Example In this section we provide an example of application of Theorem 4.3. Let us consider the following delayed non-linear optimal control problem studied by Göllmann et al. in [26]: 





min C D x(·), u(·) =

3

x 2 (t) + u 2 (t)dt,

0

s.t. x(t) ˙ = x(t − 1) u(t − 2), x(t) = 1, t ∈ [−1, 0], u(t) = 0, t ∈ [−2, 0[,

(4.4)

which is a particular case of our delayed non-linear control problem (OCPD )  optimal  with n = m = 1, a = 0, b = 3, r = 1, s = 2, g 0 x(3) = 0, f 0 (t, x, y, u, v) = x 2 + u 2 and f (t, x, y, u, v) = yv. In [26], necessary  optimality conditions were proved and applied to (4.4). The following candidate x ∗ (·), u ∗ (·) was found:

and

⎧ ⎨1, t ∈ [−1, 2], x ∗ (t) = et−2 + e4−t ⎩ , t ∈ [2, 3], e2 + 1

(4.5)

⎧ 0, t ∈ [−2, 0[, ⎪ ⎪ ⎨ t 2−t e −e u ∗ (t) = , t ∈ [0, 1], ⎪ e2 + 1 ⎪ ⎩ 0, t ∈ [1, 3].

(4.6)

It remains missing in [26], however, a proof that such candidate (4.5)–(4.6) is a solution to the problem. It follows from our sufficient optimality condition that such claim is indeed true. We denote that x0∗ (t) = 1, t ∈ [−1, 0]; x1∗ (t) = 1, t ∈ [0, 1]; x2∗ (t) = 1, t ∈ [1, 2]; t−2 4−t t 2−t ∗ x3 (t) = e e2+e , t ∈ [2, 3]; u ∗0 (t) = 0, t ∈ [−2, 0[; u ∗1 (t) = e e−e 2 +1 , t ∈ [0, 1]; +1 ∗ ∗ u 2 (t) = 0, t ∈ [1, 2] and u 3 (t) = 0, t ∈ [2, 3]. Furthermore, the corresponding adjoint function is given by

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⎧ ⎪ ⎨η1 (t), t ∈ [0, 1] η(t) = η2 (t), t ∈ [1, 2] ⎪ ⎩ η3 (t), t ∈ [2, 3] ⎧   2 e2 − 1 ⎪ ⎪ ⎪ −2t + 5 +  2 , ⎪ ⎪ ⎪ e2 + 1 ⎪  ⎪ ⎨ 2 = −  4e  + 2 t + 2 ⎪ ⎪ ⎪  e2 + 1  ⎪ ⎪ ⎪ 2 e4−t − et−2 ⎪ ⎪ ⎩ , e2 + 1

337

t ∈ [0, 1]  4 e2 − 1 e2t−2 − e6−2t  2 + 6 +  2 , t ∈ [1, 2] e2 + 1 e2 + 1 

t ∈ [2, 3].

From now on, we are going to ensure that these functions satisfy the sufficient optimality conditions studied in this section (see Theorem 4.3). So, for t ∈  [0, 3],we intend to find a function S(t, x) that is a solution of equation (4.3) with S 3, x(3) =   0. As η(t) = ∂2 S t, x(t) , we obtain that ⎧ ⎪ ⎨η1 (t)x + c1 (t), t ∈ [0, 1] S(t, x) = η2 (t)x + c2 (t), t ∈ [1, 2] ⎪ ⎩ η3 (t)x + c3 (t), t ∈ [2, 3], where ci (·) is a real function of real variable, i ∈ {1, 2, 3}. For t ∈ [2, 3], the Eq. (4.3) implies that   η˙3 (t)x ∗ (t) + c˙3 (t) − x ∗2 (t) + u ∗2 (t) + η3 (t)x ∗ (t − 1)u ∗ (t − 2) = 0   ∗ ∗ ⇔ η˙3 (t)x3∗ (t) + c˙3 (t) − x3∗2 (t) + u ∗2 3 (t) + η3 (t)x 2 (t − 1)u 1 (t − 2) = 0   4−t 2  t−2 2 e + et−2 et−2 + e4−t e + e4−t × + c ˙ (t) − ⇔− 3 e2 + 1 e2 + 1 e2 + 1 et−2 − e2−(t−2) =0 + η3 (t) × 1 × e2 + 1   5 e2t−4 + e8−2t + 2e2 ⇔ c˙3 (t) = (4.7)  2 e2 + 1   with S 3, x(3) = c3 (3) = 0. Solving the differential equation (4.7) with final condition c3 (3) = 0, we obtain that   4e2 (t − 3) + 5 e2t−4 − e8−2t c3 (t) = .  2 2 e2 + 1

