Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics: Selected Works from the BIOMAT Consortium Lectures, Rio de Janeiro, Brazil, 2022 9783031330506, 3031330501

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Rubem P. Mondaini Editor

Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics Selected Works from the BIOMAT Consortium Lectures, Rio de Janeiro, Brazil, 2022

Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics

Rubem P. Mondaini Editor

Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics Selected Works from the BIOMAT Consortium Lectures, Rio de Janeiro, Brazil, 2022

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

ISBN 978-3-031-33049-0 ISBN 978-3-031-33050-6 https://doi.org/10.1007/978-3-031-33050-6

(eBook)

Mathematics Subject Classification: 92Bxx, 92-08, 92-10 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed 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 Paper in this product is recyclable.

Preface

A BIOMAT International Series book is a collection of papers that have been selected for presentation in technical sessions at a BIOMAT Symposium, following a peer review analysis by members of the Editorial Board and other international senior reviewers. Another peer review is then performed to choose among the papers presented, which ones will be published as chapters of a BIOMAT book. We have held 22 international conferences of the BIOMAT series since the first one in 2001, held in Rio de Janeiro, Brazil. The last three, in 2020, 2021, 2022, had to be held online, either for security reasons during the pandemic or due to economic constraints generated by it. The traditional pattern of BIOMAT conferences was however fully maintained: no parallel sessions; constant recommendation to participants to attend to the lectures of all their colleagues, in the best performance of a truly interdisciplinary conference; about 40 plenary lectures delivered in 5 conference days, between 9:00am and 5:00pm: Annual Plenary Meeting of BIOMAT Consortium and Meeting of the Consortium Board of Directors, representing the different regions of the Consortium: Western Europe, Eastern Europe, North America, South America, Africa, Asia and Oceania. For this 3rd online meeting of BIOMAT Consortium, The International Symposium BIOMAT 2022, held with headquarters in Rio de Janeiro, from 07 to 11 November 2022, we had again the competent support of RNP – National Research Network, with the assistance of Dr. Beatriz Zoss, to whom once more we express our gratitude on behalf of BIOMAT Consortium. The president of the BIOMAT Consortium also express his thanks for the collaboration in the organization to professors of the Institute for High-Performance Computing and Networking (ICAR) from Naples, Italy, Professors Lucia Maddalena and Ilaria Granata. We thank Dr. Simão C. de Albuquerque Neto and Carmem Lúcia S.C. Mondaini for their always competent and helpful assistance in all stages of the conference. Rio de Janeiro, Brazil November 11, 2022

Rubem P. Mondaini

v

Editorial Board of the BIOMAT Consortium

Rubem Mondaini (Chair) Adelia Sequeira Alain Goriely Alan Perelson Alexander Grosberg Alexei Finkelstein Ana Georgina Flesia Alexander Bratus Avner Friedman Carlos Condat Denise Kirschner David Landau De Witt Sumners Ding Zhu Du Dorothy Wallace Eytan Domany Ezio Venturino Fernando Cordova-Lepe Fred Brauer Gergely Röst Hamid Lefraich Helen Byrne Jacek Miekisz Jack Tuszynski Jane Heffernan Jerzy Tiuryn John Harte John Jungck Karam Allali Kazeem Okosun Kristin Swanson

Federal University of Rio de Janeiro, Brazil Instituto Superior Técnico, Lisbon, Portugal University of Oxford, Mathematical Institute, UK Los Alamos National Laboratory, New Mexico, USA New York University, USA Institute of Protein Research, Russia Universidad Nacional de Cordoba, Argentina Lomonosov Moscow State University, Russia Ohio State University, USA Universidad Nacional de Cordoba, Argentina University of Michigan, USA University of Georgia, USA Florida State University, USA University of Texas, Dallas, USA Dartmouth College, USA Weizmann Institute of Science, Israel University of Torino, Italy Catholic University del Maule, Chile University of British Columbia, Vancouver, Canada University of Szeged, Hungary University Hassan First, Morocco University of Nottingham, UK University of Warsaw, Poland University of Alberta, Canada York University, Canada University of Warsaw, Poland University of California, Berkeley, USA University of Delaware, Delaware, USA University Hassan II, Mohammedia, Morocco Vaal University of Technology, South Africa University of Washington, USA vii

viii

Lisa Sattenspiel Louis Gross Lucia Maddalena Ludˇek Berec Maria Vittoria Barbarossa 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 Samares Pal Sandip Banerjee Seyed Moghadas Siv Sivaloganathan Sándor Kovács Somdatta Sinha Suzanne Lenhart Vitaly Volpert William Taylor Yuri Vassilevski Zhijun Wu

Editorial Board of the BIOMAT Consortium

University of Missouri-Columbia, USA University of Tennessee, USA High Performance Computing and Networking Institute, ICAR—CNR, Naples, Italy Biology Centre, ASCR, Czech Republic Frankfurt Inst. for Adv. Studies, Germany 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 University of Kalyani, India Indian Institute of Technology Roorkee, India York University, Canada Centre for Mathematical Medicine, Fields Institute, Canada Eötvos Loránd University, Hungary 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

Dynamics of an SIS Epidemic Model with No Vertical Transmission . . . . . . Sándor Kovács, Szilvia György, and Noémi Gyúró

1

Infection Spread in Populations: An Agent-Based Model . . . . . . . . . . . . . . . . . . . Adarsh Prabhakaran and Somdatta Sinha

17

Network-Based Computational Modeling to Unravel Gene Essentiality . . I. Granata, M. Giordano, L. Maddalena, M. Manzo, and M. R. Guarracino

29

Nonlinear Dynamics in an SIR Model with Ratio-Dependent Incidence and Holling Type III Treatment Rate Functions . . . . . . . . . . . . . . . . . Akriti Srivastava and Prashant K. Srivastava

57

Comparative Study of Deterministic and Stochastic Predator Prey System Incorporating a Prey Refuge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anal Chatterjee and Samares Pal

73

Mathematical Modeling and Numerical Analysis of HIV-1 Infection with Long-Lived Infected Cells During Combination Therapy and Humoral Immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zakaria Hajhouji, Majda El Younoussi, Khalid Hattaf, and Noura Yousfi

99

A Reaction-Diffusion Fractional Model for Cancer Virotherapy with Immune Response and Hattaf Time-Fractional Derivative . . . . . . . . . . . 125 Majda El Younoussi, Zakaria Hajhouji, Khalid Hattaf, and Noura Yousfi A Review of Stochastic Models of Neuronal Dynamics: From a Single Neuron to Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 M. F. Carfora Modeling the Impact of Media Coverage on the Spread of Infectious Diseases: The Curse of Twenty-First Century . . . . . . . . . . . . . . . . . . . . 153 Anal Chatterjee and Suchandra Ganguly

ix

x

Contents

Cultural and Biological Transmission: A Simple Case of Evolutionary Discrete Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Roberto Macrelli, Margherita Carletti, and Vicenzo Fano The Maximal Extension of the Strict Concavity Region on the Parameter Space for Sharma-Mittal Entropy Measures: II . . . . . . . . . . . . . . . . 181 R. P. Mondaini and S. C. de Albuquerque Neto An Eco-Epidemic Predator-Prey Model with Selective Predation and Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Sasanka Shekhar Maity, Pankaj Kumar Tiwari, Nanda Das, and Samares Pal Epidemic Patterns of Emerging Variants with Dynamical Social Distancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Golsa Sayyar and Gergely Röst On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates and Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Karam Allali Clustering of Countries Based on the Associated Social Contact Patterns in Epidemiological Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Evans Kiptoo Korir and Zsolt Vizi Multiple Predation on Prey Herding and Counteracting the Hunting . . . . . 273 Luca Bondi, Jacopo Ferri, Nicolò Giordanengo, and Ezio Venturino Benefits of Application of Process Optimization in Pharmaceutical Manufacturing: A Panoramic View . . . . . . . . . . . . . . . . . . . . . . . . 291 Antonios Fytopoulos and Panos M. Pardalos A Web-Based Non-invasive Estimation of Fractional Flow Reserve (FFR): Models, Algorithms, and Application in Diagnostics . . . . . . 305 Yuri Vassilevski, Timur Gamilov, Alexander Danilov, German Kopytov, and Sergey Simakov Perturbing Coupled Multivariable Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 A. Mukhopadhyay, Ganesh Bagler, and Somdatta Sinha Analysis of Covid-19 Dynamics in Brazil by Recursive State and Parameter Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Daniel Martins Silva and Argimiro Resende Secchi Computational Modeling of Membrane Blockage via Precipitation: A 2D Extended Poisson-Nernst-Planck Model . . . . . . . . . . . . . . . 373 H. Lefraich Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

Dynamics of an SIS Epidemic Model with No Vertical Transmission Sándor Kovács, Szilvia György, and Noémi Gyúró

1 Introduction A good 7 years ago, the World Health Organization (WHO) proposed the improvement of HIV/AIDS epidemiological rates worldwide through disease prevention and treatment as a goal to achieve. Among such goals was reaching the virtual worldwide eradication of vertical transmission, i.e., the prevention of mother-tochild transmission of the disease (cf. [21]). On the homepage of the WHO, one can also read that HIV continues to be a major global public health issue, having claimed 36.3 million [27.2–47.8 million] lives so far, but for opportunistic infections, HIV infection has become a manageable chronic health condition, enabling people living with HIV to lead long and healthy lives. From this reason, education plays an important role in slowing the spread of the virus. In modeling the spread of infectious diseases, the population is usually described by compartmental models, i.e., it is considered to be subdivided into disjoint epidemiological classes (compartments) of individuals in relation to the infectious disease: susceptible individuals, exposed individuals, infectious individuals, and removed individuals (cf. [4]). The development of the infection is represented by transitions between these classes. The number of compartments included depends on the disease being modeled. If we take into account that the acquired immunity to reinfection is virtually nonexistent and hence recovered individuals pass directly back to the corresponding susceptible class, then we deal with so-called SIS models (cf. [11, 15]).

S. Kovács () · S. György Department of Numerical Analysis, Eötvös Loránd University, Budapest, Hungary e-mail: [email protected] N. Gyúró Eötvös Loránd University, Budapest, Hungary © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_1

1

2

S. Kovács et al.

Basing on the ideas from [16] (cf. [22]) and from [17], we propose the model

.

⎫ aSI + βI − ψS − δS S =: f1 (S, E, I ), ⎪ S˙ = λ − ⎪ ⎪ ⎪ S+I ⎪ ⎪ ⎪ ⎬ ˙ E = ψS + κI − δE E =: f2 (S, E, I ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ aSI ⎪ ⎭ − κI − βI − δI I =: f3 (S, E, I ) I˙ = S+I

(1)

to describe the spread of an infectious disease. Herein, the dot means differentiation with respect to time t; .S(t) ≥ 0 and .I (t) ≥ 0 denote the number or density of the uninfected (susceptible) and infected hosts at time t, respectively. The independent variable .E ≥ 0 represents the number or density of hosts who are vaccinated or educated—by, e.g., a disease management program—according to whether the disease is considered as curable or noncurable, respectively (cf. [12, 13]). It is supposed that the death rates of the three compartments are not necessarily identical, i.e., susceptibles, educated, and infectives have .δi > 0 .(i ∈ {S, E, I }) as their death rates, respectively. This is confirmed, for instance, by the fact that hosts in the infective compartment can have more chance to die, or, e.g., in the case of a sexually transmitted disease through their illness, to become sexually inactive which leads them outcoming from the system. The birth rate .λ > 0 is considered different from the death rates; all newborns are susceptible and the education is successful, that is, the hosts in the educated compartment do not transmit the infection anymore. The parameters .a > 0, .β > 0, .ψ ≥ 0, and .κ > 0 are the average number of contacts per infective per time unit (the transmission coefficient from the infective compartment to the susceptible one), the backflow rate of healed hosts returning to the susceptible compartment, the educational rate of the susceptibles (i.e., the rate of the direct transmission from the susceptible compartment to the educated one), and the educational rate of the infected hosts (the rate of the direct transmission from the infected compartment to the educated one) for which .κ ≤ a is to be assumed. The disease transmission is assumed to be standard incidence term .

and no vertical transmission.

aSI S+I

(2)

Dynamics of an SIS Epidemic Model with No Vertical Transmission

aSI S+I

λS

δS S

3

S

I

δI I

βI κI

ψS E δE E

The aim of the present chapter is to give a detailed analysis of dynamical properties of (1). After showing the biological feasibility of the system, we perform a sensitivity analysis on the so-called basic reproduction number and show that forward bifurcation takes place: besides a unique uninfected equilibrium, an endemic equilibrium emerges if one of the system parameters crosses a critical value (cf. [1]). At the end, we show that the local stability of the equilibria implies their global stability.

2 The Biological Feasibility of the Model Due to the smoothness of the right hand side of (1), the existence and uniqueness of the appropriate initial value problem is ensured. In order to analyze the qualitative properties of the possible equilibria, one has to show that system (1) is biologically feasible. This means that the interior of the positive octant of the phase space .[S, E, I ] should be an invariant region and the solutions of (1) should exist for every .t > 0. Lemma 2.1 All solutions of (1) with positive initial conditions .S(0) > 0, .E(0) > 0 and .I (0) > 0 remain positive for all .t ≥ 0 in their domain of existence. Proof By uniqueness of solutions, since .I ≡ 0 is a solution of the third equation of (1), no solution with .I (t) > 0 at any time .t ≥ 0 can become zero in finite time. Let us assume contrary to the statement that there exists .t > 0 at which .S(t) is equal to zero and denote t ∗ := min{t > 0 : S(t) = 0}.

.

4

S. Kovács et al.

Then from the first equation of (1) we have ∗ ∗ ˙ ∗ ) = λ − aS(t )I (t ) + βI (t ∗ ) − (ψ + δS )S(t ∗ ) = λ + βI (t ∗ ) > 0. S(t S(t ∗ ) + I (t ∗ )

.

˙ ∗ ) ≤ 0, which is a contradiction Since .S(0) > 0, for .S(t ∗ ) = 0, we must have .S(t by continuity. Thus, .S(t) > 0 for all .t > 0. Similarly, the same is true for E.   We consider now system (1) restricted to .R3+ and show that all solutions stay bounded in .t ∈ [0, +∞) which implies the existence of solutions for every .t > 0. Lemma 2.2 All solutions of (1) which are initiated in .R3+ are uniformly bounded. Proof We define the function σ (S, E, I ) := S + E + I

.

(S, E, I > 0).

The time derivative along a solution of (1) is σ˙ (S, E, I ) = S˙ + E˙ + I˙ = f1 (S, E, I ) + f2 (S, E, I ) + f3 (S, E, I ) = .

= λ − δS S − δE E − δI I. For any .μ > 0 we have σ˙ (S, E, I ) + μσ (S, E, I ) = λ + (μ − δS )S + (μ − δE )E + (μ − δI )I.

.

(3)

If we choose .μ < min{δS , δE , δI }, the right-hand side of (3) is bounded in .R3+ . Then we find .K > 0 with σ˙ (S, E, I ) + μσ (S, E, I ) ≤ K.

.

Multiplying both sides of this inequality by the factor .eμt .(t ∈ [0, +∞)), we have .

σ˙ (S(t), E(t), I (t)) · eμt + μσ (S(t), E(t), I (t)) · eμt ) ≤ K · eμt   d μt ) σ (S(t),E(t),I (t))·e ( dt

(t ∈ [0, +∞)).

Integrating on both sides, we can solve the above inequality as follows. From Newton-Leibniz theorem, it follows that for every .t ∈ [0, +∞) σ (S(t), E(t), I (t)) · eμt − σ (S(0), E(0), I (0)) ≤

.

which is the same as

 K K μt e − 1 ≤ eμt μ μ

Dynamics of an SIS Epidemic Model with No Vertical Transmission

0 ≤ σ (S(t), E(t), I (t)) ≤

.

5

K + σ (S(0), E(0), I (0)) · e−μt μ

(t ∈ [0, +∞)).

As .t → +∞ we have 0 ≤ σ (S, E, I ) ≤

.

K +ε μ

for any

ε > 0.

Therefore, all trajectories initiated in .R3+ enter the region

:= (S, E, I ) ∈ R3+ : σ (S, E, I ) + ε, for any ε > 0 .

.

 

3 Basic Reproduction Ratio By solving the equation .f(S, E, I ) = 0 for .Sb , Eb , Ib where .f := (f1 , f2 , f3 ), it is easy to see that system (1) has a unique equilibrium on the boundary of the positive octant of the phase space .[S, E, I ] (uninfected equilibrium)  Eb := (Sb , Eb , Ib ) :=

.

 λψ λ ,0 , δS + ψ δE (δS + ψ)

for all parameter values. Indeed, if .Ib = 0, then the first equation in (1) implies that .Sb = λ/(δS + ψ) and the second equation implies that .Eb = λψ/(δE (δS + ψ)). The local stability of this disease-free (uninfected) equilibrium can be settled by calculating the so-called basic reproductive number .R0 (cf. [9]). This number can be interpreted as “the expected number of secondary cases produced, in a completely susceptible population, by a typical infective host” (cf. [2, 7]). The importance of .R0 in the spreading of disease is related to its value (cf. [3]). If .R0 < 1, then on average an infected host produces less than one new infected host over the course of its infectious period, and the infection cannot grow. In this case the uninfected equilibrium is locally asymptotically stable, and the disease cannot invade the population. But if .R0 > 1, then each infected host produces, on average, more than one new infection. In this case the uninfected equilibrium is unstable and invasion is always possible (cf. [14]). We shall calculate this threshold parameter using the new generation matrix method (cf. [9]). Let us therefore rewrite (1) in form of  E, I ) − G(S,  E, I ) ˙ E, ˙ I˙) =: F(S, (S,

.

where

6

S. Kovács et al.

⎡

⎤ 0  E, I ) := ⎣ 0 ⎦ .F(S, aSI S+I

and ⎡

⎤ aSI −λ + S+I − βI + (ψ + δS )S  E, I ) := ⎣ ⎦ .G(S, −ψS − κI + δE E (κ + β + δI )I  and .G  at .Eb are as functions of the variables S, E, and I . Then the Jacobians of .F ⎡

⎤ 000  b ) := ⎣ 0 0 0 ⎦ .D F(E 00a

⎡

⎤ ψ + δS 0 a−β  b ) := ⎣ −ψ δE ⎦. D G(E −κ 0 0 κ + β + δI

and

 b ) and .D G(E  b ) are partitioned as The matrices .D F(E   O 0  .D F(Eb ) =: OF

and

  J1 J2  D G(Eb ) =: O G

where 3 (Eb ) = a F := ∂3 F

and

.

3 (Eb ) = κ + β + δI . G := ∂3 G

Thus, the spectral abscissa of the next generation matrix .F G−1 (cf. [8]) is   R0 := ρ F G−1 =

.

a . κ + β + δI

(4)

In order to understand the importance of the parameters of system (1) on the value of .R0 , we perform sensitivity analysis of .R0 with respect to all parameters, i.e., we analyze the variation of it assuming that the parameter values change. We recall that the sensitivity of a variable x which depends differentiably on a parameter p can be calculated (cf. [19]) as px :=

.

Thus,

p ∂x · . x ∂p

Dynamics of an SIS Epidemic Model with No Vertical Transmission

7

λR0 =

λ(κ + β + δI ) λ ∂R0 = ·0 · R0 ∂λ a

= 0,

aR0 =

a(κ + β + δI ) 1 a ∂R0 = · · R0 ∂a a κ + β + δI

= 1,

βR0 =

−β β ∂R0 β(κ + β + δI ) −a · = , = · R0 ∂β a κ + β + δI (κ + β + δI )2

ψR0 =

ψ ∂R0 ψ(κ + β + δI ) · = ·0 R0 ∂ψ a

κR0 =

κ(κ + β + δI ) −a −κ κ ∂R0 = · = , · a κ + β + δI R0 ∂κ (κ + β + δI )2

δRS 0 =

δS ∂R0 δS (κ + β + δI ) · = ·0 R0 ∂δS a

= 0,

δRE 0 =

δE (κ + β + δI ) δE ∂R0 · = ·0 R0 ∂δE a

= 0,

δRI 0 =

−a δI ∂R0 δI (κ + β + δI ) −δI · · = = . 2 R0 ∂δI a κ + β + δI (κ + β + δI )

= 0,

.

For the values of the parameters used in (1), the following statements can be made: • the sensitivity index . aR0 is unit which means that an increase on the transmission coefficient a of .1% will result in an increase on .R0 .1%; • the sensitivity indices . ψR0 , . δRS 0 and . δRE 0 are zero which reflect the fact that .R0 doesn’t depend on the parameters .ψ, .δS , and .δE ; • the sensitivity indices . βR0 , . κR0 , and . δRI 0 are negative which has the consequence that the ratio .R0 decreases increasing the parameters: the backflow rate of the healed hosts .β, the educational rate of the susceptibles .κ, and the death rate of the infected hosts .δI . By considering the sensitivity indices in the above table, we conclude that the most sensitive parameter is the death transmission coefficient a from the infective compartment to the susceptible one. Considering Theorem 2. in [9] (cf. [7]) or calculating the Jacobian of the righthand side of (1), we can derive the following result regarding the local stability of the uninfected equilibrium .Eb . Theorem 3.1 The uninfected equilibrium .Eb is locally asymptotically stable if R0 < 1, but unstable if .R0 > 1, where .R0 is defined by (4).

.

8

S. Kovács et al.

Proof The Jacobian matrix of the left-hand side of (1) is ⎡ ⎢ Jf (S, E, I ) := ⎣

−aI 2 (S+I )2

− ψ − δS ψ

.

aI 2 (S+I )2

0 −δE 0

2

aS +β − (S+I )2 aS 2 (S+I )2

⎤

⎥ κ ⎦. − κ − β − δI

(5)

Evaulating .Jf at .Eb we have ⎤ −ψ − δS 0 −a + β ⎦. .Jf (Eb ) = ⎣ κ ψ −δE 0 0 a − κ − β − δI ⎡

Clearly, .Jf (Eb ) is a block triangular matrix and the diagonal elements are eigenvalues of the matrix. Thus, .Eb is locally asymptotically stable if and only if a − κ − β − δI < 0

.

⇐⇒

κ > a − β − δI ,

(6)

i.e., .R0 < 1 holds. If .R0 > 1 holds, then .Jf (Eb ) has a positive eigenvalue which implies that it is unstable.   Condition (6) shows that education or vaccination of the infected hosts leads to stabilization of the disease-free equilibrium, whereas the education of the susceptibles has no effect on stability of this equilibrium. Theorem 3.1 shows that .R0 is a threshold parameter for this model. In order to know what happens if .R0 is near one, we need further investigations. The following analysis of the local center manifold (cf. [6, 9]) yields the existence and local stability of a super-threshold endemic equilibrium for .R0 near one. Because .R0 is often inconvenient to use directly as a bifurcation parameter (cf. [9]), we use the parameter .κ as a bifurcation parameter and introduce .κ ∗ := a − β − δI corresponding to .R0 = 1. Clearly, ⎧ ⎨ < κ∗ .κ = κ∗ ⎩ > κ∗

⇐⇒ R0 > 1, ⇐⇒ R0 = 1, ⇐⇒ R0 < 1.

Denoting the Jacobian of the right-hand side of (1) evaluated at the critical value .κ ∗ and the boundary equilibrium .Eb by .A, i.e., ⎡

⎤ −ψ − δS 0 −a + β .A := ⎣ ψ −δE a − β − δI ⎦ , 0 0 0 one can see that the zero eigenvalue of .A is simple and the other two eigenvalues have negative real part. It is also easy to calculate that

Dynamics of an SIS Epidemic Model with No Vertical Transmission

⎡

⎤ (β − a)/(δS + ψ) .q := ⎣ −(δS (β − a) + δI (δS + ψ))/(δE (δS + ψ)) ⎦ , 1

9

⎡ ⎤ 0 ⎣ p := 0 ⎦ 1

are the right and left null vectors of .A (right and left eigenvectors corresponding to the zero eigenvalue), i.e., .Aq = 0, .AT p = 0 such that pT q ≡ p, q = 1

.

holds. Let μ :=

.

 ν := p,  B(q)

p, B(q, q) , 2

B : R3 → R3 are given by where the functions .B : R3 × R3 → R3 , . " " 3 2 ∗ ! ∂ Fi (ξ , κ ) "" .Bi (x, y) := ∂ξj ∂ξk "" j,k=1

xj yk

(i ∈ {1, 2, 3}),

ξ =Eb

" " 3 2 ∗ ! ∂ Fi (ξ , κ ) "" i (x) := .B ∂ξj ∂κ "" j =1

xj

(i ∈ {1, 2, 3})

ξ =Eb

where the right-hand side of (1) will be denoted by .(F1 , F2 , F3 )(S, E, I, κ) instead of .(f1 , f2 , f3 )(S, E, I ) in order to emphasize the dependence of the bifurcation parameter .κ. Because .p1 = 0 = p2 , the derivatives of .F1 and .F2 are not needed to calculate. All second-order derivatives of .F3 are zero, except for .

∂ 2 F3 (Eb , κ ∗ ) 2aS 2 = −2a(δS + ψ), =− 2 (S + I )3 ∂I

∂ 2 F3 (Eb , κ ∗ ) = −1. ∂I ∂κ

Hence μ=−

.

a(δS + ψ) 2 a(δS + ψ) · q3 = − 0 such that for .1 < R0 < δ system (1) has at least one locally asymptotically stable endemic equilibrium, i.e., a transcritical bifurcation takes place which is forward meaning that there is a transfer of stability from the infection-free steady state to the endemic equilibrium and vice versa (cf. [5]). The coordinates of an endemic equilibrium of (1) are determined by solving .f(S, E, I ) = 0 for .Se = 0, Ee = 0, Ie = 0. From the third equation, we get

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S. Kovács et al.

a − κ − β − δI · Se κ + β + δI

Ie =

.

which is biologically feasible for .Se > 0 (.Ie > 0) if and only if κ < a − β − δI

κ < κ∗

i.e.

.

⇐⇒

R0 > 1

(7)

holds. Substituting this in the first equation, one can see that under assumption (7) we have a unique positive .Se for which the linear expression

λ(1 + ω) + (β − a)ω + βω2 − (ψ + δS )(1 + ω) · Se = 0

.

holds, where ω :=

.

a − κ − β − δI . κ + β + δI

Direct calculation shows that under condition (7) (β − a)ω + βω2 − (ψ + δS )(1 + ω) .

= (−a) ·

(δI + κ)(a − β − δI − κ) + (δS + ψ)(β + δI + κ) κ ∗ holds, then .Jf (Eb ) has only negative eigenvalues which has (2D) is a locally asymptotically stable equilibrium of (8). In the consequence that .Eb (2D) is no longer asymptotically stable, case of .κ < κ ∗ , the boundary equilibrium .Eb (2D) emerges which is locally asymptotically and a new endemic equilibrium .Ee stable, because (2D)

12

S. Kovács et al. (2D)

Tr(J

.

(E(2D) )) = κ − a + β + δI − (ψ + δS ) < 0 e

and (2D)

det(J

(E(2D) ))) = e

.

1 a

× (a − β − δI − κ) {(δI + κ)(a − β − δI − κ) + (β + δI + κ)(δS + ψ)} is positive. We summarize the above results in the following. Theorem 4.1 If (1) .κ > κ ∗ then system (8) has only one equilibrium .Eb on the boundary of the phase space .[S, I ] which is locally asymptotically stable, and (2) the factor .κ ∗ satisfies .κ ∗ ≥ κ, then a new (endemic) equilibrium .Ee bifurcates from .Eb (as .κ crosses the value .κ ∗ ) and becomes the one locally asymptotically stable as .κ ∗ > κ, whereas .Eb is a repeller as .κ ∗ > κ. We are going to extend this local result to a global one, by showing that system (8) (and hence system (1)) has no nontrivial periodic solution. In treating this problem, we use the Poincaré criterion (cf. [18, 20]). Theorem 4.2 System (8) has no nontrivial periodic orbit. Proof Assume that (8) admits a periodic solution .(S, I ) : R → R2 with period .T > 0. It is well known that one of the characteristic multipliers is 1 (cf. [10]) and the other is given by %



T

ρ := exp

div (S(t), I (t)) dt ,

.

0

where div (S, I ) = Tr(J (S, I )) =

aS 2 −aI 2 − ψ − δS + − κ − β − δI 2 (S + I ) (S + I )2

.

=a·

S−I − (ψ + δS + κ + β + δI ) . S+I

By periodicity we have with .u ∈ {S, I } %

T

0 = ln(u(T )) − ln(u(0)) =

.

0

u(t) ˙ dt. u(t)

Dynamics of an SIS Epidemic Model with No Vertical Transmission

13

From (8) we obtain .

I˙ aS − (κ + β + δI ) . = S+I I

Thus %

T

.

0

aS = (κ + β + δI ) T S+I

and therefore % .

T

%

T

div (S(t), I (t)) dt = −

0

0

aI − (ψ + δS ) T < 0, S+I

resp. .ρ < 1. This means that the periodic solution .(S, I ) is orbitally asymptotically stable by Andronov-Witt theorem. Following from the index theory, there should be an equilibrium point in the region surrounding by the orbit of the above periodic solution. This has the consequence that there must exist an unstable periodic orbit in this region which contradicts the result above.   This means that the local asymptotic stability of both equilibria implies their global asymptotic stability. Figure 1 shows the phase portraits of system (8) when ∗ ∗ .κ > κ and .κ < κ , resp. .R0 < 1 and .R0 > 1 holds.

5 Discussion We have proposed and analyzed a mathematical model to describe the evolution of disease propagation without vertical transmission. We have computed the basic reproduction number as the largest absolute eigenvalue of the next-generation matrix, that is, we have given a threshold depending on parameters as follows: less than one, the infection cannot grow; otherwise, if it is bigger than one, then invasion is always possible. This computing shows that education or vaccination of the infected hosts leads to stabilization of the disease-free equilibrium, whereas the education of the susceptibles has no effect on stability of this equilibrium. The model is shown to have locally as well as globally asymptotically stable equilibria according to a threshold number which is less or greater than unity. After a sensitivity analysis, the occurrence of a transcritical bifurcation is proved showing the existence of an endemic equilibrium. The global stability of equilibria is shown via ruling out the existence of limit cycle of the reduced system.

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Fig. 1 A phase portrait of system (8) for .R0 < 1 and .R0 > 1

Acknowledgments The third author was supported by the ÚNKP-22-2 New National Excellence Program of the Ministry for Culture and Innovation from the source of the National Research, Development and Innovation Fund.

Dynamics of an SIS Epidemic Model with No Vertical Transmission

15

References 1. ALY, S; FARKAS, M.: Bifurcations in a predator-prey model in patchy environment with diffusion, Nonlinear Anal. Real World Appl. 5(4) (2004), 519–526. 2. ANDERSON, R. M.; MAY, R. M.: Infectious Diseases of Humans, Oxford University, Oxford, 1991. 3. BULAI, I. M.: Modeling COVID-19 considering asymptomatic cases and avoided contacts, in: Trends in biomathematics: chaos and control in epidemics, ecosystems, and cells (ed. R. Mondaini), (Springer 2021), 169–182. 4. CAPASSO, V.: Mathematical structures of epidemic systems, Lecture Notes in Biomathematics, 97. Springer-Verlag, Berlin, 1993. 5. CASTILLO-CHAVEZ, C.; COOKE, K.; HUANG, W. AND LEVIN, S. A.: On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS). I: Single population models, J. Math. Biol., 27(4), (1989), 373–398. 6. CASTILLO-CHAVEZ, C.; SONG, B.: Dynamical models of tuberculosis and their applications, Math. Biosci. Eng. 1(2) (2004), 361–404. 7. DIEKMANN, O.; HEESTERBEEK, J. A. P.: Mathematical Epidemiology of Infectious Diseases: Model Building, Analyis and Interpretation, Wiley, New York, 1999. 8. DIEKMANN, O.; HEESTERBEEK, J. A. P.; METZ, J. A. J.: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28(4) (1990), 365–382. 9. VAN DEN DRIESSCHE, P.; WATMOUGH, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48. 10. FARKAS, M.: Periodic Motions, Berlin, Heidelberg and New York: Springer-Verlag, 1994. 11. FARKAS, M.: Dynamical Models in Biology, San Diego, CA: Academic Press, 2001. 12. HADELER, K. P.; CASTILLO-CHAVEZ, C.: A core group model for disease transmission, Math. Biosci. 128(1–2) (1995), 41–55. 13. KISS, K.: On a Hiv/Aids Model, Publ. Univ. Miskolc, Ser. D, Nat. Sci., Math. 38 (1998), 51–58. 14. HETHCOTE, H. W.; VAN ARK, J. W.: Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs, Math. Biosci. 84 (1987), 85–118. 15. KOVÁCS, S.; SHABAN, A. H.: Stability of a delayed system modelling host-parasite associations, Can. Appl. Math. Q. 18(1) (2010), 59–91. 16. KOVÁCS, S.: Delay in decision making causes oscillation, Nonlinearity 17(6) (2004), 2267– 2274. 17. KOVÁCS, S.: Dynamics of an HIV/AIDS model—the effect of time delay, Appl. Math. Comput. 18(2) (2007), 1597–1609. 18. LI, BING XI.: Periodic orbits of autonomous ordinary differential equations: theory and applications, Nonlinear Anal. 5(9) (1981) 931–958. 19. NDIONE, A. B.; MENDY, A.; ONANA, C. A.: Economic development process: a compartmental analysis of a model with two delays, in: Trends in biomathematics: chaos and control in epidemics, ecosystems, and cells (ed. R. Mondaini), (Springer 2021), 355–390. 20. PERKO, L.: Differential equations and dynamical systems, Springer-Verlag, New York (1993). 21. RUBIO, E. V., AND RODOLFO G. G.: Vertical Transmission of HIV—Medical Diagnosis, Therapeutic Options and Prevention Strategy, in: Trends in Basic and Therapeutic Options in HIV Infection-Towards a Functional Cure, London, IntechOpen (2015). https://www. intechopen.com/chapters/48990 22. SCHEURLE, J., SEYDEL, R.: A model of student migration, International Journal of Bifurcation and Chaos 10(2) (2000), 477–480.

Infection Spread in Populations: An Agent-Based Model Adarsh Prabhakaran and Somdatta Sinha

1 Introduction The spread of infectious diseases in a population has been long under study from a mathematical point of view. The famous works of Daniel Bernoulli on the mathematical modeling of smallpox in 1760 [1], Kermack and McKendrick model [2–4], Anderson and May’s model [5] among others have had a strong impact on the current structure of mathematical models in epidemiology and are used for developing complicated and more realistic models to date [6]. Both within and outside the field of infectious disease modeling, the coordinated behavior of large groups and their emergent properties have always been looked into with curiosity. Such behavior, where properties unseen among the individuals constituting a group are only seen in the aggregate, is called the “collective behavior.” The study of collective behavior broadly falls under the field of complex systems. The spatiotemporal behavior exhibited by an interacting multicomponent system (“collective”) can be studied at different scales [7]. At a macroscopic scale, both space and time are continuous states, and a mean-field description is used for the

This work was part of a MS thesis carried out at the Indian Institute of Science Education and Research (IISER) Mohali, India. A. Prabhakaran Artificial Intelligence and its Applications Institute, The School of Informatics, The University of Edinburgh, Edinburgh, UK e-mail: [email protected] S. Sinha () Indian Institute of Science Education and Research (IISER), Mohali, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_2

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system with differential equations. While at a mesoscopic level, spatial description is discrete, and a lattice of coupled subsystems is used. However, at the microscopic level, both space and time are discrete, and each subsystem’s evolution in space and time can be studied using agent-based models (ABMs). Traditionally in epidemiology, macroscopic models, mainly ordinary differential equation (ODE) models, have been used to predict the outcome of an infection spreading in a population [2–5, 8, 9]. However, in recent years, researchers have improved these ODE models and developed multiple stochastic models to understand the spread of such contagion [10, 11]. For differential equation models, the population is divided into different compartments (hence, also known as “compartmental models”) corresponding to different predetermined stages of the spreading process. These compartments can be composed of susceptible individuals (S), the infected ones (I), and the recovered individuals (R). Based on the type of the infectious disease, there may be several other compartments, such as the exposed (E), and subcompartments such as the recovered with full or partial immunity, etc. These models have corresponding transmission coefficients, which denote the probability of movement of individuals from one compartment to another. Diseases are stochastic in nature when it comes to their transmission through contacts. However, the relative impact of these stochastic fluctuations reduces as the total population increases, and therefore, using deterministic models for the same is a good approximation for understanding a general trend in the dynamics. Through the recent example of COVID-19 infection, we have seen that nonpharmaceutical interventions (NPI) have a considerable role in controlling epidemics and pandemics [12, 13]. Considering structural NPIs in traditional compartmental models (and macroscopic models) makes the models very complicated, since these models do not account for physical spaces. In both macroscopic and mesoscopic models, heterogeneous interactions between individuals are not accounted for. Therefore, intervention strategies like restricting movement and interactions between individuals (e.g., quarantine or lockdown) cannot be efficiently modeled using such techniques. However, microscopic models efficiently bring together interactions between heterogeneous individuals and environments. The main tradeoff for modeling at a microscopic level comes from the limits set by computational resources. However, by adopting data-driven techniques and careful use of available data as input and output validation, researchers have successfully identified ways to reduce the level to which these limitations occur. This has led to the development of highly beneficial stochastic models [14, 15], which model systems from a microscopic level using agents (or individuals). In epidemiology, such agent-based models have heavily been used to predict and tackle the COVID-19 pandemic [16– 18]. ABMs are a class of computational techniques, which relies on local dynamical interactions between a system of autonomous, heterogeneous agents/individuals, to understand the macroscopic consequences (i.e., the “collective behaviour”) of the system due to such local interactions. ABMs have been used to study a variety of systems, from crowd behavior [19] to understanding social epidemiological [20] and evolutionary dynamics of influenza viruses [21], from ecology [22] to political sciences [23].

Infection Spread in Populations: An Agent-Based Model

19

In this paper, we develop a simple agent-based model based on the SI compartmental model. First, we study the population dynamics observed in the ABM. Next, we characterize the effect of population density on the dynamics and its effect on the time taken to infect the whole population. Finally, we incorporate a simple NPI by restricting the movement of agents using a physical barrier and identifying ways to limit the spread of the infection in the population.

2 Methods In this section, we give a detailed description of the model using the ODD protocol [24] used to describe the ABMs.

2.1 ODD Protocol 2.1.1

Purpose

The purpose of the model is to study the spread of infectious diseases in a population and characterize how physical boundaries help limit the spread and increase the time taken to infect every member of the population.

2.1.2

Entities, State Variables, and Scales

There are three agents in the system: susceptible hosts (an individual who is susceptible to the infection), the infected host (an infected individual), and a physical boundary unit (which can be placed anywhere in the environment). All three agents only have their coordinates in the environment as intrinsic variables. The environment is a mesh of 65 .× 65 patches in an enclosed boundary. Each agent performs its actions at every time step.

2.1.3

Process Overview and Scheduling

At every time step, the following processes happen: • The susceptible hosts move. • The infected hosts move. • State change condition is checked.

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2.1.4

A. Prabhakaran and S. Sinha

Design Concepts

Basic Principles In a particular time step, if a physical boundary unit does not block a patch, a maximum of two host agents (susceptible or infected) can occupy the patch. The infection spreads from an infected agent to a susceptible agent when they interact with each other. In this system, interaction occurs by default when two agents share an environment patch. At every time step, each agent moves to a different patch based on the rules mentioned below. Movement rules: Susceptible host moves randomly into a vacant patch in their Moore neighborhood. Simultaneously, infected hosts move randomly into a patch in their Moore neighborhood, which has no other infected host but may have a susceptible host. Infection rules: In the case of the SI model, a susceptible agent on getting infected stays infected. State change rules: If a susceptible agent shares a patch with any infected agent, the susceptible agent becomes an infected agent.

2.1.5

Initialization

The model starts with randomly allocating N agents in the environment (a mesh of 65 .× 65 patches in an enclosed boundary). The total population (N) is varied as 10%, 20%, 50%, 75%, and 100% of the maximum allowed population of 4225. A set of 50 different initial distributions of the agents are simulated for each parameter set. When the physical boundary is activated, individual physical boundary units are placed at coordinates (0,33) to divide the environment into two (see Fig. 1b). The

Fig. 1 The environments for the ABMs. (a) The environment mesh is not divided, and (b) there is a boundary with a slit in the middle

Infection Spread in Populations: An Agent-Based Model

21

boundary wall is punctured with a slit whose position can be varied to allow the movement of populations between the two enclosures.

2.1.6

Input Data

The model does not require any input data; only values of the above parameters are required.

3 Results and Discussion Herein, we show the results of simulating the SI ABM for 50 random initial configurations of the susceptible (S) and infected (I) agents. Specifically, we look at the population dynamics observed in the ABM for 50 instances of initial configuration by varying the total population size and the initial number of infected agent. We further study the effect of the environment (boundaries) on the spreading pattern of infection. We also look at the time taken to infect the whole population when there is a path through a boundary dividing the enclosed space. Finally, we characterize the infection time when this path is kept at different locations in the boundary and identify the optimal case. The temporal dynamics observed in the ABM implementation of the SI model is very similar to that of the classical differential equation model. Since relapsing into the susceptible state is not allowed (by design) in the SI model, the direction of the model (with respect to state changes) is toward the infected state. Therefore, all the hosts eventually get infected in the simulation. As in the case of the continuous ordinary differential equation compartmental model, the temporal dynamics here also result in an S-shaped population curve for the infected host. Figure 2 shows the temporal dynamics of S (in black) and I (in red) agent populations in 50 different initial configurations of the agents. Each row plots are for different total population sizes (10%, 25%, 50%, 75%, and 100% of the maximum possible population size), and each column is for different initial numbers (1, 5, and 10) of infected agents for the same population size—increasing from left to right. Since the total population is invariant for all 50 simulations for each case, the Xaxis is normalized with the respective total population sizes. Figure 3 summarizes the results of the 50 simulations in each parameter set (population size and initial number of infected) shown in Fig. 2 in a bar plot. Here, the X-axis shows the time taken to reach a fully infected state, for each of the total population sizes (in Y-axis). Several interesting results can be observed from Figs. 2 and 3. From the individual time evolution plots in Fig. 2 and the sizes of the standard deviation bars in Fig. 3, it can be clearly seen that as the total population size (crowding) increases, the variability observed in the temporal evolution of the S and I agent populations decreases for all the different initial infected populations (blue, yellow, and green bars) chosen. However, the initial dynamics and the variability remain

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A. Prabhakaran and S. Sinha

Fig. 2 The SI population dynamics for different total population sizes and initial number of infected agents. Each row depicts a different total population size, with the value increasing from 10%, 25%, 50%, 75%, and 100% of the maximum possible population size. Each column represents a different number of initially infected individuals (1, 5, and 10 going from left to right). From the population dynamics, we see that the total population size and the initial number of infected affect both the time taken for complete infection and the initial transient dynamics

Infection Spread in Populations: An Agent-Based Model Time taken for all agents to get infected

Infected=1 Infected = 5 Infected = 10

400 350 300 Time Taken

Fig. 3 Distribution of infection time (the time taken to infect the entire population) for different combinations of the total population and initial infected

23

250 200 150 100 50 0 10%

25%

50%

75%

100%

Percentage of area populated

the same within each of the total population sizes. So sparsely populated regions can show quite different disease evolution and spreading time compared to crowded environments, irrespective of the initial number of infected agents present. Figures 2 and 3 also show that the average time for infection decreases nonlinearly with increasing population size. This is expected in ABMs as at higher population sizes, the probability of interactions between infected and susceptible hosts increases, thus increasing the chance of infecting more individuals. Similarly, within the same total population size, with the increasing number of initial infected agents, the infection time again decreases (between blue bars (1 initial infected) and green bars (10 initial infected) for all population densities .p < 0.008; except for 10% population size, other densities show .p < 0.05 between blue and yellow (5 initial infected) bars; .p > 0.05 between the yellow and green bars for all cases). Till now, we have considered cases when initially both the S and I agents are randomly distributed in the environment. Here, we show results on the spatial spreading pattern of the agents, at different total populations, when the initial infected agent I is placed at specific locations in the environment. Figure 4 shows representative results when initially one infected agent (in red) is placed at the center (first column), in the corner (middle column), and at one edge (third column) of the environment for increasing population sizes (10%, 50%, and 100% of the total population size—from the top to the bottom rows). As seen from Figs. 2 and 3, ABMs with increased crowding (i.e., initial population size) takes decreasing times for the infection to spread completely in the environment. For instance, for Fig. 4g, h, and i the snapshots of disease evolution were taken at 25, 30, and 55 time steps to depict the different patterns of spread for different positions of the initial infected agent. With increasing crowding, the infection spread is more symmetric and the front is clearly defined. However, for less crowded space, it is difficult to predict which way the disease will spread. It may be noted that these results are valid only for the nearest neighbor movement rules. When long distance jumps are possible (presence of a bridge, air travel, etc.), the pattern of spread will be different. One of the common nonpharmaceutical ways to contain infection spread in a population is to segregate or separate the infected individuals through nonporous or semiporous boundaries such as containment zones. Obviously, if no infected agent can reach the part of the environment where there are only susceptible agents, the

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Fig. 4 Patterns of infection spread for different configurations of the initially infected individuals and total population sizes. Total population increases as 10% (top row, (a), (b), (c)), 50% (middle row, (d), (e), (f)), and 100% (bottom row, (g), (h), (i)). From left to right, the initial infected individual is located at the center (column 1), a corner (column 2), and in the middle of an edge (column 3) of the environment

disease will not spread. But that is a fairly unrealistic situation, and boundaries are generally porous. To study the effect of establishing boundaries on the infection time of the complete population, we partially restricted the ABM environment by dividing it into two parts with a nonporous boundary and placing only one opening at different positions, which allowed hosts from one side to move into the other. This was repeated 50 times by placing one infected agent in the left part of the boundary randomly. Figure 5 shows the time taken to infect the entire population when the environment is partially restricted. We simulated the effect of positioning the slit

Infection Spread in Populations: An Agent-Based Model Time taken for all agents to get infected

Center Quarter Edge

800 700 Time Taken

Fig. 5 Distribution of infection time, when the slit is kept at the edge, the center, and quarter of the length for different total population sizes

25

600 500 400 300 200 100 0 10%

25% 50% 75% Percentage of area populated

100%

for three different positions—at the middle, .1/4th the length of the boundary, and at the edge of the boundary. In general, it is seen that the infection time decreases considerably irrespective of where the opening is in the boundary, with increasing crowding. However, positioning the slit at the edge resulted in a significant increase in the infection time compared to when it was at the middle and quarter (for all population densities, the .p < 0.001, except for the case with density 10% where the .p < 0.05 for the center and quarter). Thus, we can conclude that restricting the spread of infection through semiporous (a small opening) physical boundaries not only increases the total spread time of the disease considerably, especially in less populated areas, one can also effectively contain the spread longer by designing the opening along the boundary. The openings farther away from the center of the boundary (edge and quarter length away) delay the spread of infection in all cases in a symmetric environment.

4 Conclusion In real populations, the spatiotemporal spread of any infectious disease depends on many biological, physical, economic, and social factors both at the individual level and at community level. It is not straightforward to include so much heterogeneity between individuals and the stochasticity in interactions in the continuous time models. The microscopic ABM approach for modeling infection spread in populations is a more appropriate technique as the discrete agents can be assigned individual properties, and the interactions depend on the spatial distribution of the S and I agents. This also leads to the results that the time for spreading the infection to the whole population is quite unpredictable and with large variability when population density is low. Thus, sparsely populated regions may take much longer for the disease to spread compared to crowded cities where the time for infection decreases with increasing population size and the initial number of infected. The spread of infection is also localized on fronts when the spatial environment is densely packed, thereby giving opportunities to plan concrete intervention strategies— which may not work in less crowded regions. These results demonstrate that a single

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intervention policy may not work for the entire country, which has heterogeneous distribution of population in rural and urban settings. Our study also gives interesting leads to designing containment strategies. We show that the time for full infection spread increases considerably when the movement of the agents is restricted by a boundary wall dividing the space with a passage in it. This infection time can be significantly increased by placing the passage at one of the edges compared to the wall’s center. Our results on spatial structures, like boundaries with different porosity, can suggest ways to contain infections in both heavily populated and sparsely populated areas. Thus, the position and structure of the “cordon” (isolation space) can play significant role in delaying the spread of infection in the population. This study quite effectively shows that the time taken to infect a whole population can be vastly increased with nonpharmaceutical interventions (NPIs) like planned connectivity and spatial structures. Additionally, policies incorporating such NPIs for mitigation cannot be the same across the country and may differ based on population density. More realistic environments such as the geometry of space and boundaries can be tested in future work, and the speed of infection spread can be tested with real data. Acknowledgments SS thanks the Indian National Science Academy (INSA) for the Honorary Senior Scientist award. This work was initiated while SS had the J C Bose Fellowship at IISER Mohali.

References 1. Bernoulli, D.: Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l’inoculation pour la prévenir. Histoire de l’Acad., Roy. Sci.(Paris) avec Mem, 1–45 (1760) 2. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 115 (772), 700–721 (1927) 3. Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. II.The problem of endemicity. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 138 (834), 55–83 (1932) 4. Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. III.-Further studies of the problem of endemicity. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 141 (843), 94–122 (1933) 5. Anderson, R.M., Anderson, B., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press (1992) 6. Mandal, S., Sinha, S., Sarkar, R.: A realistic host-vector transmission model for describing malaria prevalence pattern. Bulletin of Mathematical Biology 75 (12), 2499 (2013) 7. Lachowicz, M.: Microscopic, mesoscopic and macroscopic descriptions of complex systems. Probabilistic Engineering Mechanics 26 (1), 54–60 (2011) 8. Aron, J.L.: Acquired immunity dependent upon exposure in an SIRS epidemic model. Mathematical Biosciences 88 (1), 37–47 (1988) 9. May, R.M., Anderson, R.M.: Epidemiology and genetics in the coevolution of parasites and hosts. Proceedings of the Royal Society of London. Series B. Biological Sciences 219 (1216), 281–313 (1983)

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10. Volz, E., Meyers, L.A.: Susceptible–infected–recovered epidemics in dynamic contact networks. Proceedings of the Royal Society B: Biological Sciences 274 (1628), 2925–2934 (2007) 11. Saeedian, M., Khalighi, M., Azimi-Tafreshi, N., Jafari, G.R., Ausloos, M.: Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model. Physical Review E 95 (2), 022409 (2017) 12. Baker, R.E., Park, S.W., Yang, W., Vecchi, G.A., Metcalf, C.J.E., Grenfell, B.T.: The impact of COVID-19 nonpharmaceutical interventions on the future dynamics of endemic infections. Proceedings of the National Academy of Sciences 117 (48), 30547–30553 (2020) 13. Singh, S., Shaikh, M., Hauck, K., Miraldo, M.: Impacts of introducing and lifting nonpharmaceutical interventions on COVID-19 daily growth rate and compliance in the United States. Proceedings of the National Academy of Sciences 118 (12), e2021359118 (2021) 14. Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Physical Review Letters 86 (14), 3200 (2001) 15. Moreno, Y., Pastor-Satorras, R., Vespignani, A.: Epidemic outbreaks in complex heterogeneous networks. The European Physical Journal B-Condensed Matter and Complex Systems 26 (4), 521–529 (2002) 16. Kaleta, M., Lasser, J., Dervic, E., Yang, L., Sorger, J., Lo Sardo, D. R., Thurner, S., KautzkyWiller, A., Klimek, P.: Stress-testing the resilience of the Austrian healthcare system using agent-based simulation. Nature Communications 13 (1), 1–10 (2022) 17. Thurner, S., Klimek, P., Hanel, R.: A network-based explanation of why most COVID-19 infection curves are linear. Proceedings of the National Academy of Sciences 117 (37), 22684– 22689 (2020) 18. Chinazzi, M., Davis, J.T., Ajelli, M., Gioannini, C., Litvinova, M., Merler, S., Pastore y Piontti, A., Mu, K., Rossi, L., Sun, K. et al.: The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak. Science 368 (6489), 395–400 (2020) 19. Trivedi, A., Pandey, M.: Agent-based modelling and simulation of religious crowd gatherings in India. In: Advanced Computational and Communication Paradigms, pp. 465–472. Springer (2018) 20. El-Sayed, A.M., Scarborough, P., Seemann, L., Galea, S.: Social network analysis and agentbased modeling in social epidemiology. Epidemiologic Perspectives & Innovations 9 (1) (2012) 21. Roche, B., Drake, J.M., Rohani, P.: An Agent-Based Model to study the epidemiological and evolutionary dynamics of Influenza viruses. BMC bioinformatics 12 (1) (2011) 22. Grimm, V., Revilla, E., Berger, U., Jeltsch, F., Mooij, W.M., Railsback, S.F., Thulke, H.-H., Weiner, J., Wiegand, T., DeAngelis, D.L.: Pattern-oriented modeling of agent-based complex systems: lessons from ecology. Science 310 (5750), 987–991 (2005) 23. Huckfeldt, R., Johnson, P.E., Sprague, J.: Political disagreement: The survival of diverse opinions within communication networks. Cambridge University Press (2004) 24. Grimm, V., Berger, U., DeAngelis, D.L., Polhill, J.G., Giske, J., Railsback, S.F.: The ODD protocol: a review and first update. Ecological modelling 221 (23), 2760–2768 (2010)

Network-Based Computational Modeling to Unravel Gene Essentiality I. Granata, M. Giordano, L. Maddalena, M. Manzo, and M. R. Guarracino

1 Introduction 1.1 Defining the Gene Essentiality Defining what essential genes (EGs) are is one of the most intriguing challenges of life sciences. The concept of gene essentiality has been introduced for the first time in the context of minimal genome research, defined as the minimal set of genes that allow life [75]. Leaving the interesting discussion concerning the definition of life out, we can simplistically talk of a minimal genome, referring to the genes necessary for the growth, reproduction, and survival of an organism. The concept of a minimal gene set for cellular life, named essentialome, originated from the assumption that the functional parts of a living cell are protein and RNA molecules, and the instructions for making these parts are encoded in genes. As a consequence, a minimal genome can rationally be identified inactivating the genes in turn, as EGs are those that, when inactivated, cause lethal damage to the organism. Genetic inactivation or silencing is achieved through knockout (KO) experiments devoted to engineering the DNA to make one or multiple genes inoperative. Still, simply listing the identities of all genes needed in a minimal genome does not permit

I. Granata () · M. Giordano · L. Maddalena National Research Council, Institute for High-Performance Computing and Networking (ICAR), Rome, Italy e-mail: [email protected] M. Manzo University of Naples “L’Orientale”, ITS, Naples, Italy M. R. Guarracino University of Cassino and Southern Lazio, Cassino, Italy National Research University Higher School of Economics, Moscow, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_3

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understanding what the minimal set of functionalities needed is. Indeed, besides genes having a direct role in growth and viability, there is a set of genes involved in functional interactions, enhancing or inhibiting other genes’ activities. When an organism with a reduced genome is reconstructed by the results of KO experiments, a set of proteins remains of unknown functions. An example is the case of syn3.0, a synthetic Mycoplasma mycoides derivative with a genome reduced in size by nearly 50% and presenting a big initial portion of missing annotation [36]. The efforts made to annotate the proteins demonstrated a particular abundance of putative transporters and other transmembrane proteins [1], highlighting the importance of communication with the environment and the passage of nutrients and metabolites. Subsequent studies aimed at improving the reduced synthetic organism have then shown that nonessential genes (nEGs) can have a crucial role in enhancing essential functionalities, such as the growth rate [11]. The development of genome-scale models of the metabolism (GEMs), which detail all known metabolic reactions catalyzed by an organism in a reaction matrix, represents a valuable strategy for defining a minimal gene set [41] and puts the metabolism in a central role. Advances in experimental technologies for silencing genes have permitted the extension of the work for identifying EGs to more complex organisms [27]. Human EGs have been firstly associated with Mendelian diseases as a consequence of alterations that undermine human reproductive success (fitness) [5]. The concept of EGs probably need to be extended to essential genomic elements, as recent screening of the noncoding genome supports the idea that noncoding RNAs involved in regulating key processes are essential for the growth and survival of cells [5].

1.2 Experimental Procedures for Identifying Essential Genes Increasing the complexity of the organism under study automatically changes how EGs can experimentally be identified. In the case of prokaryotes or simple model organisms, such as Drosophila melanogaster or Saccharomyces cerevisiae, the viability of the whole organism following gene silencing can be assessed. In contrast, in the case of humans, the experimental techniques are obviously performed on cell lines, and it automatically translates the issue into identifying genes required for the proliferation of individual cells (cellular gene essentiality). Classical genetic approaches that allowed defining the first essentialomes in bacteria and yeasts [4, 28, 62] were based on insertional mutagenesis, where known genetic sequences are conveyed into the cells and inserted into DNA, provoking randomized mutations whose effects on the phenotype are evaluated. The recent availability of highthroughput techniques and resources (e.g., genome-wide RNA interference (RNAi), where the inserted double-stranded RNA (dsRNA) is homologous to the target locus and causes an epigenetic inactivation), genome editing technologies, and the advent of the genomic era, enabled to extend these screens to nonmodel organisms and to

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complex systems, such as mice and human cells. Indeed, regardless of the method, the availability of the genome sequence of an organism is necessary to have a full list of genes to analyze and to interpret the results. Thanks to the next-generation sequencing, we have access to 1000 and 100,000 human genomes from the homonym projects [20, 53]. This ease in acquiring sequencing data further helped improve the methodological techniques aimed at introducing tractable genetic variation. Advances in genome sequencing capabilities determined the development of genome editing technologies, such as those based on zinc-finger nucleases (ZFNs), transcription activator-like effector nucleases (TALENs), and CRISPR (clustered regularly interspaced short palindromic repeats)/Cas9 RNA. ZFNs, TALENs, and Cas9 nucleases are designed to link and cut target DNA sequences, inducing a wide range of genetic alterations [76]. In particular, CRISPR/Cas9, which is a tool of the bacterial immune system against phages, enables cost-effective and straightforward genome editing in yeasts, plants, animals, and human cells and is the method of choice for the essentialome identification. The CRISPR/Cas9 system can knock out genes at the DNA level: single-guide RNAs (sgRNAs), with a sequence complementary to the target DNA region, carry an endonuclease, Cas9, to cause a site-specific double-strand break. Then the break is repaired by a mechanism called nonhomologous end-joining (NHEJ). An engineered “dead” Cas9 (dCas9) variant carrying inactivating point mutations in the endonuclease domains does not provoke the cut of the target region but provides a platform that, depending on the mutations, attenuates the transcription or recruits activators. In this way, the dCas9 system allows genome-wide loss-of-function or gain-of-function screens to functionally annotate genomes or to study the role of EGs. Notably, both Cas9 and dCas9 have been used to map essentialomes in human cell lines with better performances of RNAi, including less data variation, more functional constructs, and fewer off-target effects [23]. All these assays measure the consequences of gene disruption in cell viability assays, but, despite the great technological improvements, they remain complex, costly, and labor- and time-intensive. Furthermore, the information retrieved from in vitro experiments does not necessarily translate to gene essentiality in vivo. Human gene essentiality in vivo may be assessed through population genome sequencing data, considering those essential genes rarely or never disrupted/truncated in the general population. Various metrics of genetic stability are proposed to measure gene essentiality from human genetic variation data, such as haploinsufficiency probability, loss-of-function intolerance probability, missense Z-score, and others [5]. However, the conversion of the scores obtained by either in vitro or in vivo approaches to binary labels (essential (E)/not essential (NE)) is not that obvious, being influenced by the choice of threshold values. Furthermore, the set of EGs determined by cell-based assays and in vivo human population studies has a poor overlapping, likely due to the different contexts: tumor cell viability versus organism fitness.

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1.3 Essential Genes: Genetic and Functional Characteristics Several gene properties are known to influence gene essentiality. All the studies, regardless of the specific organism, describe, in a general way, EGs as located in the core of developmental, metabolic, and signaling pathways. They represent about 10% of the total genes. Generally, compared with other genes, EGs are described as being broadly and strongly expressed, having higher protein abundance, being more conserved between species, having significantly fewer paralogous genes in a genome, being associated with developmental pathways and embryonic lethality, and being more interconnected in protein-protein interaction (PPI) networks [59]. Proteins encoded by EGs show more stability than other proteins [72]. EGs are generally highly expressed as fundamental for main cellular functionalities, and highly expressed proteins are more stable because they need to better tolerate translational errors that would lead to the accumulation of toxic misfolded products [43]. EGs tend to be located in the most active chromatin regions [16]. Furthermore, the expression levels seem to be particularly altered in tumor tissues compared to normal ones, as they are sensitive to tumorigenesis [57]. EGs are generally enriched in fundamental biological processes, such as rRNA processing, translational initiation, mRNA splicing, and DNA replication [16]. Furthermore, the transcription of EGs is particularly required during embryonic development [16]. DNA stabilityrelated attributes, such as gene length, guanosine/cytosine (GC) content, transcript count, and exon length, have also been investigated in EGs. Human EGs seem to be much shorter than the nEGs, and surprisingly, a wide variety of transcripts has been found, highlighting a greater variability of their mRNA [16]. Although the essentiality seems to be a species-specific prerogative, as the proportion of EGs among human-specific genes has been found to be significantly larger than that of other genes [18], several findings have demonstrated for some EGs a high conservation among species [7]. Going into details of the genetic and functional attributes of each EG, it appears obvious that essentiality cannot be intended as a static property but as a dynamic conditional property. Transcript and protein abundances, for example, show a high heterogeneity depending on the tissue or cell type [66]. The essentiality of genes encoding metabolic enzymes is clearly influenced by the availability of certain metabolites, the richness of the medium in the case of cell lines, and, therefore, the environment. CRISPR/Cas9 experiments on multiple tumor cell lines have highlighted differences in the essentialomes of different tissues and even between cancer cell lines deriving from the same tissue [6]. Genetic, epigenetic, and environmental contexts, such as culture conditions, highly influence the essentialome of human cells and help to differentiate conditional and not conditional EGs [42].

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Table 1 Some EG databases Name and Ref. Database of Essential Genes [47, 48] Online GEne Essentiality [17, 33] Essential Genes on Genome Scale Dependency Map portal

Acronym DEG

link http://www.essentialgene.org

OGEE

http://ogee.medgenius.info

EGGS

http://www.nmpdr.org/FIG/eggs.cgi

Data types Bacteria, archaea, eukaryotes Eukaryotes, prokaryotes, human Microbes

DepMap

https://depmap.org/portal/

Human

1.4 Databases of Essential/Nonessential Genes Over the years, several public databases have been developed to provide lists of EGs and related data in different organisms, as well as tools to study the essentialome. Some examples are briefly described in the following and summarized in Table 1. The Database of Essential Genes (DEG) is a comprehensive platform containing EG identity and function for 66 bacteria, 2 archaea, and 33 eukaryotes [47, 48]. It also has a separate section for noncoding RNAs, promoters, regulatory sequences, and replication origins. It provides the BLAST tool to align sequences against EGs. The Online GEne Essentiality (OGEE) Database [17, 33] contains gene essentiality experiments for 91 nonhuman species, of which 16 are eukaryotes and 75 are prokaryotes. CRISPR and RNAi experimental data of 931 cell lines from 27 human tissues are collected. The list of EGs is divided into pan-cancer and tissue-specific essential genes. Several gene attributes are collected, such as duplication status, the number of homologous genes in the same genome, the number of direct neighbors in PPI networks, the functional category of a gene, and the earliest expression stage during embryonic development. Selecting a specific dataset is possible to calculate the proportion of EGs choosing as property whether a gene is involved in the development or whether a gene is a duplicate or singleton and plot the results as a bar chart. EGGS (Essential Genes on Genome Scale) is a database that holds microbial gene essentiality data acquired from genome-wide essential gene selections. Genes are classified into three categories: essential, nonessential, and “undefined.” Essentiality data of each gene can be browsed on a gene/protein page. They can be analyzed in the context of a subsystem diagram, which helps interpret and apply essentiality data. DepMap is a portal that hosts the cancer dependency map project aimed at investigating the relationship between the genetic alterations of cancer and the dependencies they cause. It is not strictly a database of EGs, but, in the context of target discovery, it includes genome-wide RNAi and CRISPR loss-of-function screens to identify EGs across hundreds of genomically characterized cancer cell

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lines (an earlier version of this initiative has been referred to as Project Achilles). Data explorer and cell line-related tools are implemented, and the complete collection of datasets is available for download. Among the downloadable data, gene effect scores derived from CRISPR knockout screens are provided for all cell lines, as well as lists of common or pan-cancer EGs and nEGs.

1.5 Applications of Essential Gene Identification The identification of EGs has implications in several research areas. As mentioned above, EGs have been introduced in synthetic biology and are still a crucial topic of the field, for which the goal is to engineer microbes reconstructing minimal genomes and thus emphasize specific functions for the optimal production of different molecules, from chemicals to nutrients, to drugs and biofuels. As fundamental in many functions, EGs represent candidate druggable targets for antimicrobial or antitumoral therapies. It is particularly interesting that the classification of genes with different essentiality levels would serve in different types of drugs design. For instance, in the context of antimicrobial drugs, the distinction between EGs that are conserved across species or species-specific would help in the synthesis of broad- and narrow-spectrum compounds, respectively. The not context-dependent EGs would allow, for example, to reach and fight the pathogens in every context and body site, and the species-specificity would avoid damage to the commensal microbes. The context-dependent or conditional essentiality, instead, represents a prerogative for antitumoral drugs, with the aim to produce compounds that do not damage normal cells. The presence of cancer-specific mutations in certain genes renders other genes essential for the proliferation and survival of cancer cells. Patient-tailored therapy based on individualized cancer drug susceptibility profiles is already yielding promising results for precision medicine. Understanding the cellular gene essentiality can further contribute to investigating the molecular mechanisms underlying the biological processes. Furthermore, EGs and their different levels of conservation are important topics in the study of the origin and evolution of an organism.

1.6 Computational Methods for GE Classification There is a vast literature of computational methods affording the task of EGs classification, as summarized in several surveys [2, 24, 46, 60]. In their review on methods for community detection in PPI networks, Rasti and Vogiatzis [60] also consider the problem of identifying essential proteins and review more than 50 methods dated between 2001 and 2017. These are classified as either topology-based or integrating multiple sources, in the latter case distinguished by the type of information exploited (subcellular localization,

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evolutionary conservation, and gene expression) and the type of computational method employed. Li et al. [46] present a survey of more than 30 network-based methods for predicting EGs dated between 1987 and 2018. Besides topology-based methods (grouped according to their exploitation of neighborhood, path, or eigenvector information, or their combination) and methods integrating PPI networks with further biological information, they also consider methods based on machine learning (ML) and those exploiting dynamic networks. Helpful information is also provided on available databases of EGs and related biological information and tools for EG prediction based on PPI networks. Dong et al. [24] review more than 50 studies dated between 1996 and 2018 where different computational methods and biological features have been exploited to identify EGs both in prokaryotes and eukaryotes. They focus on five types of representative features (i.e., evolutionary conservation, domain information, network topology, sequence component, and expression level). These are evaluated on data of Escherichia coli MG1655, Bacillus subtilis 168, and human. The results highlight that for eukaryotes, the most suitable features include network topology and sequence component, eventually combined with expression level. Overall, the selection of features and their combination reveal crucial to improving the performance. The most recent review found in the literature, by Aromolaran et al. [2], focuses on ML approaches for EG prediction. It performs a comparative analysis of their essentiality prediction capability for Caenorhabditis elegans using different features. These include intrinsic features, which can be directly derived from gene and protein sequences, and extrinsic features, which can be computed only from the sequence’s interaction with another sequence or its environment. The main challenges are found in the incomplete and prone-to-error information coming from model organisms that affect the ML classifiers. In the following, we provide a brief overview of the most recent research, articulated into network topology-based methods, classical ML methods, and DL methods.

1.6.1

Network Topology-Based Methods

The essentiality of a gene or protein is closely related to its topological characteristics in PPI networks. Network topology-based methods for EG classification score genes or proteins by their centrality in PPI networks and use these sorting scores to determine if the genes or proteins are essential [46]. These methods are often used as a baseline for comparisons with ML methods [63, 73, 74, 78]. Typical neighborhood-based topology measures are degree centrality [37] (DC), which calculates the number of neighbors of a node in the network; local average connectivity [44] (LAC), which evaluates the local connectivity of node neighbors; and neighborhood centrality [68] (NC), which is based on the edge clustering coefficient. Path-based measures consider the global topological characteristics

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in the PPI networks. Frequently adopted examples include closeness centrality [71] (CC), which indicates how close a node is to all the other nodes, and betweenness centrality [9] (BC), which quantifies the ability of a node to monitor the communication between other nodes of the network. Eigenvector-based measures not only consider the degree of the node but also how central are their adjacent nodes. Typical eigenvector-based measures include eigenvector centrality [8] (EC), also known as eigen-centrality, which describes the importance of a node in a graph based on that of its adjacent nodes, and its variant PageRank centrality [56] (PR). Many other measures can also be considered; the interested reader can refer to [2, 46, 77].

1.6.2

Classical Machine Learning Methods

Campos et al. [12] propose a systematic analysis of EG prediction within and among eukaryotic species, including human cancer cell lines. Based on a wide set of intrinsic features, they perform feature selection (combining ElasticNet [79] and Ensemble SPLS [19]) and classification using five ML algorithms (generalized linear model, artificial neural network, gradient boosting, support vector machine [21] (SVM), and random forest [10] (RF)), also sharing their source code (https:// bitbucket.org/tuliocampos/essential). They envisage the identification of novel features and improved ML approaches as fundamental for enhanced EG prediction performance. Aromolaran et al. [3] present an ML approach to EG prediction based on the combination of a wide set of intrinsic and extrinsic features applied to Drosophila melanogaster, but also extended to human data. After feature selection (via ElasticNet) and class balancing (via SMOTE [14]), they test the same five classifiers adopted in [12]. The results show that a well-defined and elaborated assembly of intrinsic and extrinsic features considerably outperforms the approach based solely on protein sequence features. Dai et al. [22] propose an approach to human EG identification based on network embedding (see Sect. 2.2.3). Node embeddings of a human PPI network are extracted based on random walks and Word2vec [55] and then classified using state-of-the-art classifiers (SVM, deep neural network (DNN), decision trees, Naïve Bayes (NB), k-nearest neighbor, logistic regression, RF, and extra tree). Extensive experiments are provided on two different human PPI networks (from the Reactome [70] and the inBio Map [45] databases, respectively), with EGs taken from [32]. To handle the class imbalance, in cross-validation, they consider both stratified partitions (i.e., partitions having class distributions similar to the whole data) and partitions where the ratio of the two classes is 1:1, in the second case achieving better performance results. Kuang et al. [40] developed XGEP (eXpression-based Gene Essentiality Prediction), an ML approach to predict the essentiality of both protein-coding genes and lncRNAs in cancer cells through embedding applied to TCGA transcriptomic profiles. Three embedding methods were compared: collaborative filtering, which

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gives the best performance, Gene2Vec [25], and autoencoder [67]. The feature vectors derived from collaborative embedding are then adopted to build three classifier models (gradient boosting, SVM, and DNN). The genes predicted as essential by all three models are finally selected as candidate EGs. The performance of XGEP has been evaluated on two types of datasets, the TCGA pan-cancer dataset and cancer type-specific datasets, achieving, in both cases, higher performance values than the methods in [22, 32].

1.6.3

Deep Learning Methods

PPI networks and gene expression data are used as input for the DL framework proposed by Zeng et al. [74], named DeepEP. The node2vec technique [31] is used for encoding the PPI network into a low-dimensional space. The obtained embedding is concatenated with patterns extracted by a multiscale convolutional neural network from temporal gene expression profiles. The classification module consists of a fully connected layer and an output layer taking as input these two sources. To handle class unbalancing, a sampling method is proposed that samples the same number of majority and minority samples in each training epoch of the DL architecture. Comparisons are provided for S. cerevisiae data against six topology-based methods (DC, BC, CC, EC, NC, and LAC), two methods based on the integration of PPI networks and gene expression data (PeC and WDC), and other shallow ML-based methods (SVM, decision tree, RF, AdaBoost, and Naïve Bayes). The implementation of DeepEP is publicly available (https://github.com/ CSUBioGroup/DeepEP). The same group [73] later proposed a DL framework to automatically learn biological features for identifying EGs on S. cerevisiae data. Similarly to DeepEP, topological features are extracted by PPI networks using the node2vec [31] network representation learning technique. Furthermore, gene expression features are extracted via bidirectional long short-term memory (LSTM) cells, and subcellular localization information is exploited through an indicator vector. The concatenated feature vector is then fed to a fully connected layer with a sigmoid activation function to perform classification. Extensive comparisons are provided against the same methods considered in [74]. An ablation study investigating the role of the three types of biological information reveals that the PPI embedding is the most crucial component, but still, the other two sources help in enhancing performance. Schapke et al. [63] propose the EPGAT method for essentiality prediction with graph attention networks (GATs). Based on graph neural networks (GNNs) extended with an attention mechanism, EPGAT learns directly and automatically the nodes’ relations from the PPI network, integrating additional evidence from multi-omics data (gene expression profiles, orthology, and subcellular localization information) encoded as node attributes. The performance is evaluated on S. cerevisiae, E. coli, D. melanogaster, and H. sapiens and compared to that of network topology-based measures (DC, NC, and LAC) and other ML techniques (MLP, SVM, and N2VMLP). EPGAT shows performance comparable to node2vec embedding, with a

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shorter training time. The implementation of EPGAT is publicly available (https:// github.com/JSchapke/essential-gene-detection). Zhang et al. [78] describe DeepHE, a DL-based framework that integrates sequence features, both at DNA and protein level, and network features automatically extracted from the PPI network by using node2vec. These two types of feature vectors for each gene are concatenated together as the input to a classification module based on the MLP. A cost-sensitive technique is adopted during the architecture training to address the imbalanced learning problem. The embedding features learned by node2vec show better results on human datasets as compared to other ML methods (SVM, NB, RF, and AdaBoost) and network topology-based measures (DC, BC, EC, and CC).

2 Our Approach 2.1 Rationale Large international initiatives, such as the DepMap consortium, provide massive data from silencing experiments to derive knowledge about gene effects on hundreds of cell lines. Identifying EGs by experimental procedures at a genome-wide level is a cost- and time-consuming task. Consequently, data science approaches are required to complement, improve, and accelerate experimental techniques. The workflow devoted to the classification of EGs can be divided into four main tasks: (1) data structuring, (2) gene attributes, (3) class labeling, and (4) model learning. We designed this workflow taking into account the multiple issues introduced in Sect. 1. (1) Describing and representing knowledge through a network structure allows to include and connect multi-source data simultaneously. We organized the data to be used for classifying EGs in a tissue-specific integrated network. As discussed above, essentiality is a context-dependent attribute, as a gene can conditionally be essential or not depending on the organism, tissue, or even cell or environmental condition. According to this assumption, organism-specific lists of EGs take the risk of underestimating this important issue and wrongly assigning the essential label to genes. To partially overcome this limitation, we developed a tissue-specific approach based on the generation of a tissuespecific network and the use of tissue-specific attributes and labels. In particular, the results presented here regard a kidney-specific context. According to the centrality-lethality rule, genes representing central proteins in a PPI network, which interact with many other proteins, are more likely to be essential for the specific organism. Centrality metrics from the network are then used as gene attributes for essentiality prediction models. Given the wide variety of knowledge that can be extracted from network-based representations, we thought to extend the centrality concept, focused only on physical interactions from

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PPI, to a more functional point of view by including metabolic interactions. The metabolic machinery and the underlying connections have a crucial role in cellular functionalities and response to stimuli, so much that the metabolic networks are widely exploited for precision medicine purposes [30]. (2) Many studies have been conducted to determine which genetic and functional features are correlated to the essentiality, as addressed in paragraph 1.3. In the current study, given the focus on a specific tissue, we collected, for each gene representing the node of the integrated network, both tissue-specific and generic biological attributes. The chosen attributes range from gene structural characteristics to expression in healthy and tumoral tissues to involvement in biological processes and pathways. In particular, the context-dependent attributes give an important role to the prediction, as the developed model can serve to predict EGs for the specific context under study by simply changing the related attributes, effectively replacing the wet-lab procedures. (3) The tissue-specific approach imposes that also the class labels for the classification task are defined with respect to the specific tissue under study. For this reason, some predefined lists of E/NE genes were unsuitable for our work as they refer to the whole organism, such as the one provided by the DEG database. Furthermore, the definition of a gene as essential comes from experimental scores of gene-editing/deleting methods, thus strictly depending on the setting of thresholds. We defined the classes from gene effect scores of 39 kidney cell lines derived from CRISPR knockout screens published by Broad’s Achilles and Sanger’s SCORE projects downloaded from the DepMap portal. (4) In a previous study [52], we confirmed the important role of network embedding (see Sect. 2.2.3), as the features implicitly encoded in the network and learned by embedding techniques (e.g., node2vec) gave a better performance in classifying EGs than the network attributes extracted a priori (e.g., degree, strength, number of triangles, etc.). Therefore, here, we exploit only the embedding for the automatic learning of network topological information.

2.2 Materials and Methods 2.2.1

Networks

The kidney-specific network used in this study has been generated by integrating the kidney PPI and metabolic networks. The kidney PPI network has been downloaded from the Integrated Interactions Database (http://iid.ophid.utoronto.ca), one of the most comprehensive sets of context-specific human PPI networks [39]. It is made of physical connections (edges) between proteins (nodes) and consists of 11,741 nodes and 574,137 edges. The metabolic network has been obtained by extracting enzyme relationships from the kidney tissue genome-scale metabolic model [29, 51], downloaded from the Human Metabolic Atlas (HMA, https://metabolicatlas.org) repository. Two

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enzymes/genes are connected if they catalyze reactions producing and consuming a given metabolite, respectively. The metabolic network consists of 2945 nodes and 663,397 edges. In the case of enzymes involved in reversible reactions, the double direction has been considered. The integration of the two networks has been performed at the gene level, converting proteins and enzymes to the corresponding official gene symbol, thus having genes as nodes connected according to metabolic and/or physical interactions. The edges have been weighted by summing up the expression values in the kidney of the two involved genes, downloaded from the Human Protein Atlas [66] (HPA, https://www.proteinatlas.org). Since the metabolic network is naturally directed while the PPI is undirected, we created the integrated network as directed, considering the physical edges in both directions. Self-loops have been removed. To avoid disconnected paths, the largest connected component has been extracted and considered for the subsequent analysis, resulting in a network with 12,538 nodes and 1,791,462 edges, among which 1128065 are from the physical and 652,292 from the metabolic network and 11,105 belong to both networks. In the experiments (Sect. 3), we consider the integrated network, PPI .+ MET, and the single PPI and metabolic networks, PPI and MET, respectively. These are extracted from the integrated PPI .+ MET network by appropriately selecting the edges labeled as physical or metabolic.

