Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment: Selected Works from the BIOMAT Consortium Lectures, Szeged, Hungary, 2019 [1st ed.] 9783030463052, 9783030463069

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Rubem P. Mondaini Editor

Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment Selected Works from the BIOMAT Consortium Lectures, Szeged, Hungary, 2019

Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment

Rubem P. Mondaini Editor

Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment Selected Works from the BIOMAT Consortium Lectures, Szeged, Hungary, 2019

Editor Rubem P. Mondaini President, BIOMAT Consortium – International Institute for Interdisciplinary Sciences Rio de Janeiro, Brazil Federal University of Rio de Janeiro Rio de Janeiro, Brazil

ISBN 978-3-030-46305-2 ISBN 978-3-030-46306-9 (eBook) https://doi.org/10.1007/978-3-030-46306-9 Mathematics Subject Classification (2020): 92Bxx, 34-XX, 35-XX, 92B20, 92-08 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020, corrected publication 2020 This work is subject to copyright. All rights are reserved 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 is a collection of papers which have been accepted for publication after a peer-review evaluation by the Editorial Board of the BIOMAT Consortium (http://www.biomat.org) and international referees ad hoc. These papers have been presented in the technical sessions of the BIOMAT 2019 International Symposium, the 19th Symposium of the BIOMAT Series which was held at the Bolyai Institute, University of Szeged and at the Hungarian Academy of Sciences, Szeged, Hungary, October 21–25, 2019. On behalf of the BIOMAT Consortium, we thank the members of the BIOMAT 2019 Local Organizing Committee, Gergely Röst (Chair), Ferenc Bartha, Attila Dénes, János Karsai, Tibor Krisztin, Mónika Van Leeuwen-Poiner, and Gabriella Vas, for their professional expertise at following the guidelines and the fine tradition of the BIOMAT Consortium˘aduring the BIOMAT 2019 International Symposium in Szeged, Hungary We are so much indebted to all these colleagues as well as to research collaborators and Ph.D. students of the Bolyai Institute for their invaluable help since the Opening Session on Monday morning to the Closing Session on Friday evening. Financial support in terms of lunches, coffee breaks for all participants and accommodation for the invited Keynote Speakers have been provided by the Bolyai Institute at Szeged. The International Union of Biological Sciences—IUBS has also provided invaluable help with a grant for living expenses and accommodation of young research fellows coming from developing countries. On behalf of the Consortium, we thank effusively the IUBS and its Executive Director, Dr. Nathalie Fomproix, from University of Paris—Sud, Orsay, France. The BIOMAT Consortium has succeeded once more in its fundamental mission of enhancing the interdisciplinary scientific activities of Mathematical and Biological Sciences in Developing Countries with the organization of the BIOMAT 2019 International Symposium. The authors of papers from Western and Eastern Europe, North and South America, Africa, and Asia had the usual opportunity of exchanging scientific feedback of their research fields with their colleagues from Hungary and other nineteen countries: Bangladesh, Belgium, Brazil, Canada, Colombia, Czech Republic, India, Iran, Italia, Jordan, Morocco, Nigeria, Russia, Senegal, Serbia, Tanzania, USA, UK, and South Korea. v

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Preface

The Editor of the book and President of the BIOMAT Consortium is very glad for the continuous collaboration and critical support of his wife Carmem Lucia on the editorial work, from the reception of submitted papers for the peer-review procedure of BIOMAT Consortium Editorial Board to the ultimate publication of the Scientific Programme. He also thanks his former research student Dr. Simão C. de Albuquerque Neto from Federal University of Rio de Janeiro who has been nominated recently as the General Secretary of the BIOMAT Consortium for his continued collaboration with the BIOMAT Consortium since the year 2016. Szeged, Hungary

Rubem P. Mondaini

The original version of this book was revised: Chapter title of 21 was revised and also Abstract text of Chaps. 17 and 21 was replaced in the online Meta data respectively. The correction to this book is available at https://doi.org/10.1007/978-3-030-46306-9_25

Editorial Board of the BIOMAT Consortium

Rubem Mondaini (Chair) Federal University of Rio de Janeiro, Brazil Adelia Sequeira Instituto Superior Técnico, Lisbon, Portugal Alain Goriely University of Oxford, Mathematical Institute, UK Alan Perelson Los Alamos National Laboratory, New Mexico, USA Alexander Grosberg New York University, USA Alexei Finkelstein Institute of Protein Research, Russia Ana Georgina Flesia Universidad Nacional de Cordoba, Argentina Anna Tramontano University of Rome, La Sapienza, Italy Alexander Bratus Lomonosov Moscow State University, Russia Avner Friedman Ohio State University, USA Carlos Condat Universidad Nacional de Cordoba, Argentina Charles Pearce University of Adelaide, Australia Denise Kirschner University of Michigan, USA David Landau University of Georgia, USA De Witt Sumners Florida State University, USA Ding Zhu Du University of Texas, Dallas, USA Dorothy Wallace Dartmouth College, USA Eduardo Massad Faculty of Medicine, University of São Paulo, Brazil Eytan Domany Weizmann Institute of Science, Israel Ezio Venturino University of Torino, Italy Fernando Cordova-Lepe Catholic University del Maule, Chile Fernando R. Momo Universidad Nacional de General Sarmiento, Argentina Fred Brauer University of British Columbia, Vancouver, Canada Frederick Cummings University of California, Riverside, USA Gergely Röst University of Szeged, Hungary Guy Perriére Université Claude Bernard, Lyon, France Gustavo Sibona Universidad Nacional de Cordoba, Argentina Helen Byrne University of Nottingham, UK Jacek Miekisz University of Warsaw, Poland Jack Tuszynski University of Alberta, Canada vii

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Jaime Mena-Lorca Jane Heffernan Jean Marc Victor Jerzy Tiuryn John Harte John Jungck José Fontanari Karam Allali Kazeem Okosun Kristin Swanson Kerson Huang Lisa Sattenspiel Louis Gross Lucia Maddalena Ludek Berec Mario R. Guarracino Michael Meyer-Hermann Nicholas Britton Panos Pardalos Peter Stadler Pedro Gajardo Philip Maini Pierre Baldi Rafael Barrio Ramit Mehr Raymond Mejía Rebecca Tyson Reidun Twarock Richard Kerner Riszard Rudnicki Robijn Bruinsma Rui Dilão Sandip Banerjee Seyed Moghadas Siv Sivaloganathan Somdatta Sinha Suzanne Lenhart Vitaly Volpert William Taylor Yuri Vassilevski Zhijun Wu

Editorial Board of the BIOMAT Consortium

Pontifical Catholic University of Valparaíso, Chile York University, Canada Université Pierre et Marie Curie, Paris, France University of Warsaw, Poland University of California, Berkeley, USA University of Delaware, Delaware, USA University of São Paulo, Brazil University Hassan II, Mohammedia, Morocco Vaal University of Technology, South Africa University of Washington, USA Massachusetts Institute of Technology, MIT, USA University of Missouri, Columbia, USA University of Tennessee, USA High Performance Computing and Networking Institute, ICAR-CNR, Naples, Italy Biology Centre, ASCR, Czech Republic High Performance Computing and Networking Institute, ICAR-CNR, Naples, Italy Frankfurt Institute for Advanced Studies, Germany University of Bath, UK University of Florida, Gainesville, USA University of Leipzig, Germany Federico Santa Maria University, Valparaíso, Chile University of Oxford, UK University of California, Irvine, USA Universidad Autonoma de Mexico, Mexico Bar-Ilan University, Ramat Gan, Israel National Institutes of Health, USA University of British Columbia, Okanagan, Canada University of York, UK Université Pierre et Marie Curie, Paris, France Polish Academy of Sciences, Warsaw, Poland University of California, Los Angeles, USA Instituto Superior Técnico, Lisbon, Portugal Indian Institute of Technology Roorkee, India York University, Canada Centre for Mathematical Medicine, Fields Institute, Canada Indian Institute of Science, Education and Research, India University of Tennessee, USA Université de Lyon 1, France National Institute for Medical Research, UK Institute of Numerical Mathematics, RAS, Russia Iowa State University, USA

Contents

Editorial Board of the BIOMAT Consortium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolutionary Adaptation of the Permanent Replicator System. . . . . . . . . . . . . A. S. Bratus, S. Drozhzhin, and T. Yakushkina A More Realistic Formulation of Herd Behavior for Interacting Populations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Borgogni, L. Losero, and E. Venturino On Network Similarities and Their Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Granata, M. R. Guarracino, L. Maddalena, I. Manipur, and P. M. Pardalos

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Impacts of Infections and Predation on Dynamics of Sexually Reproducing Populations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ludˇek Berec

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Global Analysis of a Cancer Model with Drug Resistance Due to Microvesicle Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attila Dénes and Gergely Röst

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Contact Vaccination Study Using Edge Based Compartmental Model (EBCM) and Stochastic Simulation: An Application to Oral Poliovirus Vaccine (OPV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coura Balde, Mountaga Lam, and Samuel Bowong The Effect of Inhibitory Neurons on a Class of Neural Networks . . . . . . . . . . Márton Neogrády-Kiss and Péter L. Simon

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Pipette Hunter 3D: Fluorescent Micropipette Detection . . . . . . . . . . . . . . . . . . . . 111 D. Hirling, K. Koos, J. Molnár, and P. Horvath Delay Linear Chains in Mathematical Biology: Migratory Birds, Stem Cell Maturation, and Intracellular Chlamydia Infection . . . . . . . . . . . . . 127 Bornali Das and Gergely Röst Normalization of a Periodic Delay in a Delay Differential Equation . . . . . . . 143 K. Nah and J. Wu ix

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Competition Between Two Tufted C4 Grasses: A Mathematical Model . . . 153 D. I. Wallace Mathematical Description of Systemic and Micro Circulations . . . . . . . . . . . . 161 V. V. Kislukhin and E. V. Kislukhina The Statistical Analysis of Protein Domain Family Distributions via Jaccard Entropy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 R. P. Mondaini and S. C. de Albuquerque Neto Theoretical and Numerical Considerations of the Assumptions Behind Triple Closures in Epidemic Models on Networks . . . . . . . . . . . . . . . . . . 209 Nicos Georgiou, István Z. Kiss, and Péter L. Simon Recognition of Protein Interaction Regions Through Time-Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 A. F. Arenas, G. E. Salcedo, M. D. Garcia, and N. Arango Using a Stochastic SIR Model to Design Optimal Vaccination Campaigns via Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 A. C. S. Dusse and R. T. N. Cardoso Optimal Control Analysis of HIV-TB Co-infection Model . . . . . . . . . . . . . . . . . . 259 Tanvi and Rajiv Aggarwal A Prey–Predator Model with Pathogen Infection on Predator Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Sanchayita Pramanick, Joydeb Bhattacharyya, and Samares Pal On an Invasive Species Model with Harvesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Sándor Kovács, Szilvia György and Noémi Gyúró Generalized Linear Models to Investigate Cyclic Trends . . . . . . . . . . . . . . . . . . . . 335 Tibor András Nyári Dynamics of HIV/AIDS and TB Co-infection with Treatment Rate as Holling Type-II Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Rajiv Aggarwal, Tanvi and Tamas Kovacs Discrete and Continuum Models for the Evolutionary and Spatial Dynamics of Cancer: A Very Short Introduction Through Two Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 T. Lorenzi, F. R. Macfarlane, and C. Villa Modelling Therapeutic Vaccines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Elaheh Abdollahi, Affan Shoukat and Seyed M. Moghadas Modeling the Genetic Code: p-Adic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Branko Dragovich and Nataša Ž. Miši´c

Contents

Correction to: Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Evolutionary Adaptation of the Permanent Replicator System A. S. Bratus, S. Drozhzhin, and T. Yakushkina

1 Introduction: Revisiting Fisher’s Fundamental Theorem Starting from Fisher’s fundamental theorem of natural selection, evolutionists began to apply extreme principles to Darwinian evolution [1–3]. The theorem postulates: “The rate of increase in (mean) fitness of any organism at any time is equal to its genetic variance in fitness at that time” [4]. However, the notion of “genetic variance of fitness” was not strictly defined in the early studies. Later, Wright [5] introduced another important concept—adaptive fitness landscape, which is extensively applied in theoretical biology. Many of the extreme principles in evolutionary theory rely on the assumptions of the constant fitness landscape and steadily growing mean fitness. In biological studies, underlying understanding of the fitness landscape is often based on common visualization as a statistical hypersurface. From this point of view, the evolutionary process is depicted as a path going through the space with hills, canyons, and valleys, ending up in one of the peaks [6, 7]. For the avoidance of doubt and misinterpretation, we define the notion “fitness landscape” explicitly, providing mathematical formalization for its geometry in the case of general replicator systems.

A. S. Bratus () Department of Applied Mathematics 1, Russian University of Transport, Moscow, Russia e-mail: [email protected] S. Drozhzhin Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia T. Yakushkina School of Business Informatics, National Research University Higher School of Economics, Moscow, Russia © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_1

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From a mathematical perspective, Fisher’s fundamental theorem is correct only for systems with symmetric matrices of interaction, which corresponds to the diploid population. Moreover, from these assumptions, the maximum fitness value should be reached in the steady-state of the evolutionary system. This forms significant restrictions on the applicability, making these cases rather exceptional than realistic [8]. Various studies [9–13] were dedicated to new interpretations and re-consideration of Fisher’s postulates. Another research [11], e.g., which provides an extensive literature review on mathematical formalism for the fundamental laws in evolution, Fisher’s approach to natural selection is discussed in terms of the F-theorem. In the current study, we develop a fitness optimization technique introduced before [14].

1.1 Dynamical Fitness Landscapes: Adaptation Process One of the ways to examine fitness landscapes is to consider their fluctuations. The question arises: how the adaptive changes can be achieved in evolution. The central hypothesis of this study is that the specific time of the evolutionary adaptation of the system parameters is much slower than the time of the internal evolutionary process, which leads the system to its steady-state. Throughout the paper, we will call the first the evolutionary time. For hypercycles, we introduced a similar concept [14]. This assumption leads to the fact that evolutionary changes of the system parameters happen in a steady-state of the corresponding dynamical system. In other words, we can write an equation for a steady-state with respect to the evolutionary time over a set of possible fitness landscapes. Consider a population distribution u = (u1 , . . . , un ) representing the frequencies of different types in a replicator system. If the system is permanent over a simplex Sn (here, the notation is the same as in the previous subsection) and there is a unique internal equilibrium u ∈ intSn (stable or unstable one), then the mean integral value of the frequencies and the mean fitness value coincide with ones that reached in a steady-state. This allows examining an evolutionary process of fitness landscape adaptation using only the equation for a steady-state, where all the elements depend on the evolutionary time τ . Therefore, fitness landscape adaptation happens in time, which describes system dynamics converging to a steady-state over the set of possible fitness landscapes. Let us consider the classical evolutionary model—the replicator equation [15– 18]: u˙ i = ui ((Au)i − f (u)) , i = 1, . . . n.

(1)

Here, A = [aij ] is a given n × n matrix of fitness coefficients. Value (Au)i is i-th element of the vector Au, where vector u stands for the distribution of the species in the population over time. The term f guarantees that for any time moment t the vector u(t) belongs to the simplex Sn :

Evolutionary Adaptation of the Permanent Replicator System

u(t) ∈ Sn = {x ∈ Rn : xi  0,

n 

3

xi = 1},

(2)

i=1 n 

f (u) =

aij ui uj = (Au, u) .

i,j =1

In terms of evolutionary game theory, f (u) is the mean population fitness of the population of composition u and the payoff matrix A. The equilibria u¯ are given by the solutions to the system of algebraic equations: ¯ Au¯ = f (u)1,

1 = (1, . . . , 1)T , u¯ ∈ Sn ,

(3)

¯ is the mean fitness at the equilibrium state u¯ (which is not necessarily where f (u) stable). It is worth pointing out that the proposed approach is valid only for permanent systems. In this case, it holds:  1 T ui (t)dt, T →+∞ T 0  1 T f¯ = lim (Au(t), u(t)) dt. T →+∞ T 0 u¯ i = lim

(4)

The adaptation of the fitness landscape over the evolutionary time τ can be described by: ¯ ) = f¯(u(τ ¯ ))1, A(τ )u(τ

¯ ) ∈ Sn , u(τ

(5)

where aij ((τ )) of the matrix A are smooth functions with respect to the parameter τ ∈ [0, +∞). Each solution of the equation (5) corresponds to the dynamics of the permanent replicator system, which is characterized by (4). The possible fitness landscapes satisfy the condition: A(τ )2 =

n 

aij2  Q2 ,

τ  0, Q = const.

(6)

i,j =1

We show that the problem of evolutionary adaptation of the replicator system over the set of possible fitness landscapes (6) transforms into the problem of function f¯(τ ) maximization over the set of the solutions to (5). In a previous study [14], we obtained an expression for the mean fitness variation. Based on this result, we suggested a process of fitness optimization in the form of a linear programming problem. The numerical simulations show that during the iteration process, the fitness value increases and the systems behavior changes. For the big enough number of iterations, the steady-state of the system stays almost

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the same; however, the fitness value grows drastically at the same time. In this case, when the initial state of the system is described by the hypercycle equations, we see a qualitative transformation of the system: new connections appear, and autocatalysis can start. Increasing the number of iterations further, the coordinates of the steady-states split and, over the simplex, the system converges to a fixed fitness value. It was shown [14] that the observed effect is similar to an “error threshold” effect in the quasispecies system by Eigen [18, 19] when the eigenvalue and the eigenvector of the system stay unchanged with the mutation rate growth.

2 Hypercycle with Two Types of Behavior Let us illustrate the calculations on the basis of one the special cases of replicator systems. Here, we focus on the system that describes one of the experimental models for the RNA cooperative networks [20], which is described in Fig. 1: u˙ 1 = u1 (αu1 + βu4 − f ), u˙ 2 = u2 (αu2 + βu5 − f ), u˙ 3 = u3 (αu3 + βu6 − f ), u˙ 4 = u4 (σ u3 + γ u5 − f ),

(7)

u˙ 5 = u5 (σ u1 + γ u6 − f ), u˙ 6 = u6 (σ u2 + γ u4 − f ).

Fig. 1 Graph represents the original hypercycles (7) with matrix A (8): (a) before the evolutionary process, (b) the result of the evolutionary process after 200 iterations. Numbers on the edges correspond to the adapted fitness coefficients

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Here,   ¯ u¯ = u1 (αu1 +βu4 )+u2 (αu2 +βu5 )+u3 (αu3 +βu6 )+u4 (σ u3 +γ u5 )+ f = Au, +u5 (σ u1 + γ u6 ) + u6 (σ u2 + γ u4 ). This system describes a population, which consists of two types—“egoists” and “altruists.” We call egoists the molecules 1, 2, and 3, which are participating in autocatalysis with coefficient α, and one of the 4, 5, or 6 with coefficient σ . Altruists, in this case, are 4, 5, and 6, which enforce the catalysis of others: egoists—with βand one of the altruists γ . Consider A: ⎞ ⎛ −0.3 0 0 0.4 0 0 ⎜ 0 −0.3 0 0 0.4 0 ⎟ ⎟ ⎜ ⎟ ⎜ 0 −0.3 0 0 0.4 ⎟ ⎜ 0 (8) A=⎜ ⎟, ⎜ 0 0 0.1 0 0.05 0 ⎟ ⎟ ⎜ ⎝ 0.1 0 0 0 0 0.05 ⎠ 0 0.1 0 0.05 0 0 where α = −0.3, β = 0.4, σ = 0.1, γ = 0.05. Figure 2 shows trajectories of altruists (dotted line) and egoists (solid line) at the beginning and 125, 175, 200 steps of the evolution with A (8).

3 Conclusion In this paper, we applied an algorithm for the fitness landscape evolution of the replicator system. We defined the limitation on the sum of squares of the fitness matrix coefficients while looking at the mean integral fitness maximum. We follow our previous study, suggesting that the evolutionary time of the hypercycle adaptation is much slower than the internal system dynamics time. At the beginning of the fitness landscape adaptation process, for a significantly long period in the evolutionary timescale, the steady-state of the system remains the same. However, the structure of the transition matrix changes, which leads to the new transitions in the hypercycle system: besides the original connections, we get the backward cycle, autocatalysis, and new connections between species. This can be interpreted as a more diverse and sustainable evolutionary state.

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Fig. 2 The frequencies of the species in the hypercycle system (7) with matrix A (8) changing over system time t: (a) at the beginning, (b) at the 125-th step of the fitness landscape evolutionary process, (c) at the 175-th step of the fitness landscape evolutionary process, (d) at the 200-th step of the fitness landscape evolutionary process

Acknowledgments This work was supported by grant 19-11-00009 of the Russian Science Foundation.

References 1. G.A. Parker, J.M. Smith, Optimality theory in evolutionary biology. Nature 348(6296), 27 (1990) 2. A. Grafen, The simplest formal argument for fitness optimization. J. Genet. 87(4), 421–433 (2008) 3. A. Grafen, Optimization of inclusive fitness. J. Theor. Biol. 238(3), 541–563 (2006) 4. R.A. Fisher, The Genetical Theory of Natural Selection, ed. with a foreword and notes by J.H. Bennett (A complete variorum ed.) (Oxford University Press, Oxford, UK, 1999) 5. S. Wright, The roles of mutation, inbreeding, crossbreeding and selection, in evolution. Proc. Sixth Int. Congr. Gen. 1, 356–366 (1932) 6. F.J. Poelwijk, D.J. Kiviet, D.M. Weinreich, S.J. Tans, Empirical fitness landscapes reveal accessible evolutionary paths. Nature 445(7126), 383 (2007) 7. J. Birch, Natural selection and the maximization of fitness. Biol. Rev. 91(3), 712–727 (2016) 8. A.S. Bratus, Y.S. Semenov, A.S. Novozhilov, Adaptive fitness landscape for replicator systems: to maximize or not to maximize. Math. Model. Nat. Phenom. 13(3), 25 (2018) 9. W.J. Ewens, An interpretation and proof of the fundamental theorem of natural selection. Theor. Popul. Biol. 36(2), 167–180 (1989)

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10. S. Lessard, Fisher’s fundamental theorem of natural selection revisited. Theor. Popul. Biol. 52(2), 119–136 (1997) 11. P. Ao, Laws in Darwinian evolutionary theory. Phys. Life Rev. 2(2), 117–156 (2005) 12. P. Ao, Global view of bionetwork dynamics: adaptive landscape. J. Genet. Genomics 36(2), 63–73 (2009) 13. W.J. Ewens, S. Lessard, On the interpretation and relevance of the fundamental theorem of natural selection. Theor. Popul. Biol. 104, 59–67 (2015) 14. A.S. Bratus, S. Drozhzhin, T. Yakushkina, On the evolution of hypercycles. Math. Biosci. 306, 119–125 (2018) 15. J. Hofbauer, K. Sigmund, Dynamical Systems and the Theory of Evolution (Cambridge University Press, Cambridge, 1988) 16. P. Schuster, K. Sigmund, Replicator dynamics. J. Theor. Biol. 100(3), 533–538 (1983) 17. E.V. Koonin, The Logic of Chance: The Nature and Origin of Biological Evolution (FT Press, 2011) 18. M. Eigen, Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58(10), 465–523 (1971) 19. M. Eigen, P. Schuster, A principle of natural self-organization. Naturwissenschaften 64(11), 541–565 (1977) 20. N. Vaidya, M.L. Manapat, I.A. Chen, R. Xulvi-Brunet, E.J. Hayden, N. Lehman, Spontaneous network formation among cooperative RNA replicators. Nature 491(7422), 72 (2012)

A More Realistic Formulation of Herd Behavior for Interacting Populations D. Borgogni, L. Losero, and E. Venturino

1 Introduction Population theory is one of the most prominent fields in mathematical biology, with applications as diverse as ecology, economics, epidemiology, down to the case of bacteria or cell and tissue growth within the human or animal bodies. Classically, population interactions have been introduced about a century ago and then investigated in various contexts all along the following years. In the classical formulation, individuals of different populations are assumed to interact on a one-to-one basis, giving rise to a quadratic term in the mathematical formulation of the corresponding dynamical system, which is written as the product of the two population sizes. This, on mechanistic terms, constitutes the mass action law approach. In the predator–prey context, the interaction corresponds to the capture of prey individuals by the predator. Since biologically the unlimited ingestion of food is unrealistic, modifications have been proposed for which the mass action law gets replaced by an increasing function of the prey, bounded from above, the so-called Holling type II response function. The resulting system, called Holling–Tanner

Member of the INDAM research group GNCS. D. Borgogni · L. Losero · E. Venturino () Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Torino, Italy e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_2

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model, shows limit cycles, which in the classical quadratic model are shown never to be possible, as the coexistence equilibrium is globally asymptotically stable. In more recent years, the prey behavior has been reconsidered, especially if they gather together in flocks. This feature has been modeled via either a decreasing response function with the increasing number of the individuals forming the flock, thereby better modeling a group defense mechanism in a predator–prey situation, [6, 14, 16, 23], or via power functions, introduced already in [15], that indicate, in the same context, that the most likely individuals to be attacked are those occupying the peripheral positions in the herd. Other interpretations are possible in different contexts [5, 11, 22]. The latter approach, which for simplicity can be named the square root formulation, [4, 9], provides an interesting mathematical phenomenon, namely the possible disappearance in finite time of the herd population [24]. It can also be combined with a Holling type II response [17, 18]. From the modeling perspective, instead, the square root approach suffers from the drawback that for low numbers of individuals, the herd disappears, or better, it coincides with its boundary, and the interactions therefore must involve the whole population. The model becomes therefore inaccurate. Corrections trying to consider this aspect have been recently proposed [12, 13], also in the single population environment [19, 20]. Extensions to food chains, stochastic models, and Allee effects can be, for instance, found in [7, 8, 21]. In this paper we continue the study along the above lines, still keeping a different view from the classical predator–prey system with two populations. In particular, our target is modeling the relationship between the predators showing an individualistic behavior while prey gather together in herds, using essentially only the intuitive perspective that the bigger the herd will be, the harder will be for the predator to engage it successfully, but also that the most likely targets of the predators’ attacks are the prey that occupy the peripheral positions in the herd. This approach is in line with other similar current issues in mathematical modeling, where the uncertainties intrinsic in the functional response forms are trying to be addressed [1–3]. The main issue in this context is the structural stability of the model formulation, which may be viable even if the exact specific mathematical form of the response function is not known or not fully specified. In this presentation, we propose a novel correction of the square root response function that takes care of the changes needed in the model when the number of individuals gets small. As a result, the modeling perspective becomes more realistic, but, as it happens also for the previously conceived modifications [12, 13], the finite time extinction phenomenon is lost.

2 The Model We consider a minimal predator–prey model to describe the population interactions of interest here and the phenomenon we wish to model more appropriately. Namely

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we define the following variables dependent on time, x, the prey, and y their specialist predators, i.e. the latter do not survive in the absence of the former. These populations fluctuate in time and, for obvious biological reasons, can never become negative. The interaction model is described by the following very simple equations constituting a homogeneous system, for which the previous property also follows mathematically: x  = (x) − a(x)y

(1)



y = ag(x)y − my, where the derivative is taken with respect to time. Here the function (x) represents the growth function of the prey, which is assumed to be of logistic type, with net reproduction rate r and carrying capacity K, namely:  x . (x) = rx 1 − K The model we consider should account for the so-called herd behavior, but it should also avoid its shortcomings. Mainly, this means that when the prey population becomes small, the herd is not a herd anymore and the predator–prey interactions should revert to an individualistic 1–1 basis, as it is assumed in the classical Lotka– Volterra system. For large prey numbers instead, the approach first proposed in [4] is retained. This is reflected by the choice of the interaction function . Therefore, this function should replace the power function earlier used in the formulation of these types of problems, specifically √ x,

xγ ,

γ >

1 , 2

investigated the former in [4], while the latter constitutes the more general case considered in [10]. This problem has also been recently addressed in other papers, see, for instance, [12, 13] and also, starting from the situation of a single population, in [19], as well as in [20] for more predators attacking the same herd. In [12, 13], the herd behavior loss at low population numbers is described via a piecewise continuous function. In this paper, we want to go further and replace this assumption using a smooth function instead. To this end, the selected function  should exhibit the following properties: (x) ≈ x as x → 0,

(x) ≈

√

x as x → ∞.

(2)

We therefore choose the following function, shown in Fig. 1: (x) =

√ √ x(1 − e− x ).

(3)

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0.8 y=x y = (x) y= (x)

18

y=x y = (x) y= (x)

0.7

16 0.6 14 0.5

12 10

0.4

8

0.3

6 0.2 4 0.1

2 0

0

2

4

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0

20

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45 0.5

Fig. 1 The behavior of the function (x) on a large domain, left, and a blow up of its graph near the origin, right. The graph of (x) is compared with the square root and the bisectrix is shown as a reference frame

3 System’s Equilibria The model given by (1) and (3) has the following Jacobian J = Ji,k , i, k = 1, 2: J =



 (x) − ay (x) −a(x) , ag(x) − m agy (x)

where, observing that √

1 − e−  (x) = √ 2 x 

x

√

e− x , + 2

(4)

the entries are explicitly given by     √ 2x 1 1 √ − ay (1 − e− x ) √ + e x , J1,1 = r 1 − K 2 2 x √ √ J1,2 = −a(1 − e− x ) x,   √ 1 1 √x − x , J2,1 = agy (1 − e ) √ + e 2 2 x √ √ J2,2 = ea(1 − e− x ) x − m. It is easily seen that for the system (1) and (3) there are only three possible equilibria. Indeed, the case x = 0, y = 0 cannot arise, because the second equilibrium equation cannot be satisfied. Biologically this is evident, as the predators are specialist on this prey, and the lack of the latter prevents them from thriving. We now turn to the analysis of the remaining three situations. The possible configurations that can occur as intersections of the system’s isoclines are shown

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Fig. 2 The two possible situations that can arise for the isoclines intersections of the system (1), (3). Ignoring the origin, on the left, the coexistence equilibrium is feasible, on the right only the predator-free point appears to be viable

in Fig. 2 and correspond to the origin, the equilibrium E0 = (0, 0), the predatorfree case, equilibrium E1 = (x1 , 0), and coexistence, E ∗ = (x ∗ , y ∗ ).

3.1 The Origin For the origin, the Jacobian’s eigenvalues are easily evaluated from the diagonal entries, the only nonvanishing ones: λ1 = r > 0 and λ2 = −m. It follows that this equilibrium is unconditionally unstable.

3.2 The Predator-Free Equilibrium The predator-free point E1 = (K, 0) is obtained from (1), (3) by setting y = 0, which satisfies the second equilibrium equation. Evidently, the biology indicates that in the absence of the predators the prey settle to the environmental carrying capacity K. This equilibrium is always feasible. In this case, the Jacobian becomes an upper triangular matrix, so that once more the eigenvalues are explicitly known: −r < 0 and ag(K) − m. Imposing that the latter is negative, gives the stability condition for this equilibrium: K < −1



m ag

 .

(5)

The situation is illustrated in Fig. 2. The nontrivial point on the horizontal axis is E1 . On the left, it is unstable, on the right, it is stable, providing a convincing mathematical argument in support of the biological intuition.

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3.3 Coexistence of Both Populations Note that the same pictures in Fig. 2 allow also the discussion of the coexistence of both populations. The corresponding equilibrium is E ∗ , which is seen to be feasible on the left, while it does not exist in the first quadrant on the right. On the basis of this consideration, we will assume throughout this subsection that 

−1



m ag

 < K.

Turning to the mathematical analysis to determine this coexistence equilibrium, we solve the second equilibrium equation and find x ∗ = −1



m ag

 (6)

and substituting it into the first equation we obtain ∗   rx ∗ (1 − xK ) rg ∗ x∗ = x 1− . y = a(x ∗ ) m K

∗

In view of these results, coexistence E ∗ is feasible if x ∗ ≤ K.

(7)

To study the stability of this equilibrium, note that the Jacobian matrix evaluated at this point has one of the diagonal entries that simplifies J2,2 = 0. We then impose the Routh–Hurwitz conditions. In this way we obtain −tr(J (E ∗ )) = ay ∗  (x ∗ ) − r(1 −

2x ∗ ) = M1 , K

det(J (E ∗ )) = a 2 gy ∗ (x ∗ ) (x ∗ ) = amy ∗  (x ∗ ), having used (6) to evaluate (x ∗ ). Recalling the definition (4) of  (x), it is easily established that det(J (E ∗ )) > 0. Instead, the study of the sign of M1 requires a much deeper analysis.

3.3.1

The Sign of M1

We present now an approximate analysis of the quantity M1 , to obtain a feeling for its qualitative behavior. Observe that we have the simplifications

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√      ∗ m x∗ rg ∗ 2x ∗ ae− x 1 − x − r 1 − M1 = + 2gx ∗ 2 m K K   √   ∗ r x∗ 4x ∗ agx ∗ e− x = 1− + 1+ −2 2 m K K   √   ∗ x∗ 3 ∗ r age− x ∗ + x −1 . = x 1− 2 m K K

(8)

The explicit determination of the sign is not possible, because in the expression of x ∗ we need to calculate the inverse of the transcendental function . This inverse −1 (x) exists because (x) is a monotonically strictly increasing function. However, we can at least make a qualitative analysis of this expression: recalling that (7) holds whenever the coexistence equilibrium is feasible, a sufficient condition for M1 being positive is provided by K < 3x ∗ .

(9)

This condition appears to bear much resemblance with the one found in [4]. To gather some more qualitative information, we can fix the parameter values, for instance, as follows: a = 1, g = 1, m = 1 so that m(ag)−1 = 1, so that x ∗ = −1 (1) = 1.8224; taking r = 2 for simplicity, since this parameter does not influence the sign of M1 , compare indeed (8), we then find   5, 4672 1.8224 + − 1. M1 ≈ 0, 4725 1 − K K Letting K vary, we consider the minor to be a function of the carrying capacity, thus M1 = M1 (K). The function is shown in Fig. 3. In particular, note that M1 > 0 for K < 8.7651 and in this range the equilibrium E ∗ is stable. This result agrees with the previous discussion, since in this case K < 8.7651 = 4, 8097x ∗ , but recall that (9) represents just a sufficient condition.

3.3.2

An Approximation of −1

An alternative approach to the determination of the sign of M1 , with fixed parameter values a = 1, g = 1, m = 1, and r = 2 is now presented. Recalling that we sought (x) as a function satisfying (2), to find an approximate inverse function, we can take a similar approach. We thus seek a function (x) such that

(x) ≈ x,

as

x → 0;

(x) ≈ x 2 ,

as

x → ∞.

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Fig. 3 The function M1 (K) for the chosen parameter values: a = 1, g = 1, m = 1, and r = 2, so that m(ag)−1 = 1

A function which fits in a good way such requirements is

(x) =

x2 , x > 0. 1 − e−x

(10)

While for large x the function (x) is a good approximation of −1 (x), Fig. 4 shows its behavior near the origin. In it, one can observe that the composition of the two functions ( (x)) approximately agrees with the identity function for the smallest and the largest values of the independent variable, ( (x)) ≈ x, except for slight differences in the critical range [0.4, 3.5], where (x) deviates a bit from the graph of −1 (x). This suggests to avoid the use of this approximation in this particular range. With this approximation, the sign of M1 can be more easily assessed. Letting z = m(ga)−1 we obtain that M1 > 0 is equivalent to the condition K < (z)

√ 1 − (z)

(z) − 3 ze √ . 1 − (z)

(z) − 1 ze

(11)

As an example, let us take z = 10 to find that (11) becomes K < 300.1044, which is consistent with the previous results, as now K < 3.0009x ∗ , where x ∗ = −1 (10) ≈ (10). If we numerically evaluate −1 (10), we find that M1 > 0 for K < 300.1181, which essentially coincides with the previous value.

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5 y=x y = (x) y= (x) y= ( (x))

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 4 The behavior of (x) near the origin, compared with those of ( x), y = x, and ( (x))

To sum up these considerations, at the zero of the function M1 , the equilibrium E ∗ loses its stability and a Hopf bifurcation occurs. Thus past this threshold, the populations oscillate giving rise to persistent limit cycles, as it is shown in Fig. 5.

4 The System’s Behavior 4.1 Systems Permanence The concept of permanence implies that all the populations in the ecosystem never disappear. In order to study the permanence of the system (1), (3) we let λ = x α y β with α > 0, β > 0 and assess the sign of the following expression: λ x y =α +β λ x y

(12)

at the equilibrium point EK = (K, 0). Substituting the system equations we obtain  λ x (x) = αr 1 − − ayα + βag(x) − mβ. λ K x

(13)

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x

15 10 5 0

0

20

40

60

80

100 t

120

140

160

180

200

0

20

40

60

80

100 t

120

140

160

180

200

15

y

10

5

0

Fig. 5 The limit cycles obtained with K = 18, a = 1, r = 3, g = 0.6, m = 1. Top: prey; Bottom: predators; both shown as functions of time

Calculating this expression at EK we obtain λ = ag(K) − m λ which must be imposed to be positive. Evaluating its sign implies the requirement: K > −1



m ag

 .

(14)

This means that when the coexistence equilibrium is feasible, so that (14) holds by assumption, our system is permanent; otherwise, we would have a simple logistic growth for prey and extinction for predators, so the prey population tends to its carrying capacity.

4.2 Bifurcations The whole system’s behavior is captured in Fig. 6 obtained with the parameter values a = 1, g = 0.6, r = 3, m = 1. In such case the bifurcation diagram, obtained as function of the bifurcation parameter K, shows that the system initially settles to the predator-free equilibrium (for roughly K < 3.9), then attains coexistence at

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12 x y 10

8

6

4

2

0

0

2

4

6

10

8

12

14

16

18

k

Fig. 6 Bifurcation diagram with the parameter values a = 1, g = 0.6, r = 3, m = 1, as function of the bifurcation parameter K. The system shows initially the predator-free equilibrium (K < 3.9), then coexistence at a stable level (3, 9 < K < 14.4), finally a Hopf bifurcation occurs and persistent oscillations are observed

a stable level (for roughly 3, 9 < K < 14.4), finally a Hopf bifurcation occurs for which persistent oscillations arise. These are observed also for larger values of the bifurcation parameter with increasing amplitudes as the latter increases. It is interesting to note that when the predators invade the system, they keep on increasing their ultimate levels with an increasing value of the prey carrying capacity, until a threshold is reached, which triggers the onset of limit cycles. This is an instance of the paradox of enrichment, for which the more one feeds the prey, the larger the predator population becomes [14].