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For t ∈ [1, 2], the Eq. (4.3) implies that   η˙2 (t)x ∗ (t) + c˙2 (t) − x ∗2 (t) + u ∗2 (t) + η2 (t)x ∗ (t − 1)u ∗ (t − 2) = 0   ∗ ∗ ⇔ η˙2 (t)x2∗ (t) + c˙2 (t) − x2∗2 (t) + u ∗2 2 (t) + η2 (t)x 1 (t − 1)u 0 (t − 2) = 0     2 e2t−2 + e6−2t 4e2 ⇔ −  + 2 + + c˙2 (t) − 1 + η2 (t) × 1 × 0 = 0  2 2 e2 + 1 e2 + 1     2 e2t−2 + e6−2t − 5e2 − 3 e4 + 1 (4.8) ⇔ c˙2 (t) = −  2 e2 + 1     with η2 (2)x2∗ (2) + c2 (2) = η3 (2)x3∗ (2) + c3 (2), because S t, x(t) ∈ C 2 R2 , R . Therefore, the previous condition is equivalent to   5 1 − e4 − 4e2 c2 (2) = c3 (2) =  2 . 2 e2 + 1

(4.9)

Solving the differential equation (4.8) with the condition (4.9), we have that     2t 3e4 + 10e2 + 3 + 2 e6−2t − e2t−2 − 17e4 − 44e2 − 7 c2 (t) = .  2 2 e2 + 1 For t ∈ [0, 1], the Eq. (4.3) implies that   η˙1 (t)x ∗ (t) + c˙1 (t) − x ∗2 (t) + u ∗2 (t) + η1 (t)x ∗ (t − 1)u ∗ (t − 2) = 0   ∗ ∗ ⇔ η˙1 (t)x1∗ (t) + c˙1 (t) − x1∗2 (t) + u ∗2 1 (t) + η1 (t)x 0 (t − 1)u 0 (t − 2) = 0 2  t e − e2−t + η1 (t) × 1 × 0 = 0 ⇔ − 2 + c˙1 (t) − 1 − e2 + 1 e4−2t + e2t + 3e4 + 4e2 + 3 (4.10) ⇔ c˙1 (t) =  2 e2 + 1     with η1 (1)x1∗ (1) + c1 (1) = η2 (1)x2∗ (1) + c2 (1), because S t, x(t) ∈ C 2 R2 , R . Therefore, the previous condition is equivalent to c1 (1) = c2 (1) =

−9e4 − 24e2 − 3 .  2 2 e2 + 1

(4.11)

Solving the differential equation (4.10) with the condition (4.11), we obtain that   2t 3e4 + 4e2 + 3 + e2t − e4−2t − 15e4 − 32e2 − 9 c1 (t) = .  2 2 e2 + 1

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Concluding, the previous computations show the following result. Proposition 4.4. Function ⎧ ⎪ ⎨η1 (t)x + c1 (t), t ∈ [0, 1], S(t, x) = η2 (t)x + c2 (t), t ∈ [1, 2], ⎪ ⎩ η3 (t)x + c3 (t), t ∈ [2, 3], with

  2 e2 − 1 η1 (t) = −2t + 5 +  2 , e2 + 1   4e2 η2 (t) = −  2 + 2 t + e2 + 1   2 e4−t − et−2 , η3 (t) = e2 + 1

  4 e2 − 1 e2t−2 − e6−2t  2 + 6 +  2 , e2 + 1 e2 + 1

and   2t 3e4 + 4e2 + 3 + e2t − e4−2t − 15e4 − 32e2 − 9 c1 (t) = ,  2 2 e2 + 1     2t 3e4 + 10e2 + 3 + 2 e6−2t − e2t−2 − 17e4 − 44e2 − 7 , c2 (t) =  2 2 e2 + 1   4e2 (t − 3) + 5 e2t−4 − e8−2t , c3 (t) =  2 2 e2 + 1   is solution of the Hamilton–Jacobi equation (4.3) with S 3, x ∗ (3) = 0.

5 Conclusion In this work we did a detailed state of the art associated with optimality conditions for delayed optimal control problems. Our survey ends with sufficient optimality conditions for two different types of delayed optimal control problems that are, to the best of our knowledge, the first to give an answer to a long-standing open question. Since the proofs are long, technical, and can be found in [50, 51], we did not present them here. However, examples were provided with the purpose to illustrate the usefulness of Theorems 3.3 and 4.3. As future work, we plan to show the usefulness of our results to control infectious diseases. Acknowledgements The authors are strongly grateful to the anonymous reviewers for their suggestions and invaluable comments.

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