2.2.2

Biological Attributes

Biological attributes have been collected for each gene using generic and tissuespecific biological information. The generic biological attributes regard both genetic and functional characteristics and have been computed using data from several sources • from the Ensembl database (https://www.ensembl.org): length from gene start to end (Gene_length), Guanine-Cytosine percentage content (GC_content), and number of transcripts (Transcript_count), which are features associated with DNA stability. They have been obtained through the biomaRt R package [26] v 2.50.3; • from the DAVID bioinformatics database [34, 35] (https://david.ncifcrf.gov): annotations have been used for our gene list to perform gene enrichment analysis. This provided the counts of involved biological functions from gene ontology (GO) molecular functions (GO-MF) and biological processes (GOBP); the pathways count from BioGRID (https://thebiogrid.org), KEGG (https:// www.genome.jp/kegg/pathway.html), and Reactome [70] (https://reactome.org); the expression count from GO cellular component (GO-CC) and in tissue (UP_tissue); and the count of predicted transcription factors binding sites

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(TFBSs) from UCSC TFBS,1 as EGs likely interact with many transcription factors and have conserved motifs. • from the NCBI Gene database [69] (https://www.ncbi.nlm.nih.gov/gene): ortholog count (Orth_count) for each gene, included according to the assumption that some EGs are highly conserved; • from DisGeNET [58] (https://www.disgenet.org): number of gene-disease associations for each gene (Gene_Disease_ass_count). It is used as an indicator of gene association to human diseases, as it seems that disease-associated genes are intermediates between highly essential and nonessential genes. The tissue-specific gene attributes mainly consist of gene expression levels in kidney normal and tumor tissue, collected from different sources • from the GTEx portal [13] (https://gtexportal.org/home/): the gene median transcripts per million (tpm) counts in kidney cortex and kidney medulla (GTEx_kidney) and the gene tpm expression of 89 kidney cortex and medulla samples (GTEx-1, . . . , GTEx-89); • from the OncoDB database [65] (http://oncodb.org): the differentially expressed genes (OncoDB_DEG) in renal tumors (kidney renal clear cell carcinoma, KIRC; kidney renal papillary carcinoma, KIRP; and kidney chromophobe, KICH), selected by FDR adjusted p-value ≤ 0.05; • from the HPA: the tpm counts in kidney tissue based on RNA-seq (HPA_kidney). The above-described biological attributes sum up to 105 gene attributes (13 generic and 92 tissue-specific); in the following, they will be referred to as BIO attributes.

2.2.3

Structural Attributes

The topological attributes of the network nodes, in the following referred to as EMB attributes, are extracted by graph embedding from the networks described in Sect. 2.2.1. Graph embedding is a mechanism for learning a mapping from a graph to a vector space still preserving main graph properties [49, 50]. Let .G = (V , E) represent a graph, where .V = {vi }N i=1 is the set of nodes and .E ⊆ V × V is the set of edges, each one represented by a pair of nodes .(vi , vj ). Definition 2.1 Given a graph .G = (V , E), a graph embedding is a mapping .φ: vi ∈ V → yi ∈ Rd , i = 1, . . . , N, .d ∈ N, such that the function .φ preserves some proximity measure defined on graph G.

.

Thus, a graph embedding maps the nodes of a graph into a d-dimensional feature vector space, also known as latent space, trying to preserve local and structural

1 The UCSC TFBS Conserved Track Settings identifies motifs that are conserved across humans, mice, and rats and scores these sites based on the motif match.

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information. In this way, graphs can be represented as compact yet informative vectors in the latent space, the so-called node embeddings, suitable to be effectively and efficiently treated by nonnetwork-based machine learning methods [64]. Specifically, the EMB attributes we consider consist of node embeddings extracted using the node2vec [31] algorithm. It learns the node embeddings by utilizing the input network topology as learning paths and maximizing a neighborhood-preserving objective function. The approach is flexible because the node embedding core is a Skip-Gram neural model [54] trained on simulated biased random walks to account for different definitions of node neighborhoods.

2.2.4

Labels

The gene effect scores of kidney cells were downloaded from the DepMap portal to derive the label vector for the classification task. Negative scores of gene effects data from DepMap imply cell growth inhibition and/or death following gene knockout. The provided scores are normalized so that non-EGs have a median score of 0, and independently identified common EGs have a median score of .−1. In a previous study [52], taking inspiration from the approach presented in [15], we divided the scores into 11 CRISPR score (CS) groups, from CS0 to CS10. The label vector was obtained by assigning the label of the gene to the most frequent score group among the 39 cell lines. CS10 was not frequent enough to appear as a label. We then translated the assignment to CS groups to gene essentiality labeling by experimentally searching for the best binary grouping. In each trial, we assumed as “essential” (E) the genes belonging to the CS0 group and having the most negative values of gene effect scores, while as “nonessential” (NE) the genes from the union of groups CSx-CS9, with x varying from 1 to 7. The best overall accuracy and MCC values were obtained in the CS0 vs. CS6-CS9 classification task. The starting point of the new set of experiments proposed here is then the labeling CS0 vs. CS6-9: CS0 and CS6-9 groups include 3814 genes, of which 19.5% belong to CS0 and are thus labeled as “E,” and 80.5% are “NE.” To evaluate the behavior of genes falling in the remaining middle groups, namely, CS1-CS5, we made several experiments shifting or enlarging the window of E and NE labels (see Sect. 3.2).

2.2.5

Evaluation Procedure

We conducted a stratified fivefold cross-validation for gene essentiality classification. At each validation step, 80% of the genes were randomly selected to form the dataset for the classifier training, while the remaining 20% were input to the built model for test predictions. The chosen classifier is the light gradient boosting machine (LGBM) [38], a gradient boosting framework that adopts a tree-based learning algorithm. Unlike standard algorithms, which grow trees horizontally, it grows trees vertically, meaning a tree leaf-wise, and the leaf with max delta loss to

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grow is chosen. When growing the same leaf, a leaf-wise algorithm can reduce more loss than a level-wise algorithm. The performance is measured using several metrics: accuracy, sensitivity, specificity, balanced accuracy (BA), and Matthews correlation coefficient (MCC), defined as TP +TN . T P + FP + FN + T N TP Sensitivity = . T P + FN TN Specificity = . T N + FP Sensitivity + Specificity BA = . 2 T P × T N − FP × FN MCC = √ (T P + F P )(T P + F N )(T N + F P )(T N + F N) .

Accuracy =

(1) (2) (3) (4) (5)

where true positive (TP) is the number of positive class (E) samples the model predicted correctly, true negative (TN) is the number of negative class samples (NE) the model predicted correctly, false positive (FP) is the number of NE samples the model predicted incorrectly, and false negative (FN) is the number of E samples the model predicted incorrectly. For completeness, in Tables 2 and 3, we also report the confusion matrix (CM), defined as  T P FN , .CM = FP T N 

(6)

computed by truncating the average confusion matrices obtained over the five folds.

3 Experiments Our experiments are devoted to evaluating (1) the suitability of biological and structural gene attributes for classifying EGs; (2) the labeling strategy described in Sect. 2.2.4, also as compared with a pre-compiled list of E/NE labels provided by the OGEE public database; and (3) the contribution of the two single networks, PPI and MET. Concerning the gene attributes, we applied the classification procedure presented in Sect. 2.2.5 to the structural attributes taken alone (EMB) or together with the BIO ones (BIO + EMB), for each of the networks described in Sect. 2.2.1 (PPI, MET, PPI + MET).

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3.1 Implementation Details Only genes that are labeled according to the labeling strategy were selected for training the classifier model. All their attributes were normalized through the Zscore strategy. For the computation of the structural attributes, the embedding was performed on the whole network (including nonlabeled genes), to ensure the classifier model is trained based on complete information for each node. For the node2vec algorithm, we adopted its implementation in karateclub [61] (https://karateclub.readthedocs. io), and the embedding size was fixed to 128. For the LGBM classifier, we adopted the LightGBM Python implementation (https://lightgbm.readthedocs.io).

3.2 Analysis of Performance Results The performance results of EGs classification using the labeling CS0 vs. CS6-9 described in Sect. 2.2.4 are reported in Table 2. They confirm the importance of integrating BIO and EMB attributes (BIO + EMB) to achieve better performance. The improvement is evident in the cases of the PPI and MET + PPI networks. Herein, the EMB attributes alone, and thus the network structure, are already sufficient to discriminate the two classes, but adding BIO attributes determines a clear improvement. The BIO attributes of the labeled genes from the integrated network give an MCC comparable to that of EMB attributes. Instead, a different behavior is shown by the MET network, where an almost total contribution is given by the BIO attributes, as the embedding alone returns an almost null classification. Overall, in terms of network, the major contribution to the performance appears to be given by the PPI network, and a comparable performance can be observed when considering the integrated PPI + MET network. The performance results of EG classification using the labeling CS0-1 vs. CS2-9, with 1527 E and 10169 NE, are reported in Table 3. From these results, it appears evident that including the other CS groups into the E or NE classes, in this case, CS1 into the E class and CS2-CS5 into the NE class, remarkably decreases the performances. It is worth noticing that considering the genes belonging to CS1-5 groups as a third class that we name “intermediate genes” (IGs), the performance tested on the PPI+MET network dramatically decreases (Table 4). Splitting the multi-class classification into two binary problems made evident that the most challenging task was represented by separating the IGs from the NE. The discrimination of IGs from E was still possible, especially excluding the CS1 genes. To prove the efficacy of our labeling approach, we compared our CS-derived strategy with the E/NE tissue-specific assignment from the OGEE database. In particular, the Avana dataset was downloaded and filtered to get E/NE genes related

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Table 2 Performance results of EG classification using the labeling CS0 vs. CS6-9 BA Sensitivity Specificity MCC CM Attrib. Accuracy PPI network EMB 0.910 ± 0.011 0.807 ± 0.022 0.638 ± 0.041 0.976 ± 0.005 0.693 ± 0.039 [[475 270] [ 74 2995]] BIO + 0.926 ± 0.006 0.846 ± 0.012 0.714 ± 0.023 0.978 ± 0.005 0.753 ± 0.023 [[532 213] EMB [ 68 3001]] MET network EMB 0.809 ± 0.004 0.551 ± 0.005 0.128 ± 0.009 0.974 ± 0.004 0.194 ± 0.019 [[ 95 650] [ 79 2990]] BIO + 0.894 ± 0.004 0.789 ± 0.012 0.616 ± 0.028 0.961 ± 0.005 0.638 ± 0.015 [[459 286] EMB [120 2949]] PPI + MET network BIO 0.903 ± 0.006 0.819 ± 0.020 0.682 ± 0.044 0.957 ± 0.003 0.678 ± 0.025 [[508 237] [132 2937]] EMB 0.907 ± 0.007 0.800 ± 0.016 0.624 ± 0.033 0.976 ± 0.003 0.682 ± 0.025 [[465 280] [ 75 2994]] BIO + 0.924 ± 0.007 0.839 ± 0.022 0.699 ± 0.046 0.978 ± 0.003 0.744 ± 0.027 [[ 521 224] EMB [ 67 3002]]

Table 3 Performance results of EG classification using the labeling CS0-1 vs. CS2-9 BA Sensitivity Specificity MCC CM Attrib. Accuracy PPI network EMB 0.897 ± 0.004 0.654 ± 0.019 0.325 ± 0.038 0.982 ± 0.002 0.444 ± 0.034 [[497 1030] [179 9990]] BIO + 0.887 ± 0.006 0.639 ± 0.015 0.303 ± 0.029 0.975 ± 0.004 0.388 ± 0.038 [[462 1065] EMB [258 9911]] MET network MET 0.870 ± 0.001 0.514 ± 0.004 0.032 ± 0.008 0.996 ± 0.001 0.106 ± 0.028 [[ 49 1478] [ 43 10126]] BIO + 0.882 ± 0.007 0.611 ± 0.014 0.245 ± 0.024 0.977 ± 0.006 0.339 ± 0.042 [[374 1153] EMB [230 9939]] PPI + MET network BIO 0.887 ± 0.006 0.639 ± 0.015 0.303 ± 0.029 0.975 ± 0.004 0.388 ± 0.038 [[462,1065] [258 9911]] EMB 0.897 ± 0.006 0.665 ± 0.017 0.350 ± 0.036 0.980 ± 0.006 0.456 ± 0.037 [[534 993] [206 9963]] BIO + 0.905 ± 0.004 0.698 ± 0.015 0.418 ± 0.032 0.978 ± 0.003 0.510 ± 0.024 [[638 889] EMB [224 9945]]

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Table 4 Performance results of IG classification using the integrated PPI + MET network and varying the labeling BA Sensitivity Specificity MCC CM Attrib. Accuracy CS0 vs. CS1-5 vs. CS6-9 BIO + 0.696 ± 0.002 0.389 ± 0.167 ± 0.040 0.991 ± 0.002 0.105 ± 0.015 [[124 620 1] 0.012 [ 74 8524 126] EMB [ 13 2981 75]] CS0 vs. CS1-5 BIO + 0.923 ± 0.004 0.595 ± 0.199 ± 0.029 0.991 ± 0.002 0.338 ± 0.045 [[148 597] 0.015 [ 71 7811]] EMB CS0 vs. CS2-5 BIO + 0.915 ± 0.005 0.622 ± 0.260 ± 0.027 0.984 ± 0.004 0.367 ± 0.044 [[194 551] 0.015 [117 6983]] EMB CS2-5 vs. CS6-9 BIO + 0.687 ± 0.006 0.511 ± 0.955 ± 0.008 0.067 ± 0.007 0.048 ± 0.022 [[6782 318] 0.005 [2862 207]] EMB Table 5 Label comparison using the integrated PPI + MET network BA Attrib. Accuracy CS0 vs. CS6-9 BIO + 0.924 ± 0.007 0.839 ± 0.022 EMB CS0-1 vs. CS2-9 BIO + 0.905 ± 0.004 0.698 ± 0.015 EMB OGEE Avana dataset (E vs. NE) BIO + 0.943 ± 0.004 0.610 ± 0.020 EMB

Sensitivity

Specificity

MCC

0.699 ± 0.046 0.978 ± 0.003 0.744 0.027

CM ± [[ 521

224]

[ 67 3002]] 0.418 ± 0.032 0.978 ± 0.003 0.510 0.024

± [[638

889]

[224 9945]] 0.228 ± 0.039 0.992 ± 0.002 0.370 0.061

± [[180

610]

[ 87 11328]]

to kidney tissue. Considering our E/NE assignment, namely, CS0 vs. CS6-9, a clear difference is evident (Table 5) with a remarkably better performance achieved through our approach. Including all the genes for a balanced comparison and thus observing the results from CS0-1 vs. CS2-9, our method still showed better results.

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Fig. 1 Enrichment statistic barplot. Gene ontology biological process (GO-BP) terms enriched by the genes belonging to the different CS groups. The numbers of input genes, of genes effectively used for the enrichment, and of GO-BP significant (FDR adjusted p-value .≤ 0.05) terms are shown on the top of the bars

3.3 Analysis of Biological Results From the classification performance results, we could assign E/NE labels to CS0 and CS6-9 groups, respectively. IGs included in CS1-5 groups can undoubtedly be subdivided into two additional classes. On the one side, CS1 is essential enough to be discriminated from NE but slightly different from CS0 since it deteriorates the performance when merged into the E class. On the other side, CS2-5 is not ascribing to EGs nor nEGs, but somehow similar to the latter, as its separation was not achieved. Performing the functional enrichment of genes belonging to CS0, CS1, CS2-5, and CS6-9, we noticed that going toward NE genes, i.e., from CS0 to CS9, the number of enriched terms decreases even though the number of genes effectively used for the enrichment increases (Fig. 1). This behavior is particularly interesting as it highlights a greater variety of functions among the EGs, thus likely covering more biological processes necessary for the survival of the cell. The top enriched GO-BP terms with regard to the number of genes enriching the specific term are shown in Figs. 2 and 3, and the related Venn diagram is shown in Fig. 4. Our results, from both classification and enrichment, showed that genes belonging to CS1 are probably conditional essential, whose context is missed by the tissue specificity but needs further detail. Indeed, they share similar enriched terms of CS0 genes (10/20 terms in common) (Figs. 2a, b, and 4) and, if merged with

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Fig. 2 GO-BP terms enriched by the genes belonging to the different CS groups (Part 1). Top 20 GO-BP significant (FDR adjusted p-value .≤ 0.05) terms, with regard to the number of genes enriching the specific term (COUNT), are shown for CS0 (a) and CS1 (b) genes

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Fig. 3 GO-BP terms enriched by the genes belonging to the different CS groups (Part 2). Top 20 GO-BP significant (FDR adjusted p-value .≤ 0.05) terms, with regard to the number of genes enriching the specific term (COUNT), are shown for CS2-5 (a), and CS6-9 (b) genes

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CS2-5

9

CS0

12

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1 9

5

1

0

9

12 0 1

0 0

0

0

Size of each list 20

20

20

20

17

10 0 CS0

CS1

CS2-5

CS6-9

Fig. 4 Venn diagram of GO-BP terms. The intersection of significant GO-BP terms enriched by the genes belonging to the different CS groups

CS0 for labeling genes as E, still determine a good discrimination from NE, but with worst performance than considering only CS0 as E (PPI .+ MET; BIO .+ EMB; accuracy, .0.868 ± 0.014; BA, .0.841 ± 0.018; sensitivity, .0.760 ± 0.031; specificity, .0.922 ± 0.008; MCC, .0.697 ± 0.033). CS2-5 cannot be considered conditional or almost essential since its classification against CS6-9 was not achieved. This result highlights a similarity between the two sets of genes, but not enough to be merged to form the NE class. From the enrichment of these two groups, a slight similarity of terms (5/20 in common) emerged, as shown in Figs. 3a, b, and 4. A different set of functions is instead covered by genes belonging to CS0-1 and CS2-9 (one term in common between CS0 and CS2-5, one in common between CS1 and CS2-5, no terms shared with CS6-9 for both CS0 and CS1), as can be observed in Fig. 4.

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4 Conclusions The overview of the gene essentiality and the experimental approach we described here allowed us to draw some interesting insights that can undoubtedly serve as a basis for future work. The integration of MET and PPI networks seems to give no or poor improvement to the performance. The PPI makes a significant structural contribution. This unexpected behavior can be explained by the fact that genes whose proteins are involved in multiple physical interactions, if silenced, with the consequent absence of the relative protein, likely determine a damage that difficultly can be compensated. In contrast, the metabolism is highly interconnected and has several alternative paths that can compensate for the absence of an enzyme. Furthermore, the two networks show a poor overlap, with only 1105 joint edges, as proteins physically connected are likely to be not functionally linked in consequential paths as imposed by the nature of the MET network. Moreover, the embedding was confirmed to be crucial in improving the performance when added to the biological characteristics of genes. Besides the assignment of E/NE labels to the two extremities of CRISPR scores identifying the best grouping in CS0 for E and CS6-9 for NE, there’s still a middle group from CS1 to CS5 that is difficult to classify. From deep learning and biological enrichment, we inferred the existence of two additional classes, one represented by CS1 genes and the other by CS2-5 genes. Both are similar to the two closer extremal groups but not enough to be merged to form the E and NE classes. CS1 are likely conditional or almost essential genes whose condition is not entirely represented by the tissue specificity. They indeed share common activities and characteristics with CS0 genes, whose E nature is confirmed from the enrichment in fundamental biological processes, such as rRNA processing, translational initiation, mRNA splicing, and DNA replication [16]. CS2-5 genes, on the contrary, are separable from E genes, even if not well as CS6-9, while their discrimination from CS6-9 was not achieved. They are somehow more NE than E but probably involved in regulating essential genes. Their functional enrichment, indeed, showed the presence of transcriptionregulating terms. The effectiveness of the CS-derived labelling approach was then demonstrated by comparing the results to those achieved by using kidney-related labels from the Avana dataset (OGEE database). Although our method allowed to recognize E/NE genes, all the conclusions extracted from our experiments lead to the knowledge that the EG identification is not definable as a binary problem. Indeed, besides E and NE genes, a significant portion of genes shows an intermediate behavior and whose recognition still needs further investigation to be unraveled. Acknowledgments This work has been partially funded by the BiBiNet project (H35F21000430002) within POR-Lazio FESR 2014–2020. It was carried out also within the activities of the authors as members of the INdAM Research group GNCS and the ICAR-CNR INdAM Research Unit and partially supported by the INdAM research project “Computational Intelligence methods for Digital Health.” The work of Mario R. Guarracino was conducted within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE). Mario Manzo thanks Prof. Alfredo Petrosino for the guidance and supervision during the years of working together.

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Nonlinear Dynamics in an SIR Model with Ratio-Dependent Incidence and Holling Type III Treatment Rate Functions Akriti Srivastava and Prashant K. Srivastava

1 Introduction In the past few years, serious efforts have been made in the direction of developing plausible mathematical models for exploring the dynamics of the infectious disease transmission. When it comes to the modeling communicable diseases, the incidence rate function is considered to play a crucial role in ensuring that the model provides a realistic and comprehensive description of the transmission dynamics of the diseases. The incidence rate of a disease may be directly or indirectly influenced by several factors, such as, the density of the population, the level of media coverage, and lifestyle choices of individuals, among many others. Capasso and Serio [1] introduced a saturated incidence rate function in epidemic models to study the cholera epidemic in Bari in 1973. Saturation at high infection levels can occur due to the crowding of infected population or due to protective measures taken by susceptible population or the “psychological” effects. If the incidence function goes down when the infective population (I ) is big, it can be used to figure out what these psychological effects are. For example, if there are many infectious people, the infection force may go down as the number of infectious population increases. This is because when there are too many infectious people, the healthy population may try to reduce the number of contacts per unit time. Liu et al. [2, 3] employed a nonlinear incidence rate function to account for the behavioural kI l changes in susceptible population, given by .Sf (I ) = S 1+αI h , where kI is a measure 1 of the infection force of the disease, and . 1+αI h is a measure of the inhibition effect from the behavioural change of susceptible individuals, when the number infective individuals is higher. The parameters k and l are positive and h is a non-negative

A. Srivastava · P. K. Srivastava () Department of Mathematics, Indian Institute of Technology Patna, Patna, Bihar, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_4

57

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constant. Later, several authors made use of the different cases in which .k, l, and .α can take distinct values (for reference one can see [4–12]). In most of models, the force of infection .f (I ) is solely a function of the infected individuals. However, spreading of infectious diseases require two types of individuals: those who are healthy and can get sick, and those who can give it to others. As a result, the force of infection should be dependent on the densities of both infective and susceptible individuals and it should take the form .f (S, I ). Yuan and Li [13] have considered the force of infection function as; .f (S, I ) = f ( SI ) = k(I /S)l , 1+α(I /S)h

with the assumption that the infection force is a function of the ratio of the number of the infective to that of the susceptible and studied the model by taking .l = 1 and .h = 2. They concluded from the stability analysis that the number of infective individuals may tend to zero over time or the disease may persist. For the purpose of this study, we opted to use a ratio-dependent incidence rate function as above with .l = 1 and .h = 2. Treatment is one of the most effective interventions available for reducing the disease burden and controlling the spread of infection. However, there is no guarantee that treatment will always be accessible; and even if it is, there are limitations to the number of medical resources available. As a result, in an unexpected disease outbreak, it will probably be impossible to provide treatment to every infected person. When modeling, one way to portray this is to use of an appropriate saturation-type treatment function. If there is an ample supply of resources, then treatment may be offered to everyone, and this fact may be represented by taking a proportionate treatment function [14]. Researchers have chosen various saturation type treatment functions, such as, Holling type II [15– 17], Holling type III [18], Monod–Haldane function [19], etc. to address the issue of limited medical resources for treatment. Recently, a few more research involving saturated treatment function in epidemic models have appeared in the published literature; for further information, one may refer to the [20–26] and references contained in the aforementioned articles. The saturation in treatment is the cause of the rich nonlinear dynamics that is observed in the majority of these studies. For the purpose of our study, we chose to use a Holling type III treatment rate function. The structure of the article is as follows. In Sect. 2, we have proposed an SI R model with a ratio-dependent incidence and Holling type III treatment rate functions. The primary aim of this article is to study the nonlinear dynamical behaviour of the model system with these two nonlinear functions. The stability results of the steady states are then derived. Some types of nonlinear dynamics that are exhibited includes bi-stability, transcritical bifurcation, and hysteresis. In Sect. 3, we have developed several numerical examples to understand the significance of our theoretical findings. In Sect. 4, finally the concluding remarks and the subsequent discussion is provided.

Nonlinear Dynamics in an SIR Model. . .

59

2 SIR Mathematical Model We partition the total population (N ) into Susceptible (S), Infective (I ), and Recovered (R) sub-classes. For our model, we consider the ratio-dependent incidence rate function, .f (S, I ), defined as f (S, I ) =

.

βS 2 I . S 2 + αI 2

Further, we employ .T (I ) as the treatment rate function in our model, which is defined as T (I ) =

.

aI 2 . 1 + bI 2

Here, parameters .α and b are saturation constants, i.e., .α indicates the information induced sensitivity of individuals to the level of infection and b is related to the the limitation to the treatment availability. The parameter .β is the transmission rate of infection and a is the treatment rate. The mathematical representation of the SI R model is given by the following nonlinear ordinary differential equations. βS 2 I dS = Λ − μS − 2 , dt S + αI 2 .

βS 2 I dI aI 2 = 2 − (μ + δ)I − , 2 dt S + αI 1 + bI 2

(1)

aI 2 dR = − μR, dt 1 + bI 2 with initial conditions .(S(0), I (0), R(0))T ∈ R3+ . Here .Λ is the inflow rate in susceptible population, .μ is natural mortality rate, and .δ is the disease related death rate. Note that the model system (1) is not defined at origin, so at .(S, I, R) = (0, 0, 0) we define the model as: .

dI dR dS = Λ, = 0, = 0. dt dt dt

2.1 Positivity and Boundedness From the model system (1), we get the following:

60

A. Srivastava and P. K. Srivastava

.

| dS || = Λ > 0, dt |{S=0, I >0, R>0} | dI || = 0, dt |{S>0, I =0, R>0} | dR || aI 2 = ≥ 0. | dt {S>0, I >0, R=0} 1 + bI 2

From above, we see that all rates are non-negative on the boundary planes of the non-negative cone of .R3+ . As the vector fields direction are inward on the boundary planes, so if we start with any interior point of this cone, then the solutions will remain within this cone for all the future time. This ensures the positivity of the solutions of the system. Note that the total population .N = S + I + R. Hence we get, . dN dt = Λ − μN − δI ≤ Λ − μN. After simple calculation, we have .lim supt→∞ N(t) ≤ Λ μ . Thus, the biological feasible region for the model system is ⎫ ⎧ Λ 3 . .Г = (S, I, R) ∈ R+ : 0 ≤ S, I, R ≤ μ Since the first two equations of model system (1) are independent of the .R(t), we consider the following reduced model system for further analysis:

.

βS 2 I dS = Λ − μS − 2 , dt S + αI 2 aI 2 dI βS 2 I − (μ + δ)I − . = 2 dt S + αI 2 1 + bI 2

(2)

2.2 Existence of Disease-Free Steady State and the Basic Reproduction Number Equating the right hand side of the model system (2) to zero, we obtain the diseasefree steady state .E0 = ( Λ μ , 0) which always exists. Now, using next generation matrix method [27], we obtain the basic reproduction number .R0 for the model system (2), which is given as R0 =

.

β . μ+δ

Nonlinear Dynamics in an SIR Model. . .

61

2.3 Sensitivity Analysis of R0 Data collection and parameter estimation are inherently uncertain and never errorfree. In this case, it is worthwhile to list the parameters involved in defining epidemic threshold quantity .R0 in order of their impact on the size of .R0 . As the qualitative characteristics of the system depend on .R0 , this investigation will aid in sorting out the most influential epidemiological parameters to control the disease. We define the parameter space Ω := {(β, μ, δ) ∈ R3+ }.

.

We can easily verify that .R0 ∈ C 1 (Ω ). Now, following Chitnis et al. [28], we give the following definition. Definition 1 ([28]) The normalized forward sensitivity index of .R0 ∈ C 1 (Ω ) with respect to a parameter, say, .ψ, is defined as the following: ϒψR0 =

.

∂R0 ψ . ∂ψ R0

(3)

When a single parameter changes its value while the other parameters remain constant, then this definition provides a judgment on the normalized relative change in the quantity .R0 . A positive indexed parameter indicates that .R0 is an increasing function of that parameter, whereas a negative indexed parameter indicates that .R0 is decreasing. Using the expression of .R0 and Eq. (3), we obtain ϒβR0 = +1,

.

ϒμR0 = −

μ , μ+δ

ϒδR0 = −

δ . μ+δ

To find the sensitive indices of parameters, we take the parameter values mentioned in Table 1 and obtain sensitivity indices as given in Table 1. The .ϒ R0 = −0.8333 μ

indicates that .R0 decreases by 8.333% with an increase of 10% in the .μ-value. From Fig. 1a and Table 1, we conclude that parameter .β is very important parameter which has maximum positive correlation with .R0 , whereas .μ has maximum negative correlation with .R0 . Also .δ has significant negative correlation with .R0 . We give the two contour plots for the variation of .R0 with arguments as .(β, δ) and .(β, μ), shown in Fig. 1b and c, respectively.

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Table 1 Initial value and sensitivity indices of the associated parameters in the expression of .R0

Parameters .β .μ .δ

Initial value Sensitivity indices 0.1 +1 0.05 −0.8333 0.01 −0.1667

1

2

0.5

1

0.8

1.5

3 0.6

1

2

1

0.5

1.5

0.4

2 1

1

1.5

0.2 0.5

-1

-0.5

0

0.5

1

0

1

1.5 2

0

R0

0.5

2

0

0.2

0.4

0.6

0.8

1

(b)

(a) 1

2

0.5

1

0.8

1.5 1

0.6

1.5

1

0.5

0.4

2

1 1.5

0.5

2

0.5

0.2 1.5 1 2

0 0

0

0.2

0.4

0.6

0.8

1

(c) Fig. 1 (a) Sensitivity plot for .R0 . (b) Contour plot of .R0 a function of .β and .δ. (c) Contour plot of .R0 a function of .β and .μ. Table 1 contains the initial values for parameters

2.4 Existence of Endemic Steady States In this section, we find the endemic steady state .E ∗ = (S ∗ , I ∗ ) of the model system (2). Here S∗ =

.

xI ∗3 + yI ∗2 + zI ∗ + Λ , μ(1 + bI ∗2 )

(4)

Nonlinear Dynamics in an SIR Model. . .

63

where .x = −b (δ + μ) , y = Λ b − a, z = −(δ + μ) and .I ∗ is a positive real root of the following equation: g(I ) = AI 8 + BI 7 + CI 6 + DI 5 + EI 4 + F I 3 + GI 2 + H I + J ' = 0,

.

(5)

) ( ) ( where, .A = b3 (δ + μ) α μ2 − (δ + μ)2 (R0 − 1) , B = Λ b α b2 μ2 + x 2 − ( ) y (α b2 μ2 + 2 β b x + 3 x 2 ), C = 2 Λ b x y − β x 2 − b β y 2 + 2 x z − ) ( α b μ2 (2 x + b z) − 3 x y 2 + x z , D = Λ b (y 2 + 2 x z) − y 3 − 2 Λ x 2 − 2 x(y (β + 3 z)−2 b β( (Λ x + )y z)−2 α( b μ2 (y −)Λ b) , E = −α (x + 2 b z) μ2 − ) 2 β y + 2 x z − 3 z y 2 + x z − b β . z2 + 2 Λ y − 4 Λ x y + 2 Λ b (Λ x + y z) , ( 2 ) 2 2 2 .F = Λ b z + 2 Λ y − 2 β (Λ x + y z) − 3 y z − 2 Λ y − α μ (y − Λ b) − 2 2 2 2 4 Λ x z − 2 Λ b β z, G = ( −z (β + z) − Λ x − α) μ z − Λ b (β − 2 z) − 2 Λ y (β + 2 z) , H = Λ Λ a + 2 (δ + μ)2 (R0 − 1) and .J ' = −Λ2 (δ + μ) ' .(R0 − 1) . It is clear that .H > 0 and .J < 0 for .R0 > 1. From the second equation of the model system (2), we obtain S

.

∗2

)) ( ( I ∗2 α I ∗ a + (δ + μ) 1 + b I ∗2 ) ( = , (δ + μ) (R0 − 1) 1 + b I ∗2 − I ∗ a

(6)

We observe from the above Eq. (6) that no positive real .S ∗ exists for .R0 < 1, and hence we conclude that the model system (2) has no endemic steady state for .R0 < 1. It is very difficult to show the existence of positive real root .I ∗ of Eq. (5) analytically. We shall discuss the existence numerically later. We are able to provide the necessary condition for the existence of endemic steady state .E ∗ = (S ∗ , I ∗ ) for .R0 > 1. The necessary condition for the existence of an endemic steady state .E ∗ = ∗ (S , I ∗ ) is R0 > 1 +

.

a I∗ . (1 + b I ∗2 )(μ + δ)

2.5 Local Stability of Steady States Theorem 1 The disease-free steady state E0 of the model system (2) is locally asymptotically stable when R0 < 1 and is unstable when R0 > 1. Proof The Jacobian matrix of the model system (2) at the disease-free steady state E0 is given by ⎡ JE0 =

.

⎤ −μ −β . 0 β − (μ + δ)

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The characteristic equation of JE0 is given as (μ + λ)(β − (μ + δ) − λ) = 0.

.

Clearly, the eigenvalues are β − (μ + δ) and −μ. The eigenvalue β − (μ + δ) is negative when R0 < 1. Hence for R0 < 1, the disease-free steady state E0 is locally asymptotically stable. For R0 > 1, one of the eigenvalues becomes positive then the disease-free steady state E0 is unstable. ⨆ ⨅ Theorem 2 For R0 > 1, the unique endemic steady state E1∗ = (S ∗ , I ∗ ) is locally ∗ asymptotically stable, provided it exists, if I ∗ > √S α . Proof The Jacobian matrix of the model system (2) is given by ⎡ JE1∗ = ⎣

.

2αβS ∗ I ∗3 −βS ∗2 (S ∗2 −αI ∗2 ) (S ∗2 +αI ∗2 )2 (S ∗2 +αI ∗2 )2 2αβS ∗ I ∗3 βS ∗2 (S ∗2 −αI ∗2 ) 2aI ∗ − (μ + δ) − (1+bI ∗2 )2 (S ∗2 +αI ∗2 )2 (S ∗2 +αI ∗2 )2

−μ −

⎤ ⎦.

The characteristic equation of JE1∗ is given as λ2 + l1 λ + l2 = 0.

.

∗

(7)

( ( )) S ∗ β 2 I ∗3 α+S ∗ I ∗2 α−S ∗2

2I a Here, l1 = δ + 2 μ + + and l2 = (1+b I ∗2 )2 (S ∗2 +α I ∗2 )2 ⎞⎛ ⎛ ) ( ⎞ ∗a μ β S ∗2 S ∗2 −I ∗2 α I ∗3 S ∗ α β . The roots of (7) − δ + μ + 2 I ∗2 μ + 2 ∗2 2 2 ∗2 S +α I 1+b I ( ) ( ) (S ∗2 +α I ∗2 )2 ∗ have negative real parts provided l1 > 0 and l2 > 0. The condition I ∗ > √S α imply l1 > 0, l2 > 0 and hence the endemic steady state E1∗ is locally asymptotically ∗ ⨆ ⨅ stable, provided it exists and I ∗ > √S α .

2.6 Global Stability of Disease-Free Steady State Now, we establish the global stability of the disease-free steady state .E0 , using the method given by Castillo-Chavez et al. [29]. The model system (2) can be rewritten in the following form: dX = F (X, Y ) dt . dY = G(X, Y ) with G(X, 0) = 0, dt

(8)

here, .X ∈ Rn1 , Y ∈ Rn2 denotes the uninfected and infected individuals, respectively, where .n1 and .n2 are positive integers. Let .U0 = (X0 , 0) be the

Nonlinear Dynamics in an SIR Model. . .

65

disease-free steady state of the model system (8). We consider the following two assumptions: (H1) For . dX dt = F (X, 0), X0 is globally asymptotically stable. ˆ ˆ (H2) .G(X, Y ) = A Y − G(X, Y ), G(X, Y ) ≥ 0 for .(X, Y ) ∈ Г , where .A = DY G(X0 , 0) is a Metzler matrix (a stable matrix with non-negative off diagonal elements) and .Г is a biologically feasible region. If the model system (8) satisfies the above two conditions, then the following result holds. Theorem 3 The disease-free steady state .E0 is globally asymptotically stable when R0 < 1.

.

Proof Consider .X = S and .Y = I , by comparing system (8) with the model system (2), we obtain F (X, Y ) = Λ − μS − ⎛

.

G(X, Y ) =

βS 2 I , S 2 + αI 2

⎞ aI 2 βS 2 I . − (μ + δ)I − 1 + bI 2 S 2 + αI 2

Now, first we prove condition (H1): F (X, 0) = Λ − μS,

.

which gives .

dS = Λ − μS. dt

By solving the above two equations, we get .limt→∞ S(t) =

Λ μ

= S(0). Therefore,

X0 = = F (X, 0) as .X → X0 when is globally asymptotically stable for t → ∞ and hence (H1) is satisfied. Now, we prove condition (H2):

.

Λ μ

dX . dt

.

A = β − (μ + δ).

.

Therefore, A Y = βI − (μ + δ)I.

.

ˆ Y ) = A Y − G(X, Y ). So, Since .G(X, ˆ G(X, Y) =

.

αβI 3 aI 2 . + 1 + bI 2 S 2 + αI 2

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A. Srivastava and P. K. Srivastava

ˆ Y ) ≥ 0. Further, .A is a Metzler matrix in .Г as .β−(μ+δ) < 0 We observe that .G(X, for .R0 < 1 and thus (H2) is satisfied. Hence, the theorem follows. ⨆ ⨅

2.7 Global Stability of Endemic Steady State E1∗ Lemma 1 The model system (2) is uniformly persistent, that is, there exist a positive constant c, such that .

{ } lim inf S(t), I (t) ≥ c. t→∞

Proof It is clear from the Theorem 1 that E0 is unstable when R0 > 1. Using the uniform persistence result [30], the instability of the disease-free steady state E0 assures the uniform persistence of the system when R0 > 1. Hence the model system (2) is uniformly persistent. ⨆ ⨅ Theorem 4 The unique endemic steady state E1∗ , of the model system (2), provided it exist, is globally asymptotically stable when R0 > 1 and Δ > 0, where Δ is defined in the proof. Proof Using the geometric approach [31], we show the global stability of the unique endemic steady state E1∗ . The Jacobian matrix corresponding to the model system (2) is given as: ⎡ JE1∗ = ⎣

−βS 2 (S 2 −αI 2 ) 2αβSI 3 (S 2 +αI 2 )2 (S 2 +αI 2 )2 βS 2 (S 2 −αI 2 ) 2αβSI 3 2aI − (μ + δ) − (1+bI 2 )2 (S 2 +αI 2 )2 (S 2 +αI 2 )2

−μ −

.

⎤ ⎦.

The second additive compound matrix [32, 33] of J is J

.

[2]

)) ( ( S β 2 I 3 α + S I 2 α − S2 = −(δ + 2 μ) − ( ≡ [C]. ( )2 )2 − S2 + α I 2 1 + b I2 2I a

Assuming the function Q = Q(I, R) =

⎡S

.

I

0

0

⎤ ,

S I

hence, we get

Q

.

−1

=

⎛I S

0

0 I S

⎛

⎞ ,

Qf =

˙ I˙ I S−S I2

0

0 ˙ I˙ I S−S I2

⎞ ,

Nonlinear Dynamics in an SIR Model. . .

⎛ Qf Q

.

−1

=

S˙ S

− 0

I˙ I

67

⎞ S˙ S

0 −

I˙ I

QJ [2] Q−1 =

and

,

⎛

⎞ C0 . 0C

We define the function B = Qf Q

.

−1

+ QJ

where B11 = B22 =

.

[2]

Q

−1

⎛ =

B11 B12 B21 B22

⎞ ,

S˙ I˙ − + C, B12 = B21 = 0. S I

From the second equation of model system (2), we can easily find that .

I˙ aI βS 2 − (μ + δ) − . = 2 2 I S + αI 1 + bI 2

By using above expression in B11 , B22 , we get S˙ aI 2I a βS 2 +μ+δ+ − (δ + 2 μ) − ( − 2 )2 S S + αI 2 1 + bI 2 1 + b I2 )) ( ( S β 2 I 3 α + S I 2 α − S2 − . ( )2 S2 + α I 2

B11 = B22 =

.

Now, ν(B) ≤ sup{g1 , g2 } ≡ sup{ν1 (B11 ) + |B12 |, ν1 (B22 ) + |B21 |},

.

where ν1 denotes the Lozinskii measure with respect to the L1 norm and |B12 |, |B21 | are matrix norms with respect to the L1 vector norm. So, we have ν(B) ≤

.

⎧ ⎫ S˙ βS 2 βS(2αI 3 + S(αI 2 − S 2 ) aI (1 − bI 2 ) + + − μ+ . S (1 + bI 2 )2 S 2 + αI 2 (S 2 + αI 2 )2

(9)

Using Lemma 1, the above Eq. (9) becomes ν(B) ≤

.

Take Δ =

4αβ (1+α)2

+μ+

S˙ ⎛ 4αβ ac(1 − bc2 ) ⎞ + μ + − . S (1 + α)2 (1 + bc)2

ac(1−bc2 ) (1+bc2 )2

> 0, then we have

ν(B) ≤

.

S˙ − Δ. S

(10)

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Integrating the above Eq. (10), we get 



t

ν(B)ds ≤

.

0

=⇒

0

1 t



t

ν(B)ds ≤

0

.

=⇒

 t S˙ Δdt, dt − S 0

t

lim sup sup

t→∞

1 t



1 S(t) log − Δ, t S(0) t

ν(B)ds ≤ −Δ ≤ 0,

0

t as S(t) is bounded and Δ > 0. Hence q˜2 = limt→∞ sup sup 1t 0 ν(B)ds < 0, if Δ > 0. Thus, the system (3) is globally asymptotically stable for R0 > 1, that is, (S, I ) → (S1∗ , I1∗ ) as t → ∞. Hence, the theorem follows. ⨆ ⨅

2.8 Direction of Bifurcation at R0 = 1 Theorem 5 The model system (2) undergoes transcritical bifurcation at R0 = 1. Proof We prove the theorem by using the method of Castillo-Chavez and Song [34]. We consider β as the bifurcation parameter and by R0 = 1, we get β = β∗ = μ + δ. The model system (2) can be rewritten as:

.

dS βS 2 I + γ R := f1 , = Λ − μS − 2 dt S + αI 2 dI aI βS 2 I − (μ + δ)I − = 2 := f2 . 2 dt 1 + bI S + αI

(11)

The Jacobian matrix at (E0 , β∗ ) is given by ⎡ JE0 =

.

⎤ −μ −β∗ . 0 0

This Jacobian matrix has one negative eigenvalue −μ and one simple zero eigenvalue at R0 = 1. The right and left eigenvectors corresponding to the zero eigenvalue are u = (−(μ + δ), μ)T and v = (0, 1), respectively. Here,

.

a 1 = u21

2 ∂ 2 f2 ∂ 2 f2 2 ∂ f2 + 2u u = −2au22 > 0 + u 1 2 2 ∂S∂I ∂S 2 ∂I 2

b1 = u2 > 0.

Nonlinear Dynamics in an SIR Model. . .

69

Now using Theorem 4.1 of [34], we note that the model system (2) undergoes transcritical bifurcation at R0 = 1. Hence, the theorem follows. ⨆ ⨅

3 Numerical Simulation Example 1 In this example, we choose a set of parameters as: Λ = 0.760, μ = 0.001, β ∈ (0.0019074119, 0.0019269), a = 0.00017, δ = 0.0009151, b = 0.25, α = 0.1 so that R0 ∈ (1, 1.0061). For this set of parameters, there exists a unique endemic steady state E1∗ for R0 > 1. The Fig. 2a depicts the existence of transcritical bifurcation as R0 crosses unity in this range of β. It is observed that the endemic steady state E1∗ is stable. Further, we fix the incidence rate β = 0.001922119 (so that R0 = 1.0037) and plot the solution trajectories corresponding to different initial conditions, shown in Fig. 2b. We observe that if we start from any of these initial values, the solution trajectories converge to endemic steady state E1∗ = (759.921, 0.0413058, 0.000289926), which mimics the stability of E1∗ . Example 2 In this example, we choose a set of parameters as: Λ = 160, μ = 0.0925, β ∈ (0.189615, 0.206595), a = 0.017, δ = 0.0971, b = 0.25, α = 0.1 so that R0 ∈ (1.0001, 1.14). We divided the range of β into three sub interval such as β1 = (0.189615, 0.19026], β2 = (0.19026, 0.206595] and β3 = (0.206595, 0.206595). In the range β1 and β3 , a unique endemic steady state E6∗ and E4∗ exists, respectively, whereas in β2 , three endemic steady states E4∗ , E5∗ and E6∗ exist. The endemic steady states E4∗ , E6∗ are stable but E5∗ is unstable. These results are shown in Fig. 3a. Now, we fixed β = 0.20136, so that R0 = 1.0620 within the range of vertical line in Fig. 3a. In this case, the endemic steady states E4∗ = 0.07

0.06

0.05

R0=1 E*1 Stable

I

0.04

0.03

0.02

0.01

E0 Unstable

E0 Stable

0 0.996 0.997 0.998 0.999

1

1.001 1.002 1.003 1.004 1.005 1.006

R0

(a)

(b)

Fig. 2 (a) Existence of transcritical bifurcation. (b) Dynamics of solution trajectories for different initial values at β = 0.001922119

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A. Srivastava and P. K. Srivastava

E*4 Stable

102

E*5 Unstable

I

E*6 Stable

100

10-2

1

1.02

1.04

1.06

1.08

1.1

1.12

R0

(a)

(b)

Fig. 3 (a) Bifurcation plot on semi-log scale for the case R0 > 1 where three endemic equilibrium points exist. (b) Phase portrait of the system (1) at β = 0.20136

(663.7906, 519.6802, 0.7351), and E6∗ = (1728, 0.8034, 0.1021) are stable and E5∗ = (1718.9, 4.979, 0.6330) is unstable. Therefore, the solution trajectories starting from any point approach the stable endemic steady states, i.e., they either converge to E4∗ or E6∗ . This bi-stability behavior is seen in Fig. 3b. Interestingly, in β2 , R0 ∈ (1.0036, 1.0896), the model system exhibits “hysteresis” as shown in Fig. 3a. In this case, the upper and lower endemic steady states are stable, while the middle one is unstable. Therefore, whether there is an increase or a reduction in the R0 , the infections will continue to proceed on a stable steady state until they reach the end point of the branch and then it will make a transition to another stable branch. This important phenomenon shows that the system is sensitive to change of parameters and the interesting dynamical behaviour is observed.

4 Conclusion In this work, we have presented an SI R model, which includes a ratio-dependent incidence rate function and a Holling type III treatment rate function. The basic reproduction number, denoted by .R0 , was subjected to a sensitivity analysis, and the results show that it is particularly sensitive to the infection rate, the natural death rate, and the disease-related mortality rate. We found that when .R0 > 1, there is the possibility for multiple endemic steady states to coexist. The local and global stability of the model’s steady states have been determined. In addition, the situation of transcritical bifurcation has been achieved unconditionally. If certain conditions are met, it is determined that the unique endemic steady state is globally asymptotically stable. Further numerical explorations reveal the bi-stability and hysteresis in our model system.

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References 1. Vincenzo Capasso and Gabriella Serio. A generalization of the kermack-mckendrick deterministic epidemic model. Mathematical biosciences, 42(1–2):43–61, 1978. 2. Wei-min Liu, Simon A Levin, and Yoh Iwasa. Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models. Journal of mathematical biology, 23(2):187–204, 1986. 3. Wei-min Liu, Herbert W Hethcote, and Simon A Levin. Dynamical behavior of epidemiological models with nonlinear incidence rates. Journal of mathematical biology, 25(4):359–380, 1987. 4. Herbert W Hethcote and P Van den Driessche. Some epidemiological models with nonlinear incidence. Journal of Mathematical Biology, 29(3):271–287, 1991. 5. WR Derrick and P Van den Driessche. A disease transmission model in a nonconstant population. Journal of Mathematical Biology, 31(5):495–512, 1993. 6. Shigui Ruan and Wendi Wang. Dynamical behavior of an epidemic model with a nonlinear incidence rate. Journal of Differential Equations, 188(1):135–163, 2003. 7. ME Alexander and SM Moghadas. Periodicity in an epidemic model with a generalized nonlinear incidence. Mathematical Biosciences, 189(1):75–96, 2004. 8. Dongmei Xiao and Shigui Ruan. Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical biosciences, 208(2):419–429, 2007. 9. Dhiraj Kumar Das and TK Kar. Global dynamics of a tuberculosis model with sensitivity of the smear microscopy. Chaos, Solitons & Fractals, 146:110879, 2021. 10. Hu Zhang, V Madhusudanan, BSN Murthy, MN Srinivas, and Biruk Ambachew Adugna. Fuzzy analysis of svirs disease system with holling type-ii saturated incidence rate and saturated treatment. Mathematical Problems in Engineering, 2022, 2022. 11. Anuj Kumar and Prashant K Srivastava. Role of optimal screening and treatment on infectious diseases dynamics in presence of self-protection of susceptible. Differential Equations and Dynamical Systems, pages 1–29, 2019. 12. Yilei Tang, Deqing Huang, Shigui Ruan, and Weinian Zhang. Coexistence of limit cycles and homoclinic loops in a sirs model with a nonlinear incidence rate. SIAM Journal on Applied Mathematics, 69(2):621–639, 2008. 13. Sanling Yuan and Bo Li. Global dynamics of an epidemic model with a ratio-dependent nonlinear incidence rate. Discrete Dynamics in Nature and Society, 2009, 2009. 14. Udai Kumar, Partha Sarathi Mandal, Jai Prakash Tripathi, Vijay Pal Bajiya, and Sarita Bugalia. Sirs epidemiological model with ratio-dependent incidence: Influence of preventive vaccination and treatment control strategies on disease dynamics. Mathematical Methods in the Applied Sciences, 44(18):14703–14732, 2021. 15. Jin Gao and Min Zhao. Stability and bifurcation of an epidemic model with saturated treatment function. In International Conference on Information and Management Engineering, pages 306–315. Springer, 2011. 16. Akriti Srivastava, Prashant K Srivastava, et al. Nonlinear dynamics of a siri model incorporating the impact of information and saturated treatment with optimal control. The European Physical Journal Plus, 137(9):1–25, 2022. 17. Tanuja Das, Prashant K Srivastava, and Anuj Kumar. Nonlinear dynamical behavior of an seir mathematical model: Effect of information and saturated treatment. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(4):043104, 2021. 18. Preeti Dubey, Balram Dubey, and Uma S Dubey. An sir model with nonlinear incidence rate and holling type iii treatment rate. In Applied Analysis in Biological and Physical Sciences, pages 63–81. Springer, 2016. 19. Abhishek Kumar et al. Dynamical model of epidemic along with time delay; holling type ii incidence rate and monod–haldane type treatment rate. Differential Equations and Dynamical Systems, 27(1):299–312, 2019.

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20. Jayanta Kumar Ghosh, Uttam Ghosh, MHA Biswas, and Susmita Sarkar. Qualitative analysis and optimal control strategy of an sir model with saturated incidence and treatment. Differential Equations and Dynamical Systems, pages 1–15, 2019. 21. Anuradha Yadav and Prashant K Srivastava. The impact of information and saturated treatment with time delay in an infectious disease model. Journal of Applied Mathematics and Computing, 66(1):277–305, 2021. 22. Xu Zhang and Xianning Liu. Backward bifurcation of an epidemic model with saturated treatment function. Journal of mathematical analysis and applications, 348(1):433–443, 2008. 23. Anuj Kumar, Prashant K Srivastava, and RP Gupta. Nonlinear dynamics of infectious diseases via information-induced vaccination and saturated treatment. Mathematics and Computers in Simulation, 157:77–99, 2019. 24. Balram Dubey, Atasi Patra, PK Srivastava, and Uma S Dubey. Modeling and analysis of an seir model with different types of nonlinear treatment rates. Journal of Biological Systems, 21(03):1350023, 2013. 25. Ebenezer Bonyah, Isaac Dontwi, and Farai Nyabadza. A theoretical model for the transmission dynamics of the buruli ulcer with saturated treatment. Computational and Mathematical Methods in Medicine, 2014, 2014. 26. Abhishek Kumar, Manoj Kumar, et al. A study on the stability behavior of an epidemic model with ratio-dependent incidence and saturated treatment. Theory in Biosciences, 139(2):225– 234, 2020. 27. Pauline Van den Driessche and James Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1–2):29–48, 2002. 28. Nakul Chitnis, James M Hyman, and Jim M Cushing. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bulletin of mathematical biology, 70(5):1272–1296, 2008. 29. Carlos Castillo-Chavez, Zhilan Feng, Wenzhang Huang, et al. On the computation of r0 and its role in global stability. IMA Volumes in Mathematics and Its Applications, 125:229–250, 2002. 30. Herb I Freedman, Shigui Ruan, and Moxun Tang. Uniform persistence and flows near a closed positively invariant set. Journal of Dynamics and Differential Equations, 6(4):583–600, 1994. 31. Michael Y Li and James S Muldowney. A geometric approach to global-stability problems. SIAM Journal on Mathematical Analysis, 27(4):1070–1083, 1996. 32. William Andrew Coppel. Stability and asymptotic behavior of differential equations. Heath, 1965. 33. James S Muldowney. Compound matrices and ordinary differential equations. The Rocky Mountain Journal of Mathematics, pages 857–872, 1990. 34. Carlos Castillo-Chavez and Baojun Song. Dynamical models of tuberculosis and their applications. Mathematical Biosciences & Engineering, 1(2):361, 2004.

Comparative Study of Deterministic and Stochastic Predator Prey System Incorporating a Prey Refuge Anal Chatterjee and Samares Pal

1 Introduction Understanding the dynamical interaction between predator and prey is a major goal in ecology. In predator-prey system, there are primarily two techniques of capturing the influence of predator on prey. Predators hunt and consume prey with the first approach, which can be observed in nature [1]. In the second approach, the presence of a predator may significantly alter prey behaviour due to the fear of predation [2]. According to a recent new point of view, indirect influence can also influence the dynamics of a predator–prey system [3–5]. Because of the threat of predation, prey populations may shift their grazing zones to a safer location, sacrificing their peak intake rate areas, enhance their vigilance, adapt their reproductive tactics, and so on. The startled prey’s reproduction reduces as a result of this form of survival strategy [6–8]. Despite the fact that the fear effect has been studied for many years, the authors in [9] provided that fear of predation decreases the birth rate of prey populations in 2016. They observed that cost of fear has no influence to change the dynamical behavior for Holling I functional response, but it can stabilize the whole system by avoiding periodic solution in presence of Holling II response function. Many scientists examined other predator–prey models after this study by include the “cost of fear” term in prey’s growth function. The researchers in [10] presented a predator prey model and analyzed that at high predator density, cost of fear stabilises the dynamics. In three species food chain, the authors in [8] considered

A. Chatterjee () Department of Mathematics, Barrackpore Rastraguru Surendranath College, North 24 Parganas, Kolkata, India S. Pal Department of Mathematics, University of Kalyani, Kalyani, India © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_5

73

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two different “cost of fear” terms in growth function of prey and middle predator. Fear can stabilise a chaotic system, according to their results. Aside from the cost of fear, several other key aspects such as prey refuge [11], Allee effect [12, 13], harvesting [14] and additional food [15, 16] have a substantial impact on how a dynamical system behaves. One of the hot places in biomathematics has been the study of a prey-predator system with prey refuge, and several scientists have made significant progress in this area [17–19]. In their investigations, the majority of authors [20–22] have assumed that the prey refuge is either constant or proportionate to the prey volume. Our assumption, however, is rather different. We have assumed that the amount of prey refuge is directly proportional to both prey and predator density [23–25]. This scheme is more realistic than earlier refuge systems. The scientist in [17] has shown that the prey refuge threshold can affect the all species for long-term survival in a food chain model. The authors in [26] investigated a predator-prey model that included prey refuge and supplementary food for the predator, and discovered that somewhat higher prey refuge values allow stable species coexistence. At a strong prey refuge, however, the predator becomes extinct. The predator’s rate of prey consumption also known as the predator’s functional response, is the most important aspect in the prey predator relationship. In ecology, many forms of response functions exist in the dynamics of interacting populations. The authors in [27–29] proposed the Beddington–DeAngelis response function, which is a variant of the Holling type II functional response [30]. This included an extra concept describing predators’ mutual interference. In biology, however, the deterministic approach has some drawbacks. It is hard to precisely foresee the system’s future. In comparison to its deterministic counterpart, a stochastic model reflects a natural system more accurately. In some studies [31, 32], Gaussian white noise is included in a model of environmental fluctuations to assess the impact of noise on dynamical systems. To our knowledge, no one has explored such a model using Gaussian white noises, and color noise which have been shown to be particularly beneficial in modelling fast fluctuating phenomena in the presence of refuge, fear factor and mutual predator interference in system. In this paper, following Wiener and Ornstein-Uhlenbeck processes, these fluctuations are expressed as white and colored noises [33, 34]. A study of the model system’s mean square stability in the presence of both white and color noises was also conducted, revealing that color noise had a stabilising impact when compared to white noise. The purpose of this study is to look into the effects of hunting, the fear effect, and predator interference. To my knowledge, the combined effect of the three components mentioned above has yet to be investigated. We also, compare the model system’s stability in deterministic and stochastic situations. The major goal of this paper is to look into the following biological topics. • How does the rate at which a predator captures a prey affect the dynamics of the prey-predator system? • Can the cost of fear enhance the prey-predator system’s stability?

Comparative Study of Deterministic and Stochastic. . .

75

• Is it possible for environmental noise to alter the prey-predator system’s stability? The fear of predator hunting is considered to be lowering the birth rate of prey populations. The proposed model is extended with incorporate prey refuge and predator interference. Section 2 is depicted the construction of a mathematical model based on a set of assumptions. In Sects. 3, 4 and 5, we looked at preliminary outcomes such as boundedness, persistence as well as existence and stability of equilibria. In Sect. 6, local bifurcation analysis, such as Hopf and transcritical bifurcation analysis, are discussed. In Sect. 7, global sensitivity analysis is discussed. In Sects. 8 and 9 existence of stochastic stability and some properties are explained in presence of white and color noises respectively. Finally, Sects. 10 and 11 is illustrated numerical simulations and brief discussion.

2 Basic Assumptions and Model Formulation We develop a predator–prey model that includes the following features: (i) The response function of Holling II [35]. (ii) A prey species’ logistic growth function (iii) Prey refuge varies depending on both species (iv) Prey fear, according to the anti-predator behavior. It is also assumed that population number varies exclusively over time and is unaffected by abiotic environmental influences. Here the intrinsic growth rate and the intra-species competition rate of prey are r and .r1 , respectively. When a predator is present, intrinsic prey growth becomes a function of predator r density, such as .F (y; K) = 1+Ky where K is the level of prey fear as determined by anti-predator reaction. This above function follows some conditions: (i) .F (y; 0) = r: the prey reproduction rate remains unchanged in the absence of the fear effect. (ii) .F (0; K) = r: the prey reproductive rate remains unchanged in the absence of predators. (iii) . lim F (y; K) = 0: fearful prey is unable to reproduce. K→∞

(iv) . lim F (y; K) = 0: prey cannot reproduce when predator density is exceedy→∞

ingly high. ∂F (y;K) .(v) . < 0: with a high level of fear, the prey reproduction rate is low. ∂K ∂F (y;K) .(vi) . < 0: with a high predator density, the prey reproduction rate is low. ∂y Here, .δ1 xy be the amount of prey refuge, in which .δ1 be the refuge coefficient. Therefore, the remaining .(x − δ1 xy) prey species are exposed to predation by predators. A realistic ecosystem’s allowable range of refuge are .0 < δ1 < 1 and .0 ≤ (1 − δ1 y) ≤ 1. Here, .m1 , .a1 , .b1 represent as the rate of prey capture by the predator, half saturation parameter for prey, handling time on the feeding rate effort while .e1 and .d1 be the conversion efficiency factor of prey into new predator and predators’ natural mortality rate respectively.

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rx dx m1 (x − δ1 xy)y = − r1 x 2 − ≡ G1 (x, y) dt 1 + Ky a1 + b1 (x − δ1 xy) .

dy e1 m1 (x − δ1 xy)y − d1 y ≡ G2 (x, y). = a1 + b1 (x − δ1 xy) dt

(1)

We analyze the system (1) with the following initial conditions, x(0) ≥ 0, y(0) ≥ 0.

.

(2)

3 Mathematical Preliminaries Proposition 1 All non negative solutions (x(t), y(t)) of the system (1) initiate in 2 − {0, 0} are uniformly bounded. R+ Proof Let us choose a function  = x + y. Therefore, .

dx dy rx e1 m1 (x − δ1 xy)y d m1 (x − δ1 xy)y = + = −r1 x 2 − + −d1 y. dt dt dt 1 + Ky a1 + b1 (x − δ1 xy) a1 + b1 (x − δ1 xy)

Let us consider a positive constant ζ such that ζ ≤ d1 . Therefore, .

d m1 (1 − e1 )(x − δ1 xy)y − y(d1 − ζ ) + ζ  ≤ rx − r1 x 2 + ζ x − a1 + b1 (x − δ1 xy) dt .

By choosing  =

≤ (r + ζ )x − r1 x 2 ≤

(r+ζ )2 4r1 ,

we obtain

0 ≤ (x(t), y(t)) ≤

.

(r + ζ )2 . 4r1

 (1 − e−ζ t ) + (x(0), y(0))e−ζ t , ζ

which indicates that 0 ≤ (x(t), y(t)) ≤ ζ as t → ∞. Therefore, all non negatives 2 − {0, 0} will be restricted in the solutions of the system (1) are originated from R+  2 region ∇ = {(x, y) ∈ R+ : x(t) + y(t) ≤ ζ + ε}. It signifies that the system behaves in a specific way in ecology. The system’s boundedness implies that neither of the two interacting species grows unexpectedly or exponentially for an extended period of time. Clearly, the numbers of each species are confined due to limited resources.



Comparative Study of Deterministic and Stochastic. . .

77

4 Persistence It is necessary to demonstrate the positivity of system (1) since it suggests that the population will survive. The system is said to be persistent if a compact set .D1 ⊂ 1 = {(x, y) : x > 0, y > 0} exists in which the solutions of the system (1) ultimately enter and remain in the region. Proposition 2 The system (1) is persistence if the following conditions (i) .r > d1 (ii) .m1 > e11 [ a1 dr1 r1 + b1 d1 ] hold. Proof Take a Lyapunov function .V1 (x, y) = x ζ1 y ζ2 to demonstrate the persistent [36] where .ζ1 and .ζ2 are real constants. As a result,  the average Lyapunov function  ˙ r 1 (1−δ1 y)y + − r1 x − a1m+b looks like this: .(x, y) = VV11 = ζ1 xx˙ + ζ2 yy˙ = ζ1 1+Ky (x−δ xy) 1 1   m1 (x−δ1 xy) ζ2 ae11+b − d1 . Now, we have to show that the function is positive, at 1 (x−δ1 xy) each boundary equilibria. At, .E0 , the trivial equilibrium, the value of .(0, 0) = ζ1 r − ζ2 d1 . Let .ζ1 = ζ2 = ζ , then .(0, 0) = ζ (r − d1 ) > 0 if the condition r .r > d1 holds. Similarly, at .E1 , the predator free equilibria, we have .( , 0) = r1

m1 r ζ ( are11 +b − d1 ) > 0 if the condition .m1 > e11 [ a1 dr1 r1 + b1 d1 ] holds. It shows that 1r .(x, y) is positive at each boundary equilibria. It indicates that the system (1) is persistent. As the system is uniformly persistent there exist .ρ1 > 0 and .t > t1 such that .x(t) > ρ1 and .y(t) > ρ1 . ∀t > t1 .



5 Equilibria: Existence and Stability All possible equilibria are catalogued below: (i) The trivial equilibrium .E0 = (0, 0), (ii) The predator free equilibrium .E1 = ( rr1 , 0). (iii) The positive coexistence equilibrium .E ∗ = (e1 m1 −b1 d1 )x ∗ −a1 d1 ∗ δ1 (e1 m1 −b1 d1 )x ∗ . Also, .x is ensured by solving 4  .

(x ∗ , y ∗ ), while .y ∗

An Xn = 0.

n=0

The coefficient of .An s (n=0 to 4) are defined below. A4 A3 .A2 .A1 .A0 . .

= r1 e1 δ1 (e1 m1 − b1 d1 )2 (δ1 + K), = −d1 δ1 Ka1 r1 e1 (e1 m1 − b1 d1 ) − e1 rδ12 (e1 m1 − b1 d1 )2 , = (e1 m1 − b1 d1 )2 (δ1 + K)d1 , = −(e1 m1 − b1 d1 )[2Ka1 + a1 δ1 ]d12 , = Ka12 d13 .

=

(3)

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A. Chatterjee and S. Pal

Now, we need to find at least one positive root of the Eq. (3). In general, the Eq. (3) has at most four complex roots. Assuming one pairs of complex roots exist which are . α and it’s conjugate .α∗ . The following quadratic equations .X2 + m 1 X + n1 = (X −  α )(X − α∗ ) = X2 − 2Re( α )X + | α |2 is formed by conjugate pair in which .m 1 = −2Re( α ),  .n1 = | α |2 . Assuming that there are two real roots .x1∗ and .x2∗ of ∗ ∗ ∗ .x of the Eq. (3) such that .(x + x ) = − p1 , and .x1∗ x2∗ =  q1 . Consequently, the 1 2 factorization of Eq. (3) is as follows: 4  .

An Xn = A4 (X2 + m 1 X +  n1 )(X2 + p 1 X +  q1 ) =

n=0

  A4 X4 +( p1 + m1 )X3 +( n1 + q1 + m1 p 1 )X2 +( m1 q1 +  n1 p 1 )X +  n1 q1 . 1 = When one compares coefficients on both sides, they discover that .p and . q1 =

A0 . A4 | α |2

(4)

A3 α) A4 +2Re(

Since, . q1 > 0 then both real roots are positive if satisfy the

following conditions .p  < 0 and .p 12 − 4 q1 > 0. Therefore, there exist two positive 1 2 −4 q1 − p1 + p

− p1 − p 2 −4 q1

1 1 and .x2∗ = , only if .p 12 − 4 q1 > 0 holds real roots .x1∗ = 2 2 since as .A3 < 0. Thus the condition for the existence of the coexistence equilibrium d1 point .E ∗ (x ∗ , y ∗ ) is given by, (a) .x ∗ > e1 ma11−b 12 − 4 q1 > 0. , (b) .m1 > b1e1d1 (c) .p 1 d1 Figure 1 depicts the possibility of a coexistence equilibrium. Explicitly, general form of the Jacobian matrix at .E = (x, y) is defined as

0.45 0.4 0.35

Predator (y)

0.3 0.25

E* (0.29,0.20)

0.2 0.15 0.1 E1(5.3,0)

0.05 E0(0,0) 0 0

1

2

3

4

5

6

Prey (x)

Fig. 1 Mutual position of prey-nullclines (green) and predator-nullclines (red) of the system for the values of the reference parameters given in (33)

Comparative Study of Deterministic and Stochastic. . .



a 11 a 12 .J = a 21 a 22

79

 (5)

m1 a1 (1−δ1 y)y r (1+Ky) − 2r1 x − [a1 +b1 (x−δ1 x y)]2 , mx[a1 (1−2δ1 y)+(1−δ1 y)(bx−b1 δ1 x y)] rxK , − (1+Ky) 2 − [a1 +b1 (x−δ1 x y)]2 e1 m1 a1 (1−δ1 y)y , [a1 +b1 (x−δ1 x y)]2 e1 m1 x[a1 (1−2δ1 y)+(1−δ1 y)(b1 x−b1 δ1 x y)] − d1 . [a1 +b1 (x−δ1 x y)]2 exists a feasible trivial equilibrium .E0 of the

where .a 11 = a 12 =

.

a 21 =

.

a 22 = There system (1) which is always unstable. Since one of eigenvalue is .r > 0 which is evaluating from (5). The 1r system (1) at .E1 is unstable if .R1 = d1 (ae11rm1 +b > 1. 1 r) ∗ The Jacobian matrix at .E can be written as 

a12  a11  ∗ , .J =  a21  a22

.

m1 b1 x ∗ y ∗ (1−δ1 y ∗ )2 [a1 +b1 (x ∗ −δ1 x ∗ y ∗ )]2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ rx ∗ K 1 y )+(1−δ1 y )(b1 x −b1 δ1 x y )] ,  .a12 = − − m1 x [(a1 +c1 y )(1−2δ [a1 +b1 (x ∗ −δ1 x ∗ y ∗ )]2 (1+Ky ∗ )2 e1 m1 a1 (1−δ1 y ∗ )y ∗ ,  .a21 = a1 +b1 (x ∗ −δ1 x ∗ y ∗ )]2 e1 m1 x ∗ [a1 (1−2δ1 y ∗ )+(1−δ1 y ∗ )(b1 x ∗ −b1 δ1 x ∗ y ∗ )] − d1 . Thus the eigenvalues  .a22 = [a +b (x ∗ −δ x ∗ y ∗ )]2 ∗ where  .a11 = −r1 x +

1

1

1

in this

case are obtained as roots of the quadratic λ2 − tr(J ∗ ) + det (J ∗ ) = 0, ∗ .tr(J ) = −( a11 +  a22 ), ∗ .det (J ) =  a11 a22 −  a12 a21 . Now if .tr(J ∗ ) < 0 as well as .det (J ∗ ) > 0 then according Routh–Hurwitz criterion we can admit that .E ∗ is locally asymptotically stable which depends upon system parameters. Therefore, we satisfy the above conditions with help of numerical simulation. .

6 Local Bifurcation 6.1 Hopf-Bifurcation Proposition 3 The necessary and sufficient conditions for Hopf bifurcation of the system (1) around E ∗ at m1 = mc1 are [tr(J ∗ )]m1 =mc1 = 0, [det (J ∗ )]m1 =mc1 > 0 d and dm [tr(J ∗ )]m1 =mc1 = 0. 1 Proof The condition [tr(J ∗ )]m1 =mc1 = 0 gives

∗ ∗ ∗ ∗ ∗ ∗ m1 b1 x ∗ y ∗ (1−δ1 y ∗ )2 1 y )+(1−δ1 y )(b1 x −b1 δ1 x y )] + e1 m1 x [a1 (1−2δ − d1 = [a1 +b1 (x ∗ −δ1 x ∗ y ∗ )]2 [a1 +b1 (x ∗ −δ1 x ∗ y ∗ )]2 ∗ c Now [det (J )]m1 =m1 > 0 which is equivalent to the characteristic equation λ2 [det (J ∗ )]m1 =mc1 = 0 whose roots are purely imaginary,

−r1 x ∗ +

0. +

80

A. Chatterjee and S. Pal

For m1 = mc1 , the characteristic can be written as χ 2 + ω = 0,

(6)

.

where ω = [det (J ∗ )]m1 =mc1 > 0. Therefore, the above equation has two roots of √ √ the form χ1 = +i ω and χ2 = −i ω. Let at any neighbouring point m1 of mc1 , we can express the above roots in general form like χ1,2 = θ1 (m1 ) + ±iθ2 (m1 ), where θ1 (m1 ) =

tr(J ∗ ) 2

and θ2 (m1 ) =

det (J ∗ ) −

tr(J ∗ ) 4 .

Now it is to be verified

d the transversality condition dm (Re(χj (m1 )))m1 =mc1 = 0 for j = 1, 2. 1 Substituting χ1 = θ1 (m1 ) + iθ2 (m1 ) in (6) and calculate the derivative,

we have

2θ1 (m1 )θ1 (m1 ) − 2θ2 (m1 )θ2 (m1 ) + ω = 0,

.

2θ2 (m1 )θ1 (m1 ) + 2θ1 (m1 )θ2 (m1 ) = 0. Solving (7), we get d c dm1 (Re(χj (m1 )))m1 =m1 =

−2θ1 ω 2(θ12 +θ22 )

= 0, i.e.,

d ∗ c dm1 [tr(J )]m1 =m1

(7)

= 0, which

satisfy the transversality condition. This implies that the system undergoes a Hopf

bifurcation at m1 = mc1 .

6.2 Direction of Hopf Bifurcation By using .m1 as a bifurcation parameter in the previous theorem, we can see that the system (1) exhibits Hopf bifurcation. The direction and stability aspects of bifurcating periodic solutions arising from the coexisting equilibrium point .E ∗ via Hopf bifurcation are discussed in this paper. We first calculate the Lyapunov coefficient and then apply the theorem stated in [37] to explore the stability and direction of Hopf bifurcation. First, we turn the equilibrium point of the system (1), .E ∗ (x ∗ , y ∗ ), into the origin ∗ .z = y − y ∗ . Then the system (1) becomes .z1 = x − x ,  by allowing  2 r( z1 + x ∗ ) m1 ( d z1 z1 + x ∗ )(1 − δ1 ( z2 + y ∗ ))( z2 + y ∗ ) ∗ 2 = − r , ( z + x ) − 1 1 dt 1 + K( z2 + y ∗ ) a1 + b1 ( z1 + x ∗ )(1 − δ1 ( z2 + y ∗ )) .

d z2 z1 + x ∗ )(1 − δ1 ( z2 + y ∗ ))( z2 + y ∗ ) e1 m1 ( − d1 ( = z2 + y ∗ ). a1 + b1 ( z1 + x ∗ )(1 − δ1 ( z2 + y ∗ )) dt

The following system is obtained by expanding Taylor’s series from the above system at .( z1 , z2 ) = (0, 0) up to terms of order 3:  z2 + c20 z12 + c11 z1 z2 + c02 z22 + c30 z13 + c21 z12 z2 + c12 z1 z22 .z˙1 = c 10 z1 + c01 + c03 z23 + O(| z|4 ),

Comparative Study of Deterministic and Stochastic. . .

81

 z˙2 = d 10 z2 + d 20 z12 + d 11 z1 z2 + d 02 z22 + d 30 z13 + d 21 z12 z2 + d 12 z1 z22 z1 + d 01 + d 03 z23 + O(| z|4 )

(8)

m c2 b x ∗ y ∗ m1 c11 y ∗ r ∗ + 1 11A21 , A 1+Ky ∗ − 2r1 x − m1 (c11 x ∗ −δ1 x ∗ y ∗ ) m1 c11 x ∗ y ∗ b1 δ1 x ∗ rKx ∗ .c 01 = − − (1+Ky ∗ )2 − A A2 2 ∗ m1 c11 b1 y m1 A11 c11 x ∗ y ∗ .c 20 = −r1 + − A2 A3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ rK 1 δ1 x y −b1 δ1 x y ] .c 11 − = − (1+Ky ∗ )2 − m1 (c11A−δ1 y ) + m1 c11 [b1 (c11 x −b1 x yA)−b 2 m1 c11 A12 x ∗ y ∗ A3 ∗ ∗ ∗ ∗ ∗ ∗ δ1 x ∗ rK 2 x ∗ + m1 A − m1 (c11 x −δA1 2x y )b1 δ1 x − A22 m1Ac311 x y .c 02 = (1+Ky ∗ )3 m b c2 x ∗ y ∗ m1 c11 y ∗ A11 .c 30 = − + 1 1 A114 A3 2 x∗y∗] m [(c x ∗ −δ1 x ∗ y ∗ )A11 +c11 y ∗ A12 −2b12 δ1 c11 x∗y∗ m1 c11 b1 [(c11 −2δ1 y ∗ ) + m1 c11Ap23 .c 21 = − 1 11 4 A3 A2 m1 [b1 δ12 x ∗ y ∗ −(c11 −δ1 y ∗ )b1 δ1 x ∗ −2c11 b1 δ1 x ∗ ] m1 δ1 rK 2 + − .c 12 = + ∗ 3 A A2 (1+Ky ) m1 [c11 A22 y ∗ +A32 c11 x ∗ y ∗ +(c11 x ∗ −δ1 x ∗ y ∗ )A12 ] m1 c11 p24 x ∗ y ∗ . + , A3 A4 m1 (c11 x ∗ −δ1 x ∗ y ∗ )A22 x∗y∗ m1 δ1 x ∗ b1 δ1 x ∗ rK 3 x ∗ − .c 03 = − − + m1 c11Ap22 , 4 (1+Ky ∗ )3 A2 A3 e1 m1 c11 y ∗ .d 10 = , A c11 x ∗ −δ1 x ∗ y ∗ δ1 x ∗2 y ∗ .d 01 = e1 m1 [ + c11 b1A − e1dm1 1 ], 2 A ∗ ∗ ∗ b1 c11 y .d 20 = e1 m1 c11 [− + A11Ax3 y ], A2 ∗ ∗ c11 −δ1 y ∗ b1 c11 (c11 x ∗ −b1 x ∗ y ∗ )−c11 b1 δ1 x ∗ y ∗ −c11 b1 d1 x ∗ y ∗ .d 11 = e1 m1 [ + c11 AA123x y ], − A A2 ∗ ∗ ∗ ∗ ∗ ∗ (c11 x −δ1 x y )b1 δ1 x δ1 x ∗ .d 02 = e1 m1 [− + c11 AA223x y ], A + A2 b c2 x ∗ y ∗ c11 y ∗ A11 .d 30 = e1 m1 [ − 1 11A4 ], A3 2 x∗y∗ ∗ ∗ (c x ∗ −δ1 x ∗ y ∗ )A11 +c11 y ∗ A12 −2b12 δ1 C11 b1 c11 (c11 −δ1 y ∗ ) .d 21 = e1 m1 [− − c11 pA234x y ], + 11 A3 A2 (1−2δ1 y ∗ )b1 δ1 x ∗ +b1 δ1 (c11 x ∗ −δ1 x ∗ y ∗ ) δ1 .d 12 = e1 m1 [− A + A2 ∗ ∗ c11 A22 y ∗ +A32 c11 x ∗ y ∗ +A12 (c11 x ∗ −δ1 x ∗ y ∗ ) + − c11 PA244x y ], A3 ∗ ∗ ∗ ∗ ∗ b1 δ12 x ∗2 + ( c11 x −δA1 x3 y )A22 − c11 pA224x y ], .d 03 = e1 m1 [ A2 in which .A = a1 + b1 x ∗ − b1 δ1 x ∗ y ∗ , 2 2 2 ∗ 2 ∗ ∗ .c11 = 1 − δ1 y , .A11 = b + b δ y − 2b δ1 y , 1 1 1 1 2 2 2 ∗ ∗ ∗ .A12 = 2b δ x y − 2b δ1 x , 1 1 1 2 2 ∗2 .A22 = b δ x , 1 1 2 2 ∗ .A32 = 2b δ x , 1 1 3 2 ∗ 2 ∗ 3 ∗ .p22 = −(b1 δ1 x ) , .p23 = −3b δ1 c x , .p24 = −3b δ1 c11 x . 11 1 1

where .c10 =

System (8) can be stated in the following form if the higher-order terms are ignored: ˙ = J ∗Z    Z E + H (Z),

.

(9)

82

A. Chatterjee and S. Pal





1  z H 1 = = where .Z and .H 2  z2 H

2 2 z1 z2 + c02 z2 + c30 z13 + c21 z12 z2 + c12 z1 z22 + c03 z23 c20 z1 + c11 . The eigenvec.= d 11 z1 z2 + d 02 z22 + d 30 z13 + d 21 z12 z2 + d 12 z1 z22 + d 03 z23 c ∗ tor  .v of community matrix .J corresponding to the eigenvalues .iω0 at .m1 = m is E 1 T  . Now, define  .v = c01 , iω0 − c10

c01 0  = SY or .Y = S −1 Z,  where .Y = .S = (Re( v ), −I m( v )) = . Let .Z −c10 −ω0 (SY ), as a result of (y1 , y2 )T . The system (9) becomes .Y˙ = (S −1 JE∗ S)Y + S −1 H this transformation.