5 Conclusions We have considered a new formulation of the function modeling the predator–prey interactions when the prey gather in herds. The correction allows to overcome the shortcomings of the square root response function that arise for small numbers of individuals. Taking care of this problem allows the construction of a more realistic model. A further consideration concerns the Jacobian. In view of the construction of the system, aimed precisely at avoiding the problems found in earlier herd behavior

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models, no singularities arise in its calculation at points where the prey vanish, contrary to what found, for instance, in [4, 24]. Thus the coordinate axes are solutions of the homogeneous system (1), (3) and therefore no extinction in finite time can occur in this context. Although mathematically very interesting, this phenomenon may thus be related more to the shortcomings of the model formulation than to real biological or ecological effects. Acknowledgments This work has been partially supported by the projects “Matematica per le applicazioni” and “Questioni attuali di approssimazione numerica e loro applicazioni” of the Dipartimento di Matematica “Giuseppe Peano,” Università di Torino.

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16. S.A.H. Geritz, M. Gyllenberg, Group defence and the predator’s functional response. J. Math. Biol. 66, 705–717 (2013) 17. G. Gimmelli, B.W. Kooi, E. Venturino, Ecoepidemic models with prey group defense and feeding saturation. Ecol. Complex. 22, 50–58 (2015) 18. B.W. Kooi, E. Venturino, Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey. Math. Biosci. 274, 58–72 (2016) 19. H. Laurie, E. Venturino, I.M. Bulai, Herding induced by encounter rate, with predator pressure influencing prey response, in Current Trends in Dynamical Systems in Biology and Natural Sciences, ed. by M. Aguiar, C. Braumann, B. Kooi, A. Pugliese, N. Stollenwerk, E. Venturino (Springer-SIMAI Series, 2019) 20. H. Laurie, E. Venturino, A two-predator one-prey model of population dynamics influenced by herd behaviour of the prey. Theor. Biol. Forum 111(1–2), 27–47 (2018) 21. A. Maiti, P. Sen, D. Manna, G.P. Samanta, A predator-prey system with herd behaviour and strong Allee effect. Nonlinear Dyn. Syst. Theory 16(1), 86–101 (2016) 22. D. Melchionda, E. Pastacaldi, C. Perri, M. Banerjee, E. Venturino, Social behavior-induced multistability in minimal competitive ecosystems. J. Theor. Biol. 439, 24–38 (2018). https:// doi.org/10.1016/j.jtbi.2017.11.016 23. N.M. Oliveira, F.M. Hilker, Modelling disease introduction as biological control of invasive predators to preserve endangered prey. Bull. Math. Biol. 72, 444–468 (2010) 24. E. Venturino, S. Petrovskii, Spatiotemporal behavior of a prey-predator system with a group defense for prey. Ecol. Complex. 14, 37–47, (2013). http://dx.doi.org/10.1016/j.ecocom.2013. 01.004

On Network Similarities and Their Applications I. Granata, M. R. Guarracino, L. Maddalena, I. Manipur, and P. M. Pardalos

1 Introduction From gene regulation to protein–protein interaction and metabolism, networks enable a complex characterization of processes and functions within and among cells. The first attempts in biological network analysis have been mainly devoted to the construction of single networks and the characterization of their features in terms of local and global properties, such as modularity and resilience to failure in nodes and edges [1–4]. Comparison of complex biological networks is an ongoing area of research. Several studies have been performed to compare networks from different conditions (e.g., healthy vs. disease), through the identification of common and unique parts, or the identification of statistically different modules [5, 6]. Nevertheless, the analysis of datasets composed of networks is still in its infancy, especially in case of biological networks, whose datasets are rarely available, compared to social, economics, finance networks which are widely used by researchers to test new algorithms and approaches [7, 8]. This is mainly due to the fact that the general problem of comparing two networks is supposed to be NP-complete. In biological applications, the networks edges and nodes have a biological meaning, and therefore any pair of networks representing the same phenomenon in two different conditions are already aligned, and the similarity of networks can be computed in polynomial time. It is necessary to identify similarity measures which are particularly suited for biological network analysis. Indeed, different representations of the networks and I. Granata · M. R. Guarracino · L. Maddalena () · I. Manipur National Research Council, ICAR, Naples, Italy e-mail: [email protected] P. M. Pardalos CAO, University of Florida, Gainesville, FL, USA © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_3

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measures might highlight different aspects. For this reason, it becomes crucial to understand which are the available options and how they can be used to understand a specific problem at hand. Thus, all subsequent data analysis tasks will be based on the network representation and choices of similarity measures. Many measures for analyzing differences or similarities between graphs have been adopted in the literature [6, 9–16] and most of them have been evaluated on biological network data. Pavlopoulos et al. [9] survey several measures and analyze their role in the analysis of different biological networks, including protein–protein interaction, biochemical, transcriptional regulation, signal transduction, and metabolic networks. Also Mueller et al. [10] provide an overview of methods to compare biological networks, evaluated for the task of classifying gene networks representing benign and cancerous prostate tissue, as well as for classifying metabolic networks of organisms into the three domains of life (archae, bacterium, and eukaryote). More recently, Ruan et al. [15] perform differential analysis of DNA comethylation networks associated with ovarian cancer. These are constructed by taking each probe as a node and computing edges weights based on the Pearson’s correlation coefficient between beta observed values. Schieber et al. [12] evaluate their distribution-based measure for quantifying network dissimilarities on an undirected network representing protein interactions contained in yeast. The network contains 1846 nodes, representing proteins, and 2203 edges, representing metabolic interactions between proteins. Clemente and Grassi [13] evaluate local clustering coefficients on the neural network of the Caenorhabditis elegans worm [17]. The weighted directed network contains 297 nodes that represent neurons. A tie joins two neurons if they are connected by either a synapse or a gap junction; weights represent the number of these synapses and gap junctions. Donnat and Holmes [14] provide an overview of commonly used graph distances and a characterization of the structural changes that they are able to capture. For the evaluation, they consider a longitudinal microbiome study, consisting of a set of bacterial samples taken from different subjects at different points in time. The goal is to assess the antibiotics’ effects on microbial communities in distinct treatment phases. Raw bacterial counts are transformed into graphs by capturing interactions in microbiome samples. For each subject at a given treatment phase, they define a graph in which nodes correspond to bacteria species and edges capture pairwise affinities between bacteria. For assessing diversity in multiplex networks, Carpi et al. [16] project the genetic and protein interactions of the human immunodeficiency virus-type 1 (HIV-1) as a multiplex network and analyze its node diversity. The network consists of 1114 nodes replicated in 16 layers, where the nodes are cellular genes and proteins that have been shown to interact with those encoded by HIV-1. Here, we focus on network similarity measures in the context of directed, weighted, and structurally similar biological networks. Our evaluation is carried out on metabolic network data extracted by breast cancer patients. We focused our analyses on metabolic networks since they are gaining more and more attention in

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25

the context of systems biology and precision medicine. The metabolism influences all the physiological and pathological processes and is dramatically altered in cancer [18, 19]. The rest of this article is organized as follows. Section 2 provides basics on graphs and describes the adopted notations. A description of several measures frequently adopted for assessing similarity/dissimilarity of networks is provided in Sects. 3 through 5, subdividing them in basic measures based on the use of distance between nodes, measures based on clustering properties of the nodes, and measures based on the distance of distributions extracted from the networks, respectively. The results of those measures on tumor metabolic networks are reported and compared in Sect. 6. Section 7 draws conclusions and highlights future research directions.

2 Basic Definitions and Notation A graph G can be defined as a pair (V , E), where V is a set of vertices representing the nodes and E ⊆ V × V is a set of edges representing the connections between the nodes. The graph will be denoted as G(V , E). In the case of undirected graphs, E consists of unordered pairs of elements of V . This means that, if ei,j = (i, j ) ∈ E is an edge between nodes i, j ∈ V , then node i is connected to node j, and vice versa. Nodes i and j will be said to be neighbors or adjacent. The degree (or connectivity) of a node i is given by the number of graph edges connected to it, i.e., the cardinality of the set of its neighbors [11]. In the case of directed graphs (also called digraphs), E consists of ordered pairs of elements of V . The existence of an edge ei,j = (i, j ) ∈ E between nodes i, j ∈ V means that node i is connected to node j, but not vice versa. Node i will be said to be predecessor of node j and node j will be said to be successor of node i. The out-degree of a node i is given by the number of outgoing edges ei,k = (i, k), ∀k, i.e., the cardinality of the set of its successors. The in-degree of a node i is given by the number of incoming edges ek,i = (k, i), ∀k, i.e., the cardinality of the set of its predecessors. The degree of a node i is given by the sum of its out-degree and in-degree. A walk from node i to node j of a graph G(V , E) is a sequence p =< v0 , v1 , . . . , vk > of nodes such that i = v0 , j = vk , vi ∈ V , (vi−1 , vi ) ∈ E, i = 1, . . . , k [20]. The length of the walk p is the number k of its edges, and will be denoted as length(p). A path is a walk where all nodes and edges along it are p distinct. A path p from i to j will be denoted as i  j . Let W be a set of real numbers, called weights, such that for each e ∈ E there exists a w(e) ∈ W ; then G(V , E, W ) is called a weighted graph. Let ei,j be the edge connecting node i with node j of the weighted graph G(V , E, W ); the matrix G ∈ R|V |×|V | , whose elements are wi,j = w(ei,j ), is called the weighted adjacency matrix associated to the graph G. Here, |V | indicates the cardinality of the set of vertices V .

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For weighted graphs, the strength of a node, rather than the degree as defined above, is used [11]. For undirected weighted graphs, the strength si of node i is defined as the sum of the weights of the edges connected to it si =



wi,j .

j

Analogously, for weighted digraphs, the in-strength siin and the out-strength siout of node i are defined as   wj,i and siout = wi,j . siin = j

j

3 Nodes Distance-Based Measures Many measurements for graphs are based on parameters describing a graph in terms of the distances between its nodes [9, 11, 14, 21–23]. Here, after defining the nodes distance, we describe some measures based on it and adopted in the literature.

3.1 Distance Between Nodes in a Graph For an unweighted graph G(V , E), the distance δ(i, j ) (also called geodesic distance) of node i from node j is defined as the length of the shortest path from i to j in G; if no such path exists, then it is set to ∞ [9]  δ(i, j ) =

p

min{length(p) : i  j } if p exists . ∞ otherwise

(1)

If the graph is weighted, the same definition can be used, even though a more informative definition of distance takes into account the edge weights [11]. Given a weighted graph G(V , E, W ), with w : E → R, the weight of a path p =< v0 , v1 , . . . , vk > is given by the sum of the weights of its edges [20] weight (p) =

k 

w(evi−1 ,vi ).

i=1

The weight of the shortest path from i to j is defined as

(2)

On Network Similarities and Their Applications

 δ(i, j ) =

27 p

min{weight (p) : i  j } if p exists ∞ otherwise

(3)

and it is used as distance of node i from node j . It can be observed that, in case of unweighted graphs, the weight of a path coincides with its length (taking w:E → {1} as weight function in Eq. (2)) and the distance of Eq. (3) coincides with that given in Eq. (1). Another observation concerns the definition of weight of a path given in Eq. (2). This is the usual choice when the edge weights are proportionally related to some physical distance between the nodes, e.g., if the vertices correspond to cities and the weights to distances between these cities [11]. Indeed, in these cases, the weight to be minimized when looking for the shortest path is directly proportional to the sum of the weights of all the path edges. On the other side, often the edge weights reflect the strength of the connection between the nodes, where a larger weight corresponds to a higher reliability of the connection [9]. For example [11], if the vertices are Internet routers and the weights represent the bandwidth of the edges, the weight of the path would be better defined as the sum of the reciprocals of the edge weights weight (p) =

k 

1

i=1

w(evi−1 ,vi )

,

(4)

so that the path bandwidth is maximized when looking for the shortest path. Figure 1 provides examples of various cases.

3.2 Measures Based on Distances Between Nodes Given a weighted digraph G(V , E, W ), several properties can be considered.

Fig. 1 Distances and shortest paths. In the unweighted graph (a), the shortest path from node 1 to node 5 is p =< 1, 5 >, and δ(1, 5) = length(p) = 1. In the weighted graph (b), the shortest path from node 1 to node 5 is p =< 1, 2, 5 >, and δ(1, 5) = weight (p) = 1 + 2 = 3 when weight (p) is defined as in Eq. (2). Instead, assuming the alternative definition of weight (p) given in Eq. (4), the shortest path from node 1 to node 5 is p =< 1, 3, 4, 5 >, and δ(1, 5) = weight (p) = 1/2 + 1/3 + 1/4 = 1.083

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Average Path Length (AvgP L) [9, 11] Also called average geodesic distance, it is the average of distances δ(i, j ) taken over all pairs of distinct nodes i, j ∈ V which are connected by at least one path, computed as AvgP L(G) =

 1 δ(i, j ), |V |(|V | − 1)

(5)

i =j

where |V |(|V | − 1) is the number of possible directed edges of the graph G. Limiting the summation to connected vertices avoids divergence (i.e., adding ∞) in case of unconnected pairs of vertices [11]. Global Efficiency (GE) [21] Average of the reciprocals of the distances GE(G) =

 1 1 , |V |(|V | − 1) δ(i, j )

(6)

i =j

taken over all pairs of distinct nodes i, j ∈ V . As noted in [11], this definition avoids divergence (in case no path exists between two nodes, their distance is ∞, so that its contribution to GE is null). Harmonic Mean of Geodesic Distances (h) [11] Reciprocal of the global efficiency h(G) =

1 . GE(G)

(7)

The above properties are examples of univariate network measures, as defined in [6], that can be adopted for comparing networks if embedded in a clustering framework or if used as input to ordinary similarity measures. In the experiments, using any of the above graph properties, we consider the distance between two graphs Gq and Gr on the same set of nodes obtained as the absolute value of their difference (see Fig. 2). Further distances can be considered for two graphs Gp and Gq on the same set of nodes V , with weighted adjacency matrices W p and W q . Hamming Distance (dH am ) It is a special instance of the graph-edit distances, that measures the number of edge deletions and insertions necessary to transform one graph into another. It is defined as  dH am (Gp , Gq ) =

p

q

|wi,j − wi,j |

i,j

|V |(|V | − 1)

.

(8)

Therefore, dH am is a scaled version of the L1 norm between the weighted adjacency matrices.

On Network Similarities and Their Applications

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Fig. 2 Nodes distance-based measures. Results obtained using absolute values of the difference of the univariate network measures defined in Eqs. (5)–(7) (top row) and using the bivariate network measures defined in Eqs. (8)–(10) (bottom row)

Jaccard Distance (dJ ) Defined for the case of weighted graphs as [14] 

p

q

p

q

min(wi,j , wi,j )

i,j

dJw (Gp , Gq ) = 1 − 

.

(9)

max(wi,j , wi,j )

i,j

It assumes values in [0,1]. Edge Difference Distance (dEDD ) Defined for weighted graphs as [23] dEDD (Gp , Gq ) = W p − W q F =

 p q (wi,j − wi,j )2 . i,j

In the experiments, to bound the range of the function distance in the interval [0,1], we consider its normalized version dnEDD 

p

q

(wi,j − wi,j )2

i,j

dnEDD (Gp , Gq ) =  i,j

p

q

max(wi,j , wi,j )

.

(10)

30

I. Granata et al.

All these distances provide a local measure of dissimilarity between graphs, considering links as independent entities, while disregarding the overall structure [14, 22].

4 Clustering-Based Measures Another graph property frequently adopted for computing the dissimilarity of two graphs is the clustering coefficient. It shows the tendency of a graph to form tightly connected neighborhoods [24], i.e., to be divided into clusters, intended as subsets of vertices that contain many edges connecting these vertices to each other [9]. The global clustering coefficient (also called transitivity) is concerned with the density of triplets of nodes in a network. A triplet can be defined as three nodes that are connected by either two (open triplet) or three (closed triplet) ties. A triangle consists of three closed triplets, each centered on one node. The global clustering coefficient is defined as the number of closed triplets (or 3 x triangles) over the total number of triplets (both open and closed). It has been extended to weighted digraphs in [25]. Originally introduced for undirected graphs [26, 27], the local clustering coefficient for a node has been generalized to undirected, weighted graphs [28], as well as to binary and weighted directed graphs [13, 24]. In the experiments, we considered the local clustering coefficient CiT for node i of a weighted digraph G(V , E, W ) introduced in [13] cyc

CiT (G) = α1 Ciin (G) + α2 Ciout (G) + α3 Ci (G) + α4 Cimid (G).

(11)

It is the weighted average of four different components that separately consider different link patterns of triangles that node i can be part of. Ciin (G) deals with triangles where there are two edges incoming into node i; Ciout (G) with triangles cyc where there are two edges coming out of node i; Ci (G) with triangles where all the edges have the same direction; Cimid (G) deals with remaining triangles. Coefficients α1 , . . . , α4 are defined in terms of in- and out- strengths and degrees of node i (see [13] for the complete definition). Global values for a whole graph are obtained as the average of the above coefficients for all its nodes [13]. In the experiments, we consider the distance of two graphs computed as the absolute value of the difference of these global clustering coefficients (see Fig. 3).

5 Distribution-Based Measures Other measures used for evaluating similarities between networks are given as distances of probability distributions describing them [11, 12, 16, 29–31].

On Network Similarities and Their Applications

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Fig. 3 Clustering-based measures. Results obtained using absolute values of the difference of the average of clustering coefficients defined in Eq. (11)

5.1 Distance Measures Between Probability Distributions Several distance/similarity measures are adopted in the literature for comparing two probability distributions (see [32] for a comprehensive survey). In our experiments, given two discrete distributions P ={P1 , . . . , Pd } and Q={Q1 , . . . , Qd }, we consider the following. Euclidean distance L2   d  dEuc (P , Q) =  (Pi − Qi )2 .

(12)

i=1

Jaccard distance [33] d  (Pi − Qi )2

dJ ac (P , Q) =

i=1 d 

Pi2 +

i=1

d 

Q2i −

i=1

d 

.

(13)

Pi Qi

i=1

Hellinger distance [34] dH el (P , Q) =



1 − BC(P , Q),

(14)

where BC(P , Q) is the Bhattacharyya coefficient d   BC(P , Q) = Pi Qi . i=1

Jensen–Shannon distance (JSD) [35] dJ S (P , Q) =



J (P , Q),

(15)

32

I. Granata et al.

where J indicates the Jensen–Shannon divergence (JSd) of P and Q, defined as [36] J (P , Q) =

1 1 D(P , M) + D(Q, M), 2 2

with M = (P+Q)/2. Here, D(P , Q) indicates the Kullback–Leibler divergence [37] of P and Q, that is the expectation of the logarithmic difference between the two probabilities D(P , Q) =

d 

Pi log

i=1

Pi . Qi

All the above distances assume values in [0,1], with lower values indicating closer distributions.

5.2 Graph Probability Distributions In [29], we defined a set of probability distributions describing local and global topological properties of each node of a graph: • Nir , the node distance distribution (NDD) of node i in graph Gr : its generic element Nir (h) is the fraction of nodes in Gr having distance h from node i Nir (h) =

|{j ∈ V : δ(i, j ) ∈ [h, h + 1)}| . |V | − 1

with h=0, 1, . . . , diam, diam being the diameter (i.e., the longest shortest r } contains information about path) of Gr . The set of all NDDs {N1r , . . . , N|V | the global topology of the graph Gr . r (s) is the probability of a node • T r (s), the transition matrix (TM) of order s: Ti,j j to be reached in s steps by a random walker located in node i. Specifically, for s=1 wi,j wi,j r Ti,j (1) =  = out . si wi,k k

Therefore, T r (1) is the weighted adjacency matrix of graph Gr rescaled by the out-strength of each node. The TMs contain local information about the connectivity of the graph Gr . p

q

Given two graphs Gp and Gq , let their NDDs for node i be Ni and Ni , and their TMs be T p (s) and T q (s), s = 1, 2, respectively. Averaging their JSDs over all |V |

On Network Similarities and Their Applications

33

nodes, we defined three graph measures: |V |

M1J S (Gp , Gq ) =

1  p q dJ S (Ni , Ni ), |V |

(16)

i=1

|V |

M2J S (Gp , Gq ) =

1  p q dJ S (Ti (1), Ti (1)), |V |

(17)

i=1

and |V |

M3J S (Gp , Gq ) =

1  p q dJ S (Ti (2), Ti (2)). |V |

(18)

i=1

Moreover, we also considered two further graph distances, given as combinations of the above ones [29] 1 k MJ S (Gp , Gq ), k = 2, 3. k k

DJk S (Gp , Gq ) =

(19)

i=1

In the experiments, besides results achieved with the metrics of Eqs. (16)–(19), we also consider those achieved with analogous metrics, obtained by substituting dJ S with any of the other distribution distance measures dl , l ∈ {Euc, J ac, H el} defined in Eqs. (12)–(14). To provide a uniform notation, all these metrics will be denoted as |V |

Mkl (Gp , Gq ) =

1  k,p k,q dl (Pi , Pi ), k = 1, 2, 3 |V |

(20)

1 k = Ml (Gp , Gq ), k = 2, 3, k

(21)

i=1

and k

Dlk (Gp , Gq )

i=1

for l ∈ {Euc, J ac, H el, J S}. Here, Pi1,r , Pi2,r , and Pi3,r indicate the probability distributions Nir , Tir (1), and Tir (2) for node i in graph Gr (r = p, q), respectively (see Fig. 4). Different probability distributions, other than the NDDs or TMs above, can be adopted to describe and quantify structural information of networks. In a general f f form [30], they can be expressed as P f = {P1 , . . . , P|V | }, where the probability for node i of a graph G(V , E, W ) can be defined as

34

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(b) Fig. 4 Distribution-based measures: (a) Results using the four different distances between distributions defined in Eqs. (12)–(15) (one for each column) with the five different distributions defined in Eqs. (20) and (21) (one for each row). (b) Distances between different samples normalized with respect to the JSD distance between the two LA samples

On Network Similarities and Their Applications

f (i)

f

Pi =

|V | 

35

i = 1, . . . , |V |.

,

(22)

f (j )

j =1

Here, f : V → R+ indicates an abstract information function of G, which can be chosen in different ways. In the experiments, we consider the distribution T

T

T

C P C = {P1C , . . . , P|V | },

(23)

T

where PiC is obtained by choosing in Eq. (22) f (i) = CiT (G), the local clustering T coefficient defined in Eq. (11). Note that, in this case, P C is a single distribution in out cyc describing the whole graph G. Analogous distributions P C , P C , P C , and mid are obtained from the four components of the clustering coefficient of PC Eq. (11). Then, the distribution distance measures dl , l ∈ {Euc, J ac, H el, J S}, of Eqs. (12)–(15) are used for evaluating the distance of two graphs Gp and Gq in j terms of the distributions P C , j ∈ {T , in, out, cyc, mid}. These metrics will be denoted as j

C MC l (Gp , Gq ) = dl (P

j ,p

, PC

j ,q

),

(24)

j

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6 Experimental Results Our experiments aim at analyzing the role of the considered network distance measurements and their ability to highlight differences between networks. To this end, we have in mind a graph classification problem that in its general setting can be formalized as follows. Let (Gi , yi ), i = 1, . . . , m, be a set of m weighted digraphs over a common set of nodes in G , each associated with a class label yi ∈ K = {1, . . . , k}. Let f : G −→ K be a function such that f (Gi ) ∼ yi , for f ∈ H, the hypothesis space, and let d(f (G), y) be a metric for evaluating the difference between the predicted value f (G) and the actual value y. The aim of graph classification is to find the function f ∈ H that minimizes the empirical risk: 1  d(f (Gi ), yi ). m m

I [f ] =

i=1

(25)

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(b) Fig. 5 Distribution-based measures: clustering coefficients. (a) Results using the four different distances between distributions defined in Eqs. (12)–(15) (one for each column) with the five different distributions based on clustering coefficients defined in Eq. (24) (one for each row). (b) Distances between different samples normalized with respect to the JSD distance between the two LA samples

On Network Similarities and Their Applications

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Here, our interest is in the role of the measure d in Eq. (25), chosen among those described in the previous sections, rather than in the computation of a suitable function f , which in our case is supposed to be known (f (Gi )=yi , ∀i). Therefore, we choose the data so as to have the easiest possible graph classification problem. This is achieved by considering (1) a small number of classes (k=2), chosen among those that are most far apart, (2) a small number of samples (m=4, two samples for each class), chosen among those providing highest inter-class distance. For any measure d, each network in the dataset can be represented by the vector containing the distances from all other elements. The resulting symmetric square m × m matrix contains in each row a sample from the dataset, and it is usually called Gram matrix or Distance matrix.

6.1 Data Gene expression data of breast cancer from microarray experiments were downloaded from the NCBI Gene Expression Omnibus database (cite) (GSE78958). The metabolic networks were constructed and simplified as described in [38]. The obtained networks are made up of 3254 nodes, representing the metabolites, and 21902 edges connecting reagent and product metabolites, whose weights are combinations of expression values of the involved enzymes catalyzing the reactions in which the two metabolites are involved. To facilitate our analysis, we focused on two very different subtypes of breast cancer (see point (1) in Sect. 6): Luminal A (LA) and Basal-Like (BL) subtypes. LA is hormone-receptor positive (estrogen-receptor and/or progesterone-receptor positive) and HER2 negative. These types of cancers are low-grade, tend to grow slowly and have the best prognosis of all the subtypes. BL breast cancer is hormone-receptor negative (estrogen-receptor and progesterone-receptor negative) and HER2 negative. It is highly heterogeneous and shows very different responses to therapies. BL is often used interchangeably with Triple Negative Breast Cancer (TNBC) terminology [39, 40]. Recently, Lehmann and colleagues performed a more thorough dissection of TNBC/BL into seven distinct subtypes based on gene expression profiling [41]. Therefore, our analysis will be devoted to ascertaining the ability of the considered measures in discriminating the two classes, providing low intra-class and high inter-class distances. Moreover, we also analyze the ability of the considered measures in taking into account the subtypes of the BL class, providing variable intra-class distances for BL samples. To further facilitate the in-depth analysis of the considered network measures, we selected two samples from each of the LA and BL subtypes. These samples were chosen among those having farthest distances in the PCA plane of the whole dataset, so as to ensure that their features are as peculiar as possible of their classes (see point (2) in Sect. 6).

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6.2 Evaluation Having selected four networks, in the following named LA1 and LA2 for the Luminal A class and BL1 and BL2 for the Basal-Like class, we obtain a 4 × 4 Gram matrix containing the pairwise distances of the four networks for each of the considered measures, as shown in Figs. 2, 3, 4, 5. Here, we represent the Gram matrices through heatmaps, where diagonal elements (in white) have value zero (distance of a network from itself), while the remaining elements appear dark/light blue if their distance is high/low in the range of obtained values. Normalized Node Distance-Based Measures The first row of Fig. 2 shows that measures based on network geodesic distances produce comparable results, even considering their different value ranges. In the second row of Fig. 2, we observe that the Hamming and Jaccard distances produce similar results showing a more pronounced distance between BL samples, while the EDD measure does not capture this heterogeneity, and the distances between BL are comparable to the distance between LA samples. Clustering-Based Measures In Fig. 3, we observe that the global clustering measures C T , C out , and C cyc well discriminate the two classes, but they do not capture BL heterogeneity. Instead, C in and C mid alone lead to wrong discrimination results, as darker/lighter elements do not belong to the same class. Distribution-Based Measures Regarding the M measures, the inter-class dissimilarities obtained with these distances are comparable (see Fig. 4). With respect to within-class distances, especially for JSD distance, we can notice a sort of ranking from the most discriminative M2 to the less one M1 . Indeed, the M1 distance shows more similarity between the two LA samples and between the BL samples, but, in the case of the M2 and M3 distances, this difference is stronger, and the BL samples are more distant from each other. The range of distances seen for M1 is much greater than that for M2 and M3 ; hence, the results obtained with the D2 and D3 distances are similar to those produced by M1 . Figure 4b shows the distribution measures computed with the JSD distance normalized using the distance between the two Luminal A samples. As seen in Fig. 4a, among the distance metrics computed between the various distance distribution measures, the JSD distance performs better than the other metrics, followed by the Hellinger distance. Figure 5 reports results on the distributions based on clustering coefficients defined in Eq. (24). Here, we observe that for the measure C in , the inter-class distances and the BL intra-class distances look very similar, while for the remaining measures, results are analogous to those reported in Fig. 4. Moreover, comparing Fig. 3 with Fig. 5, we observe that global clustering values obtained through the difference of averages are much less informative than those obtained through the distribution-based measures. However, in both cases, the global measure C T based on clustering is much better suited for classification than any of its four components.

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7 Conclusions In this paper, we have addressed the research question: how can we measure the similarity of two weighted networks sharing the same labeled nodes? We believe this is a central question in biological network analysis. We provided a comparison among several network representations and similarities, highlighting their capability to capture inter- and/or intra-group heterogeneity trying to explain in what they can be useful. More work should be devoted to the subsequent data analysis step, which is greatly influenced by these choices, also extending the evaluation to a higher number of samples. A final consideration regards the limited availability of datasets of weighted directed biological networks that can be used to compare different solution strategies. We are planning to help to fill this gap, by providing the network scientific community with datasets of this specific type, extracted from publicly available data for various diseases. Acknowledgments The work was carried out also within the activities of M.R.G. and L.M. as members of the INdAM Research group GNCS. P.M.P. was supported by a Humboldt Research Award. The authors would like to thank G. Trerotola for the technical support.

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Impacts of Infections and Predation on Dynamics of Sexually Reproducing Populations Ludˇek Berec

1 Introduction Sexual reproduction is ubiquitous. A need thus arises for females and males to look for one another and mate. Mating is a complex process that starts with mate search and is often followed by mate choice. Provided that compatible mates meet, courting may follow and precede actual sexual intercourse. This sequence of events is further complicated by many existing mating systems, which determine degrees of polyandry and polygyny that females and males, respectively, adopt, duration of pair bonds within a single mating season or even across several seasons, possibility of extra-pair copulation, or any mating hierarchy that commonly occurs within both temporary leks and relatively permanent social groups [1]. It does not have to be effective to compose a population model that covers all these mating-related elements, since specific life histories may include only some of these processes while some may be irrelevant given a research question. However, most population models are not formulated with sexual reproduction and mating in mind. Curiously, it is hard to find any chapter on sex-structured models in a textbook on mathematical ecology. Such chapters are present in the books [2, 3], but even there they cover only 22 out of 652 and 12 out of 424 pages, respectively. Even though we may eventually end up with a model that does not distinguish between females and males (I call such models asexual in this article), consideration of sexual

L. Berec () Centre for Mathematical Biology, Institute of Mathematics, Faculty of Science, University of ˇ South Bohemia, Ceské Budˇejovice, Czech Republic Institute of Entomology, Department of Ecology, Czech Academy of Sciences, Biology Centre, ˇ Ceské Budˇejovice, Czech Republic e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_4

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reproduction and hence mating dynamics on the path to an eventual asexual model should in many cases be a crucial intermediate step, as I also demonstrate below. I start this article with one of the simplest sex-structured (or two-sex) population models. Then, I extend this model to investigate impacts of infections and predation on dynamics of sexually reproducing hosts or prey. In particular, I consider models of sexually transmitted infections, assuming mating-mediated consistency between the processes of host reproduction and pathogen transmission. I also consider effects of predators that attack female and male prey at different rates. All these developments require explicit consideration of females and males and hence a crucial interaction between them—mating. One of the aims of this article is to show that population models based on explicit mating dynamics may generate predictions that differ from those produced by conventional asexual models, and thus stimulate further research on sex-structured population dynamics.

2 Sex-Structured Population Models The core part of any sex-structured population model is the mating function. This function describes the rate at which females and males mate. With NF and NM denoting female and male densities, respectively, and M(NF , NM ) a generic mating function, a common two-sex modelling framework is as follows [4–7]: dNF = γ bwM(NF , NM ) − (μF + d(NF + NM ))NF , dt dNM = (1 − γ )bwM(NF , NM ) − (μM + d(NF + NM ))NM . dt

(1)

In this model, b is the number of newborns per female reproductive event, γ is the proportion of females among the offspring, w is the fraction of matings that result in reproduction, μF and μM are the intrinsic mortality rates of females and males, respectively, and d is the strength of negative density dependence in the mortality rates, assumed equally affecting both sexes. Note that both the fertilization rate wM(NF , NM ) and the reproduction rate bwM(NF , NM ) are modelled as proportional to the mating rate. I do not claim this holds universally, but may be valid for many animals where mating is commonly followed by reproduction. The model (1) is arguably one of the simplest sex-structured population models. It arises from a variety of assumptions on details of mating dynamics which are then subsumed into a particular form of the mating function M(NF , NM ) [4, 8]. This is not to say that dynamics of every sexually reproducing population can be described by the model (1). For example, formation of longer-term pair bonds or consideration of mate choice requires models that account for other population classes (pairs, or females or males with different traits to choose from, respectively) [4, 9, 10]. On the other hand, even the simple sex-structured model (1) suffices to demonstrate

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that accounting for females and males may bring about predictions that differ from those produced by conventional asexual models. Before discussing commonly used forms for the mating function M(NF , NM ), let us assume the 1:1 sex ratio at birth and sex-independent process rates, that is, γ = 0.5 and μF = μM = μ. Then, NF = NM = N/2 where N = NF + NM , and by adding equations for NF and NM the model (1) reduces to a single equation   N N dN = bwM , − (μ + dN)N. dt 2 2

(2)

A value of starting with a sex-structured model is to correctly derive a corresponding, mating-based asexual counterpart. A variety of mating functions have been proposed, most coming from the human demographic literature [11, 12]. Mating functions can roughly be divided into two broad classes. First, most sex-structured population models assume mating functions that are degree-one homogeneous, that is, satisfy M(ax, ay) = aM(x, y) for any positive x, y, and a [4, 7, 11–14]. So, doubling population density while preserving the sex ratio should double the mating rate. With a degree-one homogeneous mating function, the sex-structured model (1) becomes dNF = γ bwM(1, NM /NF )NF − (μF + d(NF + NM ))NF , dt dNM = (1 − γ )bwM(1, NM /NF )NF − (μM + d(NF + NM ))NM , dt

(3)

and its asexual counterpart (2) is dN bwM(1, 1) = N − (μ + dN)N. dt 2

(4)

Denoting β = bwM(1, 1)/2, dynamics of the model (4) is simple. The population goes extinct for β < μ, but attains the carrying capacity (β − μ)/d otherwise. The model (4) is actually a description of the logistic population growth and the sex-structured model (3) can thus be considered a two-sex generalization of logistic population dynamics (Fig. 1). Whereas the degree-one homogeneous functions are widely accepted as models of mating, they are not without drawbacks. Most importantly, they keep the per female mating rate M(NF , NM )/NF = M(1, NM /NF ) constant when the ratio NM /NF is fixed, independent of how high or low total population density is. This may be questionable, however, when members of the population challenge the socalled mate-finding Allee effect, that is, an enhanced difficulty to find mates in lowdensity populations [15–20]. To account for the mate-finding Allee effect, I use here the mating function

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A

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4 3 2 1 0 0

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4

5

0.3 0.2 0.1 0 −0.1 0

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3

Fig. 1 Dynamics of the sex-structured model (3) (a) and its asexual counterpart (4) (b). Once β > μ, the population attains a carrying capacity from any positive initial density. Parameters: γ = 0.4, w = 1, b = 2, μF = 0.2, μM = 0.3, d = 0.1, and μ = 0.25. Harmonic mean mating function M(NF , NM ) = NM NF /(NM +NF ) is used to plot this figure. The solid dot in panel A is a stable equilibrium, while the grey lines are various population density trajectories that approach it

M(NF , NM ) =

NM NF , NM + ϑ

(5)

where ϑ is a positive parameter representing the Allee effect strength [15, 21, 22]. For this function, the per female mating rate NM /(NM + ϑ) decreases as the male density NM declines, even if the ratio NM /NF remains constant. With NM = NF = N/2, the mating function (5) is  M(NF , NM ) = M

N N , 2 2

 =

N2 . 2(N + 2ϑ)

(6)

The sex-structured model (1) thus becomes dNF NM = γ bw NF − (μF + d(NF + NM ))NF , dt NM + ϑ dNM NM = (1 − γ )bw NF − (μM + d(NF + NM ))NM , dt NM + ϑ

(7)

and the asexual model (2) changes to bw N dN = N − (μ + dN)N. dt 2 N + 2ϑ

(8)

Denoting β = wb/2 and θ = 2ϑ, the population goes extinct for β < μ. However, it now goes extinct also for β > μ if its density N falls below a critical value Nc co-determined by the Allee effect strength θ and commonly referred to as

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Fig. 2 Dynamics of the sex-structured model (7) (a) and its asexual counterpart (8) (b). There is a critical manifold in the system state space (a) or a critical population density (b) below which the population goes extinct even if β > μ, and above which the population attains a carrying capacity. Parameters: γ = 0.4, w = 1, b = 2, μF = 0.2, μM = 0.3, d = 0.1, ϑ = 1, and μ = 0.25. The solid dots in panel A are alternative stable equilibria, while the grey lines are various population density trajectories that approach one or the other; the black curve separates the areas of attraction of these alternative equilibria of which one represents population extinction

the Allee threshold. If N > Nc in the latter case, the population attains a carrying capacity. The model (3) is thus a sex-structured description of the mate-finding Allee effect (Fig. 2).

3 Sexually Transmitted Infections Sexually transmitted infections (STIs) are ubiquitous among animals [23, 24]. Relative to non-sexually transmitted infections, STIs tend to cause sterility rather than increase mortality in the hosts [25]. Indeed, the association between STIs and abortion or infertility in animals is perhaps the major reason why veterinarians have studied these infections [26]. Moreover, STIs often have endemic rather than epidemic character and are more persistent in the hosts [25]. There is often either no recovery from STIs or (less often) there is recovery to the susceptible class. It is a bit of irony that while sex has an advantage in providing genetic variation to keep pace with evolving pathogens, it also has a disadvantage in that it provides a way for pathogen transmission [25]. Here, I first present a conventional model of host-STI dynamics that I then extend to sexually reproducing host populations. I consider both types of mating functions introduced in the previous section and discuss resulting host-STI dynamics. Finally, I show how the developed two-sex framework can be used to address pathogeninduced mating enhancement and mating avoidance in the hosts.

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3.1 Conventional Model of Host-STI Dynamics Basic models of an interaction between sexually transmitted pathogens and host populations have served to reveal fundamental dynamics behind such infections, as an initial step towards examining dynamics brought about by various complexities of infections or hosts [27–30]. The host population consists of susceptible and infected individuals with densities S and I , respectively. There is a possibility of sterility virulence of infection as a proportional reduction σ of host fecundity, and mortality virulence of infection as an additive increase in the host mortality rate α. Moreover, sexually transmitted pathogens are standardly assumed to be transmitted in a frequency-dependent way, with a transmission coefficient λ [25, 31–33]. With these assumptions, a basic (asexual) model of a STI is [27] SI dS = β(S + (1 − σ )I ) − λ − (μ + dN)S, dt N dI SI =λ − (μ + dN)I − αI. dt N

(9)

Dynamics of the model (9) is relatively simple [27]. If the basic reproduction number R0 = λ/(β + α) is lower than 1, the disease-free equilibrium is stable and the host population obeys a logistic growth. On the other hand, when 1 < R0 < R0c , where the critical value of R0 is R0c = 1 +

β(1 − σ )(β − μ) , (β + α)(α + μ − β(1 − σ ))

(10)

then an endemic equilibrium exists and is stable. Finally, a disease-induced extinction equilibrium exists if R0 > 1 and is stable when R0 > R0c . The argument behind frequency-dependent transmission in STIs is that the number of sexual partners per unit time is constant regardless of population density [25, 31, 32]. This actually means that at high densities the number of sexual contacts is limited only by mating opportunity or breeding season length, whereas at low densities organisms are extremely efficient in finding mates. Below, I revise this assumption in the context of sex-structured population models. Not all STIs appear to obey frequency-dependent transmission. The mite Coccipolipus hippodamiae is a parasite transmitted sexually in the two-spot ladybird Adalia bipunctata. The mean proportion of males that were found infected at the end of an experiment increased with total ladybird density rather than stayed constant as predicted by frequency dependence [34]. Moreover, the mean proportion of females that mated as a function of total ladybird density increased in a hyperbolic way [34], indicative of the mate-finding Allee effect. It is therefore likely that the transmission process in STIs is partly density-dependent because the mating rate may change with the host density; indeed, if host densities fall to such low values that the mating rate declines, it is likely that there would be a simultaneous fall in reproductive success

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[35]. Curiously, this relationship between transmission and mating in STIs has been explored only recently [36, 37]. I discuss these two studies below.

3.2 Sex-Structured STI Models Now I extend the conventional model (9) to account for both sexes, respecting the infection-free, sex-structured model (1). The extension is quite straightforward. Let M(X, Y ) be the rate at which susceptible females (X = SF ) or infected females (X = IF ) mate with susceptible males (Y = SM ) or infected males (Y = IM ). In addition, let N = NM + NF = SM + IM + SF + IF be the total population density, and αF and αM be the disease-induced mortality rates in females and males, respectively. A sex-structured host-STI model is then as follows: dSF = γ bw ( SS M(SF , SM ) + SI M(SF , IM ) + I S M(IF , SM ) dt + I I M(IF , IM )) − SI M(SF , IM ) − (μF + dN)SF , dSM = (1 − γ )bw ( SS M(SF , SM ) + SI M(SF , IM ) + I S M(IF , SM ) dt + I I M(IF , IM )) − I S M(IF , SM ) − (μM + dN)SM , dIF = SI M(SF , IM ) − (μF + dN)IF − αF IF , dt dIM = I S M(IF , SM ) − (μM + dN)IM − αM IM . dt

(11)

The remaining new symbols XY and XY (for X, Y = S, I ) represent generic, context-dependent multiplicative constants. Below, I consider three specific forms of the model (11) and specify these constants. Note that in the model (11), mating drives both host reproduction and pathogen transmission, as expected for STIs in animals. To efficiently work with the model (11), I adopt several other assumptions. First, I assume homogeneous mixing of individuals within the population. Then, M(X, Y ) = M(NF , NM )

X Y , X = SF , IF , Y = SM , IM , NF NM

(12)

where M(NM , NF ) is the total mating rate, X/NF is the proportion of females of type X among all females and Y /NM is the proportion of males of type Y among all males. Second, I assume the 1:1 sex ratio at birth and sex-independent process rates: γ = 0.5, μF = μM = μ, SI = I S = , and αF = αM = α. This implies SF = SM = S/2 where S = SF + SM , and IF = IM = I /2 where I = IF + IM . By adding equations for SF and SM and for IF and IM , the model (11) reduces to

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the asexual counterpart   dS N N SS S 2 + SI SI + I S I S + I I I 2 = bwM , dt 2 2 N2   N N SI , − 2M − (μ + dN)S, 2 2 N2   N N SI dI = 2M , − (μ + dN)I − αI. dt 2 2 N2

(13)

With no infection,   dN N N = bw SS M , − (μ + dN)N. dt 2 2

(14)

Let this model have a positive stable equilibrium (population carrying capacity) K. From the second equation of (13), it follows that R0 =

2M(N/2, N/2) 2M(N/2, N/2)/K = . μ + dK + α bw SS M(N/2, N/2) + αK

(15)

If the disease is not lethal (α = 0), we have R0 = 2/(bw SS ). Hence, for non-lethal STIs, the basic reproduction number R0 does not depend on the mating function M(N/2, N/2). With a degree-one homogeneous mating function, the model (13) becomes dS bwM(1, 1) SS S 2 + SI SI + I S I S + I I I 2 = dt 2 N SI − (μ + dN)S, − M(1, 1) N dI SI = M(1, 1) − (μ + dN)I − αI. dt N

(16)

Denoting β = bwM(1, 1)/2 and λ = M(1, 1), the eventual model is dS

SS S 2 + SI SI + I S I S + I I I 2 SI =β −λ − (μ + dN)S, dt N N dI SI =λ − (μ + dN)I − αI. dt N

(17)

Note the frequency-dependent transmission term λSI /N [36], but quite a nonstandard reproduction term relative to the conventional model (9). Substituting the Allee effect mating function (6) into the population model (13), we have

Impacts of Infections and Predation on Dynamics of Sexually Reproducing Populations

51

bw SS S 2 + SI SI + I S I S + I I I 2 SI dS = − − (μ + dN)S, dt 2 N + 2ϑ N + 2ϑ dI SI = − (μ + dN)I − αI. (18) dt N + 2ϑ Denoting β = bw/2, λ =  and θ = 2ϑ, the eventual model is dS

SS S 2 + SI SI + I S I S + I I I 2 SI =β −λ − (μ + dN)S, dt N +θ N +θ dI SI =λ − (μ + dN)I − αI. (19) dt N +θ The transmission term λSI /(N + θ ) that emerges in this model is an example of asymptotic transmission [32, 38]. This type of horizontal transmission implies density dependence at low host densities and nearly frequency dependence at high host densities [35, 39]. It appeared as a reaction to the assumption of frequency dependence at low population densities, allowing individuals search in unrealistic distances to find a mate. The mate-finding Allee effect thus implies the asymptotic transmission term. As with a degree-one homogeneous mating function, the reproduction term remains quite non-standard. Many of the results I present below are formulated in terms of the total host density N = S + I and the infection prevalence i = I /N. Indeed, it turns out that rather than to directly analyse the model (13) it is often easier to explore its transformation into the N and i variables:   N N SS (1 − i)2 + SI (1 − i)i + I S i(1 − i) + I I i 2 dN = N bwM , dt 2 2 N − (μ + dN) − αi] ,   N N SS (1 − i)2 + SI (1 − i)i + I S i(1 − i) + I I i 2 di = i −bwM , dt 2 2 N 

 N N 1−i , − α(1 − i) . (20) + 2M 2 2 N

3.3 Sterilizing Infections To study the effect of sterilization on host population dynamics, I set SS = 1,

SI = I S = 1 − σ , I I = (1 − σ )2 , and  = 1 [36, 37]. Thus, each infected individual in a mating pair reduces the reproductive output by the factor 1−σ , where σ represents the degree of disease-induced fecundity reduction.

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Table 1 Results of the model (21) under full sterilization (σ = 1) Transmission R0 = λ/(β + α) (0, 0) ((β − μ)/d, 0) (Ne , ie )

λ 1 (Fig. 7). The higher cost of mating enhancement, now in the form of increasing z, thus again lowers the likelihood that diseaseinduced mating enhancement evolves. As also shown in Ref. [43], given a cost on mating enhancement, mating enhancement is more likely to evolve if mating is enhanced in both sexes as opposed to in one sex, and with increasing polygyny in the host population (when mating is enhanced in one sex).

3.5 Mating Avoidance While host mating strategies affect the way STIs impact their host populations, STIs are likely to shape host mating strategies. By avoiding an infected mate, individuals

Impacts of Infections and Predation on Dynamics of Sexually Reproducing Populations

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Fig. 6 Regions in the β − λ parameter space in which mating enhancement evolves or does not evolve. The ‘cusp’ point at which the shaded area emerges has components β = k and λ = 2k. Darker grey in the mating enhancement area indicates higher value of mating enhancement δ attained by evolution. Parameter values: z = 2, μ = 0.1, k = 3 (a) or k = 6 (b). After Ref. [43] 10 Mating enhancement δ

Fig. 7 The value of mating enhancement δ attained by evolution as a function of the curvature z of the virulence-transmission trade-off (24). The black line connects the evolutionary attractors as they vary with the parameter z. The grey lines enclose the range of δ within which disease can invade the host population. Parameter values: β = 7, λ = 15, k = 3, μ = 0.1. After Ref. [43]

8 6 4 2 1 0 1

2

3 4 Trade−off curvature z

5

gain a direct benefit of not becoming infected and possibly also an indirect benefit if healthy individuals are more likely to possess infection resistance genes [51]. On the other hand, costs associated with mate choice, such as lost mating opportunities, may inhibit occurrence of infection avoidance strategies [52]. Surprisingly, examples of STI avoidance strategies seem virtually non-existent. I am aware of just two negative studies [53, 54]. Curiously, no theoretical study that modelled evolution of mating preferences in response to STIs ever mentioned an example system where an avoidance strategy was observed. Moreover, ecological effects of STI avoidance strategies on host population dynamics remain understudied which is why we explored these effects in Ref. [55]. I shortly present this study now. Individuals are assumed to have mating preferences driven by an infection status of their potential mating partners. Let ψSS denote a probability that two susceptible individuals mate upon (mating) encounter, ψSI a probability that a susceptible

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individual and an infected individual pair upon contact, and ψI I a probability that two infected individuals mate upon (mating) encounter. These assumptions allow for using the generic model (13) with SS = ψSS , SI = I S = ψSI , I I = ψI I , and  = λψSI . Assuming further that the mating function is degree-one homogeneous, the model (17) becomes ψSS S 2 + 2ψSI (1 − σ )I S + ψI I (1 − σ )2 I 2 dS SI =β − λψSI − (μ + dN)S, dt N N dI SI = λψSI − (μ + dN)I − αI. (27) dt N In Ref. [55], we assume that individuals are able to recognize the infection status of their potential mating partners. Susceptible individuals thus mate with all susceptible partners, while accepting infected individuals as mates only with a probability ψ. Moreover, infected individuals are assumed not to lose anything by mating with other infected individuals and thus mate unselectively. Hence, ψSS = 1, ψSI = ψ, and ψI I = 1 where one may think of the parameter ψ as willingness of an individual to accept infection risk upon mating; ψ is referred to as the mating willingness parameter further on. For ψ = 0 the infection does not spread since there is zero probability of getting infected for any susceptible individual. For ψ = 1, on the other hand, no mating preferences exist among individuals and the model (27) becomes the model (21) presented earlier. Here we study the case 0 < ψ < 1, for which the model (27) becomes [55] dS S 2 + 2ψ(1 − σ )I S + (1 − σ )2 I 2 SI =β − λψ − (μ + dN)S, dt N N dI SI = λψ − (μ + dN)I − αI. dt N

(28)

The basic reproduction number R0 here equals R0 = λψ/(β + α). As a consequence, when λ > β + α, a decrease in ψ invokes a change from R0 > 1 to R0 < 1 at some critical value ψc . Whereas for R0 > 1 the infection is able to invade the host population when rare, for R0 < 1 it is not. The infection can persist also for some R0 < 1 if its initial prevalence is sufficiently high. All these assertions are for the case of no sterilization (σ = 0) visualized in Fig. 8. Indeed, if the susceptible individuals have a low probability of accepting infection risk, the host population attains the disease-free equilibrium. Sudden drop in host density, together with sudden increase in infection prevalence occur shortly before the critical mating willingness ψc at which R0 = 1. The dynamic regime in this region of R0 < 1 corresponds to bistability between the disease-free equilibrium and an endemic equilibrium. Bistability vanishes at R0 = 1, but the host density still decreases after this point, while the infection prevalence still increases. For higher values of the disease-induced mortality rate α and lower values of the reproduction rate β,

Impacts of Infections and Predation on Dynamics of Sexually Reproducing Populations

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Fig. 8 Host-STI dynamics (host density N e scaled by the host carrying capacity K = (β − μ)/d in a and c, and infection prevalence i e in b and d) as they depend on the mating willingness ψ. Parameters: λ = 15, μ = 0.2, d = 0.1, (a, b) α = 0.1 and β = 4, (c, d) α = 1.4 and β = 1.6. Vertical line: R0 = 1, solid black line: stable disease-free equilibrium or stable endemic equilibrium, dotted black line: unstable endemic equilibrium, grey line: stable disease-induced extinction equilibrium. Bistability occurs between an endemic equilibrium and the disease-free equilibrium to the left of the R0 = 1 line. After Ref. [55]

the decrease in host abundance may even lead to disease-induced host extinction (grey line portions in Fig. 8c, d). Further increase in the mating willingness ψ causes an increase in both the host density and the infection prevalence, making the host population more resilient but with a higher proportion of infected individuals (Fig. 8). That infection prevalence increases with mating willingness is not surprising, since as ψ increases so does the infection transmission rate. But why the highest level of host population depression (possibly resulting in host extinction) is achieved for intermediate values of mating willingness? If the mating willingness is high, mating is frequent and the overall reproduction rate overcompensates negative effects of infection, despite high prevalence levels. As mating willingness declines, the mating rate initially declines at a faster rate than the infection prevalence, since most encounters of susceptible individuals are with the infected individuals.

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Hence, host density at the endemic equilibrium declines, too. However, at some point, encounters between susceptible individuals and infected individuals become rare and the overall reproduction rate starts to overcompensate negative effects of infection again. Eventually, if the probability of accepting an infected host as mating partner is very low, the infection cannot spread and the host population attains the carrying capacity. Increasing the fecundity reduction σ changes the above picture only quantitatively, in three respects. First, higher σ leads to lower host density and higher infection prevalence. Second, the effect of mating willingness ψ is inhibited for higher values of σ in that an increase in host density and infection prevalence with higher ψ is less steep. Third, as the fecundity reduction σ increases the bistability range observed for R0 < 1 expands to lower values of the mating willingness ψ until it covers the range of ψ values (α/λ, (α+β)/λ) for the case of full sterilization σ = 1. Specifically, for non-lethal STIs bistability occurs in the whole range of ψ values (0, β/λ) corresponding to R0 < 1. I note that with full sterilization (σ = 1), the model (28) is analogous to the model (21).

4 Predation on Sex-Structured Prey 4.1 Conventional Models of Predator-Prey Dynamics Despite nearly 100 years of modelling predator-prey dynamics, many current models still follow the path paved by the early fathers like Lotka and Volterra, Gause, or Rosenzweig and MacArthur. Denoting prey and predator population densities by N and P , respectively, the conventional predator-prey models are of the form: dN = Ng(N) − (N, P )P , dt dP = e (N, P )P − mP . dt

(29)

Here g(N ) is the per capita prey growth rate in the absence of predators, (N, P ) is the predator functional response or the amount of prey consumed by one predator per unit time, and e and m stand for the predator consumption efficiency and the predator per capita mortality rate, respectively. Many specific models have this form. The Lotka–Volterra model assumes g(N ) = r and (N, P ) = λN for some positive per capita prey growth rate r and the predator attack rate of prey λ, while the Rosenzweig–MacArthur model works with g(N ) = r(1 − N/K) and (N, P ) = λN/(1+hλN), where K is the prey carrying capacity in the absence of predators and h the predator handling time [3]. It is these two models that I extend here to sex-structured prey populations.

Impacts of Infections and Predation on Dynamics of Sexually Reproducing Populations

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4

Predator density P

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3

2

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0 0

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4

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8

6

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Fig. 9 Dynamics of the Lotka–Volterra predator-prey model (a) and the Rosenzweig–MacArthur predator-prey model (b). See the main text for more details. Parameters: e = 0.4, m = 1, r = 3, λ = 2, and h = 0.3 (b)

Dynamics of the Lotka–Volterra and Rosenzweig–MacArthur models are well known. In the former, the prey and predator densities oscillate in the so-called neutral cycles, the amplitude of which is determined by the initial prey and predator densities [3] (Fig. 9a). The Rosenzweig–MacArthur model is most famous for the so-called paradox of enrichment. As the prey carrying capacity increases, predators are able to invade the system at some point, the prey density declines and the predator density increases [3] (Fig. 9b). At some critical carrying capacity Kc , a supercritical Hopf bifurcation occurs and the stable limit cycles appear. These limit cycles grow in amplitude as K further increases, with the population trajectories approaching zero densities more and more closely so that stochastic extinction is eventually likely [3] (Fig. 9b). Thus, enriching the system by prey resources jeopardizes the very prey existence. This is the paradox of enrichment.

4.2 Sex-Selective Predation With help of the model (1), an extension of the Lotka–Volterra predator-prey model to sex-structured prey is straightforward [56]: dNF = γ bwM(NF , NM ) − μF NF − λF NF P , dt dNM = (1 − γ )bwM(NF , NM ) − μM NM − λM NM P , dt dP = eF λF NF P + eM λM NM P − mP . dt

(30)

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In Ref. [56] we were interested in the effect of sex-selective predation on otherwise Lotka–Volterra dynamics, so we assumed γ = 0.5 and μF = μM = μ, but left eM = eF and λM = λF . Just because of these differences in the female and male life histories, one cannot reduce the model (30) to an asexual counterpart. If there are always enough males to fertilize any female, the per female mating rate M(NF , NM )/NF = 1. On the other hand, when the mate-finding Allee effect occurs we may assume the mating function (5): M(NF , NM )/NF = NM /(NM + ϑ), where ϑ scales the Allee effect strength. Assuming λF > 0, I scale all population densities by λF and set  = λM /λF and θ = ϑλF (otherwise, assuming λM > 0, I would similarly scale all population densities by λM and set  = λF /λM and θ = ϑλM ). Through these transformations, I get the model (keeping the same notation for the rescaled state variables) dNF bw NM = NF − μNF − NF P , dt 2 NM + ϑ dNM bw NM = NF − μNM − NM P , dt 2 NM + ϑ

(31)

dP = eF NF P + eM NM P − mP . dt In this setting,  > 1 corresponds to predation biased to male prey, whereas  < 1 means female-biased predation. When θ > 0, the model (31) has three equilibria [56]: the extinction equilibrium, a prey-only equilibrium, and a coexistence equilibrium. While the extinction equilibrium is locally stable, the unstable prey-only equilibrium is a direct consequence of the mate-finding Allee effect: the prey population with high enough density would grow, but extinction would occur if prey density is low. With no Allee effect (θ = 0), the extinction equilibrium is unstable and no prey-only equilibrium exists, so populations recover from anyhow low densities. In the absence of Allee effect (θ = 0), stability of the coexistence equilibrium is primarily driven by the predation bias for one or the other prey sex: whereas malebiased predation ( > 1) leads to stable coexistence, female-biased predation ( < 1) produces increasing oscillations (Fig. 10a). These contrasting dynamic regimes are caused by emergent density dependence in prey growth. Indeed, male-biased predation gives rise to stabilizing, negative density dependence in prey growth (Fig. 10b); populations perturbed away from the coexistence equilibrium return to it (Fig. 10a). On the contrary, female-biased predation triggers destabilizing, positive density dependence (Fig. 10c): predator-prey cycles spiral away from the coexistence equilibrium (Fig. 10a). The mate-finding Allee effect (θ > 0) is itself a factor destabilizing population dynamics [22]. The domain of instability of the coexistence equilibrium thus expands with increasing θ ; higher male bias in predation is needed to overcome impacts of the Allee effect and keep the coexistence equilibrium stable (Fig. 10d). In addition, as prey oscillations expand, prey density gets closer and closer to zero.

Impacts of Infections and Predation on Dynamics of Sexually Reproducing Populations

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Fig. 10 Effects of sex-selective predation in the model (31). (a) Temporal dynamics in the absence of Allee effect (θ = 0). Male-biased predation leads to a stable coexistence equilibrium (dashed curve;  = 1.5), female-biased predation leads to expanding oscillations (solid curve;  = 0.8), and unbiased predation gives rise to neutral oscillations as in the original Lotka–Volterra model (dotted curve;  = 1). Other parameters: b = 3, d = 0.2, e1 = 0.2, e2 = 0.1, m = 1. (b) Stabilizing effect of male-biased predation; data were generated by computing trajectories for 100 random initial prey and predator conditions, selecting states with predator density close to the value P = 1.3 (results for other fixed predator densities were similar), and plotting the per capita prey growth rate (d(NM + NF )/dt)/(NM + NF ) against the prey density NM + NF ;  = 1.5, other parameters as in A. (c) Destabilizing effect of female-biased predation; data were generated as in B;  = 0.8, other parameters as in A. (d) Effect of the predation bias  and the Allee effect strength θ. The coexistence equilibrium is feasible to the left of white area and locally stable within the dark grey area. Adapted from Ref. [56]

Eventually, prey density falls below the prey-only equilibrium of the model (31) and the prey population (and consequently the predator population) goes extinct. Additional mechanisms may overshadow the effect of sex-selective predation. It is well known that logistic prey growth stabilizes Lotka–Volterra dynamics, whereas a Holling type II functional response destabilizes it [3]. Male- and female-biased predation do the same things. It should therefore come as no surprise that with a prey carrying capacity stable coexistence may be possible also for female-biased predation [56]. On the contrary, regarding a Holling type II functional response,

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when the handling time of the captured prey increases, the predator-prey equilibrium may be unstable also under male-biased predation [56].

4.3 Mating-Predation Trade-off In Ref. [8], we extended the Rosenzweig–MacArthur predator-prey model to account for sex-structured prey, and examined impacts of a trade-off between mate searching and predator avoidance on prey dynamics. More precisely, the rate at which males search for mates, denoted by λF M , was assumed to mediate that tradeoff: searching for mates at a lower rate means reduced predation due to lower exposition to predators, and vice versa. We start with the following predator-prey model: dNF = γ bwM(NF , NM ) − (μF + d(NF + NM ))NF − F (NF , NM )P , dt dNM = (1 − γ )bwM(NF , NM ) − (μM + d(NF + NM ))NM − M (NF , NM )P , dt dP = −mP + eF F (NF , NM )P + eM M (NF , NM )P , (32) dt where F (NF , NM ) and M (NF , NM ) are the rates at which a single predator consumes female and male prey, respectively; the predator functional response is then (NF , NM ) = F (NF , NM ) + M (NF , NM ). As we show in Ref. [8], consideration of a finite mate search rate λF M leads to a mate-finding Allee effect in prey, with M(NM , NF ) =

NM

NM NF , + 1/(λF M T )

(33)

where T is the length of female gestation period. Note that this expression is of the form (5), now with the process-motivated parameter ϑ = 1/(λF M T ). Moreover, assuming a type II predator functional response, we set F (NF , NM ) =

λF P NF , 1 + h(λMP NM + λF P NF )

M (NF , NM ) =

λMP NM , 1 + h(λMP NM + λF P NF )

with λF P and λMP the female-specific and male-specific attack rates and h the handling time of a single prey individual. Figure 11 exemplifies typical predator-prey dynamics. When the mate-finding Allee effect is relatively strong (low λF M ), a bifurcation diagram akin to the

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Fig. 11 Examples of typical predator-prey dynamics. Parameters: γ = 0.5, w = 1, b = 2, μF = μM = 0.1, d = 0.1, T = 1, h = 0.1, m = 0.1, e = 0.05, λF P = 0.1 and (a) λF M = 1 and (b) λF M = 4.5. Legend: solid black = stable coexistence equilibria, dashed black = unstable coexistence equilibria, dashed light grey = Allee threshold in the absence of predation, horizontal dark grey = carrying capacity in the absence of predation (solid = stable, dashed = unstable), curved dark grey = maximum and minimum of stable limit cycles, curved light grey = maximum and minimum of unstable limit cycles, H = Hopf bifurcation point. Note that in panel (a) the Hopf bifurcation is supercritical, while in (b) it is subcritical. Adapted from Ref. [8]

Rosenzweig–MacArthur model appears (Fig. 11a). There is one exception, however: stable limit cycles eventually disappear in a heteroclinic bifurcation and both populations are driven to extinction, because of the Allee effect. On the other hand, for relatively weak mate-finding Allee effect (sufficiently high λF M ), not only is the coexistence equilibrium stable for a wider range of male predation rates, but the Hopf bifurcation is now subcritical so that unstable cycles emerge (Fig. 11b). Stable limit cycles are now observed only if a fold bifurcation of limit cycles follows the subcritical Hopf bifurcation (Fig. 11b). Note that none of these behaviours occurs in the Rosenzweig–MacArthur model. Our main aim in Ref. [8] was to study evolution of λF M . We again used adaptive dynamics shortly presented in appendix. Respecting the trade-off between mating and predation, i.e., that the rate at which predators encounter male prey, λMP , is an increasing function of the male search rate for females, λF M , we considered a family of function λMP (λF M ) = λF P +

pλkF M q + λkF M

,

(34)

with p, q, and k some positive constants. All trade-offs from this family satisfy λMP (0) = λF P and λMP (λF M ) approaching λF P + p as λF M grows large. For 0 < k ≤ 1 the trade-off is concave, whereas for k > 1 it changes from convex to concave at the inflection point λF M = (q(k − 1)/(k + 1))1/k .

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As we show in Ref. [8], if the resident population with trait λrF M resides at a stable equilibrium (Nˆ F , Nˆ M , Pˆ ), the invasion fitness of the mutant male prey with trait λm F M is s(λrF M , λm FM) =

 m  λF M b Nˆ F M(Nˆ M , λrF M ) − 1 2 Nˆ M λrF M −

r λMP (λm F M ) − λMP (λF M ) (Nˆ F , Nˆ M ; λMP (λrF M ), λF P )Pˆ . r ˆ ˆ λMP (λF M )NM + λF P NF (35)

Alternatively, if the resident population with trait λrF M moves along a stable limit cycle of period TL , the invasion fitness of a mutant male prey with the trait λm F M is s(λrF M , λm FM)

b = 2



  TL λm NF (t) 1 FM −1 M(NM (t), λrF M )dt λrF M TL 0 NM (t)

r − (λMP (λm F M ) − λMP (λF M ))



TL

× 0

1 TL

(NM (t), NF (t); λMP (λrF M ), λF P ) P (t) dt. λMP (λrF M )NM (t) + λF P NF (t)

(36)

Evolution drives λF M towards the (preset) maximal mate search rate λmax F M or to an intermediate evolutionary attractor (Fig. 12). When the gestation period T is relatively low (black area in Fig. 12a, b), there is bistability between the maximal mate search rate and an intermediate evolutionary attractor. Whether prey adaptively adjust their search rate to the presence of predators (i.e., attain the evolutionary attractor) thus depends on the starting value of λF M . As T increases (dark grey area in Fig. 12a, b), an evolutionary branching point co-occurs with the maximal mate search rate as an evolutionary attractor. Once evolution brings λF M close to the branching point, disruptive selection occurs. In particular, slowly searching males coexist with fast ones. However, this dimorphism is only temporary (even if quite long) as the slowly searching males eventually go extinct, with the mate search rate evolving towards its maximum (Fig. 12c). When T (or h) increases further (light grey area in Fig. 12a, b), the maximal mate search rate is the only evolutionary outcome and the weakest possible mate-finding Allee effect thus evolves. With higher values of the trade-off curvature k, both evolutionary bistability and the biregime with evolutionary branching become more common (cf. panels A and B of Fig. 12). Despite evolutionary branching, two persistent evolutionary endpoints occur. First, there is omnipresent evolution towards the maximal mate search rate and hence the weakest possible mate-finding Allee effect, which we interpret as no adaptive response of prey to the presence of predators. Second, there is evolution towards an intermediate evolutionary attractor (i.e., the mate-finding Allee effect of

Impacts of Infections and Predation on Dynamics of Sexually Reproducing Populations

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Fig. 12 Evolutionary outcomes. (a, b) Diversity of evolutionary outcomes. The colour legend: light grey = evolution towards the maximal mate search rate λmax F M , dark grey = bi-regime at which max there is evolution towards a branching point or λmax F M , black = bistability between a CSS and λF M . Parameters are as in Fig. 11, with λF P = 0.1, λmax = 2.4, k = 5 (a) and k = 10 (b). (c) FM An evolutionary tree. After the population has arrived at a branching point, disruptive selection occurs, but just one branch eventually survives and evolution proceeds towards λmax F M . Parameters: T = 0.95, h = 0.05, λF P = 0.1, k = 10, other parameters as in Fig. 11. Adapted from Ref. [8]

an intermediate strength), which we interpret as an adaptive change in prey mating behaviour as a response to predation risk.

5 Conclusions and Extensions In this article, I present some of my earlier work on effects of sexually transmitted infections and sex-selection predation on dynamics of sexually reproducing populations. Although sex is ubiquitous, conventional population models still follow the tradition established about 100 years ago that neglects sexual reproduction and its most important process: mating. Mating is classically neglected in population modelling, by virtue of the assumption that there are always enough males to fertilize any female and hence that the (constant) birth rate subsumes a (constant) mating rate. However, human demographic literature as well as many empirical studies on animal mating show that the mating rate is generally a function of female and male densities and that it may strongly affect population dynamics [4, 11, 12, 22]. Indeed, this article more than clearly demonstrates that consideration of sexstructured population models of otherwise conventionally modelled interactions produces quite different and mostly more complex results. Mating thus not only serves to represent a bit of further reality, but also enriches the set of dynamical regimes that may occur for a given system. There is a number of avenues in which to extend the explorations summarized here. For example, mating enhancement and avoidance studies considered just degree-one homogeneous mating functions, and it is quite interesting to ask what would happen if alternatively the mate-finding Allee effect is present. On the other hand, models with sex-selective predation consider the mate-finding Allee

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effect, calling for examination of impacts of degree-one homogeneous mating functions. Also, all examined host-STI models ignore vertical transmission (a common companion of STIs [23]), recovery back to the susceptible class, and the observation that when the STIs are sterilizing the infected individuals may live longer due to using some of the saved resources, otherwise used for reproduction, to maintenance [49, 57]. Last but not least, examination of evolution of mating avoidance in STIs or of sex-selective predation are other potentially interesting topics. All in all, this article has perhaps succeeded in delivering the message that sexual reproduction should be considered more often in population modelling.

Appendix: Adaptive Dynamics Let a trait ξ be subject to evolution. The classical techniques of adaptive dynamics [44, 45] start with assuming that a resident population with trait ξr is established and a rare mutant population with trait ξm invades a stable equilibrium set by the resident. Moreover, they assume that evolution proceed in small steps, i.e., that the mutant’s ξm is close to the resident’s ξr . Then, under mild conditions which our models satisfy, successful invasion implies replacement of the resident population by the mutant one [58]. The formerly mutant population thus becomes resident one and is now challenged by a new mutant. Ecological time is thus assumed to run much faster than evolutionary time [44, 45]. In order to determine whether a rare mutant successfully invades and takes over an established resident, one needs to calculate the mutant’s invasion fitness. This is actually the initial growth rate of the mutant when the resident is at its stable equilibrium X∗ (ξ ) (can be vector if the population is structured). When this invasion fitness s(ξm , ξr |X∗ (ξ ))

(A.1)

is positive, the mutant invades and replaces the resident. On the contrary, if it is negative the mutant dies out and the resident remains at its stable equilibrium. To determine where this invasion-replacement sequence eventually stops, one calculates the selection gradient as the slope of the invasion fitness when the mutant trait is equal to the resident trait, g(ξ ) =

 ∂s(ξm , ξr )  . ∂ξm ξm =ξr =ξ

(A.2)

The value of ξ at which the selection gradient is equal to zero is referred to as the evolutionary singular point ξ ∗ . If s(ξm , ξr ) as a function of ξm is maximized at ξ ∗ , or more formally

Impacts of Infections and Predation on Dynamics of Sexually Reproducing Populations

 ∂ 2 s(ξm , ξr )  E= < 0,  ∂ξm2 ξm =ξr =ξ ∗

69

(A.3)

then this evolutionary singular point is evolutionary stable. That is, populations with trait values near ξ ∗ cannot invade the population with the trait value ξ ∗ . If it is minimized (E > 0), then ξ ∗ is evolutionary unstable. In addition, if the selection gradient is positive in a left neighbourhood of ξ ∗ and negative in a right neighbourhood of ξ ∗ , the evolutionary singular point ξ ∗ is convergence stable. That is, populations with trait values ξ closer to ξ ∗ replace those with more distant ξ values. If the opposite inequalities hold, ξ ∗ is convergence unstable. If an evolutionary singular point is both evolutionary and convergence stable, it is an evolutionary attractor. If it is convergence stable but evolutionary unstable, it is an evolutionary branching point at which a dimorphic parasite population arises. And if it is convergence unstable, it is called an evolutionary repeller. We advise the interested reader to consult Refs. [44, 45] for more details on adaptive dynamics. Acknowledgments I would like to thank Daniel Maxin, Veronika Bernhauerova, David S. Boukal, Vlastimil Krivan, Barbara Boldin, Eva Janouskova, and Michal Theuer for collaboration on the studies presented in this article.

References 1. S.M. Shuster, M.J. Wade, Mating Systems and Strategies (Princeton University Press, Princeton, 2003) 2. H. Caswell, Matrix Population Models, 2nd edn. (Sinauer Associates, Sunderland, 2001) 3. M. Kot, Elements of Mathematical Ecology (Cambridge University Press, Cambridge, 2001) 4. K.P. Hadeler, R. Waldstätter, A. Wörz-Busekros, J. Math. Biol. 26, 635 (1988) 5. J. Lindström, H. Kokko, Proc. R. Soc. Lond. B 265, 483 (1998) 6. C. Bessa-Gomes, S. Legendre, J. Clobert, Ecol. Lett. 7, 802 (2004) 7. D.J. Rankin, H. Kokko, Oikos 116, 335 (2007) 8. L. Berec, V. Bernhauerova, B. Boldin, J. Theor. Biol. 441, 9 (2018) 9. H. Kokko, E. Ranta, G. Ruxton, P. Lundberg, Evolution 56, 1091 (2001) 10. L. Berec, Theor. Ecol. 11, 225 (2018) 11. M. Iannelli, M. Martcheva, F.A. Milner, Gender-Structured Population Modeling (SIAM, Philadelphia, 2005) 12. T.E.X. Miller, B.D. Inouye, Ecology 92, 2141 (2011) 13. H. Caswell, D.E. Weeks, Am. Nat. 128, 707 (1986) 14. C. Castillo-Chavez, W. Huang, Math. Biosci. 128, 299 (1995) 15. F. Courchamp, L. Berec, J. Gascoigne, Allee Effects in Ecology and Conservation (Oxford University Press, Oxford, 2008) 16. J. Gascoigne, L. Berec, S. Gregory, F. Courchamp, Popul. Ecol. 51, 355 (2009) 17. A.M. Kramer, B. Dennis, A.M. Liebhold, J.M. Drake, Popul. Ecol. 51, 341 (2009) 18. X. Fauvergue, Entomol. Exp. Appl. 146, 79 (2012) 19. J. Régnière, J. Delisle, D.S. Pureswaran, R. Trudel, Entomol. Exp. Appl. 146, 112 (2013) 20. P.C. Tobin, K.S. Onufrieva, K.W. Thorpe, Entomol. Exp. Appl. 146, 103 (2013) 21. B. Dennis, Nat. Res. Model. 3, 481 (1989) 22. D.S. Boukal, L. Berec, J. Theor. Biol. 218, 375 (2002)

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Global Analysis of a Cancer Model with Drug Resistance Due to Microvesicle Transfer Attila Dénes and Gergely Röst

1 Introduction Resistance to chemotherapy is a major problem that current cancer research has to cope with as resistance may significantly decrease the effectiveness of the therapy [2]. There are several ways how cells may become resistant to oncologic treatment. Several studies show that tumours develop via a Darwinian evolution leading to the selection of the fittest cells. Our study is motivated by a recent discovery in cancer biology. Álvarez-Arenas et al. [1] described the following mechanism of the induction of cancer drug resistance. According to recent results, microvesicles (i.e. extracellular particles released from the cell membrane) shed by more aggressive donor cells are able to deliver some cellular components to less aggressive acceptor cells. This phenomenon can be compared to the transmission of an infectious disease. The microvesicles, among others, transport efflux membrane transporters, genetic information and transcription factors which are needed for their production in the recipient cells, this way propagating the diffusion of resistant phenotypes among the tumour cells. We construct a simplified model that still captures the key biological features. We denote by S(t) the number of sensitive cells at time t, while R(t) stands for the number of resistant cells. The notation β denotes the rate of microvesicle-mediated transfer from sensitive to resistant cells and θ is the cytotoxic action induced cell mortality of sensitive cells due to drugs. Notations ρ0 and ρr stand for birth rates of sensitive and resistant cells, respectively. Throughout this paper, we assume that ρ0 > ρr . Parameters μ0 and μr denote death rates of sensitive and resistant cells, respectively, due to apoptosis. For the tumour growth, we assume a logistic form

A. Dénes () · G. Röst Bolyai Institute, University of Szeged, Szeged, Hungary e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_5

71

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with carrying capacity K. With these notations, our model takes the form S  (t) = −βS(t)R(t) − θ S(t) + ρ0 S(t)(K − S(t) − R(t)) − μ0 S(t), R  (t) = βS(t)R(t) + ρr R(t)(K − S(t) − R(t)) − μr R(t).

(1)

Due to the biological interpretation, we consider only nonnegative solutions. In what follows we give a detailed description of the possible dynamics generated by system (1). The structure of the paper is the following. In Sect. 2, we determine the possible equilibria of (1) and study the local and global stability of them. We give a complete characterization of the global dynamics of (1), depending on the parameters. In Sect. 3, we present numerical simulations to study the possible effects of increasing drug concentration and we describe the possible bifurcation sequences.

2 Description of the Global Dynamics 2.1 Existence and Local Stability of Equilibria Depending on the parameters, system (1) has four possible equilibria: E0 = (0, 0),   0) , 0 , ES = Kρ0 −(θ+μ ρ0   r ER = 0, Kρrρ−μ , r   0 )−ρr (θ+βK+μ0 ) ρr (θ+μ0 )−β(θ−Kρ0 +μ0 )−μr ρ0 ESR = μr (β+ρβ(β+ρ . , β(β+ρ0 −ρr ) 0 −ρr ) Let us define the following threshold parameters: F0S = Kρ0 − (θ + μ0 ), F0R = Kρr − μr , FSR = ρr (θ + μ0 ) + β(Kρ0 − (θ + μ0 )) − μr ρ0 , FRS = μr (β + ρ0 ) − ρr (θ + βK + μ0 ). We first prove some statements concerning the existence and stability of the four possible equilibria, depending on the above four threshold parameters. Proposition 1 The trivial equilibrium E0 always exists. It is locally asymptotically stable if F0S < 0 and F0R < 0 and it is unstable if F0S > 0 or F0R > 0.

Global Analysis of a Cancer Model with Drug Resistance Due to Microvesicle Transfer

73

Proof The Jacobian of (1) has the form J =



−βR−θ−μ0 +(K−R−S)ρ0 −ρ0 S −βS−ρ0 S Sβ−μr −ρr R+(K−R−S)ρr βR−ρr R

 .