1 (y1 , y2 ; m1 = mc ) y˙1 0 −ω0 Q y1 1 = + This can written as . , where 2 (y1 , y2 ; m1 = mc ) Q y˙2 ω0 0 y2 1 1 and .Q 2 are nonlinear in .y1 and .y2 , given by .Q 1 (y1 , y2 ; m1 = mc ) = 1 H  .Q 1 c01 1 , c 2  (y1 , y2 ; m1 = m ) = − 1 (c10 H 2 ), with 1 + c01 H .Q 1 ω0 c01 2 2 2 2 2  .H1 = (c 20 c 01 − c 11 c 01 c 10 + c 02 c 10 )y + ω0 (2c 02 c 10 − c 11 c 01 )y1 y2 + ω c 02 y + 1 0 2 2 3 3 2 2 3 3 3 (c12 c01 c10 − c03 c10 + c30 c01 − c21 c01 c10 )y1 − ω0 c03 y2 + ω0 (2c12 c10 c01 − c21 c01 − 3c03 c210 )y12 y2 + ω02 (c12 c01 − 3c03 c10 )y1 y22 , 2 = (d 20 c2 − d 11 c01 c10 + d 02 c2 )y 2 + ω0 (2d 02 c10 − d 11 c01 )y1 y2 + ω2 d 02 y 2 + .H 10 1 01 2 0 (d 12 c01 c210 − d 03 c310 + d 30 c301 − d 21 c201 c10 )y13 − ω03 d 03 y23 + ω0 (2d 12 c10 c01 − d 21 c201 − 3d 03 c210 )y12 y2 + ω02 (d 12 c01 − 3d 03 c10 )y1 y22 . To evaluate the stability and direction of a periodic solution, we compute the first Lyapunov coefficient [37]: 1 1 1 1 1 1 2 2 2 1 2 2 .l1 = 16 [Q111 + Q122 + Q112 + Q222 ]+ 16ω0 [Q12 (Q11 + Q22 )− Q12 (Q11 + Q22 )− 1 2 1 2     Q Q 11 11 + Q22 Q22 ], 2 k 3 k k k = ∂ Q  = ∂ Q |(y ,y ;m )=(0,0;mc ) and .Q where .Q |(y ,y ;m )=(0,0;mc ) , ij

∂yi ∂yj

1

2

1

1

ij l

∂yi ∂yj ∂yl

1

2

1

1

i, j, k, l ∈ {1, 2}. If .l1 < 0 the Hopf bifurcation is supercritical, and if .l1 > 0, it is subcritical.

.

6.3 Transcritical-Bifurcation Proposition 4 When the system parameters satisfy the restriction m1 = mT1 C , the system (1) undergoes a transcritical bifurcation. The bifurcation parameter is m1 in this case. Proof The Jacobian matrix J1 of the system (1) around E1 has one zero eigenvalue for m1 = mT1 C . Let U1 and V1 be the eigenvectors of the matrix J1 and (J1 )T correspondT  m1 ing to zero eigenvalue respectively. As a result, we get U1 = −( rK r1 + a1 r1 +b1 r 1

Comparative Study of Deterministic and Stochastic. . .

83

T  (x−δ1 xy)y e1 (x−δ1 xy)y ; and V1 = (0 1)T . We have Fm1 (x, y) = − a1 +b , (x−δ xy) a +b (x−δ xy) 1 1 1 1 1   T   Fm1 E1 ; m1 = mT1 C = 0 0 and   (V1 )T Fk1 E1 ; m1 = mT1 C =  0. Also, DFm1 E1 ; m1 = mT1 C U1 = (0 − 1)T .     1r . Therefore, we obtain (V1 )T DFm1 E1 ; m1 = mT1 C (U1 ) = a1e+b  1r 2   e1 m1 r1 2a1 m1 2a1 rK T T C 2 Further, (V1 ) D F E1 ; m1 = m1 (U1 , U1 ) = − (a r +b r)2 r1 + a1 r1 +b1 r + 1 1 1  2δ1 a1 r < 0. r1 We may conclude that the system experiences a transcritical bifurcation at E1 when

m1 passes by applying Sotomayor’s theorem [38].

7 Global Sensitivity Analysis We used global sensitivity analysis (GSA) employing Latin Hypercube Sampling (LHS) with partial rank correlation coefficient (PRCC) sensitivity analysis to examine the sensitivity of each parameter. Each parameter’s sensitivity is represented in a bar graph and assessed in terms of bar length. If a parameter’s PRCC value is larger than .±0.5, it is said to be sensitive to a variable. The parameters r , K , .m1 and .d1 are all sensitive parameters for the system (1), as shown in Fig. 2.

Predator

0.6 Prey

0.6

0.4

0.4

0.2 PRCC

PRCC

0.2

0

-0.2

0

-0.2

-0.4

-0.4 -0.6

-0.6

-0.8 1

r

2

r1

3

K

4

5

m1

e1

6

7 1

a1

8

9

1

2

3

4

5

6

7

8

9

b1

d1

r

r1

K

m1

e1

1

a1

b1

d1

Fig. 2 Sensitivity analysis of estimated various parameters for prey and predator respectively

84

A. Chatterjee and S. Pal

8 The Stochastic Model with White Noise Environmental characteristics and their fluctuations are used to investigate our system. With time t , all parameters are considered to be constant. The stochastic stability of coexistence equilibrium is investigated. Two processes can be used to transform a deterministic to a stochastic system. Firstly, by replacing one of the environmental characteristics with some random parameters, and secondly, by incorporating a randomly fluctuating driving force in deterministic equations without changing any of the parameters [39]. We will use the second strategy in this case. State variable stochastic perturbations of the Gaussian white noise type around their stable values .E ∗ is a powerful tool for modelling rapidly fluctuating events that are proportional to the distances between each population’s equilibrium value .x ∗ and .y ∗ . [40]. The deterministic system (1) can be expanded to the stochastic model below based on the aforementioned assumption. dx = G1 (x, y)dt + σ1 (x − x ∗ )dξt1 ,

.

dy = G2 (x, y)dt + σ2 (y − y ∗ )dξt2 ,

(10)

where the real constant parameters .σ1 , .σ2 are the intensities of environmental fluctuations and .ξti = ξi (t), .i = 1, 2 are the standard Wiener processes that are independent of each other [41]. The stochastic system (10) can be expressed as an It.o¯ stochastic differential system in a compact form dXt = G(t, Xt )dt + g(t, Xt )dξt ,

.

Xt0 = X0 ,

(11)

The It.o¯ process is the solution of the preceding equation .Xt = (x, y)T for .t > 0. The drift coefficient, G, can be expressed as a slowly varying continuous component. Here, .g = diag[σ1 (x − x ∗ ), σ2 (y − y ∗ )] is used to indicate the diffusion coefficient which is the rapidly varying continuous random component of the diagonal matrix. Here, .ξt = (ξt1 , ξt2 )T is a two-dimensional stochastic process with scalar Wiener j process components that have increments .ξt = ξj (t + t) − ξj (t) which are free Gaussian random variables .N(0, t). Because the diffusion matrix g is dependent on the solution of .Xt , the system (10) is known as multiplicative noise.

8.1 Stochastic Stability of the Coexistence Equilibrium The coexistence equilibrium of the stochastic differential system (10) can be used to centre it. .E ∗ is obtained by introducing the perturbation vector .u(t) = (u1 (t), u2 (t))T , where .u1 = x − x ∗ and .u2 = y − y∗.

Comparative Study of Deterministic and Stochastic. . .

85

To derive the asymptotic stability in the mean square sense by the Lyapunov functions method, working on the complete nonlinear equations (10), could be attempted, following [42]. But for simplicity we deal with the stochastic differential equations obtained by linearizing (10) about the coexistence equilibrium .E ∗ . The linearized version of (11) around .E ∗ is given by du(t) = FL (u(t))dt + g(u(t))dξ(t),

(12)

.



where now .g(u(t)) = diag[σ1 u1 , σ2 u2 ] and .FL (u(t)) =

−a11 u1 − a12 u2 a21 u1 − a22 u2

 = Mu,

m1 b1 x ∗ y ∗ (1−δ1 y ∗ )2 [a1 +b1 (x ∗ −δ1 x ∗ y ∗ )]2 m1 x ∗ [a1 (1−2δ1 y ∗ )+(1−δ1 y ∗ )(b1 x ∗ −b1 δ1 x ∗ y ∗ )] rx ∗ K .a12 = − a12 = (1+Ky , ∗ )2 + [a1 +b1 (x ∗ −δ1 x ∗ y ∗ )]2 e1 m1 a1 (1−δ1 y ∗ )y ∗ .a21 = − a21 = a +b (x ∗ −δ x ∗ y ∗ )]2 , 1 1 1 ∗ ∗ ∗ ∗ ∗ ∗ 1 y )+(1−δ1 y )(b1 x −b1 δ1 x y )] .a22 = − a22 = d1 − e1 m1 x [a1 (1−2δ . and the coexis[a1 +b1 (x ∗ −δ1 x ∗ y ∗ )]2 tence equilibrium corresponds now to the origin .(u1 , u2 = (0, 0). Let . =   (t ≥ t0 ) × R 3 , t0 ∈ R + and let .(t, X) ∈ C (1,2) () be a differentiable function

a11 = − a11 = r1 x ∗ −

.

of time t and twice differentiable function of X. Let further ∂(t, u(t)) ∂(t, u) + f T (u(t)) ∂u ∂t

 2 1 ∂ (t, u) T + tr g (u(t)) g(u(t)) , 2 ∂u2

L (t, u) =

.

(13)

where .

∂ = ∂u



∂ ∂ , ∂u1 ∂u2

T ,

∂ 2 (t, u) = ∂u2



∂ 2 ∂uj ∂ui

 . i,j =1,2

With these positions, we now recall the following result, [43]. Proposition 5 Assume that the functions .(u, t) ∈ C2 () and .L satisfy the inequalities r1 |u|β ≤ (u, t) ≤ r2 |u|β ,

.

L (u, t) ≤ −r3 |u|β ,

.

ri > 0, i = 1, 2, 3, β > 0.

(14) (15)

Then the trivial solution of (12) is exponentially .β-stable for all time .t ≥ 0. Remark 1 For .β = 2 in (14) and (15), the trivial solution of (12) is exponentially mean square stable; furthermore, the trivial solution of (12) is globally asymptotically stable in probability, [43].

86

A. Chatterjee and S. Pal

Proposition 6 Assume .aij < 0, .i, j = 1, 2, and that for some positive real values of ω1 , the following inequality holds Then if .σ12 < 2a11 , it follows that

.

σ22
0,

a21 a22 > 0.

(17)

and the zero solution of system (10) is asymptotically mean square stable. Proof We consider the Lyapunov function (u(t)) =

.

 1 2 u1 + ω1 u22 , 2

(18)

where real positive constants .ω1 to be define later. It is straightforward to verify that inequalities (14) are valid for .β = 2. Moreover, L (u(t)) = (−a11 u1 − a12 u2 )u1 + (a21 u1 − a22 u2 )ω1 u2

 1 ∂ 2 + tr g T (u(t)) 2 g(u(t)) . 2 ∂u

.

(19)

Now we evaluate that ∂ 2 . = ∂u2  and

∂2 T .g (u(t)) g(u(t)) ∂u2

=

as





0 σ12 u21 0 ω1 σ22 u22

1 0 0 ω1  so that we can estimate the trace term

 ∂ 2 T g(u(t)) = σ12 u1 2 + ω1 σ 2 u2 2 . .tr g (u(t)) ∂u2 Hence from (19), we obtain .L (u(t)) = −(a11 − ω1 σ22 2 ).

If we choose .ω1 =

a12 a21 ,

then we get

L (u(t)) = −(a11 −

.

σ12 2 2 )u1 − (a12 − a21 ω1 )u1 u2 − (a22 −

a12 σ22 2 σ12 2 )u1 − (a22 − )u = −uT Qu, 2a21 2 2

(20)

Comparative Study of Deterministic and Stochastic. . . σ2

87

a σ2

12 2 )] and the diagonal matrix Q will be where .Q = diag[(a11 − 21 )u21 , (a22 − 2a 21 real symmetric positive definite matrix and hence its eigenvalues .λ1 and .λ2 will be positive real quantities iff the following conditions holds: .σ12 < 2a11 with .a11 > 0 and 2a22 a21 2 .σ < 2 a12 and .a21 a22 > 0. If .λm stands for the minimum of two positive eigenvalues .λ1 and .λ2 for the diagonal matrix. Then the previous inequality for .L (u(t)) we thus get

L (u(t)) ≤ −λm |u(t)|2 ,

.



thus completing the proof.

Remark 2 Theorem 6 provides the necessary conditions for the stochastic stability of the coexistence equilibrium .E ∗ under environmental fluctuations, [44]. Thus the internal parameters of the model together with the intensities of the environmental fluctuations help in maintaining the stability of the stochastic system.

9 The Stochastic Model with Colored Noise We perturbed the system (1) by independent white noises due to a randomly fluctuating environment in the preceding section, but in real ecosystems, random external perturbations are correlated within a finite correlation time due to interaction with the environment. External noise cannot be classified as white noise when the time scale of random fluctuations is longer than the ecosystem’s characteristic time scale. This section examines the impact of the Ornstein–Uhlenbeck colored noise perturbation, which accounts for nonzero co-relation time and so makes the system more realistic (1). As previously stated, in the presence of colored noise, the linearized version of the system (1) has the form .

du1 a12 u2 + u1 η1 (t), = a11 u1 +  dt du2 = a21 u1 +  a22 u2 + u2 η2 (t), dt

(21)

where the perturbed terms . ηi s are colored noises that are described by Ornstein– Uhlenbeck processes and must satisfy the following Langevin equations .

d ηi (t) = − αi  ηi (t) + σi ξi (t), dt

t > 0,

 ηi (0) =  ηi0 ,

i = 1, 2,

(22)

αi > 0, .σi > 0; and .ξi is a scalar white noise process with .σi2 intensity. The where . ηi (t) are given mathematical expectation and correlation functions of the processes . by

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E{ ηi (t)} = 0,

E{ ηi (s),  ηi (t)} =

.

σi2 − e αi |t−s| , i = 1, 2. 2 αi

(23)

ξi s, i=1,2 are independent standard Gaussian white noises with the expectations and correlation functions shown below:

.

E{ξi (t)} = 0,

.

E{ξi (t), ξi (s)} = δ(s − t),

i = 1, 2,

(24)

Here, the Dirac delta function is denoted by .δ(t). It is worth noting that a αi → ∞ to the possible model of .ξi (t) can be obtained by applying the limit . Ornstein–Uhlenbeck process. Markovian process refers to the four component processes .(x(t), y(t),  η1 (t),  η2 (t)). We can now deduce the sufficient conditions for exponential mean square stability. Rewriting (21) as du(t) = J ∗ u(t) +  η(t)u(t) (25) dt      a (t) 0  a  η 11 12 1 where .J ∗ = and .u(t) is defined earlier. ,  ηi (t) = 0  η2 (t)  a21  a22 Assume  .a11 +  a22 < 0, so that the trivial equilibrium of the system (21) is always locally asymptotically stable in the absence of colored noise. Let us suppose .( a11 )2 + 2 =  2 a  a − 2 a  a in the presence of parametric colored noise. This a (11 22 (22 ) 12 21 ) assumption guarantees that eigenvalues of the jacobian matrix .J ∗ are different and have negative real part. The eigenvalues of the matrix .J ∗ are denoted as  .λ1 and  .λ2 , respectively, and the matrix .J ∗ can be transformed into a .(2 × 2) diagonal matrix with the eigenvalues of ∗ .J as the primary diagonal entries using similarity transformation. It is possible to prove it in these circumstances that the system (25) is said to be exponentially stable at the pth moment surrounding its trivial equilibrium, and the following first order stochastic differential equations are similar (SDEs) .

.

dui = λi ui + ui  ηi (t), dt

i = 1, 2,

(26)

∗ where  .λ1 and  .λ2 are two eigenvalues of .J . The solution of the decoupled SDEs (26) are given by

   t ui (t) = ui (0)exp   ηi (s)ds , i = 1, 2. λi t +

.

(27)

0

Therefore,    t |ui (t)|p = |ui (0)|p exp p(Re λi )t + p  ηi (s)ds , i = 1, 2.

.

0

(28)

Comparative Study of Deterministic and Stochastic. . .

t

To .ui (t) = processes,

0

89

 ηi (s)ds, recall the features of Gaussian and Ornstein–Uhlenbeck 1

2

E(eui (t) ) = eE(ui (t))+ 2 E(ui (t)) ,

E(ui (t)) = 0

.

E(u2i (t)) =

.

σi2  αi2

t+

σi2  αi3

(exp(− αi t) − 1),

i = 1, 2.

(29)

(30)

As a result, we can calculate the pth moment of .|ui (t)|, i=1,2 using the attributes (30) and (29) in (28)    t  E{|ui (0)|p } = |ui (0)|p exp{p(Re λi )t}E exp p  ηi (s)ds

.

= |ui (0)|p exp{p(Re λi )t}exp  = |ui (0)| exp p Re( λi ) + p



pσi2



t

pE(



0

t exp

2 αi2

 ηi (s)ds +



p2 σi2 2 αi3

0

 p2 E(u2i (t)) 2 

(exp(− αi t) − 1) .

(31)

The following theorem summarises the preceding fact, Proposition 7 If  .a11 + a22 < 0 and .( a22 )2 = 2 ( a11 a22 − 2 a12 a21 ), then the a11 )2 + ( system (21) can be transformed into a set of two decoupled SDEs (26) and the null solution of system (21) is said to be exponentially stable in pth moment, if and only if Re( λi ) +

.

pσi2 2 αi2

< 0, i = 1, 2,

(32)

  where  .λ s, .σ s and . αi s are defined earlier. i i

10 Numerical Simulations We perform a numerical simulation over the set of parametric values to visualise the analytical result by using MATLAB [25]. r = 8, r1 = 1.5, K = 20, m1 = 0.39, e1 = 0.3,

.

δ1 = 0.05, a1 = 0.05, b1 = 0.06, d1 = 0.5.

(33)

It is stated that The system (1) exhibits stable behaviour around .E ∗ = (1.98, 0.80) (cf. Fig. 3a).

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0.3

0.45

0.28

0.4 0.26 0.35 E* (0,29,.20)

E*

0.3 y

y

0.24 0.22

0.25 0.2

0.2

0.15 0.18 0.1 0.16 0.14 0.1

0.05 0 0.15

0.2

0.25

0.3

0.35

x

0.4

(a)

0.45

0.5

0.55

0.6

0

0.2

0.4

0.6

0.8

x

1

1.2

1.4

1.6

(b)

Fig. 3 (a) The plot is generated for the values of the reference parameters given in (33). (b) The plot is created with .m1 = 0.6 and the other values of the reference parameters mentioned in (33)

10.1 Effect of m1 It is observed that when the capturing rate of prey by predator is high for predation, i.e., .m1 , the dynamical system switches to unstable behavior (viz. .m1 = 0.60). It is illustrated in Fig. 3b. For the parameter .m1 , Fig. 4a, b displays various steady state behaviour of prey and predator. Here, it is noted that a Hopf point are situated (H) at .m1 = 0.562887 with eigenvalue .±1.10146i and a Branch point (BP) coincide at .m1 = 0.115625 with eigenvalue .(0, −8). Figure 4c confirm that, the system experiences a series of stable limit cycles from Hopf point and having the first Lyapunov coefficient .−1.068725, encounters a supercritical bifurcation. The preceding observations guarantee that increasing the amount of .m1 can decrease both prey and predator densities and reaches a certain threshold value .m1 = 0.562887, the system (1) goes from being stable to being unstable. The branch point .(BP ) denotes the point at which the predator extends and the transcritical bifurcation occurs.

10.2 Effect of d1 The mortality rate of predator,(.d1 ), as shown in Fig. 5a, b, is key in switching the prey and predator natures. Here, we have a Branch point at .d1 = 1.686486 and a Hopf point at .0.201944 with eigenvalues .±0.689373i . The system undergoes a super critical bifurcation with the first Lyapunov coefficient .−1.045064e at that point, and each population becomes oscillate. To move forward, a family of stable limit cycle is created from H point in .d1 − x − y space (cf. Fig. 5c).

Comparative Study of Deterministic and Stochastic. . .

91

0.5 6 BP

0.4

5 4 y

x

0.3

3

0.2

H

2 0.1

1 0

0.2

0.4 m1

H 0.6 (a)

0.8

BP 0.2

0 0

0.4 m1

0.6 (b)

6

0.8

1

0.4

0.35 5 0.3 4

y

x

0.25

3

0.2

0.15 2 0.1 1 0.05

0

0 0

0.5

m

1

0

1

0.5

m

(d)

Fig. 4 (a, b) For .m1 , the trajectory illustrates the various dynamical behaviours of prey and predator. (c) The trajectory depicts a set of stable limit cycles generated from the Hopf (H) point in the .m1 − x − y plane for .m1 . (d) Bifurcation diagram for .m1

10.3 Effect of e1 Figure 6a, b depict the various steady state behaviour of each species for the parameter .e1 . These shows that a branch point at .e1 = 0.088942 in which predator goes to extinction and a Hopf point at .0.847925 with first Lyapunov coefficient −01 . This suggests that from Hopf point, a stable limit cycle bifurcates. .−5.149349e To proceed, a stable limit cycle family is formed from the H point in .e1 − x − y space (cf. Fig. 6c).

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y

0.3 0.2

LPC H

0.1 0 0

BP 0.5

1 d1

1.5

2

(b)

Fig. 5 (a, b) For .d1 , the trajectory illustrates the various dynamical behaviours of prey and predator. (c) The trajectory depicts a set of stable limit cycles generated from the Hopf (H) point in the .m1 − x − y plane for .d1 . (d) Bifurcation diagram for .d1

10.4 Bifurcation The bifurcation diagrams (see Figs. 4d, 5d, 6d) depict the whole dynamic nature of the system (1) for the effects of parameters .m1 , .d1 and .e1 respectively. To visualise the nature of predator and prey we have plotted another two bifurcations diagrams with prey fear level and intrinsic growth rate of prey separately (cf. Fig. 7a, b). Figure 8a–c display the two parameters bifurcation diagram for .m1 −d1 , .m1 −e1 and .m1 −r respectively. Bogdanov–Takens bifurcation is shown in dynamical system (1) at critical values of bifurcation parameters as .m1[bt] = 0.248381 and .d1[bt] = 0 in which both eigenvalues are zero in .m1 − d1 plane. The Bogdanov-Takens (BT) point is a location where the limit point curve and the curve corresponding to equilibria

Comparative Study of Deterministic and Stochastic. . .

93

1 0.8

y

0.6 0.4 H LPC

0.2 0 BP 0

0.5

1 e1

1.5 (b)

2

2.5

0.8

3

0.7 2.5 0.6 2

y

x

0.5

1.5

0.4

0.3 1 0.2 0.5 0.1

0

0 0

0.5

1

e1

1.5

2

0

0.5

1

e1

1.5

2

(d)

Fig. 6 (a, b) For .e1 , the trajectory illustrates the various dynamical behaviours of prey and predator. (c) The trajectory depicts a set of stable limit cycles generated from the Hopf (H) point in the .e1 − x − y plane for .e1 . (d) Bifurcation diagram for .e1

meet. In .m1 − e1 plane, GH point breaks into two subcritical and supercritical branches. All the numerical finding are summarized in Table 1.

10.5 Environmental Fluctuations Following that, we will look at the system’s dynamical behaviour in the presence of environmental perturbations. We employ the EulerMaruyama (EM) and Milstein methods using MATLAB software to numerically simulate the stochastic differential Eq. (10). Using a suitable Lyapunov function (18), we established the condition for asymptotic stability of .E ∗ in mean square sense for the system (10).

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2.5

0.8 2

3.5

1.6

3

1.4

2.5

1.2

2

1

0.7 0.6

1.5

0.4 1

y

x

y

x

0.5

1.5

0.8

1

0.6

0.5

0.4

0.3 0.2

0.5

0.1

0

0

0 0

10

20

r

30

40

0

10

20

r

30

0.2 0

40

5

10

K

(a)

15

20

0

5

10

K

15

20

(b)

Fig. 7 (a) Bifurcation diagram for r. (b) Bifurcation diagram for K Table 1 Natures of equilibrium points Parameters .m1

.e1

.d1

.(m1 , d1 ) .(m1 , e1 )

Values 0.115625 0.562887 0.088942 0.847925 1.686486 0.198071 .(0.248381, 0.0000) .(0.357081, 2.604021)

Eigenvalues .(−8, 0) .(±1.10146i) .(−8, 0) .(±1.10534i) .(−8, 0) .(±.689373i) .(≈

±0.00)

.(±1.13842i)

Equilibrium points Branch Point (BP) Hopf (H) Branch Point (BP) Hopf (H) Branch Point (BP) Hopf (H) Bogdanov-Takens (BT) Generalized Hopf (GH)

These conditions are determined by .σ1 and .σ2 and model system parameters. Using, .σ1 = 0.02 and .σ2 = 0.02, as the intensities of environmental perturbations with parameters set as apply in deterministic system, each species coexist and stochastically stable (cf. Fig. 9a). Next, we set the environmental fluctuation values to .σ1 = 0.2 and .σ2 = 0.2, coexistence equilibrium becomes unstable (cf. Fig. 9b). Further, using the same parametric settings, a numerical simulation in a stochastic system with parametric colour noise was carried out. Figure 10a, b show the results of stochastic stability and unstability, respectively, under the same noise intensity. Therefore, we discovered that by measuring the intensity of the environmental fluctuation, we can stabilize the coexistence system.

Comparative Study of Deterministic and Stochastic. . . 2

95

3 GH

2.5

1.5

1 e1

d1

2

0.5

0 0

1.5

Hopf Curve

1 0.5

BT 0.2

0.4 m1

0.6 (a)

0.8

1 0 0

0.2

0.4 m

0.6 (b)

0.8

1

20

15 Unstable Zone r

Hopf curve 10

5

0

0.2

0.4 m

0.6 (c)

0.8

1

Fig. 8 (a) Two parameters bifurcation diagram for .m1 − d1 . (b) Two parameters bifurcation diagram for .m1 − e1 . (c) Two parameters bifurcation diagram for .m1 − r

11 Discussion This present article tries to build a predator-prey relationship by incorporating prey refuge and fear effect in deterministic system and compare with stochastic system. Under the Holling II response function, it is assumed that prey population grows logistically and predators consume prey population. To begin, several basic features that are ecologically well behaved, such as boundedness and properties of existence of equilibria, are investigated and validated. The system’s local stability behavior is observed around each equilibrium. To investigate the dynamics of the given system, it was discovered that the system (1) has three equilibrium points: trivial (.E0 ), axial (.E1 ) and coexistence (.E ∗ ).

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0.7

0.3

0.5

Prey

Prey

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0.4

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0.3 0.2

0

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100

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200 Time

250

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300

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1 Predator

0.28

Predator

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0.32

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0.8 0.6 0.4

0.22 0.2

0

0

50

100

150

200 Time (a)

250

300

350

0.2

400

Fig. 9 (a) The figures depicts solution of system is stochastically stable for .σ1 = 0.02 and .σ2 = 0.02 (b) The figures depicts solution of system is stochastically unstable for .σ1 = 0.2 and .σ2 = 0.2 with white noise process 0.8

1 0.8 Prey

Prey

0.6 0.4 0.2 0

0.6 0.4 0.2

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Predator

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0.3 0.25 0.2

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Time

0

50

100

150

200

Time

250

(a)

300

350

400

0

0

50

100

150

200

Time

( b)

Fig. 10 (a) The figures depicts solution of system is stochastically stable for .σ1 = 0.02 and = 0.02. (b) The figures depicts solution of system is stochastically unstable for .σ1 = 0.2 and .σ2 = 0.2 with color noise process .σ2

The capturing rate of prey by predator, or the parameter .m1 , plays a critical role in exhibiting Hopf bifurcation and stability switching behaviour throughout the investigation. When .m1 > m1c = 0.562887, the system exhibits oscillatory behaviour, with each population exhibiting steady coexistence between .0.115625 < m1 < 0.562887. The predator will disappear and coexistence equilibrium looses stability when .m1 crosses the values 0.115625. Various two-parameter bifurcations are depicted, each with a different stability nature. Based on the findings, we can deduce that the predator’s prey capture rate, conversion efficiency factor of prey into new predator, mortality rate of predator,

Comparative Study of Deterministic and Stochastic. . .

97

intrinsic growth rate of prey and fear level should be kept within a certain range to avoid predator extinction and system instability. Environmental noise is also included in the model, and because of its low intensity, the system becomes stochastic asymptotic stable. Oscillations with large amplitudes result from high intensity values. If specific requirements regarding the maximum size of random variations in the environment and the model parameters are met, the model becomes stochastically stable.

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Mathematical Modeling and Numerical Analysis of HIV-1 Infection with Long-Lived Infected Cells During Combination Therapy and Humoral Immunity Zakaria Hajhouji, Majda El Younoussi, Khalid Hattaf, and Noura Yousfi

1 Introduction HIV stands for human immunodeficiency virus. It’s a virus that breaks down certain cells in immune system. When HIV damages the immune system, it’s easier to get really sick. Moreover, HIV infection is usually acute or chronic and can potentially lead to long-term complications such as cancers like invasive forms of cervical cancer, Kaposi’s sarcoma, and some lymphomas. According to the World Health Organization (WHO), an estimated 38.4 million people worldwide are living with HIV at the end of 2021, and nearly 1.5 million people have recently been infected with HIV, and almost 650,000 people died from HIV-related causes in 2021 [1]. There are two antigenic types of viruses: HIV-1 and HIV-2. These two viruses have many similarities, including their basic gene arrangement, modes of transmission, intracellular replication pathways, and clinical consequences. However, the major difference between of these types is that HIV-1 spreads more rapidly than HIV-2. Regarding the frequency of their occurrence, HIV-2 is very rare compared to the common HIV-1. Its occurrences are also limited to Africa, unlike HIV-1 which is widespread throughout the world. Mathematical modeling of HIV-1 infection dynamics attracts the attention of several researchers in the world. For instance, Perelson et al. [2] introduced a basic

Z. Hajhouji () · M. El Younoussi · N. Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’Sick, Hassan II University of Casablanca, Casablanca, Morocco K. Hattaf Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’Sick, Hassan II University of Casablanca, Casablanca, Morocco Equipe de Recherche en Modélisation et Enseignement des Mathématiques (ERMEM), Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Casablanca, Morocco © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_6

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mathematical model of HIV-1 infection where HIV-infected patients were given protease inhibitors and sampled frequently thereafter. Mittler et al. [3] studied a mathematical system that describes the influence of delayed viral production on viral dynamics in HIV-1 infected patients. Lai and Zou [4] extended the model of Perelson et al. [2] by adding two modes of transmission and an infinite distributed delay in cell infection. According to [5–7], there are two types of productively infected cells: short-lived productively infected cells that live for a short period of time and produce a large number of HIV-1 particles and long-lived productively infected cells that live for a long period of time and produce a small number of HIV-1 particles. Because active viral gene expression causes cell death due to viral cytopathic effects and immune response, long-lived cells likely harbor transcriptionally silent, latent provirus. Therefore, HIV-1 persistence in long-lived cellular reservoirs remains a major barrier to a cure. For these reasons, Wang and Zhou [8] presented a mathematical model of delay differential equations describing the interactions between uninfected cells, short-lived infected cells, chronically infected cells, and free virus particles. To study the process of viral infection and viral replication, Xu et al. [9] modeled a mathematical model with humoral immunity and general incidence rate that contains four different intracellular time delays in order to describe the time spent in different stages of the viral replication process, but virus mobility was ignored. In many biological systems, species may disperse spatially. So, in order to study the influences of spatial structures of virus dynamics, Wang et al. [10] studied the delay mathematical model by integrating the effect of diffusion. In addition, the general incidence rates can help us gain the unification theory by the omission of unessential details [11–14]. Recently, motivated by Wang and Zhou [8], Wang et al. [10], Geng and Xu [15] generalized the models of Wang and Zhou [8] and Wang et al. [10] by proposing a delayed and diffusive viral infection model incorporating short-lived and chronically infected cells and general nonlinear incidence function. On the other hand, it is very difficult to find the exact analytical solutions for many nonlinear models. Therefore, designing a discrete feasible scheme that efficiently preserves the same quantitative behaviors of the solutions as the corresponding continuous models is a challenging and interesting task. The model proposed in this present study will be discretized by a mixed Euler method which is a mixture of forward and backward Euler methods. The choice of discretization scheme is motivated by the works presented in [16–18]. Motivated by the above biological and mathematical considerations, we develop a mathematical model formulated by partial differential equations (PDEs) describing the HIV-1 dynamics with four discrete delays; general incidence rate; humoral immune response in the presence of therapy; two types of infected cells, namely, long-lived infected cells and chronically infected cells; free virus particles; and antibodies. To do this, Sect. 2 deals with the formulation of the developed model and the same preliminary results about the existence, uniqueness, nonnegativity and boundedness of solutions, as well as the existence of equilibria. Section 3 is devoted to mathematical analysis, while Sect. 4 is for numerical analysis. Finally, the paper ends with an application and some numerical simulations.

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2 Mathematical Model and Preliminary Results In this section, we first propose a mathematical model for HIV-1 infection with general incidence, four intracellular delays, and humoral immune response. The dynamics of this model is governed by the following equations: ⎧ ∂U ⎪ ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ∂I ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂C . ∂t ⎪ ⎪ ⎪ ⎪ ∂V ⎪ ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂W ∂t

  = λ − d1 U (x, t) − (1 − ε)f U (x, t), V (x, t) V (x, t),   = (1 − ε)(1 − ρ)f U (x, t − τ1 ), V (x, t − τ1 ) V (x, t − τ1 ) − d2 I (x, t),   = ρ(1 − ε)f U (x, t − τ2 ), V (x, t − τ2 ) V (x, t − τ2 ) − d3 C(x, t), = DV V + N¯1 d2 I (x, t − τ3 ) + N¯2 d3 C(x, t − τ4 ) − d4 V (x, t) − bV (x, t)W (x, t), = DW W + aV (x, t)W (x, t) − d5 W (x, t),

(1)

where .U (x, t), .I (x, t), .C(x, t), .V (x, t), and .W (x, t) represent the concentrations of uninfected cells, long-lived infected cells, chronically infected cells, free virus particles, and antibodies at position x and time t, respectively. .λ is the source term for uninfected cells. .ε is the efficacy of the therapy. .d1 , d2 , d3 , d4 , and .d5 are the death rates of uninfected cells, long-infected cells, chronically infected cells, virus, and antibodies, respectively. The fractions .ρ and .(1 − ρ) are the probabilities that an uninfected cell will become either chronically infected or long-lived infected. ¯1 = N1 (1 − η1 ) and .N¯2 = N2 (1 − η2 ), where .N¯1 and .N¯2 are the average .N numbers of virions produced in the lifetime of long-lived and chronically infected cells, respectively, as well as .η1 and .η2 are the efficacy of the therapy. In addition, a is the rate at which antibodies develop in response to free virus, and b is the rate of neutralization of free HIV particles by antibodies. Further, the first delay .τ1 represents the time needed for long-lived infected cells to produce virions after viral entry. The second delay .τ2 represents the time needed for chronically infected cells to produce virions after viral entry. The third delay parameter .τ3 represents the delay between viral RNA transcription and viral release from long-lived infected cells and maturation. The last delay .τ4 is the delay between viral RNA transcription and viral release from chronically infected cells and maturation. Finally, . is the Laplacian operator, and .DV and .DW are the diffusion coefficients of virions and antibodies, respectively. In this study, we assume that the general incidence function .f (U, V )V is continuously differentiable in the interior of .R2+ and satisfies the following assumptions: (H1 ) .f (0, V ) = 0, for all and .V  0; ) (H2 ) . ∂f (U,V > 0 for all .U, V  0; ∂U .(H3 ) .f (U, V ) is a monotone decreasing function with respect to V .

. .

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From a biological point of view, all three assumptions are reasonable and consistent with reality. For more details on the biological significance of these three assumptions, we refer the reader to the following works [17, 19]. It is easy to check that a class of functions .f (U, V )V satisfying .(H0 ) − (H3 ) includes some βU V common nonlinear incidence functions such as .f (U, V )V = 1+c , f (U, V )V = 1V βU V and .f (U, V )V = 1+c2 U +c for .β, c1 , c2 , c3 > 0. 1 V +c3 U V It is important to note that our model formulated by system .(1) covers several cases existing in the literature. For instance: βU V 1+c2 U +c1 V

• When we ignore the mobility of viruses, the delay effect, and the role of humoral immunity, we find the model of Wang and Zhou [8]. • The PDE model proposed by Geng and Xu [15] in 2021, when we ignore the role of humoral immunity response. The initial conditions of model (1) are given as .

U (x, θ ) = φ1 (x, θ ) ≥ 0, I (x, θ ) = φ2 (x, θ ) ≥ 0, C(x, θ ) = φ3 (x, θ ) ≥ 0, ¯ × [−τ, 0], V (x, θ ) = φ4 (x, θ ) ≥ 0, W (x, θ ) = φ5 (x, θ ) ≥ 0, ∀(x, θ ) ∈ (2)

and Neumann boundary conditions .

∂W ∂V = = 0, on ∂ × (0, +∞), ∂ν ∂ν

(3)

where .τ = max {τ1 , τ2 , τ3 , τ4 }, . is a bounded interval in .Rn with smooth boundary ∂ .∂ , and . ∂ν denotes the outward normal derivative on .∂ . Biologically speaking, the Neumann boundary conditions mean that the virions and antibodies do not have any movement across the boundary .∂ . ¯ R5 ) be the space of continuous functions from the topological Let .X = C( , ¯ space . into the space .R5 . Denote .C = C([−τ, 0], X ) be the Banach space of continuous functions from .[−τ, 0] into .C with the usual supremum normal, and .C+ = C([−τ, 0], X+ ). When convenient, we identify an element .ϕ ∈ C as a function ¯ × [−τ, 0] into .R5 defined by .ϕ(x, s) = ϕ(s)(x). We adopt the notation that from . for .σ > 0, a function .ω(.) : [−τ, σ ) −→ X induces functions .ωt ∈ C for .t ∈ [0, σ ), defined by .ωt (θ ) = ω(t + θ ) for .θ ∈ [−τ, 0]. Theorem 2.1 For any given initial data .ϕ ∈ C satisfying the condition (2), there exists a unique solution of problem (1)–(3) defined on .[0, +∞), and this solution remains nonnegative and bounded for all .t  0. Proof For any .φ = (φ1 , φ2 , φ3 , φ4 , φ5 )T ∈ C+ , we define .F = (F1 , F2 , F3 , F4 , ¯ .F5 ) : C+ −→ X as follows. For any .x ∈ ,

Mathematical Modeling and Numerical Analysis of HIV-1 Infection. . .

⎧ ⎪ F1 (φ)(x) ⎪ ⎪ ⎪ ⎪ F2 (φ)(x) ⎪ ⎪ ⎨ F3 (φ)(x) . ⎪ ⎪ F4 (φ)(x) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ F5 (φ)(x)

= = = = − =

103

  λ − d1 φ1 (x, 0) − (1 − ε)f φ1 (x, 0), φ4 (x, 0) φ4 (x, 0),   (1 − ε)(1 − ρ)f φ1 (x, −τ1 ), φ4 (x, −τ1 ) φ4 (x, −τ1 ) − d2 φ2 (x, 0),   ρ(1 − ε)f φ1 (x, −τ2 ), φ4 (x, −τ2 ) φ4 (x, −τ2 ) − d3 φ3 (x, 0), N¯1 d2 φ2 (x, −τ3 ) + N¯2 d3 φ3 (x, −τ4 ) − d4 φ4 (x, 0) bφ4 (x, 0)φ5 (x, 0), aφ4 (x, 0)φ5 (x, 0) − d5 φ5 (x, 0).

Then the system (1)–(3) can be rewritten as the following abstract functional differential equation: ⎧ ⎨ dψ = Aψ + F (ψt ), . dt ⎩ψ = φ ∈ C , 0 +

t > 0, ψt ∈ C

(4)

where .ψ = (U, I, C, V , W )T and .Aψ = (0, 0, 0, DV V , DW W )T . It is clear that F is locally Lipschitz in from [20–24] that system (4) admits a unique  .C . It follows  local solution on .t ∈ 0, Tmax , where .Tmax is the maximal existence time for the solution of system (4). In order to demonstrate the boundedness of solutions, define .T1 (x, t)

= U (x, t) + I (x, t + τ1 ) + C(x, t + τ2 ).

It follows from model (1) that .

∂T1 (x, t) = λ − d1 U (x, t) − d1 I (x, t + τ1 ) − d3 C(x, t + τ2 ) ∂t ≤ λ − δ1 U (x, t),

where .δ1 = min{d1 , d2 , d3 }. Then we have .T1 (x, t)

≤ max

λ , max{φ1 (x, 0) + φ2 (x, −τ1 ) + φ3 (x, −τ2 )} := α1 , δ1 x∈ ¯

implying U , I , and C are bounded. To show the boundness of V and W , we set .T2 (x, t)

= V (x, t) +

b W (x, t). a

Then .

∂T2 (x, t) = DV V + DW W + N¯1 d2 I (x, t − τ3 ) + N¯2 d3 C(x, t − τ4 ) ∂t bd5 −d4 V (x, t) − W (x, t). a

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When .DV = DW = D , we obtain   ∂T2 (x, t) − DG2 ≤ max N¯1 d2 + N¯2 d3 α1 − δ2 G2 (x, t), ∂t

where .δ2 = min d4 , d5 . Thus, .

.T2 (x, t)

≤ max

  ¯

N1 d2 + N¯2 d3 α1 b , max{φ4 (x, 0) + φ5 (x, 0)} := α2 , ¯ δ2 a x∈

which implies that .V (x, t) and .W (x, t) are bounded. Based on the above analysis, we have demonstrated  that .U (x, t), .I (x, t), .C(x, t),  ¯ × 0, Tmax . From the standard theory for .V (x, t), and .W (x, t) are bounded in . semilinear parabolic systems [25], we deduce that .Tmax = +∞. Next, we discuss about all the biologically feasible spatially homogeneous equilibria for our proposed model (1). Any spatially homogeneous equilibrium point .E = (U, I, C, V , W ) of the model (1) satisfies the following system of algebraic equations:   λ − d1 U − (1 − ε)f U, V V

= 0,

  (1 − ε)(1 − ρ)f U, V V − d2 I = 0, . ρ(1

  − ε)f U, V V − d3 C

= 0,

(5)

N¯1 d2 I + N¯2 d3 C − d4 V − bV W = 0, aV W − d5 W

= 0.

Clearly, from the above system of Eqs. (5), we can notice that .E0 (U0 , 0, 0, 0, 0) λ represents the unique infection-free equilibrium for the model (1) with .U0 = d1 and it always exists. Therefore, the basic reproduction number of the model (1) can be defined as  λ   ,0 (1 − ε) N¯1 (1 − ρ) + ρ N¯2 f d1 . .R0 = d4

When .W = 0, we get the following equation:   (λ − d1 U ) N¯1 (1 − ρ) + ρ N¯2 d4  , .f U, = ¯ d4 (1 − ε) N1 (1 − ρ) + ρ N¯2

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  (λ − d1 U ) N¯1 (1 − ρ) + ρ N¯2  represents the number of virions, with .V = . Since .V d4 λ . Now, we define a we need to have .V ≥ 0 and this condition leads to .U ≤ d1  λ as follows: function .G1 on the closed interval . 0, d1   (λ − d1 U ) N¯1 (1 − ρ) + ρ N¯2 d4 .  − .G1 (U ) = f U, ¯ d4 (1 − ε) N1 (1 − ρ) + ρ N¯2

Then, we have .G1 (0)

.G1 (

=−

d4  < 0,  (1 − ε) N¯1 (1 − ρ) + ρ N¯2

  d4 λ   R0 − 1 , )= ¯ ¯ d1 (1 − ε) N1 (1 − ρ) + ρ N2

and

.G1 (U )

=

 ∂f ∂f d1  ¯ . − N1 (1 − ρ) + ρ N¯2 d4 ∂V ∂U

λ ) > 1 if .R0 > 1. From the hypotheses d1 .(H2 ) on the general incidence function .f (H, V ), we obtain .G (U ) > 0 and this 1 implies that .G1 is a strictly increasing function of U. Therefore, for .R0 > 1 we have another infection equilibrium without immunity, .E1 (U1 , I1 , C1 , V1 , 0),  λ (1 − ρ)(λ − d1 U1 ) ρ(λ − d1 U1 ) where .U1 ∈ 0, , .I1 = , .C1 = , and .V1 = d d d3 1 2   ¯ ¯ (λ − d1 U1 ) N1 (1 − ρ) + ρ N2 . d4 d5 . Since W represents the number of antibody When .W = 0, we find .V = a   a(λ − d1 U2 ) N¯1 (1 − ρ) + ρ N¯2 − immune cells, we need to have .W ≥ 0. So, .W = bd4 d 4 d5 d4 d5 λ   . Let us consider − ≥ 0. This criterion leads to .U ≤ d1 ad1 N¯1 (1 − ρ) + ρ N¯2 bd5

It can be easily observed that .G1 (

.G2 (U )



d5 a(λ − d1 U ) = f U, . − d5 (1 − ε) a

∂f aλ ≥ 0. Now, using the < 0 and .G 2 (U ) = ∂U d5 (1 − ε) hypotheses .(H2 ) on the general incidence function .f (U, V ), we have .G 2 (U ) > 0, and this implies that .G2 is also a strictly increasing function of U. Now, we define

Then, we have .G2 (0) = −

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the reproduction number for humoral immunity as W

.R1

=

aV1 , d5

which denotes the average number of antibody immune cells activated by viruses. d4 d5 λ a   and Note that when .RW and .U1 < − 1 > 1, then .V1 > ¯ d1 ad1 N1 (1 − ρ) + ρ N¯2 d5

d4 λ d4 d5 λ   −  =f  − − we have .G2 d1 d1 ad1 N¯1 (1 − ρ) + ρ N¯2 (1 − ε) N¯1 (1 − ρ) + ρ N¯2

  d4 d4 d5 d5   .= 0.  , > f U1 , V1 − (1 − ε) N¯1 (1 − ρ) + ρ N¯2 ad1 N¯1 (1 − ρ) + ρ N¯2 a Therefore, the model (1) admits a unique infection equilibrium with humoral

d4 d5 λ   , immunity, .E2 (U2 , I2 , C2 , V2 , W2 ), where .U2 ∈ 0, d1 ad1 N¯1 (1 − ρ) + ρ N¯2 (1 − ρ)(λ − d1 U2 ) ρ(λ − d1 U2 ) .I2 = , .C2 = , and .W2 = d2 d3   a(λ − d1 U2 ) N¯1 (1 − ρ) + ρ N¯2 d4 d5 . − . bd5 bd4 Summarizing the above discussions, we get the following theorem. Theorem 2.2 (i) If .R0  1, then the model (1) always has one infection-free equilibrium,   λ . .E0 U0 , 0, 0, 0, 0 , where .U0 = d1 (ii) If .RW ≤ 1 < R0 , then the model (1) has an  infection equilib1  rium without humoral immunity .E1 U1 , I1 , C1 , V1 , 0 , where .U1 ∈  λ (1 − ρ)(λ − d1 U1 ) ρ(λ − d1 U1 ) , .I1 = , .C1 = , and .V1 = 0, d1 d2  d3  (λ − d1 U1 ) N¯1 (1 − ρ) + ρ N¯2 . d4 W (iii) If .R1 > 1, then the model (1) has an infection

equilibrium with humoral immu  d4 d5 λ   , nity .E2 U2 , I2 , C2 , V2 , W2 , where .U2 ∈ 0, − d1 ad1 N¯1 (1 − ρ) + ρ N¯2 (1 − ρ)(λ − d1 U2 ) ρ(λ − d1 U2 ) d5 , .C2 = .I2 = , .V2 = , and .W2 = d d a 3   2 a(λ − d1 U2 ) N¯1 (1 − ρ) + ρ N¯2 d4 d5 . − bd5 bd5

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3 Mathematical Analysis In this section, we study the global stability of the three equilibria of model (1) by constructing suitable Lyapunov functions. We first investigate the global stability of the infection-free equilibrium. Theorem 3.1 If .R0 ≤ 1, then the infection-free equilibrium .E0 of system (1) is globally asymptotically stable. Proof To study the global stability of .E0 , we consider a Lyapunov functional defined as follows:

  U   f (U0 , 0) N¯1 (1 − ρ) + ρ N¯2 U − U0 − ds + N¯1 I (x, t) L0 = U0 f (s, 0) b + N¯2 C(x, t) + V (x, t) + W (x, t) + N¯1 (1 − ρ)(1 − ) a  t  t f (U (x, s), V (x, s))V (x, s)ds + N¯2 ρ(1 − ) f (U (x, s), V (x, s))V (x, s)ds ·

.

t−τ1

+N¯1 d2



t

I (x, s)ds + N¯2 d3

t−τ3





t

t−τ2

C(x, s)ds dx, t−τ4

We denote .(x, t) =  and .(x, t − τi ) = τi for .i = 1, 2, 3, 4 and . ∈ {U, I, C, V , W } for the sake of notational simplicity. Taking the derivative of .L0 with respect to time t along the solution trajectories of the system (1)–(3), we obtain

   dL0 f (U0 , 0) N¯1 (1 − ρ) + ρ N¯2 1 − = dt f (U, 0)     d1 U0 − d1 U − (1 − ε)f U, V V         + N¯1 (1−ρ)(1−ε)f Uτ1 , Vτ1 Vτ1 − d2 I + N¯2 ρ(1 − ε)f Uτ2 , Vτ2 Vτ2 − d3 C

    ¯ + N1 (1 − ρ)(1 − ε) f U, V V − f Uτ1 , Vτ1 Vτ1 .

    + ρ N¯2 (1 − ε) f U, V V − f Uτ2 , Vτ2 Vτ2

¯ ¯ + N1 d2 Iτ3 + N2 d3 Cτ4 − d4 V − bV W

    b ¯ ¯ aV W − d5 W + N1 d2 I − Iτ3 + N2 d3 C − Cττ4 dx + a   b + DV V dx + DW W dx. a

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Then dL0 = . dt

 

   f U0 , 0 U 1− N¯1 (1 − ρ) + ρ N¯2 1 − U0 f (U, 0)

  f (U, V ) b R0 − 1 V − d5 W dx + DV + W dx. V dx + DW a f (U, 0)

   ∂V  ∂W dx = 0 and . W dx = ∂ dx = 0, we Recall that . V dx = ∂ ∂ν ∂ν obtain



   dL0 f (U0 , 0) U N¯1 (1 − ρ) + ρ N¯2 1 − ≤ . 1− + (R0 − 1) V dx. U0 f (U, 0) dt Further, from the hypothesis .(H1 ), this eventually gives rise to the following inequality: .





U f (U0 , 0) 1− 1− ≤ 0. U0 f (U, 0)

dL0 dL0 ≤ 0 if .R0 ≤ 1, and . = 0 holds if and only if .U = U0 = Hence, we have . dt dt λ , .I = 0, .C = 0, .V = 0 and .W = 0. This indicates that the singleton .{E0 } is the d1   dL0 = 0}. By LaSalle invariance largest invariant set in .{ U, I, C, V , W ∈ R5+ | dt principle, the infection-free equilibrium .E0 of model (1) is globally asymptotically stable whenever .R0 ≤ 1.  For .R0 > 1, we introduce the function .G(x) = x − 1 − ln(x) for .x > 0. Note that .G(x) = 0 if and only if .x = 1. Further, we introduce the following hypothesis: .



f (U, V ) V f (U, Vi ) ≤ 0 for all 1− − f (U, V ) Vi f (U, Vi )

U, V > 0,

(H4 )

where .Vi denote the virus components of the equilibrium .Ei for .i = 1, 2. Theorem 3.2 Assume that .(H4 ) holds for .E1 . If .RW 1  1 < R0 , then the infection equilibrium without humoral immunity .E1 of system (1) is globally asymptotically stable. Proof Constructing a Lyapunov functional .L1 as follows:     N¯1 (1 − ρ) + ρ N¯2 U − U1 − .L1 =

U U1

  f U1 , V1 I ds + N¯1 I1 G f (s, V1 ) I1

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C V b ¯ +N2 C1 G + V1 G + W C1 V1 a 

  t   f U (x, s), V (x, s) V (x, s) ¯   ds +N1 (1 − ρ)(1 − )f U1 , V1 V1 G f U1 , V1 V1 t−τ1 

  t   f U (x, s), V (x, s) V (x, s)   +N¯2 ρ(1 − )f U1 , V1 V1 ds G f U1 , V1 V1 t−τ2



 t  t I (x, s) C(x, s) ¯ ¯ +N1 d2 I1 ds dx, ds + N2 d3 C1 G G C1 I1 t−τ3 t−τ4 Taking the derivative of .L1 with respect to time t along the solution trajectories of the system (1)–(3), we obtain



   f (U1 , V1 ) U d1 U1 N¯1 (1 − ρ) + ρ N¯2 1 − 1− U1 f (U, V1 ) f (U1 , V1 ) f (Uτ1 , Vτ1 )Vτ1 I1 Iτ V1 V + + 3 + − N¯1 d2 I1 − 3 + f (U, V1 ) f (U1 , V1 )V1 I I1 V V1

f (U1 , V1 ) f (Uτ2 , Vτ2 )Vτ2 C1 f (Uτ1 , Vτ1 )Vτ1 Iτ3 + − N¯2 d3 C1 − 3 + − ln f (U, V1 ) f (U1 , V1 )V1 C f (U, V )V I



 V1 V Cτ4 V1 f (Uτ2 , Vτ2 )Vτ2 Cτ4 1− + dx + DV V dx + − ln C1 V V1 f (U, V )V C V  b + DW W dx. a

dL1 = dt

.

Since  .

V dx

= 0,

W dx

= 0,



 ∇V 2 V V dx = V 2 dx,  W  ∇W 2 W dx = W 2 dx,



we get dL1 = . dt

 

   f U1 , V1 U ¯ ¯  d1 U1 N1 (1 − ρ) + ρ N2 1 − 1−  U1 f U, V1    

f U1 , V1 f Uτ1 , Vτ1 Vτ1 I1 Iτ3 V1   +G   −N¯1 d2 I1 G +G I1 V f U1 , V1 V1 I f U, V1 



f U, V1 f (U, V ) f (U, V1 ) V ¯   +G + N1 d2 I1 1 − − f (U, V1 ) f (U, V ) V1 f U, V

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f (U1 , V1 ) f (Uτ2 , Vτ2 )Vτ2 C1 Cτ4 V1 ¯ −N2 d3 C1 G +G +G f (U, V1 ) f (U1 , V1 )V1 I C1 V





f (U, V1 ) V f (U, V ) f (U, V1 ) dx + N¯2 d3 C1 1 − − +G f (U, V ) f (U, V1 ) f (U, V ) V1   ∇V 2 bd5  R1 − 1 W − DV V1 + dx. a V2 From the hypothesis .(H1 ) on the general incidence function, we know that .f (U, V ) is a strictly increasing function with respect to U , and this eventually gives rise to the following inequality:

f (U, V ) f (U, V1 ) V 1− ≤ 0. − f (U, V1 ) f (U, V ) V1

.

dL1 ≤ 0 if .RW Hence, using the hypothesis .(H4 ), we obtain . 1 ≤ 1. Moreover, dt dL1 = 0 holds if and only if .U = U1 , .I = I1 , .C = C1 , .V = V1 , and . dt .W = 0. This indicates that the singleton .{E1 } is the largest invariant set in   dL1 = 0}. It follows from LaSalle invariance principle .{ U, I, C, V , W ∈ R5+ | dt that the infection equilibrium without humoral immunity .E1 of model (1) is globally  asymptotically stable whenever .RW 1 ≤ 1. Theorem 3.3 Assume that .(H4 ) holds for .E2 . If .RW 1 > 1, then the infection equilibrium with humoral immunity .E2 is globally asymptotically stable. Proof Constructing a Lyapunov functional .L2 as follows:  L2 =

.

f (U2 , V2 ) ds U2 f (s, V2 )

b C I V W + N¯2 C2 G + V2 G + W2 G +N¯1 I2 G I2 C2 V2 a W2

 t f (U (x, s), V (x, s)) V (x, s) ¯ dx +N1 (1 − ρ)(1 − )f (U2 , V2 ) V2 G f (U2 , V2 ) V2 t−τ1

 t f (U (x, s), V (x, s)) V (x, s) ¯ +N2 ρ(1 − )f (U2 , V2 ) V2 dx G f (U2 , V2 ) V2 t−τ2

 t  t C I ¯ ¯ +N1 d2 I2 ds dx, ds + N2 d3 C2 G G C2 I2 t−τ3 t−τ4    N¯1 (1 − ρ) + ρ N¯2 U − U2 −

U

Taking the derivative of .L2 with respect to time t along the solution trajectories of the system (1)–(3), we obtain

Mathematical Modeling and Numerical Analysis of HIV-1 Infection. . .

dL2 = . dt

111



   f (U2 , V2 ) U ¯ ¯ d1 U2 N1 (1 − ρ) + ρ N2 1 − 1− U2 f (U, V2 )





f (Uτ1 , Vτ1 )Vτ1 I2 Iτ3 V2 f (U2 , V2 ) +G +G −N¯1 d2 I2 G f (U, V2 ) f (U2 , V2 )V2 I I2 V





f (U, V ) f (U, V2 ) f (U, V2 ) V +G − + N¯1 d2 I2 1 − f (U, V2 ) f (U, V ) V2 f (U, V )





 f (Uτ2 , Vτ2 )Vτ2 C2 Cτ4 V2 f (U2 , V2 ) ¯ +G +G −N2 d3 C2 G f (U, V2 ) f (U2 , V2 )V2 I C2 V



f (U, V2 ) f (U, V ) f (U, V2 ) V ¯ +G dx + N2 d3 C2 1 − − f (U, V ) f (U, V2 ) f (U, V ) V2   ∇W (x, t)2 ∇V (x, t)2 dx − D dx. −DV V2 W W 2 V 2 (x, t) W 2 (x, t)

According to the hypothesis .(H1 ) on the general incidence function, we know that f (U, V ) is a strictly increasing function with respect to U , and this eventually gives rise to the following inequality:

.



f (U, V ) f (U, V2 ) V 1− ≤ 0. − f (U, V2 ) f (U, V ) V2

.

dL2 ≤ 0 if .RW Hence, using the hypothesis .(H4 ), we obtain . 1 > 1. Moreover, dt dL2 = 0 holds if and only if .U = U2 , .I = I2 , .C = C2 , .V = V2 , and .W = W2 . This . dt   indicates that the singleton set .{E2 } = { U2 , I2 , C2 , V2 , W2 } is the largest invariant   dL2 = 0}. By invariance principle, the infection set in .{ U, I, C, V , W ∈ R5+ | dt equilibrium with humoral immunity .E2 of model (1) is globally asymptotically  stable whenever .RW 1 > 1. This completes the proof.

4 Numerical Analysis In this section, we discretize the system (1) by using the mixed Euler. So, we obtain the following discrete model:

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⎧ m+1 U − Unm ⎪ ⎪ n ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ m+1 − I m ⎪ I ⎪ n n ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ ⎪ C m+1 − Cnm ⎪ ⎨ n t . ⎪ ⎪ m+1 ⎪ Vn − Vnm ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m+1 − W m ⎪ ⎪ n ⎩ Wn t

  = λ − d1 Unm+1 − (1 − ε)f Unm+1 , Vnm Vnm ,   = (1 − ε)(1 − ρ)f Unm−m1 +1 , Vnm−m1 Vnm−m1 − d2 Inm+1 ,   = ρ(1 − ε)f Unm−m2 +1 , Vnm−m2 Vnm−m2 − d3 Cnm+1 , = DV

m+1 m+1 Vn+1 − 2Vnm+1 + Vn−1

(x)2

+ N¯1 d2 Inm−m3 +1

+N¯2 d3 Cnm−m4 +1 − d4 Vnm+1 − bVnm+1 Wnm , = DW

m+1 m+1 Wn+1 − 2Wnm+1 + Wn−1

(x)2

+ aVnm+1 Wnm − d5 Wnm+1 . (6)

Here, we assume that .x ∈ = [xmin , xmax ] where .xmin , xmax ∈ R. Let .t be the time step size and .x = (xmax −xmin )/N be the space step size with N as a positive integer. Assume that there exist four integers .(m1 , m2 , m3 , m4 ) ∈ N4 with .τ1 = m1 t, .τ2 = m2 t, .τ3 = m3 t and .τ4 = m4 t. The space and time grid points are .xn = xmin + nx for .n ∈ {0, 1, ..., N } and .tm = mt for .m ∈ N. At each point,   we use approximations of . U (xn , tm ), .I (xn , tm ), C(xn , tm ), V (xn , tm ), W (xn , tm ) by . Unm , Inm , Cnm , Vnm , Wnm respectively. For the sake of convenience, we set all the approximation solutions at the time .tm by the .(N + 1)−dimensional vector .Z m = m )T , where .Z ∈ {U, I, C, V , W } and the notation .(.)T denotes the (Z0m , Z1m , . . . , ZN transposition of a vector. If all components of a vector Z are nonnegative, we denote it by .Z ≥ 0. The discrete initial conditions of system (6) are given as Uns = φ1 (xn , ts ) ,

.

Ins = φ2 (xn , ts ) ,

Vns = φ4 (xn , ts ) ,

.

Cns = φ3 (xn , ts ) ,

(7)

Wns = φ5 (xn , ts ) ,

for .n ∈ {0, 1, ..., N} and .s ∈ {−p, −p+1, ..., 0}, where .p = max{m1 , m2 , m3 , m4 }. Moreover, the discrete boundary conditions are given by m m m m V−1 = V0m , Vn+1 = Vnm , W−1 = W0m and Wn+1 = Wnm for m ∈ N.

.

It is not hard to show that system (6) can be rewritten as follows:

(8)

Mathematical Modeling and Numerical Analysis of HIV-1 Infection. . .

⎧ ⎪ ⎪ ⎪ Unm+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m+1 ⎪ ⎨ In .

= =

    Unm +t λ−f Unm+1 ,Vnm Vnm 1+d1 t



,

m−m1 +1 m−m1 Un ,Vn

Inm +(1−ε)(1−ρ)tf 1+d2 t



m−m2 +1 m−m2 Un ,Vn



113

m−m1

Vn

,



m−m2 Vn

C +ρ(1−ε)tf ⎪ ⎪ = n , Cnm+1 ⎪ 1+d2 t ⎪ ⎪ ⎪ m−m +1 ⎪ ⎪ Am V m+1 = V m + N¯1 d2 tIn 3 + N¯2 d3 tCnm−m4 +1 , ⎪ ⎪ ⎪  ⎩ m m+1  = 1 + tVnm+1 Wnm , B Wn m

(9)

where matrix .Am of dimension .(N + 1) × (N + 1) is given by ⎛

a0m ⎜ a ⎜ ⎜ 0 ⎜ ⎜ . ⎜ . ⎜ . ⎜ ⎜ 0 ⎜ ⎝ 0 0

Am =

.

a a1m a .. .

0 a a2m .. .

··· ··· ···

0 0 0 .. .

0 0 0 .. .

··· m a 0 0 · · · aN −2 m 0 0 · · · a aN −1 0 0 ··· 0 a

0 0 0 .. .

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ 0 ⎟ ⎟ a ⎠ m aN

  with .a = −DV t/(x)2 , .a0m = 1 + DV t/(x)2 + t d4 + bW0m ,   m .a t/(x)2 + t d4 + bWNm , and .aim = 1 + DV t/(x)2 + N  = 1 + DV  t d4 + bWim (i = 1, 2, . . . , N − 1). It is easy to show that A is a strictly diagonally dominant matrix. Hence, A is nonsingular. We then obtain that   V m+1 = (Am )−1 V m + N¯1 d2 tInm−m3 +1 + N¯2 d3 tCnm−m4 +1 ,

.

and   B m Wnm+1 = 1 + tVnm Wnm ,

.

where ⎛

B=

.

b1 ⎜b ⎜ 1 ⎜ ⎜0 ⎜ . ⎜ . ⎜ . ⎜ ⎜0 ⎜ ⎝0 0

b2 b3 b1 .. .

0 b1 b3 .. .

··· 0 0 0 ··· 0 0 0 ··· 0 0 0 . . . · · · .. .. ..

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ 0 0 · · · b1 b3 0 ⎟ ⎟ 0 0 · · · b1 b3 b1 ⎠ 0 0 · · · 0 b2 b1

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with .b1 = −DW t/(x)2 , .b2 = 1 + DW t/(x)2 + td5 , .b3 = 1 + 2DW t/(x)2 + td5 . It is easy to show that B is a strictly diagonally dominant matrix. Hence, B is nonsingular. Then we get   Wnm+1 = B −1 1 + tVnm Wnm .

.

Theorem 4.1 For any .t > 0 and .x > 0, the solutions of the discrete model (6) remain nonnegative and bounded for all .n ∈ N. Proof Obviously, .U m > 0 for all .m ∈ N. In fact, assuming the contrary and letting q m ≥ 0, C m ≥ 0, V m ≥ 0 and .q1 > 0 be the first time such that .U 1 ≤ 0, and .I m .W ≥ 0 for .m < q1 . From the first equation of (6), we have q −1

Un 1

.

 q q q q −1  q −1 . = Un 1 − t λ − d1 Un 1 − (1 − ε)f Un 1 , Vn 1 Vn 1 q

q −1

According to .(H0 ) − (H2 ) and .Un 1 ≤ 0, we get .Un 1 ≤ 0. This contradicts our assumption, and so .U m > 0 for all .m ∈ N. Now, we prove the nonnegativity of the sequences .I m , C m , .V m , and .W m by using mathematical induction. When .m = 0, we have   In0 + t (1 − ε)(1 − ρ)f Un−m1 +1 , Vnm1 Vnm1 , = 1 + d2 t   Cn0 + t (1 − ε)ρf Un−m1 +1 , Vnm1 Vnm1 1 Cn = , 1 + d3 t   Vn1 = (A0 )−1 V 0 + N¯1 d2 In−m3 +1 + N¯2 d3 Cn−m4 +1   Wn1 = B −1 1 + tVn1 Wn0 . 1 .In

Then .I 1  0 and .C 1  0. From the property of M-matrix (see, [26]), we deduce that .V 1  0 and .W 1  0. Thus, by using the induction, we get .I m  0, .C m  0, m  0, and .W m  0 for all .m ∈ N. This proves the nonnegativity of solutions. .V Next, we establish the boundedness of solutions; to this end, we define a sequence m .{G } as follows: m m+m1 Gm + Cnm+m2 , n = Un + In

.

then we have   m+1 , Gnm+1 − Gm ≤ t λ − ς G n n

.

where .ς = min(d1 , d2 , d3 ). Thus, we obtain

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λt 1 , Gm + 1 + ς t 1 + ς t n

Gnm+1 ≤

.

by using induction, we easily obtain Gm n ≤

.

1 1 + tς

m G0n +



m  1 λ , 1− ς 1 + tς

then we have .

lim sup Gm n ≤ m−→∞

λ for all n ∈ {0, 1, . . . , N }, ς

which implies that .{Gm } is bounded. Therefore, .{U m },.{I m }, and .{C m } are bounded. By the fourth equation of model (6) that N  .

Vnm+1 =

n=0

1 1 + d4 t + tbWnm N 

Vnm

+ t

n=0

N  

N¯1 d2 Inm−m3 +1 + N¯2 d3 Cnm−m4 +1

  .

n=0

Note that .{I m } and .{C m }are bounded. There exist two positive constants .M1 , M2 such that .Inm ≤ M1 , Cnm ≤ M2 for .n ∈ {0, 1, . . . , N}. Thus, we have N  .

n=0

Vnm+1

1 ≤ 1 + d4 t



N 

Vnm



+ t (N + 1) N¯1 d2 M1 + N¯2 d3 M2

 

,

n=0

By induction, we have N  n=0 .

Vnm

  N  (N + 1) N¯1 d2 M1 + N¯2 d3 M2 1 m V0 + ≤ (1 + d4 t)n d4 n=0



1 × 1− (1 + d4 t)n

≤

N  n=0

V0m

  (N + 1) N¯1 d2 M1 + N¯2 d3 M2 + , d4

implying .{V m } is bounded. It follows from the last equation of model (6) that N  .

n=0

Wnm+1 =

N  1 + atV m+1 n

n=0

1 + d5 t

Wnm .

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Since .{V m } is bounded, there exists a positive constant .M3 such that .Vnm ≤ M3 for .n ∈ {0, 1, . . . , N}, and by induction, we get N  .

1 + aM3 (N + 1)  0 Wn , d5 N

Wnm ≤

n=0

n=0

which implies that .{W m } is bounded. This completes the proof.



Theorem 4.2 If .R0 ≤ 1, then the infection-free equilibrium .E0 of system (6) is globally asymptotically stable for all .t > 0 and .x > 0. Proof Consider the following discrete Lyapunov functional:     Unm  1  ¯ f , 0) (U 0 m ds = N1 (1 − ρ) + ρ N¯2 Un − U0 − f (s, 0) t U0 m=0  b m m m m ¯ ¯ +N1 In + N2 Cn + (1 + d4 t)Vn + Wn a

Gm n

N 

.

+ N¯1 (1 − ρ)(1 − )

m−1 

  j +1 j j f Un , Vn Vn

j =m−m1

+ρ N¯2 (1 − )

m−1 

⎫ m−1 m−1 ⎬     j +1 j j j +1 j +1 f Un , Vn Vn + N¯1 d2 In + N¯2 d3 Cn . ⎭

j =m−m2

j =m−m3

j =m−m4

Then ⎧ ⎛ ⎞ N ⎨    f (U0 , 0) ⎠   λ − d1 Unm+1 Gnm+1 − Gm N¯1 (1 − ρ) + ρ N¯2 ⎝1 −  n ≤ m+1 ⎩ f Un , 0 m=0

.

m+1    V m+1 − 2Vnm+1 + Vn−1 − (1 − )f Unm+1 , Vnm Vnm + DV n+1 (x)2     + N¯1 (1 − ρ)(1 − )f Unm−m1 +1 , Vnm−m1 Vnm−m1 − d2 Inm+1     + N¯2 ρ(1 − )f Unm−m2 +1 , Vnm−m2 Vnm−m2 − d3 Cnm+1

  m−m4 +1 − d4 Vnm+1 − bVnm+1 Wnm + d4 Vnm+1 − Vnm + N¯1 d2 Inm−m3 +1 + N¯2 d3 Cm       + N¯1 (1 − ρ)(1 − ) f Unm+1 , Vnm Vnm − f Unm−m1 +1 , Vnm−m1 Vnm−m1       + N¯2 ρ(1 − ) f Unm+1 , Vnm Vnm − f Unm−m2 +1 , Vnm−m2 Vnm−m2    $ +N¯1 d2 Inm+1 − Inm−m3 +1 + N¯2 d3 Cnm+1 − Cnm−m4 +1

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Hence, m+1 .Gn

− Gm n

  N m+1    f U0 , 0 U n d1 U0 N¯1 (1 − ρ) + ρ N¯2 1 − 1 −  m+1  ≤ U0 f Un , 0 m=0

  +d4 R0 − 1 Vnm .

The last inequality is followed by the condition .(H1 )−(H3 ). Thus, if .R0 ≤ 1, then we have .Gn+1 − .Gn ≤ 0, for all .n ∈ N, which implies that .Gn is a monotone decreasing sequence. Since .Gn ≥ 0, there is a limit . lim Gn ≥ 0 n→+∞

which implies that . lim (Gn+1 − Gn ) = 0, from which we get . lim Unm = U0 n→+∞

n→+∞

and . lim Vnm (R0 − 1) = 0. We discuss two cases: .(i) if .R0 < 1, from model (6), n→+∞

we obtain . lim Inm = 0, . lim Cnm = 0, for all .m ∈ (0, 1, . . . , N); .(ii) if .R0 = 1, n→+∞

n→+∞

by . lim U nm = U0 and from model (6), we have . lim Inm = 0, . lim Cnm = 0, n→+∞

.

n→+∞

n→+∞

lim Vnm = 0. Thus, we deduce that .E0 is globally asymptotically stable. This

n→+∞



completes the proof.

Theorem 4.3 Assume that .(H4 ) holds for .E1 . If .RW 1  1 < R0 , then the infection equilibrium without humoral immunity .E1 of system (6) is globally asymptotically stable for all .t > 0 and .x > 0. m G n =

   Unm  N   m f U1 , V1 1  ¯ ¯ ds N1 (1 − ρ) + ρ N2 Un − U1 − f (s, V1 ) t U1

m=0

+ N¯1 I1 G

.



Inm I1

+ N¯2 C1 G



Cnm C1

+ (1 + d4 t)V1 G m−1 

  + N¯1 (1 − ρ)(1 − )f U1 , V1 V1

j =m−m1 m−1 

  + ρ N¯2 (1 − )f U1 , V1 V1

j =m−m2 m−1 

+ N¯1 d2

j =m−m3

Hence,



j +1

In I1 G I1

Vnm V1

+

 j +1 j  j f Un , Vn Vn   G f U1 , V1 V1

 j +1 j  j f Un , Vn Vn   G f U1 , V1 V1

+ N¯2 d3

b (1 + d5 t)Wnm a

m−1  j =m−m4



j +1

Cn C1 G C1

.



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N    f (U1 , V1 ) Unm+1 ¯ ¯ d1 U1 N1 (1 − ρ) + ρ N2 1 − 1− U1 f (Unm+1 , V1 ) m=0 ⎛ ⎞  f (U1 , V1 ) ⎠ ¯  + N1 (1 − ρ)(1 − )f (U1 , V1 ) V1 − G ⎝  f Unm+1 , V1 ⎛   ⎞ f Unm+1 , V1 ⎠ − G⎝  f Unm+1 , Vnm ⎛  m−m +1 m−m  m−m ⎞   1 1 m−m3 +1 Vn f Un 1 , Vn I1 In V1 ⎠ ⎝ −G − G f (U1 , V1 ) V1 Inm+1 I1 Vnm+1 ⎞   m+1   m+1 m   ⎛  m f U V , V , V f U V 1 1 n  ⎝  n   nm + n 1− − 1⎠ V1 f Un+1 , V1 f Unm+1 , V m V m

m nm+1 − G G n =

.

n

⎛

n

 m+1  ⎞ f U , V1 f (U1 , V1 ) ⎠ ⎠  − G⎝  n + N¯2 ρ(1 − )f (U1 , V1 ) V1 − G ⎝  m+1 m+1 f Un , Vnm f Un , V1 ⎞ ⎛  m−m +1 m−m  m−m   1 1 Vn C1 f Un 1 , Vn Cnm−m4 +1 V1 ⎠ −G − G⎝ f (U1 , V1 ) V1 Cnm+1 C1 Vnm+1 ⎞     ⎛  

f Unm+1 , V1 V1 f Unm+1 , Vnm Vnm ⎠ ⎝    m  + −1 1− V1 f Un+1 , V1 f Unm+1 , Vnm Vnm 

+

⎞

⎛

N −1 m+1  bd5  DV  (Vn+1 − Vnm+1 )2 R1 − 1 Wnm − m+1 m+1 a (x)2 Vn+1 Vn n=0

 It follows from the condition .(H1 ) − (H3 ) that   .

  f Unm+1 ,V1 V1   f Unm+1 ,Vnm Vnm

−1

.

1−

  f Unm+1 ,Vnm   f Unm+1 ,V1



≤ 0. Recall that .G(x) ≥ 0 for all .x > 0, we then

n ≤ 0, for all .n ∈ N, which implies that .G n+1 − G n is a monotone obtain .G   decreasing sequence. Since .Gn ≥ 0, there is a limit . lim Gn ≥ 0 which implies n→+∞   n = 0. Furthermore, from model (6), it can be shown that n+1 − G that . lim G n→+∞   m RW − 1 = 0, and . lim U m = U , . lim I m = I , . lim C m = C . lim Wn 1 1 1 n n n 1 n→+∞

n→+∞

n→+∞

n→+∞

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119

and . lim Vnm = V1 for all .m ∈ (0, 1, . . . , N ), which implies that .E1 of model (6) n→+∞

is globally asymptotically stable. This completes the proof. . Similarly to above ways, we can easily obtain the following result.



Theorem 4.4 Assume that .(H4 ) holds for .E2 . If .RW 1 > 1, then the infection equilibrium with humoral immunity .E2 of system (6) is globally asymptotically stable for all .t > 0 and .x > 0.

5 Application and Numerical Simulations In this section, we apply our theoretical results obtained in the previous sections to the following model: ⎧ m+1 V m Unm+1 −Unm ⎪ n m+1 − (1 − ε) κ1 Un ⎪ , = λ − d U 1 ⎪ n t m ⎪ 1 + κ V ⎪ 2 n ⎪ ⎪ m−m1 +1 m−m1 ⎪ ⎪ Vn ⎪ Inm+1 −Inm = (1 − ε)(1 − ρ) κ1 Un ⎪ − d2 Inm+1 , ⎪ m−m1 t ⎪ ⎪ 1 + κ2 Vn ⎨ κ1 Unm−m2 +1 Vnm−m2 Cnm+1 −Cnm . = ρ(1 − ε) − d3 Cnm+1 , ⎪ m−m2 t ⎪ ⎪ 1 + κ V 2 n ⎪ ⎪ ⎪ Vnm+1 −Vnm ⎪ = DV V + N¯1 d2 Inm−m3 +1 + N¯2 d3 Cnm−m4 +1 − d4 Vnm+1 ⎪ ⎪ t ⎪ ⎪ ⎪ −bVnm+1 Wnm , ⎪ ⎪ ⎩ Wnm+1 −Wnm = DW W + aVnm+1 Wnm − d5 Wnm+1 , t

(10)

where .κ1 denote the virus-to-cell infection. The nonnegative constant .κ2 measures the saturation effect. The other state variables and parameters have the same biological meanings as in system (1). As before, we consider model (10) with initial conditions .

Uns = φ1 (xn , ts ) ≥ 0, Ins = φ2 (xn , ts ) ≥ 0, Cns = φ2 (xn , ts ) ≥ 0, Vns = φ3 (xn , ts ) ≥ 0, Wns = φ4 (xn , ts ) ≥ 0, (11)

for .n ∈ {0, 1, ..., N } and .s ∈ {−p, −p+1, ..., 0}, where .p = max{m1 , m2 , m3 , m4 }. In addition, the discrete boundary conditions are given by m m m m V−1 = V0m , Vn+1 = Vnm , W−1 = W0m and Wn+1 = Wnm for m ∈ N.

.

(12)

The problem (10)–(12) is a particular case of system (1)–(3) with .f (H, V ) = κ1 U . Based on the previous sections, the system (10) has three equilibria 1 + κ2 V λ denoted: the infection-free equilibrium .E0 ( , 0, 0, 0, 0), the infection equilibrium d1

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without humoral immunity .E1 (U1 , I1 , C1 , V1 , 0), and the infection equilibrium with humoral immunity .E2 (U2 , I2, C2 , V2 , W2 ). In addition, the basic reproduction  (1 − ε) N¯1 (1 − ρ) + ρ N¯2 λκ1 . number is given by .R0 = d1 d4 On the other hand, it is clear that the hypotheses .(H0 )–.(H3 ) are satisfied. For the fourth hypothesis, we have

f (Unm+1 , Vnm ) Vnm+1 f (Unm+1 , Vi ) 1− − Vi f (Unm+1 , Vnm ) f (Unm+1 , Vi )

.

=

−κ2 (Vnm+1 − Vi )2 Vnm+1 (1 + κ2 Vnm+1 )(1 + κ2 Vi )

≤ 0,

for i = 1, 2. Hence, the last hypothesis .(H4 ) is satisfied. From Theorems 3.1, 3.2, and 3.3, we deduce the following result.

.

Corollary 5.1 (i) The infection-free equilibrium .E0 of system (10)–(12) is globally asymptotically stable if .R0 ≤ 1. (ii) The infection equilibrium without humoral immunity .E1 of system (10)–(12) is globally asymptotically stable if .RW 1  1 < R0 . (iii) The infection equilibrium with humoral immunity .E2 of system (10)–(12) is globally asymptotically stable if .RW 1 > 1. To illustrate our theoretical results, we give some numerical simulations by choosing the time interval from .t = 0 to .t = 500 with a step size .t = 0.1, and the spatial domain . = [0, 20] with a step size .x = 1. Moreover, we take .d1 = 0.01, .d2 = 0.7, .d3 = 0.07, .d4 = 13, .b = 0.0001, .ε = 0.5, .ρ = 0.195, ¯1 = 100, .N¯2 = 4.11, .τ1 = 3.5, .τ2 = 2.5, .τ3 = 1.5, .τ4 = 0.5, .κ1 = 8 × 10−7 , .N −5 and .D = D = 0.05. The values of some above parameters are taken .κ2 = 10 V W from [8, 15]. For the case .R0 ≤ 1, we take .λ = 103 , .a = 10−5 and .d5 = 0.5. By calculation, we have .R0 = 0.2502 and system (10) has a unique free-infection equilibrium point 5 .E0 (10 , 0, 0, 0, 0) which is globally asymptotically stable which biologically means that the HIV-1 is cleared and the patient will be completely cured. Figure 1 illustrates this result. If we take .λ = 104 and we keep the values of the other parameters, we find that .R0 = 2.5016 > 1 and .RW 1 = 0.0143 < 1. Therefore, the solution of system (1) converges to the infection equilibrium without humoral immunity 5 3 4 4 .E1 (5.1982 × 10 , 5.454 × 10 , 1.3211 × 10 , 2.9546 × 10 , 0) which is globally asymptotically stable. This biologically means that the HIV-1 persists and the infection becomes chronic (see Fig. 2).

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Fig. 1 Stability of the infection-free equilibrium .E0

Fig. 2 Stability of the infection equilibrium without humoral immunity .E1

To confirm numerically the third point of Theorem 3.3, we choose .a = 10−4 and we keep the same values of the other parameters as the second case. By calculation, we have .RW 1 = 2.5000 > 1. Then model (1) has an infection equilibrium with humoral immunity .E2 (8.3634×105 , 2.003×103 , 4.849×103 , 5.494×103 , 1.2725× 105 ) which is globally asymptotically stable. Figure 3 demonstrates this result.

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Fig. 3 Stability of the infection equilibrium with humoral immunity .E2

References 1. WHO. HIV/AIDS, Fact Sheet, 27 July 2022. Available online: https://www.who.int/newsroom/fact-sheets/detail/hiv-aids. 2. A. S. Perelson, A. U. Neumann, M. Markowitz, J.M. Leonard, D.D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science. 271 (1996) 1582–1586. 3. J. E. Mittler, B. Sulzer, A. U. Neumann, A. S. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci. 152 (1998) 143–163. 4. X. Lai, X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-tocell transmission. SIAM J. Appl. Math. 74 (2014) 898–917. 5. L. Rong, A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theor. Biol. 260 (2009) 308–331. 6. D. S. Callaway, A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol. 64 (2002) 29–64. 7. W. S. Hlavacek, N. I. Stilianakis, A. S. Perelson, Influence of follicular dendritic cells on HIV dynamics, Philos. Trans. R. Soc. London B Biol. Sci. 355 (2000) 1051–1058. 8. S. Wang, Y. Zhou, Global dynamics of an in-host HIV-1 infection model with the long-lived infected cells and four intracellular delays, Int. J. Biomath. 5 (2012). 9. J. Xu, Y. Zhou, Y. Li, Y. Yang, Global dynamics of a intracellular infection model with delays and humoral immunity, Math. Meth. Appl. Sci. 9 (2016). 10. K. Wang, W. Wang, S. Song, Dynamics of an HBV model with diffusion and delay, J. Theor. Biol. 253 (2008) 36–44. 11. K. Hattaf, A.A. Lashari, Y. Louartassi, N. Yousfi, A delayed SIR epidemic model with general incidence rate, J. Qual. Theor. 3 (2013) 1–9. 12. K. Hattaf, N. Yousfi, Global properties of a discrete viral infection model with general incidence rate, Math. Methods Appl. Sci. 39 (2016) 998–1004. 13. K. Hattaf, N. Yousfi, A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. Real World Appl. 13 (2012) 1866–1872. 14. G. Huang, Y. Takeuchi, W. B. Ma, Lyapunov functionals for delay differential equations model of viral infections, J. Appl. Math. 70 (2010) 2693–2708.

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15. Y. Geng, J. Xu, Global stability of a delayed and diffusive virus model with nonlinear infection function, J. Biol. Dyn. 1 (2021) 287–307. 16. K. Hattaf, A. A. Lashari, B.E. Boukari, N. Yousfi,Effect of discretization on dynamical behavior in an epidemiological model, Differential Equations Dynam. Systems.23 (4) (2015) 403–413. 17. K. Hattaf, N. Yousfi, A numerical method for a delayed viral infection model with general incidence rate, J. King Saud Univ. Sci. (2015). 18. Z. Hajhouji, M. El younoussi, K. Hattaf, N. Yousfi, A Numerical method for a diusive HBV inection moe with muti-delays and two modes of transmission. (2021) 2052–2541. 19. X. Y. Wang, K. Hattaf, H.F. Huo, H. Xiang, Stability analysis of a delayed social epidemics model with general contact rate and its optimal control, J. Ind, Manag. Optim. 12 (2016) 1267– 1285. 20. R. H. Martin, H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer.Math. Soc. 321 (1990) 1–44. 21. R. H. Martin, H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence. J. reine Angew. Math. 413 (1991) 1–35. 22. C. C. Travis, G. F. Webb, Existence and stability for partial functional diferential equations, Trans. Amer. Math. Soc. 200 (1974) 395–418. 23. W. E. Fitzgibbon, Semilinear functional differential equations in Banach space. J. Differ. Equ. 29 (1978) 1–14. 24. J. Wu, Theory and applications of partial functional diferential equations, Springer, New York. (1996). 25. D. Henry, Geometric theory of semilinear parabolic equations: Lecture notes in mathematics, springer-verlag, Berlin, New York. 840 (1993). 26. T. Fujimoto, R. R. Ranade, Two characterizations of inverse-positive matrices: the HawkinsSimon condition and the Le Chatelier-Braun principle, electron. J. Linear Algebra. 11 (2004) 59–65.

A Reaction-Diffusion Fractional Model for Cancer Virotherapy with Immune Response and Hattaf Time-Fractional Derivative Majda El Younoussi, Zakaria Hajhouji, Khalid Hattaf, and Noura Yousfi

1 Introduction Cancer is a large group of diseases that can affect any part of the body. It is the top global cause of death [1]. There are many treatments for cancer such as chemotherapy, radiotherapy, hormone therapy, virotherapy, surgery, etc. These treatments may be used alone or in combination. Virotherapy is still a novel treatment for cancer and consists in using a programmed virus to attack only cancer cells without harming the normal ones. The oncolytic M1 virus is one of the alphaviruses used in virotherapy that was isolated from culicine mosquitoes collected on Hainan Island in China. This virus has the ability to replicate specifically in cancer cells and to spread to other cancer cells to infect them [2]. Many researchers used mathematical modeling to understand the dynamic of cancer using virotherapy. For instance, an ordinary differential equation (ODE) model was suggested by Wang et al. [3] to describe the dynamic of nutrients, normal cells, tumor cells, and the M1 virus. The model given in [3] was expanded by Elaiw et al. [4] by integrating spatial effects and an antitumor immune response carried out by cytotoxic T lymphocyte (CTL) cells. To study the effect of memory on the dynamics of oncolytic M1 virotherapy, El Younoussi et al. [5] proposed a

M. El Younoussi () · Z. Hajhouji · N. Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’Sick, Hassan II University of Casablanca, Casablanca, Morocco K. Hattaf Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’Sick, Hassan II University of Casablanca, Casablanca, Morocco Equipe de Recherche en Modélisation et Enseignement des Mathématiques (ERMEM), Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Casablanca, Morocco © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_7

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fractional model for cancer therapy with M1 oncolytic virus. Since the CTL plays an important role in the adaptive immune response by invading bacteria, viruses, parasites, surgical implants, and especially cancer cells, the authors in [6] proposed a FPDE model to study the impact of CTL on virotherapy. The fractional derivative used in the above models is of Caputo with singular kernel. To avoid the singularity problem, we propose a mathematical model in order to describe the spatiotemporal dynamics of interaction between nutrient, normal cells, tumor cells, M1 virus, and CTL cells by means of the new generalized Hattaf fractional (GHF) derivative [7] that includes various forms of fractional derivatives such as the Caputo-Fabrizio fractional derivative [8], the AtanganaBaleanu fractional derivative [9], and the weighted Atangana-Baleanu fractional derivative [10]. The outline of this chapter is organized as follows. We present our reactiondiffusion fractional model as well as the steady states of the model in the second section. Section 3 deals with the stability analysis of equilibria. Finally, the paper ends with a conclusion.

2 Model Formulation and Steady States In this section, we propose a new reaction-diffusion fractional model for cancer virotherapy in presence of cellular immunity mediated by CTL cells. The reaction is modeled by the GHF derivative, while the diffusion is described by a Laplacian operator. First, we recall the definition of the GHF derivative. Definition 2.1 ([7]) Let .α ∈ [0, 1), .β, γ > 0, and .f ∈ H 1 (a, b). We define the GHF derivative of order .α in Caputo sense of the function .f (t) with respect to the weight function .w(t) as follows: C

.

α,β,γ Da,t,w f (t)

N(α) 1 = 1 − α w(t)



t

Eβ [−μα (t − x)γ ]

a

d (wf )(x)dx, dx

(1)

where .w ∈ C 1 (a, b), .w, w  > 0 on [a,b], .N(α) is a normalization function obeying +∞  α tk , and .Eβ (t) = .N (0) = N (1) = 1, .μα = is the Mittag-Leffler (βk+1) 1−α k=0 function of parameter .β. The GHF derivative introduced in the above definition generalizes and extends many special cases in literature. For instance, when .w(t) = 1 and .β = γ = 1, we get the Caputo-Fabrizio fractional derivative [8] given by C

.

α,1,1 Da,t,1 f (t)

N(α) = 1−α



t a

exp[−μα (t − x)]f  (x)dx.

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In addition, when .w(t) = 1 and .β = γ = α, we obtain the Atangana-Baleanu fractional derivative [9] given by C

.

α,α,α Da,t,1 f (t) =



N(α) 1−α

t

Eα [−μα (t − x)α ]f  (x)dx.

a

Moreover, for .β = γ = α, we get the weighted Atangana–Baleanu fractional derivative [10] given by C

.

α,α,α Da,t,w f (t) =

N(α) 1 1 − α w(t)



t

Eα [−μα (t − x)α ]

a

α,β,β

α,β

For simplicity, let .C D0,t,1 be denoted by .Dt α,β tive .∂t

d (wf )(x)dx. dx

. Hence, the time-fractional deriva-

is given for any function by α,β

∂t

.

ϕ(x, t) =

N(α) 1−α



t

Eβ [−μα (t − ξ )β ]

0

∂ϕ (x, ξ )dξ. ∂ξ

Now, we present our new model which is given by the following nonlinear system of FPDEs: ⎧ α,β ∂t S ⎪ ⎪ ⎪ α,β ⎪ ⎪ ⎪ ⎪ ∂t N ⎪ ⎨ α,β ∂t T . ⎪ ⎪ ⎪ ⎪ ⎪ ∂ α,β V ⎪ ⎪ ⎪ ⎩ tα,β ∂t Z

= DS S + A − dS(x, t) − β1 S(x, t)N (x, t) − β2 S(x, t)T (x, t), = DN N + r1 β1 S(x, t)N (x, t) − (d + 1 )N (x, t), = DT T + r2 β2 S(x, t)T (x, t) − (d + 2 )T (x, t) − β3 T (x, t)V (x, t) −β4 T (x, t)Z(x, t), = DV V + B + r3 β3 T (x, t)V (x, t) − (d + 3 )V (x, t), = DZ Z + r4 β4 T (x, t)Z(x, t) − (d + 4 )Z(x, t),

(2)

where .S(x, t), .N (x, t), .T (x, t), .V (x, t), and .Z(x, t) are the concentrations of nutrient, normal cells, tumor cells, M1 virus, and CTL cells at time t and position x , respectively. The parameters .DS , .DN , .DT , .DV , and .DZ are the diffusion coefficients for nutrient, normal cells, tumor cells, M1 virus, and CTL cells, respectively. The parameter A is the recruitment rate of nutrient. The parameter B means the minimum effective dosage of medication. .β1 SN and .β2 ST are the rates of consuming the nutrient by normal and tumor cells, respectively. .r1 β1 SN and .r2 β2 ST represent the growth rates of normal and tumor cells, respectively. The M1 virus infects and eradicates tumor cells at rate .β3 T V , while it reproduces at rate .r3 β3 T V . The constant d is the washout rate of nutrient and bacteria. CTL immune cells destroy tumor cells at rate .β4 T Z , and they replicate at rate .r4 β4 T Z . The parameters . 1 , . 2 , . 3 , and . 4 are the natural death rates of normal cells, tumor cells, M1 virus, and CTL cells, respectively. . =

n  ∂ 2 is the Laplacian operator. 2

i=1 ∂xi

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In this study, we consider system (2) with the following initial conditions: .S(x, 0)

= φ1 (x) ≥ 0, N (x, 0) = φ2 (x) ≥ 0, T (x, 0) = φ3 (x) ≥ 0, .V (x, 0)

(3)

¯ = φ4 (x) ≥ 0, Z(x, 0) = φ5 (x) ≥ 0, x ∈ .

The boundary conditions are given by the homogeneous Neumann boundary conditions: .

∂N ∂T ∂V ∂Z ∂S = = = = = 0, on ∂ × (0, +∞), ∂n ∂n ∂n ∂n ∂n

(4)

∂ is the outward where . is a bounded domain in .Rn with smooth boundary .∂ and . ∂n normal derivative on the boundary .∂ . We define the ability of absorbing nutrient by normal cells, the ability of absorbing nutrient by tumor cells, and the reproduction number for cellular immunity, respectively, by .A1

=

Ar1 β1 r β (d + 3 ) Ar2 β2 , A3 = 4 4 . , A2 = d(d + 2 ) r3 β3 (d + 4 ) d(d + 1 )

Theorem 2.1 System (2) has: (i) A competition-free equilibrium .E0 (S0 , 0, 0, V0 , 0), which always exists. (ii) A tumor-free equilibrium .E1 (S1 , N1 , 0, V1 , 0) when .A1 > 1. (iii) A treatment failure immune-free equilibrium .E2 (S2 , 0, T2 , V2 , 0) when .A2 > 1 + Bβ3 (d+ 2 )(d+ 3 ) . (iv) A partial success immune-free equilibrium .E3 (S3 , N3 , T3 , V3 , 0) when .A2 > A1 + Bβ2 ABr1 β1 β3 3) > 1 + β2r(d+

A2 β d . d(d+ )(d+ )(d+ ) and .A1 + 3

2

r3 d(d+ 2 )( A −1)

1

3 3

1

(v) A treatment failure equilibrium .E4 (S4 , 0, T4 , V4 , Z4 ) when .A3 > 1 and .A2 > 1 + β2 (d+ 4 ) B(β2 (d+ 4 )+r4 β4 d) r β d + r d(d+ )(d+ )(A −1) . 4 4

3

2

4

3

4) (vi) A coexistence equilibrium .E5 (S5 , N5 , T5 , V5 , Z5 ) when .A3 > 1, .A1 > 1 + β2r(d+

, 4 β4 d ABr1 β1 r4 β4 and .A2 > A2 + r d(d+ )(d+ )(d+ )(A −1) . 3

1

2

4

3

Proof In the absence of CTL immunity, the model (2) has four equilibrium points: • By a simple computation, system (2) has a competition-free equilibrium .E0 (S0 , 0, 0, V0 , 0), B where .S0 = A d and .V0 = d+ 3 . • Clearly, when .A1 > 1, system (2) has a tumor-free equilibrium .E1 (S1 , N1 , 0, .V1 , 0), B d 1 where .S1 = d+

r β , .N1 = β (A1 − 1) and .V1 = d+ . 1 1

1

3

3 , system (2) has another steady state called the treatment • When .A2 > 1 + (d+ Bβ 2 )(d+ 3 ) 2 , .T2 = failure immune-free equilibrium .E2 (S2 , 0, T2 , V2 , 0), where .S2 = β3 Vr2 +d+

β 2 2 √ Br2 −a2 + δ −d + and .V2 = , with .a1 = β3 (r3 β3 d + β2 (d + 3 )), .a2 = β2 β3 V2 +d+ 2 2a1 a1 2 β3 (d + 2 ) − β2 β3 (A + r2 r3 B), .a3 = −Bβ2 (d + 2 ) and .δ = a2 − 4a1 a3 .

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• By computation, system (2) has an immune-free equilibrium .E3 (S3 , N3 , T3 , V3 , 0) when β2 B ABr1 β1 β3 3) .A2 > A1 + > 1 + β2r(d+

A2 β d , where d(d+ )(d+ )(d+ ) and .A1 + 3

2

r3 d(d+ 2 )( A −1)

1

3 3

1

.S3

=

N3 =

−B + (d + 3 )V3 d + 2 A2 d + 1 , T3 = , V3 = ( − 1) r 1 β1 r 3 β3 V 3 β3 A1 Ar1 r3 β1 β3 − r3 β3 d(d + 1 ) − β2 (d + 1 )(d + 3 ) r3 β1 β3 (d + 1 ) +

Bβ2

A1 r3 β1 (d + 2 )( A − 1)

.

1

In the presence of CTL immunity, model (2) has two other steady states denoted: • The treatment failure equilibrium .E4 (S4 , 0, T4 , V4 , Z4 ), where d + 4 Br4 β4 Ar4 β4 , T4 = , V4 = β2 (d + 4 ) + r4 β4 d r 4 β4 r3 β3 (d + 4 )(A3 − 1) Ar2 β2 r3 β3 r4 β4 (d + 4 )(A3 − 1) Z4 = r3 β3 β4 (β2 (d + 4 ) + r4 β4 d)(d + 4 )(A3 − 1) .S4

=

−

(β2 (d + 4 ) + r4 β4 d)(r3 β3 (d + 2 )(d + 4 )(A3 − 1) + Bβ3 r4 β4 ) . r3 β3 β4 (β2 (d + 4 ) + r4 β4 d)(d + 4 )(A3 − 1)

which exists under the conditions .A3

>

1 and .A2

>

B(β2 (d+ 4 )+r4 β4 d) r3 d(d+ 2 )(d+ 4 )(A3 −1) .

4) 1 + β2r(d+

+ 4 β4 d

• The coexistence equilibrium .E5 (S5 , N5 , T5 , V5 , Z5 ) which exists if .A3 > 1, .A1 > ABr1 β1 r4 β4 4) 1 and .A2 > A1 + r d(d+ )(d+

, where .S5 = d+

1 + β2r(d+

r β , β d )(d+ )(A −1) 4 4

3

1

2

4

3

1 1

Br4 β4 Ar1 β1 r4 β4 −(β2 (d+ 4 )+r4 β4 d)(d+ 1 ) 4 , .V5 = r β (d+

and = d+

r4 β4 , .N5 = β1 r4 β4 (d+ 1 ) 3 3 4 )(A3 −1) r3 (r2 β2 (d+ 1 )−r1 β1 (d+ 2 ))(d+ 4 )(A3 −1)−Br1 β1 r4 β4 . . .Z5 = r β r β (d+ )(A −1)

.T5

1 1 3 4

4

3

3 Stability Analysis In this section, we will discuss the stability of all the steady states. 3 Theorem 3.1 The competition-free equilibrium .E0 is stable for .A2 ≤ 1 + (d+ Bβ and 2 )(d+ 3 ) .A1 ≤ 1.

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Proof Consider the following Lyapunov functional: 

 .L0 (t)

=



+

S0 φ

S(x, t) S0

+

 1 1 1 V (x, t) N (x, t) + T (x, t) + V0 φ r1 r2 r2 r3 V0

1 Z(x, t)dx, r2 r4

where .φ(n) = n − ln(n) − 1 for .n > 0. By applying Corollary 2 of [11], we get α,β

.∂t



L0 ≤

−d

d + 1 d + 4 (A1 − 1)N − Z r1 r2 r4  d + 3 (V − V0 )2 d + 2 Bβ3 A2 − 1 − T − + r2 (d + 2 )(d + 3 ) r2 r3 V  S 1 1 + 1 − 0 DS S + DN N + DT T r1 r2 S  V0 1 1 DV V + 1− DZ Zdx + V r2 r4 r2 r3 S

(S − S0 )2 +

By the boundary conditions (4) and the divergence theorem, we get α,β

.∂t

  d + 1 (S − S0 )2 d + 4 dx + (A1 − 1) N dx − Zdx S r1 r2 r4   d + 2 Bβ3 A2 − 1 − + T dx (d + 2 )(d + 3 ) r2   (V − V0 )2 ||∇S||2 d + 3 dx − DS S0 dx − r2 r3 V S2  ||∇V ||2 DV V 0 dx. − r2 r3 V 2 

L0 ≤ −d

α,β

3 L0 ≤ 0 when .A2 ≤ 1 + (d+ Bβ and .A1 ≤ 1. Thereby, the condition (ii) of 2 )(d+ 3 ) Bβ3 Theorem 5 of [12] is satisfied if .A2 ≤ 1 + (d+ )(d+ ) and .A1 ≤ 1. 2 3 Moreover, .φ(n) attains its global minimum at .n = 1 and .φ(1) = 0. Then .φ(n) ≥ 0 for all .n > 0. Hence, .L0 (S, N, T , V , Z) ≥ 0 and .L0 (S0 , N0 , T0 , V0 , Z0 ) = 0. Then the condition (i) of Theorem 5 of [12] is satisfied. Therefore, the competition-free equilibrium .E0 is stable 3 when .A2 ≤ 1 + (d+ Bβ )(d+ ) and .A1 ≤ 1.

Thus, .∂t

2

3

Theorem 3.2 Let .A1 > 1. The tumor-free equilibrium .E1 is stable if .A2 ≤ A1 + ABr1 β1 β3 d(d+ )(d+ )(d+ ) . 1

2

3

A Reaction-Diffusion Fractional Model for Cancer Virotherapy with Immune. . .

131

Proof Consider the Lyapunov functional 

 .L1 (t)

=



+

S1 φ

S(x, t) S1

+

 1 1 N (x, t) + T (x, t) N1 φ r1 N1 r2

 1 V (x, t) 1 + V1 φ Z(x, t)dx. r2 r3 V1 r2 r4

Obviously, .L1 (S, N, T , V , Z) ≥ 0 and .L1 (S1 , N1 , T1 , V1 , Z1 ) = 0, which satisfy the first condition of Theorem 5 in [12]. Moreover, α,β

.∂t



L1 ≤

B (V − V1 )2 (S − S1 )2 d + 4 − − Z S r r V V r2 r4 2 3 1  ABr1 β1 β3 d(d + 1 )(d + 2 ) T A2 − A1 − + Ar1 r2 β1 d(d + 1 )(d + 2 )(d + 3 ) −(d + β1 N1 )

  N S 1 1 1 − 1 DN N + DT T + 1 − 1 DS S + S r1 N r2 +

 V 1 1 1 − 1 DV V + DZ Zdx r2 r3 V r2 r4 

 (V − V1 )2 (S − S1 )2 B dx dx − S r2 r3 V V1   ABr1 β1 β3 d(d + 1 )(d + 2 ) A2 − A1 − T dx + Ar1 r2 β1 d(d + 1 )(d + 2 )(d + 3 )

= −(d + β1 N1 )

−

  d + 4 ||∇S||2 Zdx − DS S1 dx r2 r4 S2

−

  ||∇V ||2 ||∇N ||2 DV V1 DN N1 dx − dx. 2 r2 r3 V 2 r1 N

α,β

1 β1 β3 Hence, .∂t L1 ≤ 0 when .A2 ≤ A1 + d(d+ ABr . Consequently, the condition 1 )(d+ 2 )(d+ 3 ) (ii) of Theorem 5 of [12] is satisfied. Therefore, the tumor-free equilibrium .E1 is stable if ABr1 β1 β3 .A2 ≤ A1 + d(d+ )(d+ )(d+ ) . 1

2

3

A2 3 > 1 and .A3 > 1. Then Theorem 3.3 Suppose that .A2 > 1 + (d+ Bβ , .A 2 )(d+ 3 ) 1 3) the treatment failure immune-free equilibrium .E2 is stable if .1 + β2r(d+

≥ A1 + β d B(β2 (d+ 4 )+r4 β4 d) Bβ2 4) and .A2 ≤ 1 + β2r(d+

. + r d(d+

A 4 β4 d 3 2 )(d+ 4 )(A3 −1) r3 d(d+ 2 )( A2 −1) 1

Proof Consider the Lyapunov functional

3 3

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M. El Younoussi et al.



 .L2 (t)

=



+

S2 φ

S(x, t) S2

+

 1 1 T (x, t) N (x, t) + T2 φ r1 r2 T2

 1 V (x, t) 1 + V2 φ Z(x, t)dx. r2 r3 V2 r2 r4

Clearly, .L2 (S, N, T , V , Z) ≥ 0 and .L2 (S2 , N2 , T2 , V2 , Z2 ) = 0. Moreover, α,β

.∂t

 L2 ≤



−(d + β2 T2 )

−β1 (

B (V − V2 )2 (S − S2 )2 − S r2 r3 V V2

d + 1 β d + 4 − T2 )Z − S2 )N − 4 ( r 2 r 4 β4 r 1 β1

  T S DN DT 1 − 2 T + 1 − 2 DS S + N + S r1 r2 T  DZ V2 DV V + 1− + Zdx r2 r3 V r2 r4 

 B (V − V2 )2 (S − S2 )2 dx − dx r2 r3 V V2 S   β d + 4 d + 1 − T2 ) Zdx − S2 ) N dx − 4 ( −β1 ( r 2 r 4 β4 r 1 β1

= −(d + β2 T2 )



 ||∇T ||2 ||∇S||2 DT T2 dx − dx 2 r2 S T2  ||∇V ||2 DV V 2 − dx. r2 r3 V 2 −DS S2

By computation, we get α,β

.∂t



 B (V − V2 )2 (S − S2 )2 dx − dx r2 r3 V V2 S  β2 (d + 3 ) Bβ2 − A1 − −β1 (1 + N dx ) r 3 β3 d r3 d(d + 2 )( A2 − 1)

L2 ≤ −(d + β2 T2 )

A1

 B(β2 (d + 4 ) + r4 β4 d) β (d + 4 ) β + − A2 ) Zdx − 4 (1 + 2 r2 r 4 β4 d r3 d(d + 2 )(d + 4 )(A3 − 1)   ||∇S||2 ||∇T ||2 DT T2 −DS S2 dx − dx 2 r2 S T2

A Reaction-Diffusion Fractional Model for Cancer Virotherapy with Immune. . .

−

133

 ||∇V ||2 DV V2 dx. r2 r3 V 2

By Theorem 5 of [12], we deduce that the treatment failure immune-free equilibrium Bβ2 3) 4) is stable if .1 + β2r(d+

and .A2 ≤ 1 + β2r(d+

≥ A1 + A2 β d + β d

.E2

r3 d(d+ 2 )( A −1)

3 3

4 4

1

B(β2 (d+ 4 )+r4 β4 d) . r3 d(d+ 2 )(d+ 4 )(A3 −1)

ABr1 β1 β3 2 Theorem 3.4 Assume that . A A1 > 1, .A3 > 1, .A2 > A1 + d(d+ 1 )(d+ 2 )(d+ 3 ) and .A1 + Bβ2 3) > 1 + β2r(d+

A2 β d . Then, the partial success immune-free equilibrium .E3 is r3 d(d+ 2 )( A −1)

3 3

1

ABr1 β1 r4 β4 stable if .A2 ≤ A1 − r d(d+ )(d+

. 3 1 2 )(d+ 4 )(1−A3 )

Proof Let .L3 be a Lyapunov fonctional defined as follows:   1 1 S(x, t) N (x, t) T (x, t) + T3 φ + N3 φ r1 N3 r2 T3 S3  1 1 V (x, t) + + Z(x, t)dx, V3 φ V3 r2 r4 r2 r3 



.L3 (t)

=

S3 φ

then, we get α,β

.∂t



(S − S3 )2 B (V − V3 )2 − S r2 r3 V V3  S β d + 4 − T3 )Z + DS 1 − 3 S − 4( S r 2 r 4 β4   DT T N DN 1 − 3 T 1 − 3 N + + N r2 T r1  DZ V DV 1 − 3 V + + Zdx V r2 r4 r2 r3   B (V − V3 )2 (S − S3 )2 dx − dx = −(d + β2 T3 + β1 N3 ) r2 r3 V V3 S   β d + 4 ||∇S||2 − 4( − T3 ) Zdx − DS S3 dx r 2 r 4 β4 S2   DT T 3 ||∇N ||2 ||∇T ||2 DN N3 − dx − dx 2 r1 r2 T N2  DV V 3 ||∇V ||2 − dx. r2 r3 V 2

L3 ≤

−(d + β2 T3 + β1 N3 )

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 α,β A3 ) A2 − A 1 + After computation we find that .∂t L3 ≤ 0 if . d(d+ 1 )(d+ 4A)(1− Ar1 β1 r4 β4 ( A2 −1) 1 ABr1 β1 r4 β4 ≤ 0. Furthermore, it is easy to show that .L3 (S, N, T , V , Z) ≥ r d(d+ )(d+ )(d+ )(1−A ) 3

1

2

4

3

0 and .L3 (S3 , N3 , T3 , V3 , Z3 ) = 0. Then, by Theorem 5 of [12] the partial success immuneABr1 β1 r4 β4 . free equilibrium .E3 is stable if .A2 ≤ A1 − r d(d+ )(d+

)(d+ )(1−A ) 3

1

2

4

3

B(β2 (d+ 4 )+r4 β4 d) 4) Theorem 3.5 Suppose that .A3 > 1 and .A2 > 1 + β2r(d+

. Then + r d(d+

4 β4 d 3 2 )(d+ 4 )(A3 −1) β2 (d+ 4 ) the treatment failure equilibrium .E4 is stable if .A1 ≤ 1 + r β d . 4 4

Proof Consider the following Lyapunov function:  1 S(x, t) 1 T (x, t) + N (x, t) + T4 φ r1 r2 T4 S4   1 1 Z(x, t) V (x, t) + dx, + Z4 φ V4 φ V4 r2 r4 Z4 r2 r3 



=

.L4 (t)

S4 φ

then, we have 

(S − S4 )2 d + 1 B (V − V4 )2 + β1 (S4 − )N − S r 1 β1 r2 r3 V V4   Z S DN DZ 1 − 4 Z + N +DS 1 − 4 S + r2 r4 Z r1 S   DV V T DT 1 − 4 V dx 1 − 4 T + + T r2 r3 V r2   (S − S4 )2 d + 1 = −(d + β2 T4 ) ) N dx dx + β1 (S4 − S r 1 β1    (V − V4 )2 ||∇S||2 ||∇T ||2 B DT T4 − dx − DS S4 dx − dx r2 r3 V V4 r2 S2 T2   ||∇Z||2 ||∇V ||2 DZ Z4 DV V 4 dx − dx. − 2 r2 r4 Z 2 r2 r3 V

α,β .∂t L4 ≤

−(d + β2 T4 )

α,β

4) By computation we find that .∂t L4 ≤ 0 if .A1 ≤ 1 + β2r(d+

. Furthermore, we have 4 β4 d .L4 (S, N, T , V , Z) ≥ 0 and .L4 (S4 , N4 , T4 , V4 , Z4 ) = 0. Then, by Theorem 5 of [12] we 4) conclude that the steady state .E4 is stable if .A1 ≤ 1 + β2r(d+

β d . 4 4

4) Theorem 3.6 The coexistence equilibrium .E5 is stable if .A3 > 1, .A1 > 1 + β2r(d+

and 4 β4 d ABr1 β1 r4 β4 .A2 > A1 + . r d(d+ )(d+ )(d+ )(A −1) 3

1

2

4

3

Proof We consider the following Lyapunov functional: 



S(x, t) .L5 (t) = S5 φ S5



  1 1 N (x, t) T (x, t) + N5 φ + T5 φ r1 N5 r2 T5

A Reaction-Diffusion Fractional Model for Cancer Virotherapy with Immune. . .

+

135

  1 1 Z(x, t) V (x, t) + dx. Z5 φ V5 φ V5 r2 r4 Z5 r2 r3

Obviously, .L5 (S, N, T , V , Z) ≥ 0 and .L5 (S5 , N5 , T5 , V5 , Z5 ) = 0. Furthermore, 

B (V − V5 )2 (S − S5 )2 − S r2 r3 V V5    DZ DN S5 Z5 N5 S + Z + N 1− 1− +DS 1 − S r2 r4 Z r1 N   DV V T DT 1 − 5 V dx 1 − 5 T + + T r2 r3 V r2   B (V − V5 )2 (S − S5 )2 dx − dx = −(d + β1 N5 + β2 T5 ) S r2 r3 V V5   ||∇N ||2 ||∇S||2 DN N5 dx − dx −DS S5 r1 S2 N2   ||∇V ||2 ||∇Z||2 DZ Z5 DV V 5 dx − dx − 2 r2 r3 V r2 r4 Z 2  ||∇T ||2 D T T5 dx. − r2 T2

α,β

L5 ≤

α,β

L5 ≤ 0. It follows from Theorem 5 of [12] that the coexistence equilibrium .E5

.∂t

Hence, .∂t is stable.

−(d + β1 N5 + β2 T5 )

4 Conclusion In this chapter, we have proposed a reaction-diffusion model with Hattaf timefractional derivative to study the interaction between nutrient, normal cells, tumor cells, M1 virus, and CTL cells. The choice of using the GHF derivative is due to its capability to solve the problem of singularity of the kernel and also to generalize and extend the other forms of fractional derivative with nonsingular kernel existing in literature. Furthermore, the fractional derivative allows us to model the memory and the hereditary properties of cancer. In addition, we have studied the equilibria of the proposed model as well as their stability analysis. We have found that the model admits six equilibrium points denoted: (i) the competition-free equilibrium Bβ3 .E0 , that is stable if .A2 ≤ 1 + (d+ 2 )(d+ 3 ) and .A1 ≤ 1. This situation reflects the extinction of both normal and tumor cells. (ii) The tumor-free equilibrium .E1 , which 1 β1 β3 . In this case, the CTL immunity is not is stable if .A2 ≤ A1 + d(d+ ABr 1 )(d+ 2 )(d+ 3 ) active and the M1 virotherapy eradicates cancer cells and keeps normal cells. Then, the M1 virotherapy succeed and the patient’s health is improved. Moreover, the

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conditions needed to drive the system toward this equilibrium may help to improve the M1 virotherapy by calculating the minimum effective dosage needed to defeat cancer cells. (iii) The treatment failure immune-free equilibrium .E2 that is stable if β2 (d+ 3 ) B(β2 (d+ 4 )+r4 β4 d) Bβ2 4) .1 + + r d(d+

. and .A2 ≤ 1 + β2r(d+

A2 r3 β3 d ≥ A1 + )(d+ )(A −1) 4 β4 d r3 d(d+ 2 )( A −1)

3

2

4

3

1

In this case, the patient’s health is in danger because of the elimination of normal cells. (iv) The partial success immune-free equilibrium .E3 that is stable if .A2 ≤ ABr1 β1 r4 β4 . This situation reflects the capability of reducing A1 − r d(d+ )(d+

3 1 2 )(d+ 4 )(1−A3 ) cancer cells in absence of CTL immunity. (v) The treatment failure equilibrium β2 (d+ 4 ) .E4 , which remains stable if .A1 ≤ 1 + r4 β4 d . This situation reflects the fail of M1 virotherapy at eliminating cancer in presence of the CTL immunity but the normal cells are lost. (vi) The coexistence equilibrium .E5 which is stable if .A3 > 1, ABr1 β1 r4 β4 β2 (d+ 4 ) .A1 > 1+ r4 β4 d and .A2 > A1 + r3 d(d+ 1 )(d+ 2 )(d+ 4 )(A3 −1) . This situation indicates that both CTL immunity and M1 virotherapy try to eradicate cancer. By comparing this case with the second case, we can deduce that the immunity reduces the efficacy of the M1 virotherapy.

References 1. WHO, Cancer Fact Sheet, Available on https://www.who.int/news-room/fact-sheets/detail/ cancer. 2. J. Dan, L. Nie, X. Jia et al., Visualization of the oncolytic alphavirus M1 life cycle in cancer cells, Virologica Sinica, (2021) 1–12. 3. Z. Wang, Z. Guo, H. Peng, A mathematical model verifying potent oncolytic efficacy of M1 virus, Mathematical Biosciences (2016) 19–27. 4. A.M. Elaiw, A.D. Hobiny, A.D. Al Agha, Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response, Appl Math Comput, 367 (2020), Article 124758. 5. M. El Younoussi, Z. Hajhouji, K. Hattaf and N. Yousfi, A new fractional model for cancer therapy with M1 oncolytic virus, Complexity, vol. 2021, (2021) 1–12. 6. M. El Younoussi, Z. Hajhouji, K. Hattaf, N. Yousfi, Dynamics of a reaction-diffusion fractional-order model for M1 oncolytic virotherapy with CTL immune response, Chaos, Solitons and Fractals, vol. 157, p. 111957, (2022). 7. K. Hattaf, A new generalized definition of fractional derivative with non-singular kernel, Computation 8 (2020) 1–9. 8. A. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2015) 73–85. 9. A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. 20 (2016) 763–769. 10. M. Al-Refai, On weighted Atangana–Baleanu fractional operators, Adv. Differ. Equ. 2020 (2020) 1–11. 11. K. Hattaf, On some properties of the new generalized fractional derivative with non-singular kernel, Mathematical Problems in Engineering 2021 (2021) 1–6. 12. K. Hattaf, On the stability and numerical scheme of fractional differential equations with application to biology, Computation, 10 (2022) 1–12.

A Review of Stochastic Models of Neuronal Dynamics: From a Single Neuron to Networks M. F. Carfora

1 Introduction In the last decades, computational neuroscience research has been facing many pressing medical and technological challenges. Indeed, decoding the human brain is one of the most fascinating current and future research directions. In the framework of the EU’s Horizon 2020 research funding program, the Human Brain Project [2] is one of the three FET (future and emerging technology) flagship projects. This longterm and large-scale research initiative combines highly advanced methods from computing, neuroinformatics, simulation, and artificial intelligence to reach an indepth understanding of the complex structure and functions of the human brain. In such a huge project, mathematical models aiming at representing networks of neurons could be a useful tool for neuroscientists, to be fitted to the experimental recording of neuronal activity. These models are built on single neuron dynamics description, on which a wide literature exists. In particular, neuron models where action potentials are represented as events, without any attempt of characterizing their shape, are called integrate-and-fire (IF) or leaky-integrate-and-fire (LIF) models. These extremely simplified neuronal models are not intended to explain the complex biophysical and biochemical aspects of neuronal dynamics or to capture the highly nonlinear interactions between neurons [15]. However, when the dynamics of the model is enhanced, for example, by including nonlinearity, refractoriness, or adaptation, they can be a suitable tool to accurately describe the firing activity of real neurons [16]. This is the main reason for the renewed interest in the LIF model and its variants, seen as a tool to study the dynamics of networks of spiking neurons,

M. F. Carfora () Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche, Napoli, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_8

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because of their simplicity and efficient numerical implementation, along with the availability of some analytical results. Even though the understanding of neural processes starts with deterministic models, which represent neurons as dynamical systems, the many differences in neuronal responses to repeated (irregular, but also constant) input signals suggest the need for modeling stochastic effects [32]. In fact, a neuron is constantly subject to all sorts of inputs and fires irregularly or regularly according to its type and function. Such firing variability can be seen as a consequence of intrinsic neuronal noise, due to fluctuations in synaptic input or in the opening and closing of ion channels in response to electrical and chemical changes. Stochastic fluctuations in both spike intensity and timing are constantly observed also at the macroscopic level of neuronal populations, in EEGs and in in vivo measurements of the action potentials evoked by electrical shocks. In a stochastic dynamics approach, a neuron, or a population of neurons, is regarded as a dynamical system where stochastic effects are incorporated, satisfying a system of SDEs (or SPDEs in spatial models). Generally, the simplest stochastic models represent neuron dynamics as a Markov process, so that a Kolmogorov equation can be written for the evolution of its probability distribution; moreover, the analysis of the sample paths can provide insights on the process behavior. Building on the pioneering work of L.M. Ricciardi [17, 20, 34, 50, 51] who modeled the neuronal activity by a diffusion process of Ornstein-Uhlenbeck type, several of his coworkers, as well as researchers formed at his school, have given many contributions to the stochastic modeling of a single neuron [9–13, 21, 22, 30, 41] and of neuronal networks [4, 8, 18, 56, 59] providing theoretical and asymptotic results as well as algorithms for the numerical investigation of the proposed models. The aim of this review is to give a compact and clear presentation of the stochastic approach to neuronal modeling. Due to the vastness of such a field, only the main aspects of each model will be discussed; we refer the reader to the cited literature for further reading. Several other reviews exist, but they are not so recent and mainly focus on the single neuron modeling [15, 16, 53]; network dynamics is quite extensively treated in a few textbooks, such as [28]. After a short introduction on neuron physiology and a brief overview of the classical (deterministic) approach to the modeling of neuronal dynamics, the discrete stochastic models for a single neuron and their diffusion approximations through the Ornstein-Uhlenbeck (OU) process are presented in Sect. 3. Then, in Sect. 4, the firing of an action potential is studied as a first passage time (FPT) problem for a stochastic OU process: analytic results on the FPT distribution are presented, along with some asymptotic approximations; numerical techniques to evaluate this distribution are also discussed and compared. Section 5 is devoted to the modeling of interacting neurons: results for small networks are reported and the simulation of the olfactory system is presented as a working example. Finally, Sect. 6 concludes with some indication of further generalizations of this family of models.

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2 Background 2.1 An Overview of the Neuronal System Neurons are the fundamental processing units of the central nervous system. Even though they differ in properties and functions, all neurons comprise three basic components: several dendrites that are information-gathering components collecting signals coming from other neurons; a soma, or cell body, that processes these input signals; and an axon, along which the neuronal signals propagate to other neurons. The neuronal signals are short electrical pulses called action potentials (or spikes) that are delivered across junctions called synapses from any sender (presynaptic neuron) to any receiver (postsynaptic neuron). Each pulse modifies the potential difference across the neuronal membrane, either decreasing it (excitatory event) or increasing it (inhibitory event). These individual pulses are integrated both spatially and temporally in the cell body. When a sufficient number of synapses are activated within a short time interval, because of the spikes coming from presynaptic neurons, the membrane potential may reach a critical value and an action potential is delivered (a spike is fired). This neuronal output signal is, in turn, transmitted to other neurons. After a spike, there is a refractory period, during which a stronger stimulus is required to fire a second spike (refractoriness). In vertebrates, the most common synapses are the chemical ones, where the incoming action potential activates a chain of biochemical reactions: the release of a neurotransmitter leads to the opening of some ion specific channels, and this ion influx modifies the membrane potential of the postsynaptic cell. Then, in the classical theory, the membrane potential is assumed to be spatially constant and described as an electrochemical circuit.

2.2 Deterministic Models We intend just to mention here the main deterministic models of neuronal dynamics, presented as a starting point for the following stochastic models. The reader is referred to Tuckwell [62, 63] for a very detailed review, including numerical and experimental approaches. The oldest and simplest model, the deterministic leakyintegrate-and-fire (LIF, Lapicque [42]) describes in first approximation the neuronal membrane as a simple capacitor circuit, without any spatial variability of the input signal. When subject to an input current, the membrane voltage increases until it reaches a constant threshold S. Then a spike occurs and the voltage is reset to its resting value, and the process starts again to evolve. Reduced to the above circuit, the problem leads to a first-order linear differential equation for the membrane voltage: let .V (t) be the potential difference across the membrane minus its resting value. By assuming that the membrane resistance R is constant, we obtain an equation for the subthreshold voltage

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C

.

dV (t) V (t) + = I (t) dt R

t > 0;

(1)

the voltage rises or falls according to a synaptic input current .I (t) (excitatory or inhibitory). In addition, the membrane voltage spontaneously decays (this is the “leak”) with a time constant .τ = RC. The model is completed by imposing the generation of an action potential when .V (t) reaches the threshold value S; after this spike, V is reset. A generalization of this model, aimed at including spatial effects in the integration of various inputs, led to the so-called linear cable model [47], mutated by Lord Kelvin’s model for signal decay in submarine telegraphic cables. The neuron is represented as a spatially distributed circuit, and the membrane potential .V (x, t) is described by a linear partial differential equation. These simple models just describe the subthreshold dynamics, while the generation of action potentials has to be imposed by a threshold condition. More complex deterministic models, instead, succeeded in representing the neuron’s firing activity as a consequence of the biochemical reactions at the synapses. Among these models, the most important is due to Hodgkin and Huxley [33]. It uses a nonlinear reaction diffusion system to describe the modifications in the membrane voltage as provoked by variations in the conductance of the sodium and potassium ion channels and in the leakage conductance (mainly due to chloride ions). This system does not require additional threshold conditions: when the excitatory stimulus is strong enough, an action potential will be generated and propagated. A simplified version of this model has been proposed by FitzHugh [23] and Nagumo: only two equations are considered, by grouping variables having faster (voltage and sodium activation) and slower (potassium activation and sodium inactivation) time courses. Although these last two models are very powerful in describing neuron physiology, they are hardly tractable and their solutions must be obtained by numerical methods.

3 Stochastic LIF Models In a deterministic model, the voltage varies according to a synaptic rate, arising from excitatory or inhibitory inputs. In addition, a leaking term can be introduced, to describe how the voltage falls in proportion to its value above a base value, or resting potential. In stochastic IF and LIF models, these rates are replaced by stochastic point processes, so that the time evolution of the membrane potential is described by a suitable process .V (t), t > t0 with .V (t0 ) = V0 . The time intervals between action potential generations, called inter-spike intervals, or ISIs, are identified with the independent realizations of a random variable T , called first passage time (FPT) of V through the threshold .S(t), defined as .T = inf{t > t0 : V (t) > S(t)}. The probability density function (pdf) of T , when it exists, is g(t) = g(S(t), t|V0 , t0 ).

.

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3.1 Discrete Models The first stochastic IF model is according to Gerstein and Mandelbrot [26]. They described the membrane potential dynamics preceding the release of a spike by representing the incoming pulses as random jumps in the membrane voltage. Now, if the incoming pulses are frequent and of small size, the continuous limit of such a random walk is a Wiener process. Moreover, after each spike the membrane potential is instantaneously reset to its initial value. Under this model, the transition pdf of V is Gaussian, while the FPT through a constant boundary .S > V0 is distributed according to an inverse Gaussian law, which they used to fit a number of experimentally recorded ISIs. This extremely simplified model has the advantage of providing closed form solutions, so that it is still used as a basis for more realistic models. Time-varying boundaries can be introduced to account for the refractory period following a spike, as discussed in Sect. 3.4. Stein [57] introduced in 1965 a LIF model with random Poisson excitatory and inhibitory inputs. In his model, between two inputs the membrane voltage spontaneously decays (leakage) with a time constant .τ . The model for the subthreshold dynamics is 1 dV (t) = − (V (t) − ρ)dt + aE dNE (t) − aI dNI (t), τ

.

t > t0 , V (t0 ) = V0 ,

where .ρ is the resting potential, .NE , NI are independent Poisson processes of parameters .λE , λI counting excitatory and inhibitory inputs, and .aE , aI are the related amplitudes. Also in this model, the spike times are the first crossing times of the process through the boundary, and the membrane potential is instantaneously reset to its resting value after each spike. The FPT problem for this process is still unsolved; its analysis relies on simulation techniques.

3.2 Diffusion Approximation: The OU Process As for the IF model of Gerstein and Mandelbrot, where a random walk is approximated by a Wiener process, also for Stein’s model the substitution by its continuous limit increases the model tractability. The motivation for this approximation, as mentioned before, is in the huge number of synapses characterizing most neurons that induce frequent small jumps in the membrane voltage. Then, limit theorems can be applied to get a diffusion process. A diffusion process .X(t) is the solution to the stochastic differential equation (SDE) dX(t) = α(X(t), t)dt + β(X(t), t)dW (t)

.

(2)

where .W (t) is a standard Wiener process and the functions .α(x, t) and .β 2 (x, t) are called drift coefficient and infinitesimal variance (or infinitesimal moments). When .α and .β do not depend on t, the process is homogeneous. Under suitable hypotheses

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on .α, .β, the transition pdf .f (x, t|x0 , t0 ) of X satisfies two advection-diffusion PDEs: the Kolmogorov (or backward) and the Fokker-Planck (or forward) equations both having as initial condition .limt→t0 f (x, t|x0 , t0 ) = δ(x − x0 ). The diffusion approximation of Stein’s model is generally realized by considering a Gaussian process having the first two infinitesimal moments corresponding to the limiting values of those of Stein’s model, the Ornstein-Uhlenbeck (OU) process, as shown in [40], where the weak convergence for a sequence of Stein’s models .Vn to the OU process is also proved. In the following, we assume, for ease of notation, .t0 = 0, ρ = 0. Then, by setting x 1 α(x) = − x + aE λE − aI λI = − + μ, τ τ

.

2 β 2 (x) = aE λE + aI2 λI = σ 2 ,

.

we can build the continuous LIF stochastic model as an OU process .V (t) (with a slight abuse of notation we still denote by V the continuous process), solution to   V (t) + μ dt + σ dW (t). .dV (t) = − τ This process has mean mV (t) = V0 e−t/τ + μτ (1 − e−t/τ )X

.

and autocovariance cV (t, s) =

.

 σ 2  s/τ τ e − e−s/τ e−t/τ , 2

s≤t

and its transition PDF .f (y, t|x0 , 0) is known in closed form.

3.3 Generalizations of the LIF Model   Nonlinear LIF models replace the linear term . − V τ(t) + μ with a nonlinear one. The most studied nonlinear models comprise the quadratic IF model [35, 43]  1 .dV (t) = − k0 V (t)(V (t) − E0 ) + μ dt + σ dW (t) τ 

and the exponential IF model [24]

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    V (t) V (t) h0 + exp .dV (t) = − + μ dt + σ dW (t) τ τ h0 Another generalization is given by the spike response model [27], whose parameters depend on the time since the last spike, so introducing correlation between successive spikes. All these models are described in [15] and their performances in reproducing spike trains compared in [36]. Other generalizations include reversal potential models that introduce a saturation effect on the membrane sensibility (a lower bound on the membrane potential) and jump diffusion models that add stronger inputs, represented as discrete processes, to an underlying diffusion process. For these last generalizations, the reader is referred to [53].

3.4 Spike-Frequency Adaptation The simple LIF model, with its instantaneous reset of the membrane voltage after each spike, is unable to represent some phenomena observed in real neurons, such as refractoriness and spike-frequency adaptation [55]. As said, refractoriness prevents a second spike immediately after a first one was emitted. Spike-frequency adaptation that plays an important role in the transduction of a given stimulation into a spike train [54] can be defined as the slowing down in time of the firing frequency for a neuron subject to a constant stimulus. In other words, the cumulative effect of previous spikes prevents further spiking. Some ion channels have voltage-dependent sensitivity to the membrane potential: they have a faster activation at supra-threshold potentials and a slower inactivation at subthreshold potentials. Then, each action potential will induce a small increase in the number of open channels which cumulates over the spikes. To include spike-frequency adaptation, the equation for the evolution of the membrane potential is coupled to an equation describing the dynamics of a given ion species. After each spike, due to the opening and closing of specific gates, the intracellular concentration of that species is abruptly modified and then decays to its resting value; as a consequence, a resulting iondependent current affects the discharge rate [10, 44, 48]. Alternative models assume that adaptation is mimicked by a dynamically varying firing threshold .S(t) that is transiently elevated following a spike and subsequently decays until the next spike is generated [13, 19, 38, 39]

4 Estimation Methods for the FPT Distribution As already said, when we adopt the stochastic representation of the membrane voltage dynamics as an OU process, the ISI characterization corresponds to the determination of the distribution of the FPT of the process through an upper absorbing boundary .S(t). Several authors have applied the huge literature on the

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distribution of the FPT for Gaussian diffusion processes to derive theoretical results and asymptotic approximations for the firing activity of LIF neurons. We report here only the main findings and refer the reader to the excellent review [53].

4.1 Analytic Results A closed-form expression for the FPT probability density function .g(t) can be obtained only in the simplest case of a constant threshold S; this expression is given in [52]. In general, under reasonable regularity hypotheses on the threshold .S(t), .g(t) satisfies a Volterra integral equation 

t

f (x, t; x0 , 0) =

.

g(S(τ ), τ ; x0 , 0)f (x, t; S(τ ), τ )dτ,

x ≥ S(t)

(3)

0

where .f (x, t; x0 , 0) is the transition pdf of the process. For constant threshold and time homogeneous processes, the Laplace transform method can be applied and .g(t) numerically evaluated. But for .x = S(t) the equation has a weakly singular kernel for .τ approaching t, and the numerical methods become unstable. The singularity can be removed [14]: by integrating in space and differentiating in time (3), we obtain a new Volterra integral equation 

t

g(S(t), t; x0 , 0) = −2ψ(S(t), t; x0 , 0) + 2

.

g(S(τ ), τ ; x0 , 0)ψ(S(t), t;

0

(4)

S(τ ), τ )dτ, where ψ(S(t), t; x, τ ) =

.

dF (S(t), t; x, τ ) + k(t)f (S(t), t; x, τ ); dt

here .F (x, t; y, τ ) = Prob (V (t) ≤ x|V (τ ) = y) and .k(t) is an arbitrary continuous function. Then, a suitable choice for .k(t) allows to remove the kernel singularity. Another approach to obtain the transition pdf and the FPT pdf of a Gaussian diffusion process relies on suitable space-time transformations: by changing variables or measure, it is sometimes possible to trace back the process under investigation to a standard Wiener process [49].

4.2 Asymptotic Results Results on the behavior of the FPT distribution for large values of the threshold and for long times can be very useful for a first approximate description of the firing

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activity. In the first case, when the firing threshold S approaches the boundary of the diffusion interval, it has been proved [45] (for a class of diffusion processes including the OU one) that the FPT density .g(t) is well approximated by an exponential density whose mean value is the mean value of the FPT T through the threshold S:

t 1 exp − . .g(t) ≈ E(T ) E(T ) In this asymptotic approximation, .E(T ) loses the dependency upon the initial value V0 . Analogous results hold for asymptotically constant and asymptotically periodic thresholds [29]. In the same work, the asymptotic behavior for large t of the FPT density through some time-varying boundaries was also studied. It is proved that in the case of asymptotically constant boundaries, .g(t) shows an exponential behavior, while in the case of asymptotically periodic boundaries, .g(t) exhibits damped oscillations with the same period.

.

4.3 Simulations and Numerical Techniques The determination of the distribution of the ISIs, identified as the first passage times of the underlying process through the firing threshold, or at least an estimate of its main statistical indexes, is generally realized by simulating trajectories of the membrane potential through numerical methods for diffusive stochastic differential equations [31] or by exploiting the underlying known Gaussian bridge distribution [58]. The simplest algorithm, the Euler-Maruyama method, obtains an approximate solution Y of (2) on the time interval .t0 , Tmax by choosing the nodes .t0 < t1 < · · · < tn = Tmax and evaluating Yj +1 = Yj + α(Yj )(tj +1 − tj ) + β(Yj ) Wj ,

.

j = 0, . . . , n − 1

with .Y0 = V (t0 ) and . Wj = W (tj +1 ) − W (tj ). The realizations of the Wiener process W are generated for each trajectory, and the empirical distribution (histogram) of the FPT is obtained by considering a huge number of these sample trajectories. The discretization of the sample paths, however, and the related risk of not detecting the crossing of the boundary, generally leads to overestimation of the FPT [53]. An alternative algorithm, more robust to overestimation, has been proposed in [9]. A different approach is based on the numerical quadrature of the integral equation (4): when the transition density .f (x, t; y, τ ) of the process and the threshold function .S(t) are known, after choosing a suitable function .k(t), any stable quadrature algorithm can provide an approximation for .g(t). However, the computational cost of this procedure is quadratic, because of the required evaluations of the function .ψ in all nodes preceding the current one; this burden

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can be very hard when the tail of the density is very long (i.e., when the threshold value is very far from the initial value of the potential). In such cases, the asymptotic results reported in Sect. 4.2 can give a reasonable approximation of the tail of the distribution.

5 Networks of Interacting Neurons The recent research is increasingly devoted to the representation of the dynamics of interacting neurons in networks. Several aspects characterize the network representation, starting from network size, proportion of inhibitory/excitatory neurons, correlation in inputs, synchronization, and noise modeling (see [7, 46, 56] and references therein). While some approaches, based on population dynamics and mean field theory, apply limiting procedures to describe the overall behavior of the network, others describe the firing activity of each neuronal component (individual or subpopulation) as depending on the structure of the network and the kind of interaction between neurons. For instance, in the very simple model [18], the firing density of any observed neuron is represented as the FPT density through a threshold of a Gauss-diffusion process with jumps that are random in both time and amplitude. The network is then modeled by a system of time-inhomogeneous LIF equations including both external stimuli and linkage synaptic currents. In this case, the evolution of the membrane voltage of the neuron .Nij , with .i, j = 1, . . . , N, can be described by the stochastic process .Vij (t) solution of the following SDE, for .t ≥ 0,

ρij + μij θij 1 dVij (t) = − Vij (t) + + Iij (t) dt + σij (t)dWij (t) θij θij

.

with .Vij (0) = v0,ij and  Iij (t) = i0,ij e−t/αij + kij 1 − e−t/αij Hij (t),

.

for .i, j = 1, . . . , N. In this case, the linking function .Hij (t) is given by a weighted average of the pulses coming from the surrounding neurons Hij (t) = β(i−1)(j −1) H(i−1)(j −1) + β(i−1)j H(i−1)j + βi(j −1) Hi(j −1) .

.

The firing activity in the network can then be studied by any of the numerical techniques in Sect. 4.3: trajectory simulations, numerical evaluation of the integral Volterra equations, and, under hypotheses of low firing rate, asymptotic approximations. An example of such a network structure is shown in the left panel of Fig. 1. In this example, information flow is unidirectional and inputs are differently weighted, as marked on the graph edges. The right panel of the same figure shows

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Fig. 1 An example of a small neuronal network (left panel): each circle represents a neuron and each edge a synaptic input, with related weight. In the right panel, the retrieved FPT density for neuron .N22 compared with its empirical estimate Fig. 2 Schematic diagram reproducing two blocks of sensory neurons .Ni conveyed into different glomeruli (containing each a mitral cell M and a periglomerular one P G) and connected to the same granule cell G

the FPT density of neuron .N22 in case of excitatory inputs: the curve obtained by numerical quadrature of the related integral equation is superimposed to the empirical distribution (histogram) of the first firing times obtained by simulating a million trajectories of the process. To conclude this section, we present a typical example of a neuronal network: the peripherical part of the olfactory system, where sensory neurons subject to noisy external stimuli transmit information through synaptic linkages to reticular structures (glomeruli). There, the overall firing activity is modulated by selective inhibition before being conveyed to the mitral cells. Many authors have proposed models describing the processing of odor recognition and discrimination in the olfactory system [3, 6, 37], and a discussion of this literature is beyond the scope of this review. We just want to describe here a minimal firing rate model based on stochastic LIF neurons, able to qualitatively reproduce some basic interactions observed in real olfactory neurons, such as the mechanism of selective inhibition. Details of the model can be found in [4] The network structure is shown in Fig. 2: it is a small-scale block model, where in each block sensory neurons .Ni form excitatory synapses with mitral cells M that in turn form two-directional synapses with granule cells G. Periglomerular cells P G modulate the neural circuit by selective inhibition.

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Fig. 3 Histograms and numerical approximations of the FPT density functions for a generic sensory neuron (top left), a PG neuron (top right), a mitral cell (bottom left), and a granule cell (bottom right). The sensory input starts at .t1 = 20

In the reported simulation results, we assume a spontaneous state for the sensory neurons (during the first 20 time units) and then an odor-evoked state. The sharp increase in the firing activity at time .t1 = 20 can be clearly seen in the first panel of Fig. 3. Similarly in the right panel of the same figure, the simulated firing activity of the P G cell is shown. Finally, the other two panels show the M cell firing activity, suddenly varying in consequence of the direct input from sensory neurons, and the G cell response, spread over a longer time course because of its indirect connections to the sensory stimuli.

6 Current Research Directions Despite their simplicity, LIF models and their generalizations are powerful tools to investigate neuron dynamics. Very recent works [60] succeeded in fitting electrophysiological data by generalized LIF models consisting of different phe-

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nomenological mechanisms, classifying the spiking behavior of different neuron types. A very promising perspective for current research is given by fractional order models. Indeed, the enormous advances in fractional calculus, both computational and theoretical, have led to new models, called fractional-order LIF. They can easily capture adaptation effects while retaining a single variable. The model has been formulated in [61] as C

.

d α V (t) V (t) = I (t) + R dαt

where .α is the fractional order of the derivative. Once the voltage hits the threshold, it is reset. Fractional integration has been used to account for neuronal adaptation in experimental data. Fractional-order models have been studied also in [5], while in [1] a numerical method to derive the FPT distribution of a fractional LIF model by using a time-changed process has been proposed. Very recently [64], fractionalorder models have been applied to networks of neurons. This approach allows for representing neural dynamics with heterogeneous, correlated, or non-Gaussian characteristics, as widely observed in real neuron networks. Another very interesting approach is given by the model [25], closely related to both the spike response model [27] and the LIF model. In this model, a countable number of neurons interact by rare and nearly instantaneous spikes. At each moment, each neuron fires independently, with a probability depending on the history of the firings of all neurons since its last firing. Thus, the “memory” of each neuron is limited to its last firing interval.

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Modeling the Impact of Media Coverage on the Spread of Infectious Diseases: The Curse of the Twenty-First Century Anal Chatterjee and Suchandra Ganguly

1 Introduction Numerous quantitative analyses have been conducted over the last few decades to evaluate the results of media efforts for the control of infectious illnesses [1–4]. Previous studies made the assumption that the number of awareness campaigns is proportional to the number of infected people, either by treating the media as a dynamic variable or by assuming that the transmission rate is a decreasing function of the number of infected people as a result of media alerts. In the shape of COVID-19, the world currently faces one of the greatest health challenges in recorded human history. Between December 31, 2019, and July 15, 2020, the authors in [5] examined how the media and public health messaging were utilized. They examined how the media had a dual function throughout this pandemic. Through persistent advertising, they were useful for disseminating important health information, health guidelines, and aiding in the observance of hygienic habits. The media used live update dashboards to run the COVID-19 data, which was crucial for reporting on the state of the world at the time. It was also observed that more people are turning to telehealth and telemedicine. However, at the same time, false material was disseminated through numerous media channels, including unproven treatments and medications. The general public’s fear and panic were also encouraged by a number of media outlets. In particular, the authors in [6] came to the conclusion that social media has benefits and drawbacks. A quantitative approach was used to study the impact of social media on COVID-19 public health

A. Chatterjee Department of Mathematics, Barrackpore Rastraguru Surendranath College, Kolkata, India S. Ganguly () Clinical Instructor, University College of Nursing, JNMH, Kalyani, India © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_9

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metrics in Jordan [7] through public health awareness and public health behavioral changes. The analysis of a mathematical model [8] to study the impact of media awareness programs on the prevalence of infectious disease revealed that increasing the rate of implementation of media awareness programs decreases the number of infected individuals and the system remains stable up to a threshold value, after which the system oscillates. To investigate the effects of media coverage on the transmission and management of infectious diseases, the researchers in [9] created a threedimensional mathematical model. When the basic reproduction number (.R0 ) is less than unity, the disease-free equilibrium is globally asymptotically stable, according to the model’s stability analysis. The media’s impact is deemed to be sufficient when .R0 > 1. The efficacy of face masks, hospitalization, and quarantine on COVID-19 was recently evaluated using a mathematical model, according to [10]. A susceptible-exposed-infected-recovered (SEIR) model with population awareness (local and global awareness) has also been presented and examined in [11]. The findings showed that in order to quickly manage the outbreak, the aforementioned measures’ efforts should be high. Mathematical models are being investigated for several infectious diseases to determine how the implementation of awareness campaigns affects the dynamics of infectious disease [12–14]. Television and radio commercials have the power to reach a global audience with clearly defined themes and to persuade governments to alter their healthrelated regulations. Regarding this, the authors [15] have investigated a nonlinear mathematical model to observe the interaction between effects of TV and radio commercials for the prevention of infectious diseases by taking into account the necessity of disseminating information across both media as dynamic variables. The study unequivocally shows that broadcasting on TV and radio helps to lower the burden of disease. There have also been some theoretical investigations on how radio and television commercials affect the dynamics of disease. The researchers [16] have looked into the effectiveness of education and television viewing in Bangladesh in halting the spread of HIV/AIDS among married couples. In order to explore the impact of the media coverage on the spread of infectious diseases, we created and examined a four-compartment epidemiological model in this paper. In the modeling procedure, we made the assumption that the population N represents the sum of the classes of susceptible who are unaware, aware, and infected. Only via close contact with the infected class may the susceptible class, both aware and unaware, contract the disease. Under the effect of media, some members of the susceptible class will make deliberate measures to avoid coming into contact with the infected. Individuals in the aware class have a lower risk of developing an infection than those in the unaware class. Additionally, we suppose that some people will recover, and some of these recovered people will join the aware susceptible class, while the others will join the unaware susceptible class (may be due to ignorance, lacunae on their parts, etc.). Our study finds that when immigration is increased, the system becomes unstable. Further, we find that by high growth rate of broadcasting the information through smartphone and TV, the system undergoes from stable to unstable through Hopf bifurcation. We develop

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a mathematical model in the following section and look at the system’s stability and equilibrium point. In the following section, numerical results are provided. The study concludes with a brief discussion.

2 Basic Assumptions and Model Formulation B1 : Let, N(t) be the total population of the region under consideration at time t. In this case, the total population is divided into three classes: susceptible unaware (Sw ), susceptible aware (Sa ), and infective population (I). B2 : Let A be the rate of susceptible immigration. Additionally, we take into account that M is the total amount of media coverage in that area at time t that is relevant to the infectious and is driven by TV and smartphones. People can learn about illness prevention from the majority of public media, which is given via television and smartphones. We assume that diseases spread solely through contact between the susceptible and the infective. B3 : Due to awareness from media coverage, it is assumed that susceptible avoid coming into touch with the infective and make up a different class with a proportion lambda termed the aware susceptible. We suppose that some infected people recover and enter the susceptible class after receiving treatment. Following recovery, (1 − p) will enter the ignorant susceptible class, whereas a fraction of recovered individuals, p, will join the aware susceptible group. B4 : It is made aware that the cumulative media coverage’s growth rate is proportional to the disease-induced mortality rate of the affected population. Here, β denotes the proportion of unaware susceptible who come into contact with the infectious class, and λ is the proportion of susceptible who become aware through media and so belong to a different class. Here, β1 is a fraction with a value between 0 and 1 that represents the reduced probability of contracting infection. B5 : In public settings, a certain percentage of the population (cn ) constantly and appropriately wears face masks. Let the effectiveness of the face masks be n . Therefore, Fn = 1 − εn cn represents the fraction which enters the infected class. A section of the knowledgeable people (h) keeps their distance from others. Face masks can successfully stop the spread of disease when used properly. B6 : The rates of natural death, recovery, and disease-induced death are indicated by the parameters d, γ , and α respectively. The transfer rate of aware individuals to the unaware susceptible class is represented by λ0 in this instance. B7 : We have concentrated on two media, namely, television and smartphones, to make awareness more accessible. Although self-quarantine and social isolation may rise as a result of awareness spreading, people will still employ these tactics to lessen their overall susceptibility. Since TV coverage receives most of the public’s attention, it stands to reason that as more people become aware of it, smartphone coverage will likely develop less quickly. However, the growth rate of Sa smartphone coverage is reduced by the factor f (Sa ) = θ ω+S . The growth rate a of cumulative TV and smartphone coverage is assumed to be proportionate to the

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infected populations. As a result, the net growth rate of smartphone-related media Sa coverage will be μ1 (1 − θ ω+S ). Here, let θ be the decay of smartphone coverage a as the number of aware individuals increases. In the present article, we consider media coverage as a convex combination of TV and smartphone to propagate the process of mitigating the transmission of the infectious disease. Additionally, let M0 be the constant baseline amount of advertisements in the investigated region. The effectiveness of TV and smartphone coverage is substantially reduced over time because of the inefficiency and psychological barriers; thus, advertisements through TV and smartphone become less influential and resulting in media decline with disappearing rate δ. Advertising on TV and posting/sharing clip through smartphone are believed to have an impact on susceptible individuals, and therefore it is believed that they form an isolated class known as the aware class in order to protect themselves from infection. Here, μ1 and μ2 represent the growth rate of broadcasting the information through smartphone and TV, respectively. In addition, λ1 and λ2 represent the convex combination constant. B8 : The susceptible population is thought to only be somewhat affected by television advertisements and information shared via smartphones, so awareness M spreads among them at a rate of λ pS1w+M . The constant p1 designates the halfsaturation point for the impact of advertisements on unaware susceptible people using two platforms, and it achieves half of its maximum value λSw when the cumulative number of advertisements to the aware population reaches at p1 . Every dynamic variable is described in Fig. 1. With these above assumptions, our model system is

.

⎫ dSw Sw M ⎪ = A − βFn Sw I − λ − dSw + λ0 Sa + (1 − p)γ I ⎪ ⎪ ⎪ dt p1 + M ⎪ ⎪ ⎪ ⎪ ⎪ dSa Sw M ⎪ ⎪ + pγ I − β1 β(Fn − h)Sa I − dSa − λ0 Sa ⎪ =λ ⎬ p +M dt 1

dI = βFn Sw I + β1 β(Fn − h)Sa I − γ I − αI − dI dt dM Sa = α1 μ1 (1 − θ )I + α2 μ2 I − δ(M − M0 ). ω + Sa dt

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(1)

The system (1) has to be analyzed with the following initial conditions: Sw (0) > 0, Sa (0) ≥ 0, I (0) ≥ 0, M(0) ≥ 0.

.

(2)

Using the fact that N = Sw + Sa + I , the system (1) transform to the following system:

Modeling the Impact of Media Coverage on the Spread of Infectious Diseases. . .

Death rate (d)

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Death rate (d)

SwM

Susceptible unaware (Sw)

λ

p1+M

Susceptible aware (Sb)

λ0

μ2 I Fra

(p) γ Fra ctio n

β1β (1–εncn–h)

-p)γ

n (1

ctio

Immigration (A)

μ1 (1–θ

Sa ω + Sa

Media (M) )I

β (1–εncn)

δ(M – M0)

Recovery rate (γ)

Death rate (d)

Infected(I)

Disease related death rate (γ)

Fig. 1 Schematic diagram

dI = βFn (N − (1 − β1 )Sa − I )I − β1 βhSa I − (γ + α + d)I dt ≡ G1 (I, Sa , N, M) dSa = λ(N − Sa − I )M + pγ I − β1 β(Fn − h)Sa I − dSa − λ0 Sa dt .

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = A − dN − αI ≡ G3 (I, Sa , N, M) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Sa ⎪ ⎭ = α1 μ1 (1 − θ )I + α2 μ2 I − δ(M − M0 ) ≡ G4 (I, Sa , N, M).⎪ ω + Sa (3) ≡ G2 (I, Sa , N, M)

dN dt dM dt

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

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Now it is sufficient to discuss system (3) rather that system (1). Here the region of 4 : 0 ≤ S ,I ≤ N ≤ attraction which is given by the set  = {(N, Sa , I, M) ∈ R+ w α1 μ1 +α2 μ2 A , 0 ≤ M ≤ + M . According to existence and uniqueness theorem, 0 d dδ the trajectories cannot approach to unfeasible domain from positive octant which indicates that solution remain in positive octant. This ensures that the system is well defined. Explicitly, the Jacobian matrix at E = (N, Sa , I , M) can be defined as ⎡

J11 ⎢ J21 .J = ⎢ ⎣ −α J41

J12 J22 0 J42

J13 J23 −d 0

⎤ 0 J24 ⎥ ⎥, 0 ⎦ −δ

(4)

where J11 = βFn N − βFn (1 − β1 )S a − 2βFn I − β1 βhS a − (γ + α + d), J12 = −βFn (1−β1 )I −β1 βhI , J13 = βFn I , J21 = −β1 β(Fn −h)S a +pγ − J22 = − J41 =

γM p1 +M

− β1 β(Fn − h)I − (d + λ0 ), J23 =

α1 μ1 (1 − θ S a ) + α2 μ2 , ω+S a

J42 =

γM , p1 +M

J24 =

γM , p1 +M λp1 (N −S a I ) , (p1 +M)2

− α1 μ1 θωI2 . (ω+S a )

3 Some Preliminary Results 3.1 Equilibria The system (1) possesses the following equilibria: disease-free equilibrium AλM0 (DFE) .E0 = (0, d(λM0 +(d+λ , A , M0 ) and endemic equilibrium .E ∗ = 0 )(p1 +M0 )) d ∗ ∗ ∗ ∗ (N , Sa , I , M ). 3.1.1

Disease-Free Equilibrium and Basic Reproduction Number

There is always disease-free equilibrium (DFE) .E0 in the system. We obtain the expression for the basic reproduction number using the next-generation operator approach. The basic reproduction number, .R0 , is an index often used by public health organizations to estimate the severity of a particular epidemic. ⎛

⎞ βI Fn (N − (1 − β1 )Sa I ) ⎜ ⎟ 0 ⎟, .ζ = ⎜ ⎝ ⎠ 0 0

Modeling the Impact of Media Coverage on the Spread of Infectious Diseases. . .

159

⎛

⎞ (γ + α + d)I ⎜ − λ(N −Sa −I ) − pγ I + β β(F − h)S I + (d + λ )S ⎟ 1 n a 0 a⎟ ⎜ p1 +M η=⎜ ⎟, ⎝ ⎠ dN + λI − A Sa δ(M − M0 ) − α1 μ1 (1 − θ ω+Sa )I − α2 μ2 I  (of new infection terms) and .V  (of the transition terms) Now, we find the matrix .F as ⎤ ⎡ βFn A (d+λ0 )(p1 +M0 )+β1 λM0 d [ λM0 +d+λ0 )(p1 +M0 ) ] 0 0 0 ⎥ ⎢ 0 0 0 0⎥ = ⎢ .F ⎥, ⎢ ⎣ 0 0 0 0⎦ 0 000 ⎡ ⎤ γ +α+d 0 0 0 0 ⎢ λM0 λp1 ( A d −Sa ) ⎥ 0 0 ⎢ p +M − pγ + β1 β(Fn − h)Sa0 p λM + (d + λ0 ) − p1λM +M0 (p1 +M0 )2 ⎥ 1 0 1 +M0 ⎢ ⎥,  V =⎢ ⎥ λ 0 d 0 ⎣ ⎦ Sa0 0 0 δ −α1 μ1 (1 − θ ω+S 0 ) − α2 μ2 a

AλM0 where .Sa0 = d(λM0 +(d+λ . .R0 = ρ(F V −1 ) represents the basic repro0 )(p1 +M0 )) duction number, where .ρ is the spectral radius of the next-generation matrix V −1 . Thus, from system (3), we determine the expression for .R0 as .R0 = .F β1 λM0 (Fn −h)+βFn (d+λ0 )(P1 +M0 ) βA . Furthermore, the Jacobian matrix evaluated [λM0 +(d+λ0 )(p1 +M0 )] d(α+λ+d) 0 at .E0 has four eigenvalues, namely, .−d , .−δ , .−( p1λM +M0 + (d + λ0 )), and .(d + α + γ )(R0 − 1) (Fig. 2).

Proposition 3.1 For system (3), the disease-free equilibrium .E0 is locally asymptotically stable if .R0 < 1 and unstable if .R0 > 1.

3.1.2

Feasibility and Stability of Endemic Equilibrium

For system (3), an endemic equilibrium is represented by .E ∗ = (I ∗ , Sa∗ , N ∗ , M ∗ ), the components of which are positive solutions to the system’s equilibrium equations. We have the following from the third equilibrium equation: .N

−I =

A − (α + d)I d

(5)

Applying (5), from the first equation of system (3), we can derive .Sa

=

βFn (A − (α + d)I ) − (γ + α + d)d . d[βFn (1 − β1 ) + β1 βh]

(6)

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60

50

I

40

30

20

10

0 0

0.5

1

1.5

2

2.5

3

3.5

4

R0

Fig. 2 One parameter bifurcation diagram with .R0 as a bifurcation parameter illustrating a transcritical bifurcation at .R0 = 1 when the rate of immigration of susceptible A crosses the critical value .A = 1.262656. For .R0 < 1, the system is stable at the DFE (.E0 ), whereas for ∗ .R0 > 1, the system is stable at the endemic equilibrium (.E ). Herein, blue, green, and red stars ∗ indicate that the system (3) is stable at .E0 , stable at .E , and unstable at .E0 respectively

If .Sa is positive, we must have .I < .A

>

d(γ +α+d) . βFn

βFn A−d(γ +α+d) βFn (α+d)

= Ib (say). Therefore, .Ib > 0 if

From (6), we have .H (I )

=

 = 1−θ

Sa ω + Sa



ωd[βFn (1 − β1 ) + ββ1 h] + (1 − θ )[βFn (A − (α + d)I ) − (γ + α + d)d] . ωd[βFn (1 − β1 ) + β1 βh] + βFn [A − (α + d)I ] − (γ + α + d)d

(7) Differentiating with respect to I , we obtain .H



(I ) =

ωθβ 2 d(α + d)[Fn (1 − β1 ) + β1 h] . [ωd[βFn (1 − β1 ) + β1 βh] + βFn [A − (α + d)I ] − (γ + α + d)d]2

(8)

Clearly .H  (I ) > 0. Further from the fourth equation of system (3), we obtain .

M δM0 + (α1 μ1 H (I ) + α2 μ2 )I = p1 + M δ(p1 + M0 ) + (α1 μ1 H (I ) + α2 μ2 )I

(9)

Modeling the Impact of Media Coverage on the Spread of Infectious Diseases. . .

161

Now substituting (5), (6), and (9) in the second equation of system (3), we get a equation in I as .P (I )

=

λ[(γ + α + d) − [A − (α + d)I ]ββ1 (Fn − h)] d[βFn (1 − β1 ) + ββ1 h] δM0 + (α1 μ1 H (I ) + α2 μ2 )I (I ). +N δ(p1 + M0 ) + (α1 μ1 H (I ) + α2 μ2 )I

×

(10)

)−(γ +α+d)d where .Nˆ (I ) = pγ I − [ββ1 (Fn − h)I + (d + λ0 )] βFn (A−(α+d)I . d[βFn (1−β1 )]+ββ1 h From Eq. (10), we have the following:      βFn A−(γ +α+d)d (γ +α+d)d−Aββ1 (Fn −h) 0 − (d + λ ) (i) .βP (0) = λ p1M 0 d[Fn (1−β1 )+β1 h] d[Fn (1−β1 )+β1 h] < 0 if +M0 .R0 > 1. λ[(γ +α+d)−[A−(α+d)Ib ]ββ1 (Fn −h)] δM0 +(α1 μ1 H (Ib )+α2 μ2 )Ib (ii) .βP (Ib ) = d[Fn (1−β1 )+β1 h] δ(p1 +M0 )+(α1 μ1 H (Ib )+α2 μ2 )Ib + pγβIb > 0, (iii) .P  (I ) > 0 in .(0, Ib ). Thus, there exists a unique positive root of P(I) in the interval .(0, Ib ), say .I = I ∗ , if .R0 > 1. Now, using the value of this .I = I ∗ in Eqs. (5), (6), and (9), we get positive values of .N ∗ , .Sa∗ , and M. Therefore, the endemic equilibrium .E ∗ = (I ∗ , Sa∗ , N ∗ , M ∗ ) is feasible in .(0, Ib ), provided .R0 > 1.