J evaluated at the trivial equilibrium gives 

 0 Kρ0 − (θ + μ0 ) , 0 Kρr − μr  

from which we directly obtain the assertion of the proposition.

Proposition 2 The boundary equilibrium ES exists if and only if F0S > 0. It is locally asymptomatically stable if FSR < 0 and unstable if FSR > 0. Proof It is obvious that ES exists if and only if F0S > 0. The Jacobian evaluated at ES has the form  θ + μ0 − Kρ0 0

(β+ρ0 )(θ+μ0 −Kρ0 ) ρ0 ρr (θ+μ0 )+β(Kρ0 −(θ+μ0 ))−μr ρ0 ρ0

 ,

which has the two eigenvalues θ + μ0 − Kρ0 and ρr (θ+μ0 )+β(Kρρ00−(θ+μ0 ))−μr ρ0 . The first eigenvalue is always negative whenever ES exists, while the second one is negative if FSR < 0 and it is positive if FSR > 0. Here we also take into account that Kρ0 − (θ + μ0 ) is always positive in case of existence of ES .   Proposition 3 The boundary equilibrium ER exists if and only if F0R > 0. It is locally asymptomatically stable if FRS < 0 and unstable if FRS > 0. Proof Clearly, ER exists if and only if F0R > 0. J evaluated at the equilibrium ER gives μ

r (β+ρ0 )−(Kβ+θ+μ0 )ρr ρr (β−ρr )(Kρr −μr ) ρr

0 μr − Kρr

 ,

0 )ρr and μr − Kρr . From this we which has the two eigenvalues μr (β+ρ0 )−(Kβ+θ+μ ρr obtain the assertion of the proposition.  

Proposition 4 The following statements hold for the coexistence equilibrium ESR . (i) The coexistence equilibrium ESR exists if and only if FSR > 0 and FRS > 0. (ii) F0S > 0 is a necessary condition for the existence of ESR , i.e. whenever FSR > 0 and FRS > 0 are satisfied, then F0S > 0 also holds.

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Proof The first statement of the proposition is obvious considering the two threshold parameters FSR and FRS , as well as the formulas in the numerators of the two coordinates of ESR . To show the second statement of the proposition, we estimate S  (t) as S  (t) ≤ S(t)(Kρ0 ) − (θ + μ0 ) = F0S S(t), hence, if F0S < 0, then S(t) → 0 as t → ∞, and thus ESR cannot exist.

 

2.2 Global Dynamics To study the global dynamics of (1), we will apply the Dulac–Bendixson criterion. Let us choose D(S, R) = 1/(SR) as a Dulac function. Then, ∂ −βSR−θS+ρ0 S(K−S−R)−μ0 S ∂S SR

+

∂ βSR+ρr R(K−S−R)−μr R ∂R SR

= − ρR0 −

ρr S ,

which is clearly nonpositive in the nonnegative quadrant. Hence, using the Bendixson–Dulac theorem, we obtain that there is no periodic solution of (1). Applying the Poincaré–Bendixson theorem, it follows that all solutions tend to one of the equilibria. Based on this result and the local stability properties of the equilibria, we can give a complete characterization of the dynamics of system (1), depending on the threshold parameters F0S , F0R , FSR , FRS . The following lemma excludes some of the possibilities regarding the signs of the threshold parameters. Lemma 1 (i) If F0S > 0 and F0R < 0, then FRS > 0. (ii) If F0S < 0 and F0R > 0, then FRS < 0. (iii) If F0S > 0 and F0R > 0, then at least one of FSR > 0 and FRS > 0 holds. Proof The first two assertions can be proved in a similar way, here we only show the proof of the first statement. Let us suppose that F0S > 0 and F0R < 0. Then we have the following estimations: μr (β + ρ0 ) > Kρr (β + ρ0 ) > Kρr β + (θ + μ0 )ρr , which is exactly what we wanted to show. Here the first estimation is based on F0R < 0 and the second estimation is based on F0S > 0. Let us now turn to the third statement of the lemma. Let us suppose that this assertion does not hold, i.e. F0S > 0, F0R > 0, FSR < 0 and FRS < 0 hold at the same time. Applying the results obtained from the linearization, we can see that if these conditions hold, then E0 is unstable with both eigenvalues of the Jacobian evaluated in E0 being positive, ES and ER are both locally asymptotically stable and ESR does not exist. From this we obtain that there exists a separatrix which separates

Global Analysis of a Cancer Model with Drug Resistance Due to Microvesicle Transfer

75

the basins of attraction of the two boundary equilibria. Let us start a solution from a point of the separatrix. The limit set of this solution is an equilibrium of (1). However, we can exclude all equilibria being the limit of this solution, from which we obtain that such a combination of the signs of the four threshold parameters is not possible.   To formulate our main theorem, we introduce the extinction spaces   2 XS = (S, R) ∈ (R+ ) : S = 0 0 and   2 XR = (S, R) ∈ (R+ 0) :R =0 . Based on the discussions above, we obtain the following characterization of the global dynamics. Theorem 1 (i) If F0S < 0 and F0R < 0, then the trivial equilibrium E0 is globally asymptotically stable. (ii) If F0S > 0, F0R < 0, FSR < 0 and FRS > 0, then ES is globally asymptotically stable on X \ XS . E0 is globally asymptotically stable on XS . (iii) If F0S > 0, F0R < 0, FSR > 0 and FRS > 0, then ESR is globally asymptotically stable on X \ XS . E0 is globally asymptotically stable on XS . (iv) If F0S < 0 and F0R > 0, then ER is globally asymptotically stable on X \ XR . E0 is globally asymptotically stable on XS . (v) If F0S > 0, F0R > 0, FSR < 0 and FRS > 0, then ES is globally asymptotically stable on X \ XS . ER is globally asymptotically stable on XS . (vi) If F0S > 0, F0R > 0, FSR < 0 and FRS > 0, then ES is globally asymptotically stable on X \ XS . ER is globally asymptotically stable on XS . (vii) If F0S > 0, F0R > 0, FSR > 0 and FRS < 0, then ER is globally asymptotically stable on X \ XR . ES is globally asymptotically stable on XR . (viii) If F0S > 0, F0R > 0, FSR > 0 and FRS > 0, then ESR is globally asymptotically stable on X \ (XS ∪ XR ). ES is globally asymptotically stable on XR and ER is globally asymptotically stable on XS . In Table 1, we summarize the stability properties of the four possible equilibria for all possible cases, depending on the four reproduction numbers. We note that all the combinations of the signs of the four threshold parameters given in Theorem 1 and Table 1 are possible as can also be seen from Examples 1 and 2.

> 0,

> 0,

F0S

>0

> 0,

FSR

F0S , F0R , FSR , FRS

> 0,

> 0,

FSR

F0R

F0R

< 0,

> 0,

> 0,

FRS

FRS

FRS

>0

0

FRS

Unstable, GAS on XS

F0S > 0, F0R , FSR , FRS < 0

FSR

GAS

F0S , F0R < 0

F0R

E0

Reproduction numbers

Table 1 Stability of equilibria depending on the reproduction numbers

× × ×

GAS on X \ XR GAS on X \ XR GAS on X \ XR Unstable, GAS on XS

Unstable, GAS on XR GAS on X \ XS Unstable, GAS on XR

Unstable

Unstable

Unstable, GAS on XS

GAS on X \ (XS ∪ XR )

×

Unstable, GAS on XR

×

×

GAS on X \ XS

ESR

GAS on X \ (XS ∪ XR )

×

×

ER ×

ES ×

76 A. Dénes and G. Röst

Global Analysis of a Cancer Model with Drug Resistance Due to Microvesicle Transfer

77

3 The Effect of Drug Concentration In this section, we present numerical simulations to study the effect of changing the drug concentration c. The cytotoxic action induced cell mortality of sensitive cells θ clearly depends on the drug concentration, and here we assume that microvesicle production also depends on the concentration, thus θ = θ (c) and β = β(c). In the following, we assume that both of these parameters are monotonically increasing in the concentration c. We fix the values of the remaining parameters and study how the dynamics of system (1) changes due to modifications of drug dosage. Example 1 We fix the following parameter values: ρ0 = 0.0582,

ρr = 0.01,

μ0 = 0.01,

μr = 0.01.

K = 0.927,

For these values, the threshold parameter F0R has the fixed value F0R = −0.00073 < 0. For different values of the free parameters β and θ , we can experience different global dynamics depending on the sign of the remaining threshold parameters. Figure 1 shows the plain (θ, β), divided into five regions by the level curves F0S = 0, FSR = 0, FRS = 0. Using Theorem 1, we can determine which of the equilibria is globally asymptotically stable on that region. Hence, one can see that ES is globally asymptotically stable in the bottom left region, ESR is globally asymptotically stable in the top left region, while E0 is globally asymptotically stable in the remaining three regions. Assuming different functional forms for the dependence of β and θ on the drug concentration, we show four possible sequences of transitions among the different regions depicted in Fig. 1. Figure 2 shows the total cancer mass, i.e. TCM = S + R, as a function of drug concentration for the four different functional forms of the increase of the concentration. Example 2 In the second example, we fix the parameter values as ρ0 = 0.0517,

ρr = 0.0172,

μ0 = 0.01,

μr = 0.01.

K = 0.823,

With these values, the second threshold parameter is fixed at F0R = 0.0041556 > 0. Again, the plain (θ, β) is divided into five regions by the level curves F0S = 0, FSR = 0, FRS = 0. Using Theorem 1, one can see that ES is globally asymptotically stable in the region in the bottom left corner and ESR is globally asymptotically stable in the left top region, while ER is the globally asymptotically stable equilibrium in the remaining three regions. Again, we show four examples of possible transitions among the different regions, applying four different functional forms for the dependence of the parameters β and θ on the drug concentration c.

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0.10

0.08

0.06

0.04

0.02

0.00 0.00

0.02

0.04

0.06

0.08

0.10

Fig. 1 Possible scenarios for the global dynamics for different drug concentrations in the (θ, β)plain for Example 1. The coloured curves represent possible transitions due to change in drug concentration, assuming different functional responses to the concentration

One might observe that, in this case, it is not possible to completely eliminate the tumour as the threshold parameter F0R remains positive, independently of the drug concentration. In this case, the three possible scenarios are the following: all cells are sensitive, all cells are resistant or both types of cells are present in the tumour. The related sequences of transitions are depicted in Fig. 3. Analogously to Figs. 2 and 4 shows the total cancer mass.

4 Discussion We have established a mathematical model to describe the evolution of chemotherapy-resistant and sensitive tumour cells, considering that resistance may emerge as a result of the therapeutic drug. According to some recent discoveries, this might take place via the transfer of microvesicles from resistant to sensitive cells, similarly to the spread of an infectious disease. First, we calculated four threshold parameters; these determine which of the four possible equilibria is

TCM

TCM

0.5 0.6 0.4 0.4 0.3 0.2

0.2

0.05

0.10

0.15

0.20

0.25

c

0.02

0.04

0.06

0.08

0.10

0.03

0.04

0.05

c

(b)

(a)

TCM

TCM

0.6

0.6

0.4

0.4

0.2

0.2

0.01

0.02

0.03

0.04

0.05

c

0.01

0.02

c

(d)

(c)

Fig. 2 Total cancer mass (TCM) as a function of drug concentration (c) with different functional forms of θ(c) and β(c). Black dots denote a concentration level where another of the equilibria becomes globally asymptotically stable. (a) θ(c) = c2 /(1 + c2 ), β(c) = 0.6c2 /(1 + c2 ). (b) θ(c) = 5c/(1 + 150c), β(c) = c. (c) θ(c) = c, β(c) = 0.2c. (d) θ(c) = c, β(c) = 0.4c/(1 + 50c) 0.10

0.08

0.06

0.04

0.00 0.00

0.02

0.04

0.06

0.08

0.10

Fig. 3 Possible scenarios for the global dynamics for different drug concentrations in the (θ, β)plain for Example 2. The coloured curves represent possible transitions due to change in drug concentration, assuming different functional responses to the concentration

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A. Dénes and G. Röst

TCM

TCM

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.05

0.10

0.15

c

0.002

0.004

(a)

0.006

0.008

0.010

0.012

c

(b)

TCM

TCM 0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.005

0.010

0.015

(c)

0.020

0.025

c

0.005

0.010

0.015

0.020

0.025

c

(d)

Fig. 4 Total cancer mass (TCM) as a function of drug concentration (c) with different functional forms of θ(c) and β(c). (a) θ(c) = c2 /(1 + c2 ), β(c) = 0.6c2 /(1 + c2 ). (b) θ(c) = 5c/(1 + 150c), β(c) = c. (c) θ(c) = c, β(c) = 0.2c. (d) θ(c) = c, β(c) = 0.4c/(1 + 50c)

globally asymptotically stable. We showed numerical examples to study how an increase in drug concentration might affect the sensitive and resistant cells. As a future work, we plan to include Lamarckian transition of sensitive cells to resistant cells. Further, we would like to consider temporally changing drug concentrations to mimic realistic chemotherapy. Acknowledgments A. Dénes was supported by the Hungarian National Research, Development and Innovation Office grant NKFIH PD_128363 and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. G. Röst was supported by EFOP-3.6.1-16-2016-00008 and by the Hungarian National Research, Development and Innovation Office grant NKFIH KKP_129877 and 20391-3/2018/FEKUSTRAT.

References 1. A. Álvarez-Arenas, A. Podolski-Renic, J. Belmonte-Beitia, M. Pesic, G.F. Calvo, Sci. Rep. 9, Article No. 9332 (2019) 2. C. Holohan, S. Van Schaeybroeck, D.B. Longley, P.G. Johnston, Nat. Rev. Cancer 13, 714 (2013)

Contact Vaccination Study Using Edge Based Compartmental Model (EBCM) and Stochastic Simulation: An Application to Oral Poliovirus Vaccine (OPV) Coura Balde, Mountaga Lam, and Samuel Bowong

1 Introduction The use of vaccines has been a successful strategy in preventing major disease outbreaks. Generally, vaccines contain an inactivated or attenuated agent responsible for the disease. Individuals who received the vaccine become protected, immunized against the disease. The main strategy against Polio disease is vaccination with Oral Polio Vaccine (OPV) which contains live attenuated Wild Polio Virus (WPV). It provides lifelong immunity to the vaccinated individuals. Generally vaccinated individuals shed these live attenuated WPV for 3–4 weeks through their feces after immunization (see Duintjer Tebbens et al. [2]). Then these viruses could spread to their non-immunized close contacts producing little or no illness. These nonimmunized individuals may become immunized like they have received OPV. This is commonly known as contact immunity (see Duintjer Tebbens et al. [2]). Contact immunity is helpful in increasing the herd immunity by increasing the amount of vaccinated individuals in a population. This is one of the beneficial facts which makes OPV the vaccine of choice for the Global Polio Eradication Initiative (GPEI). As in WPV, the transmission of OPV virus is fecal-oral involving any route, direct or indirect, from feces to mouth. These viruses can also transmit through droplets. Generally viruses settle quickly out of the air and contact must usually be within 1.5 m to be effective. These viruses can also infect fomite or surface. In populations with poor hygiene and sanitation, OPV virus could spread to a neighborhood

C. Balde () · M. Lam Department of Mathematics and Computer Sciences, Faculty of Sciences and Technology, Cheikh Anta Diop University, Dakar, Senegal e-mail: [email protected] S. Bowong Faculty of Science, University of Douala, Douala, Cameroon © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_6

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through sewage water. Recently immunized children shed live virus in their feces for a few days after immunization. About 25% of people coming into contact with someone immunized with OPV gained protection from polio through this form of contact immunity. Thus, contact pattern is a relevant fact in the transmission dynamic of OPV virus. Epidemic spread on Networks has been extensively studied in the last decade [10, 12, 15]. The importance of assumptions about network structure in the epidemic spread of Polio disease has been analyzed in (Duintjer Tebbens et al. [14]). Investigating contact vaccination on a network can be really useful for policy maker. In a contact network, the population is represented by nodes connected by edges. This models the contact pattern that exists in human populations so that individuals have varying number of contacts and, if an epidemic occurs, infected individuals can only transmit to a limited number of other individuals. In this paper we consider a population whose partnerships are static; that is, each individual has a fixed number of contacts from a degree distribution P (k). The population has the Configuration Model (CM) structure (see Molloy and Reed [11]): each individual is assigned a number of stubs (its degree, k), and then finds partners for each stub randomly from the available stubs. So if P (k) is the probability a random individual u has k partners, the probability that a partner v has k  partners is k  P (k  )/ where is the average degree. We assume that infected individuals with WPV and OPV transmit to each partner as an independent Poisson process of rate τ1 and τ2 , respectively, and recover as an independent Poisson process of rate γ and α, respectively. There exist a couple of methods for modelling spread process in a configuration model network: percolation theory, pairwise model, effective degree, and edge based compartmental model (see Miller [9]). The aim of this study is to investigate contact vaccination in a population where a proportion of individuals is infected with WPV using Edge Based Compartmental Model (EBCM) framework as it provides a simple derivation with low-dimensional dynamical system and Stochastic Simulation. We study contact vaccination in three theoretical networks: Erd˝os–Rényi, Barabàsi–Albert, and a geometric random graph as depicted in Fig. 1. These networks are known as general networks which depict approximately well real contact patterns among individuals in a population. Erd˝os–Rényi random graph or G(N, p) random graph is constructed by connecting N nodes randomly. Each edge is included in the graph with probability p independent from every other edge. This gives rise to a random network with Poisson degree distribution of nodes. In the Barabàsi–Albert graph nodes are connected following a power law distribution and preferential attachment. This is a scale free network where most nodes have a few links to other nodes, but a small number of nodes are highly connected and have a huge number of links to the other nodes. In network epidemics theory highly connected individuals are called super-spreaders. In a geometric graph individuals are placed randomly over a two-dimensional plane then are connected randomly uniformly if their distance is at most radius. This is a spatial network with degree distribution of nodes approximately Poisson. The paper is organized as follows: in Sect. 2 we develop the EBCM framework and stochastic simulation. In Sect. 3 we compute epidemic threshold. In Sect. 4 we show the results of numerical simulation

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Fig. 1 Example of networks of 100 nodes. (a) Erd˝os–Rényi random graph, (b) Barabàsi–Albert graph, (c) Geometric graph

and then analyze insight properties of Networks which could get OPV transmission efficient. In Sect. 5 we present a summary of our work and give a conclusion.

2 Method Our derivation is based on the framework of EBCM by Miller [8–10]. This approach relies on creating compartments of partnerships (or edges) rather than compartments of individuals. We assume that epidemic occurs deterministically which means that the population is very large enough and the initial number of infected individuals is large enough for the disease to behave deterministically. We recall that the population is connected according to the configuration model. We define a test individual u as follows: u is a test individual if u is randomly selected from the population and prevented from transmitting to its neighbors. Let v be a neighbor of u. We focus on calculating the probability u is susceptible, infected, vaccinated, or recovered. Prevented u from transmitting to any of his neighbors v has no impact on the probability u is in any given state and therefore, it does not affect the calculation of the proportion of the population in each state (Miller [9]).

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Fig. 2 Flowchart of Polio disease in Network contact. The first diagram shows transition of individuals. The second diagram shows transitions of the status of a partner v of a randomly chosen individual u as well as whether v has transmitted to u ignoring transmissions from u to v

2.1 Edge Based Compartmental Model (EBCM) Each individual is in one of the four states: susceptible (S), vaccinated (V ) that is infected with OPV, infected (I ) infected with WPV, and recovered from infection with OPV or WPV (R). The per-individual transition rate from I to R is γ , and the per-individual transition rate from V to R is α. Infected individual with WPV and infected individual with OPV transmit to their neighbors at rate τ1 and τ2 = κτ1 , respectively, where κ is the relative transmission rate of OPV virus compared to WPV. Figure 2 shows the corresponding flow diagrams for individuals and neighbors. The variables θ , φS , φV , φI , φR give information about the probability a partner v of u has a given status and the probability the partner v has not transmitted to u. All variables and parameters are defined in Table 1. The probability distribution P gives information about the possible degrees of u or its partners. In a population with arbitrary initial fraction ρ infected with WPV and  arbitrary initial fraction infected with OPV, the probability that u is still susceptible, which means that u was initially susceptible at time t = 0 and has not received any infection (no infection with OPV, no infection with WPV) from his neighbors at time t, is the sum over all k of the product of the probability u has degree k with the probability u is still susceptible θk  k given it was initially susceptible, 1−ρ −. We have S(t) = (1−ρ −) P (k)θ = (1 − ρ − )ψ(θ ). Thus we conclude S(t) = (1 − ρ − )ψ(θ ).

(1)

An infected partner with WPV or OPV transmits to u at a rate τ1 and τ2 , respectively. Then the flux from φI and φV to 1 − θ is τ1 φI and τ2 φV , respectively. Thus the dynamic of θ is θ˙ = −τ1 φI − τ2 φV .

(2)

From (1) and (2) the proportion of infected individuals through WPV and OPV is ˙ ), − S˙ = (1 − ρ − )(τ1 φI + τ2 φV )ψ(θ

(3)

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Table 1 Variables and description Variable S V I R Test individual u θ(t) φS (t), φV (t), φI (t), φR (t)

P (k)  = k kP (k)  ψ(θ) = k P (k)θ k τ1 κ τ2 γ α  ρ

Description Susceptible individuals Vaccinated individuals Infected individuals Recovered individuals A randomly chosen member of the population who is prevented from causing infection The probability a random partner v of u which did not transmit to u by t = 0 has not transmitted to u by time t The probabilities that a random partner v of u is susceptible, infected by WPV, infected by OPV virus or recovered and has not transmitted infection to u by time t ≥ 0 The probability an individual has degree k The average degree The probability that the test individual u is susceptible at time t Transmission rate of WPV Relative transmission rate of OPV compared to WPV Transmission rate of OPV The per-individual transition rate from I to R The per-individual transition rate from V to R The proportion of vaccinated individuals at t = 0 The proportion of infected individuals at t = 0

˙ ) are new infected individuals and (1 − ρ − )τ2 φV ψ(θ ˙ ) where (1 − ρ − )τ1 φI ψ(θ are new vaccinated individuals. Following the flowchart in Fig. 2 it is easy to see that R˙ = αV + γ I . Thus we have the following set of equations for individuals: ˙ ), S˙ = (1 − ρ − )θ˙ ψ(θ ˙ ) − αV , V˙ = (1 − ρ − )τ2 φv ψ(θ ˙ ) − γ I, I˙ = (1 − ρ − )τ1 φI ψ(θ

(4)

R˙ = αV + γ I. To close the system we must find equation for φI and φV . Following the flowchart Fig. 2 we only need to derive the flux from φS to φV and φS to φI . Let us calculate φS explicitly. A partner v of the test individual u has degree k with probability kP (k)/. Thus the probability that v is susceptible is the sum over k of the probability that v was initially susceptible, 1 − ρ − , has k neighbors and the product of the probability that v has not received any transmission from any of his neighbors which is θ (k−1) . So we have

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φS = (1 − ρ − )

 kP (k)θ (k−1) k

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˙ ) ψ(θ .

(5)

Then the test individuals u become infectious through contact with a vaccinated individuals or infected individuals at rate − φ˙ S = (1 − ρ − )(τ1 φI + τ2 φV )

¨ ) ψ(θ ,

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¨ ) ¨ ) ψ(θ ψ(θ is the flux from φS to φV and (1 − ρ − )τ1 φI where (1 − ρ − )τ2 φV

is the flux from φS to φI . Following the flowchart in Fig. 2 we have the following equations: ¨ ) ψ(θ ,

¨ ) ψ(θ φ˙ V = (1 − ρ − )τ2 φV − αφV − τ2 φV ,

¨ ) ψ(θ φ˙ I = (1 − ρ − )τ1 φI − γ φI − τ1 φI ,

φ˙ R = αφV + γ φI , φ˙ S = (1 − ρ − )θ˙

(7)

θ˙ = −τ1 φI − τ2 φV . This summarizes the following EBCM model: ˙ ), S˙ = (1 − ρ − )θ˙ ψ(θ ˙ ) − αV , V˙ = (1 − ρ − )τ2 φv ψ(θ ˙ ) − γ I, I˙ = (1 − ρ − )τ1 φI ψ(θ R˙ = αV + γ I, ¨ ) ψ(θ ,

¨ ) ψ(θ φ˙ V = (1 − ρ − )τ2 φV − αφV − τ2 φV ,

¨ ) ψ(θ − γ φI − τ1 φI , φ˙ I = (1 − ρ − )τ1 φI

φ˙ R = αφV + γ φI , φ˙ S = (1 − ρ − )θ˙

θ˙ = −τ1 φI − τ2 φV .

(8)

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2.2 Stochastic Simulation We perform stochastic simulations by considering infection and recovery events as Poisson processes which leads to a continuous-time Markov chain. This leads to a stochastic process which describes that infection events and recovery events evolve randomly with time. Our stochastic simulation derive from the event-driven model in Kiss et al. [6]. The algorithm is based on the use of a priority queue Q. We first generate semi-random networks using the configuration model (Molloy and Reed [11]). For convenience we will say a node is infected if it receives transmission from WPV and vaccinated if it receives transmission from OPV. The key of this simulation is when a node becomes infected or vaccinated we can calculate the time when it recovers or transmits to its neighbors. Thus these events are inserted into Q ordered by time. At each step of the simulation, the next event in the list is removed and processed. If it is a transmission (OPV or WPV), new events will be added to the queue corresponding to the recovery of the node and any transmissions from that node. This process iterates until the time specified or no events remain in Q. The epidemic simulations then proceed as follows: • initial infected nodes and initial vaccinated nodes are selected randomly from the population. • when a node v is infected at time t, a time (t)I of infection is drawn from an exponential distribution (parameter τ1 ) for each of its susceptible neighbors. Then a time of recovery from infection (t)RI is drawn from an exponential distribution (parameter γ ) and is assigned to v. The time t + (t)RI and the time t + (t)I are added to Q. • when a node is vaccinated at time t, a time (t)V at which it will transmit its immunity is drawn from an exponential distribution (parameter τ2 ) for each of its susceptible neighbors. Then a time at which it will enter the recover states (t)RV is drawn from an exponential distribution (parameter α) and is assigned to v. The time t + (t)RV and the time t + (t)V are added to Q. • when t  = t + (t)I is the earliest time Q, an infection transmission event will occur, providing v has not recovered. Then v transmits to whatever of its susceptible neighbors which change status to I . Then a new time t  + (t)I is added to Q. • when t  = t + (t)V is the earliest time in Q, a transmission of OPV virus event will occur, providing v can still transmit these viruses. Then v transmits to whatever of its susceptible neighbors which change status to V . Then a new time t  + (t)V is added to Q. The process continues until there are no more events in the queue.

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and Rv0 3 Reproductive Number Rw 0 3.1 Reproductive Number The reproductive number is the expected average number of infections caused by a node infected early in an epidemic but not the very first case in the population. This is different from the more commonly known quantity, basic reproductive ratio (R0 ) [6, 16]. Here we compute the reproductive number for both OPV transmission (Rv0 ) and WPV transmission Rw 0 . Consider a node v with degree k infected by a neighbor early in the epidemic. Then the expected number of additional infections caused by v, given its degree k is (k − 1)τ1 /(τ1 + γ ). Thus the expected number of infections caused by v averaged over all k among all susceptible individuals (1 − ) is Rw 0 = (1 − )

τ1  kP (k) , (k − 1) τ1 + γ

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(9) (10)

The same derivation leads to the OPV reproductive number Rv0 = (1 − ρ)

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(11)

v Note that epidemics of WPV and OPV are possible only if Rw 0 and R0 are greater w than one. Thus a strategy to stop WPV transmission is to drive R0 below one by vaccinating initially the required number of susceptible individuals. This is commonly known as the critical vaccination rate. The critical vaccination rate follows easily:

crit = 1 −

τ2 + α . τ2 < K 2 − K >

(12)

4 Results and Discussions Here we perform simulation in the three types of network: Erd˝os–Rényi graph, Barabàsi–Albert, and geometric graph. All three networks contain 10,000 individuals. The parameters of the degree distributions were chosen so that each network has an identical average degree of < K >= 6. That is, the density of connections in each network is the same. It is well known that the degree distribution has a great impact on the dynamical behavior and the final size of the epidemic [7, 12, 15]. Figure 3 illustrates the dynamical behavior of infected individuals with WPV virus (I ) and infected individuals with OPV virus (V ) in Erd˝os–Rényi graph,

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Parameters τ1 κ τ2 γ α

References Assume Duintjer Tebbens et al. [2] Duintjer Tebbens et al. [3] Duintjer Tebbens et al. [2]

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Barabàsi–Albert graph, and geometric graph. The infected individuals and vaccinated individuals are initially in the same proportion. Figure 3b shows that in the Barabàsi–Albert graph the expansion phase (the time at which cumulative incidence increases at its maximum rate) of infected individuals with WPV virus is reached very quickly in contrast with the Erd˝os–Rényi graph and the geometric graph where the expansion phase of the epidemic is reached lately. The same dynamical behavior is observed for infected individuals with OPV in Fig. 3a. Figure 4b illustrates the well-known result about the final size of epidemic (Meyers et al. [7], Newmann [12] and Eric Volz [15]). That is when the degree distribution changes from Poisson to exponential to power law the final size of epidemic is largest in Poisson distribution. The same observation is done here, the final size of WPV transmission is largest in the Erd˝os–Rényi graph and geometric graph. However, in contrast, Fig. 4a shows that the final size of OPV transmission

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is largest in Barabàsi–Albert graph. This result is consistent with findings in Kim and Rho [5] where it was shown that by shifting network from Poisson to power law the final size of OPV viruses is largest if degree distribution is power law. That is, vaccinated individuals may be large in a population where contacts between people follow power law distributions. A relevant fact is the dynamical behavior and final size of WPV and OPV transmission is identical in the Erd˝os–Rényi graph and the geometric graph. This might be due to the fact that the geometric graph has degree distribution approximately Poisson such as Erd˝os–Rényi graph. This highlights the fact that the degree distribution is a relevant property of networks. The result in Eq. (12) shows that there is a critical vaccination rate crit above which epidemic of WPV is impossible. In Erd˝os–Rényi graph and geometric graph the critical vaccination rate is equal crit = 75%. In Barabàsi–Albert graph crit = 92%. Figure 5 illustrates the result in (12). Figure 5a, c shows that in Erd˝os– Rényi graph and geometric graph randomly vaccinated over 75% of the population leads to no WPV epidemic. Figure 5b shows that randomly vaccinated over 92% of the population in Barabàsi–Albert graph leads to no WPV epidemic. That is, in a population where contact pattern is Barabàsi–Albert if polio outbreak occurs almost the total population should be randomly vaccinated to stop the transmission of polio. Generally for many diseases we must immunize 75–100% of the population to eradicate the pathogen. Generally difficult to achieve, contact immunity could play a major role by increasing the amount of vaccinated individuals when only a relatively small proportion of the population could be vaccinated. Prior studies show that in the preferential attachment model of Barabàsi–Albert [1], there is no

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level of random vaccination that is sufficient to prevent an epidemic (Albert et al. [1], Pastor-Satorras and Vespignani [13], and Keeling and Eames [4]). However, we can decrease the impact of the epidemic by increasing the amount of vaccinated individuals through contact vaccination. Figure 6 shows that, for random vaccination, vaccinated individuals could be dominant in Erd˝os–Rényi graph, Barabàsi–Albert graph, and geometric graph. It shows that there is a threshold of initial vaccinated individuals depending on the initial infected individuals above which more susceptible individuals are reached by contact immunity such that vaccinated individuals become dominant. Figure 6a, c shows that the threshold of initial vaccinated individuals is identical in Erd˝os– Rényi graph and geometric graph. Figure 6b shows that in Barabàsi–Albert graph the threshold of initial vaccinated individuals is higher compared to Erd˝os–Rényi graph and geometric graph. It means that if a population is infected by WPV, the proportion of susceptible individuals which might be vaccinated to improve contact vaccination depends on the proportion of infected individuals and the contact pattern of the population. Nevertheless the proportion of susceptible individuals might be highest if contact pattern is Barabàsi–Albert. A relevant fact is for each graph the threshold of initial vaccinated individuals is over the proportion of initial infected

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individuals. Nevertheless it might be close to the proportion of initial infected individuals if contact pattern is Erd˝os–Rényi or geometric. Figure 7 illustrates the existence of a threshold for initial vaccinated individuals for different values of initial vaccinated individuals in the Barabàsi–Albert graph. Figure 7a shows that when initial vaccinated individuals are below the proportion of infected individuals remains dominant. Figure 7b, c shows cases where initial vaccinated individuals is over the proportion of infected individuals but it is not sufficient to get more vaccinated individuals such that the final size of vaccinated individuals is larger. Figure 7d shows a case where the proportion of initial vaccinated individuals is over the proportion of initial infected individuals and sufficient such that the final size of vaccinated individuals is larger. Figure 8 illustrates target immunization. That is, individuals are vaccinated following their degree. In all three Networks most connected individuals are targeted for vaccination. Thus individuals of degree over or equal to the average degree are vaccinated. Figure 8a, c shows that in Erd˝os–Rényi graph and geometric graph targeting individuals of degree over or equal to the average degree leads to almost 60% vaccinated individuals in the population which is sufficient to increase the amount of individuals reached by contact immunity such that vaccinated individuals are dominant over time and epidemic of WPV is impossible. Figure 8b targeting individuals with contact over the average degree in Barabàsi Albert graph leads to more than 40% vaccinated individuals in the population which is sufficient to stop

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WPV transmission such that epidemic of WPV is impossible. It is well known that target vaccination is really efficient in scale free network. The existence of highly connected individuals (super-spreaders) plays two major roles. It is important to realize that having many contacts has two effects; it means that the individuals are at greater risk of infection and can transmit to many contacts. Thus by targeting these individuals for vaccination we make barrier of the disease and also give protection to their contacts through contact vaccination. Thus heterogeneity of contact really favors contact immunity. Figure 9 shows the degree distribution of infected individuals and vaccinated individuals compared to the degree distribution of the entire Network. In the three networks the degree distribution of infected individuals and vaccinated individuals follow the same shape as the degree distribution of the entire network. Further Fig. 9a, b shows that the proportion of infected individuals tends to be higher in degree where the density of the population is higher. The same is observed in Fig. 9c for Barabàsi Albert graph.

5 Summary and Conclusion In this paper we investigate contact immunity using the framework of edge based compartmental model (EBCM) and Stochastic Simulation. Investigations are done

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Fig. 9 Comparison of the degree distribution of the Network (histogram) with the degree distribution of infected individuals with WPV and OPV (solid lines). Each graph contains 5000 nodes. We perform 100 simulations in each graph. (a) Erd˝os–Rényi graph, (b) geometric graph, (c) Barabàsi–Albert graph. All the parameters are as in Table 2

among three types of networks: Erd˝os–Rényi, Barabàsi–Albert, and a geometric random graph. We have shown that the dynamical behavior and final size of infected individuals and vaccinated individuals depend on the degree distribution. From Poisson degree distribution to power law degree distribution infected individuals and vaccinated individuals behave differently and the final size is highest when degree distribution is Poisson. We also shown that for random immunization there exists a critical vaccination coverage rate above which transmission of WPV no longer progress in the population. But when epidemic occurs there is a threshold for initial vaccinated individuals above which vaccinated individuals become dominant. For both the threshold might be highest in the Barabàsi–Albert graph. Moreover we show that target vaccination also helps contact immunity. A relevant fact is, despite the difference between the geometric graph and the Erd˝os–Rényi graph, the dynamical behavior, the final size of epidemic, the threshold for vaccinated individuals are the same in both graphs. This highlights the fact that the degree distribution is the relevant property in studying contact vaccination in spatial network such as the geometric graph. In summary the relevant network property which might help to improve contact vaccination might be the degree distribution of the Network. However, knowledge of infected and vaccinated component (isolated subgraph of the entire network where nodes are infected or vaccinated individuals) might also help. Indeed from bond percolation theory it is well known that if an epidemic starts on a node which resides on a finite component it quickly terminates after infecting a small, non-extensive, number of nodes. In contrast, an epidemic which starts on a node which resides on the giant component (most

Contact Vaccination Study Using Edge Based Compartmental Model (EBCM). . .

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bigger component) endangers a significant fraction of the entire network. Therefore, the quantities of interest in the context of contact vaccination might be those that characterize the giant component rather than the overall network such as the degree distribution of the giant component. This will reveal the degree distribution of infected and vaccinated individuals before any transmission occurs and thus help to plan adequate vaccination strategy. However, our study has many limitations. Indeed we have assumed only simple statics networks with Poisson or power law degree distributions. Although real-world contact networks, they are generally dynamic and contain many short cycles. Assuming simple networks, however, they facilitate the analysis of the model. Despite these limitations, we believe that this study can serve as a good starting point from which more detailed models can be developed to plan a policy to get more efficient vaccination strategy. Acknowledgments Coura Balde thanks the International Union of Biological Sciences (IUBS) for partial support of living expenses in Szeged, during the 19th BIOMAT International Symposium, October 20–26, 2019.

References 1. R. Albert, H. Jeong, A.-L. Barabàsi, Error and attack tolerance of complex networks. Nature 406, 378–381 (2000) 2. R.J. Duintjer Tebbens, et al., Review and assessment of poliovirus immunity and transmission: synthesis of knowledge gaps and identification of research needs. Risk Anal. 33(4), 606–646 (2013) 3. R.J. Duintjer Tebbens, M. A. Pallansch, K.M. Chumakov, N.A. Halsey, T. Hovi, P.D. Minor, J.F. Modlin, P.A. Patriarca, R.W. Sutter, P.F. Wright, S.G.F. Wassilak, S.L. Cochi, J.-H. Kim, K.M. Thompson, Expert review on poliovirus immunity and transmission. Risk Anal. 33(4), 544–605 (2013) 4. M.J. Keeling, K.T. D. Eames, Networks and epidemic models. J. R. Soc. Interface 2, 295–307 (2005) 5. J.-H. Kim, S.-H. Rho, Transmission dynamics of oral polio vaccine viruses and vaccine-derived polioviruses on networks. J. Theor. Biol. 364, 266–274 (2015) 6. I.Z. Kiss, J.C. Miller, P.L. Simon, Mathematics of Epidemics on Networks (Springer, Berlin, 2017) 7. L.A. Meyers, B. Pourbohloul, M.E.J. Newman, D.M. Skowronski, R.C. Brun-Ham, Network theory and SARS: predicting outbreak diversity. J. Theor. Biol. 232, 71–81 (2005) 8. J.C. Miller, Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes. Infect. Dis. Model. 2(1), 35–55 (2017) 9. J.C. Miller, I.Z. Kiss, Epidemic spread in networks: existing methods and current challenges. Math. Model. Nat. Phenom. 9(2), 4–42 (2014) 10. J.C. Miller, A.C. Slim, E.M. Volz, Edge-based compartmental modelling for infectious disease spread. J. R. Soc. Interface 9, 890–906 (2012) 11. M. Molloy, B. Reed, A critical point for random graphs with a given degree sequence. Random Struct. Algoritm. 6(2–3), 161–180 (1995) 12. M.E.J. Newman, The spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002) 13. R. Pastor-Satorras, A. Vespignani, Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001)

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14. H. Rahmandad, K. Hu, R.J. Duintjer Tebbens, K.M. Thompson, Development of an individualbased model for polioviruses: implications of the selection of network type and outcome metrics. Epidemiol. Infect. 139, 836–848 (2011) 15. E. Volz, SIR dynamics in random networks with heterogeneous connectivity. J. Math. Biol. 56, 293–310 (2008) 16. E. Volz, L.A. Meyers, Epidemic thresholds in dynamic contact networks. J. R. Soc. Interface 6, 233–241 (2009)

The Effect of Inhibitory Neurons on a Class of Neural Networks Márton Neogrády-Kiss and Péter L. Simon

1 Introduction The dynamical behaviour of recurrent neural networks has been intensively studied since the publication of the pioneering works [4, 6] of the field. The widely used ODE model of a recurrent neural network takes the form x˙ = −Ax + Wy + b,

yi = f (xi + i ),

where x(t), y(t), b ∈ RN , A, W ∈ RN ×N , A is diagonal, f is a bounded, increasing function and i ∈ R. The state variable x(t) is the vector representing the activity of the neurons. The initial results were achieved for the case of symmetric weight matrix W , see e.g. [4, 6]. The main goal of research was first to describe the synchronization of neurons that was investigated by many authors. Since the focus of our study is not on synchronization, we only refer to some review articles [1, 7, 8]. Synchronization has also been studied for systems containing delays, see e.g. [8]. The qualitative behaviour of the above ODE system has also been studied. The dynamical behaviour for small networks with one, two or three neurons is presented in [2], where bifurcation diagrams are determined. The dynamics of networks composed of homogeneous populations of excitatory and inhibitory neurons is investigated in [3]. In that paper, only four different weights occur in the matrix W enabling the authors to determine local bifurcation diagrams analytically and computing global bifurcation diagrams by using numerical toolboxes. Those

M. Neogrády-Kiss () · P. L. Simon Institute of Mathematics, Eötvös Loránd University Budapest, Budapest, Hungary Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Budapest, Hungary © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_7

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bifurcation diagrams were computed in the parameter space of the inputs (b) for fixed value of the weights. In our study, instead of the inputs, the weights are the parameters. The purpose of this paper is to show that further analytical results are available when the weights belonging to excitatory neurons are equal to each other, this value will be denoted by wE , and similarly the weights belonging to inhibitory neurons are equal to a value denoted by wI . (No self-edges are considered, that is the diagonal of the matrix W is zero.) This somewhat plausible assumption leads to the order preserving property of the system that is explained in Sect. 2. The order preservation enables us to describe the dynamical behaviour of the system when the activation is piecewise constant, i.e. when f is a step function, as it is shown in Sect. 3. A complete characterization of the dynamical system is given in Sect. 3 for the case with step functions having only two values. An extension of this study is shown in Sect. 4 for step functions with three values leading to periodic orbits even in a simple case with only three neurons. Our analytical results, achieved for piecewise constant activation functions and homogeneous populations of neurons, may pave the way for the characterization of possible qualitative behaviours in the general case. A natural extension of our results, that will be the subject of further study, is for the case when some entries in the matrix W are zeros, that is a neuron is not necessarily connected to all other neurons.

2 The Order Preservation Property Consider the following system of differential equations: x(t) ˙ = W · f (x(t)) − x(t),

(1)

where W ∈ RN ×N , and f : R → (0, 1] is a bounded monotone increasing function. From now on we assume that in a column of W , all the coordinates are either equal with a given positive number wE or all are equal with a negative number wI , except for the main diagonal, which is assumed to be zero. That is the weight matrix has the form ⎛

0 ⎜ wE ⎜ W =⎜ . ⎝ ..

wE 0 .. .

··· ··· .. .

⎞ wI wI ⎟ ⎟ .. ⎟ . . ⎠

(2)

w E wE · · · 0 This means that a neuron is either excitatory or inhibitory. Let us divide the state vector x into an excitatory and inhibitory part as follows: x = (x E , x I )T , where

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the numbers of excitatory and inhibitory neurons are NE and NI , respectively, and NE + NI = N. Since we will use step functions as activation functions, first rewrite (1) in the form of an integral equation: 

t

x(t) = x(0) +

(W · f (x(s)) − x(s)) ds.

(3)

0

It is known that this integral equation has a continuous solution even when the nonlinearity is a step function. Hence, during the further study we will consider the integral equation instead of the differential equation. Our analytical results are based on the following order preservation property. Theorem 1 Let W have the form (2) and f be a positive bounded monotone increasing activation function. Then the followings are true: (1) The order of inhibitory neurons is preserved, that is xiI (0) < xjI (0) implies xiI (t) < xjI (t) for all nonnegative t. (2) After a while, all excitatory neurons will be smaller than any inhibitory neurons, that is there exist T > 0, such that xiE (t) < xjI (t) for all t > T and for all i, j . (3) Any two excitatory neurons tend to each other asymptotically, that is |xiE (t) − xjI (t)| → 0 as t → ∞. Proof 1. Consider two arbitrary inhibitory neurons: xiI and xjI . Then using (3) and the special structure of W we get xiI (t) − xjI (t) = xiI (0) − xjI (0) + − xjI (s))

 ds.

 t 0

− wI (f (xiI (s)) − f xjI (s))) − (xiI (s) (4)

Suppose that xiI (0) > xjI (0). Since the solution is continuous, xiI (t) > xjI (t) will hold for a small t. Denote t ∗ the smallest time (if any)  when the two variables are equal (otherwise t ∗ = ∞). Then in the interval 0, t ∗ ), the function z(t) = t xiI (t) − xjI (t) satisfies z(t) ≥ z(0) − 0 z(s)ds. A standard comparison theorem, see [5] implies that z(t) can be estimated from below by z˜ (t), the solution of 

t

z˜ (t) = z˜ (0) +

−˜z(s)ds,

(5)

0

z˜ (0) = z(0), which is z˜ (t) = e−t z(0). Therefore z(t) ≥ z˜ (t) > 0 so z(t ∗ ) = 0 is a contradiction. 2. Let xI and xE be two arbitrary inhibitory and excitatory neuron. Then similarly to 1, we can write

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xI (t) − xE (t) = xI (0) − xE (0) +

 t

− wI f (xI (s)) + wE f (xE (s))

0

 − (xI (s) − xE (s)) ds.

(6)

First, it is easy to see from the boundedness of f that the solution is also bounded. Since the solution x is bounded, f (x) has a positive minimum, there exists an ε > 0 number such that −wI f (xI (t)) + wE f (xE (t)) ≥ ε for all t and solutions. Then the equation written for z(t) = xI (t) − xE (t) can be estimated from below by  z˜ (t) = z˜ (0) +

t

(ε − z˜ (s))ds,

(7)

0

where z˜ (0) = z(0). The solution is z(t) ≥ z˜ (t) = e−t z(0) + ε(1 − e−t ), which will be positive after some time. 3. This is also quite similar to the previous ones. Suppose xiE (0) > xjE (0). We know that if at some point their values were the same, then the two functions would be the same from there. So suppose there is no such time. Then, since −wE (f (xiE (t)) − f (xjE (t))) ≤ 0, z(t) = xiE (t) − xjE (t) can be estimated from   above by z˜ (t) = e−t xiE (0) − xjE (0) . From the assumption z(t) is positive, therefore it tends to zero.   The above statements apply also to step functions that will be considered from now on.

2.1 Step Function as Activation Function Denote by fa,c the following step function, where a ∈ Rm and c ∈ Rm+1 (we assume that the coordinates of a and c are in ascending order): fa,c (z) = ci

if ai−1 < z < ai ,

where a0 = −∞ and am+1 = ∞. E Definition 1 For any vectors nE , nI ∈ Nm+1 where nE 1 + . . . + nm+1 = NE and  E I I I N n1 + . . . + nm+1 = NI denote n , n ⊂ R the subset of the state space, which consists of vectors p ∈ RN , p = (pE , pI )T where there are nE i coordinates of pE in the interval (ai−1 , ai ) and there are nIi coordinates of pI in the interval (ai−1 , ai ).

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The global behaviour of the solutions is determined by the possible transitions between these domains, hence we introduce the following notations for the transitions. Definition 2 For given vectors nE , nI , n˜ E , n˜ I ∈ Nm+1 let us denote by  E I n , n → n˜ E , n˜ I the statement that for all states p ∈ nE , nI there exists  time t ∗ , such that x(0) = p and x(t ∗ ) ∈ n˜ E , n˜ I and for t < t ∗ the solution x(t)  E I  E I can only be in n , n or n˜ , n˜ .   Denote by nE , nI  the statement that for all states p ∈ nE , nI the solution starting from p stays in the same subset for all t > 0. Define S : RN → RN as S(p) = W · fa,c (p). We note that if in a given time none of the variables are equal to any of ai , then the solution is differentiable and (1) give the derivatives so it can be written as x(t) ˙ = S(x(t)) − x(t).

(8)

3 Two-Valued Case Let us consider now the case when the activation function has only two values, that is we are given a1 ∈ R, 0 < c1 < c2 and our activation function takes the form ! fa,c (z) =

c1 if z < a1 c2 if z > a1 .

 In this case, the sets nE , nI have the general form: [(NE −kE , kE ), (NI −kI , kI )], where 0 ≤ kE ≤ N E and 0 ≤ kI ≤ N I . Recall that the subset [(NE −kE , kE ), (NI − kI , kI )] of the state space consists of those vectors (pE , pI )T = p ∈ RN , for which there are NE − kE coordinates of pE and NI − kI coordinates of pI below a1 , and kE coordinates of pE and kI coordinates of pI above a1 . The second statement of Theorem 1 implies that not all of these domains play role in the asymptotic behaviour of system (8), since after a finite time, the trajectories will enter the regions specified in the lemma below. Lemma 1 Starting from any initial value, the solution can be only in one of the following subsets after a finite time: [(NE , 0), (NI , 0)], [(NE , 0), (NI − 1, 1)], . . . , [(NE , 0), (0, NI )], [(NE − 1, 1), (0, NI )], . . . , [(0, NE ), (0, NI )]. Remark 1 Moreover, we can assume, based on Theorem 1 that in each domain the activation rates of excitatory neurons are below those of the inhibitory neurons.

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Taking a point p from one of the above domains, the coordinates of S(p) can have four different values, for which we introduce the following notations. Definition 3 For given 0 ≤ kE ≤ N E , 0 ≤ kI ≤ N I we define := wE {(NE − kE − 1)c1 + kE c2 } + wI {(NI − kI )c1 + kI c2 } , SkE− E ,kI SkE+ := wE {(NE − kE )c1 + (kE − 1)c2 } + wI {(NI − kI )c1 + kI c2 } , E ,kI SkIE−,kI := wE {(NE − kE )c1 + kE c2 } + wI {(NI − kI − 1)c1 + kI c2 } , SkIE+,kI := wE {(NE − kE )c1 + kE c2 } + wI {(NI − kI )c1 + (kI − 1)c2 } . Using these notations, the differential equations in (8) take a very simple form. For example, if an excitatory neuron is below a1 , that is xiE (t) < a1 for some coordinate i and for some time t, then x˙iE (t) = SkE− − xiE (t). This means that xiE E ,kI is a strictly monotone function tending to SkE− until all neurons remain in the same E ,kI side of a1 , that is until x(t) ∈ [(NE − kE , kE ), (NI − kI , kI )]. Simple calculation shows that the following inequalities hold. Lemma 2 For any 0 ≤ kE ≤ N E and 0 ≤ kI ≤ N I , the following inequalities hold. (1) SkE+ < SkE− < SkIE−,kI < SkIE+,kI . E ,kI E ,kI (2) SkIE+,kI +1 = SkIE−,kI . The transitions between the above domains can be characterized by the values defined in Definition 3. Lemma 3 For a given 0 ≤ kI ≤ NI the followings hold. I− I+ (1) If S0,k < a1 < S0,k then [(NE , 0), (NI − kI , kI )]  and every solution I I starting in this set converges to the unique equilibrium point S(p) where p ∈ [(NE , 0), (NI − kI , kI )]. I− (2) If a1 < S0,k , then [(NE , 0), (NI − kI , kI )] → [(NE , 0), (NI − kI − 1, kI + 1)]. I I+ (3) If S0,kI < a1 , then [(NE , 0), (NI − kI , kI )] → [(NE , 0), (NI − kI + 1, kI − 1)].

Proof 1. We prove that the domain [(NE , 0), (NI − kI , kI )] is positively invariant and that all solutions starting in this domain tend to the unique equilibrium. Let x(0) ∈ [(NE , 0), (NI − kI , kI )], then xiE (0) < a1 for all i, i.e. for all excitatory neurons. E− I− According to the first statement of Lemma 2, we have S0,k < S0,k < a1 , hence I I E− E E E xi satisfies the differential equation x˙i (t) = S0,kI − xi (t) until all neurons remain in the same side of a1 as they were at the initial instant. Therefore xiE is strictly monotone and xiE (t) < a1 until all neurons remain in the same side of E− a1 . If they stay in the same domain for all t > 0, then xiE (t) → S0,k as t tends I to infinity.

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Consider now an inhibitory neuron with xjI (0) > a1 , then xjI satisfies the

I+ − xjI (t) until all neurons remain in the same differential equation x˙jI (t) = S0,k I side of a1 as they were at the initial instant. Therefore xjI is strictly monotone and xjI (t) > a1 until all neurons remain in the same side of a1 . If they stay in the I+ as t tends to infinity. same domain for all t > 0, then xjI (t) → S0,k I Finally, consider an inhibitory neuron with xjI (0) < a1 , then xjI satisfies the

I− − xjI (t) until all neurons remain in the same differential equation x˙jI (t) = S0,k I side of a1 as they were at the initial instant. Therefore xjI is strictly monotone and xjI (t) < a1 until all neurons remain in the same side of a1 . If they stay in the

I− as t tends to infinity. same domain for all t > 0, then xjI (t) → S0,k I 2. Let x(0) ∈ [(NE , 0), (NI − kI , kI )], we prove that there is a time t ∗ > 0, such that x(t ∗ ) ∈ [(NE , 0), (NI − kI − 1, kI + 1)] and for all t ∈ (0, t ∗ ) the point x(t) is in one of these domains. (Except a single point that is on the boundary of the two domains.) Consider first an inhibitory neuron, for which xjI (0) > a1 . I+ I− > S0,k > a1 , and xjI satisfies the differential equation Since we have S0,k I I

I+ − xjI (t) until all neurons remain in the same side of a1 as they were x˙jI (t) = S0,k I at the initial instant. Therefore xjI is strictly monotone and xjI (t) > a1 until all neurons remain in the same side of a1 . Consider now an inhibitory neuron, for I− > a1 , and xjI satisfies the differential which xjI (0) < a1 . Since we have S0,k I

I− − xjI (t), one of these functions will cross a1 first, that is equation x˙jI (t) = S0,k I for some j and for some t ∗ we will have xjI (t ∗ ) > a1 . Observe that before this change all excitatory neurons remain below a1 , because of Remark 1. 3. This statement can be verified similarly, we do not present the details here.  

The next lemma is a consequence of the trivial inequalities SkE− < SkE− E ,NI E +1,NI E+ E+ and SkE ,NI < SkE +1,NI . Lemma 4 The following transitions hold. E− (1) If S0,N > a1 , then I

[(NE , 0), (0, NI )] → [(NE − 1, 1), (0, NI )] → . . . → [(0, NE ), (0, NI )]. E+ < a1 , then (2) If SN E ,NI

[(0, NE ), (0, NI )] → [(1, NE − 1), (0, NI )] → . . . → [(NE , 0), (0, NI )]. Theorem 2 If the activation function is a step function consisting of two parts, then for any initial condition the solution of (8) converges to an equilibrium.

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Proof According to Lemma 2, the following inequalities hold:

=

=

=

E− I− I− I− I− S0,N < S0,N < S0,N < S0,N · · · < S0,0 I I I −1 I −2 I+ I+ I+ I+ S0,N < S0,N · · · < S0,1 < S0,0 . I I −1 E+ plays also a crucial To determine the behaviour of the solutions, the value SN E ,NI role. According to its position, the following cases are possible: E+ 1. SN < a1 . E ,NI I− I− ∗ Let 0 ≤ kI∗ ≤ NI be the index, for which S0,k ∗ < a1 < S0,k ∗ −1 . For kI = 0 I

I

I− > or NI the inequalities should be changed appropriately. If kI < kI∗ , then S0,k I a1 and because of the second statement of Lemma 3 we have [(NE , 0), (NI − I+ < a1 kI , kI )] → [(NE , 0), (NI − kI − 1, kI + 1)]. If kI > kI∗ , then S0,k I and because of the third statement of Lemma 3 the transition [(NE , 0), (NI − kI , kI )] → [(NE , 0), (NI − kI + 1, kI − 1)] holds. Using Lemma 4 yields

 [(NE , 0), (NI , 0)] → . . . → (NE , 0), (NI − kI∗ , kI∗ ) ← . . . ← [(0, NE ), (0, NI )] ,  which means that every solution enters the domain (NE , 0), (NI − kI∗ , kI∗ ) . Finally, we can use the first statement of Lemma 3 for kI∗ , implying that the solutions starting in (NE , 0), (NI − kI∗ , kI∗ ) will remain there and tend to an equilibrium. E+ E− 2. SN > a1 and S0,N > a1 . E ,NI I In this case we have [(NE , 0), (NI , 0)] → . . . → [(NE , 0), (0, NI )] → . . . → [(0, NE ), (0, NI )]  . This means that every solution enters the domain [(0, NE ), (0, NI )], where there is a unique stable equilibrium attracting all solutions. We note that the symbol  is defined in Definition 2. E+ E− 3. SN > a1 and S0,N < a1 . E ,NI I Then similarly to 1, there exists an index 0 ≤ kI∗ ≤ NI , such that  [(NE , 0), (NI , 0)] → . . . → (NE , 0), (NI − kI∗ , kI∗ ) ← . . . ← [(NE , 0), (NI , 0)] and [(0, NE ), (0, NI )]  .

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 So every solution enters the domain (NE , 0), (NI − kI∗ , kI∗ ) or the domain [(0, NE ), (0, NI )]. In these two domains the solutions tend to the unique steady state there. So there are two stable equilibrium points attracting all solutions.  

4 Oscillatory Behaviour in the Three-Valued Case In this section the activation function takes the form: ⎧ ⎨ c1 if z < a1 , f (z) = c2 if a1 < z < a2 , ⎩ c3 if a2 < z. Our goal here is to show that periodic solutions may occur in this case. This can happen with only three neurons, hence we restrict ourselves to the case when there are two excitatory and one inhibitory neuron. Then system (8) takes the form x˙1E = wE f (x2E ) + wI f (x I ) − x1E ,

(9)

x˙2E = wE f (x1E ) + wI f (x I ) − x2E ,

(10)

x˙ I = wE f (x1E ) + wE f (x2E ) − x I .

(11)

For simplicity let us consider the special initial condition x1E (0) = x2E (0). Then x1E (t) = x2E (t) for all t. Theorem 1 shows that this is a weak restriction because the distance of excitatory neurons tends to zero. In this case the model can be reduced to a two dimensional problem: x˙ E = wE f (x E ) + wI f (x I ) − x E ,

(12)

x˙ I = 2wE f (x ) − x .

(13)

E

I

We will show that this model can exhibit a stable limit cycle. Using the notations of Definition 1 the state space can be partitioned into the following domains. [(2, 0, 0), (1, 0, 0)], [(2, 0, 0), (0, 1, 0)], [(2, 0, 0), (0, 0, 1)], [(0, 2, 0), (0, 1, 0)], / [(0, 2, 0), (0, 0, 1)], [(0, 0, 2), (0, 0, 1)]. Now we can write S(p) = [S E (p), S I (p)] for a p ∈ R2 and similarly to the second and third statement of Lemma 3, we can prove the next Lemma. Lemma 5 Assume that the following conditions hold S E (p) = c2 wI + c1 wE > a1 if p ∈ [(2, 0, 0), (0, 1, 0)], S I (p) = 2c2 wE > a2 if p ∈ [(0, 2, 0), (0, 1, 0)],

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S E (p) = c3 wI + c2 wE < a1 if p ∈ [(0, 2, 0), (0, 0, 1)], S I (p) = 2c1 wE < a2 if p ∈ [(2, 0, 0), (0, 0, 1)]. Then we have the transitions [(2, 0, 0), (0, 1, 0)] → [(0, 2, 0), (0, 1, 0)] → [(0, 2, 0), (0, 0, 1)] → [(2, 0, 0), (0, 0, 1)] → [(2, 0, 0), (0, 1, 0)] The following lemma describes how the solution travels between these sets. Lemma 6 Take p ∈ R2 so that p1 = a1 and p2 = a2 . Then a solution starting from p will move on a straight line to S(p) until the first coordinate passes a1 or the second passes a2 . Proof If x(0) = p, then until x E (t) = a1 or x I (t) = a2 , x˙ E (t) = S E (p) − x E (t), x˙ I (t) = S I (p) − x I (t). Therefore x E (t) = e−t x E (0) + (1 − e−t )S E (p), x I (t) = e−t x I (0) + (1 − e−t )S I (p), so x(t) = S(p) + e−t (x(0) − S(p)).   The transitions above show that the trajectory moves periodically among the domains and now we show that each trajectory tends to a stable periodic orbit. To prove that we introduce the following notations, see Fig. 1. Let S1 = (c3 wI + c2 wE , 2c2 wE ), S2 = (c3 wI + c1 wE , 2c1 wE ), S3 = (c2 wI + c1 wE , 2c1 wE ) and S4 = (c2 wI + c2 wE , 2c2 wE ). The distance of these points from the two lines are (1) (1) (2) denoted by d1 = |c3 wI + c2 wE − a1 |, d2 = |2c2 wE − a2 |, d1 = |c3 wI + c1 wE − a1 |, d2(2) = |2c1 wE − a2 |, d1(3) = |c2 wI + c1 wE − a1 |, d2(3) = |2c1 wE − a2 |, (4) (4) d1 = |c2 wI + c2 wE − a1 |, d2 = |2c2 wE − a2 |. Fig. 1 The direction field of system (12)–(13)

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Fig. 2 The trajectories of system (12)–(13)

Now take an arbitrary point (a1 + x1 , a2 ), where x1 > 0, see Fig. 2. We know from Lemma 5 that starting a solution from this point, it will eventually return to a point (a1 +x1∗ , a2 ) with some x1∗ > 0. Now we determine the function f associating x1∗ with x1 , that is f (x1 ) = x1∗ . Define the values x2 , x3 , x4 > 0 where (a1 , a2 + x2 ) is the point where the solution first crosses the half line (a1 , a2 + y), y > 0, (a1 − x3 , a2 ) is the point where the solution first crosses the line (a1 − x, a2 ), x < 0 and (a1 , a2 −x4 ) is the point where the solution first crosses the line (a1 , a2 −y), y < 0, (a1 +x1∗ , a2 ) is the point where the solution first crosses the line (a1 , a2 +x), x > 0. Using triangle similarities we can determine the values x2 , x3 , x4 and x1∗ as d d d x∗ d x3 x4 x2 = (1) 2 , = (2) 1 , = (3) 2 , 1 = (4) 1 . x1 x x x d1 + x1 2 d2 + x2 3 d1 + x3 4 d2 + x4 (1)

(2)

(3)

(4)

After subsequent substitutions we get x1∗ =

(4) (3) (2) (1)

d1 d2 d1 d2 x1 (4) (3) (2) (1)

(4) (3) (2)

(4) (3) (1)

(4) (2) (1)

(3) (2) (1)

d2 d1 d2 d1 +(d2 d1 d2 +d2 d1 d2 +d2 d1 d2 +d2 d1 d2 )x1

.

So f : (0, ∞) → (0, ∞) takes the form ax/(b + cx), where a, b, c > 0. The first and second derivatives of f are f  (x) =

ab −2abc > 0, f  (x) = < 0. 2 (b + cx) (b + cx)3

Since f is concave, it can be shown that it has a fixed point in (0, ∞) if and only if f  (0) > 1, that is when a/b > 1. Simple calculation shows that

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Fig. 3 The time dependence of solutions of system (12)–(13) with parameter values c1 = 0.1, c2 = 0.5, c3 = 1, a1 = −1, a2 = 1, wE = 1.15, wI = −1.6. The blue and red curves show the excitatory and inhibitory solutions

(4) (3) (2) (1)

d d d d a = 1(4) 2(3) 1(2) 2(1) b d2 d1 d2 d1 =

|c2 wI + c2 wE − a1 | · |2c1 wE − a2 | · |c3 wI + c1 wE − a1 | · |2c2 wE − a2 | |2c2 wE − a2 | · |c2 wI + c1 wE − a1 | · |2c1 wE − a2 | · |c3 wI + c2 wE − a1 |

=

|c2 wI + c2 wE − a1 | · |c3 wI + c1 wE − a1 | . |c2 wI + c1 wE − a1 | · |c3 wI + c2 wE − a1 |

The shape of the graph of f also shows that this fixed point is globally asymptotically stable, which means that there exists a globally asymptotically stable limit cycle in the system (12)–(13). So we proved the following proposition. Proposition 1 In system (12)–(13) there exists a stable limit cycle if the followings hold: c2 wI + c1 wE > a1 , 2c2 wE > a2 , c3 wI + c2 wE < a1 , 2c1 wE < a2 , |c2 wI + c2 wE − a1 | · |c3 wI + c1 wE − a1 | > 1. |c2 wI + c1 wE − a1 | · |c3 wI + c2 wE − a1 | Figure 3 shows an example that satisfies these conditions.

5 Concluding Remarks In this paper we considered the Cohen–Grossberg or Hopfield model of recurrent neural networks when the weight matrix W of the graph has a special structure. Namely, in a column of W , all the entries are either equal to a given positive number

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wE or all are equal to a negative number wI , except for the main diagonal, which is assumed to be zero. In Sect. 2, we proved an order preservation property for this class of fully connected networks. This made the network’s dynamics analysable. We also introduced step functions as activation functions enabling us to study the dynamical behaviour analytically. In Sect. 3, we presented the full characterization of the dynamical behaviour for step functions with only two values. In this case two different behaviours may occur. The network has either one globally asymptotically stable steady state or there are two different stable steady states. In Sect. 4, we investigated the question whether networks with three-valued step functions have oscillatory behaviour. We examined a network with two excitatory and one inhibitory neurons and proved that periodic solution can exist even in this small network. The examination of neural network’s dynamics is hard due to its highly nonlinear activation function and the complexity of the network’s connections. We made simplifications that can pave the way towards the study of more complex networks. Acknowledgments The project has been supported by the European Union, co-financed by the European Social Fund (EFOP-3.6.3-VEKOP-16-2017-00002). Péter L. Simon acknowledges support from Hungarian Scientific Research Fund, OTKA, (grant no. 115926).

References 1. P. Ashwin, S. Coombes, R. Nicks, Mathematical frameworks for oscillatory network dynamics in neuroscience. J. Math. Neurosci. 6(1), 2 (2016) 2. R.D. Beer, Parameter space structure of continuous-time recurrent neural networks. Neural Comput. 18(12), 3009–3051 (2006) 3. D. Fasoli, A. Cattani, S. Panzeri, The complexity of dynamics in small neural circuits. PLoS Comput. Biol. 12(8), e1004992 (2016) 4. S. Grossberg, Nonlinear neural networks: principles, mechanisms, and architectures. Neural Netw. 1(1), 17–61 (1988) 5. J. Hale, Ordinary Differential Equations (Dover Publications, New York, 2009) 6. J.J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. 81(10), 3088–3092 (1984) 7. L.M. Pecora, F. Sorrentino, A.M. Hagerstrom, T.E. Murphy, R. Roy, Cluster synchronization and isolated desynchronization in complex networks with symmetries. Nat. Commun. 5, 4079 (2014) 8. H. Zhang, Z. Wang, D. Liu, A comprehensive review of stability analysis of continuous-time recurrent neural networks. IEEE Trans. Neural Netw. Learn. Syst. 25(7), 1229–1262 (2014)

Pipette Hunter 3D: Fluorescent Micropipette Detection D. Hirling, K. Koos, J. Molnár, and P. Horvath

1 Introduction Segmentation of objects with well-defined geometries is a fundamental problem in image analysis. Several methods were proposed to detect lines, ellipses, or rectangles to identify roads [7], trees [12], or houses [8], respectively, using marked point processes. Another way is to compromise strict geometries and use variational methods. For example, higher order active contours (HOAC) can describe various objects with defined shapes while allowing slight variations on the boundaries. HOACs were successfully used to model circular objects [10] or complex road structures [13]. Recently a family of hybrid variational models was proposed [14, 15] that is capable of capturing circular and elliptical objects by minimizing only a few parameters. Based on this, we have already presented a model to detect elongated straight object pairs that have a common reference point to detect edges of glass micropipettes in 2D label-free microscopy images [6]. Here, as an extension to three dimensions, we present a variational method to detect a truncated cone in an image stack, allowing each of its parameters to evolve. We use this model to segment pipette tips in fluorescence microscopy images and automatically navigate these tips with micrometer precision for patch clamp recording and measure the properties of neuron cells.

D. Hirling · K. Koos · J. Molnár Biological Research Centre of Szeged, Szeged, Hungary e-mail: [email protected]; [email protected]; [email protected] P. Horvath () Biological Research Centre of Szeged, Szeged, Hungary Institute for Molecular Medicine Finland, University of Helsinki, Helsinki, Finland e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_8

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Fig. 1 Schematic whole-cell patch clamp process

Patch clamp recording is a technique to study ion channels in cells. The technique was invented by Erwin Neher and Bert Sakmann in the early 1980s who received the Nobel Prize in Physiology or Medicine in 1991 for their work. Although the technique can be applied to a wide variety of cells, it is especially useful for measuring the electrophysiological properties of nerve cells (neurons). The setup of the patch clamp process is illustrated in Fig. 1. First, a glass pipette is pulled onto an electrode. The tip of the glass pipette is open, thus the measured signal originates only from the pipette tip, because the glass does not transfer electricity. Then, the pipette is pushed next to a cell. When a tight connection, called “gigaseal,” is formed between the cell and the pipette, the cell membrane is broken by vacuum or relatively high voltage pulses. This way the whole-cell patch clamp configuration is established, the electric signal is passed to an amplifier and then it is ready to be recorded. The patch clamp process has to be repeated manually for every target cell. Experienced biologists can usually perform only 10–30 successful patch clamp recording a day. The process is repetitive and monotonous, thus error prone as the researchers get fatigued. Recently, efforts have been made to automate the technique. Early automatic patch clamp setups were used for in vivo applications [3]. A similar technique was extended to a multi-electrode system [11] using up to 12 pipettes. A MATLAB implementation of an automatic patch clamp software is publicly available [2]. A detailed description of building an automatic patch clamp setup can also be found [4]. Furthermore, automatic patch clamp recording has been successfully used on cardiomyocytes [1]. An issue of automatic patch clamp systems is that glass pipettes have to be changed after every attempt, which limits the throughput. A process was developed to clean the pipettes [5] which allows them to be used up to 10 times. Patch clamp technique is often used in tissue slices when there is an imaging modality to see the target cells and the pipette, unlike to in vivo applications. However, changing the pipettes introduces another problem. The pipettes are not perfectly identical and the tip can be slightly translated after the change. A method has already been proposed for automatic pipette tip detection in fluorescence images [9], however it is based on a simple thresholding technique and its robustness can be improved. Fluorescent materials can damage cell functions, thus sometimes label-free techniques have to be applied.

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Recently, a method for pipette detection in label-free microscopy was proposed in combination with fluorescent cell detection in tissues [16]. The method is used in low magnification (4x) which provides sharp image of the pipette due to the relatively thick focus plane, even if it is tilted. The detection is based on finding intersecting lines using Hough transform, which calculates the lateral position of the tip. The Z coordinate is refined using a focus detection algorithm. In this paper we propose a novel three-dimensional pipette tip detection algorithm using energy minimization. The method works on optically sliced images of a pipette acquired using fluorescence microscopy. Such microscopy techniques require the pipette to be filled with a fluorescent material which provides bright image intensities in the region of the pipette, while the background will be dark. The idea of fitting a primitive shape to the image is inspired by the Snakuscule [14] algorithm that segments circular objects. Besides the exact location of the tip’s endpoint in 3D, our detection algorithm determines the orientation and tilt angle as well. The algorithm can be extended to three-dimensional label-free pipette detection.

2 Methods The mechanism is modeled as a truncated cone as shown in Fig. 2. Its nappe is intended to be aligned with a rotationally symmetric image of a similar object. It has 8 degrees of freedom (DOF), specifically 3 DOF for R, then ϕ 1 , ϕ 2 for the angles, and r, l, m for the bases. Note that upper indices indicate variables on which the energy depends and are not powers. The two independent angles (e.g., the spherical angles) determine its orientation. However, the symmetry is not utilized a priori in the energy, but taken into account later, during the analysis of the associated extreme-valued equation. Fig. 2 The 3D fluorescent pipette model

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2.1 Euler-Angles (Z-Y-Z) Configuration One natural way to describe the orientation of a unit direction vector is by using the spherical angles, where ϕ 1 is the angle measured from the standard basis vector i, ϕ 2 is the angle measured from the standard basis vector k. This can be described with two rotations, first around axis Y, then around axis Z (using extrinsic description). The resulting rotation sequence applicable to (row) vectors from the left is: Rϕ 1 Rϕ 2 . Allowing full degrees of rotational freedom, the natural extension is to enable the object spinning around itself by a third angle ϕ 3 , hence using the complete Eulerangles configuration Z-Y-Z or Rϕ 1 Rϕ 2 Rϕ 3 : ϕ 1 = αspherical , ϕ 2 = βspherical , ϕ 3 = spin, ⎡

Rϕ 1 Rϕ 2

⎤ ⎤⎡ cos ϕ 1 − sin ϕ 1 0 cos ϕ 2 0 sin ϕ 2 = ⎣ sin ϕ 1 cos ϕ 1 0⎦ ⎣ 0 1 0 ⎦ 2 0 0 1 − sin ϕ 0 cos ϕ 2 ⎡ ⎤ cos ϕ 2 cos ϕ 1 − sin ϕ 1 sin ϕ 2 cos ϕ 1 = ⎣ cos ϕ 2 sin ϕ 1 cos ϕ 1 sin ϕ 2 sin ϕ 1 ⎦ − sin ϕ 2 0 cos ϕ 2 ⎡

Rϕ 1 Rϕ 2 Rϕ 3

⎤⎡ cos ϕ 2 cos ϕ 1 − sin ϕ 1 sin ϕ 2 cos ϕ 1 cos ϕ 3 − sin ϕ 3 2 1 1 2 1 ⎣ ⎦ ⎣ = cos ϕ sin ϕ cos ϕ sin ϕ sin ϕ sin ϕ 3 cos ϕ 3 2 2 − sin ϕ 0 cos ϕ 0 0  = e1 e2 n

⎤ 0 0⎦ 1 (1)

where ⎡

⎤ cos ϕ 1 cos ϕ 2 cos ϕ 3 − sin ϕ 1 sin ϕ 3 e1 = ⎣sin ϕ 1 cos ϕ 2 cos ϕ 3 + cos ϕ 1 sin ϕ 3 ⎦ − sin ϕ 2 cos ϕ 3 ⎡

⎤ − cos ϕ 1 cos ϕ 2 sin ϕ 3 − sin ϕ 1 cos ϕ 3 e2 = ⎣− sin ϕ 1 cos ϕ 2 sin ϕ 3 + cos ϕ 1 cos ϕ 3 ⎦ sin ϕ 2 sin ϕ 3 ⎡ ⎤ cos ϕ 1 sin ϕ 2 n = ⎣ sin ϕ 1 sin ϕ 2 ⎦ cos ϕ 2

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In the last line, the rotated coordinate system’s basis vectors e1 , e2 , n are defined. Later on we will need the derivatives of these basis vectors w.r.t. the angles. These are for every basis vector: ∂e1 ∂e1 ∂e1 = cos ϕ 2 e2 − sin ϕ 2 sin ϕ 3 n, 2 = − cos ϕ 3 n, 3 = e2 1 ∂ϕ ∂ϕ ∂ϕ ∂e2 ∂e2 ∂e2 = − cos ϕ 2 e1 − sin ϕ 2 cos ϕ 3 n, 2 = sin ϕ 3 n, 3 = −e1 ∂ϕ 1 ∂ϕ ∂ϕ

(2)

∂n ∂n ∂n = sin ϕ 2 (sin ϕ 3 e1 + cos ϕ 3 e2 ), 2 = cos ϕ 3 e1 − sin ϕ 3 e2 , 3 = 0 1 ∂ϕ ∂ϕ ∂ϕ

2.2 The System Energy and the Energy-Gradient First, we determine a point inside the truncated cone using the spatial coordinates of the pivot point of the capturing mechanism and its local coordinates in the e1 , e2 , n basis: P = R(x 1 , x 2 , x 3 ) + ζ n(ϕ 1 , ϕ 2 , ϕ 3 ) + ρ[cos φe1 (ϕ 1 , ϕ 2 , ϕ 3 ) + sin φe2 (ϕ 1 , ϕ 2 , ϕ 3 )] ⎡ 1⎤ ⎡ ⎤ x cos ϕ 1 sin ϕ 2 P = ⎣x 2 ⎦ + ζ ⎣ sin ϕ 1 sin ϕ 2 ⎦ x3 cos ϕ 2 ⎛ ⎡ ⎤ cos ϕ 1 cos ϕ 2 cos ϕ 3 − sin ϕ 1 sin ϕ 3 + ρ ⎝cos φ ⎣sin ϕ 1 cos ϕ 2 cos ϕ 3 + cos ϕ 1 sin ϕ 3 ⎦ sin ϕ 2 cos ϕ 3 ⎡ ⎤⎞ − cos ϕ 1 cos ϕ 2 sin ϕ 3 − sin ϕ 1 cos ϕ 3 + sin φ ⎣− sin ϕ 1 cos ϕ 2 sin ϕ 3 + cos ϕ 1 cos ϕ 3 ⎦⎠ sin ϕ 2 sin ϕ 3

(3)

Then the normalized energy of the capturing mechanism can be defined as follows: E(x 1 , x 2 , x 3 , ϕ 1 , ϕ 2 , ϕ 3 , r, m, l)= ˙ =

1 V



2π



ζ1 +l

φ=0 ζ =ζ1



1 V

 I (P)dV V

r+m(ζ −ζ1 )