Now, the Jacobian matrix at the endemic equilibrium is represented by the symbol .J ∗ . ⎡

−βFn I ∗ −βFn (1 − β1 )I − β1 βhI ∗ βFn I ∗ ⎢ ∗ ∗ −J22 J23 ⎢ −J21 ∗ .J = ⎢ ⎣ −α 0 −d ∗ ∗ J41 J42 0 ∗

⎤ 0 ⎥ J24 ⎥ ⎥, 0 ⎦ −δ

(11)

λM ∗ ∗ p1 +M ∗ +β1 β(Fn −h)I +(d +λ0 ), ∗ Sa α1 μ1 θ ωI ∗ ∗ α1 μ1 (1 − θ ω+S . ∗ ) + α2 μ2 , .J42 = (ω+Sa∗ )2 a

∗ ∗ = β β(F −h)S ∗ + λM where .J21 1 n a p1 +M ∗ −pγ , .J22 = ∗ 23

=

λM ∗ p1 +M ∗ , .J24

=

λp1 (N ∗ −Sa∗ −I ∗ ) ∗ , .J41 (p1 +M ∗ )2

= We now utilize the Routh-Hurwitz criterion to show that the endemic equilibrium ∗ ∗ .E is locally stable. The characteristic equation of the matrix .J is given as follows:

.J

. ω

4

+ Q1  ω 3 + Q2  ω 2 + Q3  ω + Q4 = 0,

(12)

∗ + δ + βF I ∗ + d > 0, where .Q1 = J22 n ∗ δ+J ∗ J ∗ )+(βF I ∗ +d)(J ∗ +δ)+βF dI ∗ −[βF (1−β )I ∗ +β βhI ∗ ]J ∗ + = (J22 n n n 1 1 24 42 22 21 βFn αI ∗ > 0, ∗ δ+J ∗ J ∗ )+(βF I ∗ +d)(J ∗ +δ)+βF dI ∗ > [βF (1−β )I ∗ +β βhI ∗ ]J ∗ + if .(J22 n n n 1 1 24 42 22 21 βFn αI ∗ . ∗ + d)(J ∗ d + J ∗ J ∗ ) + βF I ∗ d(J ∗ + δ) − [βF (1 − β )I ∗ + .Q3 = (βFn I n n 1 22 24 42 22 ∗ ∗ α − J ∗ J ∗ ] + βF αI ∗ (J ∗ + δ) > 0, ∗ β1 βhI ][J21 (d + δ) + J23 n 24 41 22 .Q2

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∗ d + J ∗ J ∗ ) + βF I ∗ d(J ∗ + δ) + βF αI ∗ (J ∗ + δ) > [βF (1 − if .(βFn I ∗ + d)(J22 n n n 24 42 22 22 ∗ (d + δ) + J ∗ α − J ∗ J ∗ ]. ∗ ∗ β1 )I + β1 βhI ][J21 23 24 41 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ .Q4 = βFn dI (J δ +J J )−[βFn (1−β1 )I +β1 βhI ][J dδ +J αδ −J J d]+ 22 24 42 21 23 24 41 ∗ ∗ ∗ βFn αI (J22 δ + J24 J42 ) > 0 ∗ δ + J ∗ J ∗ ) + βF αI (J ∗ δ + J ∗ J ∗ ) > [βF (1 − β )I ∗ + if .βFn dI ∗ (J22 n n 1 24 42 22 24 42 ∗ ∗ αδ − J ∗ J ∗ d]. ∗ β1 βhI ][J21 dδ + J23 24 41

In addition, .Q1 Q2 −Q3 > 0 if .Q1 Q2 > Q3 as well as .Q3 (Q1 Q2 −Q3 )−Q21 Q4 > 0 if .Q3 (Q1 Q2 − Q3 ) > Q21 Q4 . Then, by the Routh-Hurwitz criterion, we can state say that all the eigenvalues of the Jacobian matrix .E ∗ will be located in the complex plane’s left-half and endemic equilibrium, .E ∗ , is locally asymptotically stable depending upon system parameters.

3.2 Hopf Bifurcation at Coexistence In characteristic Eq. (3), we express the coefficients as a function of A: . ω

4

+ Q1 (A) ω3 + Q2 (A) ω2 + Q3 (A) ω + Q4 (A) = 0,

(13)

Since all .Qi, i = 1, 2, 3, 4 are positive, the following condition applies when .A = Ac : .ψ1 (A)

= Q1 (A)Q2 (A)Q3 (A) − Q23 (A) − Q21 (A)Q4 (A) = 0.

(14)

Then, at .A = Ac , Eq. (13) is written as 2

.( ω

+

Q3 Q1 Q4 ). )( ω 2 + Q1  ω+ Q3 Q1

(15)

Let . ωi (i=1,2,3,4) be the roots of above equation. Thus,it is evident that Eq. (13) Q3 has two imaginary roots like . ω1,2 = ±i ω0 , where . ω0 = Q . For Hopf bifurcation 1 ω0 must lie in the left half of the complex plane. To to exist, all roots except .±i ω3,4 ), we have the following: ascertain the nature of the remaining two roots (i.e., . . ω3

+ ω4 = −Q1 ,

ω3  ω4 = Q2 ,  ω02 +   ω02 ( ω3 +  ω4 ) = −Q3 , ω3  ω4 = Q4 .  ω02 

(16)

Now, first we check if . ω3 ,  ω4 are real. Then from the fourth relation of Eq. (16), we ω3 and . ω4 are of the same sign as .Q4 , and from the first relation of (16), it get that . implies that they should be negative, i.e., . ω3 , . ω4 . Further, if . ω3 and . ω4 are complex

Modeling the Impact of Media Coverage on the Spread of Infectious Diseases. . .

163

conjugate, then .2Re( ω3 ) = −Q1 , i.e., . ω3 and . ω4 have negative real parts. Thus, in both cases, . ω3 and . ω4 lie in the left half of the complex plane. To determine the interval in which Hopf bifurcation takes place, we now confirm the transversality condition. For this let at any point .A ∈ (Ac − , Ac + ) and . ω1,2 =  κ ± η, putting this values in Eq. (13), we get 4  .κ + Q1 κ 3 + Q2 κ 2 + Q3 κ + Q4 +  η4 − 6 κ 2 η2 − 3Q1 κ η2 − Q2 η2 = 0,

.4 κ η( κ

2

− η2 ) − Q1 η3 + 3Q1 κ 2 η + 2Q2 κ η + Q3 η = 0.

(17)

(18)

η(A) = 0, from Eq. (18), we get Since, . .

− (4 κ + Q1 ) η2 + 4 κ 3 + 3Q1 κ 2 + 2Q2 κ + Q3 = 0.

(19)

η2 in Eq. (17), we have Putting the value of . .

− 64 κ 6 − 96Q1 κ 5 − 16(3Q21 + 2Q2 ) κ4

−8(Q31 + 4Q1 Q2 ) κ 3 − 4(Q22 + 2Q21 Q2 + Q1 Q3 − 4Q4 ) κ2 −2Q1 (Q22 + Q1 Q3 − 4Q4 ) κ − (Q1 Q2 Q3 − Q23 − Q21 Q4 ) = 0.

(20)

c Differentiating above equation with respect to A and by using  .κ (A ) = 0, we have

 .

d κ dA



 = A=Ac



d 2 2 dA (Q1 Q2 Q3 − Q3 − Q1 Q4 ) −Q1 (Q22 + Q1 Q3 − 4Q4 ) A=Ac

(21)

  Q1 (Ac )Q2 (Ac )Q3 (Ac )−Q23 (Ac ) from Eq. (14) we obtain using the value of .Q4 (Ac ) = Q21 (Ac )   d 2 2  dκ  d dA (Q1 Q2 Q3 −Q3 −Q1 Q4 ) = 0. If . dA (Q1 Q2 Q3 −Q23 −Q21 Q4 ) = 0, . Q3 dA A=Ac = 2 −2Q1 (Q1 Q3 +(2 Q −Q2 ) ) 1

A=Ac

the transversality condition holds. Hence, we have the following proposition for Hopf bifurcation existence. Proposition 3.2 In the simplified model system (3), the Hopf bifurcation arises around the endemic equilibrium .E ∗ if .A = Ac exists such that 2 2 c c c c c c (a) .Q  1 (A )Q2(A )Q3 (A ) − Q3 (A ) − Q1 (A )Q4 (A ) = 0, d i (A) (b) . Re dωdA (Q1 Q2 Q3 − Q23 − Q21 Q4 ) = 0. = 0 for .i = 1, 2 i.e., . dA c A=A

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1

Susceptible aware

1

0.8

0.8 0.6

0.4

0.4

0.2

0.2 PRCC

PRCC

0.6

0

0

-0.2 -0.2

-0.4 -0.4

-0.6 -0.6

-0.8

-0.8

-1 0

5 1

d p

10 0

A

n

15

cn h p1

1 2 1 2

20

M0

-1 0

5 1

d

10

p

0

A

n

15

cn h p1

1

20 2

1 2

M0

Fig. 3 Sensitivity analysis of estimated various parameters for susceptible aware and infective population respectively

4 Global Sensitivity Analysis We used global sensitivity analysis (GSA) employing Latin hypercube sampling (LHS) with partial rank correlation coefficient (PRCC) sensitivity analysis to examine the sensitivity of each parameter. Each parameter’s sensitivity is represented in a bar graph and assessed in terms of bar length. If a parameter’s PRCC value is larger than .±0.5, it is said to be sensitive to a variable. The parameters .β , .γ , .λ0 , .α1 , .μ1 , and A are all sensitive parameters for the system (1), as shown in Fig. 3.

5 Numerical Simulations With the aid of numerical simulation, we examine the effects of awareness campaigns in this section. Here, using MATLAB, we look into how different factors affect the system’s qualitative behavior. We begin with the parametric values[15] .A

= 3, β = 0.00001, β1 = .01, λ = 0.05, λ0 = .0032, γ = 0.25, p1 = 3000,

d = 0.00004, μ0 = 0.06, n = 0.25, cn = .05, h = 0.05, p = 0.05, μ1 = 0.08, μ2 = 0.003, α1 = 0.6, α2 = 0.4, δ = 0.06, M0 = 10, θ = 0.005, ω = 60.

(22) We see that the system is locally asymptotically stable at endemic equilibrium ∗ = (35793, 5791, 31, 35) when dealing with the abovementioned collection of parametric values (cf. Fig. 4a).

.E

Modeling the Impact of Media Coverage on the Spread of Infectious Diseases. . . 10

3.6

4

10

5

4

10

7000

Sa

1.5

N

Sa

N

4.5

6000

4

3.56

4

2

8000 3.58

165

1

5000 3.54

3.5

4000 0

0.5

1

1.5

2

0

0.5

10 4

time

1

1.5

200

0.5 0

2

0.5

10 4

time

1

1.5

150

600

100

400

2

0

1

1.5

2 10 4

time 400 300 M

100

I

M

150 I

0.5

10 4

time

50

200

0

0

200 100

50 0 0

0.5

1

time

1.5

2 10

4

0

0.5

1

time

1.5

(a)

2 10

4

0 0

0.5

1

time

1.5

2 10

4

0

0.5

1

time

1.5

(b)

2 10

4

Fig. 4 (a) The equilibrium point .E ∗ is stable for the parametric values as given in (22). (b) The figure depicts oscillatory behavior around the coexistence (endemic) equilibrium point .E ∗ of system (1) for .A = 8. (c, d) The figure depicts oscillatory behavior around coexistence (endemic) equilibrium point .E ∗ of system (3) for .μ1 = 0.3 and .μ2 = 0.3 respectively

Taking .A = 8, the system exhibits oscillations around .E ∗ (cf. Fig. 4b). Figure 5a illustrates the different steady state behaviors of infected class in the system (3). Herein, we can identify Hopf bifurcation points at .A = 6.18 (denoted by a red star .(H )) with eigenvalues .−0.0651356, .−0.000381901, .±0.0196374i and first Lyapunov coefficient being .−1.086052e−10 that forms a family of stable limit cycle bifurcates from the H and loses its stability. Herein, .A = 1.26(BP ) denotes the branch point of the system (3) with eigenvalues .0, 0, −0.06, −0.0034. We obtain the critical value of immigration rate .A = Ac = 1.26, above which the endemic equilibrium exists. It is interesting to see that high growth rate of broadcasting the information through smartphone (.μ1 ) and TV (.μ2 ) plays a big impact to destabilize the whole system (cf. Fig. 4c, d) respectively. Now for clear understanding of dynamic change, we plot a

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I

150

100

50 H LPC 0 0

0.05

0.1

0.15 1

0.2

0.25 (b)

0.3

0.35

140 120

I

100 80 60 40 LPC H 20 0

0.1

0.2 2

0.3 (c)

0.4

Fig. 5 (a–c) The trajectory illustrates the various dynamical behaviors of infected population for A, .μ1 and .μ2 respectively

bifurcation diagram with respect to .μ1 . From Fig. 5b, it follows that for lower values of .μ1 , the system is stable, but when above a threshold value of .μ1 = μc1 = 0.231715, the system loses its stability and periodic solution arises through Hopf bifurcation. Further, we also vary .μ2 as a free parameter; a bifurcation diagram (cf. Fig. 5c) indicates that the system loses its stability for high value of .μ2 after it crosses the critical value .μ2 = μc2 = 0.229434. The bifurcation diagrams (cf. Fig. 6a, b) depict the whole dynamic nature of the system (1) for the effects of parameters A and .μ2 . Figure 7a–c presents two-parameter bifurcation diagrams for .A −μ1 , .A −μ2 , and .μ1 − μ2 respectively. All the numerical finding are summarized in Table 1.

Modeling the Impact of Media Coverage on the Spread of Infectious Diseases. . .

167

Fig. 6 (a) Bifurcation diagram for A. (b) Bifurcation diagram for .μ2

0.5

0.5

0.4

0.4

0.3 Hopf Curve

0.3

Unstable Zone at E *

0.2

0.2

0.1

0.1

0 0

1

2

3

4 (a)

A

5

Hopf Curve

2

1

Unstable Zone at E *

6

7

0 0

1

2

3 A

4 (b)

5

6

7

0.5 0.4 Hopf Curve

2

0.3 0.2

Unstable Zone around at E *

0.1 0 0

0.1

0.2

0.3 1

0.4 (c)

0.5

Fig. 7 (a–c) Two-parameter bifurcation diagram for .A − μ1 , .A − μ2 , and .μ1 − μ2 respectively

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Table 1 Natures of equilibrium points Parameters A .μ1 .μ2

Values 1.262656 6.183250 0.231715 0.229434

Eigenvalues .(−0.06, −0.0034, 0, 0) .(−0.0651, −0.00038, ±0.0196374i) .(−.0645623, −.000175605, ±0.0172407i) .(−.06456, −.000176, ±0.0172407ii)

Equilibrium points Branch point (BP) Hopf (H) Hopf (H) Hopf (H)

6 Discussion The importance of media coverage in spreading knowledge and raising awareness of infectious disease prevention plan is emphasized heavily. Therefore, we examined a four-compartment mathematical model in our paper. Pathogens are thought to spread through direct contact between the susceptible and the infected. Under such conditions, the model displays two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. First, the model is examined analytically, demonstrating that the system shows disease-free equilibrium with the basic reproduction number .R0 < 1. It results in the presence of an endemic equilibrium for .R0 > 1 (cf. Fig. 2). Our study indicates that when constant immigration rate of suspected population is high, the number of infected individuals increases. But after crossing the threshold value, system becomes unstable. Further, we observe that lower value of immigration rate results in disease-free equilibrium. Moreover, the growth rate of broadcasting the information through smartphone and TV helps in keeping the system stable.

References 1. X. Yuan, Y. Xue and M. Liu. Analysis of an epidemic model with awareness programs by media on complex networks. Chaos, Solitons & Fractals, 48, 1–11 (2013). 2. A. K. Misra, A. Sharma and J. B. Shukla. Stability analysis and optimal control of an epidemic model with awareness programs by media. Biosystems, 138, 53–62 (2015). 3. B. Dubey, P. Dubey and U. S. Dubey. Role of media and treatment on an SIR model. Nonlinear Analysis: Modelling and Control, 21(2), 185–200 (2016). 4. F. A. Basir, E. Venturino and P. K. Roy. Effects of awareness program for controlling mosaic disease in Jatropha curcas plantations. Mathematical Methods in the Applied Sciences, 40(7), 2441–2453 (2017). 5. A. Anwar, M. Malik, V. Raees and A. Anwar. Role of mass media and public health communications in the COVID-19 pandemic. Cureus, 12(9) (2020). 6. D. A. González-Padilla and L. Tortolero-Blanco. Social media influence in the COVID-19 pandemic. International braz j urol, 46, 120–124 (2020). 7. H. Al-Dmour, A. Salman, M. Abuhashesh and R. Al-Dmour. Influence of social media platforms on public health protection against the COVID-19 pandemic via the mediating effects of public health awareness and behavioral changes: integrated model. Journal of medical Internet research, 22(8), e19996 (2020).

Modeling the Impact of Media Coverage on the Spread of Infectious Diseases. . .

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8. S. Samanta, S. Rana, A. Sharma, A.K. Misra and J. Chattopadhyay. Effect of awareness programs by media on the epidemic outbreaks: A mathematical model. Applied Mathematics and Computation, 219(12), 6965–6977 (2013). 9. J. Cui, Y. Sun, and H. Zhu. The impact of media on the control of infectious diseases. Journal of dynamics and differential equations, 20(1), 31–53 (2008). 10. A. K. Srivastava, P. K. Tiwari, P. K. Srivastava, M. Ghosh and Y. Kang. A mathematical model for the impacts of face mask, hospitalization and quarantine on the dynamics of COVID-19 in India: deterministic vs. stochastic[J]. Mathematical Biosciences and Engineering, 18(1): 182– 213 (2021). 11. G. O. Agaba, Y. N. Kyrychko and K. Blyuss. Mathematical model for the impact of awareness on the dynamics of infectious diseases. Mathematical biosciences, 286, 22–30 (2017). 12. A. K. Misra, A. Sharma and V. Singh. Effect of awareness programs in controlling the prevalence of an epidemic with time delay. Journal of Biological Systems, 19(02), 389–402 (2011). 13. D. Greenhalgh, S. Rana, S. Samanta, T. Sardar,S. Bhattacharya, and J. Chattopadhyay. Awareness programs control infectious disease–multiple delay induced mathematical model. Applied Mathematics and Computation, 251, 539–563 (2015). 14. S. Samanta. Effects of awareness program and delay in the epidemic outbreak. Mathematical Methods in the Applied Sciences, 40(5), 1679–1695 (2017). 15. A. K. Misra, R. K. Rai and Y. Takeuchi. Modeling the control of infectious diseases: Effects of TV and social media advertisements. Mathematical Biosciences & Engineering, 15(6), 1315 (2018). 16. M. S. Rahman and M. L Rahman. Media and education play a tremendous role in mounting AIDS awareness among married couples in Bangladesh. AIDS Research and Therapy, 4(1), 1–7 (2007).

Cultural and Biological Transmission: A Simple Case of Evolutionary Discrete Dynamics Roberto Macrelli, Margherita Carletti, and Vicenzo Fano

1 Introduction An investigated issue in biological and social sciences is understanding how behavioral and cultural traits, inherited from previous generations or produced in response either to changes in socioeconomic or ecological conditions, are spread in populations. Transmissions of behavioral traits have been observed in different animals like reef fish [1] and birds [2] and also between parents and their offspring [3]. Transmissions are due to social interactions and they are possible either between parents and their own offspring (vertically transmissions), between members of a cohort (horizontal transmissions), or between parents and offspring of others (oblique transmissions) [4, 5]. Modeling this kind of evolution results in the use of the basic format of the one-locus theory [6] in which the trait is a cultural one. Following the formalism of Rice [6], we consider an extension of one-locus theory in which a cultural (behavioural) trait is vertically transmitted from parents to their descendants.

2 Material and Methods In order to illustrate our technique for finding how behavioral and cultural traits are spread in populations starting from an inherited one either from previous generations or produced in response to changes in socioeconomic or ecological conditions, we

R. Macrelli () · M. Carletti · V. Fano DISPeA Department, University of Urbino, C. Bo, Urbino, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_10

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Table 1 Summary table of our model Parents and .A1 .A1 and .A2 .A2 and .A1 .A2 and .A2

.A1

Frequency = u2 .u12 = u(1 − u) .u21 = u(1 − u) 2 .u22 = (1 − u)

.u11

Probability of a .A1 offspring

Fitness

.a11

.w11

.a12

.w12

.a21

.w21

.a22

.w22

consider a simple biological model and we address several parameters. We are interested in mating rates, as well as fitness of individuals with different genotypes. Our technique is based on simple discrete mathematics. Evolutionary dynamics involves the interaction between individuals, assuming them to be expressed by concentration of populations. In our framework we consider a cultural-behavioural trait A, for example, a trait corresponding to being inclined to altruism [7] that is vertically culturally transmitted and its two possible values .A1 (altruistic) and .A2 (not altruistic) in an individual. Let .u1 = u denote the frequency of .A1 , whereas .u2 = 1 − u the frequency of .A2 . We denote .aij the probability for an offspring, which contains .A1 trait, descending from the mating of its parents. Let .wij be the fitness, that is, the capability to adapting to environmental changes and to have descendants. We sum up the situation in the following table (Table 1). Consider the fitness .wij , .∀i, j frequency-independent and let’s assume mean population fitness as the following: w¯ =



.

uij wij .

(1)

i,j

We evaluate the frequency of u in the next generation, .u , summing over the frequency of each mating type multiplied by its probability of producing a .A1 offspring [6] and by the value of the fitness divided by the mean population fitness: u =



.

i

j

uij aij

wij . w¯

(2)

Associating values with the parameters in Table 1, we can assume .a11 = 1, .a12 = r, a21 = r, .a22 = 0, .w12 = w21 = wE , where r is a dominance value. Therefore, the frequency model (2) is simplified in

.

u =

.

u u2 w11 + (1 − u)2rwE . w¯ w¯

(3)

At equilibrium of (3), that is, .u = u, we observed that w¯ = (1 − u)2rwE + uw11 .

.

(4)

Cultural and Biological Transmission: A Simple Case of Evolutionary Discrete. . .

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Several results for the frequency of the trait at the equilibrium are examined manipulating the values of the fitness .w11 , .wE , .w22 , of the probability .a11 , r, .a22 and of the initial frequency of the trait .u0 . .w11 is the fitness of altruistic organisms, .w22 that of not altruistic organisms, and .wE is an intermediate fitness. Results are exposed in different cases. All the numerical experiments presented in this paper were done on a last-generation PC and coded in MATLAB (R2021b) environment running under Windows 10.

3 Results 3.1 Case 1 Let .w22 = 0, in this case the frequency of individual that doesn’t carry .A1 is equal to zero. This is the case in which not altruistic organisms have no fitness. Evolutionary dynamics (3) results in the fixation of the frequency of the trait at equilibrium. Some examples in Fig. 1a, b illustrate that equilibrium frequency is .u = 1, providing .w11 is greater than .wE . If .w11 is less than .wE –the intermediate is fitter than the altruist– we have different values at equilibrium. For instance, in Fig. 2a we obtain .u = 0.13, in Fig. 2b .u = 0.11. In general, assuming .w11 = kwE ; .k ∈ R, .k ≥ 0, evolutionary dynamics (3) results in u =

.

u(k − 2r) + 2r . u(k − 2) + 2

(5)

Evolutionary dynamics (5) results in a fixed value for the frequency u at equilibrium: 2r , in particular, we have two possible values at equilibrium, .u = 1, .∀k, r or .u = 2−k if .0 ≤ k < 2(1 − r). The convergency to one or other fixed equilibrium points of the frequency 2r depends on k, r: if .0 ≤ k < 2(1 − r), the value .u = 2−k is an attractor; otherwise, if .k ≥ 2(1 − r), .u = 1 is an attractor. To sum up, .k = 2(1 − r) is the threshold for having an altruistic population. Values of k lightly smaller than this threshold result in a substantially altruistic population.

3.2 Case 2 When the fitness .w11 decreases and the fitness .w22 increases –with respect to “0”– we observed some difference in the evolutionary dynamics. In particular, let us consider a situation in which the fitness of two parents that carry .A1 , .w11 , tends to zero, whereas the fitness of two individuals that don’t carry .A1 , .w22 , tends to one.

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un+1

0.7 0.6 0.5 0.4 0.3 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

un

(a) 1 0.9 0.8

un+1

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.2

0.4

0.6

0.8

1

un

(b) Fig. 1 (a) Evolutionary dynamics for the frequency of the altruistic trait with .w11 = 0.8, .wE = 0.1, .w22 = 0, .a11 = 1, .r = 0.1, .u0 = 0.1 after .N = 10,000. (b) Evolutionary dynamics for the frequency of the altruistic trait with .w11 = 0.5, .wE = 0.25, .w22 = 0, .a11 = 1, .r = 0.1, .u0 = 0.01 after .N = 10,000

This is the situation in which to be non-altruistic is fitter. In a nutshell, .w11 → 0; wE > 0; .wE  w22 ; .w22 → 1.

.

Cultural and Biological Transmission: A Simple Case of Evolutionary Discrete. . .

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0.135 0.13

un+1

0.125 0.12 0.115 0.11 0.105 0.1 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.5

0.6

0.7

0.8

un

(a) 0.4

0.35

un+1

0.3

0.25

0.2

0.15

0.1 0.1

0.2

0.3

0.4 un

(b) Fig. 2 (a) Evolutionary dynamics for the frequency of the altruistic trait with .w11 = 0.2, .wE = 0.4, .w22 = 0, .a11 = 1, .r = 0.1, .u0 = 0.01 after .N = 10,000. (b) Evolutionary dynamics for the frequency of the altruistic trait with .w11 = 0.1, .wE = 0.45, .w22 = 0, .a11 = 1, .r = 0.1, .u0 = 0.8 after .N = 10,000

We also assume that the probability of a .A1 offspring for two individuals which don’t carry .A1 , .a22 , is different from zero. This is possible if the trait is a cultural one [7]. For instance, assume .a22 = 0.01. In this case the frequency of the trait at the equilibrium tends to fixation value different from zero. An example is shown in

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un+1

0.0112 0.011

0.0108 0.0106 0.0104 0.0102 0.01

0

0.1

0.2

0.3

0.4

0.5

un

Fig. 3 Evolutionary dynamics for the frequency of the altruistic trait with .w11 = 0.001, .wE = 0.005, .w22 = 0.989, .a11 = 1, .r = 0.1, .a22 = 0.01, .u0 = 0.5 after .N = 10,000

Fig. 3; we obtain .u = 0.01. We see that the survival of altruism is of the same order of magnitude than the probability that two non-altruistic organisms be parents of an altruistic offspring.

3.3 Case 3 Assume that the probability of an .A1 offspring for two individuals that don’t carry A1 increases toward 1 (.a22 → 1) and a high value for the fitness .w22 (next to one). This situation is made possible if there is a strong effort from the outside to reward individuals who are selected to learn the cultural trait. In this case we noticed an asymptotic behavior for the frequency u at the equilibrium (Fig. 4). We emphasize two interesting features of this result: (1) altruism oscillates strongly in the population for a while and (2) at the asymptotic equilibrium, the population is more altruistic than not.

.

3.3.1

Case 4

Suppose that two individuals who carry the .A1 trait cannot mate. This could be due to the fact that non-altruistic organisms tend to deceive collaborative individuals. Assume .a11 → 0; .w11 = 0; .wE > 0; .wE  w22 . Suppose that r changes between generations and it is proportional to the mean fitness. In rough terms, with

Cultural and Biological Transmission: A Simple Case of Evolutionary Discrete. . .

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1 0.9 0.8

un+1

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

un

Fig. 4 Evolutionary dynamics for the frequency of the altruistic trait with .w11 = 0.0001, .wE = 0.0500, .w22 = 0.8999, .a11 = 1, .r = 0.1, .a22 = 1, .u0 = 0.1 after .N = 10000

the increasing of offspring the dominance of .A1 increases as well. This could be due to the fact that in more numerous offspring, the drift due to the influence of the environment could augment as well. That is the probability of an .A1 offspring for two individuals which only one of them carrying .A1 depends on the mean fitness. Assume .r = w/w ¯ E . It is easy to demonstrate that this kind of assumption is reasonable (r is between 0 and 1). In this case we noticed the fixation of the frequency u at the equilibrium. An example is shown in Fig. 5a. Increasing the value of r, .r = 1.5w/w ¯ E , the frequency of the trait at the equilibrium tends to a limit cycle. Examples are shown respectively in Fig. 5b, c. Finally, assuming .r = 1.95w/w ¯ E , we observe a chaotic behaviour for the trait .A1 at the equilibrium. An example is shown in Fig. 5d. To sum up, if the dominance of .A1 is strongly pushed by the mean fitness, then the presence of altruist organism in a population is chaotic, even if the fitness of non-altruistic organisms is high. This is the last case we investigated. Furthermore, we notice that the frequency at the equilibrium is not a local maximum for the mean fitness: an increase of the frequency does not imply a decrease of the mean fitness. This is due to the presence of the dominance value r. In particular by algebraic manipulation, we can derive equation (4) when .w22 = 0: .

dw¯ w11 . = −2rwE + w11 < 0 ↔ r > du 2wE

(6)

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Fig. 5 (a) Evolutionary dynamics for the frequency of the altruistic trait with .w11 = 0, .wE = 0.05, .w22 = 0.90, .a11 = 0, .r = ww¯E , .a22 = 1, .u0 = 0.1 after .N = 10000. (b) Evolutionary dynamics for the frequency of the altruistic trait with .w11 = 0, .wE = 0.05, .w22 = 0.90, .a11 = 0, 1.5w¯ .r = wE , .a22 = 1, .u0 = 0.5 after .N = 10000. (c) Evolutionary dynamics for the frequency of

w¯ the altruistic trait with .w11 = 0, .wE = 0.05, .w22 = 0.90, .a11 = 0, .r = 1.5 wE , .a22 = 1, .u0 = 0.1 after .N = 10000. (d) Evolutionary dynamics for the frequency of the altruistic trait with .w11 = 0, 1.95w¯ .wE = 0.05, .w22 = 0.90, .a11 = 0, .r = wE , .a22 = 1, .u0 = 0.1 after .N = 10000

Providing .r > w11 /2wE , we can conclude that .w¯ decreases if u increases. Otherwise, the mean fitness is not a local maximum for the fitness at the equilibrium. Similar results are possible if we consider .w22 = 0, for example, .w11 = 0.0001, .wE = 0.0500, .w22 = 0.8999, .a11 = 1, .r = 0.1, .a22 = 1, .u0 = 0.1. We observe that the mean fitness is not a local maximum for the fitness at the equilibrium (Fig. 6).

4 Discussions In this article, we provided a simple one-locus model to offer insights into the spread of a cultural trait. The importance of studying of a cultural trait is crucial for the role of education and teaching.

Cultural and Biological Transmission: A Simple Case of Evolutionary Discrete. . .

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0.14 0.13

mean wn+1

0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05

0.06

0.07

0.08

0.09 0.1 mean wn

0.11

0.12

0.13

0.14

Fig. 6 Evolutionary dynamics for the mean fitness with .w11 = 0.0001, .wE = 0.0500, .w22 = 0.8999, .a11 = 1, .r = 0.1, .a22 = 1, .u0 = 0.1 after .N = 10000

We showed a simple model, revisited from Rice [6], to study the evolutionary dynamics resulting from the transmission of this trait in the population. The dynamics depends on several parameters: in particular, the value of the fitness of the organisms, the probability of successful transmission, and the possible changes of social and environmental contest. We generally observed the fixation of the frequency of the trait to a stable equilibrium value. Nevertheless, for some particular values of the parameters, the dynamics produced a stable limit cycle or chaotic behavior of the frequency of the trait. The values that are assumed by the parameters have a correspondence in biological and human sciences. It is remarkable that the dominance factor r is crucial to the dependence of the dynamics of the model and to understand the evolution of the dynamics. Furthermore, we showed that when the equilibrium of the frequency is achieved, the mean population fitness did not necessarily reach a local maximum of the fitness. Our model opens to the scenarios in which the best possible value of the fitness was not reached by means of the equilibrium of the frequency of the trait. It is also quite general and can be applied to different examples, not only in life sciences. Unfortunately, at the moment, its applicability is conditioned by its simplicity: in this paper, we study a model of vertical transmission only. Further studies are required for the interactions between individual and to understand the evolution of a cultural trait in larger groups: migration or other social dynamics are not taken into account in our model. This is a challenge due to the difficult-to-manipulate mathematical framework for a correct representation of the reality. All of these issues are being addressed in a new research article in preparation.

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References 1. Helfman, G.S., Schultz, E.T.: Social transmission of behavioral traditions in a coral reef fish, Anim Behav. 32 (2), 379–384 (1984) 2. Fisher, J., Hinde, R.A.: The opening of milk bottles by birds, British Birds 42, 347–357 (1949) 3. Cavalli-Sforza, L.L., Feldman W.: Cultural Trasmission and Ecolution: A Quantitative Approach, Princeton University Press, Princeton, NJ (1981) 4. Feldman, M.W., Cavalli-Sforza, L.L.: Cultural and biological evolutionary processes, selection for a trait under complex transmission. Theor. Popul. Biol., 9 (2), 238–259 (1976) 5. Feldman, M.W., Cavalli-Sforza, L.L.: Cultural and biological evolutionary processes: geneculture disequilibrium. Proc. Natl. Acad. Sci. USA, 81 (5), 1604–1607 (1984) 6. Rice, S.H.: Evolutionary Theory: Mathematical and Conceptual Foundations. Sinaurer (2004) 7. Fehr, E., Fischbacher, U.: The nature of human altruism. Nature, 425 (6960), 785–791 (2003)

The Maximal Extension of the Strict Concavity Region on the Parameter Space for Sharma-Mittal Entropy Measures: II R. P. Mondaini and S. C. de Albuquerque Neto

1 Introduction. The Construction of the Probabilistic Space In recent decades, we have seen a proliferation of several proposals [1] for entropy measures in the literature that are essentially associated with the concept of probabilities of occurrence. We consider the information obtained from statistical data as given in two-dimensional arrays of m rows and n columns. The probabilities of occurrence are then considered as geometric objects associated with these arrays, and they result to be very convenient for unveiling fundamental properties of the probabilistic distribution in an objective and straightforward way. We then define the probability of occurrence of a set of t variables .a1 , . . . , at corresponding to t ordered columns .j1 , . . . , jt , respectively, to be chosen among the n ordered columns .j1 , . . . , jn of the array. Each variable .a1 , . . . , at is able to assume any of the values .1, . . . , W : .1 ≤ a1 , . . . , at ≤ W . pj1 ...jt (a1 , . . . , at ) =

.

Nj1 ...jt (a1 , . . . , at ) , 1 ≤ a1 , . . . , at ≤ W , m

(1)

where .Nj1 ...jt is the number of occurrences of the t-set of values .a1 , . . . , at in the t-set of ordered columns, .j1 , . . . , jt , respectively. The ordering of columns [2] is realized through j1 < j2 < . . . < jt ,

.

1 ≤ t ≤ n,

(2)

R. P. Mondaini () · S. C. de Albuquerque Neto COPPE, Centre of Technology, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_11

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

.

(3)

jt−1 = jt−2 + 1, . . . , n − 1 , jt = jt−1 + 1, . . . , n .   n! We then have . nt = t!(n−t)! geometrical objects .pj1 ...jt , and each of them has .(W )t components .pj1 ...jt (a1 , . . . , at ). The corresponding probabilities of occurrence of t-sets could be written as Q

Q

.

Q

pj1 ...jt (a1 μ , . . . , at μ ) =

Q

Nj1 ...jt (a1 μ , . . . , at μ ) , m

(4)

μ = 1, . . . , M ; 1 ≤ M ≤ m . Q

Q

where .(a1 μ , . . . , at μ ) stands for .qμ identical (to each .μ value, there are .qμ Q

Q

identical t-sets .(a1 μ , . . . , at μ )), according to Q1 = 1, . . . , q1 Q2 = q1 + 1, q1 + 2, . . . , q1 + q2 Q3 = q1 + q2 + 1, q1 + q2 + 2, . . . , q1 + q2 + q3 .

(5)

.. . QM = q1 + q2 + . . . + qM−1 + 1, q1 + q2 + . . . + qM−1 + 2, . . . , q1 + q2 + . . . + qM

Generically Qμ = q1 + . . . + qμ−1 + 1, q1 + . . . + qμ−1 + 2, . . . , q1 + q2 + . . . + qμ , .

μ = 1, . . . , M . (6) We then have trivially for Eq. (4): Q

Q

Q

pj1 ...jt (a1 μ , . . . , at μ ) =

.

Q

Nj1 ...jt (a1 μ , . . . , at μ ) qμ , = m m

(7)

The Maximal Extension of the Strict Concavity Region on the Parameter Space. . .

183

and for all .μ-sets: 

Q

Qμ

M  Nj1 ...jt (a Qμ , . . . , atQμ )  qμ 1 = ≡1 m m Q

Q

pj1 ...jt (a1 μ , . . . , at μ ) =

.

Qμ

Qμ

a1 ,...,at

a1 ,...,at

μ=1

μ

(8) The structure of a .m × t array is then given by ⎛  ⎞ Q Q  a1 1 , . . . , at 1 q ×t 1 ⎟ ⎜  Q2 ⎜ a , . . . , atQ2 ⎟ 1 q2 ×t ⎟ ⎜ .⎜ ⎟ .. ⎜ ⎟ . ⎝ ⎠ QM QM  a1 , . . . , at q ×t M

, 1 ≤ M ≤ m.

(9)

m×t

Some useful applications of this definition to the probabilistic space are working with the values .(a1 , . . . , at ) assuming the representative letters A, C, T, and G of the nucleotide bases of DNA .(W = 4), or on the one-letter code for the amino acids A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, and Y of proteins .(W = 20) Q Q [2–4], i.e., the t-set .(a1 μ , . . . , at μ ) does correspond to a sequence of amino acids Q

obtained from { A, C, .. . ., W, Y } or .aj μ ∈ { A, C, .. . ., W, Y }, .∀μ, .1 ≤ j ≤ t. In order to fix ideas, we now present examples of .m × t hypothetical arrays, corresponding to the occurrence of t-sets of amino acids: In Fig. 1 there are six different groups of three sets for the .8 × 3 array, four different groups of four sets for the .7 × 4 array, and four different groups of six sets for the .6 × 6 array. We then have from Eq. (8): (I) pj1 j2 j3 (AQ1 , CQ1 , DQ1 ) + pj1 j2 j3 (AQ2 , CQ2 , EQ2 ) + pj1 j2 j3 (AQ3 , EQ3 , CQ3 )

.

+ pj1 j2 j3 (AQ4 , DQ4 , CQ4 ) + pj1 j2 j3 (EQ5 , CQ5 , AQ5 ) + pj1 j2 j3 (DQ6 , AQ6 , EQ6 ) =

6  qμ = 1. m

(10)

μ=1

Fig. 1 (I), (II), and (III), a × 3, .7 × 4, and .6 × 6 arrays of t-sets of amino acids, respectively

.8

A A A A A A E D

C C C E D C C A

D D E C C D A E

(I)

A D A A D E A

C C C D C D D

D E D C E C C

(II)

E F E F F A F

A A A C D D

C D C E C C

D C D A E E

E E E G F F

( III)

F G F F G G

G F G D A A

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The number of t-sets on each .μ-group is q1 = 3 ; q2 = q3 = q4 = q5 = q6 = 1 ,

(11)

.

and the sum of their probabilities of occurrence is

.

6  qμ 3 1 1 1 1 1 = + + + + + = 1. 8 8 8 8 8 8 m

(12)

μ=1

Analogously, (II) pj1 ...j4 (AQ1 , CQ1 , DQ1 , EQ1 ) + pj1 ...j4 (DQ2 , CQ2 , EQ2 , FQ2 )

.

+pj1 ...j4 (AQ3 , DQ3 , CQ3 , FQ3 ) + pj1 ...j4 (EQ4 , DQ4 , CQ4 , AQ4 ) =

4  qμ = 1, m

μ=1

(13) q1 = q2 = q3 = 2 ; q4 = 1 ,

(14)

4  qμ 2 2 2 1 = + + + = 1. m 7 7 7 7

(15)

.

and

.

μ=1

(III) pj1 ...j6 (AQ1 , CQ1 , DQ1 , EQ1 , FQ1 , GQ1 )

.

+ pj1 ...j6 (AQ2 , DQ2 , CQ2 , EQ2 , GQ2 , FQ2 ) + pj1 ...j6 (CQ3 , EQ3 , AQ3 , GQ3 , FQ3 , DQ3 ) + pj1 ...j6 (DQ4 , CQ4 , EQ4 , FQ4 , GQ4 , AQ4 ) 4  qμ = 1, = m

(16)

μ=1

q1 = q4 = 2 ; q2 = q3 = 1 ,

(17)

4  qμ 2 1 1 2 = + + + = 1. m 6 6 6 6

(18)

.

and

.

μ=1

The Maximal Extension of the Strict Concavity Region on the Parameter Space. . .

185

2 The Sharma-Mittal Class of Entropy Measures In this section, in order to alleviate the notation, we do not take in consideration the superindices .Qμ . We now introduce the two-parameter Sharma-Mittal class of entropy measures [4] through

(SM)j1 ...jt =

 1−r 1−s (s) αj1 ...jt −1

.

1−r

(19)

.

(s)

The symbols .αj1 ...jt are given by (s)

αj1 ...jt =

.

  s pj1 ...jt (a1 , . . . , at ) ,

(20)

a1 ,...,at

where .pj1 ...jt (a1 , . . . , at ) is given by Eq. (1). The elementary analysis of the strict concavity requirement to be done without the consideration of .Qμ -values will lead to .

s−1  s−r ∂(SM)j1 ...jt s   = αj1 ...jt 1−s pj1 ...jt (a1 , . . . , at ) , 1−s ∂ pj1 ...jt (a1 , . . . , at ) ∂ 2 (SM)j1 ...jt

.

 ∂ pj1 ...jt (a1 , . . . , at )

(21)

s−r  s−2 1−s (s) · pj1 ...jt (a1 , . . . , at ) = s α 2 j1 ...jt 

 s(s − r) · pˆ j ...j (a1 , . . . , at ) − 1 < 0 , (1 − s)2 1 t

(22)

where .pˆ j1 ...jt (a1 , . . . , at ) is the escort probability associated to .pj1 ...jt (a1 , . . . , at ) pˆ j1 ...jt (a1 , . . . , at ) =

.

s  pj1 ...jt (a1 , . . . , at ) αj(s) 1 ...jt

,

(23)

with .1 ≤ a1 , . . . , at ≤ W , and  .

pj1 ...jt (a1 , . . . , at ) =

a1 ,...,at



pˆ j1 ...jt (a1 , . . . , at ) = 1 .

(24)

a1 ,...,at

From Eq. (22), we get [3] r ≥ s > 0.

.

(25)

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

However, we have to note that we are working with a multivariate function and Eq. (22) alone is clearly not sufficient to characterize its strict concavity. We have to proceed with the derivation through an efficient analysis based on the sign of the principal minors of a Hessian matrix [5]. We do this in Sects. 3 and 4. The special cases of the Sharma-Mittal entropies are known in the literature: (a) The Havrda-Charvat entropy measure .(r = s) [6] (s)

Hj1 ...jt =

.

αj1 ...jt − 1 1−s

(26)

.

(b) The Landsberg-Vedral entropy measure .(r = 2 − s) [1] Lj1 ...jt =

.

−1 αj(s) 1 ...jt (1 − s)αj(s) 1 ...jt

=

Hj1 ...jt αj(s) 1 ...jt

(27)

.

(c) The Renyi entropy measure [7] (s)

Rj1 ...jt = lim (SM)j1 ...jt =

.

log αj1 ...jt

r→1

1−s

.

(28)

(d) The “non-extensive Gaussian” entropy measure [8] Gj1 ...jt = lim (SM)j1 ...jt =

.

s→1

e(1−r)Sj1 ...jt − 1 , 1−r

(29)

where Sj1 ...jt = −



.

pj1 ...jt (a1 , . . . , at ) log pj1 ...jt (a1 , . . . , at ) ,

(30)

a1 ,...,at

is the Gibbs-Shannon entropy measure, which is given by the convenient limit of all previous entropy measures, or, .

lim Hj1 ...jt = lim Lj1 ...jt = lim Rj1 ...jt = lim Gj1 ...jt = Sj1 ...jt .

s→1

s→1

s→1

r→1

(31)

According to inequality (25), the region of the parameter space corresponding to the strict concavity of the Sharma-Mittal class of entropies, or, C = {(s, r) | r ≥ s > 0} ,

.

(32)

is generally regarded in the literature as the hatched region in Fig. 2. This is the epigraph of the curve .r = s which is depicted in brown. The unidimensional

The Maximal Extension of the Strict Concavity Region on the Parameter Space. . .

187

r

Fig. 2 Strict concavity region of the .(s, r)-parameter space of the Sharma-Mittal class of entropy measures. Special cases of one-parameter entropies are depicted in color according to the text

3

2

1

0

1

2

3

s

regions corresponding to the Landsberg-Vedral .(r = 2 − s), Renyi .( lim ) and r→1

“non-extensive” Gaussian .( lim ) entropies are depicted in green, blue, and red, s→1

respectively. The point .(r = 1, s = 1) does correspond to the Gibbs-Shannon entropy.

3 The Maximal Extension of the (s, r) Parameter Space First of all, we should notice that Eq. (20) should be written after substituting Eqs. (1) and (8) as (s) .α j1 ...jt

=

m   qμ s μ=1

m



=

Qμ Qμ a1 ,...,at



Q

Q

Nj1 ...jt (a1 μ , . . . , at μ ) m

s .

(33)

For the .8 × 3, .7 × 4, and .6 × 6 arrays of Fig. 1, we then get, respectively (s) .α j1 ...j3

=

6   qμ s μ=1

(s)

m

αj1 ...j4 =

.

 s  s  s  s  s  s 3 1 1 1 1 1 = + + + + + , 8 8 8 8 8 8

4   qμ s μ=1

m

=

 s  s  s  s 2 2 1 2 + + + , 7 7 7 7

(34)

(35)

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

and (s) .α j1 ...j6

=

4   qμ s μ=1

m

 s  s  s  s 2 1 1 2 = + + + . 6 6 6 6

(36)

We should also remember that the negative definiteness of the quadratic form associated with the Hessian matrix of a multivariate function is the requirement for its strict concavity. This means that the principal minors of the Hessian matrix of odd order should be negative and those of even order, positive [5]. We then define the Hessian matrix: ⎛ (H)M×M = (HQμ Qν ) = ⎝ ⎛ .



⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝

⎞

∂ 2 (SM)j1 ...jt Qμ

Qμ

∂pj1 ...jt (a1 , . . . , at

∂ 2 (SM)j1 ...jt

Q Q ∂ pj1 ...jt (a1 1 ,...,at 1 )

···

2

. . .

..

∂ 2 (SM)j1 ...jt

Q Q Q Q ∂pj1 ...jt (a1 M ,...,at M )∂pj1 ...jt (a1 1 ,...,at 1 )

Q

⎠ ) M×M ⎞

∂ 2 (SM)j1 ...jt

⎟ Q Q Q Q ∂pj1 ...jt (a1 1 ,...,at 1 )∂pj1 ...jt (a1 M ,...,at M ) ⎟ . . .

.

···

Qν

)∂pj1 ...jt (a1 ν , . . . , at



∂ 2 (SM)j1 ...jt

Q Q ∂ pj1 ...jt (a1 M ,...,at M )

2

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ M×M

(37)

We then have analogously to Eq. (21) for the first derivative: .

∂(SM)j1 ...jt  s−r s  Qμ Qμ s−1 αj1 ...jt 1−s pj1 ...jt (a1 , . . . , at ) , =  Qμ Qμ  1 − s ∂ pj1 ...jt (a1 , . . . , at )

(38)

and a generic element of the Hessian matrix will be written as HQμ Qν =

.

∂ 2 (SM)j1 ...jt Qμ Qμ Q Q ∂pj1 ...jt (a1 , . . . , at )∂pj1 ...jt (a1 ν , . . . , at ν )

  s−r   Qμ Qμ s−2 = s αj1 ...jt 1−s pj1 ...jt (a1 , . . . , at ) ⎡ ⎤ Qμ Qμ pj1 ...jt (a1 , . . . , at ) s(s − r) Q Q ν ν pˆ j ...j (a , . . . , at ) − δμν ⎦ , ·⎣ (1 − s)2 pj ...j (a Qν , . . . , a Qν ) 1 t 1 1

t

1

t

where .δμν stands for the Kronecker symbol. The principal minors of the Hessian matrix (Eq. (37)) are given by

(39)

The Maximal Extension of the Strict Concavity Region on the Parameter Space. . .

189

 k(s−r) (s) 1−s det HQμ Qν (μ, ν = 1, . . . , k) = (−1)k−1 s k αj ...j 1

⎡ ·⎣

t

⎤s−2

k 

Qμ Qμ pj1 ...jt (a1 , . . . , at )⎦

μ=1

.

⎤ k  s(s − r) Qμ Qμ ·⎣ pˆ j1 ...jt (a1 , . . . , at ) − 1⎦ , (1 − s)2 ⎡

(40)

μ=1

1 ≤ k ≤ m,

where  Qμ Qμ s pj1 ...jt (a1 , . . . , at ) Qμ Qμ , .pˆ j ...jt (a 1 , . . . , at ) = 1 (s) αj ...j t 1

(41)

Q

with .aj μ , .1 ≤ j ≤ t ∈ {A, C, . . . , W, Y}, stands for the escort probability associated to Qμ

Qμ

) analogously to Eq. (23). We then have analogously to Eq. (8):

.pj ...jt (a 1 1

, . . . , at

 .

Qμ

Qμ

pj1 ...jt (a1 , . . . , at

Qμ Qμ a1 ,...,at



)=

Qμ

Qμ

pˆ j1 ...jt (a1 , . . . , at

) = 1.

(42)

Qμ Qμ a1 ,...,at

The alternance of signs of the principal minors is guaranteed by the negativity of the second square bracket in Eq. (40), or, .

k s(s − r)  Qμ Qμ pˆ j1 ...jt (a1 , . . . , at ) − 1 < 0 . 2 (1 − s)

(43)

μ=1

This means that . det HQμ Qν (μ, ν

= 1) < 0 , .

(44)

det HQμ Qν (μ, ν = 1, 2) > 0 , .

(45)

det HQμ Qν (μ, ν = 1, 2, 3) < 0 , .

(46)

. .. . . . det HQμ Qν (μ, ν = 1, . . . , m) < 0 → m odd, > 0 → m even.

(47)

From Eqs. (33) and (40), we can write for Eq. (43): .

s(s − r) σk (s) − 1 < 0 , (1 − s)2

(48)

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where k  q s  μ .σk (s)

m

μ=1

= m  ,  qμ s

σm (s) = 1 .

(49)

m

μ=1

The k -curves of the regions .Ck

    (1 − s)2 , 1 ≤ k ≤ m, = (s, r)  s > r > s − sσk (s)

(50)

will be given by .rk (s)

=s−

(1 − s)2 , 1 ≤ k ≤ m, sσk (s)

(51)

1 , s

(52)

with .rm (s)

=2−

as the greatest lower bound (g.l.b) of the epigraph regions and is depicted in black in Fig. 3. The hatched region in Fig. 3 is the maximal extremum of the strict concavity region and it is given by .Cmax

Fig. 3 The region of maximal extension of strict concavity for the Sharma-Mittal class of entropy measures and its .k = m curve .(r = 2 − 1/s) in black. The other colors are related to unidimensional regions corresponding to special cases of the Sharma-Mittal class as in Fig. 2

= Cm ∪ C ,

(53)

r

3

2

1

0

0.5

1

2

3

s

The Maximal Extension of the Strict Concavity Region on the Parameter Space. . .

191

where C is the C -region given by Eq. (32).

4 Some Essential Results Obtained from Arrays of Fig. 1 We now present the results of our calculations of the principal minors for the Hessian matrices corresponding to the arrays .8×3, .7×4, and .6×6 of Fig. 1 by using Eqs. (34)– (36) and (41)–(43), respectively: (I) .8 × 3 array; .μ = 1, . . . , k , . k = 1, . . . , 8. There are six .μ-groups (Fig. 4), and the number of three sets on each group is given by .q1

= 3; q2 = q3 = q4 = q5 = q6 = 1.

(54)

For completeness, we also write: .q7 = q8 = 0. ⎧ (q /8)s (q /8)s +(q2 /8)s σ = α1 ; σ2 = 1 α ; ⎪ ⎪ j1 ...j3 j1 ...j3 ⎪ 1 ⎪ ⎪ s s s ⎪ (q /8) +(q2 /8) +(q3 /8) ⎪ ⎪ ; σ3 = 1 ⎪ αj1 ...j3 ⎪ ⎪ ⎪ s +(q /8)s +(q /8)s +(q /8)s ⎪ (q /8) ⎪ 1 2 3 4 ⎨ σ4 = ; αj1 ...j3 .σk (s)-factors s +(q /8)s +(q /8)s +(q /8)s +(q /8)s ⎪ (q /8) 2 3 4 5 ⎪ ; σ5 = 1 ⎪ ⎪ αj1 ...j3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ6 = 1 . ⎪ ⎪ ⎪ ⎪ ⎩ σ 7 = σ8 = 1 .

Fig. 4 k-curves for a .8 × 3 hypothetical array. Six .μ-groups

(55)

2

1

0

0.5 1

2

3

4

5

6

7

8

9

10

192

R. P. Mondaini and S. C. de Albuquerque Neto ⎧ ⎪ r1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r ⎪ ⎪ 3 rk (s) ⎨ . r4 curves ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r5 ⎪ ⎪ ⎪ ⎪ ⎪ r6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r7

2 αj ...j = s − (q 1/8)3s (1−s) ; s 1 2 αj ...j ; = s − (q /8)s1+(q3 /8)s (1−s) s 1 2 2 αj ...j = s − (q /8)s +(q 1/8)3s +(q /8)s (1−s) ; s 1 2 3 2 αj ...j ; = s − (q /8)s +(q /8)s1+(q3 /8)s +(q /8)s (1−s) s 1 2 3 4

(56)

2 αj ...j ; = s − (q /8)s +(q /8)s +(q 1/8)3s +(q /8)s +(q /8)s (1−s) s 1 2 3 4 5

= 2 − 1s . = r8 = 2 − 1s .

(II) .7 × 4 array; .μ = 1, . . . , k , . k = 1, . . . , 7. There are four .μ-groups (Fig. 5) and the number of FOUR sets on each group is given by = q2 = q3 = 2; q4 = 1.

.q1

(57)

We also write .q5 = q6 = q7 = 0.

.σk (s)-factors

Fig. 5 k-curves for a .7 × 4 hypothetical array. Four .μ-groups

⎧ (q /7)s +(q2 /7)s (q /7)s ⎪ σ = α1 ; σ2 = 1 α ; ⎪ ⎪ 1 j1 ...j4 j1 ...j4 ⎪ ⎪ s s s ⎪ ⎪ ⎨ σ3 = (q1 /7) +(q2 /7) +(q3 /7) ; α j1 ...j4

(58)

⎪ ⎪ σ4 = 1 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ 5 = σ 6 = σ7 = 1 .

2

1

0

0.5 1

2

3

4

5

6

7

8

9

10

The Maximal Extension of the Strict Concavity Region on the Parameter Space. . . ⎧ ⎪ ⎪ ⎪ r1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r ⎪ ⎪ 2 rk (s) ⎨ . r3 curves ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r4 ⎪ ⎪ ⎪ ⎩ r5

193

2 αj ...j = s − (q 1/7)4s (1−s) ; s 1 2 αj ...j = s − (q /7)s1+(q4 /7)s (1−s) ; s 1 2 2 αj ...j = s − (q /7)s +(q 1/7)4s +(q /7)s (1−s) ; s 1 2 3

(59)

= 2 − 1s . = r6 = r7 = 2 − 1s .

(III) .6 × 6 array; .μ = 1, . . . , k , . k = 1, . . . , 6. There are four .μ-groups (Fig. 6), and the number of six sets on each group is given by .q1

= q4 = 2; q2 = q3 = 1.

(60)

We also write .q5 = q6 = 0.

.σk (s)-factors

⎧ (q /6)s +(q2 /6)s (q /6)s ⎪ ; σ2 = 1 α ; σ = α1 ⎪ ⎪ 1 j1 ...j6 j1 ...j6 ⎪ ⎪ s +(q /6)s +(q /6)s ⎪ (q /6) ⎪ 2 3 ⎨ σ3 = 1 ; α ⎪ ⎪ σ4 = 1 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ5 = σ6 = 1 .

j1 ...j6

⎧ 2 αj ...j ⎪ ; r1 = s − (q 1/6)6s (1−s) ⎪ s ⎪ 1 ⎪ ⎪ ⎪ 2 α ⎪ j ...j ⎪ ⎪ r2 = s − (q /6)s1+(q6 /6)s (1−s) ; ⎪ s ⎪ 1 2 ⎨

rk (s) 2 αj ...j ; r3 = s − (q /6)s +(q 1/6)6s +(q /6)s (1−s) s 1 2 3 curves ⎪ ⎪ ⎪

.

(61)

(62)

⎪ ⎪ ⎪ ⎪ r4 = 2 − 1s . ⎪ ⎪ ⎪ ⎪ ⎩ r5 = r6 = 2 − 1s .

We now proceed to check the application of the method summarized in this section to real data obtained from the protein family PF01926 from the Pfam database [9, 10], versions 27.0 and 35.0. We have used .80 × 80 arrays of amino acids, and we apply the method to three sets of amino acids in 80 rows (.80 × 3 arrays). The results are depicted in Figs. 7, 8, and 9.

5 Concluding Remarks The consideration of .qμ -values corresponding to .μ-groups of t -sets provides a rigorous derivation of the maximal strict concavity region of the .(s, r)-parameter space. The requirement for fully synergetic distributions will reduce this region

194 Fig. 6 k-curves for a .6 × 6 hypothetical array. Four .μ-groups

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

2

1

0

0.5 1

2

3

4

5

6

7

8

9

10

Fig. 7 k-curves for a .80×3 array, columns: 3, 4, 5, obtained from protein domain family PF01926, (a) Pfam database, version 27.0; 43 .μ-groups; (b) Pfam database, version 35.0; 41 .μ-groups

of maximal extension deduced in this paper. The specification of these synergy conditions on the distributions of entropy values of amino acid occurrences in protein domain families is now in progress [11, 12] and is based on the methods and results presented here. Figures 4, 5, 6, 7, 8 and 9 will show the existence of a general patterning scheme to identify the evolution of the protein domain database in terms of k -curves associated with .m × t arrays .(1 ≤ t ≤ n). All these developments are then supposed to be able to characterize the evolution of protein domain families through the evolution of entropy configurations driven by Fokker-Planck equations. This evolution has been already detected in some published versions of the Pfam database [9, 10] and will appear in a forthcoming publication.

The Maximal Extension of the Strict Concavity Region on the Parameter Space. . .

195

Fig. 8 k-curves for a .80 × 3 array, columns: 30, 31, 32, obtained from protein domain family PF01926, (a) Pfam database, version 27.0; 44 .μ-groups; (b) Pfam database, version 35.0; 44 .μgroups

Fig. 9 k-curves for a .80 × 3 array, columns: 71, 72, 73, obtained from protein domain family PF01926, (a) Pfam database, version 27.0; 56 .μ-groups; (b) Pfam database, version 35.0; 58 .μgroups

References 1. Landsberg, P.T., Vedral, V.: Distributions and Channel Capacities in Generalized Statistical Mechanics. Physics Letters A 247 (3), 211–217 (1998) https://doi.org/10.1016/S03759601(98)00500-3 2. Mondaini, R.P., de Albuquerque Neto, S.C.: The Statistical Analysis of Protein Domain Family Distributions via Jaccard Entropy Measures. In: Mondaini, R.P. (ed.) Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment: Selected Works from the BIOMAT Consortium Lectures, Szeged, Hungary, 2019, pp. 169–207. Springer International Publishing, Cham (2020) https://doi.org/10.1007/978-3-030-46306-9_13

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3. Mondaini, R.P., de Albuquerque Neto, S.C.: Khinchin–Shannon Generalized Inequalities for “Non-additive” Entropy Measures. In: Mondaini, R.P. (ed.) Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics: Selected works presented at the BIOMAT Consortium Lectures, Morocco 2018, pp. 177–190, Springer International Publishing, Cham (2019) https://doi.org/10.1007/978-3-030-23433-1_13 4. Mondaini, R.P., de Albuquerque Neto, S.C.: Alternative Entropy Measures and Generalized Khinchin-Shannon Inequalities. Entropy 23, 1618 (2021) https://doi.org/10.3390/e23121618 5. Marsden, J.E., Tromba, A.: Vector Calculus, 6th Edition. W. H. Freeman and Company Publishers, New York, NY, USA (2012) 6. Havrda, J., Charvát, F.: Quantification Method of Classification Processes. Concept of Structural α-entropy. Kybernetika 3 (1), 30–35 (1967) 7. Rényi, A.: On Measures of Entropy and Information. In: Neyman, J. (ed.) Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, pp. 547–561, University of California Press, Berkeley, California, USA (1961) 8. Oikonomou, T.: Properties of the “non-extensive Gaussian” entropy. Physica A: Statistical Mechanics and its Applications 381, 155–163 (2007) https://doi.org/10.1016/j.physa.2007.03. 010 9. Finn, R.D., Bateman, A., Clements, J., Coggill, P., Eberhardt, R.Y., Eddy, S.R., Heger, A., Hetherington, K., Holm, L., Mistry, J., Sonnhammer, E.L.L., Tate, J., Punta, M.: Pfam: the protein families database. Nucleic Acids Research 42 (D1), D222–D230 (2013) https://doi. org/10.1093/nar/gkt1223 10. Mistry, J., Chuguransky, S., Williams, L., Qureshi, M., Salazar, G.A., Sonnhammer, E.L.L., Tosatto, S.C.E., Paladin, L., Raj, S., Richardson, L.J., Finn, R.D., Bateman, A.: Pfam: The protein families database in 2021. Nucleic Acids Research 49 (D1), D412–D419 (2020) https:// doi.org/10.1093/nar/gkaa913 11. Mondaini, R.P., de Albuquerque Neto, S.C.: The Maximal Extension of the Strict Concavity Region on the Parameter Space for Sharma-Mittal Entropy Measures. In: Mondaini, R.P. (ed.) Trends in Biomathematics: Stability and Oscillations in Environmental, Social, and Biological Models: Selected Works from the BIOMAT Consortium Lectures, Rio de Janeiro, Brazil, 2021, pp. 265–286. Springer International Publishing, Cham (2022) https://doi.org/10.1007/978-3031-12515-7_15 12. Mondaini, R.P., de Albuquerque Neto, S.C.: Essential Conditions for the Full Synergy of Probability of Occurrence Distributions. Entropy 24, 993 (2022) https://doi.org/10.3390/ e24070993

An Eco-Epidemic Predator-Prey Model with Selective Predation and Time Delays Sasanka Shekhar Maity, Pankaj Kumar Tiwari, Nanda Das, and Samares Pal

1 Introduction Parasites affect the food web characteristics strongly and play an impactful role in trophic interactions. It can impact the potential of its host and affect the abundance of a predator [1]. Many researchers have explored predator-prey-parasite phenomenon [2–4] after the seminal work of Anderson and May [5]. Indeed, predator-prey-parasite models exhibit more complex and interesting dynamical behaviors than the conventional ecological or epidemiological models [6]. A plethora of eco-epidemiological models have been proposed and analyzed with an aim to control the disease in ecological community [7–10]. Several studies have demonstrated that the predators tend to forage for prey having high parasite burden [11–16]. In [14], the impact of preference has been investigated in an eco-epidemiological system. Their results show that for lower values of the force of infection and predator’s reproductive gain, parasites and predators both go to extinction irrespective of predator’s preference. The predator’s preference greatly influences the survival as well as extinction of species. Biswas et al. [15] investigated the selective feeding behavior of zooplankton on phytoplankton in the presence of free-viruses. They observed that on increasing the intensity of preference of zooplankton on infected phytoplankton, the system exhibited transition from stable coexistence to oscillations around coexistence equilibrium

S. S. Maity · S. Pal () Department of Mathematics, University of Kalyani, Kalyani, India P. K. Tiwari Department of Basic Science and Humanities, Indian Institute of Information Technology, Bhagalpur, India N. Das Department of Mathematics, Maulana Azad College, Kolkata, India © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_12

197

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S. S. Maity et al.

to oscillations around disease-free equilibrium. Bairagi et al. [16] showed that the dynamics of phytoplankton-zooplankton interaction strongly depends on the selective predation of zooplankton. Almost every biological situation is mediated by some time delay, which regulate the dynamical stability of the retarded systems [17]. Logistic delay or delay in growth term is the time delay concealed during maturation from juvenile to adult. When immature animals experience environmental stresses such as malnutrition or disease, maturation is delayed until conditions improve and normal growth resume [18–20]. Moreover, a fixed time is elapsed to make a susceptible prey into an infectious one after successful contact between an infective prey and a susceptible prey, known as incubation delay. In [21], authors have investigated the effects of time delays involved in the transmission of disease and the reproduction of predator on consuming prey items. They observed that stability switching occurs when the time delays surpass some critical values. Ghosh et al. [22] studied the effects of logistic delay and gestation delay in an eco-epidemiological system. They observed that increase in gestation delay leads to chaotic oscillations, whereas if the maturation delay exceeds its critical value, the system exhibits limit cycle oscillations. In the present study, our aim is to understand the combined impacts of maturation delay and incubation delay in an eco-epidemiological system with selective predation. We rigorously analyze the delayed and non-delayed systems with mathematical as well as numerical techniques. The results obtained in this study may provide the some better knowledge on species abundance and disease outbreak.

2 The Mathematical Model Let N and P be the prey and predator populations, respectively, at any time .t > 0. Suppose the prey population is infected by some microparasites. So, the prey population is divided into susceptible prey, S, and infected prey, I . The susceptible prey is supposed to reproduce and become infected by the pathogen horizontally. The infected prey is assumed to suffer parasitic effects due to castration, conspicuousness, behavior modification, morbidity, etc., [11, 23]. Thus, the infected prey do not reproduce but, contribute to the carrying capacity and die at a higher rate. The growth of susceptible prey is according to the logistic law with intrinsic growth rate r and carrying capacity K. Further, we consider that the disease is of SI.−type and infection spreads following the mass action law, which is governed as .λSI , where .λ stands for the rate of infection. The predator population predate susceptible prey as well as infected prey. The selectivity/preference of the predators to susceptible and infected preys plays significant role in the system dynamics [13]. Following [14], nS nθ I we consider multiple-prey type II response functions . and . a + S + θI a + S + θI for susceptible and infected preys, respectively. Here, the parameter .θ measures the selectivity/preference of the predator to infected prey over the susceptible one or

An Eco-Epidemic Predator-Prey Model with Selective Predation and Time Delays

199

Fig. 1 Schematic diagram of the systems (1) and (4). Here, the terms in blue and red colors represent the time lags involved in the process of maturation and transmission of disease, respectively

vice versa. If .θ > 1, predators prefer infected preys, .θ = 1 implies equal preference to susceptible and infected preys, and .0 < θ < 1 means that predators prefer susceptible preys. Let d be the death rate of infected prey, which includes the death rate of infected prey by natural mortality and virulence of the disease. Let .γ and .μ be the natural death rate of predator and mortality due to intraspecies competition, respectively. On the basis of the above assumptions, a schematic diagram is depicted in Fig. 1. Thus, we have the following mathematical model:   dS S+I nSP . = rS 1 − − λSI − , dt K a + S + θI nθ I P dI = λSI − − dI, dt a + S + θI dP b2 nθ I P b1 nSP + − γ P − μP 2 . = a + S + θI a + S + θI dt

(1)

System (1) is to be analyzed with the following initial conditions: S(0) = S0 > 0, I (0) = I0 > 0, P (0) = P0 > 0.

.

(2)

Since all the parameters of system (1) are nonnegative, the right hand side is a smooth function of the variables S, I , and P in the positive region . = {(S, I, P ) : S, I, P ∈ R+ }. The biological meanings of parameters involved in system (1) are given in Table 1.

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Table 1 Biological meanings of parameters in the model (1) and their values used for numerical simulations Parameters r K .λ n a .θ d .b1 .b2

.γ .μ

Biological meanings Intrinsic growth rate of susceptible prey Carrying capacity of the environment for the prey Infection rate in prey populations Susceptible prey capture rate Half-saturation constant in prey population Preference of the predator Mortality rate of infected prey Growth in the predator population due to consumption of susceptible prey Growth in the predator population due to consumption of infected prey Natural mortality rate of predator population Mortality rate of predator population due to crowding

Values 1 140 0.011 0.4 15 0.1 0.24 0.15

Units Day.−1 Number Number.−1 day.−1 Day.−1 Number – Day.−1 –

0.015

–

0.002 0.002

Day.−1 Number.−1 day.−1

3 System’s Equilibria and Stability System (1) has the following five biologically feasible equilibria: (1) The population-free equilibrium .E0 = (0, 0, 0), which always exists. (2) The susceptible prey only equilibrium .E1 = (K, 0, 0), which always exists. d and .I2 = (3) The predator-free equilibrium .E2 = (S2 , I2 , 0), where .S2 = λ r(Kλ − d) . The equilibrium .E2 exists provided .Kλ − d > 0. λ(r + Kλ)   1 b1 nS3 −γ (4) The infection-free equilibrium .E3 = (S3 , 0, P3 ), where .P3 = μ a + S3 and .S3 is (are) positive root(s) of the cubic equation .S 3 + A2 S 2 + A1 S − A0 = 1 n−γ ) 0 with .A2 = 2a − K, .A1 = Kn(brμ − (2aK − a 2 ) and .A0 = a 2 K + anKγ . Notably, this cubic rμ   1 arμ(2K−a) + γ . n Kn

equation has a positive root provided .K < 2a, b1 >

B 3 − B1 S ∗ (5) The coexistence equilibrium .E ∗ = (S ∗ , I ∗ , P ∗ ), where .I ∗ = , B2   ∗ ∗ λ r 1 n(b1 S + b2 θ I ) r ∗ − γ , where .B1 = + , B2 = λ + , B3 = .P = ∗ ∗ a + S + θI θ K K μ d ∗ r + , and .S is (are) a positive root(s) of the equation: θ C3 S 3 + C2 S 2 + C1 S − C0 = 0,

.

(3)

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with  C3 = λ 1 +

θ 2 B12 B22

C2 = 2aλ − d − .

2B1 θ − B2

,

θ 2 (2B1 B3 λ + dB12 )

C1 = a(aλ − 2d) +

B22

+

2θ B1 (aλ − d) 2λθ B3 − , B2 B2

(aλ − d) θ θ 2 (2dB1 B3 + λB3 ) + 2θ B3 + 2adB1 2 B2 B2 B2

nθ (b2 n − γ ) nθ (b1 n − γ ) , − μ B2 μ  θ 2 B32 B3 nθ γ a 2 C0 = d a + . + 2aθ − 2 B2 μ B2 + B1 θ

Regarding local stability of equilibria of the system (1), we have the following theorem. Theorem 3.1 (1) The equilibrium .E0 is always unstable. (2) The equilibrium .E1 is stable provided .Kλ − d < 0, b1 nK − γ (a + K) < 0. aγ − (b1 n − γ )S2 . (3) The equilibrium .E2 , if exists, is stable if .θ < θ1 = I2 (b2 n − γ ) (4) The equilibrium .E3 , if exists, is locally asymptotically stable if and only if .r(a + S3 )2 − nKP3 > 0, nθ P3 + (d − λS3 )(a + S3 ) > 0. (5) The equilibrium .E ∗ , if exists, is locally asymptotically stable if and only if .x1 > 0, x3 > 0, x1 x2 − x3 > 0, where .

x1 = V1 − V5 + V9 , x2 = V9 (V1 − V5 ) + V3 V7 + V2 V4 + V6 V8 − V1 V5 , x3 = V2 (V4 V9 − V6 V7 ) + V3 (V4 V8 − V5 V7 ) + V1 (V6 V8 − V5 V9 )

with .

V1 = −

nS ∗ P ∗ rS ∗ nθ S ∗ P ∗ rS ∗ − λS ∗ + + , V2 = − , ∗ ∗ 2 K K (a + S + θ I ) (a + s ∗ + θ I ∗ )2

V3 = −

nθ I ∗ P ∗ nS ∗ , V4 = λI ∗ + , ∗ ∗ a + S + θI (a + S ∗ + θ I ∗ )2

V5 =

nθ I ∗ nθ 2 I ∗ P ∗ , V = − , 6 a + S∗ + θ I ∗ (a + S ∗ + θ I ∗ )2

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V7 =

nθ P ∗ {b2 (a + S ∗ ) − b1 S ∗ } b1 nP ∗ + nθ I ∗ P ∗ (b1 − b2 ) , V = , 8 (a + S ∗ + θ I ∗ )2 (a + S ∗ + θ I ∗ )2

V9 = −μP ∗ . Proof Can be easily proved by using a simple stability analysis technique.

 

4 Effects of Time Delays In this section, we modify model system (1) by considering time delays involved in the process of maturation and transmission of disease. For simplicity, we assume that the time lags involved in the process of maturation for the susceptible and infected preys are equal. Let .τ1 and .τ2 be the time lags involved in the process of maturation and the incubation of the disease. With these modifications, system (1) becomes   nSP S(t − τ1 ) + I (t − τ1 ) dS − λI S − , = rS 1 − . K a + S + θI dt dI nθ I P − dI, = λI (t − τ2 )S(t − τ2 ) − a + S + θI dt dP b1 nSP b2 nθ I P = + − γ P − μP 2 . dt a + S + θI a + S + θI

(4)

Initial conditions for the system (4) take the form S(φ) = ψ1 (φ), I (φ) = ψ2 (φ), P (φ) = ψ3 (φ), −τ ≤ φ ≤ 0,

.

(5)

where .ψ = (ψ1 , ψ2 , ψ3 )T ∈ C+ such that .ψi (φ) ≥ 0, i = 1, 2, 3 ∀ φ ∈ [−τ, 0] and .C+ denotes the Banach space .C+ ([−τ, 0], R3+0 ) of continuous functions mapping the interval .[−τ, 0] into .R3+0 . Denote the norm of an element .ψ in .C+ by . ψ = sup {| ψ1 (φ) |, | ψ2 (φ) |, | ψ3 (φ) |}, where .τ = max{τ1 , τ2 }. For −τ ≤φ≤0

biological feasibility, we further assume that .ψi (0) ≥ 0 for .i = 1 − 3. By the fundamental theory of functional differential equations [24], there is a unique solution .(S(t), I (t), P (t)) to the system (4) with initial conditions (5).

4.1 Stability Analysis in the Presence of Time Delays To study the local stability behavior of the equilibrium .E ∗ in the presence of time delays, we linearize the model system (4) about the equilibrium .E ∗ , and get

An Eco-Epidemic Predator-Prey Model with Selective Predation and Time Delays

.

dY = LY (t) + MY (t − τ1 ) + NY (t − τ2 ), dt

203

(6)

where ⎛

⎞ ⎛ l11 l12 l13 m11 m12 .L = ⎝ l21 l22 l23 ⎠ , M = ⎝ 0 0 l31 l32 l33 0 0

⎞ ⎛ ⎞ ⎞ ⎛ s 0 0 0 0 0 ⎠ , N = ⎝ n21 n22 0 ⎠ , Y (t) = ⎝ i ⎠ 0 0 0 p 0

with .

  nP ∗ (a + θ I ∗ ) S∗ + I ∗ − λI ∗ − l11 = r 1 − , K (a + S ∗ + θ I ∗ )2 l12 = −λS ∗ −

nθ S ∗ P ∗ , (a + S ∗ + θ I ∗ )2

l13 = −

nS ∗ nθ I ∗ P ∗ , , l = 21 a + S∗ + θ I ∗ (a + S ∗ + θ I ∗ )2

l22 = −

nθ P ∗ (a + S ∗ ) − d, (a + S ∗ + θ I ∗ )2

l31 =

b2 nθ I ∗ P ∗ b1 nP ∗ (a + θ I ∗ ) , − (a + S ∗ + θ I ∗ )2 (a + S ∗ + θ I ∗ )2

l32 = −

b1 nθ S ∗ P ∗ b2 nθ P ∗ (a + S ∗ ) + , (a + S ∗ + θ I ∗ )2 (a + S ∗ + θ I ∗ )2

l33 = −μP ∗ , m11 = m12 = −

rS ∗ , n21 = λI ∗ , n22 = λS ∗ . K

Here, s, i, and p are small perturbations around the equilibrium .E ∗ . The characteristic equation for the linearized system (6) is obtained as D(ξ, τ1 , τ2 ) ≡ P (ξ ) + Q(ξ )e−ξ τ1 + R(ξ )e−ξ τ2 + T (ξ )e−ξ(τ1 +τ2 ) = 0,

.

where .P (ξ ), .Q(ξ ), .R(ξ ) and .T (ξ ) are the polynomials in .ξ and are given by .

P (ξ ) = ξ 3 + A2 ξ 2 + A1 ξ + A0 , Q(ξ ) = B 2 ξ 2 + B 1 ξ + B 0 , R(ξ ) = C 2 ξ 2 + C 1 ξ + C 0 , T (ξ ) = D 1 ξ + D 0

with .

A2 = −(l11 + l22 + l33 ), A1 = l11 (l22 + l33 ) + (l33 l22 − l23 l32 ) − (l21 l12 + l31 l13 ), A0 = l12 (l33 l21 − l23 l31 ) − l11 (l22 l33 − l23 l32 ) − l13 (l32 l21 − l22 l31 ),

(7)

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B 2 = −m11 , B 1 = m11 (l22 + l33 ) − m12 l21 , B 0 = m12 (l33 l21 − l23 l31 ) − m11 (l22 l33 − l23 l32 ), C 2 = −n22 , C 1 = l11 n22 + l33 n22 − l12 n21 , C 0 = l33 (l12 n21 − l11 n22 ) − l13 (l32 n21 − l31 n22 ), D 1 = m11 n22 − m12 n21 , D 0 = l33 (m12 n21 − m11 n22 ). The local stability behavior of the equilibrium .E ∗ is discussed in the following three different cases. Case I .τ1 = τ2 = 0. In the absence of time delays, the stability behavior of the equilibrium .E ∗ has already been discussed in Theorem 3.1. Case II .τ1 > 0 and .τ2 = 0. In this case, the characteristic equation (7) becomes D(ξ, τ1 ) ≡ ξ 3 + (A2 + C 2 )ξ 2 + (A1 + C 1 )ξ + A0 + C 0

.