I (P)ρdρdζ dφ ρ=0

(4)

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where V is the volume encompassed by the mechanism (using cylindrical coordinates which will have some advantages). The expression includes the radii of the truncated cones as the function of the variables r, m and the length l: r(ζ ; r, m) = r + m(ζ − ζ1 ) = (r − mζ1 ) + mζ ⇒



1 V = π l r + mlr + l 2 m2 3

(5)



2

The gradient-components of the energy (4) are as follows: ∂E 1 = i· V ∂x 1 ∂E 1 = j· 2 V ∂x







ζ1 +l



φ=0 ζ =ζ1



∂E 1 = k· V ∂x 3 ∂E 1 = k ∂ϕ V

2π

2π



2π



ζ1 +l



ζ1 +l

ζ1 +l

φ=0 ζ =ζ1



r+m(ζ −ζ1 )

I ∇(P)ρdρdζ dφ

ρ=0

φ=0 ζ =ζ1 2π

I ∇(P)ρdρdζ dφ

ρ=0

φ=0 ζ =ζ1



r+m(ζ −ζ1 )





r+m(ζ −ζ1 )

I ∇(P)ρdρdζ dφ

ρ=0 r+m(ζ −ζ1 )

ρ=0

I ∇(P) ·

∂P ρdρdζ dφ, ∂ϕ k

   1 ∂V 2π ζ1 +l r+m(ζ −ζ1 ) ∂E =− 2 I (P)ρdρdζ dφ ∂r V ∂r φ=0 ζ =ζ1 ρ=0     r+m(ζ −ζ1 ) 1 2π ζ1 +l ∂ I (P)ρdρ dζ dφ + V φ=0 ζ =ζ1 ∂r ρ=0    1 ∂V 2π ζ1 +l r+m(ζ −ζ1 ) ∂E =− 2 I (P)ρdρdζ dφ ∂m V ∂m φ=0 ζ =ζ1 ρ=0     r+m(ζ −ζ1 ) 1 2π ζ1 +l ∂ I (P)ρdρ dζ dφ + V φ=0 ζ =ζ1 ∂m ρ=0    1 ∂V 2π ζ1 +l r+m(ζ −ζ1 ) ∂E =− 2 I (P)ρdρdζ dφ ∂l V ∂l φ=0 ζ =ζ1 ρ=0  ζ1 +l    1 2π r+m(ζ −ζ1 ) ∂ I (P)dζ ρdρdφ + V φ=0 ρ=0 ∂l ζ =ζ1

k = 1, 2, 3

(6)

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The following calculations specify the constituents designated by these equations. First we tackle with the last three components (descriptors of the truncated cone). Component r:  2π  ζ1 +l  r+m(ζ −ζ1 ) πl ∂E = − 2 (2r + ml) I (P)ρdρdζ dφ ∂r V φ=0 ζ =ζ1 ρ=0     1 2π ζ1 +l + dζ dφ [r + m(ζ − ζ1 )] I  V φ=0 ζ =ζ1 ρ=r+m(ζ −ζ1 )

(7)

In the second term, the weighted intensities are integrated over the nappe points. Component m:    2π  ζ1 +l  r+m(ζ −ζ1 ) 2 π l2 ∂E = − 2 r + lm I (P)ρdρdζ dφ ∂m 3 V φ=0 ζ =ζ1 ρ=0   *  1 2π ζ1 +l ) 2 + r(ζ − ζ1 ) + m(ζ − ζ1 ) I  dζ dφ V φ=0 ζ =ζ1 ρ=r+m(ζ −ζ1 )

(8)

In the second term, the weighted intensities are integrated over the nappe points. Component l:   2π  ζ1 +l  r+m(ζ −ζ1 ) π  ∂E = − 2 r 2 + 2lmr + l 2 m2 I (P)ρdρdζ dφ ∂l V φ=0 ζ =ζ1 ρ=0    1 2π r+m(ζ −ζ1 )  ρI  dρdφ + V φ=0 ρ=0 ζ1 +l

(9)

In the second term, the weighted intensities are integrated over the closing circular disc points. Volume expressed by the descriptors (variables) of the truncated cone is given in Eq. (5). Second, the derivatives by the Euler-angles, using (2) and (3) also expressed in the local basis e1 , e2 , n: I∇ ·

∂P = (I ∇ · e1 )(ζ sin ϕ 2 sin ϕ 3 − ρ sin φ cos ϕ 2 ) ∂ϕ 1 + (I ∇ · e2 )(ζ sin ϕ 2 cos ϕ 3 + ρ cos φ cos ϕ 2 ) + (I ∇ · n)ρ(− cos φ sin ϕ 2 sin ϕ 3 − sin φ sin ϕ 2 cos ϕ 3 )

I∇ ·

∂P = (I ∇ · e1 )(ζ cos ϕ 3 ) + (I ∇ · e2 )(−ζ sin ϕ 3 ) ∂ϕ 2 + (I ∇ · n)ρ(− cos φ cos ϕ 3 + sin φ sin ϕ 3 )

I∇ ·

∂P = (I ∇ · e1 )ρ(− sin φ) + (I ∇ · e2 )ρ(cos φ) ∂ϕ 3

(10)

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Here one can make two important observations: (1) Q=ρ(−I ˙ ξ sin φ + Iη cos φ) = t · I ∇, where t is the tangent vector to a circle with radius ρ laying in the plane perpendicular to local basis vector n. The line integral of this expression is zero for complete circles (φ ∈ [0, 2π ]). It follows from the gradient theorem which is valid for any closed curve. As a consequence, the integral of Q cancels out for any rotationally symmetric object. The result does not depend on the spin parameter ϕ 3 , hence arbitrary. Particularly choosing ϕ 3 = 0, the rotational energy-gradient components are as follows: 

∂E sin ϕ 2 = − V ∂ϕ 1

∂E 1 = V ∂ϕ 3



2π

0

 0



0

cos ϕ 2 + V ∂E 1 = V ∂ϕ 2

2π



ζ1 +l

2π



r+m(ζ −ζ1 )

(Iζ ρ sin φ − ζ Iη )ρdρdζ dφ

0

ζ1 ζ1 +l



r+m(ζ −ζ1 )

Qρdρdζ dφ 0



ζ1 +l

ζ1





ζ1 +l

ζ1

0

r+m(ζ −ζ1 )

(11) (ζ Iξ − Iζ ρ cos φ)ρdρdζ dφ

0

ζ1 2π





r+m(ζ −ζ1 )

Qρdρdζ dφ = 0

0

(2) Introducing the “local position vector” q=ξ ˙ e1 + ηe2 + ζ n, ξ = ρ cos φ, η = ρ cos φ, all integrands can be expressed by the torque q × I ∇ if expressing the derivatives along the local coordinates instead of basis vectors: ∂E sin ϕ 2 = − V ∂ϕ 1

 e1 · (q × I ∇)dV V

+

cos ϕ 2

 n · (q × I ∇)dV

V V

∂E 1 = 2 V ∂ϕ



(12) e2 · (q × I ∇)dV

V

∂E 1 = V ∂ϕ 3

 n · (q × I ∇)dV V

Finally, the non-zero Euler-angle equations can be summarized as follows:

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∂E sin ϕ 2 = − V ∂ϕ 1

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 (ηIζ − ζ Iη )dV V

∂E 1 = 2 V ∂ϕ

(13)

 (ζ Iξ − ξ Iζ )dV V

Third, the common factor of energy-gradient for the pivot point expressed again in the local system (again, ϕ 3 = 0 is assumed, Eqs. (1) and (6) are used): ∂E 1 = 1 V ∂x

 (Iξ cos ϕ 1 cos ϕ 2 − Iη sin ϕ 1 + Iζ cos ϕ 1 sin ϕ 2 )dV V

∂E 1 = 2 V ∂x

 (Iξ sin ϕ 1 cos ϕ 2 − Iη cos ϕ 1 + Iζ sin ϕ 1 sin ϕ 2 )dV

(14)

V

∂E 1 = 3 V ∂x

 (−Iξ sin ϕ 2 + Iζ cos ϕ 2 )dV V

After rearranging, we get: ∂E cos ϕ 1 cos ϕ 2 = V ∂x 1



sin ϕ 1 Iξ dV − V

V

+

cos ϕ 1 sin ϕ 2 V

 Iη dV V

 Iζ dV V

∂E sin ϕ 1 cos ϕ 2 = V ∂x 2



cos ϕ 1 Iξ dV + V

V

+

sin ϕ 1 sin ϕ 2

 Iη dV V

 Iζ dV

V V

∂E sin ϕ 2 = − V ∂x 3

 V

cos ϕ 2 Iξ dV + V

 Iζ dV V

All integrals in (12) and (14) are calculated in the local system.

(15)

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2.3 Tait–Bryan Angles (X-Y-Z ) Configuration Another way to describe the orientation of a unit direction vector is the usage of angles around all axes. If ϕ 1 is the angle measured from the first standard basis vector i, ϕ 2 is the angle measured from the third standard basis vector k we get the spherical description. This can be described with two rotations, first around axis Y, then around axis Z (using extrinsic description). The resulting rotation sequence applicable to (row) vectors from the left is: Rϕ 1 Rϕ 2 . Now we complete the rotational freedom by enabling the object rotating first around the X axis with a third angle ϕ 3 , hence using the complete Tait-Bryan (Roll-Pitch-Yaw) angles configuration, X-Y-Z (extrinsic) or Rϕ 1 Rϕ 2 Rϕ 3 : ϕ 1 = α, ϕ 2 = β, ϕ 3 = xrot ⎡

Rϕ 1 Rϕ 2 Rϕ 3

⎤⎡ ⎤ cos ϕ 2 cos ϕ 1 − sin ϕ 1 sin ϕ 2 cos ϕ 1 1 0 0 = ⎣ cos ϕ 2 sin ϕ 1 cos ϕ 1 sin ϕ 2 sin ϕ 1 ⎦ ⎣0 cos ϕ 3 − sin ϕ 3 ⎦ − sin ϕ 2 0 cos ϕ 2 0 sin ϕ 3 cos ϕ 3  = e1 e2 n (16)

where ⎤ cos ϕ 1 cos ϕ 2 e1 = ⎣ sin ϕ 1 cos ϕ 2 ⎦ − sin ϕ 2 ⎡

⎡

⎤ − sin ϕ 1 cos ϕ 3 + cos ϕ 1 sin ϕ 2 sin ϕ 3 e2 = ⎣ cos ϕ 1 cos ϕ 3 + sin ϕ 1 sin ϕ 2 sin ϕ 3 ⎦ cos ϕ 2 sin ϕ 3 ⎡

⎤ sin ϕ 1 sin ϕ 3 + cos ϕ 1 sin ϕ 2 cos ϕ 3 e3 = ⎣− cos ϕ 1 sin ϕ 3 + sin ϕ 1 sin ϕ 2 cos ϕ 3 ⎦ cos ϕ 2 cos ϕ 3 In the last line, the rotated coordinate system’s basis vectors e1 , e2 , n are defined. Later on we will need the derivatives of these basis vectors w.r.t. the angles. These are specified below, for every basis vector:

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∂e1 ∂e1 ∂e1 = cos ϕ 2 (cos ϕ 3 e2 − sin ϕ 3 n), 2 = − sin ϕ 3 e2 − cos ϕ 3 n, 3 = 0 ∂ϕ 1 ∂ϕ ∂ϕ ∂e2 ∂e2 ∂e2 = − cos ϕ 2 cos ϕ 3 e1 − sin ϕ 2 n, 2 = sin ϕ 3 e1 , 3 = n 1 ∂ϕ ∂ϕ ∂ϕ ∂n ∂n ∂n = cos ϕ 2 sin ϕ 3 e1 + sin ϕ 2 e2 , 2 = cos ϕ 3 e1 , 3 = −e2 ∂ϕ 1 ∂ϕ ∂ϕ (17) The calculation of the integrand-gradients w.r.t. the angles: P = R(x 1 , x 2 , x 3 ) + ζ n(ϕ 1 , ϕ 2 , ϕ 3 ) * ) + ρ cos φe1 (ϕ 1 , ϕ 2 , ϕ 3 ) + sin φe2 (ϕ 1 , ϕ 2 , ϕ 3 ) ⇒ I∇ ·

∂P = (Iξ ζ − Iζ ρ cos φ) sin ϕ 3 cos ϕ 2 ∂ϕ 1 + (Iη ζ − Iζ ρ sin φ) sin ϕ 2 + cos ϕ 3 cos ϕ 2 Q

I∇ ·

∂P = (Iξ ζ − Iζ ρ cos φ) cos ϕ 3 − sin ϕ 3 Q ∂ϕ 2

I∇ ·

∂P = −(Iη ζ − Iζ ρ sin φ) ∂ϕ 3

(18)

Similarly to the previous configuration, these equations can be expressed by the torque, reduced the image gradient to the pivot point (q × I ∇) as: I∇ ·

∂P = sin ϕ 3 cos ϕ 2 (ζ Iξ − ξ Iζ ) − sin ϕ 2 (ηIζ − ζ Iη ) ∂ϕ 1 + cos ϕ 3 cos ϕ 2 (ξ Iη − ηIξ ) ) * = cos ϕ 2 sin ϕ 3 e2 · (q × I ∇) + cos ϕ 3 n · (q × I ∇) − sin ϕ 2 e1 · (q × I ∇) (19)

∂P I ∇ · 2 = cos ϕ 3 (ζ Iξ − ξ Iζ ) − sin ϕ 3 (ξ Iη − ηIξ ) ∂ϕ = cos ϕ 3 e2 · (q × I ∇) − sin ϕ 3 n · (q × I ∇) I∇ ·

∂P = ηIζ − ζ Iη = e1 · (q × I ∇) ∂ϕ 3

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For rotationally symmetric objects, the energy-gradients w.r.t. the angles are: ∂E sin ϕ 3 cos ϕ 2 = V ∂ϕ 1



sin ϕ 2 (ζ Iξ − ξ Iζ )dV − V



V

∂E cos ϕ 3 = V ∂ϕ 2

(ηIζ − ζ Iη )dV V

 (ζ Iξ − ξ Iζ )dV V

∂E 1 = V ∂ϕ 3

 (ηIζ − ζ Iη )dV V

(20) Note that only two equations are independent, for example: ∂E cos ϕ 3 = V ∂ϕ 2

 (ζ Iξ − ξ Iζ )dV V

∂E 1 =− V ∂ϕ 3

 (ηIζ − ζ Iη )dV

(21)

V

⇒ ∂E ∂E ∂E = tan ϕ 3 cos ϕ 2 2 − sin ϕ 2 3 ∂ϕ 1 ∂ϕ ∂ϕ The pivot-point equations can be calculated as: ∂E 1 = V ∂x 1

 Iξ (i · e1 ) + Iη (i · e2 ) + Iζ (i · n)dV V

∂E 1 = V ∂x 2

 Iξ (j · e1 ) + Iη (j · e2 ) + Iζ (j · n)dV

(22)

V

∂E 1 = V ∂x 3

 Iξ (k · e1 ) + Iη (k · e2 ) + Iζ (k · n)dV V

Note that the expressions for e1 , e2 , n in (16) depend only on two independent angles, which dependency is given in (21). Also note that the radial components of the energy-gradient are independent of the angles, hence calculated by (8).

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3 Results The input images to this algorithm were taken of pipettes filled with fluorescent material as shown in Fig. 3. The result of the algorithm was compared to manually picked ground truth tip positions. The test set contained 8 image stacks. The results were expressed in voxels and the Z dimension was scaled by a factor of 10 to have nearly cubic voxels. The average error was 75.66 ± 30.97 voxels. This quality is appropriate for the method to be used in automatic patch clamp systems. There are a few observations that should be considered when analyzing the inaccuracy of the approach. As the example images show in Fig. 3, the model fits the data points well but the tip looks shifted along its middle axis. This is due to the low concentration of fluorescent dyes in the tip of the pipette which appear as less bright intensity regions in the image. On the other hand, where the radius of the pipette is bigger and more material condenses, the image is brighter and attracts the model. Another negative effect is that when the exact distances are not known in the lateral plane and along the Z-axis, then the pipette is not interpreted as a (truncated) cone. However, the model does not allow such deformations and fits the data anyway (see Fig. 3d), leading to errors compared to the manually selected tip positions. Finally, the 8

Fig. 3 Example images during the iterations of the Pipette Hunter 3D algorithm. The point cloud is the slice view of the image data without the zero elements. The blue truncated cone is the current state of the model. (a) Initial state of the model. (b) Initial state of the model from another angle. (c) Result of the algorithm. (d) Result of the algorithm from an angle that shows that the circular base of the cone differs from the elliptical nature of the data

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DOFs almost make the algorithm unstable and it is challenging to find the best parameters for the different cases.

4 Discussion In this paper an energy minimization framework has been proposed for micropipette detection in 3D fluorescence images. The main idea is to fit a truncated cone on a bright image region, which is the inner part of the pipette. Our model has 8 degrees of freedom, and the final result includes the orientation of the pipette as well as its size characteristics. Due to the nature of gradient descent which requires many iterations for a reasonable result, and the amount of data, our 3D case requires a deliberate implementation to keep the runtime low. The equations are highly parallelizable and can be derived to matrix operations which makes Matlab a good language choice. In an object fitting task the underlying image does not change, thus only the voxel positions have to be determined and indexed which are covered by the model. The image gradient can be precomputed. The code is vectorized in the following way. (Vectorization is a Matlab terminology and technique for optimizing loop-based, scalar-oriented code.) First, the possible parameter combinations are computed without the variables (that change). Then in every iteration, the variables are applied to the equations and the resulting voxel positions are stored using linear indexing. This results in a “flat” (vectorized) code where the only loop variable is the iteration number. A stopping criteria is still beneficial to the algorithm. In every 100th iteration it is checked whether the variables changed more than a threshold value. If they did not change or the sign of the change alternates, then the algorithm terminates and the result is the last point. The algorithm is evaluated on real image stacks of pipettes. The detected tip positions are compared to manually determined ground truth positions. The method can be extended to three-dimensional label-free pipette detection. In this case, the model would consist of two cylinders that face towards a shared reference point. This model would cover the dark image regions in the stack, which are introduced by the edges of the pipette, similarly to our previous 2D model. Acknowledgments We thank Attila Ozsvár and Gábor Tamás for the images. We acknowledge the financial support from the LENDULET-BIOMAG Grant (2018-342); and the European Regional Development Funds (GINOP-2.3.2-15-2016-00006, GINOP-2.3.2-15-2016-00001, GINOP-2.3.215-2016-00026, GINOP-2.3.2-15-2016-00037). This is part of ATTRACT that has received funding from the European Union’s Horizon 2020 Research and Innovation Programme.

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References 1. B. Amuzescu, O. Scheel, T. Knott, Novel automated patch-clamp assays on stem cell-derived cardiomyocytes: will they standardize in vitro pharmacology and arrhythmia research? J. Phys. Chem. Biophys. 4(4), 1000153, 1–4 (2014) 2. N.S. Desai, J.J. Siegel, W. Taylor, R.A. Chitwood, D. Johnston, Matlab-based automated patchclamp system for awake behaving mice. J. Neurophysiol. 114(2), 1331–1345 (2015) 3. S.B. Kodandaramaiah, G.T. Franzesi, B.Y. Chow, E.S. Boyden, C.R. Forest, Automated wholecell patch-clamp electrophysiology of neurons in vivo. Nat. Methods 9(6), 585–587 (2012) 4. S.B. Kodandaramaiah, G.L. Holst, I.R. Wickersham, A.C. Singer, G.T. Franzesi, M.L. McKinnon, C.R. Forest, E.S. Boyden, Assembly and operation of the autopatcher for automated intracellular neural recording in vivo. Nat. Protoc. 11(4), 634–654 (2016) 5. I. Kolb, W.A. Stoy, E.B. Rousseau, O.A. Moody, A. Jenkins, C.R. Forest, Cleaning patch-clamp pipettes for immediate reuse. Sci. Rep. 6, Article number: 35001 (2016) 6. K. Koos, J. Molnár, P. Horvath, Pipette Hunter: patch-clamp pipette detection, in Scandinavian Conference on Image Analysis (Springer, Berlin, 2017), pp. 172–183 7. C. Lacoste, X. Descombes, J. Zerubia, Point processes for unsupervised line network extraction in remote sensing. IEEE Trans. Pattern Anal. Mach. Intell. 27(10), 1568–1579 (2005) 8. F. Lafarge, X. Descombes, J. Zerubia, M. Pierrot-Deseilligny, Automatic building extraction from DEMs using an object approach and application to the 3D-city modeling. ISPRS J. Photogramm. Remote Sens. 63(3), 365–381 (2008) 9. B. Long, L. Li, U. Knoblich, H. Zeng, H. Peng, 3D image-guided automatic pipette positioning for single cell experiments in vivo. Sci. Rep. 5, 18426 (2015) 10. C. Molnar, I.H. Jermyn, Z. Kato, V. Rahkama, P. Östling, P. Mikkonen, V. Pietiäinen, P. Horvath, Accurate morphology preserving segmentation of overlapping cells based on active contours. Sci. Rep. 6, Article number: 32412 (2016) 11. R. Perin, H. Markram, A computer-assisted multi-electrode patch-clamp system. J. Vis. Exp. (80), e50630 (2013) 12. G. Perrin, X. Descombes, J. Zerubia, Tree crown extraction using marked point processes, in Proceedings of the European Signal Processing Conference (University of Technology, Vienna, 2004) 13. M. Rochery, I.H. Jermyn, J. Zerubia, Higher order active contours. Int. J. Comput. Vis. 69(1), 27–42 (2006) 14. P. Thevenaz, M. Unser, Snakuscules. IEEE Trans. Image Process. 17(4), 585–593 (2008) 15. P. Thevenaz, R. Delgado-Gonzalo, M. Unser, The ovuscule. IEEE Trans. Pattern Anal. Mach. Intell. 33(2), 382–393 (2011) 16. Q. Wu, I. Kolb, B.M. Callahan, Z. Su, W. Stoy, S.B. Kodandaramaiah, R. Neve, H. Zeng, E.S. Boyden, C.R. Forest, A.A. Chubykin, Integration of autopatching with automated pipette and cell detection in vitro. J. Neurophysiol. 116(4), 1564–1578 (2016)

Delay Linear Chains in Mathematical Biology: Migratory Birds, Stem Cell Maturation, and Intracellular Chlamydia Infection Bornali Das and Gergely Röst

1 Introduction Compartmental models are consistently used to describe the evolution of biological systems which can be partitioned into separate compartments. Dynamics of such processes are characterized by the movement of particles between the compartments and by specifying probable input and output factors from the environment of the system [1]. Our particles will be various biological entities, such as the population of cells, bacteria, or animals. They are capable of reproducing and performing other biological functions, yet for the sake of simplicity we refer to them as particles throughout this paper. Detailed discussions of multi-compartmental models used as mathematical descriptions of biological systems can be found in various books on mathematical models [2, 3]. In many compartmental systems, ordinary differential equations are predominantly used to describe the flow of particles for it is assumed that the time taken for the particles to move from one compartment to another is zero [4]. However, in many biological systems, the time necessary for the particles to move between the compartments is not negligible [5]. Compartmental systems with pipes are a concrete example of a compartmental model with time delay [6]. Such a system is best envisioned as one in which particles are moving through hypothetical pipes joining the compartments and the transit time is associated with the length of the pipes. The lags of time caused by the pipes give rise to a system of model equations formulated as differential equations with time delays [7]. The structure of the differential equations describing compartmental systems is specific to the biological and physical aspects. The model variables are assigned to

B. Das () · G. Röst Bolyai Institute, University of Szeged, Szeged, Hungary e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_9

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Fig. 1 Sketch of a linear chain describing transition and growth of particles through successive compartments

compartments to express the number of particles which behave homogeneously. The transition from one compartment to another takes place when the particles undergo some physical or biological transformation or change their spatial location [8]. In our case, we construct a model with a finite number of compartments where the rates of inflow and outflow are governed by physical and biological mechanisms. In this paper, we study a linear system of delay differential equations suited for mathematical representation of some compartmental models in mathematical biology and present three different applications. First, we develop a model of seasonal bird migration. This describes the movement of the bird population during a full cycle of migration: it begins with the spring migration and ends on reaching the final wintering destination. The second application deals with a model to study stem cell maturation in the absence of regulatory feedback and cell death. We consider a stem cell population that has the potential to either self-renew or differentiate into progenitor cells that eventually become mature cells. The last application presents a growth model of the chlamydia bacteria inside human cells.

2 The Model Description Consider a delayed linear chain as shown in Fig. 1, illustrating particles moving through a number of successive compartments before reaching a final stage. The multiplicative rates represent growth between the compartments. All the compartments have inflow and outflow terms except the first and the last: the first compartment has only outflow and the last compartment has only inflow. Time delay signifies the time needed to complete the transition of particles between successive compartments. Let the number of particles in the ith compartment at time t be yi (t) (i = 0, 1, . . . , n). The rate at which the particles are moving out of the ith compartment is denoted by ai for i = 0, 1, . . . , n − 1 and bi−1 is the rate at which the particles are entering the ith compartment for i = 1, 2, . . . , n. Also, we assume that the particles are arriving with a time delay τi−1 into the ith compartment for i = 1, 2, . . . , n. We describe such a process by a system of delay differential equations as follows: y0 (t) = −a0 y0 (t), yi (t) = bi−1 yi−1 (t − τi−1 ) − ai yi (t), yn (t)

= bn−1 yn−1 (t − τn−1 ),

i = 1, 2, . . . , n − 1,

(1)

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129

where ai > 0, bi > 0, and τi ≥ 0 for all i. The natural phase space for our system is C([−τ, 0], Rn+1 ), where τ = max{τ0 , . . . , τn−1 }. Initial conditions for this system are given by yi (θ ) = ϕi (θ ) for θ ∈ [−τ, 0], i = 0, . . . , n,

(2)

where ϕ := (ϕ0 , . . . , ϕn ) ∈ C([−τ, 0], Rn+1 + ). It is well known that the initial value problem (1)–(2) is well-posed. Non-negativity of the initial data is a natural requirement for the biological systems we consider, and from the non-negativity of the rates it follows that solutions remain non-negative for all future time. For the biological problems we consider later, it is paramount to predict the eventual state of the system. This is addressed in the following proposition, giving an explicit expression for the limit of each compartment. Proposition 2.1 Solutions of problem (1) has the following limits: lim yi (t) = 0 for i = 0, 1, 2, . . . , n − 1

t→∞

and lim yn (t) =

t→∞

b0 . . . bn−1 b1 . . . bn−1 bn−1 ϕ0 (0) + ϕ1 (0) + . . . + ϕn−1 (0) a0 . . . an−1 a1 . . . an−1 an−1 b0 . . . bn−1 +ϕn (0) + a1 . . . an−1 +... +

bn−2 bn−1 an−1





0

b1 . . . bn−1 ϕ0 (s)ds + a2 . . . an−1 −τ0 

0

−τn−2

ϕn−2 (s)ds + bn−1



0 −τ1

ϕ1 (s)ds

0 −τn−1

ϕn−1 (s)ds.

Proof Solving the first equation gives us y0 (t) = ϕ0 (0)e−a0 t . Thus, limt→∞ y0 (t) = 0. Next, we assume that limt→∞ yk (t) = 0 for some k. Since for k + 1 we have  yk+1 (t) = bk yk (t − τk ) − ak+1 yk+1 (t), the assumption that limt→∞ yk (t) = 0 implies limt→∞ yk+1 (t) = 0. By induction, we find that for i = 0, 1, . . . , n − 1, each compartment has limit zero. ∞ 0 Now, for k = 0, 1, . . . , n, we define Ik := 0 yk (t)dt and Jk := −τk ϕk (s)ds. Integrating the first equation from zero to infinity, since limt→∞ y0 (t) = 0, we have 

∞

−a0 I0 = 0



∞

−a0 y0 (t)dt = 0

y0 (t)dt = −y0 (0) = −ϕ0 (0).

Hence, I0 = a10 ϕ0 (0). Integrating the kth equation of the system (1) from zero to infinity for k = 1, . . . , n − 1, and using limt→∞ yk (t) = 0, we have

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 −yk (0) =

∞

[bk−1 yk−1 (t − τk−1 ) − ak yk (t)]dt

0



= bk−1



0

−τk−1

∞

ϕk−1 (s)ds +



∞

yk−1 (s)ds − ak

0

yk (t)dt 0

= bk−1 Jk−1 + bk−1 Ik−1 − ak Ik . Rearranging the terms, for Ik we find the recursive relation Ik =

bk−1 bk−1 1 Ik−1 + Jk−1 + ϕk (0) for k = 1, 2, . . . , n − 1. ak ak ak

(3)

Substituting k = 1 in the above relation, we have b0 b0 1 b0 b0 I1 = I0 + J0 + ϕ1 (0) = ϕ0 (0) + a1 a1 a1 a0 a1 a1



0

−τ0

ϕ0 (s)ds +

1 ϕ1 (0). a1

Iteratively, from the above relation, substituting k = n − 1, we have In−1 =

b0 . . . bn−2 b1 . . . bn−2 1 ϕ0 (0) + ϕ1 (0) + · · · + ϕn−1 (0) a0 . . . an−1 a1 . . . an−1 an−1 b0 . . . bn−2 + a1 . . . an−1 +

bn−2 an−1





0

b1 . . . bn−2 ϕ0 (s)ds + a2 . . . an−1 −τ0



0

−τ1

ϕ1 (s)ds + . . .

0

−τn−2

ϕn−2 (s)ds.

Finally, integrating the last equation of the system (1) from zero to infinity, we have  ∞ lim yn (t) = bn−1 yn−1 (t − τn−1 )dt + yn (0) t→∞

0

= bn−1



0

−τn−1



∞

ϕn−1 (s)ds +

yn−1 (s)ds + ϕn (0)

0

= bn−1 Jn−1 + bn−1 In−1 + ϕn (0). Substituting the value of In−1 and replacing Jn−1 by the integral, we have the limit as lim yn (t) =

t→∞

b0 . . . bn−1 b1 . . . bn−1 bn−1 ϕ0 (0) + ϕ1 (0) + . . . + ϕn−1 (0) a0 . . . an−1 a1 . . . an−1 an−1

Delay Linear Chains in Mathematical Biology: Migratory Birds, Stem Cell. . .

+ϕn (0) +

+... +

b0 . . . bn−1 a1 . . . an−1

bn−2 bn−1 an−1





0

−τ0

0

−τn−2

ϕ0 (s)ds +

b1 . . . bn−1 a2 . . . an−1 

ϕn−2 (s)ds + bn−1

131



0 −τn−1

0 −τ1

ϕ1 (s)ds

ϕn−1 (s)ds.  

3 Mathematical Modeling for Bird Migration Bird migration is a seasonal movement of birds and is an integral part of the ecological process. Examples of such species of birds include the white stork, pelicans, loggerhead shrike, Eurasian spoonbill, mountain bulbul, etc. The act of migration of birds is primarily triggered by availability of food, temperature, and accessibility to safe breeding grounds [9]. During migration, birds may fly over long distances, often covering several thousand kilometers. There can be a series of potential stopovers between the departure patch and wintering grounds [10]. This movement between distant stopovers along migratory routes is illustrated in Fig. 2. The aim of our model is to predict the expected number of birds after a full migration cycle during a single year. The mathematical model for the bird migration is formulated as follows: we denote the number of birds in the patch pi at time t by yi (t) for i = 0, 1, 2, . . . , k − 1, k, k + 1, . . . , n − 1, n. As the spring migration is initiated, the flock of birds begin their migration from the departure patch p0 . We assume that they make stop at the patches pi , where i = 1, 2, . . . , k − 1 before they arrive at the summer breeding ground pk . We denote by γ the birth rate in the summer breeding

Fig. 2 Schematic representation of the migratory route. The birds begin their migration from the departure patch p0 and make k − 1 stopovers before reaching the summer breeding ground pk . For the autumn migration, they make another n + 1 − k stopovers before reaching the final wintering ground pn

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ground. For the autumn migration, the birds make the stopovers at pi , where i = k + 1, k + 2, . . . , n − 1 before arriving at the wintering ground pn . We also assume that the birds arrive at each of the patches pi with a time delay τi−1 after leaving patch pi−1 , where i = 1, 2, . . . , n, representing the time needed to fly such a long distance between the patches. The departure rates from each of the patches pi are assumed to be mi , where i = 0, 1, . . . , n − 1. The number of birds in each of the patches is also affected by the local death rates di , where i = 0, 1, . . . , n − 1. Also, assuming a bird mortality rate μi−1 during the flight between patches pi−1 and pi , the survival probability from the patch pi−1 to patch pi during the flight is e−μi−1 τi−1 for i = 1, 2, . . . , n − 1. Then, the population of birds in each patch can be expressed by following system of delay differential equations: For the bird population in the initial patch p0 we have y0 (t) = −m0 y0 (t) − d0 y0 (t),

(4)

for the compartments i = 1, 2, . . . , k − 1, k + 1, . . . , n − 1 corresponding to the potential stopovers, we have yi (t) = mi−1 e−μi−1 τi−1 yi−1 (t − τi−1 ) − mi yi (t) − di yi (t),

(5)

the bird population in the summer breeding ground is governed by yk (t) = mk−1 e−μk−1 τk−1 yk−1 (t − τk−1 ) + γ yk (t) − mk yk (t) − dk yk (t),

(6)

and the number of birds in the final wintering ground is yn (t) = mn−1 e−μn−1 τn−1 yn−1 (t − τn−1 ).

(7)

Here, mi > 0, di > 0 for i = 0, 1, . . . , n and the bird reproduction rate on the breeding site is γ > 0. For the model to be realistic, we assume that md + dk > γ ; otherwise, the bird population will have uncontrolled growth. As the summer migration begins we assume that there is an exodus of birds from the initial patch p0 , i.e. we consider the bird population which are ready to take the flight. Hence, we have initial functions, y0 (0) = ϕ 0 > 0, y0 (t) = 0 for t < 0, and yi (t) = 0, t  0 , for i = 1, 2, . . . , n. (8) We can also write equations to track the number of birds flying between the patches, for example, if we denote by fk (t) the number of birds flying between patches pk and pk+1 , then fk (t) = mk yk (t) − μk fk (t) − mk yk (t − τk ). These equations decouple from the rest; hence, we can ignore them.

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Proposition 3.1 The compartments of the system (4)–(8) have the following limits: lim yi (t) = 0 for i = 0, 1, . . . , n − 1

t→∞

and lim yn (t) =

t→∞

m0 m1 e−μ1 τ1 mk−1 e−μk−1 τk−1 mk e−μk τk ... m0 + d0 m1 + d1 mk−1 + dk−1 mk + dk − γ ·

mk+1 e−μk−1 τk−1 mk−1 e−μn−1 τn−1 ... ϕ . mk+1 + dk+1 mn−1 + dn−1 0

Proof We rearrange the above set of equations in the system (4)–(7) as follows: y0 (t) = −(m0 + d0 )y0 (t), yi (t) = mi−1 e−μi−1 τi−1 yi−1 (t − τi−1 ) − (mi + di )yi (t), yk (t) = mk−1 e−μk−1 τk−1 yk−1 (t − τk−1 ) − (mk + dk − γ )yk (t), yn (t) = mn−1 e−μn−1 τn−1 yn−1 (t − τn−1 ). The second equation holds for i = 1, 2.., k −1, k +1, . . . n−1. This system of equations then becomes equivalent to the system (1) with the following substitutions: mi + di = ai , for i = 0, 1, . . . , k − 1, k + 1, . . . , n − 1, mi−1 e−μi−1 τi−1 = bi−1 , for i = 0, 1, . . . , n, mk + dk − γ = ak . Since all ai and bi are positive, making use of the above substitutions and using Proposition 2.1, we have that lim yn (t) =

t→∞

m0 m1 e−μ1 τ1 mk−1 e−μk−1 τk−1 mk e−μk τk ... m0 + d0 m1 + d1 mk−1 + dk−1 mk + dk − γ ·

mk+1 e−μk−1 τk−1 mk−1 e−μn−1 τn−1 ... ϕ mk+1 + dk+1 mn−1 + dn−1 0

and lim yi (t) = 0 for i = 0, 1, . . . , n − 1.

t→∞

 

Thus, the model gives us an estimate of the final size of the birds after a full migration circle. We can use this to find a critical breeding rate γ∗ which is a threshold between growth and decay of the yearly bird population. Assume that the survival probability during the winter at the wintering ground is σ . Let

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Fig. 3 Example of a bird population dynamics along a migration route with twelve stopovers. Only the populations at the departure patch, the breeding ground, and the wintering ground are plotted during the year

C=

m0 m1 e−μ1 τ1 mk−1 e−μk−1 τk−1 ... m0 + d0 m1 + d1 mk−1 + dk−1 ·

mk+1 e−μk−1 τk−1 mk−1 e−μn−1 τn−1 ... ϕ . mk+1 + dk+1 mn−1 + dn−1 0 −μk τk

mk e Then the yearly replacement ratio of the birds is σ C m . This is equal to one k +dk −γ −μ τ when γ = γ∗ = mk e k k (1 − σ C) + dk , which gives us the critical birth rate of the birds at breeding site which is necessary to maintain their population. An illustration of a yearly population dynamics is shown in Fig. 3.

4 Mathematical Modeling of Stem Cell Maturation Without Regulatory Feedback Stem cells are found in most of the multi-cellular organisms. They divide through mitosis into new daughter cells and have the potential to either self-replicate or to differentiate into diverse specialized cells. Cells that are already more specific than a stem cell and pushed to differentiate into its target cell are called progenitor cells. The self-renewal and the differentiation process of stem cells and progenitor cells are thought to be governed by some regulatory feedback from the mature cell population (Fig. 4) [11]. Regulatory feedback is an external mechanism which regulates the self-renewal rates or enhances/inhibits proliferation. There are mathematical models for stem cell maturation in the presence of regulatory feedback and cell death [12]. In this paper, we develop a model for stem cell maturation with the assumption that there is no regulatory feedback and death of cells. We show that even with this assumption, the system can reach a finite final state.

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Fig. 4 Schematic illustration of stem cell renewal and differentiation in the absence of regulatory feedback and death. Stem cells can either renew themselves and stay in the same compartment or differentiate into progenitor cells moving on to the next compartments. The progenitor cells eventually differentiate into mature cells

We denote the number of stem cells by y0 (t). The stem cells are assumed to enter division by mitosis at the rate γ0 and the daughter cells in this compartment can either replenish this compartment with a probability of self-renewal p0 or differentiate into progenitor cells and enter the next compartment with a probability 1 − p0 . We will denote by yi (t) the number of progenitor cells in the ith stage of the maturation process for i = 1, 2, . . . , n − 1. The population of cells in these compartments are effected by the influx of the differentiated cells in the preceding compartments, also the flux from the divided cells in the compartment that renew themselves and the outflow of the differentiated cells into the succeeding compartment. For the progenitor cells in the ith compartment, the proliferation rate is denoted by γi , their self-renewal probability is pi , and 1 − pi is their probability of differentiating, where i = 1, 2, . . . , n − 1. The number of the matured cells in the final compartment is denoted yn (t) and the influx to this compartment is from the differentiated cells from the compartment n − 1 of the progenitor cells. We allow a time delay τi−1 for the cells arriving in the compartment i for i = 1, 2, . . . , n, which can be interpreted as the time needed to complete the cell mitosis. Based on this discussion, we formulate our model as follows. For the stem cell population we have y0 (t) = 2p0 γ0 y0 (t − τ0 ) − γ0 y0 (t),

(9)

for the progenitor compartments i = 1, 2.., n − 1, we have yi (t) = 2γi−1 (1 − pi−1 )yi−1 (t − τi−1 ) − γi yi (t) + 2γi pi yi (t − τi ),

(10)

and the mature population is governed by yn (t) = 2γn−1 (1 − pn−1 )yn−1 (t − τn−1 ).

(11)

Here, the self-renewal probabilities pi ∈ [0, 1], and the division rates γi > 0 for i = 0, 1, 2, . . . , n−1. We consider a situation where initially there are no progenitor

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and matured cells, and the process starts with a pool of stem cells. Then the system corresponds to the initial conditions y0 (0) = ϕ 0 > 0, y0 (t) = 0 for t < 0, and yi (t) = 0, t  0 , for i = 1, 2, . . . , n. (12) Similarly as in the case with bird migration, we can have equations for the number of cells undergoing division or differentiation, but since the equations decouple from the rest, we ignore them in this case too. Proposition 4.1 Assume that pi < 12 for i = 0, 1, . . . , n − 1, then for the system (9)–(12) the compartments have the following limits: lim yi (t) = 0 for i = 0, 1, . . . , n − 1

t→∞

and lim yn (t) =

t→∞

(1 − p0 )(1 − p1 ) . . . (1 − pn−2 )(1 − pn−1 ) ϕ × 2n . (1 − 2p0 )(1 − 2p1 ) . . . (1 − 2pn−2 )(1 − 2pn−1 ) 0

Proof The proof is analogous to the proof of Proposition 2.1. Due to the assumption that pi < 12 , limt→∞ yi (t) = 0 holds for i = 0, 1, . . . , n − 1. Integrating Eq. (9) from zero to infinity, since limt→∞ y0 (t) = 0 and y0 (t) = 0 for t < 0, we have 

∞

−y0 (0) =

[2p0 γ0 y0 (t − τk−1 ) − γ0 y0 (t)]dt

0



= 2p0 γ0

0 −τ0



∞

y0 (s)ds +

 y0 (s)ds − γ0

0

∞

y0 (t)dt

0

= 2p0 γ0 I0 − γ0 I0 . Hence, I0 = a10 ϕ 0 , where a0 = γ0 (1 − 2p0 ). Integrating Eq. (10) for k and using limt→∞ yk (t) = 0, and since yk (t) = 0, t  0 we have  ∞ 0= [2γk−1 (1 − pk−1 )yk−1 (t − τk−1 ) − γk yk (t) + 2γk pk yk (t − τk )]dt 0



∞

= 2γk−1 (1 − pk−1 ) 0

 yk−1 (s)ds − γk

∞

0

 yk (t)dt + 2γk pk

∞

yk−1 (s)ds 0

= 2γk−1 (1 − pk−1 )Ik−1 − γk Ik + 2γk pk Ik . Hence, we have the relation analogous to the recursive relation (3) in Proposition 2.1 as

Delay Linear Chains in Mathematical Biology: Migratory Birds, Stem Cell. . .

Ik =

137

bk−1 Ik−1 for k = 1, 2, . . . , n − 1, ak

where bk−1 = 2γk−1 (1 − pk−1 ) and ak = γk (1 − 2pk ). Substituting k = 1 in the above relation, we have b0 b0 2γ0 (1 − p0 ) I0 = ϕ0 = ϕ . a1 a0 a1 γ0 (1 − 2p0 )γ1 (1 − 2p1 ) 0

I1 =

Iteratively, from the above relation, substituting k = n − 1, we have In−1 =

2γ0 (1 − p0 )2γ1 (1 − p1 ) . . . 2γn− (1 − pn−2 ) ϕ . γ0 (1 − 2p0 )γ1 (1 − 2p1 ) . . . γn−1 (1 − 2pn−1 ) 0

We now integrate Eq. (11) from zero to infinity and since yn−1 (t) = 0, t  0 and yn (t) = 0 

∞

lim yn (t) =

t→∞

2γn−1 (1 − pn−1 )yn−1 (t − τn−1 )dt

0



= 2γn−1 (1 − pn−1 )

0 −τn−1

 yn−1 (s)ds +

∞

yn−1 (s)ds 0

= 2γn−1 (1 − pn−1 )In−1 . Substituting the value of In−1 , we have lim yn (t) =

t→∞

2γ0 (1 − p0 )2γ1 (1 − p1 ) . . . 2γn−1 (1 − pn−1 ) ϕ . γ0 (1 − 2p0 )γ1 (1 − 2p1 ) . . . γn−1 (1 − 2pn−1 ) 0

Since the proliferation rates cancel out, we find that lim yn (t) =

t→∞

(1 − p0 )(1 − p1 ) . . . (1 − pn−2 )(1 − pn−1 ) ϕ × 2n . (1 − 2p0 )(1 − 2p1 ) . . . (1 − 2pn−2 )(1 − 2pn−1 ) 0

 

This result shows that even in the absence of regulatory feedback, it is possible for the system to reach a finite final state. Note that the size of the final mature population is independent of the division rates. This final state can be reached by different combinations of the parameters, corresponding to different transient dynamics. To illustrate, we consider a three compartmental model consisting of stem cells, progenitor cells, and mature cells. In Fig. 5, we compare two scenarios with different tendencies of self-renewal and differentiation of the stem cells and the progenitor cells. The exact parameter values in the simulations are (p0 , γ0 , p1 , γ1 ) = (0.2, 0.5, 0.4, 0.5) in the left and (p0 , γ0 , p1 , γ1 ) = (0.4, 0.5, 0.2, 0.5) in the right in Fig. 5. The initial pool in both cases consists of 500 stem cells. We can see that the same final mature population size can be reached with very different intermediate sizes of the progenitor population.

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Fig. 5 In the left: stem cell maturation process where the stem cell self-renew less frequently than the progenitor cells. In the right: Stem cell maturation process where the stem cell self-renew more frequently than the progenitor cells

5 Mathematical Formulation of the Intracellular Chlamydia Development Cycle Chlamydia, an obligate intracellular bacterial parasite is responsible for causing important disease in human and animals. Among the species known to cause human disease (namely Chlamydia trachomatis, Chlamydia pneumoniae, and Chlamydia psittaci), C. trachomatis is sexually transmitted and is the most common cause of sexually transmitted infections (STI) in humans [13]. C. trachomatis is also significant for causing Trachoma, which is the leading cause of preventable blindness worldwide [14]. Infection caused by Chlamydia is mostly asymptomatic in nature and in the majority of the cases may remain undetected for months to years and consequently remain untreated. Untreated C. trachomatis infection can lead to serious sequelae such as pelvic inflammatory disease (PID), infertility, ectopic pregnancy, and chronic pelvic pain [15]. C. pneumoniae plays a role in acute respiratory disease and is associated with pneumonia and bronchitis [16]. Furthermore, C. pneumoniae is implicated to play a potential role in cardiovascular disease and pathogenesis of atherosclerosis [17, 18]. Currently, there is no effective vaccine against Chlamydia infection in humans and thus it remains a major public health problem throughout the world. The intracellular development of Chlamydia is governed by a very distinct life cycle (Fig. 6). It alternates between a non-replicating, infectious, extracellularly viable elementary bodies (EBs) and the non-infectious, intracellularly replicating reticulate bodies (RBs). EBs have the ability to attach to and invade susceptible cells. These EBs then transform into the metabolically active form of RBs, which replicate by undergoing repeated cycles of binary fission within an intracytoplasmatic parasitophorous vacuole called inclusion. After secondary transformation of the RBs back to EBs, the host cell lyses releasing a large number of new EBs that infect neighboring cells [19].

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Fig. 6 Life cycle of Chlamydia

Microbiological investigation of the intracellular development of Chlamydia is an extensive field of research [20]. Recently, a number of dynamic models of the disease transmission as well as the intracellular growth have been developed [21]. In this paper, we present a model for a laboratory experiment of Chlamydia infecting human cells [22]. The basic model for the intracellular development of the Chlamydia cells in vivo is formulated as follows. Let y0 (t) denote the number of Chlamydia EBs outside human cells at time t, y1 (t) denotes the number of Chlamydia EBs attached to the human cells at time t. Upon infection of the healthy human cells, the intracellular EBs transform to RBs upon which the RBs undergo repeated cycles of division. Hence, we will denote by y2 (t) the number of Chlamydia EBs that have transformed to RBs and yi (t) will denote the number of Chlamydia RBs after the ith cycle of replication for i = 3, 4, . . . , n − 1. The RBs then reorganize back to the EBs, following which the host cell lyses releasing the newly formed EBs. The yn (t) will denote the number of RBs converting back to EBs. We impose time delays to account for the time needed for transforming between RBs and EBs, as well as for completing cell division. According to these assumptions and interactions, the population dynamics of EBs and RBs can be described in mathematical terms as follows: y0 (t) = −a0 y0 (t)y0 (t), y1 (t) = a0 y0 (t − τ0 ) − a1 y1 (t), y2 (t) = a1 y1 (t − τ1 ) − a2 y2 (t), yi (t) = 2ai−1 yi−1 (t − τi−1 ) − ai yi (t), yn (t) = an−1 yn−1 (t − τn−1 ).

i = 3, 4, .., n − 1 (13)

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Here, a0 is the rate at which the EBs enter the human cell and they are assumed to differentiate to RBs at a rate a1 inside the human cells. Parameter ai−1 is the rate at which the RBs enter the ith cycle of replication for i = 3, 4, . . . , n − 1, and the RBs will convert back to the EBs with the rate an−1 . In consistency with the laboratory experiment[21] we have the initial conditions y0 (0) = 100, y0 (t) = 0 for t < 0 and yi (t) = 0 for t  0, where i = 1, 2, . . . , n. As in the previous two cases, we can have equations for the number of cells undergoing transformation or differentiation, but since the equations decouple from the rest, we ignore them in this case too. Proposition 5.1 The compartments of the system (13) have the following limits: lim yi (t) = 0 for i = 0, 1, . . . , n − 1

t→∞

and lim yn (t) = 100 × 2n−3 .

t→∞

Proof The system of Eqs. (13) is a special case of the system (1) with ai = bi for i = 0, 1, bj = 2aj for i = 1, 2, . . . , n − 2, and an−1 = bn−1 , with τ0 = 0 and initial conditions y0 (0) = 100, y0 (t) = 0 for t < 0 and yi (0) = 0 for t ≤ 0, i = 1, 2, . . . , n. Making these substitutions in Proposition 2.1, we have that lim yn (t) =

t→∞

a0 a1 2a2 . . . 2an−2 an−1 100 a0 a1 a2 . . . an−1

= 100 × 2n−3 . and lim yi (t) = 0 for i = 0, 1, . . . , n − 1.

t→∞

 

The result of the proposition simply states that there are n − 3 replication cycles. However, the model can accurately reproduce the empirical findings of the laboratory experiments, in particular it can predict the number of EBs at any given time. Figure 7 shows that, after fitting our model parameters, we could generate time curves that match the laboratory measurements [21]. In the future we plan to extend this model to include antibiotics and to adapt it to within-host environment.

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Fig. 7 Model based curves after parameter fitting. The colored dots are taken from laboratory measurements. It shows the growth cycle of two different strains of Chlamydia bacteria, the fast-replicating C. trachomatis and slow-replicating C. pneumoniae as reflected in the figure

References 1. G.L. Atkins, Multicompartment models for biological systems, 153 Seiten. Methuen Co. Ltd., London 1969. Geb. 35 s. Food/Nahrung 15(2), 225–225 (1971) 2. B.C. Patten, Systems Analysis and Simulation in Ecology (Elsevier, New York, 2013) 3. J.N.R. Jeffers, Mathematical Models in Ecology (Blackwell Scientific Publications, Oxford, 1972) 4. D. Elliott, Introduction to Mathematical Biology: By SI Rubinow (Wiley-Interscience, New York, 1975), p. 386 5. A. Mazanov, Stability of multi-pool models with lags. J. Theor. Biol. 59(2), 429–442 (1976) 6. I. Gy˝ori, J. Eller, Compartmental systems with pipes. Math. Biosci. 53(3–4), 223–247 (1981) 7. I. Gy˝ori, J. Wu, A neutral equation arising from compartmental systems with pipes. J. Dyn. Diff. Equat. 3(2), 289–311 (1991) 8. D.H. Anderson, Compartmental Modeling and Tracer Kinetics (Springer Science & Business Media, New York, 2013) 9. T. Piersma, Y. Verkuil, I. Tulp, Resources for long-distance migration of knots Calidris canutus islandica and C. c. canutus: how broad is the temporal exploitation window of benthic prey in the western and eastern Wadden Sea? Oikos, 71(3), 393–407 (1994) 10. L. Bourouiba, J. Wu, S. Newman, J. Takekawa, T. Natdorj, N. Batbayar, C.M. Bishop, L.A. Hawkes, P.J. Butler, M. Wikelski, Spatial dynamics of bar-headed geese migration in the context of H5N1. J. R. Soc. Interface 7(52), 1627–1639 (2010) 11. A. Marciniak-Czochra, T. Stiehl, A.D. Ho, W. Jäger, W. Wagner, Modeling of asymmetric cell division in hematopoietic stem cells – regulation of self-renewal is essential for efficient repopulation. Stem Cells Dev. 18(3), 377–386 (2009) 12. P. Getto, A. Marciniak-Czochra, Mathematical modeling as a tool to understand cell selfrenewal and differentiation, in Mammary Stem Cells (Springer, New York, 2015), pp. 247–266 13. D. Wilson, P. Timms, D. McElwain, A mathematical model for the investigation of the Th1 immune response to Chlamydia trachomatis. Math. Biosci. 182(1), 27–44 (2003) 14. S.J. Bhosai, R.L. Bailey, B.D. Gaynor, T.M. Lietman, Trachoma: an update on prevention, diagnosis, and treatment. Curr. Opin. Ophthalmol. 23(4), 288 (2012) 15. J. Paavonen, W. Eggert-Kruse, Chlamydia trachomatis: impact on human reproduction. Hum. Reprod. Update 5(5), 433–447 (1999) 16. C.C. Kuo, L.A. Jackson, L.A. Campbell, J.T. Grayston, Chlamydia pneumoniae (TWAR). Clin. Microbiol. Rev. 8(4), 451–461 (1995)

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17. L.A. Campbell, C.-C. Kuo, J.T. Grayston, Chlamydia pneumoniae and cardiovascular disease. Emerg. Infect. Dis. 4(4), 571 (1998) 18. J.A. Ramirez, Isolation of chlamydia pneumoniae from the coronary artery of a patient with coronary atherosclerosis. Ann. Intern. Med. 125(12), 979–982 (1996) 19. Y.M. AbdelRahman, R.J. Belland, The chlamydial developmental cycle. FEMS Microbiol. Rev. 29(5), 949–959 (2005) 20. I. Eszik, I. Lantos, K. Önder, F. Somogyvári, K. Burián, V. Endrész, D.P. Virok, High dynamic range detection of chlamydia trachomatis growth by direct quantitative PCR of the infected cells. J. Microbiol. Methods 120, 15–22 (2016) 21. M.M. Rönn, E.E. Wolf, H. Chesson, N.A. Menzies, K. Galer, R. Gorwitz, T. Gift, K. Hsu, J.A. Salomon, The use of mathematical models of chlamydia transmission to address public health policy questions: a systematic review. Sex. Transm. Dis. 44(5), 278–283 (2017) 22. K. Siewert, J. Rupp, M. Klinger, W. Solbach, J. Gieffers, Growth cycle dependent pharmacodynamics of antichlamydial drugs. Antimicrob. Agents Chemother. 49(5), 1852–1856 (2005)

Normalization of a Periodic Delay in a Delay Differential Equation K. Nah and J. Wu

1 Introduction In this paper, we consider the scalar delay differential equation (DDE) with timedependent delay, x  (t) = f (t, x(t), x(t − τ (t)).

(1)

In Brunner and Maset’s paper[1], the authors suggested a method to construct a time-transformation function t = h(s) such that x(t) is the solution of an initial value problem with DDE Eq. (1) and y = x ◦ h is a solution of the initial value problem with the DDE with constant delay y  (s) = h (s)f (h(s), y(s), y(s − τ ∗ )),

(2)

where τ ∗ > 0. With certain conditions on the delay function, they showed that the constructed time-transformation is strictly increasing, continuous, rightdifferentiable. With a periodic delay function τ (t), Eq. (1) is used in modeling the population dynamics of a single species. Wu et al. [2], motivated by the physiological characteristics of the populations of some ectothermic insects such as deer ticks (Ixodes scapularis), a vector of Lyme disease, formulated the growth of such tick populations as a system of delay differential equations with the time-periodic delay capturing the seasonality in the development of ticks. Linearization of the system at

K. Nah () · J. Wu Department of Mathematics and Statistics, York University, Toronto, ON, Canada e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_10

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the trivial equilibrium gives the scalar periodic delay differential equation decoupled from the rest of the system. If Eq. (1) is a periodic DDE and the derivative of a time-transformation h is periodic, then the reduced DDE Eq. (2) is a periodic DDE with a discrete delay. With a special type of the periodic DDE with a discrete delay, Röst[3] studied Neimark–Sacker bifurcation phenomenon when the delay is an integer multiple of the period. The same condition is used to prove the existence, uniqueness, and global attractivity of the periodic solution of a periodic logistic equation with a discrete delay[4], while the existence and stability of a periodic solution have remained as open problems when the delay term is time-dependent[5]. In this paper, we prove the existence of a time-transformation h which is continuously differentiable. We identify conditions for h to be continuously differentiable and find a quadratic polynomial which satisfies the conditions. The iterated extension of the polynomial forms h. In the second part of the paper, we show that when the delay τ (t) is a periodic function, the derivative of the time-transformation cannot be a periodic function with a period P , where P = τ ∗ /k, k ∈ Z+ . We find a numerical example of the delay function which makes the derivative of the corresponding time-transformation to be a periodic function with the period P where P = τ ∗ /k. We also present a numerical example showing that the reduced DDE Eq. (2) can be an asymptotically singular perturbation problem. In Sect. 4, we introduce conditions for the existence of a periodic solution of DDEs with timedependent delay, Eq. (1).

2 Time-Transformation of Time-Dependent Delay Differential Equations Theorem 1 Let τ ∈ C 1 (R, R+ ) be a bounded function with τ  < 1. For given τ ∗ ∈ R+ and s0 ∈ R, there exists a strictly increasing function h ∈ C 1 ([s0 − τ ∗ , ∞), R) which satisfies h(s) − τ (h(s)) = h(s − τ ∗ ).

(3)

Moreover, if x(t) is the solution of !

x  (t) = f (t, x(t), x(t − τ (t))), x(t) = ψ(t), t < t0

t ≥ t0

(4)

for the given initial function ψ ∈ C, then y = x ◦ h is the solution of !

y  (s) = h (s)f (h(s), y(s), y(s − τ ∗ )), y(s) = ψ(h(s)), s < s0 ,

where s = h−1 (t) and s0 = h−1 (t0 ).

s ≥ s0

(5)

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145

Proof Let g(t) = t −τ (t). Since τ is bounded, g(R) = R. Since g  (t) = 1−τ  (t) > 0, g is strictly increasing on R. Therefore, the inverse function g −1 : R → R is welldefined and strictly increasing. Since g is continuously differentiable with g  > 0, g −1 is continuously differentiable. Let φ ∈ C 1 ([s0 − τ ∗ , s0 ], R) be a strictly increasing function satisfying φ(s0 ) = t0 ,

(6)

φ(s0 ) = g −1 (φ(s0 − τ ∗ )) and φ  (s0 ) =

φ  (s0 − τ ∗ ) . 1 − τ  (t0 )

(7)

Existence of such a function φ is guaranteed by the following form: φ(s) = c0 (s − s0 ) + c1 (s − (s0 − τ ∗ )) + c2 (s − s0 )(s − (s0 − τ ∗ )),

(8)

where c0 = −(t0 − τ (t0 ))/τ ∗ , c1 = t0 /τ ∗ , and c2 =

τ (t0 )τ  (t0 ) ∗ (τ )2 (2 − τ  (t

0 ))

.

Note that c2 is well-defined because τ ∗ > 0 and τ  < 1. We can easily see that φ satisfies the three conditions. We will show that φ is strictly increasing on [s0 − τ ∗ , s0 ). Case 1. τ  (t0 ) ≥ 0: If τ  (t0 ) ≥ 0, c2 ≥ 0. Since s ≥ s0 − τ ∗ , φ  (s) = c0 + c1 + 2c2 s − c2 (2s0 − τ ∗ ) ≥ c0 + c1 + 2c2 (s0 − τ ∗ ) − c2 (2s0 − τ ∗ ) = c0 + c1 − c2 τ ∗ =2

τ (t0 ) 1 − τ  (t0 ) τ ∗ 2 − τ  (t0 )

> 0. Case 2. τ  (t0 ) < 0: Since τ  (t0 ) < 0, c2 < 0. Since s < s0 , φ  (s) = c0 + c1 + 2c2 s − c2 (2s0 − τ ∗ ) ≥ c0 + c1 + 2c2 s0 − c2 (2s0 − τ ∗ ) = c0 + c1 + c2 τ ∗

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=2

1 τ (t0 ) τ ∗ 2 − τ  (t0 )

> 0. We construct a function h : [s0 − τ ∗ , ∞) → R by h(s) = φ(s) f or

s0 − τ ∗ ≤ s < s0 ,

h(s) = g −1 (h(s − τ ∗ ))

s0 + (k − 1)τ ∗ ≤ s < s0 + kτ ∗ ,

f or

(9)

where k = 1, 2, 3 . . . . It follows that h(s) = (g −1 )k ◦ φ(s − kτ ∗ ) f or

s0 + (k − 1)τ ∗ ≤ s < s0 + kτ ∗ .

Also, h satisfies h(s) − τ (h(s)) = h(s − τ ∗ )

f or

s ≥ s0 .

Since φ and g −1 are strictly increasing, h is strictly increasing. Note that h is continuous on [s0 + (k − 1)τ ∗ , s0 + kτ ∗ ), k = 0, 1, 2 . . . . Also, h is continuous at s = s0 since lim h(s) = lim φ(s) = φ(s0 ),

s→s0−

t→s0−

h(s0 ) = g −1 (h(s0 − τ ∗ )) = g −1 (φ(s0 − τ ∗ )) = φ(s0 ), by assumption. For n ∈ N, lim

s→(s0 +nτ ∗ )−

h(s) =

lim

s→(s0 +nτ ∗ )−

(g −1 )n (φ(s − nτ ∗ ))

= (g −1 )n (φ(s0 )) = (g −1 )n (g −1 (φ(s0 − τ ∗ ))) = (g −1 )n+1 (φ(s0 − τ ∗ )) = h(s0 + nτ ∗ ). Therefore, h is continuous on [s0 − τ ∗ , ∞). Now, we show that h is differentiable on [s0 − τ ∗ , ∞). Since φ and g −1 are continuously differentiable, h is continuously differentiable on [s0 + (k − 1)τ ∗ , s0 + kτ ∗ ), k = 0, 1, 2 . . . . Note that h is continuously differentiable at s = s0 , because lim h (s) = lim φ  (s) = φ  (s0 ),

s→s0−

s→s0−

Normalization of a Periodic Delay in a Delay Differential Equation

lim h (s0 ) =

s→s0+

147

φ  (s0 − τ ∗ ) φ  (s0 − τ ∗ ) φ  (s0 − τ ∗ ) = , = 1 − τ  (φ(s0 )) 1 − τ  (t0 ) g  (g −1 (φ(s0 − τ ∗ ))

and φ  (s0 ) =

φ  (s0 − τ ∗ ) 1 − τ  (t0 )

by Eq. (7). For n ∈ N, lim

s→(s0 +nτ ∗ )−

h (s) =

lim

s→(s0 +nτ ∗ )−

= +n k=1

φ  (s − nτ ∗ )  −1 ∗ k=1 g (g (h(s − kτ )))

+n

φ  (s0 )  −1 g ((g )(n−k)+1 (φ(s

0 ))

,

and lim

s→(s0 +nτ ∗ )+

h (s) =

lim

s→(s0 +nτ ∗ )+

= +n+1 k=1

= +n+1 k=1

= =

φ  (s − (n + 1)τ ∗ ) +n+1  −1 ∗ k=1 g (g (h(s − kτ ))) φ  (s0 − τ ∗ )

g  ((g −1 )n−k+2 (φ(s0 − τ ∗ ))) φ  (s0 − τ ∗ ) g  ((g −1 )n−k+1 (φ(s0 )))

g  (φ(s0 ))

φ  (s0 − τ ∗ )  −1 n−k+1 (φ(s ))) 0 k=1 g ((g )

+n

φ  (s0 − τ ∗ ) + . (1 − τ  (φ(s0 ))) nk=1 g  ((g −1 )n−k+1 (φ(s0 )))

By the continuous differentiability condition Eqs. (7) and (6), h is continuous at s = s0 + nτ ∗ for all n = 0, 1, 2 . . . . Let x(t) be the solution of the initial value problem (4). Let y = x ◦ h. Then, y  (s) = x  (h(s))h (s) = h (s)f (h(s), x(h(s)), x(h(s) − τ (h(s)))) = h (s)f (h(s), x(h(s)), x(h(s − τ ∗ ))) = h (s)f (h(s), y(s), y(s − τ ∗ )). Since x(t) = ψ(t) for t ≥ t0 , y(s) = x(h(s)) = ψ(h(s)) for s ≥ s0 .

 

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Note that a time-transformation satisfying the conditions in Theorem 1 is not unique. For example, when τ (t) = 1 and τ ∗ = 1, a transformation t = h(s) with h(s) = s satisfies those conditions. Indeed, the extension h(s) of any strictly increasing function φ ∈ C 1 ([−1, 0)) which satisfies φ(0) − 1 = φ(−1) and φ  (0) = φ  (−1) can be a such time-transformation where h(s + n) = φ(s) + n for s ∈ [−1, 0) and n ∈ Z. If the lag function τ (t) is lower-bounded by a positive value, we can infer the long-term dynamics of the original DDE from the long-term dynamics of the reduced DDE with a discrete delay, see the following Proposition. One can prove it using Eq. (3) and h(s0 ) = t0 . Proposition 1 For a lag function τ (t) satisfying the conditions in Theorem 1, if there exists τˆ > 0 such that τ (t) ≥ τˆ for all t ∈ R, then h([s0 , ∞)) = [t0 , ∞). More specifically, h(s) ≥ t0 + (n − 1)τˆ for s ∈ [s0 + (n − 1)τ ∗ , s0 + nτ ∗ ). Even though h is unbounded, h can approach zero as s → ∞ making Eq. (5) an asymptotically perturbation problem. We will show a numerical example of this case in the next section.

3 Normalization of a Periodic Delay In this section, we observe the properties of the time-transformation h when the lag function is a periodic function. The following proposition shows that when τ (t) is a periodic function, h (s) cannot be a periodic function with a period P where P = τ ∗ /k for k ∈ Z+ . Proposition 1 Assume that a lag function τ (t) is a non-constant function. Then, there exists no time-transformation t = h(s) such that h (s) is a periodic function with period P , where P = τ ∗ /k for k ∈ Z+ . Proof Assume that h(s) is a time-transformation corresponding to a non-constant lag function τ (t) such that h (s + P ) = h (s) for all s ∈ [s0 , ∞), where P = τ ∗ /k for some k ∈ Z+ . Integrating both sides of h (s + P ) = h (s) yields 

s+s0 +(n−1)P s0

h (u + P )du =



s+s0 +(n−1)P

h (u)du

s0

for any n ∈ Z. That is, h(s + s0 + nP ) − h(s0 + P ) = h(s + s0 + (n − 1)P ) − h(s0 ) for all n ∈ Z. Therefore, h(s + s0 ) − h(s + s0 − kP ) = k(h(s0 + P ) − h(s0 )).

(10)

Normalization of a Periodic Delay in a Delay Differential Equation 1.3

100

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Fig. 1 Graphs of h (s), h(s), and s − h(s) when h (s) is periodic. In this example, the lag function is given by τ (t) = a sin(t) + c, where a = 1/5 and c = 1. The time-transformation t = h(s) is constructed by Eq. (9) with τ ∗ = 1, s0 = 0, t0 = 0, and the initial function given in Eq. (8). The graph of s − h(s) is presented to show that h(s) in this example is non-linear

Since τ ∗ = kP , by Eq. (3), h(s + s0 ) − τ (h(s + s0 )) = h(s + s0 − kP ).

(11)

Then, by Eqs. (10)–(11), τ (h(s + s0 )) = h(s + s0 ) − h(s + s0 − kP ) = k(h(s0 + P ) − h(s0 )). Therefore, τ ◦ h is constant. Since h is strictly increasing, τ should be constant. This contradicts the assumption that τ is non-constant.   However, h can be a periodic function with period P when the delay is not an integer multiple of the period, i.e., when P = τ ∗ /k for some k ∈ Z+ . A numerical example of such case is presented in Fig. 1. In this example, the delay function is given by τ (t) = 1/5·sin(t)+1 which satisfies the conditions of Theorem 1, i.e., τ (t) is strictly positive and τ  (t) = 1/5 · cos(t) < 1. A strictly increasing transformation function t = h(s) is presented in the right panel of the figure. It is constructed by Eq. (9) with τ ∗ = 1, s0 = 0, t0 = 0, and the initial function in Eq. (8). As shown in the left panel, h is a periodic function with a period P > 1. Note that τ ∗ = 1 is not an integer multiple of P .

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8

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Fig. 2 Graphs of h(s), h (s), and τ  (h(s)) when h approaches zero. In this example, the lag function is given by τ (t) = a sin(2π t) + c, where a = 1/2π − 0.001 and c = 1/2π + 0.001. The time-transformation t = h(s) is constructed by Eq. (9) with τ ∗ = 1, s0 = 0, t0 = 0, and the initial function given in Eq. (8)

Figure 2 shows another example of the time-transformation h, where h (s) approaches zero as s → ∞. In this example, the lag function τ (t) is given as a periodic function in C 1 (R, R+ ) satisfying the conditions τ  < 1 and τ (t) ≥ τˆ for some τˆ > 0. In the top panels, we observe that h is unbounded and h (s) approaches zero as s → ∞. The bottom panel shows the graph of τ  (h(s)). Differentiation of Eq. (3) yields h (s) 1 = . − τ ∗) 1 − τ  (h(s))

h (s

Normalization of a Periodic Delay in a Delay Differential Equation

Note that

h (s) h (s−τ ∗ )

151

< 1 if and only if τ  (h(s)) < 0.

4 Existence of a Periodic Solution In this section, we show that under some condition, the existence of a periodic solution of the discrete delay differential equation Eq. (5) can lead to the existence of a periodic solution of the delay differential equation Eq. (4) with a periodic delay. Proposition 1 For a given function τ satisfying the conditions in Theorem 1, let h be a corresponding time-transformation. Let y(s) be a solution of Eq. (5). Then, x = y ◦ h−1 is a solution of Eq. (4). Moreover, if both h and y are periodic with period P , then x is periodic with period h(s0 + P ) − h(s0 ). Proof The first part of the proposition can be easily proven by the chain rule. Now, if we assume that h and y are periodic with period P , then 

s

h(s) − h(s0 ) = 

h (u)du

s0 s

=

h (u + P )du

(12)

s0

= h(s + P ) − h(s0 + P ) for all s ≥ s0 . Using Eq. (12) and that y is periodic with period P , we get x(t) = x(h(s)) = y(s) = y(s + P ) = x(h(s + P )) = x(h(s) + h(s0 + P ) − h(s0 )) = x(t + h(s0 + P ) − h(s0 )).

 

Acknowledgments Kyeongah Nah thanks the International Union of Biological Sciences (IUBS) for partial support of living expenses in Szeged, during the 19th BIOMAT International Symposium, October 20–26, 2019. We thank Dr. Dimitri Breda (University of Udine) for comments that greatly improved the manuscript.

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References 1. H. Brunner, S. Maset, Time transformations for delay differential equations. Discrete Contin. Dyn. Syst. Ser. A. 25 (2009), 751–775 2. X. Wu, F.M. Magpantay, J. Wu, X. Zou, Stage-structured population systems with temporally periodic delay. Math. Method Appl. Sci. 38(16), 3464–3481 (2015) 3. G. Röst, Neimark-Sacker bifurcation for periodic delay differential equations. Nonlinear Anal. Theory Methods Appl. 60(6), 1025–1044 (2005) 4. B.G. Zhang, K. Gopalsamy, Global attractivity and oscillations in a periodic delay-logistic equation. J. Math. Anal. Appl. 150, 274–283 (1990) 5. S. Ruan, O. Arino, M.L. Hbid, E.A. Dads (eds.), Delay Differential Equations and Applications. NATO Science Series, Springer. Series II: Mathematics, Physics and Chemistry, vol. 205 (2006), pp. 477–517

Competition Between Two Tufted C4 Grasses: A Mathematical Model D. I. Wallace

1 Introduction The native coastal prairies of Texas are dominated by several species of grasses, the distributions of which depend on multiple factors such as nutrient levels, pH, intraand inter-species competition, as well as grazing and burning [1]. Understanding the nature of plant competition under different conditions would inform decisions about grazing strategies and burn control [2–4]. Models of competition among grasses may also inform our understanding of the observed impact of climate on plant communities [5]. Several models have been developed to quantify competition, with varying emphasis on nutrient availability, seed production, and other factors [6–9]. The model developed here is for vegetative reproduction only, and takes advantage of the particular growth pattern of tufted grasses, specifically Schizachyrium scoparium and Paspalum plicatulum, both native to coastal Texas prairies. These two perennial grasses are of ecological interest, as they represent vegetative growth that can resprout from the crown after grazing or light burning. Competition between these grasses is either measured indirectly in the field or directly through controlled experiment. Plants of the same biomass are grown in pots, singly, in monoculture pairs, or in inter-species pairs. At the end of the growth period the dry mass of each tuft is measured. The experimenter then compares the biomass of a species grown in monoculture to that same species grown with the other species. The differences in final dry mass provide a measure called “absolute competition intensity” [1]. The experiment may be repeated under different growth conditions.

D. I. Wallace () Department of Mathematics, Dartmouth College, Hanover, NH, USA e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_11

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For a single growth condition an experiment of this sort cannot identify reasons for competition. It is possible that one species grows faster than the other, using up available resources faster. If so, that growth may be vertical or horizontal. It is also possible that there is direct competition, for example the production of growth inhibitors by the roots of one of the two species. In this study we separate these sources of competition and compare model outcomes with experimental data to see if any direct competition is likely to be present.

2

Model Development

Bunch grasses do not produce rhizomes or stolons that allow other types of grass to propagate away from the mother plant. Rather, they grow outward from a central mother plant, maintaining a vertical tussock, or bunch [10]. Leaves arise from a well defined “crown” that spreads laterally. The aboveground height of both species considered here is about 1 m at maturity. The plants maintain a characteristic root to shoot ratio that determines the maximum depth belowground of the roots. Thus the model includes three constraints on growth: a maximum, genetically determined below and aboveground height, a constraint based on available shared resources, and a constraint arising from direct inter-species competition. Data from a study of S. scoparium is used to parametrize its growth over time as a single tuft [11]. Constants are then scaled to match P. plicatulum data from a second study [1]. An example of intra- and inter-species competition is then investigated.

2.1 Vegetative Growth of One Tuft of Grass Grass tufts grow upward and outward. The final height of the tuft is genetically determined but it is assumed in this model that the crown may grow arbitrarily large if sufficient nutrient is present to sustain the entire plant. Therefore, we model growth of the tuft as a cylinder, as in Fig. 1. Two differential equations are needed: one describing height (both below- and aboveground) of the tuft, H , and one describing surface area occupied, S, which is assumed to be roughly a disk. This disk shaped cross section is measured as the area of the “crown area,” which is the growing mass just at the surface of the soil. Biomass, that is, the total mass of the plant is then proportional to the volume of the cylinder, H S. It is assumed in this model that a carrying capacity limits total biomass. It is assumed that horizontal growth is not only limited by biomass, but at low biomass is proportional to the area of the disk. This simple proportionality reflects the way growth rates are measured in a study by Wallace et al. [11], from which some model parameters are drawn. In Fig. 1 both above and belowground height are illustrated. The ratio of these is measured by something called “root to shoot ratio,” which is reported in the study by Van Auken and Bush [1]. Note that the

Competition Between Two Tufted C4 Grasses: A Mathematical Model

155

Fig. 1 Diagram of a bunchgrass. A. Aboveground height, B. belowground height, C. crown area. Total height is A + B

height reported in [11] is only aboveground, and the root to shoot ratio determines what the full height ought to be. The model described in Eqs. (1) and (2) includes logistic limit on the growth of H S in terms of biomass. Height is also limited with a second logistic factor, reflecting the fact that these plants never exceed a certain genetically determined height. It was assumed that the natural growth at low height is proportional to the height itself. This simplifying assumption is justified by the data in [11]. With these assumptions the model describes the growth of a tuft of height H and crown area S by Eqs. (1) and (2): 

H H = aH 1 − m 



 HS 1− , k

  HS S  = bS 1 − k

(1)

(2)

2.2 Competition Between Two Tufts of the Same or Different Species To model competition between two types of grasses, we will assume that this competition constrains the growth of the neighbor’s cross-sectional area rather than height. There are two kinds of constraints—resource driven and direct interference. We will model resource driven competition by including both species in the biomass carrying capacity. This is a sort of “fair fight” for resources to grow, a constraint that would apply whether the two tufts were the same or different species of grass. In the following model given by Eqs. (3), (4), (5), and (6), H and S are the height and crown area of the first tuft and G and R are the height and crown area of the second tuft. Parameters f and g describe the presence of direct competitive forces

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in addition to those arising from shared resources. When the two species are the same or if two species are simulated without direct competition, f and g are set to zero. When the two species differ, H and S shall refer to S. scoparium and G, R to P. plicatulum.    HS GR H H  = a1 H 1 − 1− − m1 k k

(3)

  GR HS S  = b1 S 1 − − − f SH GR k k

(4)

   HS GR G G = a2 G 1 − 1− − m2 k k

(5)

  HS GR R = b2 R 1 − − − gSH GR k k

(6)





2.3 Parameters Relative rates a1 , b1 for S. scoparium are taken from the June-July measurements in an early paper of Wallace [11]. The June/July growth period was chosen because temperature is similar to the conditions in a study by Van Auken and Bush study, which compares its growth to that of P. plicatulum [1]. Both authors use months as their time unit. Maximum aboveground height for S. scoparium is commonly estimated as 100 cm. To get belowground height, we use the root–shoot ratio given in the control experiment (pH = 7 and no added nutrient) in [1]. The maximum height, m1 , is the sum of the aboveground and belowground heights. Based on comparing initial conditions in both studies, one of which gives initial crown and aboveground height measurements, and the other gives initial dry weight, a multiplier, r was constructed to convert volume to dry mass. The Van Auken and Bush study reports a total dry mass for SS in monoculture of 3 g at 4.5 months, and for PP in monoculture 1.25 g at 4.5 months, with two plants in a container [1]. The resource related carrying capacity, k, was chosen so that the growth of S. scoparium matched data for this study. For P. plicatulum there is less detailed data. The aboveground terminal height of P. plicatulum is also approximately 100 cm, but the root–shoot ratio is different, and the terminal height, m2 , is adjusted accordingly. Initial conditions for crown and height were scaled from [11] according to dry mass ratios in [1] to get initial conditions for P. plicatulum. The parameters a1 and b1 are scaled to get a2 and b2 by using the data for a four and a half month growth period of P. plicatulum grown singly in [1], until Eqs. (1) and (2) matched the results of that study. Table 1 summarizes all parameters and Fig. 2 shows the results of modeling a monoculture of two plants of each species.