+{B 2 ξ 2 + (B 1 + D 1 )ξ + B 0 + D 0 }e−ξ τ1 = 0, which can be rewritten as ξ 3 + b2 ξ 2 + b1 ξ + b0 + (d 2 ξ 2 + d 1 ξ + d 0 )e−ξ τ1 = 0,

(8)

.

where .b2 = A2 + C 2 , b1 = A1 + C 1 , b0 = A0 + C 0 , d 2 = B 2 , d 1 = B 1 + D 1 , d 0 = B 0 + D 0 . For .τ1 = 0, Theorem 3.1 provides the conditions under which all the roots of Eq. (8) are either negative or have negative real parts, while for .τ1 > 0, Eq. (8) has infinitely many roots. By Rouche’s Theorem and continuity in .τ1 , sign of roots of Eq. (8) will change if it crosses imaginary axis, i.e., if Eq. (8) has purely imaginary roots. Hence, putting .ξ = iω (.ω > 0) in Eq. (8) and separating real and imaginary parts, we get .

ω3 − b1 ω = d 1 ω cos(ωτ1 ) − (d 0 − d 2 ω2 ) sin(ωτ1 ), .

(9)

b2 ω − b0 = (d 0 − d 2 ω ) cos(ωτ1 ) + d 1 ω sin(ωτ1 ).

(10)

2

2

Squaring and adding Eqs. (9) and (10), we get 2

(ω3 − b1 ω)2 + (b2 ω2 − b0 )2 = (d 0 − d 2 ω2 )2 + d 1 ω2 .

(11)

.

Simplifying Eq. (11) and substituting .ω2 = ψ, we get the following equation: (ψ) = ψ 3 + a 2 ψ 2 + a 1 ψ + a 0 = 0,

(12)

.

2

2

2

2

2

2

where .a 2 = b2 − 2b1 − d 2 , a 1 = b1 − 2b0 b2 + 2d 0 d 2 − d 1 , a 0 = b0 − d 0 . We discuss about the existence of positive root(s) of Eq. (12) in the following lemma.

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Lemma 4.1 The polynomial Equation (12) has at least one positive root provided a 2 < 0, a 1 > 0, a 0 < 0 and exactly one positive root if either .a 2 < 0, a 1 < 0, a 0 < 0 or a 2 > 0, a 1 < 0, a 0 < 0 or a 2 > 0, a 1 > 0, a 0 < 0. For other choices of the coefficients of Eq. (12), the existence of positive root cannot be guaranteed.

.

Now, we have the following theorem. Theorem 4.1 Suppose that the equilibrium .E ∗ exists and is locally asymptotically 2 be a positive root of Eq. (12). Then, stable for .τ1 = τ2 = 0. Also, let .ψ0 = ω10 0 there exists .τ1 = τ1 such that the equilibrium .E ∗ of the system (4) is asymptotically stable when .0 ≤ τ1 < τ10 and unstable for .τ1 > τ10 , where k .τ1

  3 2 2 1 kπ −1 d 1 ω10 (b0 − b2 ω10 ) + (d 0 − d 2 ω10 )(ω10 − b1 ω10 ) tan = + 3 2 2 ω10 ω10 d 1 ω10 (ω10 − b1 ω10 ) − (d 0 − d 2 ω10 )(b0 − b2 ω10 )

for .k = 0, 1, 2, 3 · · · . Further, the system (4) will undergo a Hopf bifurcation at the 2 ) > 0. equilibrium .E ∗ when .τ1 = τ10 provided .  (ω10 2 is a solution of Eq. (12), the characteristic Equation (8) has Proof Since .ψ = ω10 2 . It follows from Eqs. (9) and (10) that .τ k a pair of purely imaginary roots .±iω10 1 2 is a function of .ω10 for .k = 0, 1, 2, · · · . Therefore, the system (4) will be locally asymptotically stable around the equilibrium .E ∗ for .τ1 = τ2 = 0, if the conditions stated in Theorem 3.1 hold. In that case, by Butler’s lemma, the equilibrium .E ∗ will remain stable for .τ < τ10 , where .τ10 = min τ1k , and the equilibrium .E ∗ will be k≥0  d(Re(ξ )) 0 = 0. Differentiating Equation (8) unstable for .τ1 ≥ τ1 provided .sgn dτ1 0 τ1 =τ1

with respect to .τ1 , we get .

dξ ξ(d 2 ξ 2 + d 1 ξ + d 0 )e−ξ τ1 = . dτ1 3ξ 2 + 2b2 ξ + b1 + (2d 2 ξ + d 1 )e−ξ τ1 − τ1 (d 2 ξ 2 + d 1 ξ + d 0 )e−ξ τ1 

This gives .  .

sgn

dξ dτ1

−1

=



3ξ 2 + 2b2 ξ + b1 + (2d 2 ξ + d 1 )e−ξ τ1

d(Re(ξ )) dτ1 τ1 =τ10



= sgn

ξ(d 2 = sgn



ξ2

+ d 1ξ + d 0

−1 d(Re(ξ )) dτ1 τ1 =τ10

4 +2a ω2 +a 3ω10 1 2 10 2 2 2 )2 d 1 ω10 +(d 0 −d 2 ω10



= sgn

)e−ξ τ1

−

τ1 . Now, ξ

    −1 dξ = sgn Re dτ 1 

2 )   (ω10 2 2 2 ) d 1 ω10 +(d 0 −d 2 ω10

 . 2

ξ =iω10

2 ) = 0 if Lemma 4.1 holds. Hence, the transversality It may be noted that .  (ω10 condition is satisfied. Thus, a Hopf bifurcation occurs at .τ1 = τ10 i.e., a family of

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periodic solutions bifurcate from the equilibrium .E ∗ as .τ1 passes through .τ10 [25].   Case III .τ1 = 0 and .τ2 > 0. In such a case, the characteristic Equation (7) takes the form D(ξ, τ2 ) ≡ ξ 3 + (A2 + B 2 )ξ 2 + (A1 + B 1 )ξ + A0 + B 0

.

+{C 2 ξ 2 + (C 1 + D 1 )ξ + C 0 + D 0 }e−ξ τ2 = 0, which can be rewritten as ξ3 +  b2 ξ 2 +  b1 ξ +  b0 + (d2 ξ 2 + d1 ξ + d0 )e−ξ τ2 = 0,

.

(13)

b1 = A1 +B 1 ,  b0 = A0 +B 0 , d2 = C 2 , d1 = C 1 +D 1 , d0 = where . b2 = A2 +B 2 ,  C 0 + D 0 . Following the similar analysis as in Case II, we can state the following theorem. Theorem 4.2 Suppose that the equilibrium .E ∗ exists and is locally asymptotically 2 be a pair of purely imaginary roots of stable for .τ1 = τ2 = 0. Also, let .±iω20 0 Eq. (13). Then, there exists .τ2 = τ2 such that the equilibrium .E ∗ of the system (4) is asymptotically stable when .0 ≤ τ2 < τ20 and unstable for .τ2 > τ20 , where   2 ) + (d  0 − d2 ω2 )(ω3 +  1 ω20 ( b − b ω b ω ) d kπ 1 0 2 1 20 20 20 20 k + tan−1 , .τ2 = 3 2 2       ω20 ω (d0 − d2 ω20 )(b0 − b2 ω20 ) + d1 ω20 (ω20 + b1 ω10 ) 20 for .k = 0, 1, 2, 3 · · · . Moreover, the system (4)will undergo  a Hopf bifurcation at d(Re(ξ )) 0 ∗ = 0. the equilibrium .E when .τ2 = τ2 provided .sgn dτ2 τ2 =τ 0 2

Proof Proof is similar to the Case II.

 

5 Numerical Simulations Here, we conducted some numerical computations of the systems (1) and (4) by choosing important parameters values given in Table 1. Unless it is mentioned, the parameter values will be same as in the Table 1.

5.1 Simulation Results of System (1) We see time series behaviors of system (1) by changing some of the parameters. We observe that the system (1) shows stable focus for the parameter values in

An Eco-Epidemic Predator-Prey Model with Selective Predation and Time Delays 55

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Fig. 2 System (1) shows (a) stable interior, (b) unstable interior at n = 0.9, (c) unstable diseasefree at n = 0.9 and θ = 0.5, (d) stable predator-free at n = 0.9, θ = 0.5 and γ = 0.2, and (e) susceptible only equilibrium at n = 0.9, θ = 0.5, γ = 0.2 and λ = 0.0011. Other parameters are at the same values as in Table 1

Table 1, Fig. 2a. On increasing the value of n from .0.4 to .0.9, the system exhibits limit cycle oscillations, Fig. 2b. But, at .n = 0.9, on increasing the value of .θ from .0.1 to .0.5, we observe that the infected prey population extinct from the system, and the system shows limit cycle oscillations around the infection-free equilibrium (see Fig. 2c). Further, we fix .n = 0.9 and .θ = 0.5 and increase the value of .γ from .0.002 to .0.2. We find that the predator population disappear and the system settles to stable predator-free equilibrium, Fig. 2d. That is, on increasing the natural mortality rate of predator, the system’s dynamics changes from oscillations around infection-free equilibrium to stable predator-free equilibrium. Now, we fix .n = 0.9, .θ = 0.5, and .γ = 0.2 at which the system is stable at predator-free equilibrium and decrease the rate of infection .λ from .0.011 to .0.0011. For this parametric setup, both infected prey and predator population extinct from the system and the system

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Fig. 3 System (1) shows (a) stable interior at .n = 0.9 and .μ = 0.02, (b) unstable interior at = 0.3, (c) unstable interior at .b2 = 0.95, (d) chaos at .μ = 0.0002, and (e) unstable predatorfree at .n = 0.9 and .λ = 3. Other parameters are at the same values as in Table 1

.b1

is stable at susceptible prey only equilibrium, Fig. 2e. Thus, by lowering the rate of infection and increasing the preference of selectivity and natural mortality of predator population, the infection dies out and also the predator extinct leaving the system stable with susceptible prey only. Note that at .n = 0.9 and .μ = 0.002, the system (1) shows limit cycle oscillations around the coexistence equilibrium (see Fig. 2b), but on increasing the value of .μ from .0.002 to .0.02, the system shows stable coexistence of prey and predator, Fig. 3a. On increasing the value of .b1 from .0.15 to .0.3, the dynamics of system (1) changes from stable coexistence to limit cycle oscillations around the coexistence equilibrium, Fig. 3b. Similarly, on increasing the value of .b2 from .0.015 to .0.95, the coexistence equilibrium becomes unstable and limit cycle oscillations appear around the coexistence equilibrium, Fig. 3c. These indicate that if the reproductive

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.μ.

gains in predator population on consumptions of susceptible and infected prey are very high, the system becomes unstable around the coexistence equilibrium. Further, we note that the system shows chaotic dynamics on decreasing the value of .μ from .0.002 to .0.0002, Fig. 3d. The occurrence of chaotic oscillation may be explained through incommensurate limit cycles around the coexistence equilibrium [26, 27]. Thus, for very lower values of predator death due to intraspecies competition, the system enters into chaotic regimes. Again, we keep the system in oscillatory state, i.e., we choose .n = 0.9 and increase the value of .λ from .0.011 to 3. We note that for this large value of infection rate, the predator population extinct and the system oscillates around predator-free equilibrium. Next, we draw bifurcation diagrams of the system (1) by varying some of the key parameters: .θ , n, .b1 , .b2 , and .μ, Fig. 4. We see that the coexistence equilibrium is

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unstable for lower values of .θ , but at a threshold value of .θ , the system undergoes a Hopf bifurcation, and the system settles to stable coexistence, Fig. 4a. Thus, the parameter .θ has stabilizing effect on the dynamics of system (1). The system is stable at the coexistence equilibrium for lower ranges of n and undergoes a Hopf bifurcation as the value of n crosses a critical threshold, Fig. 4b. On further increasing the values of n, the infected prey extinct from the system after critical value of n. Thus, the parameter n destabilizes the system through limit cycle oscillations, and an unstable infection-free equilibrium appears for larger values of n. The parameters .b1 and .b2 , representing growths in predator population due to consumptions of susceptible and infected prey, respectively, destabilize the system at the coexistence equilibrium through Hopf bifurcations (see Fig. 4c and d). On increasing the death of predator population due to intraspecies competition, the oscillations are killed out, and the system settles to stable coexistence, Fig. 4e.

5.2 Simulation Results of Delayed System (4) Now, we see how presence of time delays regulate the dynamics of system (4), Fig. 5. We use the parameter values of Table 1 at which the system is at stable coexistence in the absence of both time delays. First, we keep .τ2 = 0 and gradually increase the values of .τ1 . We see that at .τ1 = 1.49, the system shows stable focus at the coexistence equilibrium, Fig. 5a. On increasing the value of .τ1 to .τ1 = 1.6, we find that the system exhibits limit cycle oscillations around the coexistence equilibrium, Fig. 5b. However, at .τ1 = 2.5, the system generates period doubling oscillations around the coexistence equilibrium, Fig. 5c. Thus, for .τ2 = 0, the dynamics of delayed system (4) changes from stable coexistence to limit cycle oscillation around the coexistence equilibrium to period doubling oscillation on gradual increase in the values of delay parameter .τ1 . Next, we fix .τ1 = 0 and see the effect of .τ2 on the dynamical behavior of system (4). We see that the system shows limit cycle oscillations around the coexistence equilibrium at .τ2 = 3.5, Fig. 5d. Now, we increase the value of .μ from .0.002 to .0.02 and also increase the value of .τ2 to .τ2 = 35 and see that the system shows double periodic oscillations, Fig. 5e. To get clearer view on the effects of time delays and change in their critical values on varying other parameters of the system, we draw some two parameters bifurcation diagrams of the system (4), Fig. 6. In the figures, .∗ represents stable coexistence region of the system (4), whereas .∗ corresponds to unstable domain for the coexistence equilibrium. First, we draw the bifurcation diagram by varying the delay parameters. From Fig. 6a, we see that on increasing the values of either .τ1 or .τ2 , the system loses its stability at the coexistence equilibrium and the coexistence equilibrium becomes unstable; regions of instability for the coexistence equilibrium increase on the diagonal. Next, we couple the delay parameters with the preference of selectivity. Up to a certain range of .τ1 , the system is stable at the coexistence equilibrium for all values of values of .θ ; after a critical value of .τ1 , the coexistence equilibrium remains unstable for all .θ . We find that there is a range of .τ2 such that if

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Fig. 5 Simulation results of system (4) for (a) τ1 = 1.49 and τ2 = 0, (b) τ1 = 1.6 and τ2 = 0, (c) τ1 = 2.5 and τ2 = 0, (d) τ1 = 0 and τ2 = 3.5, and (e) τ1 = 0 and τ2 = 35. Parameters are at the same values as in Table 1 except in (a)–(d) K = 180 and μ = 0.002, and (e) μ = 0.02

the coexistence equilibrium is unstable due to .τ2 , then the coexistence equilibrium gets stabilized on increasing the values of .θ . Similarly, there are certain ranges of .τ1 and .τ2 for which the system dynamics changes from unstable coexistence to stable coexistence on increasing the values of .μ. That is, increasing the death of predator population due to intraspecies competition stabilizes an otherwise unstable coexistence equilibrium. Further, we observe that on coupling n with .τ1 or .τ2 , the system is stable at the coexistence equilibrium only for lower values of n and becomes unstable after threshold values of n. On the other hand, the stable coexistence equilibrium becomes unstable for increasing the values of either .τ1 (see Fig. 6f) or .τ2 (see Fig. 6g). For a certain range of .τ1 , stability switches occur on increasing the values of .λ; the coexistence equilibrium is unstable for lower and higher values of .λ, but stable coexistence is achieved for moderate values of .λ. For

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Fig. 6 Bifurcation diagrams of the system (4) in (a) .τ1 − τ2 , (b) .θ − τ1 , (c) .θ − τ2 , (d) .μ − τ1 , (e) − τ2 , (f) .n − τ1 , (g) .n − τ2 , (h) .λ − τ1 , and (i) .λ − τ2 planes. Parameters are at the same values as in Table 1, and .K = 180 and .μ = 0.002. Here, blue asterisks and red asterisks represent stable and unstable regions for the coexistence equilibrium .E ∗

.μ

very lower values of .τ1 or .τ2 , the system is stable at the coexistence equilibrium for all values of .λ. There is a range of .λ for which the system is stable at the coexistence equilibrium for all values of .τ2 . On the other hand, there is a range of .τ2 for which the system changes its dynamics from stable coexistence to unstable coexistence on increasing the values of .λ.

6 Results and Discussion Here, we have investigated dynamical behaviors of a predator-prey system with disease in the prey population only. We assumed that the predators feed upon both the susceptible and infected preys with some preference to infected over susceptible prey or vice versa. Further, we have considered death in the predator population due to intraspecies competition apart from the natural death. The numerical results showed that disease can be controlled by lowering its transmission rate. For higher values of preference of predator on infected prey, the infected prey may extinct

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from the system. Increasing predation rate destabilizes the system through limit cycle oscillation around the coexistence equilibrium. However, the oscillations may be terminated, and the system becomes stable at the coexistence equilibrium with the increment in preference of predation of infected prey. Moreover, the system oscillates around infection-free equilibrium on higher preference of predation on infected prey. We observed that for the lower ranges of intraspecies competition among predator population, the system behaves chaotically but, settles at the stable coexistence if the intraspecies competition is very high. Predator population can extinct from the system if the rate of infection and rate of predation on infected prey are sufficiently larger. Further, we consider the effects of maturation and incubation delays in the system. We found that the system can be destabilized on increasing the values of either of the time delays. Double periodic solutions are observed for certain ranges of time delays. We saw that for the lower ranges of time delays, the system is stable at the coexistence equilibrium for all values of selectivity parameter, but the coexistence equilibrium becomes unstable on increasing the predation rate. If the rate of infection is low, then the system is stable at the coexistence equilibrium for all values of incubation delay, while the coexistence equilibrium becomes unstable on increasing the infection rate. On the other hand, if the logistic delay is small, the system is stable at the coexistence equilibrium irrespective of the values of infection rate. But, for certain ranges of logistic delay, the system dynamics changes from unstable coexistence to stable coexistence to unstable coexistence forever. The intraspecies competitions do not alter the stability behavior of system for lower values of time delay; instability occurs only for higher values of time delay. If the delay parameters are at higher values, the system remains unstable even if the intraspecies competition is very large. Compliance with Ethical Standards Conflicts of Interest The authors declare that there is no conflict of interests regarding the publication of this article. Ethical Standard The authors state that this research complies with ethical standards. This research does not involve either human participants or animals.

Acknowledgment The research of Samares Pal is partially supported by Science and Engineering Research Board, Government of India (Grant No. CRG/2019/003248).

References 1. C. MacNeil, J.T. Dick, M.J. Hatcher, R.S. Terry, J.E. Smith, A.M. Dunn, Parasite-mediated predation between native and invasive amphipods, Proc. R. Soc. Lond. B 270 (2003) 1309– 1314. 2. J. Chattopadhyay, N. Bairagi, Pelicans at risk in Salton Sea - an eco-epidemiological model, Ecol. Model. 136 (2001) 103–112.

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3. Y. Xiao, L. Chen, Modelling and analysis of a predator-prey model with disease in the prey, Math. Biosci. 171 (2001) 59–82. 4. E. Venturino, Epidemics in predator-prey models: disease in the predators, IMA J. Math. Appl. Med. Biol. 19 (2002) 185–205. 5. R.M. Anderson, R.M. May, The invasion, persistence, and spread of infectious diseases within animal and plant communities, Philos. Trans. R. Soc. Lond. B 314 (1986) 533–570. 6. K. Hadeler, H.I. Freedman, Predator-prey population with parasite infection, J. Math. Biol. 27(6) (1989) 609–631. 7. O. Arino, J. Mikram, J. Chattopadhyay, Infection on prey population may act as a biological control in ratio-dependent predator-prey models, Nonlinearity 17 (2004) 1101–1116. 8. D. Greenhalgh, M. Haque, A predator-prey model with disease in the prey species only, Math. Meth. Appl. Sci. 30 (2006) 911–929. 9. F. Hilker, K. Schmitz, Disease-induced stabilization of predator-prey oscillations, J. Theo. Biol. 255 (2008) 299–306. 10. B.W. Kooi, G.A. van Voorn, K.P. Das, Stabilization and complex dynamics in a predator-prey model with predator suffering from an infectious disease, Ecol. Compl. 8 (2011) 113–122. 11. K.D. Lafferty, A.K. Morris, Altered behaviour of parasitized killfish increases susceptibility to predation by bird final hosts, Ecology 77 (1996) 1390–1397. 12. W.R. DeMott, Optimal foraging theory as a predictor of chemically mediated food selection by suspension-feeding copepods, Limnol. Oceanogr. 34(1) (1989) 140–154. 13. S.R. Hall, M.A. Duffy, C.E. Caceres, Selective predation and productivity jointly drive complex behavior in host-parasite systems, Am. Nat. 165 (2005) 70–81. 14. N. Bairagi, D. Adak, Complex dynamics of a predator-prey-parasite system: An interplay among infection rate, predator’s reproductive gain and preference, Ecol. Compl. 22 (2015) 1–12. 15. S. Biswas, P.K. Tiwari, Y. Kang, S. Pal, Effects of zooplankton selectivity on phytoplankton in an ecosystem affected by free-viruses and environmental toxins, Math. Biosci. Eng. 17(2) (2020) 1272–1317. 16. N. Bairagi, S. Saha, S. Chaudhuri, S.K. Dana, Zooplankton selectivity and nutritional value of phytoplankton influences a rich variety of dynamics in a plankton population model, Phy. Rev. E 99 (2019) 012406. 17. S. Samanta, P.K. Tiwari, A.K. Alzahrani, A.S. Alshomrani, Chaos in a nonautonomous ecoepidemiological model with delay, Appl. Math. Model. 79 (2020) 865–880. 18. J.M. Tennessen, C.S. Thummel, Coordinating growth and maturation-insights from drosophila, Curr. Biol. 21 (2011) R750–R757. 19. G.E. Hutchinson, Circular casual systems in ecolgy, Ann. NY Acad. Sci. 50 (1948) 221–240. 20. G.E. Hutchinson, An introduction to population ecology, Princeton University Press, Princeton, (1978). 21. D. Adak, N. Bairagi, Stability and Hopf bifurcation in a multi-delayed eco-epidemiological model, Mathematical Biology and Biological Physics 72 (2017). 22. K. Ghosh, S. Biswas, S. Samanta, P.K. Tiwari, A.S. Alshomrani, J. Chattopadhyay, Effect of multiple delays in an eco-epidemiological model with strong Allee effect, Int. J. Bifur. Chaos 27(11) (2017) 1750167. 23. K.D. Lafferty, Foraging on prey that are modified by parasites, Am. Nat. 140 (1992) 854–867. 24. J.K. Hale, Functional differential equations (Springer Berlin Heidelbergi), (1971). 25. K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics. Mathematics and its applications, 74, Kluwer Academic Pub. Dordrecht (1992). 26. Hastings, A., Powell, T.: Chaos in a three-species food chain. Ecology 72(3), 896–903 (1991) 27. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer Science & Business Media, 42 (2013)

Epidemic Patterns of Emerging Variants with Dynamical Social Distancing Golsa Sayyar and Gergely Röst

1 Introduction COVID-19 is a highly contagious viral disease caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The disease quickly spread worldwide, resulting in the COVID-19 pandemic that has been declared the most significant global health crisis since the influenza pandemic of .1918 [1]. Since it was initially detected in late December 2019 in Wuhan, Hubei Province, China, as of October 2022, more than 600 million confirmed cases, including over 6.5 million deaths, have been reported on the World Health Organization (WHO) coronavirus (COVID-19) dashboard. Mathematical models have been commonly applied throughout the pandemic to inform decision makers and help the pandemic response. In the early phase, the potential for global spread was the main concern [2, 3]. Control measures, including social distancing measures that reflect a strong effort to suppress or at least slow down the spread of the virus, began in mid-March 2022 in most European countries. When countries introduced non-pharmaceutical measures, mathematical models were used to evaluate the various disease control strategies [4, 5]. Compartmental models describing the transmission dynamics of the infection have been extended by considering social distancing measures as manipulable control inputs to devise intervention strategies with optimal timing and intensity [6]. Mathematical models were also useful to give insights about the potential and optimization of testing

G. Sayyar Bolyai Institute, University of Szeged, Szeged, Hungary e-mail: [email protected] G. Röst () National Laboratory for Health Security, Bolyai Institute, University of Szeged, Szeged, Hungary e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_13

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strategies [7], including group testing [8]. The impact of vaccination and the waning and boosting of immunity have also been evaluated by models [9, 10] With the emergence of new strains of the virus, researchers considered the spread of multiple lineages [11], the potential for new waves [12], and the adaptation of test and trace strategies [13]. Since the onset of the SARS-CoV-2 pandemic, multiple new variants of concern (VOC) have emerged as follows [14–18]: • Alpha (B.1.1.7): It was first detected in the United Kingdom in late December 2020. This variant contains several key mutations in the spike protein that distinguish it from the original Wuhan strain. The WHO monitored the spread of Alpha variants containing an additional E484K mutation, which may help the virus to escape the body’s immune defenses by evading neutralizing antibodies generated through vaccination or previous infection. • Beta (B.1.351): This variant first reported in South Africa in December 2020. It was more transmissible than previous variants and was considered a concerning variant in terms of reducing neutralization by antibodies generated through previous infection, as well as vaccine efficacy. It implies that people who have already recovered from COVID-19 are at risk of being reinfected, or vaccination may be less effective against it. • Gamma (P.1): It was detected in Brazil in early January 2021. This variant is associated with increased transmissibility due to its ability to evade humoral immunity and cause reinfections. It was estimated to result in virus levels 3–4 times higher than earlier variants and responsible for 1.1–1.8 times more deaths. • Delta (B.1.617.2): This variant was first reported in India in December 2020. It is estimated to be more contagious than the Alpha variant and roughly twice as transmissible as the original Wuhan strain of SARS-CoV-2. Epidemiological studies have highlighted that mutations in the spike of the Delta variant may increase infectivity and reduce neutralization to sera from individuals infected with prior variants. These escape mutations are implicated in reinfection, but the observed reduction in effectiveness of immunization has been modest, with continued strong protection against hospitalization, severe disease, and death. • Omicron (B.1.1.529): It was initially detected in South Africa in November 2021. It has a large number of mutations, including in the receptor-binding domain of the spike protein which causes a 13-fold increase in viral infectivity and is 2.8 times more infectious than the Delta variant. As these variants circulated around the globe, a central topic of discussion emerged about the future directions of the evolution of the virus [19]. A motivation of our work is the paper by G. Lobinska et al. [20], where the authors studied the evolution of resistance to COVID-19 vaccination in presence of social distancing. They have derived a formula for the probability of the emergence of vaccine resistance over time for a model with two strains, WT (wild-type virus) and vaccineresistant mutant virus (MT). In their simulations the social activity level (contact numbers) is adjusted such that the number of infected individuals remains constant in time (i.e., the effective reproduction number is modulated to one). They found that

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under slow vaccination, resistance is more likely to emerge even if social distancing is maintained, while in the case of rapid vaccination, the emergence of mutants can be prevented if social distancing is observed during vaccination. Here we construct a somewhat similar model but one that includes the emergence of n (.n ∈ N) new strains via mutations. Our focus is to investigate the emerging patterns with specific cross-immunities between the strains, when disease spread is constrained by dynamically changing social distancing. This work is outlined as follows: Sect. 2 introduces our general model. Then two different scenarios are proposed for this model. The one-way cross-immunity scenario is considered in Sect. 3. In this scenario, recovered individuals gain immunity against the strain that infected them and all older strains, but they are still susceptible to newly emerging strains. We compute the time varying effective reproduction number for each strain (Sect. 3.3) and discuss the patterns of consecutive wave obtained numerical simulations. In Sect. 4 the second scenario is introduced and compared with the first one. Here recovered individuals gain immunity only against the one strain that infected them, and they are assumed to be susceptible to both new and previous strains. This situation is leading to more complex dynamics compared to the first scenario. Finally, we compare our results with the collected data on COVID-19 variants in the Netherlands in Sect. 5.

2 Model Description We construct a compartmental model to describe a general model of an infectious disease with multiple variants. In our model, the population N is divided into the following three main classes, tracking the disease status of individuals: S denotes susceptible individuals, i.e., those who can be infected by the disease, I denotes infected individuals, and R represents the population of recovered individuals. This classical SI R-setting is extended now by multiple strains. We use the notation .Ik for the class of individuals who are infected by strain k (.k = 1, 2, . . . , n). Upon recovery, individuals move to compartment .Rk . As a simplification, for those who went through more than one infections, possibly by different strains, their immune status is associated to the last recovery. Hence, individuals in .Rk are those who recovered from infection k, and they all have the same type of immunity, irrespective of they might have been infected in the past by some other strains. The model neglects any changes in the population due to birth, death, or migration during the period of consideration. Thus, the total population (N ) would be invariant, and for all t, S(t) +

n 

.

k=1

(Ik (t) + Rk (t)) = N.

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We introduce the general cross-immunity matrix .C = ci,j (i, j = 1, 2, ..., n) as the relative immunity to strain i for an individual that has been last recovered from strain j and thus being in .Rj . With this notation, .ci,j = 0 denotes full immunity of .Rj individuals to strain i, and .ci,j = 1 means no immunity at all (full susceptibility) to the strain. If the value of .ci,j is between 0 and 1, then there is partial immunity to strain i. In the model equations, the coefficients .ci,j will appear as a reduction factor in the transmission rate between compartments .Rj and .Ii . The strains are assumed to share the same epidemiological parameters (transmission and recovery rates); they only differ in terms of population immunity against them. The proposed model is governed by the following system of differential equations: ˙ = −βσ (t)S(t) S(t)

n 

Ii (t),

i=1

I˙1 (t) = βσ (t)(1 − μ)S(t)I1 (t) + βσ (t)(1 − μ)I1 (t) R˙ 1 (t) = aI1 (t) − βσ (t)R1 (t)

n 

n 

c1,i Ri (t) − aI1 (t),

i=1

ci,1 Ii (t),

i=1

.. . I˙k (t) = βσ (t)(1 − μ)S(t)Ik (t) + βσ (t)(μ)S(t)Ik−1 (t) − aIk (t) n n   +βσ (t)(1 − μ)Ik (t) ck,i Ri (t) + βσ (t)(μ)Ik−1 (t) ck−1,i Ri (t) . i=1 n 

R˙ k (t) = aIk (t) − βσ (t)Rk (t)

i=1

ci,k Ii (t),

i=1

.. . I˙n (t) = βσ (t)μS(t)In−1 (t) + βσ (t)S(t)In (t) − aIn (t) n n   +βσ (t)In (t) cn,i Ri (t) + βσ (t)(μ)In−1 (t) cn−1,i Ri (t) i=1

R˙ n (t) = aIn (t) − βσ (t)Rn (t)

n 

i=1

ci,n Ii (t),

i=1

(1) where .k = 2, 3, ..., n − 1. In the above system, parameters .β and a are the average transmission rate and recovery rates, respectively. In this model a mutation occurs (with a small mutation probability .μ) when exposure to an infected individual with strain i results in an infected individual transmitting further strain .i + 1. Hence, mutation is assumed to be sequential, always producing the next variant, where .i = 1, 2, . . . , n. Moreover, recovered individuals from strain i are fully protected from .Ii (those infected with strain i) and susceptible or immune to others as given by the elements of matrix C. In System (1), parameter .σ = σ (t) is a social distancing measure that varies over time and ranges in .[0, 1], representing the reduction in contacts between individuals

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Table 1 Parameters and values applied in the simulations. The parameters are set to give .R0 = 3 Parameter .β

a .μ .σ

N L

Interpretation Transmission rate Recovery rate (day.−1 ) Mutation probability Social distancing parameter (reduction in contacts) Total population Number of daily new infected (allowed incidence)

Values × 10−7 .0.25 −6 .10 .[0, 1] 6 .10 .900, 1500, 3500 .7.5

compare to baseline. Here, .σ = 1 indicates no social distancing, and .σ = 0 means complete lockdown. Social distancing has a key role in our model since it is a control measure to keep the infected population within an acceptable level. Denoting by L this daily new allowed infection threshold, .σ (t) is adjusted such that the infectious population does not exceed .L/a (see Table 1 for parameter references). In the following, we focus on two scenarios for the matrix C: .(i) if recovery from strain i provides immunity against old strains, then C is a triangular matrix (discussed in Sect. 3), and .(ii) if recovery from strain i provides immunity only against i, but such individuals remain susceptible to both earlier and later strains, then C is a diagonal matrix (discussed in Sect. 4).

3 Scenario One: One-Way Cross-Immunity Toward Earlier Variants 3.1 Model Equations and the Cross-Immunity Matrix In this case, recovered individuals from strain .i (i = 1, 2, ..., n) are fully immune to any strain j , where .j ≤ i, but have no immunity to the upcoming strains j , where .j > i. The cross-immunity matrix below, C, represents the above description. For instance, it could be understood that the recovered individual from strain three is immune to strain three and the previous strains as well, (.c3,j = 0, .j = 1, 2, 3), but they are still susceptible to the new strains, therefore can be infected by them with the same rate as susceptible individuals (.c3,j = 1, .j = 4, ..., n). Hence,

Cn,n

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⎛ 0 ⎜0 ⎜ ⎜ = ⎜0 ⎜. ⎝ ..

1 0 0 .. .

1 ··· 1 ··· 0 ··· .. . . . .

⎞ 1 1⎟ ⎟ 1⎟ ⎟ .. ⎟ .⎠

0 0 0 ··· 0 is an upper triangular cross-immunity matrix, with zeros in the diagonal as well.

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In order to derive the model equations, keep in mind a series of assumptions: • Each strain can infect susceptible individuals S. • Recovered individuals from strain i are susceptible to strain .j, j > i, and immune to previous ones. • Individuals infected by strain i can transmit strain .i + 1 due to mutation with mutation probability .μ. Based on these assumptions, the system of differential equations for the model is given by ˙ = −βσ (t)S(t) S(t)

n 

Ii (t),

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I˙1 (t) = βσ (t)(1 − μ)S(t)I1 (t) − aI1 (t), n  Ii (t), R˙ 1 (t) = aI1 (t) − βσ (t)R1 (t) i=2

.. . I˙k (t) = βσ (t)(1 − μ)S(t)Ik (t) + βσ (t)(μ)S(t)Ik−1 (t) − aIk (t) k−1 k−2   Ri (t) + βσ (t)(μ)Ik−1 (t) Ri (t), +βσ (t)(1 − μ)Ik (t) . i=1

R˙ k (t) = aIk (t) − βσ (t)Rk (t)

n 

(2)

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Ii (t),

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.. . I˙n (t) = βσ (t)μS(t)In−1 (t) + βσ (t)S(t)In (t) − aIn (t) n−1 n−2   Ri (t) + βσ (t)(μ)In−1 (t) Ri (t) +βσ (t)In (t) R˙ n (t) = aIn (t),

i=1

i=1

where .k = 2, 3, ..., n − 1.

3.2 Epidemiological Dynamics We have numerically solved System (2), and Fig. 1 shows the number of infected individuals in time for each strain. Initially, we consider 1000 infected individuals by strain 1, while the whole population is considered .N = 106 . In the early stage of the epidemic, there is no social distancing, therefore, .σ = 1. Then, when the infected population reach the level .L/a (corresponding to daily incidence L), we apply social distancing to the population and set it so that the infected population remains this n  Ij = L/a. Under this assumption, the effective reproduction fixed number, . j =1

number during these times would be equal to one (it will be explained in details

Epidemic Patterns of Emerging Variants with Dynamical Social Distancing

221

Fig. 1 Infections behavior over time of scenario (1); N = 106 , L = 1500, a = 0.25, β = 7.5 × 10−7

in a next subsection). The number of infections of the first strain decreases due to recovery, and since we considered a constant infected population, it can be observed in Fig. 1 that the newly emerged strains will sequentially replace the previous ones in dominance, having a similar slope to the old declining strains. Figure 2 represents how the number of allowed new infections per day affects the imposed social distancing over time: a high number of newly infected individuals per day leads to higher .σ (t), corresponding to milder interventions for social distancing. It is also noticeable that when a dominant strain is in decline, we can relax the measures to some extent, but later we need to make it more stringent again, giving rise to an oscillatory pattern as newer and newer strains taking over. Looking at Fig. 1, a natural question is whether the dominance period of newer strains is the same as earlier ones, since on the graph they look very similar. Hence, we compare the dominance periods in the population for each strain with three different values of L. The interesting outcome is represented in Fig. 3. This figure reflects the fact, on one hand, that the more new individuals are allowed to be infected by a strain, the shorter that strain remains in dominance and the faster it fades out. To calculate the period of dominance of a variant in the population, the distance between two different times is calculated when the number of infected individual with each strain is .L/(2a). As seen in Fig. 1, the number of people infected with strain .i .(i = 1, 2, ..., n) reaches .L/(2a) at two different times, when strain i is emerging (strain .i−1 is dying out) and fading out (strain .i+1 is emerging). Then, the dominance period is defined as the time difference between these two points, i.e., the time duration for a strain having higher number of infections than

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0.9

Social distancing

0.8

0.7

0.6

0.5

0.4

0.3 0

0.5

1

1.5

2

2.5

3

3.5

Time (year)

Fig. 2 Relation between social distancing parameter, .σ (t), and L for ten strains of first model

L/(2a). As Fig. 3 shows, in this scenario each subsequent strain has shorter period of dominance than the previous ones. On the other hand, higher allowed incidence corresponds to shorter dominance periods and faster emergence of novel strains.

.

3.3 Reproduction Numbers Figure 4 exhibits the effective reproduction number for each strain as it changes over time. Initially, all strains can potentially infect the entire population, and as time goes by, each strain infects fewer individuals due to recovery from the infection by that strain. New mutations can still potentially infect every individual in the population since recovered individuals from old strains are susceptible to the new ones. Since, according to our basic assumption, the social distance parameter is n  manipulated so that we have a fixed number of infected populations (. Ij = L/a), j =1

the effective reproduction remains at 1 for the model 1 (dashed line in Fig. 4). The overall effective reproduction number is the weighted average of all strain reproduction numbers, weighted by the number of infected individuals by that strain. To calculate these specific reproduction numbers, we create the next-generation matrix .F V −1 , where matrices F and V are defined as follows [21]: Let .I = (I1 , I2 , ..., In )T be the number of individuals in infection compartments in System (2). We rewrite the corresponding equations in the form of .I˙i = Fi (I ) −

Epidemic Patterns of Emerging Variants with Dynamical Social Distancing

223

300 L=900 L=1500 L=3500

280

260

Duration (day)

240

220

200

180

160

140

120 1

2

3

4

5

6

7

8

9

Strains

Fig. 3 Relation between new infections per day .(L(t)) and duration of persistence in the population

Fig. 4 Effective reproduction number for each strain of system (1)

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Vi (I ) for .i = 1, 2, ..., n. Here, .Fi is the term for the appearance of new infections in compartment i, and .Vi is the rate of transitions between compartment i and other ) infected compartments. Define nonnegative matrix .F = ∂F∂Ii (I and non-singular j matrix .V =

∂Vi (I ) ∂Ij

for .1 ≤ i, j ≤ n. This way, we obtain

⎛

Fn,n

.

0 0 (1 − μ)A¯ 1 ⎜ μA¯ ¯ (1 − μ) A 0 ⎜ 1 2 ⎜ 0 μA¯ 2 (1 − μ)A¯ 3 ⎜ =⎜ ⎜ 0 0 μA¯ 3 ⎜ . . .. ⎜ .. .. ⎝ . 0 0 0

··· ··· ··· ··· .. .

0 0 0 0 .. . ¯ · · · μAn−1

⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟ 0 ⎟, ⎟ .. ⎟ . ⎠ A¯ n

and ⎛

Vn,n

.

where .A¯ 1 = βσ S and .A¯ i = βσ (S + generate the next-generation

⎞ 0 0⎟ ⎟ 0⎟ ⎟, .. ⎟ .⎠ 0 0 0 ··· a

a ⎜0 ⎜ ⎜ = ⎜0 ⎜. ⎝ ..

0 a 0 .. .

0 0 a .. .

··· ··· ··· .. .

i−1 

Rk ) for .i k=1 matrix .F V −1 as

⎛

F V −1 n,n =

.

= 2, 3, ..., n. At the next step, we

(1−μ)A¯ 1 0 0 ⎜ μaA¯ 1 (1−μ)A¯ 2 ⎜ 0 a ⎜ a ⎜ 0 (1−μ)A¯ 3 μA¯ 2 ⎜ a a ⎜

⎜ ⎝

.. .

.. .

.. .

0

0

0

··· ··· ··· ···

0 0 0 .. .

0 0 0 .. .

μA¯ n−1 A¯ n a a

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

The eigenvalues of the next-generation matrix give us the effective reproduction numbers for each strain, varying with time. The eigenvalues R(1) =

.

and

(1 − μ)βσ (t)S(t) a

Epidemic Patterns of Emerging Variants with Dynamical Social Distancing

βσ (t)(S(t) +

n−1 

225

Rk (t))

k=1

R(n) =

.

a

are effective reproduction numbers corresponding to the first and last strains, respectively, and (1 − μ)βσ (t)(S(t) +

i−1 

Rk (t))

k=1

R(i) =

.

a

is the effective reproduction number for infection compartments .Ii , .i = 2, 3, ..., n − 1. Moreover, the overall effective reproduction number for the first scenario of our model can be obtained by n 

Rscenario1 =

.

j =1

Ij (t)R(j )

n  j =1

. Ij (t)

Note that in general, the next-generation matrix is calculated at a steady state. Here, to calculate the effective reproduction numbers to every t, for a given t, we freeze the values of .S(t), σ (t), Rk (t) for the time period the subsequent infected generation being created, ignoring short-term changes in these values and treating them as steady states, and we perform the same formal calculations as in the classical case in a true steady state.

4 Second Scenario: Absence of Cross-Immunity 4.1 Model Equations and the Cross-Immunity Matrix Now, let us assume protection upon recovery from one strain holds only against that particular strain; thus they are susceptible to old and new strains, e.g., in the cross-immunity matrix below the diagonal elements are zero which represents full immunity to the strain that recovered individuals have been recovered from, and other elements are one which conveys no protection against the novel and earlier strains. Then,

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Cn,n

.

⎛ 0 ⎜1 ⎜ ⎜ = ⎜1 ⎜. ⎝ ..

1 0 1 .. .

1 1 0 .. .

··· ··· ··· .. .

⎞ 1 1⎟ ⎟ 1⎟ ⎟. .. ⎟ .⎠

1 1 1 ··· 0 The second scenario is based on the assumptions that are listed below: • Each strain can infect susceptible individuals S. • Recovered individuals from strain i are only resistant to this strain and not protected against other strains. • Individuals infected by strain i can transmit strain .i + 1 due to mutation. The dynamics of the epidemic model we described can be summarized by the following equations for .k = 2, 3, ..., n − 1: ˙ = −βσ (t)S(t) S(t)

n 

Ii (t),

i=1

I˙1 (t) = βσ (t)(1 − μ)S(t)I1 (t) + βσ (t)(1 − μ)I1 R˙ 1 (t) = aI1 (t) − βσ (t)R1 (t)

n 

n 

Ri (t) − aI1 (t),

i=2

Ii (t),

i=2

.. . I˙k (t) = βσ (t)(1 − μ)S(t)Ik (t) + βσ (t)(μ)S(t)Ik−1 (t) − aIk (t) n n   +βσ (t)(1 − μ)I (t) R (t) + βσ (t)(μ)I (t) k i k−1 . i=1=k n 

R˙ k (t) = aIk (t) − βσ (t)Rk (t)

i=1=k−1

i=1=k

Ii (t),

.. . I˙n (t) = βσ (t)μS(t)In−1 (t) + βσ (t)S(t)In (t) − aIn (t) n−1 n   Ri (t) + βσ (t)(μ)In−1 (t) +βσ (t)In (t) i=1

R˙ n (t) = aIn (t) − βσ (t)Rn (t)

n−1 

Ri (t),

i=1=n−1

Ri (t),

Ii (t).

i=1

(3)

Epidemic Patterns of Emerging Variants with Dynamical Social Distancing

227

6000

I1 I2 I3 I4

5000

I5

Number of infected individual

I6 I7

4000

I8 I9 I 10

3000

2000

1000

0 0

2

4

6

8

10

12

14

16

18

Time (year)

Fig. 5 Infected population of each strain of Scenario 2 that are approaching .L/(an)

4.2 Epidemiological Dynamics In this case, unlike in Scenario (1), recovered individuals are not immune to old variants, so the infected populations in Fig. 5 do not have as regular behavior as those in Fig. 1. As we can see in Fig. 5, for each strain the infections settle around L the value (. an ) over time, In other words, for large t Ii (t) ≈

.

L , an

i = 1, 2, ...n.

This can be explained intuitively as follows: in the first scenario, we have susceptibility only to the new strains, that is, recovered people cannot be infected with old strains and there is no mutation from new strains to the previous strains, so the earlier strains cannot compete with newer ones and converge to zero, while the new ones rise to .L/a. However, in Scenario 2, each strain infects all recovered individuals, who were recovered from new and old strains (except from the very same strain); hence their potential pools are equalize. Moreover, the total amount of infected is forced to be .L/a, which are now distributed among n equally competitive strains. Consequently, the effective reproduction numbers go rounding around one since the number of infected individuals neither increases nor decreases during these times (Fig. 6).

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G. Sayyar and G. Röst 3 R strain 1 R strain 2

Effective reproduction number for each strain

R strain 3 R strain 4

2.5

R strain 5 R strain 6 R strain 7 R strain 8

2

R strain 9 R strain 10

1.5

1

0.5 0

2

4

6

8

10

12

14

16

18

Time (year)

Fig. 6 Effective reproduction numbers for each strain approach one as the infected population approaches .L/(an)

4.3 Effective Reproduction Numbers To compute the effective reproduction number for each strain in this scenario, we proceed as in Sect. 3.3: we first generate the next-generation matrix and then calculate its eigenvalues which are the effective reproduction numbers corresponding to each strain. We rewrite the equations for the infected compartments .I = (I1 , I2 , ..., In )T in System (3), as .

I˙i = Fi (I ) − Vi (I ),

i = 1, 2, ..., n.

Fi and .Vi are defined the same way as in Sect. 3.3. Define the nonnegative matrix ) F = ∂F∂Ii (I , and the non-singular matrix .V = ∂V∂Ii j(I ) for .1 ≤ i, j ≤ n as follows: j

. .

Fn,n

.

⎛ 0 0 (1 − μ)B1 ⎜ μB (1 − μ)B2 0 ⎜ 1 ⎜ 0 μB2 (1 − μ)B3 ⎜ =⎜ ⎜ 0 0 μB3 ⎜ .. .. .. ⎜ ⎝ . . . 0 0 0

··· ··· ··· ··· .. .

0 0 0 0 .. .

0 0 0 0 .. .

· · · μBn−1 Bn

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

Epidemic Patterns of Emerging Variants with Dynamical Social Distancing

⎛

Vn,n

.

a ⎜0 ⎜ ⎜ = ⎜0 ⎜. ⎝ ..

0 a 0 .. .

0 0 a .. .

··· ··· ··· .. .

229

⎞ 0 0⎟ ⎟ 0⎟ ⎟, .. ⎟ .⎠

0 0 0 ··· a where .Bi = βσ (S +

n  k=1=i

Rk ) for .i = 1, 2, ..., n. Then we create the matrix .F V −1

⎛ (1−μ)B F V −1 n,n

.

⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

a μB1 a

0 .. . 0

1

0

0 0

(1−μ)B2 a μB2 (1−μ)B3 a a

.. . 0

.. . 0

··· ··· ··· ···

0 0 0 .. .

0 0 0 .. .

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

μBn−1 Bn a a

The eigenvalues of this matrix give us the effective reproduction numbers for each strain. In particular, the eigenvalue βσ (t)(S(t) + R(n) =

.

n−1 

Rk (t))

k=1

a

is the effective reproduction number that corresponds to the last strain, and (1 − μ)βσ (t)(S(t) + R(i) =

.

n  k=1=i

Rk (t))

a

is the effective reproduction number corresponding to strain i, .i = 1, 2, ..., n − 1.

5 Discussion The way in which novel mutations of an infectious disease, particularly COVID-19, emerge and persist in the population prompts us to implement a general model with multiple strains emerging via mutations. For the sake of simplicity, here we assumed that the strains have identical epidemiological parameters, and they differ only in the target population which they can infect, and this is determined by a cross-immunity matrix. For the purposes of this paper, we considered two different scenarios which

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focus only on immune evasion of newer strains emerging by virus mutations. Recovery from a strain in the first scenario confers full protection against previous strains and no immunity to the novel strains. From the simulation results, we can observe a highly structured sequential behavior of strain replacement where newer strains predominate over older strains but last in the population for an ever shorter period of time. However, since in the second scenario, immunity upon recovery holds only against a given strain and none of the others, the population infected by each strain has an erratic pattern over time, with recurrence of past strains, and co-circulation of many strains without any of them being clearly dominant. In the following example, we compare our results with the reported data on the coronavirus SARS-CoV-2 variants in the Netherlands. Figure 7 represents the frequency of COVID-19 variants since 2021 in the Netherlands, based on the data from [22]. In this graph, variants Alpha and Delta are emerging after each other and are fully dominant, showing a very similar picture to our Scenario 1, where even the periods of dominance are of similar length in the model and in the data. But almost after June 2022, most of the variants of SARS-CoV-2 are from the Omicron lineage, from subvariants BA.1 through BQ.1. While the BA.5 subvariant is still dominant, some new subvariants such as BQ.1 are gaining. The recently emerging Omicron subvariants are alarmingly immune-evasive [23], and they could further compromise the effectiveness of current COVID-19 vaccines, as well as curtailing prior natural immunity. This, in turn, could increase cases of infections and reinfections. The recent situation, 1 0.9 0.8

Proportion of each variant

0.7 0.6

Alpha Delta BA.1 BA.2 BA.5 BA.4.6 BF.7 BQ.1 BA.1+BA.2

0.5

0.4 0.3

0.2 0.1 0 01/Jan

01/Apr

01/Jul

01/Oct

01/Jan

01/Apr

01/Jul

01/Oct

week

Fig. 7 Variants of the coronavirus SARS-COV-2 in the Netherlands from .30/11/2020 to .14/12/2022

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231

regarding the cross-immunity between novel strains is more similar to our Scenario 2, where, after a while we can see irregular circulation patterns in Fig. 5, just as in Fig. 7. Thus, this phenomenon is more similar to the Omicron-era Netherlands data. This confirms that our model captures some of the essence of the variant dynamics that we have seen during COVID-19. Acknowledgments GS has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 955708, EvoGamesPlus. GR was supported by Hungarian grants NKFIH KKP 129877, RRF-2.3.1-21-2022-00006, and TKP2021-NVA-09.

References 1. M. Cascella, M. Rajnik, A. Aleem, S. C. Dulebohn, and R. Di Napoli, “Features, evaluation, and treatment of coronavirus (COVID-19),” Statpearls [internet], 2022. 2. P. Boldog, T. Tekeli, Z. Vizi, A. Dénes, F. A. Bartha, and G. Röst, “Risk assessment of novel coronavirus covid-19 outbreaks outside china,” Journal of Clinical Medicine, vol. 9, no. 2, p. 571, 2020. 3. C. R. Wells, P. Sah, S. M. Moghadas, A. Pandey, A. Shoukat, Y. Wang, Z. Wang, L. A. Meyers, B. H. Singer, and A. P. Galvani, “Impact of international travel and border control measures on the global spread of the novel 2019 coronavirus outbreak,” Proceedings of the National Academy of Sciences, vol. 117, no. 13, pp. 7504–7509, 2020. 4. S. Flaxman, S. Mishra, A. Gandy, H. J. T. Unwin, T. A. Mellan, H. Coupland, C. Whittaker, H. Zhu, T. Berah, J. W. Eaton, et al., “Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe,” Nature, vol. 584, no. 7820, pp. 257–261, 2020. 5. G. Röst, F. A. Bartha, N. Bogya, P. Boldog, A. Dénes, T. Ferenci, K. J. Horváth, A. Juhász, C. Nagy, T. Tekeli, et al., “Early phase of the COVID-19 outbreak in Hungary and postlockdown scenarios,” Viruses, vol. 12, no. 7, p. 708, 2020. 6. T. Péni, B. Csutak, G. Szederkényi, and G. Röst, “Nonlinear model predictive control with logic constraints for COVID-19 management,” Nonlinear Dynamics, vol. 102, no. 4, pp. 1965–1986, 2020. 7. T. Péni, B. Csutak, F. A. Bartha, G. Röst, and G. Szederkényi, “Optimizing symptom based testing strategies for pandemic mitigation,” IEEE Access, vol. 10, pp. 84934–84945, 2022. 8. T. Tekeli, A. Dénes, and G. Röst, “Adaptive group testing in a compartmental model of COVID19,” Mathematical Biosciences and Engineering, vol. 19, no. 11, pp. 11018–11033, 2022. 9. L. Childs, D. W. Dick, Z. Feng, J. M. Heffernan, J. Li, and G. Röst, “Modeling waning and boosting of COVID-19 in canada with vaccination,” Epidemics, p. 100583, 2022. 10. D. W. Dick, L. Childs, Z. Feng, J. Li, G. Röst, D. L. Buckeridge, N. H. Ogden, and J. M. Heffernan, “COVID-19 seroprevalence in Canada modelling waning and boosting COVID-19 immunity in Canada - a Canadian immunization research network study,” Vaccines, vol. 10, no. 1, p. 17, 2021. 11. E. Ari, B. M. Vásárhelyi, G. Kemenesi, G. E. Tóth, B. Zana, B. Somogyi, Z. Lanszki, G. Röst, F. Jakab, B. Papp, et al., “A single early introduction governed viral diversity in the second wave of SARS-CoV-2 epidemic in Hungary,” Virus Evolution, vol. 8, no. 2, p. veac069, 2022. 12. F. A. Bartha, P. Boldog, A. Dénes, T. Tekeli, Z. Vizi, and G. Röst, “Potential severity, mitigation, and control of omicron waves depending on pre-existing immunity and immune evasion,” Chapter 22 in: Trends in Biomathematics: Stability and Oscillations in Environmental, Social, and Biological Models (ed. R. Mondaini), Springer 2022.

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On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates and Therapy Karam Allali

1 Introduction The infectious diseases such as dengue fever, tuberculosis, human immunodeficiency virus, and coronavirus disease COVID-19 can experience the coexistence of multiple strains [1–5]. In order to check the conditions permitting the coexisting of those strains, many mathematical models have been developed. Blocking the strains mutation processes remains one of the hard tasks in epidemiology. In general, two ways are used to model the latent infection period; the first is to include the exposed class .(E) or to take into account explicitly the time delay. The mathematical analysis of single-strain susceptible-exposed-infectious-recovered (SEIR) have been studied in many works [6–8]. Recent other works have tackled the same issue but for twostrain infection. For instance and when both the incidences are bilinear, [9] studied mathematically an SEIR two-strain epidemic model. With two general incidence rates, [5] have studied the same issue, and numerical simulations are carried out with different incidence rates. The time delay can be also used to reflect the infection latency in modeling an infectious disease. The global analysis of delayed two-strain epidemic model have been tackled in [10]. Here, the authors consider that both incidence functions are bilinear. Later and recently, the same issue was tackled in [11] by considering the fact that the two incidence rates are in general form. The authors have studied the global stability of the different equilibria and performed numerical simulations by choosing several specific incidence rates. Motivated by the previous papers, we will consider in this work a delayed two-strain infection model under the effect of

K. Allali () Laboratory of Mathematics, Computer Science and Applications, Faculty of Sciences and Technologies, Hassan II University of Casablanca, Mohammedia, Morocco e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_14

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generalized incidence rates and two therapies. Accordingly, we will consider the following nonlinear system of four differential equations: ⎧ dS(t) ⎪ ⎪ dt =  − (1 − η1 )F(S(t); I1 (t))I1 (t) − (1 − η2 )H(S(t); I2 (t))I2 (t) − ρS(t), ⎪ ⎪ ⎪ dI ⎨ 1 (t) = (1 − η1 )F(S(t − τ1 ); I1 (t − τ1 ))I1 (t − τ1 )e−ρτ1 − (ρ + d1 + ν1 )I1 (t), .

dt

dI2 (t) −ρτ2 − (ρ + d + ν )I (t), ⎪ ⎪ 2 2 2 ⎪ dt = (1 − η2 )H(S(t − τ2 ); I2 (t − τ2 ))I2 (t − τ2 )e ⎪ ⎪ ⎩ dR(t) dt = ν1 I1 (t) + ν2 I2 (t) − ρR(t), (1)

where S represents the susceptible individuals, .I1 the first strain infected individuals, I2 the second strain infected individuals, and R the removed individuals. The parameters of problem are described as follows: . is the recruitment rate, .ρ is the death rate of the population, .ν1 (respectively, ν2 ) is the transfer rates from first strain (respectively, second strain) infected individuals to recovered, .di (i = 1, 2) is in the infection-induced death rate of the i strain .(i = 1, 2); .F(S; I1 ) is the rate of transmission of susceptible individuals to strain 1 infected individuals, .H(S; I2 ) is the rate of transmission of susceptible individuals to strain 2 infected individuals, .τ1 is the time delay describing the strain 1 incubation period, and .τ2 is the time delay describing the strain 2 incubation period. The terms .e−ρτ1 and .e−ρτ2 stand for the each strain individual survival probability from time .t − τ1 to time t and from time .t − τ2 to time t, respectively. The parameter .η1 (.η2 ) represents the efficiency of the first strain therapy (the efficacy of the second strain treatment, respectively). The incidence rates functions .F(S; I1 ) and .H(S; I2 ) are assumed to be continuously differentiable in the interior of .R2+ and satisfies the following hypotheses as in [5, 12]: .

F(0; I1 ) = 0, H(0; I2 ) = 0, ∀Ii  0, i = 1, 2.

(H1 )

.

∂F ∂H (S; I2 ) > 0, ∀S > 0, ∀Ii ≥ 0, i = 1, 2, (S; I1 ) > 0, ∂S ∂S

(H2 )

.

∂F ∂H (S; I2 ) ≤ 0, ∀S ≥ 0, ∀Ii ≥ 0, i = 1, 2, (S; I1 ) ≤ 0, ∂I2 ∂I1

(H3 )

.

The three above properties .(H1 ), .(H2 ), and .(H3 ), for the both functions .F and .H, are straightforwardly verified by many classical biological incidence rates such as the αS bilinear incidence function .αS [13–15], the saturated incidence function . or 1 + σ1 S αS αS . [18– [16, 17], Beddington-DeAngelis incidence function . 1 + σ1 S + σ2 I 1 + σ2 I αS [21–23], and 20], Crowley-Martin incidence function . 1 + σ1 S + σ2 I + σ1 σ2 SI αS non-monotone incidence function . [24–28]. 1 + σI2

On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates. . .

235

This work is organized as follows. Section 2 will be devoted to the wellposedness of our problem as well as the problem steady states. The global stability of the equilibria is fulfilled in Sect. 3. Followed in Sect. 4 by some numerical simulations in order to validate the theoretical findings. The last section concludes the work.

2 Wellposedness Result and Problem Equilibria 2.1 Existence, Positivity, and Boundedness of Solutions   First, let .τ = max τ1 , τ2 ; we define .C = C([−τ, 0]; R) the functional space of continuous functions equipped with the following norm .·∞ . The set .C+ is defined by   C+ = φ ∈ C/φ(θ )  0, −τ  θ  0 .

.

 We assume that the initial condition of system (1) verify .φ = φ1 , φ2 , φ3 , φ4 ∈ (C+ )4 . Proposition 1 For the initial conditions .φi ∈ C+ for .i = 1, 2, 3, 4, the problem (1) has a unique, positive, and bounded solution .(S(t), I1 (t), I2 (t), R(t)) for all .t ≥ 0. Moreover we have • .T (t)  T (0) +

 , ρ ν ρ I1 ∞

• .R(t)  R(0) + + I2 ∞ ,   such that .ν = sup ν1 , ν2 and .T (t) = S(t) + eρτ1 I1 (t + τ1 ) + eρτ2 I2 (t + τ2 ). Proof By applying the theory in [29, 30], we can deduce that there exists a unique positive local solution .(S(t), I1 (t), I2 (t), R(t)) to problem (1). Define now T (t) = S(t) + eρτ1 I1 (t + τ1 ) + eρτ2 I2 (t + τ2 )

.

therefore, .

dT (t) ≤  − ρT , then dt T (t)  T (0)e−ρt +

.

 . ρ

So, .S(t), I1 (t), and .I2 (t) are bounded on .[0, tφ ). From the last equation of the system (1), we have:

(2)

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K. Allali

.

dR(t)  ν(I1 (t) + I2 (t)) − ρR(t). dt

Hence, R(t)  R(0)e−ρt +

.

ν I1 ∞ + I2 ∞ ρ

(3)

Then, .R(t) is bounded on .[0, tφ ). From the inequalities (2)–(3), we have the fact that the solution can be extended to any time .tm > 0. We conclude that the positive solution of system (1) exists globally on .[0, +∞[ 

2.2 The Problem Equilibria Since the first four equations of the system (1) are independent of R, we can omit the fourth equation, and the system (1) can be reduced to ⎧ dS(t) ⎪ ⎪ =  − (1 − η1 )F(S(t); I1 (t))I1 (t) − (1 − η2 )H(S(t); I2 (t))I2 (t) − ρS(t), ⎪ ⎪ dt ⎪ ⎪ ⎨ dI1 (t) . = (1 − η1 )F(S(t − τ1 ); I1 (t − τ1 ))I1 (t − τ1 )e−ρτ1 − (ρ + d1 + ν1 )I1 (t), ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dI2 (t) = (1 − η2 )H(S(t − τ2 ); I2 (t − τ2 ))I2 (t − τ2 )e−ρτ2 − (ρ + d2 + ν2 )I2 (t). dt

(4)

The basic reproduction number is the spectral radius of the next-generation matrix F V −1 [31]: .R0 = (F V −1 ), where F is the nonnegative matrix of the new infection terms, and V is the matrix of the associated infection transitions. In our model, the infected compartments are .I1 and .I2 , and then the matrices F and V are

.



−ρτ1 (1 − η1 )F(  0 ρ , 0)e .F = −ρτ2 0 (1 − η2 )H(  ρ , 0)e

0 ρ + d1 + ν1 V = 0 ρ + d2 + ν2

 and

The basic reproduction number is the dominant eigenvalue of the matrix .F V −1 . So 1 2 .R0 = max{R , R }, where 0 0 R01 =

.

−ρτ1 (1 − η1 )F(  ρ , 0)e

ρ + d1 + ν1

and R02 =

−ρτ2 (1 − η2 )H(  ρ , 0)e

ρ + d2 + ν2

.

On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates. . .

237

In the sequel and for simplicity of expressions, let the two following numbers .a = −ρτ1 /a ρ + d1 + ν1 , .b = ρ + d2 + ν2 . Then we will have .R01 = (1 − η1 )F(  ρ , 0)e

−ρτ2 /b and .R02 = (1 − η2 )H(  ρ , 0)e

Theorem 1 The system (4) has a unique disease-free equilibrium .E0 and three endemic equilibria under the following conditions: • If .R01 > 1, then the strain 1 endemic equilibrium .E1 exists. • If .R02 > 1, then the strain 2 endemic equilibrium .E2 exists. • If .R01 > 1 and .R02 > 1, then the both strains endemic equilibrium .Et exists. Proof In order to find the steady states of the system (4), we solve the following:  − (1 − η1 )F(S(t); I1 (t))I1 (t) − (1 − η2 )H(S(t); I2 (t))I2 (t) − ρS(t) = 0, . (5)

.

(1 − η1 )F(S(t − τ1 ); I1 (t − τ1 ))I1 (t − τ1 )e−ρτ1 − (ρ + d1 + ν1 )I1 (t) = 0, . (6) (1 − η2 )H(S(t − τ2 ); I2 (t − τ2 ))I2 (t − τ2 )e−ρτ2 − (ρ + d2 + ν2 )I2 (t) = 0. (7) From where, we obtain • If .I1 = 0 and .I2 = 0, we obtain the disease-free equilibrium .E0 = (  ρ , 0, 0). • If .I1 = 0 and .I2 = 0, we will find the strain 1 endemic equilibrium defined as follows  ∗ ∗ e−ρτ1 and .I ∗ = 0. ∗ ∗ ∗ ∗ .E1 = (S , I 1 1,1 , I2,1 ) with .S1 ∈ [0, ρ ], .I1,1 = ( − ρS1 ) a 2,1 This endemic equilibrium exists when .R01 > 0. Indeed, from Eqs. (5)–(6) and ∗ = 0, we have knowing that .I2,1  .

∗ )I ∗ − ρS ∗ = 0,  − (1 − η1 )F(S1∗ ; I1,1 1,1 1 ∗ )I ∗ e−ρτ1 − aI ∗ (1 − η1 )F(S1∗ ; I1,1 1,1 1,S1 = 0.

From where, we will have ∗ I1,1 = ( − ρS1∗ )

.

e−ρτ1 . a

∗ ≥ 0, we will have the condition .S ∗ ≤ However, from the positivity result .I1,1 1

 ρ.

Now, let us define on .[0,  ρ ] the following function: .ϕ(S) = (1−η1 )F(S, (− ∂ϕ(S) −ρτ ρS) e a 1 ) − aeρτ1 . Thus from the conditions .(H2 ) and .(H3 ), we have . = ∂S −ρτ 1 ∂F(S, I1 ) ∂F(S, I1 ) ρ(1 − η1 )e > 0. (1 − η1 ) − ∂S a ∂I1

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However, .(1 − η1 )F(0,  e

−ρτ1

a

) − aeρτ1 = −aeρτ1 < 0. Therefore, .ϕ(  ρ) =

(1 − − = − 1) is positive for .R01 > 1. Hence, the strain 1 endemic equilibrium .E1 exists when .R01 > 1. • If .I2 = 0 and .I1 = 0, we will find the strain 2 endemic equilibrium defined as follows:  ∗ ∗ e−ρτ2 and .I ∗ = 0. ∗ ∗ ∗ ∗ .E2 = (S , I 2 1,2 , I2,2 ) with .S2 ∈ [0, ρ ], .I2,2 = ( − ρS2 ) b 1,2 This endemic equilibrium exists when .R02 > 1. Indeed, from Eqs. (5)–(7) and ∗ knowing that .I1,S = 0, we have: 1 η1 )F(  ρ , 0)

aeρτ1 (R01

aeρτ1

 .

∗ )I ∗ − ρS ∗ = 0,  − (1 − η2 )H(S2∗ ; I2,2 2,2 2 ∗ )I ∗ e−ρτ2 − bI ∗ = 0. (1 − η2 )H(S2∗ ; I2,2 2,2 2,2

From where, we will have ∗ I2,2 = ( − ρS2∗ )

.

e−ρτ2 . b

∗ ≥ 0, we will have the condition .S ∗ ≤ However, from the positivity result .I2,2 2

 ρ.

Let us define on .[0,  ρ ] the following function: .ψ(S) = (1 − η2 )H(S, ( − −ρτ 2 ∂ψ(S) e = )−beρτ2 . Thus from the conditions .(H2 ) and .(H3 ), we have . ρS) ∂S b ∂H(S, I2 ) ρ(1 − η2 )e−ρτ2 ∂H(S, I2 ) (1 − η2 ) − > 0. ∂S b ∂I2 e−ρτ2 ) − beρτ2 = −beρτ2 < 0. Therefore, .ψ(  However, .(1 − η2 )H(0,  ρ)= b ρτ2 = beρτ2 (R 2 − 1) is positive for .R 2 > 1. Hence, the (1 − η2 )H(  0 0 ρ , 0) − be strain 2 endemic equilibrium .E2 exists when .R02 > 1. • If .I1 = 0 and .I2 = 0, we obtain the both  strains endemic equilibrium  1 ∗ eρτ1 − bI ∗ eρτ2 ∗ ∗ ∗ ∗ = .ESt = (St , I1,t , I2,t ), where .St = ρ  − aI1,t 2,t     (1−η1 )F( ρ ,0) ∗ (1−η2 )H( ρ ,0) ∗ 1 I1,t . − I2,t . In this last case, we have 1 2 ρ − R0



.

(1 − η1 )F(  ρ , 0) R01

R0

∗ I1,t +

(1 − η2 )H(  ρ , 0) R02

∗ I2,t , .R01 > 1 and .R02 > 1.



3 Global Stability of the Problem Equilibria In this section, we will prove the global stability of the equilibrium points. To this end, some suitable Lyapunov functionals will be used to prove the global stability.

On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates. . .

239

3.1 Global Stability of Disease-Free Equilibrium Theorem 2 The disease-free equilibrium E0 is globally asymptotically stable if R0 ≤ 1. Proof Consider the following Lyapunov function: L0 = S − S0∗ −

.

 +  +

t



S S0∗

(1 − η1 )F(S0∗ , 0) dX + eρτ1 I1 F(X, 0)

(1 − η1 )F(S(θ ), I1 (θ ))I1 (θ ) dθ + eρτ2 I2

t−τ1 t

(1 − η2 )H(S(θ ), I2 (θ ))I2 (θ ) dθ,

t−τ2

where S0∗ =

 ρ.

The derivative of L0 is given by

F(S0∗ , 0) S˙ + eρτ1 I˙1 + (1 − η1 )F(S, I1 )I1 F(S, 0)   −(1 − η1 )F S(t − τ1 ), I1 (t − τ1 ) I1 (t − τ1 )   +eρτ2 I˙2 + (1 − η2 )H(S, I2 )I2 − (1 − η2 )H S(t − τ2 ), I2 (t − τ2 ) I2 (t − τ2 )   F(S, I ) F(S0∗ , 0) S  1 + aeρτ1 I1 R01 − 1 = ρS0∗ 1 − ∗ 1 − S0 F(S, 0) F(S, 0)  F(S ∗ , 0) H(S, I ) 2 0 R2 − 1 +beρτ2 I2 F(S, 0) H(S0∗ , 0) 0   F(S0∗ , 0) S  + aeρτ1 I1 R01 − 1 ≤ ρS0∗ 1 − ∗ 1 − F(S, 0) S0  F(S ∗ , 0) H(S, I ) 2 0 2 +beρτ2 I2 − 1 . (8) R F(S, 0) H(S0∗ , 0) 0

L˙0 = S˙ −

.

We discuss two cases: – If S  S0∗ , from (H2 ), we have

H(S,I2 ) H(S0∗ ,0)

 1, then

  F(S ∗ , 0)  S  F(S0∗ , 0) 0 L˙0 ≤ ρS0∗ 1− ∗ 1− +aeρτ1 I1 R01 −1 +beρτ2 I2 R02 −1 . F(S, 0) F(S, 0) S0

.

F(S0∗ ,0) 2 F(S,0) F(S0∗ ,0)  1, we have F(S,0) R0 − 1  0.   F(S0∗ ,0) F(S0∗ ,0) Moreover, 1 − F(S,0)  0, thus ρS0∗ 1 − SS∗ 1 − F(S,0) 0

Since R02 

 0.

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K. Allali

– If S > S0∗ , from (H2 ), we have ˙0 .L

> 1, and

F(S0∗ ,0) F(S,0)

< 1 then

   H(S, I ) F(S0∗ , 0) S  2 ≤ ρS0∗ 1− ∗ 1− +aeρτ1 I1 R01 −1 +beρτ2 I2 R02 −1 . ∗ S0 F(S, 0) H(S0 , 0) H(S0∗ ,0) H(S,I2 )

H(S,I2 ) 2 R0 − 1 < 0. < 1, we have H(S ∗ 0 ,0)   F(S0∗ ,0)  0. < 1, we obtain ρS0∗ 1 − SS∗ 1 − F(S,0)

From R02 < Since,

H(S,I2 ) H(S0∗ ,0)

F(S0∗ ,0) F(S,0)

0

From the above discussion, we deduce that, if R01  1 and R02  1, then L˙0  0. We conclude that the disease-free equilibrium E0 is globally asymptotically stable when R0  1. 

3.2 Global Stability of Strain 1 Endemic Equilibrium For the global stability of .E1 , we assume that the function f satisfies the condition: ∗  I1 F(S, I1 )  F(S, I1,1 ) − 1−  0, ∀S, I1 > 0 ∗ ) ∗ F(S, I1,1 F(S, I1 ) I1,1

.

(H4 )

Theorem 3 The strain 1 endemic equilibrium .E1 is globally asymptotically stable if .R01 > max{1, R02 }. Proof Consider the following Lyapunov function: L1 = S 

.

·

t

 t−τ1

− S1∗

 −

S

S1∗

∗ ) F(S1∗ , I1,1 ∗ ) F(X, I1,1

∗ dX + eρτ1 I1,1 (

I1 ∗ ∗ ∗ ∗ ) + (1 − η1 )F(S1 , I1,1 )I1,1 I1,1

 t  F(S(θ ), I (θ ))I (θ ) 1 1 ρτ2 dθ + e I + (1 − η2 )H(S(θ ), I2 (θ ))I2 (θ ) dθ. 2 ∗ )I ∗ F(S1∗ , I1,1 t−τ2 1,1 (9)

where .(x) = x − 1 − ln x, (x > 0). Obviously, .(x) is positive in .R+ and has its strict global minimum .(1) = 0. The derivative of .L1 is given by ∗ )   I∗ F(S1∗ , I1,1 ˙ + eρτ1 1 − 1,1 I˙1 + eρτ2 I˙2 + (1 − η2 )H(S, I2 )I2 S L˙1 = 1 − ∗ ) I1 F(S, I1,1   ∗ ∗ −H S(t − τ2 ), I2 (t − τ2 ) I2 (t − τ2 ) + (1 − η1 )F(S1∗ , I1,1 )I1,1   F(S, I )I 1 1 ×  ∗ ∗ ∗ F(S1 , I1,1 )I1,1

.

On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates. . .

  F S(t − τ1 ), I1 (t − τ1 ) I1 (t − τ1 )  . − ∗ )I ∗ F(S1∗ , I1,1 1,1

241

(10)

∗ )I ∗ +ρS ∗ and .(1−η )F(S ∗ , I ∗ )I ∗ = aeρτ1 I ∗ . We have . = (1−η1 )F(S1∗ , I1,1 1 1,1 1 1 1,1 1,1 1,1 Then ∗ )  F(S1∗ , I1,1 S  L˙1 = ρS1∗ 1 − ∗ 1 − ∗ ) F(S, I1,1 S1

.

∗ +aeρτ1 I1,1 ∗ −aeρτ1 I1,1

+

∗ )  I F(S, I ) F(S, I1,1 I1 1 1 + − 1 − ∗ ∗ F(S, I ∗ ) F(S, I1 ) I1,1 I1,1 1,1

 F(S ∗ , I ∗ ) 1 1,1 F(S, I1 )

+

∗ ) F(S, I1,1 F(S, I1 ) F(S, I1 ) + −3− ∗ ) ∗ ) F(S1∗ , I1,1 F(S, I1 ) F(S1∗ , I1,1

 F S(t − τ1 ), I1 (t − τ1 ) I1 (t − τ1 ) ∗ )I F(S1∗ , I1,1 1

− ln

  F S(t − τ1 ), I1 (t − τ1 ) I1 (t − τ1 )  F(S, I1 )I1

 F(S ∗ , I ∗ ) 1 1,1 +beρτ2 I2 ∗ ) F(S, I1,1

(1 − η2 )H(S, I2 ) − 1 , beρτ2

(11)

then ∗ )  F(S1∗ , I1,1 S  L˙1 = ρS1∗ 1 − ∗ 1 − ∗ ) S1 F(S, I1,1

.

∗ +aeρτ1 I1,1

∗ )  I F(S, I ) F(S, I1,1 I1 1 1 + − 1 − ∗ F(S, I ∗ ) ∗ I1,1 F(S, I1 ) I1,1 1,1

 F(S, I ∗ )   F(S ∗ , I ∗ ) 1 1,1 1,1 ∗ +   −aeρτ1 I1,1 ∗ ) F(S, I1 ) F(S, I1,1   F S(t − τ1 ), I1 (t − τ1 ) I1 (t − τ1 )  + ∗ )I F(S1∗ , I1,1 1 +beρτ2 I2

 F(S ∗ , I ∗ ) H(S, I ) 2 1 1,1

∗ ) H(S ∗ , 0) F(S, I1,1 0

H(S,I2 ) From (H2 ) and (H3 ), we have . H(S ∗ ,0)  1, then 0

R02 − 1 .

(12)

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K. Allali ∗ )  F(S1∗ , I1,1 S  L˙1  ρS1∗ 1 − ∗ 1 − ∗ ) F(S, I1,1 S1

.

∗ +aeρτ1 I1,1

∗ )  I F(S, I ) F(S, I1,1 I1 1 1 + −1− ∗ ∗ ∗ F(S, I1 ) I1,1 I1,1 F(S, I1,1 )

 F(S, I ∗ )   F(S ∗ , I ∗ ) 1,1 1 1,1 ∗ +   −aeρτ1 I1,1 ∗ ) F(S, I1,1 F(S, I1 )   F S(t − τ1 ), I1 (t − τ1 ) I1 (t − τ1 )  + ∗ )I F(S1∗ , I1,1 1 +beρτ2 I2

 F(S ∗ , I ∗ ) 1 1,1 ∗ ) F(S, I1,1

From (H4 ), we have I1 . ∗ I1,1

F(S,I1 ) ∗ ) F(S,I1,1

+

∗ ) F(S,I1,1 F(S,I1 )

−1−

I1 ∗ I1,1

R02 − 1 .

 = 1−

F(S,I1 ) ∗ ) F(S,I1,1

(13)

 F(S,I ∗

1,1 ) F(S,I1 )

−

I1 ∗ I1,1



 0.

We discuss two cases: F(S ∗ ,I ∗ )

1 1,1  1, and for .R02  1, we have – If .S1∗  S, from (H2 ), we have . F(S,I ∗ 1,1 )   ∗ ∗ ∗ ∗ ) F(S1 ,I1,1 ) 2 F(S1 ,I1,1 S ˙ 1 − . R − 1  0, and . 1− ∗ ∗ ∗ 0 F(S,I ) S F(S,I )  0 , hence, .L1  0. 1,1

–

1

1,1

∗ ) ∗ ) F(S1∗ ,I1,1 F(S,I1,1 2 If .S  S1∗ , from (H2 ), we have . F(S,I ∗ )  1, since .R0  F(S ∗ ,I ∗ )  1 we 1 1,1 1,1 F(S ∗ ,I ∗ )  ∗ ) F(S1∗ ,I1,1 1 1,1 S 2 1 − F(S,I ∗ )  0 , hence, .L˙1  0. . ∗ ) R0 − 1  0, and . 1 − S ∗ F(S,I1,1 1 1,1

have

From the above discussion, we deduce that, if .R02  1, then .L˙1  0. We conclude that the strain 1 endemic equilibrium .E1 is globally asymptotically stable when .R02  1 and .R01 > 1. 

3.3 Global Stability of Strain 2 Endemic Equilibrium For the global stability of .E2 , we assume that the function g satisfies the condition: ∗  I2 H(S, I2 )  H(S, I2,2 ) − ∗  0, ∀S, I2 > 0 1− ∗ H(S, I2 ) I2,2 H(S, I2,2 )

.

(H5 )

Theorem 4 The strain 2 endemic equilibrium .E2 is globally asymptotically stable if .R02 > max{1, R01 }. Proof Consider the following Lyapunov function:

On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates. . .