Competition Between Two Tufted C4 Grasses: A Mathematical Model

157

Table 1 Parameters Parameter Description S. scoparium: a1 Intrinsic growth of height m1 Genetic height limit b1 Intrinsic growth of crown f Competition (intra, inter) H (0) Initial height S(0) Initial crown P. plicatulum: a2 Intrinsic growth of height m2 Genetic height limit b2 Intrinsic growth of crown g Competition (intra, inter) G(0) Initial height R(0) Initial crown Both: k Shared resource limit r Volume to dry weight conversion factor

Units

Source

0.9

% per month

[11]

150 0.75 (0, 0.0021) 5.28 1.32

cm % per month None cm cm2

[1] [11] Matched [1] [11] [11]

0.0855

% per month

Scaled to match [1]

125 0.07125 (0, 0.00009) 6.35 1.58

cm % per month None cm cm2

[1] Scaled to match [1] Matched [1] Scaled [1, 11] Scaled [1, 11]

75 0.04

cm3 No units

Matched [1] [1, 11]

b

14 height S.s. crown area S.s. dry weight S.s.

12

dry weight, crown, height

dry weight, crown, height

a

Value

10 8 6 4 2 0

0

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10 8

height P.p. crown area P.p. dry weight P.p.

6 4 2 0 0

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Fig. 2 S. scoparium and P. plicatulum grown in monoculture with two identical plants sharing nutrient. (a) Crown area, height, and dry weight of a single S. scoparium grown in a two-plant monoculture, (b) Crown area, height, and dry weight of a single P. plicatulum grown in a two-plant monoculture

2.4 Modeling Competition Competition between the two species was simulated without assuming direct competition, i.e. f = g = 0. The result is shown in Fig. 3.

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a

b 8

height S.s. crown area S.s. dry weight S.s.

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Fig. 3 S. scoparium and P. plicatulum grown in mixed culture with two plants sharing nutrient. No direct competition is assumed (f = g = 0). (a) Crown area, height, and dry weight of a single S. scoparium grown in mixed culture, (b) Crown area, height, and dry weight of a single P. plicatulum grown in mixed culture, (c) Dry weights of each plant and total dry weight, (d) Percent of total biomass for each species. Data points indicate null hypothesis for each species

The conclusions of [1] were that growth of individual plants was not significantly different between monoculture and mixed culture in control (pH = 7, no added nutrient) conditions. This conclusion is represented by two data points included at the 4.5 month time point in Figs. 3 and 4, panels C and D. In Fig. 4, the parameters for direct competition, f and g were adjusted to match the data in panel D. All simulations were run on Matlab software [12].

3 Results and Discussion The model developed here is successful at reproducing the results of [1] for both mono and mixed culture growth of two bunchgrasses, S. scoparium and P. plicatulum. However, the experiment in [1] did not include sufficient detail to

Competition Between Two Tufted C4 Grasses: A Mathematical Model

b

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Fig. 4 S. scoparium and P. plicatulum grown in mixed culture with two plants sharing nutrient. Direct competition is assumed (f = 0.0021, g = 0.00009). (a) Crown area, height, and dry weight of a single S. scoparium grown in mixed culture, (b) Crown area, height, and dry weight of a single P. plicatulum grown in mixed culture, (c) Dry weights of each plant and total dry weight, (d) Percent of total biomass for each species. Data points indicate null hypothesis for each species

parametrize the model, and so a second study on S. scoparium, [11], was also used, leading to potential errors in parametrizing intrinsic growth rates. In addition to biomass, the model produces estimates of crown area and height, which could be compared in a future study with more detailed data on initial and final conditions. The authors of [1] make the point that considerable apparent competition could arise from the naturally differing growth rates of two species, as the slower growing species leaves extra resources for the faster one to use. This phenomenon is seen in Fig. 3, which shows the results of inter-species growth with no direct competition. The data points marked in Fig. 3(c and d) represent the values these species attained at 4.5 weeks when grown with the same kind. In a simulation of mixed culture without direct competition, the faster growing S. scoparium attained higher biomass at the expense of the slower growing P. plicatulum. The difference, had it been observed in experiment, would have been described as “absolute competition” even though no direct competition was included in this simulation.

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Interestingly, the experiment in [1] shows little difference in final biomass of these two species, whether in intra-species or inter-species competition. It appears from the figures that monocultures produced slightly more biomass than mixed cultures, at most half a gram, but the difference was not statistically significant for the control experiment in that study. In Fig. 4, direct competition parameters were fitted to the expect percent biomass for the two species in mixed culture, as seen in Fig. 4d. Figure 4c shows both species to have final biomass slightly below what was expected, on the order of 0.1–0.3 g. If this had occurred in the experiment, it is unlikely it would have been found to be statistically significant. The model illustrates that significant direct competition must be occurring to counterbalance the natural advantage one of these species has over the other. It also illustrates that the experimental measure of “absolute competition” is deceptive, as resource based and direct competition are not separated and may compensate for each other. Future directions for this work would include conducting experiments in sufficient detail to verify that the time series in Figs. 2, 3, and 4 are correct at multiple time points for both crown area and height, developing a model that tracks shoots and roots separately, and including externalities such as grazing and burning. Acknowledgments The author wishes to acknowledge her Dartmouth differential equations class in Spring 2014 for testing an early version of the model presented here and Dr. Paul Gagnon for suggesting this problem.

References 1. O.W. Van Auken, J.K. Bush, Int. J. Plant Sci. 158, 325–331 (1997) 2. K.N. Suding, J. Veg. Sci. 12, 849–856 (2001) 3. P.R. Gagnon, H.A. Passmore, W.J. Platt, J.A. Myers, C.E.T. Paine, K.E. Harms, Ecology 91(12), 34813486 (2010) 4. P.R. Gagnon, K.E. Harms, W.J. Platt, H.A. Passmore, J.A. Myers, PLoS One 7(1), e29674 (2012). https://doi.org/10.1371/journal.pone.0029674 5. P.B. Adler, J. HilleRisLambers, P.C. Kyriakidis, Q. Guan, J.M. Levine, Proc. Natl. Acad. Sci. 103(34), 12793–12798 (2006) 6. R. McMurtrie, L. Wolf, Ann. Bot. 52(4), 449–58 (1983) 7. V. Halty, M. Valds, M. Tejera, V. Picasso, H. Fort, Ecol. Appl. 27(8), 2277–2289 (2017) 8. H. Fort, Ecol. Model. 10;387, 154–162 (2018) 9. L.H. Uricchio, S.C. Daws, E.R. Spear, E.A. Mordecai, Am. Nat. 193(2), 213–26 (2019) 10. Forage Information System, Bunch and Sod-forming Grasses, Oregon State University (2019) 11. L.L. Wallace, Oecologia 72(3), 423–428 (1987) 12. MATLAB (The MathWorks Inc. Natick, 2016)

Mathematical Description of Systemic and Micro Circulations V. V. Kislukhin and E. V. Kislukhina

1 Introduction Two pathological conditions, a retention of nitrogen during dialysis session, and a decompression sickness have their origin in microcirculation disorder. The essence of a disorder was revealed in the 1940s during blood compensation therapy. After emergency steps one was looking for blood volume (BV) values. Measurement of BV by an indicator dilution, see Fig. 1, reveals that for many patients the time for complete mixing of N particles demands up to 40 min [1]. Two-compartment presentation for cardiovascular system (CVS) was introduced, Fig. 2, with first order exchange between active and slow (ACV and SCV) circulating blood volumes, thus on the exponential back extrapolation of the concentrations of an indicator [2]. Thus the first aim of the manuscript is to present a mathematical model of the passage of a tracer throughout CVS. The model is based on an observation made in the 1930s by Romanovsky [3] that CVS is an oriented closed graph, with orientation given by flow, Fig. 3. The main feature of A, due to a numeration that starts from right atrium, RA, is aij = 0 if j < i. Exceptions am1 and an1 , they correspond to a connection of superior and inferior vena cava with RA. Math model based on a graph of CVS reveals that rebound and bends are the consequences of microcirculation disorder that in math terms means the presence of non-zero elements on main diagonal of A. This statement is the consequence of the observation: if all elements from main V. V. Kislukhin () Medisonic, Moscow, Russia e-mail: [email protected] E. V. Kislukhina Sklifosovsky Institute for Emergency Medicine, Moscow, Russia © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_12

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Fig. 1 Concentration of indicator in the samples (dark circles) as a function of time. The last two measurements point out that the mixing is reached. The C is the concentration used for BV determination, BV=N /C, and the H , back extrapolation, is for ACV = N /H Fig. 2 Two-compartment presentation for CVS

Fig. 3 Graph CVS and its transport matrix A. Dot-dash line passes diagonal elements

diagonal are zero the only real characteristic number (roots of det(sA − E) = 0) is s1 = 1, by other words no monotones drop, as on Fig. 1. One (and effective) tool to estimate the state of microcirculation is LDF-meter. In experiment it is possible to check any tissue. Theory of LDF states that timerecoding signal is proportional to the blood flow under probe [4]. In turn, Krogh and followers established that flow through capillaries is, mainly, proportional to the number of open capillaries [5–7]. Thus recorded by LDF sequence, Fig. 4 reflects variation of the number of open microvessels. Stochastic scheme for behavior of microvessels is presented in Fig. 6. The sum of probabilities for microvessels to change their state is the characteristic of microcirculation, R = β + μ, where β for open microvessels to become close, and μ for closed microvessels to become open. Particularly, variation of R can vary a

Mathematical Description of Systemic and Micro Circulations

163

#8-IK

7 6.5 6 5.5 5 4.5 4

3

1 84 167 250 333 416 499 582 665 748 831 914 997 1080 1163 1246 1329 1412 1495 1578 1661 1744 1827 1910 1993 2076 2159 2242 2325 2408 2491 2574 2657 2740 2823 2906 2989 3072 3155 3298 3321 3404 3487 3570 3653 3736 3819 3902 3985 4068

3.5

Fig. 4 LDF recording

delivery of O2 to tissues [8]. Thus the second aim of the manuscript is to present the ability to obtain R from LDF-time series.

2 Mathematical Model of an Indicator Passing Throughout CVS CVS is the set of the next three edges with different pattern of blood flow: (1) Heart. Chambers are pumps. + The generating function of the distribution of time to pass any chamber is 4j =1 bj s/(1 − aj s), with bj as ejection fraction and aj + bj = 1; (2) Connective vessels, diameter 200–300 μm and up. They connect heart and microcirculation. Generating function for time to pass is s n , where n is the time to pass the given vessel; (3) Microcirculation. It has two properties (a) there is dispersion in length of microvessels. It can be approximated by binominal distribution: (ps + q)k where p · k is the mean and pq · k is the dispersion [9], (b) the flow in each microvessel is irregular. Simplest math description is to accept: all microvessels can be open (No ) or closed (Nc ) and as functions of time they follow the next matricial equation:       No (t + 1) αμ No (t) = · Nc (t + 1) Nc (t) β ν

(1)

where α is probability for open vessel to remain open, and (α + β = 1), and ν is probability for closed vessel to remain closed (ν + μ = 1). A scheme for stochastic exchange between open and closed microvessels is shown at Fig. 5. Generating function for distribution of probability to pass microcirculation as combination of two properties could be written as [10]:

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Fig. 5 Two type of edges within microcirculation: open and closed microvessels

 ) *k q + p αs + βμs 2 /(1 − νs)

(2)

The edges that correspond to “closed microvessels” are the edges with mean time to be in as 1/μ thus we have non-zero elements on main diagonal. Direct calculation [10] for evolution of indicator injected into RA, z1 (0) = 1, and recorded also in RA, z1 (t) leads to Eq. (3), if R is small. z1 (t) ≈ d11 + d11 s2−t (1)

(2)

(3)

s2 ≈ 1 + μ · BV/ACV > 1 + R

(4)

With, at least, two real characteristic numbers: s1 = 1;

Thus we have by exponential extrapolation on zero time the volume within heart, conductive vessels, and open microvessels (1)

d11 =

SV 1 = ; T BV

(1) (2) d11 + d11 =

1 SV ; = To ACV

BV = SV · T ACV = SV · To

(5) (6)

3 Math Equations for LDF Typical recording of LDF-meter is given on Fig. 4. Since variations of flow are proportional to the number of open microvessels, No , it is No as the subject of variations with time as on Fig. 4. We accept the scheme for variations of the state of microvessels as on Fig. 6. Thus probability to change from No open microvessels to any m microvessels is:  No Nc  α k β No −k μl ν Nc −l pm = k l m=k+l

(7)

Mathematical Description of Systemic and Micro Circulations Fig. 6 Stochastic exchange between open and close microvessels

165

α μ ν=1−μ β=1−α

N=100, a =0.95, m=0.25

95 90 85 80 75 70

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235 248 261 274 287 300 313 326 339 352 365 378 391 404 417 430 443 456 469 482 495 508

65

Fig. 7 Example of model time series

Since



pm = 1 we can create time series on a math model:

(a) interval [0, 1] is divided into N + 1 (N = No + Nc ) [11] intervals with length equal to pm ; (b) if evenly distributed quantity on [0, 1] belongs to interval m, then we have moved from No to m open microvessels. Repetition of (a) and (b) generates time-sequence on math model of LDF, see Fig. 7. Experiments on model reveal that (a) Two time series, with equal R, are statistically identical; (b) Between R and mean spectral frequency, or M 

Fm =

SPD(k) · k · 

k=1 M 

SPD(k)

k=1

there is a functional relation, see Fig. 8. Thus we have next scheme for treatment real LDF: we made Fourier transformation, got the spectral power density, thus, Fm , and calculate the accumulation curve. Using the Fm we obtained R, see Fig. 8, then we take model curve with this R and made the same treatment for it. If difference between two accumulation curves is less than 7% we take R as rate of vasomotion for real curve (Fig. 9).

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y = 2018x5 – 2510x4 + 1049.9x3 – 162x2 + 11.54x – 0.3311 R2 = 0.9998

1.2 0.8 0.4 0 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Fig. 8 Dependence between Mean spectral frequency Fm (Ox) and R (Oy)

Fig. 9 Treatment of real LDF series. Left: from up to down—time-series, spectrum, accumulation curve; Right: from up to down—model-time series, spectrum, and two accumulation curves

There are three main reasons for insufficiency of the R to describe real time LDF series and they are given on Fig. 10: 1. No stable time series (5–8%). 2. Spectrum included significant breathing waves (20%). 3. Spectrum included significant Meyer waves (∼0.08 Hz) (20%).

Mathematical Description of Systemic and Micro Circulations

167

Fig. 10 Left: spectra of different situations. Right: accumulation curves, model and real LDF, with the same R

4 Discussion Urea and other nitrogen derivatives are byproducts of the proteins metabolism. Urea is generated in the liver, freely penetrated into all tissues, and removed by the kidney. Patients with not working kidneys regularly clean their blood, mainly from urea, by the artificial kidney. Dialysis session is usually 3–4 h, and in majority of the sessions this time is enough for the blood cleaning. However, in some cases, after the session is finished, the concentration of urea is increasing significantly faster than should be expected by the regular generation. There are two main explanations for urea rebound. The low flow explanation states that the urea from poorly perfused regions enters the active circulation, thus relatively rapidly increases concentration. Another explanation is based on the assumption that the permeability for urea is low in some parts of tissue and therefore urea appears in blood slowly. This explanation can be called the low permeability explanation. Described on previous pages the pathological state of microcirculation with a low rate of exchange between closed and open microvessels leads to the explanation based on microcirculation disorder: from tissue with a low rate of vasomotion removing of urea (as well as other metabolites) demands time. Since urea freely dissolved in water the low

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permeability explanation does not work: a temporal decrease of permeability is usually due to edema. Analogous situation takes place with the bends.

5 Conclusion (a) Math analysis of transport matrix for an indicator passing throughout CVS reveals that slow exchange between two states of microvessels (open and closed) is the cause for the urea or the rebound. (b) Treatment of LDF recording leads to the estimation of the rate of exchange between open and closed states of microvessels.

References 1. J.G. Gibson 2nd, W.C. Peacock, A.M. Seligman, T. Sack, Circulating red cell volume measured simultaneously by the radioactive iron and dye methods. J. Clin. Invest. 25(126), 838–847 (1946) 2. C.J. Wiggers, Physiology of Shock (Commonwealth Fund, New York, 1950) 3. V.I. Romanovsky, Discrete Markov Chains (Moscow-Leningrad, Groningen), p.158 [in Russian]. V.I. Romanovsky, Discrete Markov Chains (Wolter-Nordhoff, Groningen, 1970, in English) 4. I. Fredriksson, C. Fors, J.D. Johansson, Laser Doppler Flowmetry—A Theoretical Framework, Department of Biomedical Engineering (Linköping University, Linköping, 2007). www.imt. liu.se/bit/ldf/ldfmain.html 5. A. Krogh, The Anatomy and Physiology of Capillaries (Hafner, New York, 1959) 6. B.W. Zweifach, Functional Behavior of the Microcirculation (Charles C. Thomas Publisher, Springfield, 1961) 7. G.I. Mchedlishvili, Cerebral circulation regulatory mechanisms. IV. Physiologic mechanisms for correlating blood flow with brain function. Tr. Inst. Fiz. Akad. Nauk Gruz. Ssr 15, 235–250 (1968) 8. V.V. Kislukhin, Stochasticity of flow through microcirculation as a regulator of oxygen delivery. Theor. Biol. Med. Model. 7(1), 29 (2010) 9. C.R. Honig, M.L. Feldstein, J.L. Frierson, Capillary lengths, anastomoses, and estimated capillary transit times in skeletal muscle. Am. J. Physiol. 233(1), H122–H129 (1977) 10. V.V. Kislukhin, Actively circulating volume as a consequence of stochasticity within microcirculation. Appl. Math. 2(4), 508–513 (2011) 11. V.V. Kislukhin, Vasomotion model explanation for urea rebound. ASAIO J. 48(3), 296–299 (2002)

The Statistical Analysis of Protein Domain Family Distributions via Jaccard Entropy Measures R. P. Mondaini and S. C. de Albuquerque Neto

1 Introduction: The Concepts of Protein Domain Families and Clans We start from the summary of a procedure for introducing the concepts of protein domain families and family clans [1–6]. In a set of proteins, stable structures are identified which have similar 3-D structures and identical protein function. These structures are the protein domains. In Fig. 1, we depict the protein domains on four fictitious protein sequences. The amino acids of the protein domain families are represented by the symbols: ◦ × ∗ . We define partial alignments of (◦), (×), (∗), ()-protein domains of Fig. 1, which are usually called “seeds”. An alignment process will produce the “seeds” given in Figs. 2 and 3. There are 6! 6! × 2!4! 3!3! 8! 3!5! 7! 7! × 0!7! 3!4! 8! 8! × 0!8! 5!3!

6! 1!5! 8! × 4!4! 7! × 1!6! 8! × 3!5! ×

6! 0!6! 8! × 0!8! 7! × 2!5! 8! × 4!4! ×

= 15 × 20 × 6 × 1 = 1800 (◦)-seeds = 56 × 70 × 1 = 3920 (×)-seeds = 1 × 35 × 7 × 21 = 5145 (∗)-seeds = 1 × 56 × 56 × 70 = 219520 ()-seeds

like those given by Figs. 2 and 3.

R. P. Mondaini () · S. C. de Albuquerque Neto Federal University of Rio de Janeiro, Centre of Technology, COPPE, Rio de Janeiro, RJ, Brazil © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, https://doi.org/10.1007/978-3-030-46306-9_13

169

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Fig. 1 Protein domains in four fictitious protein sequences Fig. 2 Examples of “seeds” of (◦), (×), (∗), ()-protein domains as obtained from Fig. 1

Fig. 3 Other “seeds” of (◦), (×), (∗), ()-protein domains

There is an alignment corresponding to each of these “seeds”. A hidden Markov model (HMM) profile [7] (with its intrinsic thresholds of acceptance and negation) is determined for each one of these “seeds”. The best HMM profiles corresponding to the best seeds are able to recognize the (◦)-domains of the original (◦)-domain

The Statistical Analysis of Protein Domain Family Distributions via Jaccard. . .

171

Fig. 4 An example of full alignment of (◦)-domain families as obtained from Fig. 1

family of Fig. 1 within their intrinsic thresholds. The (◦)-domains of these best seeds should be tested as the similarity of their 3-D structures are concerned. The false positives do correspond to different 3-D structures. The seeds corresponding to false positives should be discarded. The domains corresponding to the remaining seeds will be considered as members of the (◦)-domain family. The association of the (◦), (×), (∗), ()-domain families is the domain family of the set of four proteins. We can repeat this procedure for any set of protein sequences and their domain families. The domains identified on these sequences through the HMM process are included in a full alignment of the corresponding family. This full alignment contains the domains already used in the seed alignment and all other non-redundant domains recognized by using the HMM process. For all the seeds of the (◦)-domain 6! families, we then have 1800 × 2!4! = 1800 × 15 = 27000 full alignments. An example is depicted in Fig. 4. These domain families are then classified hierarchically according to the comparison of 3-D structures, the comparison of protein functions, the comparison of best HMM profiles of new prospective families to the best HMM profiles of families already recognized. This procedure will lead to the recognition of clans of protein domain families. In order to comment on a practical methodology to derive the classification of domain families and their clans, let us assume that we have two sets (S1 ), (S2 ) of domain families and the corresponding sets of alignments with their HMM profiles if there is an alignment. With a profile for set (S1 ), HMM(1) and an alignment with a profile for set (S2 ), HMM(2). If HMM(1) = HMM(2), this means that these alignments will recognize protein domains of the same family. If HMM(1) recognizes some domains of family (2) and HMM (2) some domains of family (1), this means that the corresponding alignments will recognize domain families of the same clan. Otherwise, if HMM(1) does not recognize any domain from family (2) and HMM (2) any domain from family (1), this situation means that the domain families (1) and (2) will belong to different clans. The concepts introduced up to this point may not coincide with those of the literature [2, 5]. Our aim was to summarize a systematic procedure of classifying protein domains into families and clans. However, we believe that this classification should be undertaken by a consequent statistical approach and the results of the literature should be tested by

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this approach. This has been introduced on some previous publications [8, 9] and we now give an elementary introduction of the statistical techniques already adopted.

2 The Construction of the Sample Space for Statistical Analysis The statistical approach to biological events needs a construction of a sample space in order to derive results related to random variables like probabilities of occurrence and the analysis of the behaviour of the corresponding entropy measures. In order to motivate the introduction of a candidate for sample space, we consider representative games associated to the tossing of fair dice. A first game like that will consist of n consecutive tossings of m fair dice of W faces with the results to be stored on n sets of m boxes each. This process of storage is a Poisson process and it has been treated on a previous publication [10, 11]. A second game will consist of n(n−1) consecutive tossings of 2m fair dice of W faces with the results to be 2 , n(n−1) stored on 2 sets of 2m boxes. A general game will consist of nt , 1 ≤ t ≤ n tossings of t · m fair dice of W faces and the results will be stored on ,consecutive nsets of t · m boxes. We can think on the fair dice as regular polyhedra of W = 4, t 6, 8, 12, 20 faces. Nature has chosen W = 4, 20 to play the games with tetrahedra and icosahedra, respectively. A nice scientific dream would be the description of the genetic code by the analysis of unfair dice formed by gluing together faces of tetrahedra to faces of icosahedra. For the first game above, we get for the probability of finding the value a in the j th set of m boxes is pj (a) =

nj (a) , m

j = 1, 2, . . . , n;

a = 1, . . . , W

(1)

where nj (a) is the number of occurrences of the value a on the j th set of m boxes and we have W 

nj (a) = m

⇒

a=1

W 

pj (a) = 1 , ∀j

(2)

a=1

For an equiprobable probability distribution we would have W nj (a) = m , ∀j , ∀a

⇒

pj (a) =

1 , ∀j , ∀a W

(3)

A useful representation of the probabilities pj (a) is given by introducing n vectors of components pj (a) or

The Statistical Analysis of Protein Domain Family Distributions via Jaccard. . .

⎛ ⎛ ⎞ ⎞ ⎞ p1 (1) p2 (1) pn (1) ⎜ p1 (2) ⎟ ⎜ p2 (2) ⎟ ⎜ pn (2) ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ p 1 = ⎜ . ⎟ ; p 2 = ⎜ . ⎟ ; . . . ; pn = ⎜ . ⎟ ⎝ .. ⎠ ⎝ .. ⎠ ⎝ .. ⎠

173

⎛

p1 (W )

p2 (W )

(4)

pn (W )

For the second game above, the joint probability of finding the pair of values a, b in the j th and kth sets, respectively, of m boxes each is given by pj k (a, b) =

nj k (a, b) , m

a, b = 1, . . . , W

(5)

where nj k (a, b) is the number of occurrences of the pair a, b in the j th, kth sets of m boxes each and we can write W W  

nj k (a, b) = m

⇒

a=1 b=1

W W  

pj k (a, b) = 1 , ∀j, k

(6)

a=1 b=1

For an equiprobable joint probability distribution, we have W 2 nj k (a, b) = m , ∀j, k , ∀a, b

⇒

pj k (a, b) =

1 , ∀j, k , ∀a, b W2

(7)

The conditional probability of finding the values a on j th box if the values b are given a priori on kth box, pj k (a|b) is given by pj k (a, b) = pj k (a|b) pk (b)

(8)

with W 

pj k (a|b) = 1 ,

∀j, k , ∀b

(9)

a=1

which is obtained from Eqs. (2) and (6). From Bayes’ law, we can write pj k (a|b) pk (b) = pkj (b|a) pj (a)

(10)

and we have from Eq. (8), pj k (a, b) = pkj (b, a)

(11)

One can think on Bayes’ law according to the scheme of Fig. 5. From right to left on Fig. 5, we have that if we close the kth set of boxes, and we toss m dice, we obtain the distribution pj (a). We then close the j th set with the dice

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R. P. Mondaini and S. C. de Albuquerque Neto

Fig. 5 The application of Bayes’ law for tossing 2m dice in 2m boxes

inside and we open the kth set. After tossing other m dice, we obtain the distribution pj k (a, b). The same result is obtained according to Eq. (10) if we close first the j th set and we toss m dice to obtain pk (b). We then open the j th set and toss other m dice with the k set closed. A representation of the probabilities pj k (a, b) can be introduced by n(n−1) 2 square matrices pj k ⎞ ⎛ pj k (1, 1) . . . pj k (1, W ) ⎟ ⎜ .. .. .. (12) pj k = ⎝ ⎠ . . . pj k (W, 1) . . . pj k (W, W ) where j = 1, 2, . . . , n − 1 , k = j + 1, j + 2, . . . , n. In the general game, the joint probability of finding a number t of values a1 , a2 , . . ., at in the j1 th, j2 th, . . ., jt th sets, respectively, of m boxes each is given by pj1 j2 ...jt (a1 , a2 , . . . , at ) =

nj1 j2 ...jt (a1 , a2 , . . . , at ) m

(13)

We have 1≤t ≤n j1 < j2 < . . . < jt ,

a1 , a2 , . . . , at = 1, 2, . . . , W

(14) (15)

with j1 = 1, . . . , (n − t + 1) j2 = (j1 + 1), . . . , (n − t + 2) .. . jt = (jt−1 + 1), . . . , n

(16)

The Statistical Analysis of Protein Domain Family Distributions via Jaccard. . .

175

and nj1 j2 ...jt (a1 , a2 , . . . , at ) is the number of occurrences of the values a1 , a2 , . . ., at in the j1 th, j2 th, . . ., jt th sets, respectively, of m boxes each and we can write W W   a1 =1 a2 =1

...

W 

nj1 j2 ...jt (a1 , a2 , . . . , at ) = m

(17)

at =1

For an equiprobable joint probability distribution, we have (W )t nj1 j2 ...jt (a1 , a2 , . . . , at ) = m , ⇒ pj1 j2 ...jt (a1 , a2 , . . . , at ) =

∀j1 , . . . , jt , ∀a1 , . . . , at

1 (W )t

(18)

, There are then nt objects pj1 j2 ...jt and each of them has (W )t components to be given by Eq. (13). These objects are generalized joint probabilities. Their probabilistic nature can be confirmed from the successive application of Bayes’ law, Eq. (10), or pj1 j2 ...jt−1 jt (a1 , a2 , . . . , at−1 |at ) pjt (at ) = pjt j1 j2 ...jt−1 (at |a1 , a2 , . . . , at−1 ) pj1 j2 ...jt−1 (a1 , a2 , . . . , at−1 )

(19)

where pj1 j2 ...jt (a1 , a2 , . . . , at−1 |at ) stands for the generalized conditional probability of occurrences of the ordered set of symbols a1 , a2 , . . ., at−1 in columns j1 , j2 , . . ., jt−1 , respectively, if the symbol at is allocated a priori in column jt . We can write, analogously to Eq. (8), pj1 j2 ...jt−1 jt (a1 , a2 , . . . , at−1 , at ) = pj1 j2 ...jt−1 jt (a1 , a2 , . . . , at−1 |at ) pjt (at ) (20) We then have from Eqs. (19) and (20): W 

pj1 j2 ...jt−1 jt (a1 , a2 , . . . , at−1 , at )

at =1

=

W 

pj1 j2 ...jt−1 jt (a1 , a2 , . . . , at−1 |at ) pjt (at )

at =1

=

W 

pjt j1 j2 ...jt−1 (at |a1 , a2 , . . . , at−1 ) pj1 j2 ...jt−1 (a1 , a2 , . . . , at−1 )

at =1

= pj1 j2 ...jt−1 (a1 , a2 , . . . , at−1 )

(21)

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Fig. 6 Protein domain families as obtained from protein domains of Fig. 1 by the full alignment process of Fig. 4

where in the last equality we have used the generalization of Eq. (9): W 

pjt j1 j2 ...jt−1 (at |a1 , a2 , . . . , at−1 ) = 1 , ∀j1 , j2 , . . . , jt−1 , jt at =1 ∀a1 , a2 , . . . , at−1 , at

(22)

In order to continue our construction of the sample space, we remove all the gaps introduced by the full alignment process as depicted in Fig. 4 and we obtain for (◦), (×), (∗), ()-protein domain families the representative regions of Fig. 6: We now consider rectangular regions of m × n amino acids, with m = 5 (rows) and n = 3 (columns) as the representative of protein domain families of Fig. 6. We call these regions “5 × 3 blocks”. In general we consider at least one m × n block as the representative of each protein domain family. The construction of the m × n blocks will follows the guideline given in Ref. [9]: Let us suppose that a protein domain family has m rows with nL amino acids, L = 1, 2, . . . , m on the Lth row. A representative block of this protein domain family will be obtained as a member of the sample space by discarding firstly all the rows such that nL < n. We then discard (nL − n) amino acids for the rows with nL > n. In Fig. 7, we give an example of block formation for the case W = 20 where the symbol a in Eq. (1) or the symbols a1 , . . . , at in Eq. (13) will run over the one-letter code for the amino acids, i.e., or a1 , . . . , at = A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y. After applying this rule to the protein domain families of Pfam 27.0 database [12] we get the generic structure of Fig. 7. Probabilities of occurrence of amino acids based on 1, 2, . . . , t columns pj1 j2 ...jt (a1 , a2 , . . . , at ) as defined in Sect. 2, Eqs. (1)–(16), will be then calculated and listed for the subsequent calculation of generalized entropy measures to be used on the generalized statistical analysis of the research programme to be undertaken in the present work.

The Statistical Analysis of Protein Domain Family Distributions via Jaccard. . .

177

Fig. 7 A (m × n) block as a representative of a protein domain family. The protein domains in red and the remaining part of protein domains in blue should be discarded

Fig. 8 Values of entropy measures associated to the j th columns of (m × n) representative ϕl blocks for the lth clan

3 The Fisher–Snedecor Distribution as Applied to Protein Domain Families and Clans: A Theoretical Derivation Let us assume that protein domain families are associated on N clans. The lth clan, l = 1, . . . , N , has ϕl protein domain families. This lth clan does correspond to ϕl blocks of size (m × n) as depicted in Fig. 7. We have at least one block to represent one of the ϕl families in the sample space. In order to fix ideas as the characterization of the entropy measure distribution is concerned, we take t = 1 in Eq. (20), which means that we are now considering only the entropy measure distribution of amino acids per column of the (m × n) blocks and we associate the values of these entropy measures Sj p ϕl , j = 1, . . . , n, p = 1, . . . , ϕl to them. In Fig. 8 we show the scheme of association of entropy values Sj p ϕl to the j th columns for the representative (m × n) blocks of the ϕl families. The entropy measures Sj p ϕl to be adopted in the subsequent development are assumed to be normally distributed with mean Sj ϕl  and variance σj2ϕl , or

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R. P. Mondaini and S. C. de Albuquerque Neto

NSj ϕ  , σ 2 (Sj p ϕl ) = . l

j ϕl

1 2π σj2ϕl

e

−(Sj p ϕl −Sj ϕl )2 /2σj2 ϕ

l

(23)

There is a global maximum value for the entropy which corresponds to an equiprobable probability distribution on each column. Its dependence on W , the number of available states, is then given by the special form of this entropy. We call the local maximum Max Sj p ϕl and we write 0 ≤ Sj p ϕl ≤ Max Sj p ϕl

(24)

We now define the variable λj p ϕl by λj p ϕl ≡ Sj p ϕl − Sj ϕl 

(25)

− Sj ϕl  ≤ λj p ϕl ≤ Max Sj p ϕl − Sj ϕl 

(26)

and we have

We then consider the normal distribution of null mean and equal variances, N0 , σ 2 (λj p ϕl ) = √

1 2π σ 2

e

−

(λj p ϕ )2 l 2σ 2

(27)

The cumulative distribution function (cdf) of the probability density functions (pdf) [13] of Eq. (27) is given by  cdf(j ϕl ) = P (j ϕl ≤ λj p ϕl ) =

λj p ϕl

−∞

N0 , σ 2 (y) dy

(28)

where j ϕl is the random variable and λj p ϕl the values which it assumes. From Eq. (28), the Eq. (27) can be also written as N0 , σ 2 (λj p ϕl ) =

d cdf(j ϕl ) dλj p ϕl

(29)

We now introduce the “χ -square” distribution [13] through the definition of a new random variable: Xj ϕl ≡ (j ϕl )2 The associated cumulative distribution function is then given by

(30)

The Statistical Analysis of Protein Domain Family Distributions via Jaccard. . .

179

, , , cdf Xj ϕl = (j ϕl )2 = P Xj ϕl ≤ χj p ϕl = P (j ϕl )2 ≤ χj p ϕl , √ √ = P − χj p ϕl ≤ j ϕl ≤ χj p ϕl  √χ j p ϕ l = N0 , σ 2 (y) dy =2

√ − χj p ϕl  √χ j p ϕ l 0

N0 , σ 2 (y) dy

(31)

where we have used Eq. (28) and Xj ϕl stands for the random variable with χj p ϕl as the values which it assumes. , The probability density function (pdf) associated to cdf Xj ϕl = (j ϕl )2 is then given by

γX

j ϕl

, d cdf Xj ϕl = (j ϕl )2 dχj p ϕl  √χj p ϕ l d =2 N0 , σ 2 (y) dy dχj p ϕl 0 √  √χj p ϕ d χj p ϕ l l d =2 √ N0 , σ 2 (y) dy d χj p ϕl dχj p ϕl 0

(χj p ϕl ) =

1 √ =√ N 2 ( χj p ϕ l ) χj p ϕ l 0 , σ

(32)

Or, from Eqs. (27) and (30):

γX

j ϕl

(χj p ϕl ) = √

1 2π σ 2

√

χj p ϕ l 1 − e 2σ 2 ; χj p ϕ l

0 ≤ χj p ϕ l < ∞

(33)

Eq. (33) is a gamma distribution [14]: 1 α β ωβ−1 e−αω , α > 0 , β > 0 γ α β (ω) = !(β)

(34)

where α=

1 1 , β= 2 2σ 2

(35)

and  !(β) =

∞

ωβ−1 e−ω dω ,

0

is the Euler Gamma function.

β > 0,

!(1/2) =

√

π

(36)

180

R. P. Mondaini and S. C. de Albuquerque Neto

We follow the same argument as above and we introduce a distribution Zj ϕl to be given by the sum of two independent χ -square distributions, with random variables X1 j ϕl and X2 j ϕl : Zj ϕl = X1 j ϕl + X2 j ϕl = (1 j ϕl )2 + (2 j ϕl )2

(37)

The associated cdf is given by , (38) cdf Zj ϕl = (1 j ϕl )2 + (2 j ϕl )2 = P (X1 j ϕl + X2 j ϕl ≤ zj p ϕl )  ∞  zj p ϕ −χ1 j p ϕ l l = X1 j ϕl (χ1 j p ϕl ) X2 j ϕl (χ2 j p ϕl )dχ2 j p ϕl dχ1 j p ϕl

γ

0

γ

0

The corresponding pdf is then obtained from

γZ



d

j ϕl (zj p ϕl ) =

dzj p ϕl

∞

γX

1 j ϕl

0

1 j ϕl

(χ1 j p ϕl )

(χ2 j p ϕl )dχ2 j p ϕl dχ1 j p ϕl

2 j ϕl

=

γX

0

0

γX

· 

∞  zj p ϕl −χ1 j p ϕl

(χ1 j p ϕl )

γX

2 j ϕl

(zj p ϕl − χ1 j p ϕl )dχ1 j p ϕl (39)

Since the pdf ∞, the pdfs

γZ

j ϕl

γX γX

j ϕl

(χj p ϕl ) of Eq. (33) is defined on the interval 0 ≤ χj p ϕl