L2 = S − S2∗ − 

.

t

·

 t−τ2



S

S2∗

∗ ) H(S2∗ , I2,2 ∗ ) H(X, I2,2

∗ dX + eρτ2 I2,2 (

243

I2 ∗ ∗ ∗ ∗ ) + (1 − η2 )H(S2 , I2,2 )I2,2 I2,2

 t  H(S(θ ), I (θ ))I (θ ) 2 2 ρτ1 dθ + e I + (1 − η1 )F(S(θ ), I1 (θ ))I1 (θ ) dθ. 1 ∗ )I ∗ H(S2∗ , I2,2 t−τ1 2,2

(14) where .(x) = x − 1 − ln x, (x > 0). The derivative of .L2 is given by ∗ )   I∗ H(S2∗ , I2,2 ˙ + eρτ2 1 − 2,2 I˙2 + eρτ1 I˙1 + (1 − η1 )F(S, I1 )I1 L˙2 = 1 − S ∗ ) H(S, I2,2 I2   H(S, I )I   2 2 ∗ ∗ ∗  − F S(t − τ ), I (t − τ ) I (t − τ ) + (1 − η )H(S , I )I 1 1 1 1 1 2 2 2,2 2,2 ∗ )I ∗ . H(S2∗ , I2,2 2,2   H S(t − τ2 ), I2 (t − τ2 ) I2 (t − τ2 )  . − ∗ )I ∗ H(S2∗ , I2,2 2,2

(15) ∗ )I ∗ + ρS ∗ and .(1 − η )H(S ∗ , I ∗ )I ∗ We have . = (1 − η2 )H(S2∗ , I2,2 2 2,2 2 2 2,2 2,2 = ∗ . Then beρτ2 I2,2 ∗ ) ∗ )   I H(S, I ) H(S2∗ , I2,2 H(S, I2,2 S  2 2 ρτ2 ∗ + be L˙2 = ρS2∗ 1 − ∗ 1 − I + 2,2 ∗ ) ∗ H(S, I ∗ ) S2 H(S, I2,2 I2,2 H(S, I2 ) 2,2 ∗ )  H(S ∗ , I ∗ ) H(S, I2,2 I2 H(S, I2 ) H(S, I2 ) 2 2,2 ρτ2 ∗ − be I + + −3− 2,2 ∗ ) ∗ ) ∗ H(S, I2 ) H(S2∗ , I2,2 H(S, I2 ) H(S2∗ , I2,2 I2,2    H S(t − τ2 ), I2 (t − τ2 ) I2 (t − τ2 )  H S(t − τ2 ), I2 (t − τ2 ) I2 (t − τ2 ) + − ln ∗ )I H(S2∗ , I2,2 H(S, I2 )I2 2

−1−

.

+ aeρτ1 I1

 H(S ∗ , I ∗ ) (1 − η )F(S, I ) 1 1 2 2,2 −1 , ∗ ρτ 1 ae H(S, I2,2 )

(16) then ∗ ) ∗ )  I H(S, I )  H(S2∗ , I2,2 H(S, I2,2 S  2 2 ρτ2 ∗ + be L˙2 = ρS2∗ 1 − ∗ 1 − I + 2,2 ∗ ) ∗ H(S, I ∗ ) S2 H(S, I2,2 I2,2 H(S, I2 ) 2,2

.

 H(S, I ∗ )   H(S ∗ , I ∗ ) I2 2,2 2 2,2 ρτ2 ∗ +  − be  I 2,2 ∗ ) ∗ H(S, I2,2 H(S, I2 ) I2,2   H S(t−τ2 ), I2 (t−τ2 ) I2 (t − τ2 )   H(S ∗ , I ∗ ) F(S, I ) 1 2 2,2 + + aeρτ1 I1 R01 − 1 . ∗ ∗ ∗ ∗ H(S, I2,2 ) F(S0 , 0) H(S2 , I2,2 )I2 −1−

F(S,I1 ) From (H2 ) and (H3 ), we have . F(S ∗ ,0)  1, then 0

(17)

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K. Allali

∗ ) ∗ )  I H(S, I )  H(S, I2,2 H(S2∗ , I2,2 S  2 2 ρτ2 ∗ + be + L˙2  ρS2∗ 1 − ∗ 1 − I 2,2 ∗ ) ∗ H(S, I ∗ ) H(S, I2,2 I2,2 H(S, I2 ) S2 2,2

.

 H(S, I ∗ )   H(S ∗ , I ∗ ) I2 2,2 2 2,2 ρτ2 ∗ +  − be  I 2,2 ∗ ) ∗ H(S, I2,2 H(S, I2 ) I2,2   H S(t − τ2 ), I2 (t − τ2 ) I2 (t − τ2 )   H(S ∗ , I ∗ ) 2 2,2 R01 − 1 . + + aeρτ1 I1 ∗ ∗ ∗ H(S, I2,2 ) H(S2 , I2,2 )I2 −1−

(18) From (H5 ), we have I2 . ∗ I2,2

H(S,I2 ) ∗ ) H(S,I2,2

+

∗ ) H(S,I2,2 H(S,I2 )

−1−

I2 ∗ I2,2

 = 1−

H(S,I2 ) ∗ ) H(S,I2,2

 H(S,I ∗

2,2 ) H(S,I2 )

−

I2 ∗ I2,2



 0.

We discuss two cases: – If .S2∗  S, from (H2 ), we have ∗ ) H(S2∗ ,I2,2 . ∗ H(S,I2,2 )

 . 1−

.

∗ ) H(S2∗ ,I2,2 ∗ H(S,I2,2 )

 1, and for .R01  1, we have

R01 − 1  0, and  ∗ ) H(S2∗ ,I2,2 S ˙ 1 − ∗ ∗ S H(S,I )  0 , hence, .L2  0. 2

2,2

H(S ∗ ,I ∗ )

H(S,I ∗ )

2 2,2 – If .S  S2∗ , from (H2 ), we have . H(S,I  1, since .R01  H(S ∗ ,I2,2  1 we ∗ ∗ 2,2 ) 2 2,2 )   ∗ ∗ ∗ ∗ H(S2 ,I2,2 ) 1 H(S2 ,I2,2 ) S 1 − H(S,I  0 , hence, .L˙2  0. have . H(S,I ∗ ) ∗ ) R0 − 1  0, and . 1 − S ∗ 2,2

2

2,2

From the above discussion, we deduce that,  1 , then .L˙2  0. We conclude that the strain 2 endemic equilibrium .E2 is globally asymptotically  stable when .R01  1 and .R02 > 1. if .R01

3.4 Global Stability of Both Strains Endemic Equilibrium For the global stability of .Et , we assume that the functions f and g satisfy the following conditions:  .

1−

∗ ∗ ∗ ∗ ∗ I2 H(S, I2 ) F(St , I1,t )  H(St , I2,t ) F(S, I1,t ) −  0, ∀S, I1 , I2 > 0 ∗ ) ∗ ∗ ) F(S, I ∗ ) H(S, I2 ) F(St∗ , I1,t I2,t H(St∗ , I2,t 1,t (H6 ) ∗ )  H(S ∗ , I ∗ ) ∗, I ∗ )  F(S F(S, I1,t t t 1,t 2,t −  0, ∀S, I1 , I2 > 0 (H7 ) . 1− ∗ ) ∗ ) H(S, I2 ) F(S, I1,t F(St∗ , I1,t

On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates. . .

245

Theorem 5 The both strains endemic equilibrium .Et is globally asymptotically stable if .R0 > 1. Proof Consider the following Lyapunov function: Lt = S − St∗ −  .

·

t

 t−τ1



S St∗

∗ ) F(St∗ , I1,t ∗ ) F(X, I1,t

∗ dX + eρτ1 I1,t 

I 1 ∗ ∗ + (1 − η1 )F(St∗ , I1,t )I1,t ∗ I1,t

 F(S(θ ), I (θ ))I (θ ) I 1 1 2 ∗ dθ + eρτ2 I2,t  ∗ ∗ ∗ ∗ F(St , I1,t )I1,t I2,t

∗ ∗ + (1 − η2 )H(St∗ , I2,t )I2,t



t

 t−τ2

 H(S(θ ), I (θ ))I (θ ) 2 2 dθ. ∗ )I ∗ H(St∗ , I2,t 2,t (19)

where .(x) = x − 1 − ln x, (x > 0). The derivative of .Lt is given by ∗ )    I∗ I∗ F(St∗ , I1,t ˙ + eρτ1 1 − 1,t I˙1 + eρτ2 1 − 2,t I˙2 S L˙ t = 1 − ∗ ) F(S, I1,t I1 I2

  F(S, I )I 1 1 ∗ ∗  + (1 − η1 )F(St∗ , I1,t )I1,t ∗ )I ∗ F(St∗ , I1,t 1,t

.

 FS(t − τ1 ), I1 (t − τ1 )I1 (t − τ1 )  − ∗ )I ∗ F(St∗ , I1,t 1,t

(20)

  H(S, I )I 2 2 ∗ ∗ + (1 − η2 )H(St∗ , I2,t  )I2,t ∗ )I ∗ H(St∗ , I2,t 2,t  HS(t − τ2 ), I2 (t − τ2 )I2 (t − τ2 )  − . ∗ )I ∗ H(St∗ , I2,t 2,t ∗ )I ∗ + (1 − η )H(S ∗ , I ∗ )I ∗ + ρS ∗ , It is easy to show that . = (1 − η1 )F(St∗ , I1,t 2 t t 1,t 2,t 2,t ∗ ∗ ∗ ∗ ∗ ρτ ∗ ∗ ρτ2 I ∗ . 1I .(1 − η1 )F(St , I 1,t )I1,t = ae 1,t and .(1 − η2 )H(St , I2,t )I2,t = be 2,t Thus

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∗ )  F(St∗ , I1,t S  L˙ t = ρSt∗ 1 − ∗ 1 − ∗ ) F(S, I1,t St

  F(S ∗ , I ∗ ) t 1,t ∗ − aeρτ1 I1,t  ∗ ) F(S, I1,t +

.

 F(S, I ∗ ) 1,t F(S, I1 )

+

 F(S(t − τ ), I (t − τ ))I (t − τ )  1 1 1 1 1 ∗ )I F(St∗ , I1,t 1

∗ − beρτ2 I2,t

 H(S(t − τ ), I (t − τ ))I (t − τ )    H(S ∗ , I ∗ ) t 2 2 2 2 2 2,t +  ∗ )I H(St∗ , I2,t H(S, I2 ) 2

∗ + aeρτ1 I1,t

∗ )  I F(S, I ) F(S, I1,t I1 1 1 + − 1 − ∗ ∗ F(S, I ∗ ) F(S, I1 ) I1,t I1,t 1,t

∗ + beρτ2 I2,t

 H(S ∗ , I ∗ ) F(S, I ∗ ) t 1,t 2,t H(S, I2 )

∗ ) F(St∗ , I1,t

+

∗ ∗ H(S, I2 )I2 F(St , I1,t ) I2 − ∗ −1 ∗ ∗ ∗ ∗ H(St , I2,t )I2,t F(S, I1,t ) I2,t

∗ ) ∗ ) ∗ ) F(S, I ∗ )  H(St∗ , I2,t H(St∗ , I2,t F(St∗ , I1,t 1,t ∗ + . − 1− + beρτ2 I2,t ∗ ) ∗ ) H(S, I2 ) F(St∗ , I1,t H(S, I2 ) F(S, I1,t

(21) Using the following trivial inequalities .1 − ∗ ) F(St∗ ,I1,t ∗ ) F(S,I1,t

∗ ) F(St∗ ,I1,t ∗ ) F(S,I1,t



< 0 for .S < St∗ . From (H4 ) we have .  F(S,I ∗ )  F(S,I1 ) I1 1,t 0 1 − F(S,I ∗ ) F(S,I1 ) − I ∗ 1,t

From 6 ) we have  H(S ∗ (H ,I ∗ ) F(S,I ∗ )

.



.

t

1−

I1 F(S,I1 ) ∗ F(S,I ∗ ) I1,t 1,t

+

∗ ) F(S,I1,t F(S,I1 )

and .1 − − 1 − II∗1 = 1,t

1,t

∗ ∗ H(S,I2 )I2 F(St ,I1,t ) 1,t ∗ ) ∗ )I ∗ F(S,I ∗ ) F(St∗ ,I1,t H(St∗ ,I2,t 2,t 1,t ∗ ) F(S,I ∗ ) ∗ ∗ H(St∗ ,I2,t H(S,I2 ) F(St ,I1,t ) 1,t ∗ ∗ ∗ ) ∗ ∗ H(S,I2 ) F(St ,I1,t H(St ,I2,t ) F(S,I1,t )

2,t H(S,I2 )

≥ 0 for .S ≥

St∗ ,

+



−1 = − II∗2  0,

−

I2 ∗ I2,t

2,t

also (H7 ), we have   from ∗ ) F(S,I ∗ ) ∗ ) ∗ ) ∗ )  H(S ∗ ,I ∗ ) H(St∗ ,I2,t H(St∗ ,I2,t F(S,I1,t F(St∗ ,I1,t t 2,t 1,t = 1 − . 1− − + ∗ ∗ ∗ ∗ ∗ ) H(S,I2 ) F(St ,I1,t ) H(S,I2 ) (1−η2 )H(S,I2 ) − F(S,I1,t ) F(St ,I1,t ∗ F(St∗ ,I1,t ) F(S,I ∗ )  0. 1,t

We conclude that the steady state .Et is globally asymptotically stable when .R0 > 1. 

4 Numerical Simulations In this section, numerical simulations will be carried out for two cases; the first one is to consider the problem (1) under the simplest two bilinear incidence rates .F(S, I1 ) = α1 S and .H(S, I2 ) = α2 S and the second case considers two non-

On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates. . . 5

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Fig. 1 The dynamics of the infection showing .Ef stability for .η1 = 0.9 and .η2 = 0.9

α2 S α1 S . For all and .H(S, I2 ) = 2 1 + κ2 I22 1 + κ1 I1 our numerical simulations, we will choose the following initial conditions: .S(θ ) = 3; .I1 (θ ) = 1; .I2 (θ ) = 1 and .R(θ ) = 0 for .θ ∈ [−τ, 0]; with .τ = max (τ1 , τ2 ). Also we will choose for all our numerical tests the following parameters: . = 1, .α1 = 0.65, .α2 = 0.6, .ν1 = 0.4, .ν2 = 0.25, .ρ = 0.22, .d1 = 0.1, .d2 = 0.07, .κ1 = 4.5, .κ2 = 3.5, .τ1 = 3 and .τ2 = 3. For each numerical simulation, we will change only the therapy efficiencies .η1 and .η2 . Figure 1 shows the time evolution of the infection for .η1 = 0.9 and .η2 = 0.9. Within these chosen parameters, the strain reproduction numbers are .R01 = 0.212 and .R02 = 0.261. Both of the reproduction numbers are less than unity which makes us predict, theoretically, the disease extinction. This figure confirms the theoretical prediction, since, the curves converge toward the disease-free equilibrium .Ef =  ( ≈ 4.545; 0; 0; 0). Indeed, the susceptible individuals reach their maximal ρ values, while each strain infected persons vanish. In this numerical result, the treatment efficiency is very high; therefore, to eradicate the infection, it would preferable to give treatment to the entire population with high efficiency. Figure 2 shows the time evolution of the infection for .η1 = 0.9 and .η2 = 0.001. Within these chosen parameters, the strain reproduction numbers are .R01 = 0.212 and .R02 = 2.607. Here, the first strain reproduction number is less than unity, while the other strain reproduction number remain greater than one. From the theoretical point of view, only the first strain infection will know an extinction. This figure confirms the theoretical prediction, since, the curves converge toward the first strain endemic equilibrium .E1 = (2.324; 0; 0.350; 0.398) for the bilinear monotonic incidence rates .F(S, I1 ) =

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1.2 1

2.5 0.8 2

0.6 0.4

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Fig. 2 The dynamics of the infection showing .E1 stability for .η1 = 0.9 and .η2 = 0.001 2

3 2.5

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Fig. 3 The dynamics of the infection showing .E2 stability for .η1 = 0.001 and .η2 = 0.9

case and toward .E1 = (2.884; 0; 0.262; 0.298) for the non-monotonic case. Indeed, the first strain individuals (second component) vanish. In this numerical result, the treatment efficiency is very high only for the first strain individuals. Figure 3 shows the time evolution of the infection for .η1 = 0.001 and .η2 = 0.9. Within these chosen parameters, the strain reproduction numbers are .R01 = 2.118 and 2 .R = 0.261. Here, the second strain reproduction number is less than unity, while 0

On Time-Delayed Two-Strain Epidemic Model with General Incidence Rates. . . 3

2.5

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Fig. 4 The dynamics of the infection showing .Et stability for .η1 = 0.001 and .η2 = 0.001

the other strain reproduction number remains greater than one. From the theoretical point of view, only the second strain infection will know an extinction. This figure confirms the theoretical prediction, since the curves converge toward the second strain endemic equilibrium .E2 = (2.145; 0.379; 0; 0.689) for the bilinear case and toward .E2 = (2.84; 0.268; 0; 0.488) for the non-monotonic case. Indeed, the first strain individuals (second component) vanish. In this numerical result, the treatment efficiency is very high only for the first strain individuals. Finally, Fig. 4 shows the infection dynamics for .η1 = 0.001 and .η2 = 0.001. In this case, both the strains’ reproduction number are greater than unity, .R01 = 2.118 and .R02 = 2.607. Here, it corresponds to the persistence to disease in both strains. The curves of the figure confirm that by converging toward a strictly positive levels. Hence, the therapy efficiency can be considered as an important strategy to control the infection spread.

5 Conclusion Time-delayed two-strain infection model with the presence of two generalized incidence rates and therapy have been studied in this work. The wellposedness of our problem is established in terms of proving the existence, the positivity, and boundedness of solution. The problem admits four steady states, namely, the disease-free equilibrium, the first strain endemic equilibrium, the second strain endemic equilibrium, and the both strains endemic equilibrium. The global stability of the problem has been proven by constructing some suitable Lyapunov functional. It was remarked that the global stability of the equilibria depends mainly on the

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each strain basic reproduction number. Numerical tests are carried out in order to show the stability of the equilibria for two cases of incidence functions, namely, two bilinear incidence functions and two non-monotonic incidence rates. It was remarked that therapy efficiency plays an essential role in decreasing the number infection individuals. More precisely, when the first strain therapy efficiency is increased, the first strain infected individuals are reduced considerably. Similarly, the second strain therapy efficiency when is increased the second strain infected individuals are diminished remarkably. In addition, it was observed that any change of the stain therapy efficiency may change the nature of the equilibrium stability. In order to control the spread of the infection in a two-strain environment, it would be important to act on both strains treatment efficiencies.

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Clustering of Countries Based on the Associated Social Contact Patterns in Epidemiological Modelling Evans Kiptoo Korir and Zsolt Vizi

1 Introduction The spread of respiratory infections such as measles, influenza, and the recent COVID-19 occur predominantly through social contacts between susceptible and infected persons. Quantification of these contacts relevant is significant to predict disease dynamics and effect of introduced strategies that aim to control and curb further spread [15]. Epidemic models (especially deterministic ones) are widely used to understand and predict the disease dynamics and the impact of new strategies and interventions such as lockdown, school closure, or vaccination to control infectious disease transmission. Using these models, individuals are assumed to follow certain age-specific contact patterns [13, 19, 22, 25, 28]. These patterns are often inferred from diary-based surveys of self-recorded contacts which captures demographic information (age, household size, gender, and occupation) [12]. Such surveys are rare and limited in nature due to economic cost involved. The rate of contacts between people is not same as it depends on factors such as individual behaviors, age groups, gender, and location of the contact (home, school, workplace, community, or other setting) [12, 13, 16, 19, 22, 25]. Age group is a critical determinant of disease transmission, and it is usually represented by contact matrices. The elements of these matrices are the average number of contacts an individual in some age group i makes with other individuals belonging to each age group j within a specified period of time, and they are commonly assortative [12, 13, 16, 19, 22, 25]. Age-structured models provide an understanding of age groups that are key epidemic drivers and the portion of people who are vulnerable to the disease. For an outbreak of infectious diseases such as COVID-

E. K. Korir · Z. Vizi () University of Szeged, Bolyai Institute, Szeged, Hungary e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_15

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19 and influenza, public health introduces non-pharmaceutical interventions (school closure, contact tracing, social distancing, lockdowns, travel restrictions), or public health interventions such as vaccination and antiviral use to alter the pattern of contacts [19]. Multiple studies targeting to estimate social contacts have been published. The most referred study, POLYMOD, estimated contact matrices in 8 European countries based on surveys that involved 7290 participants with reported 97,904 contacts [19]. Authors in [21] estimated synthetic contact matrices for 152 countries in 2017 based on [19], demographic data, surveys, and other sources. The contact matrices were then updated in 2021 with recent data to include 177 countries with introduction of countries such as Hong Kong and the People’s Republic of China [22]. Fumanelli et al. [6] estimated the contact matrices by constructing virtual populations in 26 European countries using detailed census and demographic data, an approach that can be adopted in the absence of specific experimental data. Iozzi et al. in their paper [10] entitled “little Italy” used agent-based model to compute social contact matrices by simulating individual-based model using demographic and Italian use data. Country-specific contact matrices have also been estimated, for instance, in Uganda [15], Kenya [11], Zimbabwe [17], India [14], China [23], Vietnam [9], Peru [7], and Russia [2]. As social contact patterns depend on the region of investigation, there is no evidence on spatial comparison of social contacts relevant for transmission between countries. Countries with unfavorable social structures and large population with risky behaviors are likely to have high transmission rates. Understanding this behavior is critical for timely introducing non-pharmaceutical interventions, monitoring the impacts of such decisions, and possible reintroduction of new measures in the event of cases resurgence in the countries [16]. During the COVID-19 world pandemic, decision-makers in countries, where an outbreak occurred later in time, monitored the actively intervening countries and analyzed the applied strategies to make efficient decisions with proper timing. In these situations, it is crucial to know which regions have to be monitored and which practices are recommended to be adapted. For this end, social contact patterns made in different countries are grouped into clusters and then examined to understand which countries have similar mixing patterns. Here, we employ hierarchical clustering technique to determine the clusters of countries based on only the social contact patterns. The approach can be extended to include country characteristics such as gross domestic product (GDP), population density, life expectancy, total fertility rate, mortality rates, and other metrics. Since optimization of learning algorithms are highly affected by dimensionality of the data, we handle this problem with usage of dimension reduction techniques. Additionally, for ensuring comparability of the underlying contact matrices, we use epidemic models to normalize these matrices as a preprocessing step before the abovementioned steps. The structure of the paper is the following: in the second chapter, we introduce the underlying data and the different pieces of the proposed framework, namely, the concept of an age-structured epidemic model, the dimension reduction to avoid

Clustering of Countries Based on the Associated Social Contact Patterns in. . .

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curse of dimensionality, and possible approaches for clustering the data points represented by a reduced feature vector. The third section gives a demonstration about how the pipeline works with specific setup of the framework. In the last section, we discuss the considerable options for modifying and improving the presented methods.

2 Methods and Materials 2.1 Data Prem et al. [22] updated synthetic contact matrices based on [19] with the most recent data from sources such as the United Nations Population Division, International Labor Organization, World Bank databases, demographic data, and surveys. The authors estimated contact patterns in four different locations (home, work, school, and other locations) for non-POLYMOD countries and made comparison to prior estimated contact matrices. Contact patterns were then projected for a total of 177 countries. We used prior results of this constructed social contact matrices in [22] but targeting only all the European countries. The social contact matrices are of size .16 × 16 representing four settings (home, school, work, and other location) among 16 age groups: 0–4, 5–9, 10–14, .. . . , .75+. A total of 39 countries in Europe were available for analysis (countries are listed in Appendix). We considered the complete social contact mixing in the form of a full social contact matrix: MC = MC,H + MC,S + MC,W + MC,O

.

where .MC,X is the social contact matrix with type .X ∈ {H, W, S, O} from country C ∈ {1, ..., 39}. When social contacts are measured empirically, reciprocity is usually reported [12, 13, 16, 19, 22, 25, 28] since contacts are mutual. We corrected each matrices .MC to ensure that the total number of contacts from age group j to i is equal to the total number of contacts from age group i to age group j : .

(i,j )

MC

.

(i,j )

(j,i)

Pi = MC

Pj

(1)

where .MC is the element of .MC in the ith row and the j th column .Ni and .Nj represent the number of individuals in age class i and j , respectively. Specifically, we want to work with contact matrices satisfying Eq. (1); thus we applied symmetrization for the underlying data to ensure the consistency of total number of contacts between two age groups:

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Fig. 1 Age-specific contact matrices in Hungary. Mixing patterns by age at home, school, work, and other places. The full matrix is obtained as the sum of the matrices represented in the four settings (home, school, work, and other places). In the horizontal axis, we have the age of the participant that was involved in the survey, and vertical axis shows the age of the reported contact. The elements of the matrices are the average number of contacts in the age group of the participant, which increases from dark blue to red on the images

MC =

.

 1  (i,j ) (j,i) MC Pi + MC Pj 2Pi

where .MC is the full contact matrix now being corrected for reciprocity. Its elements are average number of contacts between age group i and j with very strong contacts along the diagonals characterizing strong age-assortative mixing (interactions between people of same age group). The off-diagonals capture intergenerational mixing (interactions between people of different age groups, for instance, children and their parents). Figure 1 shows the contact matrices obtained for Hungary as an example of the social mixing patterns in European countries. The matrices are highly structured and varied considerably between settings. Reported contacts at home in Fig. 1 show a dominant diagonal (assortative pattern) representing contacts with people of the same age group such as young individuals. The off-diagonals reports intergenerational mixing contacts, for instance, between children and their parents. Notably, these contacts are least pronounced for people aged over 60. The contacts at school in Fig. 1 is evident as young people (people with age 0–19 years) interact mostly with other students of the same age and school. In the workplaces displayed in Fig. 1, we can see that most contacts occur between people in working population age (individuals with age group 20–64 years). These contacts are less assortative than at home. Overall contacts in other places (excludes home, work, and school) shows age assortative for younger groups and less assortative for older groups (Fig. 1). The full contacts in Fig. 1 which combines all the contacts in the four settings shows strong contacts between people of age 15–59 years in the country.

2.2 Epidemic Model Epidemic models particularly age-structured models provide a better understanding of global spread and enable designing public health interventions. Deterministic compartmental models such as variants of SI R, SEI R, and SI S are formulated as a system of ordinary differential equations. The proposed framework can consider any

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age-structured model for analyzing the similarities between countries with different contact matrices. As a demonstration, we consider the model in Fig. 2 from [25]. The model has 15 compartments. S denotes the susceptibles, i.e., those who are at the risk of contracting the disease. Latents .(L) are infected individuals but do not show symptoms and cannot infect. .Ip compartment contains the pre-symptomatic individuals who do not show symptoms but are able to infect. Asymptomatic (or mildly symptomatic) and symptomatic infectious individuals, denoted by .Ia and .Is , respectively, are divided into three groups (enabling the modelling of recovery time by an Erlang-distributed variable). Individuals from the last stage of asymptomatic compartment will all recover and hence proceed to the recovered class R, while those in the last stage of symptomatic compartment may recover without the need for hospital treatment (and thus move to R) or become hospitalized, either with normal hospital care (.Ih ) or intensive critical care (.Ic ). Those who need normal hospital care will recover, and those at the intensive care units may overwhelm the disease proceeding first to a rehabilitation unit (.Icr ) before complete recovery, or for them fatal outcome might occur (D). The example model has an age-structured setup considering heterogeneity of contacts in the population via contact matrices and introduces age-specific parameters for some transitions in the transmission diagram. For the governing equations, see Appendix 2. In this demonstration, we assume that the model parameters are the same for all countries except the baseline transmission rate, which is strongly related to the corresponding contact matrix; thus we denote by .(β0,C ) the baseline transmission rate for country C. An important quantity to calculate in epidemiology is the basic reproduction number .R0 , which represents the number of secondary infections caused by a single infective introduced into a wholly susceptible population [5, 12, 13, 16, 25]. For an age-stratified model, the basic reproduction number can be obtained via using the Next-Generation Matrix (shortly NGM) method. As a summary (see details

Fig. 2 Flowchart of the transmission model from [25] used for demonstrating the proposed framework

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in Appendix 2), .R0 can be calculated as the dominant eigenvalue of the modelspecific NGM and particularly baseline transmission rate can be take out in the computations. Since calculation of the NGM depends on the underlying contact matrix, thus for a fix value of .R0 , we can determine the baseline transmission rate .β0,C for each country .C ∈ {1, 2, . . . , 39} (assuming that model parameters and contact matrix are known for the calculations). Using these transmission rates, we apply a standardization to all full contact matrices enabling comparability of the countries for a fixed value of .R0 , and we denote this new matrix by .SC : SC = β0,C · MC .

.

This standardization is used as a normalization for transforming matrices to be distributed on the same interval. Significantly large differences can appear for the matrices depending on how much data was available for the estimation in [22]. This data preprocessing step is needed for the next data science-specific parts of the framework.

2.3 Dimensionality Reduction Data with high-dimensional input vectors (i.e., order of sample size is at least the order of the dimensionality) pose problems to learning methods, since training performance of these methods is affected by the curse of dimensionality [3]. This severe problem can be solved by the introduction of dimension reduction technique or feature selection [1, 8]. In the classical theory of clustering, the features are arranged into vectors, and we often use projection-based algorithms for reducing the dimensionality of this vector. Principal component analysis (shortly PCA) is a popular technique to project the data vectors onto an affine hyperplane. The principal components are the vectors spanning an optimal projection plane and can be computed effectively using covariance matrix or SVD (former solves the optimization problem to maximize explained variance; latter gives the optimal low-rank approximation for the data matrix). We will refer to the classical PCA as 1D PCA. Matrix-valued data are becoming increasingly common with advancement in the technology, e.g., images in computer vision problems. Driven by 1D PCA, several techniques that aim to preserve the 2D structure of the matrix have been developed [27]. These includes 2D variants of PCA, which consider projection along rows/columns of the matrices and .(2D)2 PCA, which applies projection along both dimensions. Recent approaches were developed such as the population value decomposition (PVD) that is precisely equivalent to a two-step SVD [29]. Using .(2D)2 PCA approach, we concatenate each country-specific full contact  ] ∈ R(39·16)×16 . For the matrices to obtain the matrix .Acol = [M1 , M2 , . . . , M39 demonstration, in the column direction, two principal components were retained by projecting A onto the plane spanned by the columns of .Q ∈ R16×2 , thus having

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Aˆ col = Acol Q ∈ R(39·16)×2 .

.

In the next step, along the row direction, we retained two principal components as well and projected the concatenation of the transposed matrices .Arow = [M1 , M2 , . . . , M39 ] ∈ R(39·16)×16 using matrix .P ∈ R16×2 obtaining Aˆ row = Arow P ∈ R(39·16)×2 .

.

Using the two projection matrices P and Q, we project all the matrices .MC with size .16 × 16 applying both P and Q to get the reduced matrix Mˆ C = Q MC P ∈ R2×2 ,

.

where .C ∈ {1, 2, . . . , 39}. In the clustering, for country C we use the vector .mC ∈ R4 calculated by flattening the .Mˆ C matrix as a feature vector. Since the contact matrices satisfy Eq. (1), the lower and upper triangular parts of a contact matrix are not independent from each other. For applying 1D PCA technique, it is beneficial to consider only the independent elements of the contact matrix; thus we will take only the upper triangular part consists of 136 elements. In summary, the data matrix for 1D PCA will be .A ∈ R39×136 , where Cth row of A is the flattened upper triangular part of matrix .MC . In the demonstration, we can compare the result from 1D PCA projecting this data matrix to .A˜ ∈ R39×4 and use   ∈ R4 associated to country C in the clustering. the row vector . A˜ (C)

2.4 Clustering Algorithm Grouping of data points has become significant in various fields such as health, medicine, social, and spatial [24]. Solving this problem requires, for instance, a clustering technique to identify groups within a data set such that the similarity within the cluster is high and is low in-between [1]. Notably, there are several methods for clustering such as distance-based, connectivity-based methods, densitybased, and probabilistic techniques [1, 3, 8]. Since we have a very small data set (i.e., number of data points is very low), the approaches considering dense set of data points do not work well now. In this study, agglomerative hierarchical clustering algorithm has been used due to its wide range of applications, simplicity, and ease of implementation relative to other clustering methods. This technique is expected to give better results in comparison to other methods, where large amount of data is considered. In hierarchical clustering, the closest sets of clusters are merged at each level using a linkage method, and then the dissimilarity matrix is updated correspondingly [1, 4, 18, 20, 24, 26]. This iterative process continues until only one maximal cluster remains.

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Several proximity measures are available for this clustering technique, e.g., single, complete, average, and ward’s criterion. We have used complete linkage method as it takes the cluster structure into consideration (nonlocal in behavior) and generally obtains compact-shaped clusters [1, 18]. For any clusters .Ci and .Cj , this linkage measures the similarity of clusters as the similarity of their most dissimilar members, that is, it considers the longest distance among all their observations: Dcomplete (Ci , C j ) = max{d(x, y) : x ∈ Ci , y ∈ Cj }

.

where d is a chosen distance measure. In this work, we have considered the Euclidean distance, but other .Lp norms or specialized distance metrics can be defined (in many applications, the distance metric is the key to a good clustering). Using the dissimilarity matrix and the linkage method, we can construct a dendrogram showing the agglomeration process during the running of the algorithm. The optimal number of clusters is obtained by manually cutting the tree at any given level and obtaining the clusters correspondingly without rerunning the algorithm. The rule of thumb we applied considers the longest vertical distance without any horizontal line passing through it, i.e., cutting the dendrogram where we have the largest gap between two successive merges [1].

2.5 The Proposed Framework The complete pipeline of the framework can be seen in Fig. 3. The main input of the pipeline is the list of country-specific contact matrices (or any list of contact matrices for which the grouping needs to be executed). The symmetrization of the matrices is done at loading the data to ensure consistent total number of contacts between the age groups. The resulted matrices inputted to an epidemic model, which is specified by the user/domain expert, and then the base transmission rate .β0 is calculated along a concept. The concept determines the indicator for the calculation of .β0 : in this paper, we use .R0 to calculate this parameter, but a viable concept would be to obtain that .β0 for which the final epidemic size or final number of deaths is a pre-defined fraction of the population (this can be done for all countries). Using the symmetrized matrices and the previously calculated .β0 s, we apply the standardization for the matrices via scaling the elements of them. In the next phase of the procedure, we apply the data science-related steps on the standardized matrices. First a dimension reduction is applied to avoid curse of dimensionality in the next steps. The outputted low-dimensional vectors (i.e., the feature vectors) are used in the clustering algorithm, which is preferred to be a connectivity-based approach for very small datasets. In Fig. 3, yellow-colored shapes show the generic parts of the pipeline (i.e., these parts are determined by the expert specifying the choices based on the needs and goals). The rectangular shapes give the output data from the modules depicted by ellipsoids.

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Concept

MC

Model

Base transmission rate

Standardization

SC

Dimension reduction

Parameters

Cluster ID

Clustering

Feature vector

Fig. 3 Flowchart showing a summary of the proposed framework. Rectangles show the output of a module depicted by ellipsoids, which contain function calls executing a specific part of the pipeline. The yellow shapes are generic and can be replaced by considering improved/applicationspecific alternatives. As it can be seen, the input of the procedure is the contact matrices and cluster correspondence computed at the output. The concept associated with the model describes along which indicator the standardization has to be done

The proposed framework is implemented in Python using object-oriented programming to keep the code generic and easily extendable. We used open source packages to solve differential equations and eigenvalue problems and execute dimensionality reduction and clustering. The code is publicly available in Github [30].

3 Results In this section, we demonstrate the previously introduced framework on the agestructured model of [25] with conceptually simple and correct algorithms for dimensionality reduction and clustering. Since [22] offers contact matrices for 16 age groups, we aligned the age-dependent parameters of the model. Here we investigate a total of 39 countries in Europe (see Appendix 1) to avoid significant differences between the countries from viewpoint of economics, location, demographics, and culture. Following the methodology, both 1D PCA and .(2D)2 PCA approaches were applied to project into .R4 to highlight importance of the proper reduction for matrix-valued data. The clustering algorithm was chosen to be a connectivity-based one with agglomerative execution and using Euclidean distance and complete linkage for merging the clusters during the procedure. For

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Fig. 4 The pairwise distance matrices computed for all countries based on the feature vector from the dimensionality reduction step. The left figure corresponds to 1D PCA, while the right is calculated from the result of the .(2D)2 PCA approach. Here we used Euclidean distance during the computations. We applied the dendrogram from the hierarchical clustering to arrange the rows and columns of the matrices providing a nicer presentation of the elements

the simulation we chose .R0 = 2.2 for calculating country-specific transmission rates. Applying standardization for the matrices and executing dimensionality reduction, the computed feature vectors can be used to calculate pairwise Euclidean distance for the countries. For a nicer representation, the rows and columns of the distance matrix can be permuted using the dendrogram from the hierarchical clustering. The left figure in Fig. 4 shows the distance matrix computed from the result of 1D PCA: as it can be seen, countries with dark-blue color entries such as Bulgaria, Latvia, and Belarus have very close social mixing patterns, and such countries are expected to belong to the same cluster. Notably, countries such as Netherlands, Albania, and Bosnia have orange and red cell colors, respectively, with most countries indicating somehow different social contact patterns. Such countries can have different cluster and seemingly different intervention strategies. The same intuition applies to .(2D)2 PCA approach in Fig. 4. The advantage of using hierarchical clustering method is that it allows for manually cutting the hierarchy at any given level and obtaining the clusters correspondingly just by defining a threshold. The choice of threshold depends completely on the user. Dividing the dendrogram from 1D PCA at a cluster distance of 0.25, three separate clusters were formed (Fig. 5), Netherlands became a single cluster. In the same way, cutting the dendrogram at a distance of 5.5 between the clusters calculated on reduced data from .(2D)2 PCA technique, the dendrogram formed three meaningful clusters (Fig. 6). The final list of clusters is provided in Tables 1 and 2 (for the former one, the one-element cluster with Netherlands was omitted from the table). Notice, that cluster sizes are highly imbalanced for 1D PCA, which seems to show the flaw of this reduction approach compared to the result using .(2D)2 PCA, where cluster sizes are more comparable, which indicates better separation after the projection.

Fig. 5 Dendrogram associated with the clustering of countries using feature vector generated by 1D PCA. Vertical axis indicates the distance between the two clusters being connected. The agglomerative manner of the algorithm can be read from following the tree from bottom to top

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Fig. 6 Dendrogram associated with the clustering of countries using feature vector generated by .(2D)2 PCA. Vertical axis indicates the distance between the two clusters being connected. The agglomerative manner of the algorithm can be read from following the tree from bottom to top

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Table 1 Clusters produced using 1D PCA. Netherlands is one element cluster, and thus it is omitted in this listing. Clusters produced by the agglomerative hierarchical clustering algorithm based on social contact patterns for the 39 European countries applying projection 1D PCA. Clusters 1 and 2 have 5 and 33 elements, respectively. Netherlands is one element cluster, and thus it is omitted in this listing Cluster 1 Albania, Bosnia, Armenia, Malta, North Macedonia

Cluster 2 Luxembourg, Ireland, Italy, Lithuania, Montenegro, Denmark, Austria, Iceland, France, Switzerland, Estonia, Czechia, Slovenia, Finland, Spain, Greece, Portugal, Belgium, Germany, Latvia, Russia , Poland, Belarus, Bulgaria, Ukraine, Cyprus, Sweden, United Kingdom, Croatia, Hungary, Serbia, Romania, Slovakia

Table 2 Clusters produced by the agglomerative hierarchical clustering algorithm based on social contact patterns for the 39 European countries applying projection .(2D)2 P CA). Clusters 1, 2, and 3 have 16, 12, and 11 elements, respectively Cluster 1 Albania, Ireland, Cyprus, Bulgaria, Slovakia, Belarus, Iceland, Russia, Armenia, North Macedonia, Poland, Romania, Ukraine, Montenegro, Lithuania, Austria.

Cluster 2 Italy, Serbia, Portugal, Greece, Croatia, Belgium, Denmark, Germany, Finland, Malta, Hungary, United Kingdom.

Cluster 3 Bosnia, Spain, Latvia, Slovenia, Czechia, Estonia, Netherlands, Sweden, Switzerland, France, Luxembourg.

Fig. 7 Standardized contact matrices correspond to Armenia (left), Belgium (center), and Estonia (right), selected from different clusters resulted by clustering based on the feature vectors generated by the .(2D)2 P CA technique. These countries are located “in the middle” of the ordered list of the cluster elements, based on the dendrograms. Differences between these matrices appear along the elements in the main diagonal (larger number for younger and old age groups for Belgium), in the elements along diagonals parallel to the main one (expressing higher contact rates between “adult children” and older/retired parents) and middle square of the matrices (which come mainly from work contact)

Figure 7 shows one element from each of the clusters calculated from reduced vectors resulted by .(2D)2 PCA. These countries (Armenia, Belgium, and Estonia) were chosen, since they are located in the middle of the blocks along the horizontal axis. Considering these matrices, we see that differences occur between the matrices along the main diagonal (for Armenia, the elements for the older age groups are not as strong as the rest of the elements are not that large), along the diagonal parallel to the main diagonal (these express the contacts between the adult parents and retired

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grandparents; here Belgium and Estonia show more concentrated rates) and in the middle square of the matrix (this part comes mainly from workplace contacts, the distribution of the elements differ for all three of the samples). It is important to mention that distribution of the elements is directly related to the statistical method described in [22], since stronger smoothing effect can be recognized for estimation with less amount of data; thus distance measuring will be highly affected by the quality of the estimations. Nevertheless, the dimensionality reduction techniques developed for matrices aim to preserve the patterns in the structure of the matrices, for which the previously mentioned set of elements (diagonals, middle submatrix) give examples.

4 Discussion Since emerging infectious diseases do not occur at the same time in all countries, it is important to understand which countries are affected in the same way with a non-pharmaceutical intervention. The government of a country where the outbreak happens later in time can benefit from this information to make better decision in prevention and damage control. This paper proposes a framework which can support decision-makers in determining, which other countries have to be primarily monitored and analyzed based on social mixing patterns. The underlying pipeline assumes only the contact matrices in the population and transforms it via an epidemic model and the concept of the analysis and then applies dimension reduction on this data and executes clustering on the projected data points associated with the considered countries. A realization of the framework was demonstrated with simple and correct approaches for the generic parts such as the epidemic model, the dimensionality reduction and the clustering. In this procedure we can replace the concept of the analysis, since .R0 describes only the transmission spread of a disease, but if a country primarily wants to minimize the number of fatal outcomes or the number of patients admitted to the hospital, these aspects can be also used for determining the country-specific scaling parameter in the abovementioned standardization. As an example, for all countries we can determine that .β0 , for which the ratio of fatal outcomes is a fixed proportion. Clearly, the epidemic model can be replaced by any pre-defined model written for any infectious disease. For dimensionality reduction we applied a simple technique developed for matrices, but other approaches might be suitable for this part; see [27] for similar but more sophisticated methods, but convolution-based techniques can be developed, or modified optimization problem can be written down for the matrix reconstruction technique. For very small data, connectivity-based algorithms perform well, and many different approaches are available (agglomerative and divisive types, different similarity measures, and linkage methods). Globally, we can extend the feature vectors after the dimension reduction with other descriptors about the countries, which can be relevant in the analysis, e.g., some economical indicators, which can be relevant for the NPI strategy. Finally, the full contact matrix

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used as a main input to the procedure can also be computed with other weighting in the summation to give more importance to specified type of contacts, e.g., home contacts are really important, since typical NPIs reducing contacts (school closure, home office, closing malls) leave this contact unchanged (or even slightly increase them). Acknowledgments This research was supported by project TKP2021-NVA-09 and implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.

Appendix 1: First Appendix List of the European Countries We considered the following European countries in the demonstration: Albania, Armenia, Austria, Belarus, Belgium, Bosnia, Bulgaria, Croatia, Cyprus, Czechia, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Montenegro, Netherlands, North Macedonia, Poland, Portugal, Romania, Russia, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Ukraine, United Kingdom.

Appendix 2: Second Appendix The Governing Equations of the Epidemic Model The governing equations of the disease model described in Sect. 2.2 take the form ⎤ ⎡ 16 3 3   S i (t)  (k,i) ⎣ k k k · M Ia,j (t) + Is,j (t)⎦ .S (t) = − β0 Ip (t) + infa Ni i

k=1

⎡

j =1

j =1

⎤ 16 3 3 i (t)    S  k k · M (k,i) ⎣Ipk (t) + infa Ia,j (t) + Is,j (t)⎦ − 2αl Li1 (t) Li1 (t) = β0 Ni k=1

 Li2 (t)

= 2αl Li1 (t) − 2αl Li2 (t),



Iai (t) = 2αl Li2 (t) − αp Ipi (t) 

i i (t) = p i αp Ipi (t) − 3γa Ia,1 (t) Ia,1 

i i i (t) = 3γa Ia,1 (t) − 3γa Ia,2 (t) Ia,2

j =1

j =1

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i i i Ia,3 (t) = 3γa Ia,2 (t) − 3γa Ia,3 (t) i  Is,1 (t)

(2)

i = (1 − p i )αp Ipi − 3γs Is,1 (t)



i i i (t) = 3γs Is,1 (t) − 3γs Is,2 (t) Is,2 

i i i (t) = 3γs Is,2 (t) − 3γs Is,3 (t) Is,3 

i (t) − γh Ihi (t) Ihi (t) = hi (1 − ξ i )3γs Is,3 

i (t) − γc Ici (t) Ici (t) = hi ξ i 3γs Is,3 

Icri (t) = (1 − μi )γc Ici (t) − γcr Icri (t) 

i i R i (t) = 3γa Ia,3 (t) + (1 − hi )3γs Is,3 (t) + γh Ihi (t) + γcr Icri (t) 

D i (t) = μi γc Ici (t),

where the index .i ∈ 1, ..., 16 represents the corresponding age group. As it can be seen, the model contains age-dependent parameters (probabilities p, h, .ξ , .μ, for which upper index shows the age group) and age-independent ones (fraction .infa and transition parameters .α and .γX , where .X ∈ {a, s, h, c, cr}). Notation here is aligned with the parameter file located in the repository of the framework. Here .infa denotes the relative infectiousness of .Ia compared to .Is , for more details about the other parameters and the methodology for parametrization, see [25].

Next-Generation Matrix To calculate .R0 for the previously mentioned epidemic model, we consider the infectious subsystem for i

i

i

i

.L1 (t), L2 (t), Ip (t), Ij (t),

with .j ∈ {a, s} × {1, 2, 3}, i ∈ {1, ..., 16}, thus .X(t)

 i (t) I i (t) I i (t) I i (t) I i (t) I i (t) = Li1 (t) Li2 (t) Ipi (t) Ia,1 a,2 a,3 s,1 s,2 s,3

and linearization gives .X



(t) = (β0 · T + ) · X(t),

where .T ∈ R144×144 is the transmission part and . ∈ R144×144 represents the transition mechanisms in the model. The matrix . is a block-diagonal matrix, where blocks have size of .9 × 9 containing transition parameters related to the linear

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terms of the system. On the other hand, the transmission matrix T is partitioned into blocks of size .9 × 9, and each of this blocks have nonzero elements only in their first rows, since transmission between individuals affects only the classes i .L , i ∈ {1, 2, . . . , 16}, and these nonzero elements are related to the corresponding 1 elements of the contact matrix and the transmission-related parameter .infa . The Next-Generation Matrix (shortly NGM) can be calculated as .NGM

= −β0 · T  −1 ,

and the basic reproduction number is the dominant eigenvalue of the NGM, i.e., .R0

= β0 · ρ(−T  −1 ).

On one hand, the model parameters are the same for all countries in the paper; on the other hand, this calculation has to be executed for each countries separately, since social contact matrices (thus T matrices) are different. For more details about NGM method, see [5].

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Multiple Predation on Prey Herding and Counteracting the Hunting Luca Bondi, Jacopo Ferri, Nicolò Giordanengo, and Ezio Venturino

1 Introduction In this paper we continue to analyze possible social behavior in animals. Specifically, this is accounted for by their possible gathering together in herds. Some previous investigations have taken into consideration various situations that can arise, which encompass all possible two populations mutual interactions: predatorprey, symbiosis, and competition. In a more recent investigation, [1], the focus has been on the possible defensive attitude that a large pack of prey may have in response to predators’ attacks. We extend now this feature to explore a richer system, a trophic chain encompassing three populations. The origin of this field of research dates back to the paper [2]. Various extensions of these ideas to the case of ecoepidemic situations, namely, interacting populations among which a transmissible disease is found, are contained in [3, 4, 8, 10, 13]. An extension to space of these ideas is found in [19] and in other more recent investigations [12, 18]. In [19] a shortcoming of these models near the origin is emphasized, and a way of overcoming it is investigated in [6], where a reformulation through a better function is proposed.

Work partially supported by the project “Metodi numerici per l’approssimazione e le scienze della vita” of the Dipartimento di Matematica “Giuseppe Peano”. L. Bondi · J. Ferri · N. Giordanengo Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Torino, Italy E. Venturino () Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Torino, Italy INdAM Research Group GNCS, Torino, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_16

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The most striking remark on these models of interaction type is contained in [16], where tristability of a model with two competing populations is shown to arise. The specific choice of a square root behavior is removed, allowing the consideration of herds even of fractal dimension, in [7, 9]. Also, generic response functions have been used in this framework, [20], as well as other functional responses, [11]. In [14, 15], the mechanism of herd formation is investigated from the single individual perspective. These ideas are finally revisited in terms of a mechanistic approach in [5]. In this paper we consider interactive populations, of predator-prey type. In particular in [1], the following model is considered for the prey N and the predator P : √   dN N a NP . = rN 1 − − dt KN 1 + bN √   dP P ea NP = sP 1 − + KP 1 + bN dt

(1)

It exhibits two main features. As in [2] the prey are assumed to gather in herds and are hunted by individual predators, thereby explaining the origin of the square root term. It expresses the fact that the predators mainly hunt the prey visible to them, which are on the boundary of the herd. The second feature has to do with another prey behavior that here is assumed to have a response to predators’ attacks. The prey in large numbers are able to defend themselves, and therefore the success rate of the predators attacks decreases with the size of the prey herd. The fraction in the last terms of both equations exhibits the feature of a decrease to zero when N tends to infinity. Here we consider a modification of the above model, namely, a chain, where a specialist superpredator on both species hunts both the prey and the intermediate predator. The latter instead is a generalist predator, feeding on other sources modeled implicitly by the logistic term in the second equation. The paper is organized as follows. In the next section, we formulate the model. In the following one, the equilibria of the system are determined, and their feasibility and stability are assessed. The paper concludes with some remarks comparing these results with other ones of similar models.

2 The Model Continuing to name the prey N and their predators P , we introduce also the superpredator population H , specialist on both N and P , hunting them on an individual basis. Furthermore, we assume here also a feeding saturation when the superpredator has too many intermediate predators available, this fact being expressed by the Holling type II (HTII) terms in the second and third equations.

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Table 1 Model parameters, their interpretation, and their units of measure Dimensions

a

Description Hunting rate of P on N

.

1 [t]

c

Hunting rate of H on N

.

1 [t]

f

Hunting rate of H on P

.

1 [t]

r

Reproduction rate of N

.

1 [t]

s

Reproduction rate of P

.

1 [t]

.μ

Mortality rate of H

.

1 [t]

e l h .KN , .KP b, d, g

Conversion coefficient of P on hunting N Conversion coefficient of H on hunting N Conversion coefficient of H on hunting P Carrying capacities of N e P Handling times

– – – – –

The model is a natural generalization of (1): √ √   dN c NH N a NP . − = rN 1 − − KN 1 + bN 1 + dN dt √   dP P fPH ea NP = sP 1 − − + dt KP 1 + bN 1 + gP √ dH c NH fPH +l − μH =h 1 + gP 1 + dN dt

(2)

Here all parameters are taken nonnegative, and in particular the intermediate predator reproduction rate is positive, .s > 0. Table 1 contains a list of the model parameters and their meaning. The first equation for the prey at the bottom trophic level and gathered in a herd states their logistic growth and the predation occurring only on the boundary individuals, expressed by the last two terms. Both the intermediate predator P and the top one H exert this pressure, with a rate that decreases the larger the size of the prey herd. The second equation contains the dynamics of the intermediate predators that have food sources expressed by the first logistic term, other than the prey modeled in the system. The second term contains the benefit for hunting the prey; the third one is their own damage suffered by the action of the top predators. In this latter case, the functional response is a HTII function, for feeding satiation. The third equation captures the behavior of the top predators, specialist on both prey and intermediate predators, with functional responses that are taken from the previous equations, via suitable conversion coefficients. The last term expresses their natural mortality.

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3 Analysis 3.1 The Equilibria and Their Feasibility 3.2 The One-Population Equilibria The admissible equilibria of model (2) are the origin .E0 = (0, 0, 0), the prey-only equilibrium .E1 = EN = (KN , 0, 0), and the intermediate predator-only point .E2 = EP = (0, KP , 0), which are all unconditionally feasible. Note that the superpredator, being specialist on both of N and P , cannot thrive in the absence of these food sources.

3.3 The Two-Population Equilibria Equilibrium E5 = EN P The case in which the superpredator disappears E5 = EN P = (N5 , P5 , 0) corresponds to the coexistence equilibrium analyzed in [1]. Through a saddle node bifurcation, two such equilibria are indeed possible. This is completely analyzed in [1] and will not be reported here. Equilibrium E3 = EN H We further find the point E3 = EN H = (N3 , 0, H3 ) where the intermediate predator P vanishes. The equilibrium equation for H gives √ .lc N − μ − μdN = 0. It is equivalent to a quadratic equation, leading to the roots:  .

lc ±

N3± =

 l 2 c2 − 4μ2 d , 2μd

(3)

which are real provided that l 2 c2 − 4μ2 d ≥ 0

.

(4)

and N3± coincide when (4) vanishes. The values of the superpredator at this equilibrium follow by substitution into the equilibrium equation for P :   r N3± .H (N3± ) = N3± 1 − (1 + dN3± ). c KN

(5)

They are nonnegative if N3± < KN .

.

Thus E3 is feasible for (4) and (6).

(6)

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Equilibrium E4 = EP H = (0, P4 , H4 ) Next, we have the point E4 = (0, P4 , H4 ) where the prey at the bottom trophic level N vanishes. This point is admissible in principle since the intermediate predator is assumed to be generalist and therefore has other food sources. From the equilibrium equation of H the value of P4 follows P4 =

.

μ hf − μg

(7)

while the equilibrium equation for P we find the corresponding value of the superpredator as function of P4 H4 =

.

KP − P4 s (1 + gP4 ) f KP

as well as in explicit form H4 = sh

.

KP (hf − μg) − μ . KP (hf − μg)2

(8)

Feasibility is ensured by the nonnegativity of the two populations, which leads to the condition 0 < μ < (hf − μg)KP .

.

(9)

The Coexistence Equilibrium E∗ = EN P H = (N∗ , P∗ , H∗ ) Coexistence E∗ = (N∗ , P∗ , H∗ ) is instead investigated numerically later. Here however we describe a possible geometric approach to show its feasibility. We consider the corresponding equilibrium equations as surfaces:    √ N 1 + dN aP N 1− .H1 (P , N ) = r , − KN 1 + bN c   √   P ea N 1 + gP H2 (P , N ) = s 1 − , + KP 1 + bN f √ μ(1 + dN) − lc N P (N) = √ (hf − μg)(1 + dN) + lcg N 

(10)

and study their possible intersections in the positive cone, thereby yielding points for coexistence. Now, P (N) is a cylinder with the axis parallel to the H axis. Figure 1 shows that the intersection may indeed not exist. Intersecting H1 (P , N ) and H2 (P , N ) gives a line that lies above the height of the cylinder P (N ), in the figure represented just by the orange line for better reading. Instead, Fig. 2 illustrates that

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Fig. 1 The intersection for coexistence is not feasibly attained. This occurs for the parameters a = 0.5, b = 0.02, c = 0.5, d = 0.02, e = 0.3, f = 0.5, g = 0.02, h = 0.3, l = 0.3, r = 2, s = 2, KN = 100, KP = 40, μ = 0.3

Fig. 2 The intersection for coexistence exist. The points are feasible for the parameter choice a = 0.5, b = 0.2, c = 0.5, d = 0.2, e = 0.3, f = 0.5, g = 0.2, h = 0.3, l = 0.3, r = 2, s = 2, KN = 100, KP = 40, μ = 0.3

coexistence may indeed be attained. The two curves obtained from the intersection of H1 (P , N) and H2 (P , N) do indeed meet the cylinder at two points, respectively, with abscissa close to P = N = 0 and close to P = 0, N = 100.

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In particular from Fig. 2, note that if the slope along the N axis of the surface H1 is lowered, the intersections of the three surfaces move closer to each other and ultimately may coalesce and disappear, through a saddle-node bifurcation. In this numerical experiment, the experience gained indicates that the most important parameters for the intersection to exist appear to be b, d, and g, while the carrying capacities KN and KP seem to be scantly influential.

3.4 Equilibria Stability For the stability, the Jacobian is needed: √ √ ⎤ a N c N − − J1,1 ⎥ ⎢ 1 + bN 1 + dN ⎥ ⎢ ⎢ eaP (1 − bN) fP ⎥ ⎥ J2,2 − .J = ⎢ √ ⎢ 2 N(bN + 1)2 1 + gP ⎥ ⎥ ⎢ ⎦ ⎣ clH (1 − dN) f hH J √ 3,3 2 2 2 N (dN + 1) (1 + gP ) ⎡

where     2N 1 aP (bN − 1) cH (dN − 1) + J1,1 = r 1 − + √ , KN (1 + dN)2 2 N (1 + bN)2 √   2P fH ae N , − + J2,2 = s 1 − KP 1 + bN (1 + gP )2 √ cl N f hP J3,3 = − μ. + 1 + gP 1 + dN

.

We partition the equilibria among those in which the prey N vanishes, because the Jacobian then has a singularity in the first column. In such case the analysis is performed by observing the dominant terms in the system (2).

3.4.1

Equilibria with N = 0

Equilibrium E0 For E0 , looking at the dominant behavior of the system near this point, we find .

√ √   √ √ dN c NH N a NP − ≈ N [r N − (aP + cH )], = rN 1 − − KN 1 + bN 1 + dN dt √   dP fPH P ea NP − = sP 1 − + KP 1 + bN 1 + gP dt

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√ ≈ P [s + (ae N − f H )] ≈ sP > 0,

√ √ dH c NH fPH +l − μH ≈ H [f hP + cl N − μ)] ≈ −μH < 0, =h 1 + gP 1 + dN dt from which its instability follows from the sign of the second equation. Equilibrium E2 = EP For equilibrium EP we have √ √   √ √ dN c NH N a NP . − ≈ N [r N − aP − cH ] = rN 1 − − KN 1 + bN 1 + dN dt √ √ √ ≈ N [r N − a(P − KP + KP ) − cH ] ≈ −aKP N < 0. Along the N axis the behavior is stable. For the remaining minor of the Jacobian evaluated at (KP , 0) an upper triangular matrix is obtained: ⎤ f KP ⎢ 1 + gKP ⎥ .⎣ ⎦ f hKP −μ 0 1 + gKP ⎡

−s −

from which the eigenvalues are immediate, −s and the element 2, 2 of the above matrix. The former cannot vanish unless s = 0, a case that we do not consider. The latter provides the stability condition μ>

.

f hKP . 1 + gKP

(11)

Note that in the limit KP → +∞ condition (11) simply becomes gμ > f h,

.

i.e., the superpredators H mortality times the handling time must exceed the benefit gained by hunting the intermediate predators. Figure 3 shows that this point can indeed be stably achieved, when condition (11) holds. The condition (11) combined with (9) hints to the onset of a transcritical bifurcation between E2 and E4 . This is now rigorously proven by means of Sotomayors’ theorem, [17]. Take as bifurcation parameter μ. The critical value for which the second eigenvalue annihilates is μ† =

.

f hKP . 1 + gKP

(12)

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Fig. 3 The equilibrium EP is obtained; the intermediate predator population P attains its carrying capacity KP

The right v and left w eigenvectors of the Jacobian evaluated at this point, for the critical parameter value μ† , are  v=

.

f KP ,s 1 + gKP

T ,

w = (0, 1)T .

Denoting by F the last two right hand sides of (2), upon suitable differentiation, we find   ∂F 0 . = Fμ = −H ∂μ from which we obtain Fμ (E2 , μ† ) = 0, from which wT Fμ (E2 , μ† ) = 0 follows. Further,  DFμ =

.

0 0 0 −1



implying in turn DFμ (E2 , z† )v = (0, −s)T and therefore no saddle-node bifurcation can occur, since wT [DFμ (E2 , z† )v] = −s = 0. To assess the onset of other possible bifurcations, we need to evaluate wT [D 2 F(E2 , z† )(v, v)]. In particular we need the quadratic forms D 2 F(E2 , z† )(v, v), but since the only nonvanishing component of w is the second one, we can just evaluate its second component: D 2 F2 (E2 , z† )(v, v) =

.

But

∂ 2 F2 ∂ 2 F2 ∂ 2 F2 v2 v1 + 2 v1 v2 + 2 ∂P ∂H ∂H 2 ∂P

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.

∂ 2 F2 2f gH , = 2 (1 + f P )2 ∂P

∂ 2 F2 2f , = ∂P ∂H (1 + f P )2

∂ 2 F2 = 0. ∂H 2

Evaluating at the equilibrium and for the critical threshold, we find D 2 F2 (E2 , z† )(v, v) = −s

.

2f v1 = 0 (1 + f P )2

showing thus the existence of the transcritical bifurcation. Equilibrium E4 = EP H For the point EP H , as in the previous case, the equation for N becomes √ √   √ √ dN c NH N a NP . − ≈ N [r N − aP − cH ] = rN 1 − − KN 1 + bN 1 + dN dt √ √ √ ≈ N [r N − a(P − P4 + P4 ) − c(H − H4 + H4 )] ≈ −(aP4 + cH4 ) N < 0. giving stability along the N direction. Considering the remaining part of the system, we obtain upon evaluation in P4 e H4 , the Jacobian becomes:

JP H

.

⎡  ⎤  2P fP fH − − ⎢s 1 − K (1 + gP )2 1 + gP ⎥ P ⎥ =⎢ ⎣ ⎦ f hH 0 2 (1 + gP )

Applying the Routh-Hurwitz conditions, for tr(JP H ) < 0, we explicitly find fgKP H4 < s(1 + gP4 )2

.

(13)

while the determinant turns out to be always positive: .

det(JP H ) =

f 2 hH4 P4 >0 (1 + gP4 )3

therefore not contributing to the stability of the point. For EP H the fact that condition (13) can indeed be satisfied is shown in Fig. 4.

3.4.2

Equilibria with N = 0

Equilibrium E1 = EN For Equilibrium EN where N = KN , P = H = 0, the Jacobian becomes an upper triangular matrix, whose eigenvalues are easily determined:

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Fig. 4 For the parameter choice a = 0.5, b = 0.2, c = 0.5, d = 0.2, e = 0.4, f = 0.2, g = 0.02, h = 0.2, l = 0.3, r = 2, s = 2, KN = 50, KP = 50, μ = 0.4, the point EP H is stably achieved

.

− r < 0,

√ ae KN > 0, s+ 1 + bKN

√ cl KN −μ 1 + dKN

and from the second one, unconditional instability follows. Equilibrium E5 = EN P For the equilibrium EN P , the discussion is already contained in [1]. Equilibrium E3 = EN H Assuming feasibility (4) for the equilibrium EN H , the Jacobian factorizes, to give one explicity eigenvalue, J22 , and the minor ⎡  ⎤ √  2N3 c N3 cH (N3 )(dN3 − 1) r 1 − − + √ ⎢ KN 1 + dN3 ⎥ 2 N3 (dN3 + 1)2 ⎥ = ⎢ .J ⎣ ⎦ clH (N3 )(1 − dN3 ) 0 √ 2 2 N3 (dN3 + 1) The Routh-Hurwitz condition for the trace, using (5), reduces to the requirement tr(J) = J1,1 (N3 , H (N3 )) =

.

r(N3 (d(3KN − 5N3 ) − 3) + KN )

.

3dKN − 3 +

√ dKN (9dKN + 2) + 9 10d

For the determinant, we find .

c2 lH (N3 )(dN3 − 1) det(J) = −J1,3 J3,1 = − 2(1 + dN3 )3

(14)

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so that assuming feasibility, for which H (N3 ) > 0, the second stability condition becomes N3
N3 > d

√ dKN (9dKN + 2) + 9 := (d) 10d

(16)

Observing that for (5) to be nonnegative, we need N3 < KN ; in reality in place of (16) we require the more stringent conditions: .

√   1 3dKN − 3 + dKN (9dKN + 2) + 9 min KN , > N3 > := (d). d 10d

If 1 < dKN , observe that the function (d) is a sigmoid, such that .

lim (d) =

d→∞

3 KN , 5

lim (d) =

d→0+

KN . 3

Conversely, .

1 = 0+ , d→∞ d lim

lim

d→0+

1 = +∞. d

Hence (16) can be satisfied only for very small values of d, namely, 1 3dKN − 3 + . > d

√ dKN (9dKN + 2) + 9 10d

from which  9d 2 KN2 + 2dKN + 9 < 13 − 3dKN

.

and 9d 2 KN2 + 2dKN + 9 < 169 − 78dKN + 9d 2 KN2

.

finally yielding dKN < 2.

.

If instead 1 > dKN , it is easily seen that (d) < KN holds for all d ∈ R. Indeed

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 9d 2 KN2 + 2dKN + 9 < 7dKN + 3

(d) < KN ⇐⇒

.

is equivalent to 2 9d 2 kN + 2dKN + 9 < 49d 2 KN2 + 42dKN + 9,

.

which in turn yields 40(dKN + 1) > 0. In summary, det(J) > 0 holds only for 

 1 .(d) < N3 < min KN , , d

KN
(N3 ),

.

where   r N3 .(N3 ) = 1− (1 + dN3 ), KN c

1 (N3 ) = f



s ea +√ 1 + bN3 N3

 .

Now,  is a concave parabola, with roots N3 = −

.

and maximum located at NM =

1 < 0, d

dKN −1 2d ,

N3 = KN > 0, which is positive only if

dKN > 1,

(19)

.

so that the interval in which to seek its intersections with  reduces to N3 ∈ [0, KN ]. The function (N3 ) is defined on the interval (0, +∞) with negative derivative,  (N3 ) = −

.

eab s dKN . Left to right, the frames are respectively obtained for KN = 20, 40, 100. The other parameters used are r = 2, d = 0.002, c = 0.7, a = 0.5, e = 0.5, f = 0.3, s = 2, b = 1. Note that in this case the eigenvalue J22 is negative only in the right frame, in the interval where the blue curve  lies above the red curve 

1 N3 →+∞ f lim



s ea +√ 1 + bN3 N3



= 0+ .

Figures 5 and 6 show the behavior of the two functions (N3 ) and (N3 ). The former can lie above the latter at most only in a relatively narrow interval for the values of N3 , or not at all; see Fig. 5. A sufficient condition, for a positive abscissa NM of the maximum of the parabola given by (19), would be guaranteed by the condition (NM ) > (NM ).

(20)

√ f r(d 2 KN2 − 1) 2ead s 2d . > . +√ 2d + b(dKN − 1) 4cdKN dKN − 1

(21)

.

Explicitly,

In case that the abscissa of the maximum of  is negative, the possible intersections of (N3 ) and (N3 ) are represented in Fig. 6. The latter, in which the abscissa of the maximum of  is positive in the left and central frames, corresponding to the condition 1 < dKN being satisfied, shows other possible crossings among the two functions in terms of the parameter d.

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Fig. 6 Graphical representation of the possible behaviors of the functions (N3 ) and (N3 ) in case 1 < dKN , as function of the parameter d. Left to right, the frames are respectively obtained for d = 0.2, 0.02, 0.002. The other parameters used are r = 2, KN = 100, c = 0.7, a = 0.5, e = 0.5, f = 0.3, s = 2, b = 1 Note that the maximum of the parabola here has a positive abscissa only in the left and center frames. Here in all frames intervals exist for which  >  entailing the negativity of J22 Fig. 7 The point EN H is attained by the parametr choice a = 0.09, b = 10−6 , c = 0.08, d = 10−6 , e = 0.5, f = 0.08, g = 1e − 6, h = 0.2, l = 0.8, r = 3, s = 1, KN = 20, KP = 100, μ = 0.2

In summary, stability of E3 is ensured by (14) and (17), coupled with (18). In particular a sufficient condition for ensuring that the latter is satisfied is given by (21). The fact that these conditions can all simultaneously satisfied is apparent from Fig. 7. However, the parameter d in all these cases must be chosen sufficiently small. The Coexistence Equilibrium E∗ = EN P H = (N∗ , P∗ , H∗ ) As for coexistence EN P H , Fig. 8 shows that it can be attained. Here the choice of parameter values for b, d, and g that are one order of magnitude smaller than the other ones is necessary.

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Fig. 8 Coexistence is here attained for the parameters a = 0.09, b = 10−6 , c = 0.08, d = 0.1, e = 0.5, f = 0.08, g = 10−6 , h = 0.2, l = 0.8, r = 3, s = 1, KN = 20, KP = 100, μ = 0.2

Fig. 9 A Hopf bifurcation originating from the coexistence equilibrium has here taken place, as persistent oscillations are discovered for the parameters a = 0.4, b = 0.2, c = 0.2, d = 0.2, e = 0.5, f = 0.5, g = 0.2, h = 0.8, l = 0.2, r = 2, s = 2, KN = 100, KP = 20, μ = 0.5

At this point, a Hopf bifurcation is seen to take place, as persistent oscillations are obtained in Fig. 9.

4 Conclusions Table 2 summarizes the system behavior providing the conditions for the equilibria feasibility and stability. This model allows the prey gathering in herds and, if in large numbers, counteracting the predator’s pressure, to thrive in some circumstances even under predators’ attacks. In fact, in addition to the findings of [1] which here corresponds to equilibrium .E5 where the prey survives together with the intermediate predator P at the immediate higher trophic level, the prey can thrive with the top predator as well, both in the absence of the intermediate predators, equilibrium .E3 , and in their presence, coexistence point .E∗ . In this last case, the three populations may also coexist through persistent oscillations.

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Table 2 Feasibility and stability conditions for all the equilibria of the system (2) Equilibria = (0, 0, 0) .E1 = EN = .(KN , 0, 0) .E2 = EP = .E0

Feasibility – – –

.

.(0, KP , 0) .E3

= EN H =

.l

2 c2

.(N3 , 0, H (N3 ))

.N3

= EP H = .(0, P4 , H (P4 )) .E5 = EN P .(N5 , P5 , 0) .E∗ = EP N H

.0

.E4

Stability Unstable Unstable

− 4μ2 d ≥ 0,

< KN

< μ < (hf − μg)KP

f hKP (N3 ) .(d)

.fgKP H4

< s(1 + gP4 )2

see [1]

See [1]

Intersections in (10)

Simulations

References 1. Francesca Acotto, Ezio Venturino, Modeling the herd prey response to individualistic predators attacks, submitted. 2. V. Ajraldi, M. Pittavino, E. Venturino, Modelling herd behavior in population systems, Nonlinear Analysis Real World Applications, 12 (2011) 2319–2338. 3. Malay Banerjee, Bob W. Kooi, Ezio Venturino, An ecoepidemic model with prey herd behavior and predator feeding saturation response on both healthy and diseased prey, Mathematical Models in Natural Phenomena, 12 (2), 133–161, 2017; https://doi.org/10.1051/mmnp/ 201712208 4. S. Belvisi, E. Venturino, An ecoepidemic model with diseased predators and prey group defense, SIMPAT 34, 144–155, 2013. https://doi.org/10.1016/j.simpat.2013.02.004 5. Cecilia Berardo, Iulia Martina Bulai, Ezio Venturino, Interactions obtained from basic mechanistic principles: prey herds and predators, Mathematics (MPDI), 9(20), 2555 (2021). https://doi.org/10.3390/math9202555 6. Davide Borgogni, Luca Losero, Ezio Venturino, A more realistic formulation of herd behavior for interacting populations Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment Selected Works from the BIOMAT Consortium Lectures, Szeged, Hungary, 2019 (Rubem P. Mondaini, Editor), Springer 2020, 9–21. 7. Iulia Martina Bulai, Ezio Venturino, Shape effects on herd behavior in ecological interacting population models, Mathematics and Computers in Simulation, 141 (2017) 40–55 https://doi. org/10.1016/j.matcom.2017.04.009 8. Elena Cagliero, Ezio Venturino, Ecoepidemics with infected prey in herd defense: the harmless and toxic cases, IJCM, 93(1), 108–127, 2016. https://doi.org/10.1080/00207160.2014.988614 9. Salih Djilali, Impact of prey herd shape on the predator-prey interaction Chaos, Solitons & Fractals 120, (2019), 139–148 10. Giacomo Gimmelli, Bob W. Kooi, Ezio Venturino, Ecoepidemic models with prey group defense and feeding saturation, Ecological Complexity, 22 (2015) 50–58. 11. González-Olivares, E., Rivera-Estay, V., Rojas-Palma, A., Karina Vilches-Ponce, A Leslie– Gower type predator-prey model considering herd behavior. Ricerche di Matematica (2022). https://doi.org/10.1007/s11587-022-00694-5

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12. Heping Jiang, Xiaosong Tang. Hopf bifurcation in a diffusive predator-prey model with herd behavior and prey harvesting, Journal of Applied Analysis & Computation, 2019, 9(2): 671– 690. https://doi.org/10.11948/2156-907X.20180142 13. Bob W Kooi, Ezio Venturino, Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey, Math. Biosc 274 58–72 2016. 14. Henri Laurie, Ezio Venturino, A two-predator one-prey model of population dynamics influenced by herd behaviour of the prey, Theoretical Biology Forum, 111(1–2) 27–47, 2018. 15. Henri Laurie, Ezio Venturino, Iulia Martina Bulai, Herding induced by encounter rate, with predator pressure influencing prey response, in “Current Trends in Dynamical Systems in Biology and Natural Sciences”, Maira Aguiar, Carlos Braumann, Bob Kooi, Andrea Pugliese, Nico Stollenwerk, Ezio Venturino (Editors) Springer-SIMAI series, 2019, 63–92, 2020. 16. D. Melchionda, E. Pastacaldi, C. Perri, M. Banerjee, E. Venturino, Social behavior-induced multistability in minimal competitive ecosystems, J. Theoretical Biology 439, 24–38, 2018. https://doi.org/10.1016/j.jtbi.2017.11.016 17. L. Perko, Differential Equations and Dynamical Systems, Springer, New York, USA, 2011. 18. Fethi Souna, Salih Djilali, Fayssal Charif Mathematical analysis of a diffusive predator-prey model with herd behavior and prey escaping, Math. Model. Nat. Phenom. 15 (2020) 23 https:// doi.org/10.1051/mmnp/2019044 19. E. Venturino, S. Petrovskii, Spatiotemporal behavior of a prey-predator system with a group defense for prey, Ecological Complexity 14, 37–47, 2013. http://dx.doi.org/10.1016/j.ecocom. 2013.01.004 20. Karina Vilches, Eduardo González-Olivares, Alejandro Rojas-Palma Prey herd behavior modeled by a generic non-differentiable functional response, Math. Model. Nat. Phenom. 13 (2018) 26

Benefits of Application of Process Optimization in Pharmaceutical Manufacturing: A Panoramic View Antonios Fytopoulos and Panos M. Pardalos

1 Introduction During the last decades, regulatory bodies have rapidly changed their demands leading to requirements such as better quality of the final products, application of green processes, minimization of waste during manufacturing, and creation of a more sustainable pharmaceutical sector. At the same time, industries should produce products in higher yields, reduce production times, and create more efficient and optimized—continuous—processes working in a more structured framework [1]. The achievement of the aforementioned is not always trivial in the pharmaceutical industry, since most of the times, it is very difficult to recognize or model the exact dynamics that occur in a pharmaceutical manufacturing process, while often, unwanted phenomena happen that jeopardize the quality of the final product. In this context, optimization plays a crucial role in finding the available best feasible solutions of continuous manufacturing (CM) processes, leading to more efficient and robust designs [2]. One of the main concerns in pharmaceutical manufacturing has been the switch from batch to continuous processes. Among the main disadvantages of batch processes is that they are often costly and time-consuming, with increased unproductive time and significant variations in quality between batches. Thus, it seems that the first step toward optimization could be the conversion of batch processes to continuous ones. In addition, the use of Process Analytical Technology

A. Fytopoulos () Department of Chemical Engineering, KU Leuven, Leuven, Belgium School of Chemical Engineering NTUA, Athens, Greece e-mail: [email protected] P. M. Pardalos Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_17

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(PAT) framework can help in controlling production quality during processing by providing timely measurements. Hence, understanding of the key elements and basic mechanisms and dynamics of a process together with accurate information about the critical quality attributes (CQA) and critical material attributes [3] can be helpful for optimization of final product [1]. Finally, optimization in models that map the exact interplay and dynamics in a process help in quantification and can be used as the necessary framework for the application of optimization techniques. In this brief survey, the use of optimization techniques in pharmaceutical applications will be discussed. The survey is structured as follows: we first briefly present the pharmaceutical regulatory environment that ensures the production of high-quality products. Subsequently, we discuss the important role of continuous process as a necessary framework for easier application of optimization techniques. Then, we review some of the applications of optimization techniques in several pharmaceutical manufacturing processes. Finally, we conclude by summarizing the potential applications of optimization in the pharmaceutical sector and the perspectives for future research.

1.1 Regulatory Environment The large amount of manufacturing failures and recalls [4] by the year 2000 led regulatory authorities to give the proper attention to the matter of quality. Emphasis has been given by all regulatory bodies for the production of safer, reliable products that will satisfy customers. To this end, regulatory bodies exercised pressure to pharmaceutical industry to improve pharmaceutical manufacturing processes. More efficient, cheaper, robust, and trustworthy green manufacturing processes had to be designed and applied in order to reduce waste, minimize impurities, and verify perfect quality of the final product. Therefore, pharmaceutical companies should work in a systematic framework that could verify the aforementioned demands. Quality by Design (QbD) concept is defined—according to ICH Q8 guideline— as “a systematic approach to development that begins with predefined objectives and emphasizes product and process understanding and process control, based on sound science and quality risk management.” Traditionally, quality used to be verified in final products often leading to trial-and-error processes with the concomitant waste production. According to the new framework, quality is built into the product in the first place [5], i.e., if the design of a pharmaceutical product is poor in the first place, the result will not comply with approved specifications no matter how many different analytical tests are applied in the final product. The concept of QbD was published in 2004 [6] with a view to create a holistic common approach. However, even if the concept of QbD seems to be reasonable, many companies resist to recognize the potential benefits, claiming that bureaucracy is increased and additional time is needed until the final market launch of the product. Of course, these delays lead to loss of profits, and this makes them show increased skepticism. The truth is that since 2004 there were strong evidence of

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significant progress in quality of the final product, leading to the conclusion that QbD concept should be enhanced [7]. The advantages of implementing Quality by Design framework can be summarized as follows [5]: • The application of QbD need (and enhances) significant scientific understanding of the pharmaceutical process. • The adjustment of variables within the design space is easy. • Involves risk assessment. • The critical quality attributes (CQA) are identified and should be met in order to ensure the safety of the final pharmaceutical product—focuses on safety. • Trial and error-based production is avoided along with the concomitant waste production. • It is a general framework in which all pharmaceutical companies can follow universal guidelines toward the cheaper production of safer products, faster, and with increased profits.

1.2 The Need for Green Engineering In 2005 the ACS GCI Pharmaceutical Roundtable was founded with a view to change the culture and establish an overall greener way of thinking. This led to the development of a list of “key research areas” [8–10], among which continuous processing, biotechnology, improvement of separations, solvent selection, optimization, and intensification are of major importance. Further research in the aforementioned areas can minimize waste and energy requirements, develop realtime process monitoring and enhance the application of optimization methods in pharmaceutical manufacturing [11]. Traditionally, pharmaceutical manufacturing uses batch chemistry. Typically, unit operations in the pharmaceutical sector consist of segmented steps where the product of one reactor is used as an input to the next reactor and so on. Among others, one of the main drawbacks of these technics is the demand for more energy leading to an increase in greenhouse emissions and waste. On the contrary, the use of continuous processes can lead to minimization of waste, process safety, and efficiency in terms of energy, without compromising the final result, i.e., the manufacturing of effective medicines with minimum environmental footprint [12]. In addition, the enhanced use of biotechnology for the production of biopharmaceuticals can become a great opportunity to minimize economic cost and foster more sustainable engineering. Furthermore, the selection of green solvents and reagents and the recovery or recycling of used ones can significantly decrease the volumes of process waste. Finally, the use of Process Analytical Technology and the intensification of unit operations used in pharmaceutical processes (e.g., in separations) can pave the way toward improved outcome.

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As a result of the abovementioned guidelines, the pharmaceutical industries adopted metrics that help in mapping the sustainability of processes, including minimization of environmental footprint and energy consumption. For instance, among these metrics, Process Mass Intensity (PMI) tool, which is available online (https://acsgcipr-predictpmi.shinyapps.io/pmi_calculator/), helps user to select a “green-by-design” approach in developing efficient processes. Finally, E-factor and life cycle assessment (LCA) methodologies can be useful tools in evaluation of and assessment of the environmental impact and sustainability of pharmaceutical production.

1.3 Continuous vs Batch Processes In a batch pharmaceutical manufacturing process, the raw materials are initially supplied to the system and discharged at the end of the process without any exchange of ingredients throughout the process. On the other hand, in continuous processes a constant and simultaneous charging and discharging of raw materials and products occurs during the process. Traditionally, the pharmaceutical sector is dominated by batch processes, where the product of one batch can be the input of another and so on, creating series of batch unit operations that work in a raw. Regarding quality assurance, what should be mentioned is that in batch processes, the product is removed, analyzed, and then used as input—if it meets the quality standards— for the next process, while the final product is extensively tested as well. This leads often in variations in quality of the final product [13]. On the contrary, in continuous manufacturing quality assurance is checked in real time, leading to superior product stability in terms of quality and characteristics. Several other advantages of continuous processes can be mentioned compared to batch processes. Among them, the minimized cost due to demand of smaller equipment, investment cost, and space are of outmost importance. Furthermore, unproductive time between the consecutive steps is minimized—in the order of hours [1]—compared to batch production, where the transport of materials is often intermittent, leading to process times of days or weeks and having negative effects on the quality of medicines, that can easily degrade during time. On the top of that, small scales and equipment utilized in continuous processes (e.g., smaller reactors, volumes of solvents, etc.) enhance safety [9] since volumes of hazardous materials are smaller. Furthermore, continuous recycling that can be applied in continuous processes leads to “greener” production and minimizes waste. This fact, in combination with the use of microreactors can be of great importance for the production of pharmaceuticals, leading to high-quality medicines that can satisfy public demands, but also enhance the green processes [14]. Ergo, end-to-end sustainable production can be feasible by combining continuous optimized processes with “green mindset” with a final outcome that is economically attractive for industries and efficient for people. In addition, regulatory authorities and bodies enhance continuous

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manufacturing, which in turn leads industries to support this mentality. Finally, several universities have already established collaborations with pharmaceutical companies oriented to the enhancement of continuous processes (e.g., Novartis— MIT center for Continuous Manufacturing). With regard to optimization, continuous processes can be characterized as ideal for the application of optimization techniques, since in-line measurements and process monitoring can provide real-time data for the product. These data can be used as input for the optimization—control procedure that will be able to adjust the process parameters in real time, with a view to achieving the desired final assurance of quality. From the aforementioned, it is evident for the reader that Process Systems Engineering (PSE) is the necessary adhesive for plant-wide application of optimization and control techniques that will pave the way toward integrated pharmaceutical processes, which will not have the need for trial-and-error-based processes. Finally, the introduction of Process Analytical Technology (PAT) concept enhanced the need for application of PSE tools in the pharmaceutical sector.

1.4 Process Analytical Technology (PAT) PAT is a framework issued by FDA in 2004, which became one of the most influential trends in pharmaceutical manufacturing. Over the years, PAT became a system closely engaged with the design, analysis, and control of pharmaceutical manufacturing, and its principles helped in further understanding of many processes. According to this idea, real-time monitoring of specific and most important parameters of a product is performed aiming at verifying the quality of the final medicine. These important critical quality attributes (CQA) are properties (physical, chemical, etc.) identified by critical analysis and keeping them between specific limits ensures that the final medicine will meet the requirements for the public release. These critical properties are closely connected to critical process parameters (CPP), which in turn strongly affect directly their values. Therefore, continuous manufacturing (CM) in combination with PAT can lead to continuous optimized processes, where the quality of the product is ensured and real-time release [15] of the product occurs. PAT tools help in quantification of the product CQA’s during the process. Four main categories of tools can be used according to FDA [6], namely, (1) multivariate tools for analysis, design, and acquisition, (2) process control tools, (3) process analyzers, and (4) continuous improvement and knowledge management tools. A critical description of these tools and their subcategories can be found elsewhere [16]. PAT measurements can be categorized based on four different methods that are depicted in Fig. 1. In “in-line” methods real-time measurements take place without the removal of product. These methods are of outmost importance, especially for the application of control during processes, since they provide real-time data of the system and control is performed at once. In “online” methods, the product is diverted from the reactor, measured, and then charged again in the reactor, while for “at-line” methods, the product is discharged and measured near the process

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Fig. 1 Illustration of the different methods that can be applied for measurements in PAT framework

On - line

In - line

Off - line LAB

At line Analysis

stream. Finally, “off-line” methods are conducted far from the process unit in a laboratory specialized in quality control measurements. These methods are the most time-consuming. In addition, the product is destroyed (as in at-line methods as well), and the results are representative of the exact time that the sampling was performed, and no real-time reaction is feasible if a divergence is verified. All in all, the Process Analytical Technology framework is necessary for realtime monitoring, which is essential for the application of optimization and control. Ergo, Process Systems Engineering methods are expected to play an increasingly critical role in the future of pharmaceutical manufacturing, but in order to effectively manipulate processes, quantitative models able to describe the exact interplay and dynamics of the mechanisms that occur [17] are needed.

1.5 Model-Based Optimization Optimization techniques can be used to aid the pharmaceutical production, but the truth is that mathematical optimization needs a specific framework to work with [18]. In most of the pharmaceutical processes, a complex interplay occurs among dynamics that are desired or not. This complicates the application of optimization methods, as it suggests that attention should be given in another domain, namely, process modelling. Using mathematical models to describe processes is necessary for the application of optimization techniques greatly contributes to the efficient design and can help avoid trial-and-error experiments that are extremely timeconsuming and costly in pharmaceutical sector. All the aforementioned, indicate that process modelling is the necessary substrate for the application of optimization techniques. Apart from the models used in specific pharmaceutical production processes, model-based optimization and control can be applied holistically [19, 20] in series of processes [21], in plant design, planning, scheduling [22]. or logistics formulating cost-efficient processes [23]. In this context, several methodologies have been applied, such as gradient-based methods (saddle-point method, Newton’s method, etc.), surrogate-based methods (artificial neural networks (ANNs), etc.), and heuristics.

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In the last decade, machine learning algorithms and emerging technologies such as artificial intelligence [24] tend to transform all scientific domains, among them pharmaceutical industry, as well. Along these lines, the need for better—more efficient—optimization methods and process modelling is much more demanding than in the past, since more data are available and more parameters have to be adjusted.

2 Optimization in Pharmaceutical Processes 2.1 Crystallization One of the most promising—yet difficult for optimization to be applied—unit operation is that of crystallization. This is due to the fact that in crystallization processes, many different phenomena can occur, including undesired ones due to the unexpected character of nucleation, breakage, etc. of crystals that can compromise the quality of the produced medicine. Real-time measurements should be performed (e.g., of concentration, number of crystals, etc.), analyzed, and used as inputs for mechanistic models to extract the kinetics of the sub-processes in the process. One of the most widely used model-based methods to treat these problems is Population Balance Modelling (PBM) approach. Population Balance Models [25, 26] use partial integro-differential equations coupled with ordinary ones (mass balances) to map the evolution of crystal size distributions (CSD’s) over time. Of the main benefits of this technique is that these models incorporate information for many parameters in the processes than traditional kinetic models. This is important, since it provides the necessary framework for the application of well-known optimization techniques. The main drawback of these models is its complexity and the increased computational cost needed for the calculations. In this context, Population Balance Models that describe growth, dissolution and agglomeration [27, 28], breakage [29], or nucleation were formulated in the past. Finally, there are cases where the mechanisms that occur are not always known a priori or, even when are known, it is difficult to incorporate them in the models. In this way, in order to map complex phenomena, such as agglomeration, the mathematical formulations of PBM’s become even more computationally demanding and often simplifications are needed. On the other hand, data-driven models are based basically to previously available data. Neural networks exhibit universal characteristics and can be used to model crystallization process without previous knowledge of the system and the mechanisms that interact, as long as there are enough experimental (or mathematically generated by using a PBM) data to train the model. Optimization is achieved by minimization of the predictions of neural network model [30], and the real value of target property and optimal control can be ensured. Finally, hybrid models [31] that implement both mechanistic models (basically PBM’s) and data-driven methods

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were used in the past, a framework that combines both the improved knowledge for the process and the emerging technologies as well, leading to superior results.

2.2 Filtration Since the process of filtration is an intermediate one between the upstream and downstream parts, only few researchers have paid attention to improve and optimize this process, and this is why, even nowadays, this process often continues to be performed in a semiempirical way, rather than based on complete process understanding. The main idea is to formulate continuous filtration that could easily be incorporated in a complete unit, where all sub-processes could work smoothly, aiming at producing the proper medicine of high quality. In order to achieve this, material properties and process parameters should be adjusted according to the requirements. Traditionally, crystal size distribution (CSD) and purity of the final product along with the minimization of granulation or dissolution of the crystals and clogging are of crucial importance in the process of filtration [32]. Among the important parameters of the process that should be adjusted are the CSD, the viscosity of the feed, and the porosity of the filter. In that way the viscosity of the product combined with the filter porosity could potentially minimize the flow compromising the reduction of process time and cost. On the other hand, the use of a high-porosity filter could lead to insufficient filtering, leading to impurities in the final product. Along these lines, approaches regarding the complete modelling and combination of the filtration process with crystallization performed in the past [33]. However, even if these works showed the potential of a fully understood and incorporated filtration sub-process, further research is needed in this way in order to completely incorporate filtration in a continuous manufacturing framework.

2.3 Extraction Through extraction, the active constituents can be isolated using a selective solvent. Basic extraction types that are typically used in the pharmaceutical sector are liquidliquid extraction (LLE), solid-liquid extraction (SLE), or solid phase extraction (SPE). Regarding optimization of the process, several studies have been performed, mostly based on process modelling. For example Diab et al. [34] performed a process systems engineering approach to optimize the manufacturing of atropine with liquid-liquid extraction and cost minimization as target. A nonlinear optimization algorithm was used, and the estimation of kinetics was performed. Finally, except for optimization techniques, process intensification methods [35]—alternative techniques, such as the use of microwaves (e.g., microwave assisted extraction) or ultrasounds [36] and use of supercritical fluids (sonication extraction, supercritical fluid extraction)—could pave the way toward a highly

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efficient separation and the concomitant cost effective continuous manufacturing process.

2.4 Powder Feeding A critical part of the pharmaceutical production is the feeding process, where powders are fed in the process in a precise manner. When a continuous manufacturing approach is used, the optimal operation of feeding process verifies the requirements that should be met and are important for the critical quality attributes of final product. In this context, the selection of the proper type of feeder is crucial, and there are feeders that are based on vibratory channels, rotary valves or screw (e.g., loss-inweight feeders [37]) that are commonly used in pharmaceutical manufacturing. In addition to equipment, the properties of materials that are to be fed crucially impact the rate of flow of powders, as it often can lead to variabilities in volumetric flow (e.g., if the material is cohesive it can stick to the screws). During the years, control systems were used in order to optimize the powder feeding procedure, mainly by using PID controllers, e.g., for loss-in-weight feeders. According to this method, a sensor continuously measures the weight of the powder in the feeder and provides the necessary data for the calculation of the exact feed rate. When the controller detects a variability in the flow, it automatically regulates the screw speed (increase or decrease) in order to accurately keep the desired flow rate constant. In addition, there are studies where control is achieved, based on partial least squares (PLS) regression [38], where a correlation between the material properties and outputs, such as the feed rate, is performed.

2.5 Granulation In order to improve particle properties and achieve the critical quality attributes required, granulation process is used to verify the homogeneity of the powders. In this way, better and constant flow is achieved, which helps to improve the performance of other process, as well, such as tableting. It is actually a process that enlarges particles through agglomeration [39], improving the homogeneity of the API, reducing dust and creating larger particles that can be easily compressed. There are two types of granulation: (1) wet granulation, where a binder is added and volatile solvents are used, which helps particles agglomerate in the granulator and (2) dry granulation, where no liquid is required, but high pressure is excreted and dry binder can be used; and finally, melt granulation, where the binder is meltable (e.g., twin screw granulation [39]). The optimization of this process can be enhanced by the thorough knowledge of the principal phenomena that occur during the process, i.e., particle nucleation, growth, agglomeration, and breakage.

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Based on the knowledge of these phenomena and the exact interplay among them, that can be observed in experimental work, mechanistic models such as Population Balance Models can be used as a basis for optimization of the process [40]. Different parameters of granules can be used, such as strength, particle size distribution characteristics, and/or compressibility as important parameters that affect tableting. Finally, in a Process Systems Engineering approach, several different optimization-control techniques can be applied toward scaling-up. Among those, one could mention population balance-based methods, ARX and ARMAX models for linear Model Predictive Control, open-loop dynamic optimization for MPC, and real-time measurements for control and optimization [40].

2.6 Tableting Granulation process produces the desired granules, and compression of those granules to tablets is the following step. In that way, one of the most usual drug formulations nowadays is represented by tablets [41] (a proportion of about 70%) [42], since tablets are easy to swallow and the dose can be easily quantified. Thus, the production of tablets with desired stable properties is of major importance for the pharmaceutical companies. During the tablet formulation, several parameters such as friability, hardness, thickness and precise mass, and capping coefficient (CC) can be set as targets for optimization models to fulfill the desired requirements of the final product. Regarding optimization techniques, several methods were used in the past for smooth and optimal tableting. For example, Beli˘c et al. [43, 44] used artificial neural networks (ANN) and fuzzy models to model and optimize the effect of particle size on the capping coefficient (CC). In that way, available data are used as inputs to a nonlinear model, and only one output is exported, while the model uses backpropagation training algorithms. In addition, fuzzy models are based on fuzzy sets, where an element can be a member of the set or not and can be based on piecewise linear models. Finally, partial least square models were used for modelling and optimization of tablet manufacturing lines [45].

2.7 Coating Coating of tablets is performed in order to produce more stable products, mask the unpleasant odors or taste, increase drug’s shelf life, and minimize the dust generation during the production. The most commonly used type of coating is that of film coating according to which a uniform thin film is produced on the surface of the pill, by spraying a rotating tablet. Several different types of thin film coating can be applied, such as organic solvent-based, aqueous, solvent-free, etc. A more detailed approach on these types can be found in Seo et al. [46].

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The most common parameters that can be optimized during the coating process are, among others, the curing time, droplet size, spray properties, viscosity of the coating solution, and equipment used. For example, Kureck et al. [47] used a discrete element method (DEM) to improve simulations and detect contacts between biconvex tablets with spherical band shape. Finally, Wong et al. [48] used an ultrasonic nozzle to directly optimize a bed fluidized tablet coating.

3 Conclusions and Future Perspectives In this short survey, the impact and advantages of optimization methods on pharmaceutical manufacturing processes were presented. The facts indicate that an increasing demand for better understanding of the underlying mechanisms, formulation of better models, and application of optimization methods and control in each manufacturing step is required for the production of better, more stable, and efficient medicines. In addition, a holistic approach is required, and application of models able to predict and optimize end-to-end manufacturing processes is essential for the pharmaceutical production. The establishment of a more standard mathematical framework, in which pharmaceutical processes will be modelled and optimized, is not only a necessity in terms of cost-effectiveness but also a requirement by regulatory bodies. In this context, process design and Process Systems Engineering approach are expected to play a crucial role in the near future and be combined with QbD framework for the formulation of a more structured approach in pharmaceutical manufacturing.

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A Web-Based Non-invasive Estimation of Fractional Flow Reserve (FFR): Models, Algorithms, and Application in Diagnostics Yuri Vassilevski, Timur Gamilov, Alexander Danilov, German Kopytov, and Sergey Simakov

1 Introduction Evaluation of stenosis severity is an important part of the decision-making process for coronary lesions treatment. Fractional flow reserve (FFR) is a golden standard for evaluating hemodynamic importance of coronary stenosis [1]. It is defined to be the ratio of the average pressure distal to the stenosis to the average aortic pressure, during maximum hyperemia. Computed (virtual) FFR [2–4] has emerged as an effective computational tool for non-invasive FFR evaluation. Virtual FFR evaluation provides a nontraumatic and cheap diagnostic tool for patients with the ischemic heart disease. In this work we present a new web-based computational technology for noninvasive estimation of FFR based on patient-specific data. In Sect. 2 we briefly

Y. Vassilevski () INM RAS, Moscow, Russia Sirius University of Science and Technology, Sochi, Russia T. Gamilov · A. Danilov Sechenov University, Moscow, Russia Sirius University of Science and Technology, Sochi, Russia e-mail: [email protected] G. Kopytov Baltic Federal University, Kaliningrad, Russia INM RAS, Moscow, Russia e-mail: [email protected] S. Simakov MIPT, Dolgoprudny, Russia Sirius University of Science and Technology, Sochi, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_18

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present the 1D mathematical model of coronary haemodynamics. In Sect. 3 we describe algorithms for contrast enhanced CT (ceCT) data segmentation and construction of a 1D core network of individual coronary vasculature. For important details of the algorithms we refer to [5, 6]. In Sect. 4 we present our novel web-based software. It provides 3D visualization and a graphical user interface. We also discuss and describe a web protocol for efficient data exchange and processing. Finally, in Sect. 5 we demonstrate performance of our technology and discuss remaining issues and a future work.

2 Mathematical Model and Its Numerical Implementation This section describes a coronary hemodynamics model which underlies our software. To calculate FFR, the model uses a 1D mesh on a vascular graph extracted from patient’s CT scans. The model describes unsteady axisymmetric flows of Newtonian viscous incompressible fluid (blood) through a network of elastic tubes representing patient’s vasculature. Below we present a brief summary, for details we refer to [3, 5, 7]. The flow in every vessel (graph edge) is described by the mass and momentum balance equations .

 V=

.

∂V ∂F(V) = G(V) , + ∂x ∂t

(1)

     0 A Au u , , G(V) = , F(V) = −8π μ u u2 /2 + p(A) /ρ A

where t is the time, x is the distance along the vessel counted from the vessel’s junction point, .ρ = 1.060 kg/m3 is the blood density, .A(t, x) is the vessel crosssection area, p is the blood pressure, .u(t, x) is the linear velocity averaged over the cross-section, and .μ = 2.5 mPa · s is the dynamic viscosity of the blood. Relation between the pressure and the cross-sectional area is defined by the constitutive equation     A −1 −1 , p(A) = ρw c2 exp A˜

.

(2)

where .ρw = 1.060 kg/m3 is the density of the vessel wall material, c is the velocity of small disturbance propagation in the material of the vessel wall (velocity of the pulse wave propagation), and .A˜ is the rest cross-sectional area of the vessel at zero transmural pressure and zero flow. At the vessel’s junction points, we impose the mass conservation condition and the total pressure continuity (Bernoulli’s law):

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εk Ak (t, x˜k ) uk (t, x˜k ) = 0,

.

(3)

k=k1 ,k2 ,...,kM

pk (t, x˜k ) +

.

ρu2 (t, x˜k ) ρu2 (t, x˜k+1 ) , k = k1 , k2 , . . . , kM−1 , = pk+1 (t, x˜k+1 ) + 2 2 (4)

where k is the index of the vessel, M is the number of connected vessels, {k1 , . . . , kM } is the range of indices of the connected vessels, .ε = 1, x˜k = Lk for incoming vessels, and .ε = −1, x˜k = 0 for outgoing vessels. The boundary conditions at the aortic root simulate the blood flow from the heart, which is set as a function .QH (t):

.

u(t, 0) A(t, 0) = QH (t) ,

.

  ⎧ ⎨SV π sin π t , τ 2τ .QH (t) = ⎩ 0, τ < t  T ,

0  t  τ,

(5)

(6)

where SV is the stroke volume of the left ventricle, T is the period of the cardiac cycle, and .τ is the duration of the systole. Parameters .SV (τ ), .T (τ ) and .τ can be extracted from patient’s data [8]. The boundary condition at the outlet of terminal vessel k involves constant outflow pressure .Pout = 25 mmHg and resistance .Rk : Pt − Pout = Qt /Rk .

.

(7)

where .Pt and .Qt are pressure and flow at the terminal point. .Rk is set according to the Murray’s law through an iterative algorithm described in [3]. The computational domain (graph) consists of the aortic root (represented by a graph edge), the aorta (a graph edge), and two patient-specific graphs for the left coronary artery (LCA) and the right coronary artery (RCA) with their branches. We imitate a stenosis of coronary arteries as a separate vessel with decreased diameter according to the patient’s data. Section 3 describes an algorithm for automatic generation of the computational graph and evaluation of the stenosis geometry. FFR is computed as the ratio between the averaged (in time) pressure distal to stenosis and the averaged (in time) aortic pressure during hyperemia. Hyperemia is imitated by 70% reduction of .Rk in (7) which increases coronary flow by 150–250% [1]. Equation (1) at the internal (for each vessel or graph edge) grid nodes is discretized and solved by the explicit grid-characteristic method [9]. The boundary conditions at the junction nodes involve compatibility conditions of the hyperbolic system (1) along characteristics leaving the integration domain. Together with Eqs. (3) and (4), the boundary condition yields a system of algebraic nonlinear equations solved by the Newton method. Numerical method allows us to split

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calculations for internal mesh points and boundary points, processing each branch and each junction separately. Such splitting grants efficient parallelization.

3 Algorithms for ceCT Image Processing Given a contrast enhanced CT (ceCT)1 image of coronary vessels, we perform the 1D vessel network reconstruction in three major steps: vessel segmentation, thinning-based extraction of centerlines, and graph reconstruction. We utilize semiautomatic user-supervised algorithms for segmentation of the aorta and coronary arteries. Skeletonization and graph reconstruction are performed in an automatic way. The methods and algorithms are presented in details in our previous work [6]. Once the 1D vessel network graph is constructed, the user may define stenotic regions in order to estimate their hemodynamic significance. In our framework of the web-based simulation, all the steps are performed on the server side and are supervised and guided by the user on the client side.

3.1 Segmentation The client uploads the input data in an anonymized format to the server. The user may crop and/or resample the image in order to reduce the computational cost of the image segmentation. The first stage of the segmentation is detection of the aorta on the transversal ceCT slices through Hough circleness transform [6]. This algorithm is used to detect the largest bright disk corresponding to contrast enhanced blood inside the circular-shaped aorta (Fig. 1, stage 2). The user may override the detected slice and/or location of the aorta circle. The center of the aorta circle is used as a seed point in the isoperimetric distance trees (IDT) method for aorta segmentation [6, 10] (Fig. 1, stage 3). Depending on the quality of the ceCT image, the user may tune the default value of the ratio coefficient used in the IDT method in order to improve the quality of the segmentation. Since the actual 3D shape of the aorta is not used in the virtual FFR computation, the final quality of the aorta segmentation is not crucial. Once the rough shape is captured, the user is advised to proceed to the next stage (Fig. 1, stage 4) where additional morphological operations [11] are applied in order to hide pulmonary arteries and smooth the aorta. After that, the Frangi vesselness filter [12] is used to segment the coronary arteries [6] (Fig. 1, stage 5). Depending on the quality of the input ceCT images, the user may tune the default values of

1 More

precisely, coronary CT angiography (cCTA) with contrast.

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1. Input DICOM image

2. Aorta slice detection

3. IDT method

4. Smoothed aorta

5. Frangi segmentation

6. Skeletonization

7. Graph contruction

8. Stenosis definition

9. Numerical calculation

Fig. 1 Flowchart of the segmentation and modelling pipeline stages: (1) input ceCT data as DICOM image, (2) aorta slice detection by Hough transform filter, (3) aorta segmentation by IDT method, (4) smoothing of aorta by mathematical morphology, (5) coronary arteries segmentation by Frangi filter, (6) skeletonization of coronary arteries, (7) initial 1D graph network, (8) modified graph with marked stenoses, and (9) virtual FFR calculation

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coefficients used in the Frangi filter in order to improve the quality of the coronary artery segmentation. However, in practice the default values are good enough for almost all ceCT images.

3.2 Skeletonization and Graph Construction Once the ceCT image is segmented, the centerlines are extracted from the coronary arteries binary masks through skeletonization process with the help of the distanceordered homotopic thinning [13] modified method [6] and through false twig elimination post-processing [6] (Fig. 1, stage 6). These methods reduce the diameter of the vessels down to one voxel while preserving the topology of the network. Minor false twigs occurring at the first step are identified and removed during the second step. Once the skeletonization is done, all voxels in the resulting skeleton are classified in three types: endpoints, midpoints, and junctions. The endpoint voxels have only one neighbor in the skeleton. The midpoint voxels have exactly two neighbors in the skeleton. The junction voxels have more than two neighbors in the skeleton. This voxel classification is used to reconstruct a graph representation of the vessel network. The endpoint voxels become endpoints in the graph. The groups of junction voxels become bifurcation (junction) points. The groups of midpoint voxels become edges of the graph (Fig. 1, stage 7).

3.3 Processing of Stenoses The user defines a stenotic region basing on a vessel profile histogram which represents the vessel diameter along its centerline. For vessel diameter calculation, we use a method of inflating ball. If voxels of the ceCT image are cubic, we locate the center of the ball with the initial diameter of three voxels at the centerline of the processed vessel. This ball is assumed to contain only voxels masked as belonging to the vessel. Then we inflate the ball until it catches a background (non-masked) voxel closest to the ball center. The diameter (in millimeters) of the inflated ball is the vessel diameter. We note that ceCT images are given in the DICOM format, and Z-spacing between DICOM slices can differ from XY-spacing so voxels are not exactly cubic. For example, for DICOM data with XY-spacing 0.405 mm and Zspacing 1 mm, the cube in the Euclidean space is represented by a set of 7 .× 7 .× 3 voxels. To take into account the geometric anisotropy of the voxel grid, we compute the Euclidean distance from each voxel center to the ball center.

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4 User’s Interface The browser-based application virtual FFR allows for non-invasive estimation of FFR on the basis of the conventional CT angiography (cCTA) with contrast.

4.1 General Concept The general idea is to use a browser as a visualization and interaction tool for scientific modeling systems. In other words, a browser becomes a front end or a user interface (UI) for a remotely executed program. This does not mean using a browser for trivial visualization but rather utilizing browser capabilities for rich 3D visualization and interaction with the user. This approach has an obvious advantage that there is no need to install a remote program on the user’s computer so the user can run it faster and can access from everywhere where you can find a browser. This fits ideally to the software as a service (SaaS) concept. In particular, our web application virtual FFR does not need to be installed on a local machine (user’s computer) and requires only a browser to get full access to application capabilities. DICOM data is uploaded from the local machine to a remote server and processed there, and results of virtual FFR calculation are visualized in a browser at a local machine. All data are anonymized before transferring to a remote server. This approach requires solution of three tasks: (1) how to visualize in a browser, (2) how to interact with a remote program, and (3) how to make a remote program capable of interacting with a browser. For the first task, we suggest using THREE.JS2 library which is a de facto standard for developing web-based applications with sophisticated visualization. The second task can be accomplished in different ways. One can use HTTP protocol as a natural way of communication with a browser, or use existing frameworks like Google’s gRPC, or build custom mechanisms. The text-based nature and simple request-response schemes of HTTP protocol do not allow for more elaborated schemes like client-server streaming interactions. Google’s gRPC provides a variety of interaction types including streaming interactions, but its deploying requires significant efforts both on client and server sides. By that reason we decided to develop our own transport mechanism based on web sockets and custom data serialization similar to gRPC Protocol Buffers. The mechanism was tailored for transferring and processing of DICOM data although the result turned out to be equally applicable for any type of interaction between a browser and a remote program.

2 https://threejs.org/.

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To accomplish the third task, we use docker multistage builds and CRADLE containers, a special technique of compiling and linking service executable with no external dependencies. We also use web socket stubs for implementing remote procedure call (RPC) interface via web sockets.

4.2 Transport Protocol and Data Serialization Data exchange between a browser and a remote program occurs via web socket, a two-way communication channel on top of TCP/IP connection. Like HTTP protocol, the web socket protocol WS uses the same port 80 that eliminates a problem of traffic being blocked by a firewall. The web socket supports both text and binary messages. Virtual FFR application usually uses the text mode for sending short messages with a few input or output parameters. Binary messages are sent only if there is a big payload in a message. The web socket protocol easily distinguishes between binary and text mode so there is no confusion. The web socket binary format supports sending all types of integers (signed and unsigned), floats and doubles. On top of that a serialization protocol was developed that introduces new data types of one-, two-, and three-dimensional arrays of basic types. Strings are represented as 1D arrays of UTF-8 characters. All data transferred in messages are tagged with one-byte data descriptor which allows for unambiguous parsing. Besides that, serialization protocol introduces three new types tailored specifically for DICOM data. These types are bit mask, mask with ROI, and bit mask with ROI. Type bit mask is a 3D Boolean array packed as a bit string. Type mask with ROI is a 3D Boolean array that contains additional indices of a region of interest (ROI). ROI is a subvolume that encompasses foreground (masked) voxels. Type bit mask with ROI is a packed version of the second type. Voxel masks (volumes containing 0 or 1) often occur in DICOM processing. More often voxel masks are sparse when regions with “ones” occupy only a small part of the original volume. Introducing new types of data allows us to reduce greatly (up to 50 times) the amount of transferred data and storage data. Packed/unpacked data is a trade-off between memory and speed of data access, and only a programmer decides what is better. Transport protocol implements both simple RPC and server streaming RPC calls. The latter are used for long-lasting calculations like coronary arteries segmentation for reporting back a percentage of operation execution which is visualized in UI as a waiting cursor or progress bar. So far there was no need in client streaming or bidirectional RPC calls.

4.3 Software, Algorithms, and Libraries Virtual FFR application is written on pure JavaScript and uses Google Closure Compiler (GCC) for JS minification and bundling. Custom web components are

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built upon Google Closure Library. GCC is apparently the best JS minifier so far and produces a bundle two times less than Webpack and UglifyJS. GCC library has a variety of components including sliders, dialogues, splitters, etc. but requires lowlevel programming when using. For rendering of DICOM data, the application uses AMI Toolkit developed by radiology research Center FNNDSC (Fetal-Neonatal Neuroimaging Developmental Science Center) at Boston Children’s Hospital. AMI Toolkit is based on THREE.JS library. The application uses NODE.JS3 both for building WEB UI and server part of the application. To avoid “dependency hell” problem when different services require different versions of libraries, we use docker multistage builds for compiling and linking service executables. Every service has a pre-configured docker container called CRADLE with all compilers and libraries installed. For example, CRADLE container for virtual FFR application is based on Ubuntu 20.04 with specific versions of ITK and LWS libraries installed. It has a footprint of 1.13 Gb. The application building system uses CRADLE container for compiling and linking static service executable with no external dependencies that can be executed in any lightweight container such as alpine Unix. Later we found that a no-externaldependencies requirement allows for service execution even on a SCRATCH docker container, i.e., on a bare Unix kernel with 77 byte footprint. Such extremely lightweight service containers can be distributed via dockerhub and executed on any docker-enabled computer rather than a dedicated server.

5 Application Example In this section we present a short illustration of the stenosis markup process and virtual FFR application user interface (Fig. 2). The user uploads DICOM data from the local machine to the remote server and gets back aorta slice located by the application. Then the application performs aorta and coronary vessel segmentation using segmentation threshold and other parameters given by the user if necessary. For stenosis markup one has to select consecutive histogram bars on vessel profile histogram and input stenosis percentage (vascular occlusion factor (VOF)) before sending the request to the remote server to calculate the virtual FFR. The resulting FFR is visualized for the coronary tree (Fig. 3). The developed approach was tested on a variety of clinical cases. It was discovered that the main factors affecting the virtual FFR values are the degree (VOF) and the length of each stenosis followed by the hyperemia model, the cardiac output, the vessel’s elasticity, and the outflow pressure. With proper parameters identification, this technology provided deviation of the computed FFR from invasive FFR measurement about 0.05 (around 6% of FFR value) [14]. This

3 https://nodejs.org/.

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Fig. 2 Various stages of segmentation process. Top-left, detection of the aorta; top-right, segmentation of the aorta and coronary arteries; bottom-left, extraction of coronary arteries and centerlines; bottom-right, stenoses demarcation Fig. 3 Calculated FFR along coronary arteries

accuracy is good enough for most of the cases, but stenoses with FFR values close 0.8 may be misdiagnosed for stenting.

6 Discussion and Future Research The developed software provides a user-friendly interface which allows for the manual control of the data preparation process, computations, and analysis. The web-based architecture grants easy installation in any medical center on any known operating system. The proposed methodology provides sufficient accuracy for patient-specific noninvasive FFR estimation at computational time acceptable for clinical usage (few minutes). Our approach was compared with computed FFR based on 3D simulations

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[4]. We demonstrated that 1D and 3D approaches provide similar sensitivity and specificity. Correlation coefficient between 1D and 3D models were rather high, while the demand for computational resources for 1D models was significantly lower. These results correspond to findings of other scientific groups [15]. However, some cases require additional computational analysis of various pathophysiological factors including multivascular stenosis [16], increased heart output due to a physical load, mental stress, [17], blood viscosity [18], variable heart rate [8, 19], and myocardial perfusion damage [20]. There exist other hemodynamic indices (CFR, iFR) which are important for analyzing coronary stenosis severity as well. Their virtual counterparts have been studied by computational models in [3, 21–23]. These computational models should be clinically tested and implemented in future releases of our virtual FFR application. The use of neural networks and machine learning technique may decrease substantially the computational time and increase robustness of the software in detecting stenoses [24] and evaluating FFR [25]. Acknowledgment The work was supported by Russian Science Foundation Grant No.21-7130023.

References 1. Gould, K.L. Kirkeeide, R.L., Buchi, M.: Coronary flow reserve as a physiologic measure of stenosis severity. J. Am. Coll. Cardiol. 15(2), 459–474 (1990) 2. Morris, P.D., van de Vosse, F.N., Lawford, P.V., Hose, D.R., Gunn, J.P.: “Virtual” (Computed) Fractional Flow Reserve: Current Challenges and Limitations. JACC Cardiovasc Interv. 8(8), 1009–1017 (2015) https://doi.org/10.1016/j.jcin.2015.04.006 3. Simakov, S., Gamilov, T., Liang, F., Kopylov, P.: Computational Analysis of Haemodynamic Indices in Synthetic Atherosclerotic Coronary Netwroks. Mathematics. 9(18), 2221 (2021) https://doi.org/10.3390/math9182221 4. Gognieva, D. G., Pershina, E. S., Mitina, Yu. O., et.al.: Non-Invasive Fractional Flow Reserve: a Comparison of One-Dimensional and Three-Dimensional Mathematical Modeling Effectiveness. Cardiovascular Therapy and Prevention. 19(2):2303 (2020) (In Russian) https:// doi.org/10.15829/1728-8800-2020-2303 5. Vassilevski, Yu., Olshanskii, M., Simakov, S., Kolobov, A., Danilov, A.: Personalized Computational Hemodynamics: Models, Methods, and Applications for Vascular Surgery and Antitumor Therapy. Academic Press (2020) 6. Danilov, A., Ivanov, Yu., Pryamonosov, R., Vassilevski, Yu.: Methods of graph network reconstruction in personalized medicine. Clinical implications of dysregulated cytokine production. Int. J. Numer. Method. Biomed. Eng. (2015) https://doi.org/10.1002/cnm.2754 7. Simakov, S.: Spatially averaged haemodynamic models for different parts of cardiovascular system. Russian J. of Num. Anal. and Math. Mod. 35(5), 285–294 (2020) 8. Gamilov, T., Kopylov, P., Serova M., et.al.: Computational Analysis of Coronary Blood Flow: The Role of Asynchronous Pacing and Arrhythmias. Mathematics. 8(8), 1205 (2020) https:// doi.org/10.3390/math8081205 9. Magomedov, K. M., Kholodov, A. S. Grid-characteristic numerical methods. Nauka, (2018) [in Russian].

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10. Grady, L.: Fast, quality, segmentation of large volumes – Isoperimetric distance trees. Comput. Vis. – ECCV 2006, 3953, 3711–3723 (2007) 11. Yang, G., Kitslaar, P., Frenay, M., Broersen, A., Boogers, M.J., Bax, J.J., Reiber, J.H.C., Dijkstra, J.: Automatic centerline extraction of coronary arteries in coronary computed tomographic angiography. Int. J. Cardiovasc. Imaging, 28, 921–933 (2012) 12. Frangi, A.F., Niessen, W.J., Vincken, K.L., Viergever, M.A.: Multiscale vessel enhancement filtering. Medical Image Computing and Computer-Assisted Interventation – MICCAI98, 1496, 130–137 (1998) 13. Pudney, C.: Distance-ordered homotopic thinning: A skeletonization algorithm for 3D digital images. Comput. Vis. Image Underst., 72, 404–413 (1998) 14. Carson J.M., Pant, S., Roobottom, C., et.al.: Non-invasive coronary CT angiography-derived fractional flow reserve: A benchmark study comparing the diagnostic performance of four different computational methodologies. Int. J. Numer. Method. Biomed. Eng., 35(10), e3235, (2019) https://doi.org/10.1002/cnm.3235 15. Blanco, P.J., Bulant, C.A., Müller, L.O., et.al.: Comparison of 1D and 3D Models for the Estimation of Fractional Flow Reserve. Sci Rep 8, 17275 (2018). https://doi.org/10.1038/ s41598-018-35344-0 16. Simakov, S.S., Gamilov, T.M., Koylov, F.Yu., Vasilevkii, Yu.V.: Evaluation of Hemodynamic Significance of Stenosis in Multiple Involvement of the Coronary Vessels by Mathematical Simulation. Bulletin of Experimental Biology and Medicine, 162(1), 111–114, (2016). 17. Gamilov, T., Simakov, S.: Blood Flow Under Mechanical Stimulations, Advances in Intelligent Systems and Computing, 1028, 143–150 (2020) 18. Simakov, S., Gamilov, T.: Computational Study of the Effect of Blood Viscosity to the Coronary Blood Flow by 1D Haemodynamics Approach. Smart Innovation, Systems and Technologies, 214, 237–248 (2021) 19. Ge, X., Simakov, S., Liu, Y., Liang, F.: Impact of Arrhythmia on Myocardial Perfusion: A Computational Model-Based Study. Mathematics, 9(17), 2128 (2021) 20. Simakov, S., Gamilov T., Liamf, F., et.al.: Numerical evaluation of the effectiveness of coronary revascularization. Russian J. of Num. Anal. and Math. Mod., 36(5), 303–312 (2021) 21. Ge, X., Liu, Y., Tu, S., et. al.: Model-Based Analysis of the Sensitivities and Diagnostic Implications of FFR and CFR Under Various Pathological Conditions. Int. J. Numer. Method. Biomed. Eng., 37(11), e3257, (2021) https://doi.org/10.1002/cnm.3257 22. Ge, X., Liu, Y., Yin, Z., et. al.: Comparison of Instantaneous Wave-Free Ratio (iFR) and Fractional Flow Reserve (FFR) With Respect to Their Sensitivities to Cardiovascular Factors: A Computational Model-Based Study. Journal of Interventional Cardiology, 2020, 40941421 (2020) https://doi.org/10.1155/2020/4094121 23. Carson, J.M., Roobottom, C., Alcock, R., Nithiarasu, P.: Computational Instantaneous WaveFree Ratio (IFR) for Patient-Specific Coronary Artery Stenoses Using 1D Network Models, Int. J. Numer. Method. Biomed. Eng., 35(11), e3255, (2019) https://doi.org/10.1002/cnm.3255 24. Jones, G., Parr, J., Nithiaasu, P., Pants, S.: Machine Learning for Setection of Stenoses and Aneurysms: Application in a Physiologically Realistic Virtual Patient Database. Biomechanics and Modeling in Mechanobiology, 20(6), 2097–2146 (2021) 25. Itu, L., Rapaka, S., Passerini, T., et.al.: A Machine-Learning Approach for Computation of Fractional Flow Reserve From Coronary Computed Tomography. J. Appl. Physiol., 121, 42– 52 (2016)

Perturbing Coupled Multivariable Systems A. Mukhopadhyay, Ganesh Bagler, and Somdatta Sinha

1 Introduction Most natural and artificial systems are organised assemblies of several interacting variables and parameters. The functional dynamics exhibited by such systems are emergent properties arising due to nonlinear interactions of these variables. This is particularly evident in biological systems. For example, electrical activity in neural or cardiac cell is the result of a large number of ionic currents and signalling events [5]. Even the simplest model of the cardiac/neural cell is a two variable Fitzhugh-Nagumo model [10]. Processes during development [20] or gene regulation networks [19] are composed of and regulated by a large and highly interconnected network of genetic and biochemical processes. A reasonably realistic model for the synthesis of the amino acid Tryptophan in bacteria is a four-variable model where several molecular reactions have been summarised to few variables [18]. Natural systems experience perturbations of different time scales—including noise—all the time [9]. The emergent behavioural response of multivariable systems to any systemic or environmental perturbations can be both complex and nonintuitive. General physicochemical or biological perturbations such as thermal stress, electromagnetic fields, mechanical pressure, chemical concentration, or population density can influence the system behaviour in different ways based on the variables

A. Mukhopadhyay MorphoSys AG, Planegg, Germany G. Bagler Department of Computational Biology, IIIT-Delhi, New Delhi, India S. Sinha () Indian Institute of Science, Education and Research Kolkata, Mohanpur, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_19

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on which the perturbation may act and their nonlinear interaction. For example, electrical shocks used for elimination of spiral waves in cardiac tissue can act both at the cellular system level—of transmembrane currents and the interacting signalling processes in the cells—and also at a larger spatial scale (spanning several cells) of the tissue for disturbing the spiral waves [16]. In recent times, large-scale experiments involving genomics, proteomics, and mathematical modelling are done to unravel the interaction within and between biochemical pathways in cells by systematically perturbing each pathway component or interactions either genetically or by changing the environment [4]. Several unknown interactions in the pathway have been unravelled by detecting and quantifying the corresponding global cellular response to each perturbation. Efficient ways of cellular perturbations and global measurements will help increase knowledge of these large pathways considerably. External influences can also come from the extrinsic sources of noise. How random variations in cellular components can lead to very different developmental path in cells are known in biological systems [9, 17]. The dependence of a multivariable-coupled nonlinear system’s dynamical behaviour to perturbations that may interact with different variables has not received much attention. Simple nonlinear systems such as one-dimensional maps having different functional forms (density dependence) used as models for single discrete populations are considered to belong to the same universality class of single hump maps [3, 7, 8] as they exhibit similar bifurcation behaviour—from stability to chaos—with the variation of the growth parameter. It has been shown [15, 21– 23, 25] that the dynamical response of these “equivalent” systems to constant external perturbation (e.g. migration or harvesting) are quite different. Thus, the same perturbation elicits different response in otherwise-equivalent maps due to differences in the details of the nonlinear evolution functional form [15]). This has been shown to be true for spatio-temporal systems for both single and multivariablecoupled map lattice systems [11–14]. In this paper we investigate the role of external perturbation on model multivariable-coupled dynamical systems that show complex dynamics, e.g. the discrete two-variable host-parasite system [24] exhibiting quasi-periodic dynamics. We study the system dynamics by applying fixed perturbation to the variables individually or simultaneously. We show that the response of the same system can be quite different depending on which of its variable is perturbed. Also, not all variables can respond equally well to perturbation for controlling the complex dynamics in the system. When external perturbations are applied to a system that is governed by nonlinear multivariable processes, the response can differ based on how the perturbation interacts with each variable. This is important in the context of real physical and biological systems where factors in the environment can perturb each of the system variables differentially, and hence the net observable dynamical response of the system will not be easily predictable.

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2 Model and Method Our method of applying constant perturbation to the state variables in a coupled multivariable discrete system is as follows: .

X(n + 1) = F (X(n)) + L,

where .X represents the vector of N state variables, n is the time steps in discrete systems, .F denotes the vector of the nonlinear functions, and the vector .L represents the strength of constant perturbation applied to each variable which can assume positive and negative values. We have chosen the two variable coupled host-parasite (HP) model system [24] in ecology as the prototype of the discrete systems, which is given by H (n + 1) ≡ F1 (H (n), P (n)) = rH (n)[1 − H (n)] Exp[−βP (n)],

.

P (n + 1) ≡ F2 (H (n), P (n)) = H (n){1 − Exp[−βP (n)]},

(1)

where .H (n) and .P (n) are the densities of the host and parasite population at the nth generation. The growth of the host population follows the logistic map, which is modulated by parasitism .(exp[−βP (n)]). Parasite grows only by infecting the hosts. The parameters r and .β represent the intrinsic growth rate of the host and the searching efficiency of the parasite. The trivial steady state .(0, 0) and the axial steady state .((1 − 1r ), 0) always exist for the model system (1). The other nontrivial steady states .(H ∗ , P ∗ ) of the system (1) can be found from the following equations:

.

 . 1−

     1 1 ∗ ,. = 1 − Exp −β H 1 − r(1 − H ∗ ) r(1 − H ∗ )   1 ∗ ∗ . P = H 1− r(1 − H ∗ )

(2) (3)

On designating the left- and right-hand sides of Eq. (2) as f and g, respectively, we find .H ∗ ∈ (0, 1) where f and g intersect in (.H, (f, g)) plot. The corresponding .P ∗ can be found from the Eq. (3). It is clear from Eq. (3) that .P ∗ ≥ 0 implies .g ≥ 0 which also implies (from Eq. (2)) .f ≥ 0. Thus for positive steady states .(H ∗ , P ∗ ), one should consider the value of H for which the intersections of f and g lie in the positive quadrant of the (.H, (f, g)) plot. Depending upon the parameter values of r and .β, the system (1) has unique/multiple nontrivial steady state/states. In the plot (r and .β) vs .H ∗ , the surfaces I and I I show steady-state (.H ∗ ) for different values of r and .β (see Fig. 1). The surface I shows that for a particular value of r, the axial steady state remains constant for all .β and it increases as r increases. For a given value of r, up to certain value of .β(≈1.4), the surface I I indicates that there is no interior

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steady state and interior steady state appears for .β > 1.4. The curve I I I in .(r, β) plane shows the set of values of .(r, β) from where one gets interior steady state. In particular, for the parameters .r = 4 and .β = 3.5, the steady states are .(0, 0), .(0.75, 0), and .(0.42, 0.24). By substituting .H (n) = H ∗ + hn , P (n) = P ∗ + pn in the Eq. (1) and retaining only linear terms, the stability of the system around the steady state .(H ∗ , P ∗ ) can be determined [1] by solving x 2 − Bx + C = 0,

.

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using .hn = ax n . Here r + (β − 2r)H ∗ , r(1 − H ∗ )   H∗ ∗ C = βH 1 − . r(1 − H ∗ )2

B=

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The magnitude of roots .x1 and .x2 of the Eq. (4) determines the stability of the steadystate .(H ∗ , P ∗ ). It is sufficient to test .|xi | < 1, (i = 1, 2) by the inequalities [1]: |B| < 1 + C < 2.

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Fig. 2 The stable and unstable regions in the .(r, β) plane. (a) The region with dot .(·) indicates stable axial steady state. (b) The stable region of the interior steady state is marked by cross .(×), and (c) the region with .(o) indicates where ==the system (1) is unstable r }; (e.g. for .r = 4 and .β = 3.5, .B > 0 Note that .B > 0 when .0 < H ∗ < min{1, 2r−β ∗ when .H < 0.89). Figure 2 shows the stability region in the .(r, β) plane. The region with dot .(·) indicates where the axial steady state is stable. The stable region of the interior steady is marked by cross .(×), and the remaining region with .(◦) indicates the values of r and .β where the system (1) is unstable. It is important note that the region above dash and solid lines together indicates the values of r and .β where the interior steady state exists. Moreover, the interior steady state is stable in the entire region enclosed by the lines except for a small portion of it. It is observed from Fig. 2 that the axial steady state is unstable for .r > 3 as desirable for logistic map. The dynamical behaviour of the HP system is depicted in the bifurcation diagram (see Fig. 3) of H with respect to .β for four different values of r. For example, in absence of parasite, at .r = 4, the host shows chaotic population variation with density of .H ∈ [0, 1]. In presence of P with .β = 3.5, this HP system shows quasi-periodic dynamics. At this state the HP system has three steady states, e.g. .(0, 0), .(0.75, 0), and .(0.42, 0.24)—all unstable (for details see Appendix A). As our method stated earlier, constant perturbation is applied to the HP system Eq. (1) in the following manner:

H (n + 1) = rH (n)[1 − H (n)] Exp[−βP (n)] + L1 ,

.

P (n + 1) = H (n){1 − Exp[−βP (n)]} + L2 ,

(7)

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L1 and .L2 are strengths of constant perturbation to the host and parasite populations, respectively. We studied the response of this system when perturbation is applied to either host (H ), or parasite (P ), or to both the populations simultaneously.

.

3 Results 3.1 Steady-State Analysis The steady states of the Eq. (7) cannot be obtained analytically. The nontrivial steady states .(H ∗ , P ∗ ) can be found from the following equations: .

 1−

      H ∗ − L1 H ∗ − L1 ∗ , . (8) 1 − + L = 1 − Exp −β H 2 rH ∗ (1 − H ∗ ) rH ∗ (1 − H ∗ )   H ∗ − L1 ∗ ∗ + L2 . P = H 1− (9) rH ∗ (1 − H ∗ )

As in section (2), designating the left and right-hand sides of Eq. (8) as f and g , respectively, .H ∗ can be found from the intersections of f and g for specific values of r , .β , and .L1 , .L2 . The corresponding .P ∗ can be found from Eq. (9). Here, it is also clear from Eq. (9) that .P ∗ ≥ 0 implies .g ≥ 0 and .f ≥ 0 (from Eq. (8)). Thus for positive steady states, the intersections of f and g should lie in the positive quadrant of the (.H, (f, g)) plot for specific values of r , .β , and .L1 , .L2 .

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0.1

Fig. 4 The steady states .(H ∗ , P ∗ ) of the system (7) for different values of .L1 and .L2 and for specific values of .r = 4 and .β = 3.5

We plot .H ∗ and .P ∗ for different values of .L1 and .L2 and for specific values of .r = 4 and .β = 3.5. It is observed that for all values .L1 , there are two nontrivial steady states for .L2 ≤ 0, and for all values of .L1 and .L2 > 0, there is one interior steady state. It is interesting to note that as .L2 decreases from zero, two steady states come closer to each other. The above discussion was depicted in Fig. 4. For example, Fig. 5a–c show the steady states obtained for the HP system (.β = 3.5, and .r = 4) for three cases—(a) no perturbation .(L1 = L2 = 0) and simultaneous (b) negative and (c) positive perturbation of .L(= L1 = L2 ) = ∓0.05. The unperturbed (.L = 0) HP system (Fig. 5a) shows the two nonzero .H ∗ values (.0.422 and .0.75), and as mentioned earlier the positive steady state for this system is .(0.422, 0.239). For .L = −0.05 (Fig. 5b), there are two positive (.(0.491, 0.175) and .(0.692, 0.040)) steady states. For .L = 0.05 (Fig. 5c), there is a unique positive steady state ∗ .(0.383, 0.298). It is important to note, for .L < 0, the axial steady state .(H = 0.75) moves close to the interior steady state, whereas for .L > 0 it moves away from the interior steady state as observed in the Fig. 4.

3.2 Linear Stability Analysis Similar to the unperturbed system linearised, the system of the Eq. (7) around the steady state .(H ∗ , P ∗ ) can be obtained by substituting .H (n) = H ∗ + hn , P (n) =

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(a)

f&g

0.6 0.4 g

f

0.2 0 −0.2

(b)

f&g

0.6 0.4 0.2

f

g

0 −0.2

(c)

f&g

0.6 g

0.4

f

0.2 0 −0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

H

Fig. 5 Steady state of the host (.H ∗ ) of the H P system obtained from the intersection of the two graphs (cf., Eq. (8)) when both variables are perturbed equally and simultaneously (.L1 = L2 = L)—(a) .L = 0, (b) .L = −0.05, (c) .L = 0.05. Parameter values are .r = 4 and .β = 3.5

P ∗ + pn in the Eq. (7) and retaining only linear terms. For stability we have similar equation (cf., Eq. (4)). Here, (H ∗ − L1 )(r + (β − 2r)H ∗ ) , rH ∗ (1 − H ∗ )   H ∗ − L1 ∗ . C = β(H − L1 ) 1 − r(1 − H ∗ )2

.B

=

(10)

The magnitude of roots .x1 and .x2 of the Eq. (4) with reference to the values of B and C in Eq. (10) determines the stability of the steady state .(H ∗ , P ∗ ). It is sufficient to test the inequalities Eq. (6) with B and C given in Eq. (10). For the parameter values chosen (.r = 4, .β = 3.5), .B > 0 if .L1 < H ∗ < 0.89. For small perturbations considered here (.−0.05 ≤ L1 , L2 ≤ 0.05), .H ∗ < 0.89, and hence, B is always positive in this case. For particular values of .r = 4 and .β = 3.5, if plotted .(L1 , L2 ) vs .(|B| & 1 + C), it is found that .|B| < 1 + C (cf., Fig. 6). Figure 6 shows .1 + C crosses the .Z = 2 plane giving the range of values of .L1 and .L2 where the interior steady state is stable. It is also observed that the other steady state (when there exist two interior steady states) is unstable. In the parameter space ((.L1 , L2 )—plane), the plot (see Fig. 7) shows the unstable .(+) and stable .(·) regions for different values of .L1 , .L2 , and .L1 = L2 with same parameter values .r = 4, .β = 3.5.

Perturbing Coupled Multivariable Systems

2.2

325

Z=2

2 1.8

1+C 1.6 1.4 1.2

|B|

1 0.8 0.6

0.05 0.05

0.03

0.01

0.0 −0.01

−0.03

−0.05

−0.05

L1

L2

Fig. 6 Graphical study of the local stability analysis of the perturbed H P system. Functions .|B| and .1 + C (in Eq. (10)) plotted for .(L1 , L2 ) with .r = 4 and .β = 3.5 0.05

0.04

0.03

0.02

L2

0.01

0

−0.01

−0.02

−0.03

−0.04

−0.05 −0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

L

1

Fig. 7 In the parameter space ((.L1 , L2 )—plane), the unstable .(+) and stable .(·) regions for different values of .L1 , .L2 , and .L1 = L2 with same parameter values .r = 4, .β = 3.5

We studied the dynamic response of the quasi-periodic HP system when perturbation is applied to either the host H , or to the parasite P , or to both H and P populations simultaneously.

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Specific Perturbation on Host (L2 = 0)

3.2.1

In this case Eqs. (8) and (9) take the form as .

 1−

     H ∗ − L1 H ∗ − L1 ∗ = 1 − Exp −β H ,. 1 − rH ∗ (1 − H ∗ ) rH ∗ (1 − H ∗ )   H ∗ − L1 P∗ = H∗ 1 − . rH ∗ (1 − H ∗ )

(11) (12)

is embedded in the expressions of .H ∗ and .P ∗ , and so when .L2 = 0, there are no apparent change in the expressions of B and C in Eq. (10). Solving Eqs. (11) and (12) in the range .−0.05 ≤ L1 ≤ 0.05 yielded two positive steady states of which only one satisfied the condition in Eq. (6). It is clear from the Fig. 6 that .1 + C is always greater than B . Along the line .L2 = 0, Fig. 7 shows that the system will be stable for .L1 > 0.008. This implies that the stability of this steady state changes from unstable to stable for a small positive perturbation. Thus, the quasi-periodic dynamics of the H and P is stabilised with positive perturbation applied to H alone.

.L2

3.2.2

Specific Perturbation on Parasite (L1 = 0)

On perturbing the parasite (.L2 ∈ [−0.05, 0.05]), the Eqs. (8) and (9) take the form (for nonzero steady states) as  . 1−

      1 1 ∗ + L2 , . = 1 − Exp −β H 1 − r(1 − H ∗ ) r(1 − H ∗ )   1 P∗ = H∗ 1 − + L2 . r(1 − H ∗ )

(13) (14)

The expressions for B and C in Eq. (10) take the form as in Eq. (5). A similar analysis showed that only one positive steady state satisfied the condition in Eq. (6). Figure 7 shows the stable and unstable regions of the system along the .L1 = 0 line. It is found that this steady state is stable for .L2 < −0.013 and .L2 > 0.035 indicating that both positive and negative perturbation to the parasite alone can stabilise the HP system.

3.2.3

Same Perturbation on to Both Host and Parasite Simultaneously

In this case both host and parasite are perturbed equally and simultaneously with ∈ [−0.05, 0.05]. For the steady states, we have (from Eqs. (8) and (9))

.L

 . 1−

      H∗ − L H∗ − L ∗ + L , . (15) = 1 − Exp −β H 1 − rH ∗ (1 − H ∗ ) rH ∗ (1 − H ∗ )

Perturbing Coupled Multivariable Systems

 P∗ = H∗ 1 −

327

H∗ − L rH ∗ (1 − H ∗ )

 + L.

(16)

Here we have got the expressions for B and C from Eq. (10) by replacing .L1 by L. Similar to the case of perturbation of the parasite alone, here also the stability of the steady state changes from stable to unstable then back to stable for .L < −0.03 and .L > 0.009 where Eq. (6) is also satisfied (see Fig. 7 along the .L1 = L2 line). Thus, both positive and negative perturbations can stabilise the HP dynamics when both the variables are perturbed simultaneously.

3.3 Numerical Simulation of HP Dynamics Under Perturbation Figure 8 shows the population dynamics of the host and parasite obtained by numerically simulating Eq. (7) for different strengths of perturbation. These results corroborate the analytical work presented earlier. The quasi-periodic dynamics of the HP system is suppressed to equilibrium dynamics with positive perturbation (.L1 ≈ 0.008) of the host alone (cf., Fig. 7 along .L2 = 0 line), and the dynamics shows higher amplitude complex oscillations for negative perturbation (Fig. 8a) indicating enhancement in the amplitude of the complex oscillation. Figure 8b shows the case when the parasite is perturbed alone, and the quasi-periodic dynamics is suppressed to equilibrium for both positive (.L2 > 0.035) and negative 1

(a) H, P

H

0.5

P

L

0

1

1

H, P

(b) H

0.5

P

L

0

2

1

H, P

(c) H

0.5 P

0 −.05

L −.04

−.03

−.02

−.01

0

.01

.02

.03

.04

.05

Fig. 8 Bifurcation behaviour of H and P populations when negative and positive perturbation is applied to (a) H , (b) P , and (c) both H and P , with .r = 4 and .β = 3.5

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(.L2 < −0.013) perturbation (cf., Fig. 7 along .L1 = 0 line). Similar behaviour of the HP system is observed (see Fig. 8c) when both H and P are perturbed equally and simultaneously. The quasi-periodic dynamics is suppressed for both positive (.L > 0.009) and negative (.L < −0.03) perturbation. This is in accordance to Fig. 7 along the .L1 = L2 line. Thus we have shown that perturbation of the two variable host-parasite system can yield differential response in dynamics based on the specific variable being perturbed.

4 Conclusions We have shown that the emergent dynamical response of nonlinearly interacting multivariable systems to external perturbation depends, along with the nature of the perturbation, also on how each of the underlying variables of the dynamical system responds to such perturbation and on the nature of the coupling among the variables. This implies that local dynamical systems, which are regulated by coupled processes of multiple variables, may respond quite differently to external perturbations specifically applied to their evolution, compared to when the whole coupled multivariable system encounters a general perturbation. Hence the emergent system dynamics will depend crucially on the nature of perturbation and its site of action. Thus a “general” perturbation may or may not mimic behaviour obtained under “specific” perturbations. Modulating the variable in a discrete system shows that controllability of complex dynamics, using this approach, depends on the local functional form and opposite signs of the perturbation do not always lead to opposing types of control (i.e. control or anti-control). Even when qualitative response of the system may be similar for some “specific” and “general” perturbations, it can induce very different response for some other “specific” perturbations to the same variables. This indicates that the emergent behaviour of dynamical systems, which are regulated by coupled processes of multiple variables, will depend crucially on the specific variable perturbed. Our results on the host-parasite system imply that, in adopting control measures in a diseased population, killing the parasites and removing the hosts (quarantine) may have opposite effects. This is a nonintuitive result. Thus, the same vaccination strategy, harvesting policy, or conservation measures of removal/introduction of individuals in a population may not yield the same results for all interacting populations in nature. This is important in the context of several real social and biological systems where many different factors in a complex environment can perturb each of the underlying variables differentially. At the cellular level, there is much interest in the differential role of intrinsic and extrinsic fluctuations in the levels of the cell components in deciding the emergent expression of genes [2, 6, 9, 26, 27]. An example of variable extrinsic noise by inducing a repressor protein that affects gene expression [2] showed nonlinear response of the gene expression system. If such a protein controls several downstream steps in a regulatory cascade, then the effect

Perturbing Coupled Multivariable Systems

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of external noise shows nonlinear response [27]. It is not clear how the system behaviour would change if the external perturbation is in any of the other levels of the cascade or is acting at all the levels simultaneously. The cell encounters different types of stress signals from its external environment. Though we are still far from understanding the internal organisation of a whole cell that guides its response to external environment, the multiple genome projects and their functional analysis have resulted in the building of the preliminary networks of regulatory interactions that govern the biochemical functions in the cell. On studying the well-characterised transcription regulation network in bacteria E.coli, it has been found [19, 28] that a modular organisation of few patterns of interactions recur, which regulate the temporal programme of genes and also govern the responses to external signals. There are generalised perturbations, such as osmotic stress, drug, and superoxides, that involve interaction of many genes and factors at different levels. In this scenario, our results, using simpler model systems, show that the cell can exhibit different behaviour based on the level at which the generalised perturbations affect, and it may not be obvious by looking at the overall cellular behaviour to predict the level of influence of the perturbation. Acknowledgments SS thanks Indian National Science Academy for the Honorary Senior Scientist award. AM and GB thank the Department of Biotechnology and CSIR India for fellowships during the preliminary part of the work.

Appendix: Stability Analysis HP System Without Perturbation For stability of the system (1), we have to examine the inequality given in Eq. (4). In parametric form we have the following inequalities: .

B 0.00003 | N0,i − Nep,3 > 0 instead of   . Ti (N0,i ) > 0.00003 | Tp − Nep,3 − N0,j , ≥ 0 used in [14]. Finally, some states, such as Roraima (RR) and Paraíba (PB), had manual adjustments of the initial time .t (N0,j ) to satisfy the prevalence constraints defined by Eq. 5 for .j = {1, 2, 3}: ui (t) =

.

.

dIprev,i = γI,i Ii + γQ,i Qi + γA,i Ai + γR,i Ri + σi Ti − κprev,i Iprev,i dt P rei,j,min ≤

.

1 Ntotal,j

(4)

Ntotal,j −1



Iprev,i (Nep,j + k) ≤ P rei,j,max

(5)

k=0

where the bounds .P rei,j,min and .P rei,j,max of the Brazilian federative units are given in Table 1 according to the formulation proposed by Marra and Quartin [28]. .t (Nep,1 ) = 14/May/2020, .t (Nep,2 ) = 4/June/2020, .t (Nep,3 ) = 21/June/2020 are initial dates from the first, second and third phases of EPICOVID19-BR, respectively, while .Ntotal,1 = 8 and .Ntotal,2 = Ntotal,3 = 4 correspond to their respective duration in days. The end of the model identification was kept at .t (Nf,i ) = 1/October/2020 for comparison purposes concerning the fine-tuning change of the state estimator. It

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Table 1 Prevalence bounds from seroprevalence survey EPICOVID19-BR. Unbounded prevalences for the model identification problems are described by “–” i RO AC AM RR PA AP TO MA PI CE RN PB PE AL SE BA MG ES RJ SP PR SC RS MS MT GO DF

.P revi,1,min

.P revi,1,max

.P revi,2,min

.P revi,2,max

.P revi,3,min

.P revi,3,max

0 0.0193 0.0844 – 0.0900 0.0526 0 – – 0.0427 0.0065 0 0.0105 – 0 0.0006 – 0 – 0.0073 – – – – – – 0

0.0200 0.0690 0.1600 – 0.1500 0.1300 0.0094 – – 0.1000 0.0420 0.0370 0.0470 – 0.0140 0.0180 – 0.0270 – 0.0430 – – – – – – 0.0130

0.0061 0.0362 0.1037 0.1929 – 0.0957 0 0.0454 0 0.0940 0.0147 0.0297 0.0010 0.0320 0 0.0211 0.0005 0.0040 0.0310 0 0 – 0 – 0 0 0

0.0430 0.0860 0.1800 0.2900 – 0.1900 0.0140 0.0940 0.0870 0.1700 0.0570 0.0720 0.0390 0.0860 0.0180 0.0660 0.0089 0.0280 0.0890 0.0280 0.0170 – 0.0093 – 0.0190 0.0110 0.0200

0.0298 0.0295 0.0488 0.1635 0.0440 0.0862 0.0030 0.0509 0.0573 0.1427 0.0286 0.0030 0 0.0500 0.0104 0.0060 0 0.0142 0.0482 0 0.0014 0.0012 0.0001 0 0 0 0

0.0860 0.0770 0.1100 0.2600 0.0840 0.1800 0.0230 0.0960 0.1100 0.2200 0.0760 0.0330 0.0190 0.0770 0.0550 0.0420 0.0130 0.0430 0.1200 0.0190 0.0150 0.0110 0.0093 0.0140 0.0140 0.0220 0.0200

was discussed by Silva and Secchi [14] that the formulation of the testing dynamics in the function of time also infers in the transmission dynamics; however, an important factor was neglected. The government policies in the first epidemic wave of COVID-19 do not strictly follow the linear correlation defined in Eq. 1 for .νi ; thus, model parameters regarding testing are subject to significant uncertainty, especially during the relaxation of the restrictive social distancing policies. Most units of the federative units had already established a significant reopening of the economic activities on .t (Nf,i ) = 1/October/2020; however, further relaxation of the restrictive policies continued over time. The subsequent outcomes in the estimated transmissibility are observed in several case studies further in Sect. 5. The aforementioned dynamic does not affect the results of the previous lethality analyses; however, it increases the uncertainty in the relative estimation of the transmissibility from the Zeta variant. The assumption of constant testing during

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341

the simulation with the state and parameter estimation is certainly violated as there was a lack of rapid tests available following the initial circulation of the Omicron variant. In addition, the longer period of analysis progressively increases the risk of violating this constraint. Nonetheless, the results in Sect. 5 showed the robustness of the proposed methodology concerning this uncertainty. The acknowledgment of the uncertainty related to the testing dynamics led to the addition of the constraint Eq. 6 to guarantee a smooth derivative of the testing dynamic at the end of the model identification.   aζ,i bζ,i exp(−bζ,i (Nf,i − cζ,i )) aζ,i d   = ≤ 10−4 . 1− dt (1 + exp(−bζ,i (Nf,i − cζ,i )))2 1 + exp −bζ,i (t − cζ,i ) (6) The basic reproduction number .R0,i was not explicitly calculated in our previous work [14]; however, it was calculated in this work as a fundamental epidemiological property for tuning the state estimator. Meta-analyses of this property have been performed for different variants of concern during the COVID-19 pandemic, such as the ancestral [29], Delta [30], and Omicron [31] variants. Estimations were highly heterogeneous, explained by different mathematical designs, population behavior, government policies, and further dynamics correlated to the disease transmission among the samples. However, the range defined for .R0,i corresponds to a realistic target zone for the model identification and, subsequently, for tuning the state estimators. In this work, the basic reproduction number is calculated according to the next-generation matrix following the methodology proposed by Van den Driessche and Watmough [32]. First, the matrix .Fi and .Vi are defined as ⎡

0 α0,i (1 − ui ) 0 α0,i (1 − ui ) 0 ⎢0 0 0 0 0 .Fi = ⎢ ⎣0 0 0 0 0 0 0 0 0 0 ⎡

ρ 0 ⎢ pρ −(λi + i ) ⎢ ⎢ 0 i ⎢ .Vi = ⎢ ⎢(1 − p)ρ 0 ⎢ ⎣ 0 0 0 0 Thus

⎤ 0 0⎥ ⎥ 0⎦ 0

⎤ 0 0 0 0 ⎥ 0 0 0 0 ⎥ ⎥ −λi 0 0 0 ⎥ ⎥ ⎥ 0 0 0 −(θi + μi + κi ) ⎥ ⎦ −(μi + κi ) 0 0 θi μi −(σi + τi ) 0 μi

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D. M. Silva and A. R. Secchi

⎡

Fi (−Vi )−1

.

r1,i ⎢ 0 ⎢ ⎢ ⎢ 0 =⎢ ⎢ 0 ⎢ ⎣ 0 0

r2,i 0 0 0 0 0

0 0 0 0 0 0

r3,i 0 0 0 0 0

⎤ 00 0 0⎥ ⎥ ⎥ 0 0⎥ ⎥ 0 0⎥ ⎥ 0 0⎦ 00

α0,i (1 − ui )p α0,i (1 − ui )p α0,i (1 − ui )(1 − p) , .r2,i = , and + θi + μi + κi λ i + i λ i + i α0,i (1 − ui )(1 − p) .r3,i = . The largest eigenvalue of the next-generation matrix θi + μi + κi −1 is given by .r ; thus, the .R .Fi (−Vi ) 1,i 0,i corresponding to Eq. 1 is defined as where .r1,i =

R0,i =

.

α0,i (1 − ui )p α0,i (1 − ui )(1 − p) + λ i + i θi + μi + κi

(7)

Model identification is constrained to an upper bound of the .R0,i relative to the ancestral lineages in the early stages of the COVID-19 pandemic. A meta-analysis  of this dynamic [29] inferred the bound . R0,i (k) ≤ 3.44 |k = N0,i , N0,i + 1, . . . ,  . Nf,i . However, the start of the model identification is not limited by a lower bound because most studies comprise periods under severe restrictive social distancing policies. Finally, an integral squared error criterion is defined for the model identification to match the estimation error with the addition of quadratic penalty functions. The dependence on the initial guess of the methodology is removed by performing a nondeterministic optimization followed by a deterministic optimization under more severe tolerances. The hybrid optimization is defined an optimization problem:

min

identi

Nf,i 

zi (k) − yi (k)2Qid,i +

k=N0,i

ng,i 

max {0, gi (xi , ui )}2Wg,i

k=1

Subject to:  .

k+1

xi (k + 1) = xi (k) +

f(xi (t), ui (t)) dt

(8)

k

yi (k) = h(xi (k)) Si (t0,i ) ∈ [0.9, 1], α0,i ≥ 0, xc,i ∈ [0, 1], xm,i ∈ [0, 1], wu,i ∈ [0, 1] axs,i ∈ [0.9, 1], aζ,i ∈ [0, 1], bζ,i ∈ [0, 5], cζ,i ∈ [0, Nf,i − N0,i ]

 where .identi = Si (t0,i ) α0,i xc,i xm,i axs,i aζ,i bζ,i cζ,i wu,i is the optimization variable, .zi are the measured variables, and .Qid,i is a dense    weight matrix calculated by the covariance matrix of . zi (k) − zi (k − 1) | k ∈ N0,i , N0,i + 1, . . . , Nsim

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343

calculated by the routine cov from MATLAB where .t (Nsim ) = 1/October/2022. The tuning weight .Wg,i = 1014 Ing,i is set to ensure minimal constraint violation, and .gi (xi , ui ) is defined by rewriting Eqs. 5, 6, .axs,i + xc,i ≤ 1, and    . R0,i (k) ≤ 3.44 |k = N0,i , N0,i + 1, . . . , Nf,i in the form .gi ≤ 0. In addition, the previously identified parameter .axa,i was simplified to .axa,i = 1 to reduce a constraint in .gi and match an improved definition of the initial state .xi (t0,i ). The change in the performance criterion and the smaller values of .Nf,i − N0,i for some case studies in the model identification was balanced by defining .xi (t0,i ) =

   Si (t0,i ) 1 − Si (t0,i ) − zi,1 (t0,i ) 0 0 0 zi,1 /2 zi,3 (t0,i ) zi,1 /2 zi,2 (t0,i ) zi,4 (t0,i ) where .zi,1 = zi,1 (t0,i ) − zi,2 (t0,i ) − zi,3 (t0,i ) − zi,4 (t0,i ). First, the optimization defined in Eq. 8 was solved by a particle swarm optimization (PSO) algorithm [46] for an objective function tolerance of .10−4 following the MATLAB implementation in the routine particleswarm. The solution provided by the PSO algorithm is used as the initial guess .ident0,i of an SQP optimization implemented according to the routine fmincon from MATLAB, using a tolerance of .10−6 . The tolerance of the first optimization was relaxed to reduce the computational cost of the model identification while providing an initial guess in the neighborhood of a local solution for the deterministic optimization. The integration of Eqs. 1 and 4 were numerically solved by the Runge-Kutta method according to the ode45 implementation from MATLAB, using relative and absolute tolerances of .10−6 and .10−3 , respectively. The results of the model identification for all the case studies are presented in Table 2. Spatial independent parameters follow the values defined in our previous work [14]. The relaxation of the prevalence constraints and the consequent delay in the initial moment of model identification directly affected .cζ,i as seen in Table 2. The time t in Eq. 1 is defined as .t (N0,i ) = 0; thus, the model identifications that started in earlier stages of the COVID-19 pandemic presented higher values, while smaller values were found for later .t (N0,i ). Overall, the inflection point, which depends only on .cζ,i , was found at the end of May, June, or July from 2020, which is acceptable since the first epidemic wave had different starting dates and durations among studied cases. In addition, Table 2 shows the lack of observability for some model parameters. The solutions found for .axs,i are either at the lower bound .axs,i = 0.9 or the upper bound .axs,i + xc,i = 1, which also happens for .wu,i ∈ [0, 1]. The first should describe the testing patterns, while the latter should describe the heterogeneous population behavior in function of the Google mobility data. Nonetheless, the high number of fitted initial conditions correlated to the virus spread, which also includes .Si (t0,i ), .α0,i , .aζ,i , .bζ,i , and .cζ,i , makes the less sensitive optimization variables .axs,i and .wu,i unidentifiable. The definition of a smaller number of parameters correlated to the viral transmission would achieve a more descriptive model; however, they were not studied because they are not essential for the main objective of this work. The data preprocessing and the generalization of the tuning strategy were more relevant extensions of our previous work [14] aimed at improving the recursive state and parameter estimation method for analyzing further case studies under a longer analysis period.

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Table 2 Solution of the model identification for each Brazilian federative unit i i RO AC AM RR PA AP TO MA PI CE RN PB PE AL SE BA MG ES RJ SP PR SC RS MS MT GO DF

.t0,i

.Si (t0,i )

.α0,i

.xc,i

.xm,i

.axs,i

.aζ,i

.bζ,i

.cζ,i

.wu,i

28/04/2020 23/04/2020 03/04/2020 10/04/2020 31/03/2020 27/04/2020 13/05/2020 13/04/2020 04/05/2020 07/04/2020 06/05/2020 29/04/2020 10/04/2020 25/04/2020 04/05/2020 12/05/2020 03/06/2020 10/04/2020 02/04/2020 27/03/2020 03/06/2020 10/06/2020 27/05/2020 11/06/2020 19/05/2020 03/06/2020 02/05/2020

0.972 0.955 0.911 0.921 0.918 0.900 0.985 0.929 0.923 0.921 0.978 0.970 0.986 0.967 0.994 0.974 0.988 0.991 0.948 0.993 0.992 0.984 0.993 0.973 0.990 0.982 0.993

0.208 0.255 0.204 0.333 0.193 0.269 0.221 0.253 0.254 0.278 0.181 0.170 0.218 0.192 0.409 0.188 0.250 0.350 0.234 0.378 0.201 0.267 0.221 0.185 0.210 0.183 0.359

0.0116 0.0126 0.0139 0.0057 0.0126 0.0100 0.0176 0.0056 0.0096 0.0088 0.0310 0.0314 0.0581 0.0208 0.0458 0.0141 0.0359 0.0300 0.0152 0.0931 0.0506 0.0314 0.0507 0.0206 0.0528 0.0278 0.0729

0.412 0.625 0.391 0.553 0.462 0.556 0.409 0.556 0.313 0.455 0.472 0.408 0.486 0.405 0.525 0.413 0.280 0.690 0.440 0.274 0.288 0.292 0.309 0.266 0.416 0.356 0.237

0.973 0.987 0.964 0.992 0.987 0.900 0.901 0.934 0.968 0.990 0.953 0.919 0.942 0.979 0.900 0.900 0.900 0.900 0.985 0.900 0.900 0.969 0.949 0.900 0.900 0.972 0.900

0.120 0.177 0.089 0.450 0.273 0.224 0.300 0.076 0.091 0.065 0.315 0.428 0.392 0.282 0.500 0.171 0.254 0.500 0.085 0.500 0.323 0.420 0.282 0.122 0.500 0.237 0.500

0.0725 0.0517 0.1090 0.0730 0.0510 0.1520 0.0397 0.0402 0.0513 0.2498 0.2463 0.0404 0.0300 0.0623 0.0713 0.0880 0.0595 0.0657 0.2500 0.0401 0.0946 0.0374 0.0526 0.0377 0.0723 0.0504 0.1089

31.6 17.0 31.6 98.0 72.1 23.3 38.8 64.6 57.3 47.9 35.0 27.5 59.2 40.8 39.9 46.7 35.9 39.1 58.4 75.6 29.8 0.0346 0.256 11.7 45.3 22.9 29.3

0.000 1.000 0.000 0.802 0.770 0.955 0.340 0.931 0.551 0.000 0.000 0.0312 0.523 0.251 1.000 0.000 1.000 0.975 0.236 1.000 0.155 0.925 0.0360 0.315 0.0179 0.148 0.443

3 Data Preprocessing Data preprocessing comprises algorithms or operations applied to raw data before a later use in the designed goal. It is an essential step depending on the data quality or the desired result since it might improve overall results. In this work, data preprocessing was performed to acquire data from hospitalized and recovered after hospitalization, to normalize the data into the final output measurements, and to perform data cleansing.

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3.1 Confirmed Cases and Deceased The measurements of the cases and deceased clinically diagnosticated with COVID19 were obtained from the Ministry of Health of Brazil [33], which has multiple outliners during the pandemic. Several federative units in Brazil reviewed underreported notifications by the end of an epidemic wave; however, there were also reviews for overreporting at a lesser extent for some of case studies. In our previous work, outliers in the raw data were remarked as the main factor for limiting aggressive tuning of the state estimators [14], which limits the performance of the overall methodology. Several model uncertainties were supposed to be mitigated or described by the state estimator for a realistic estimation; however, the targets might be unfeasible due to tuning bounds imposed by the data outliers. Hence, additional steps of data preprocessing were applied to remove the outliers to improve a posterior fine-tuning of the state estimator. The realistic estimations for each federative unit i were defined by zone tracking some epidemiological properties in an optimization problem detailed in Sect. 4. Both measurements from the Ministry of Health of Brazil [33] are given in absolute numbers at a national, federative unit, and municipality level; thus, the data is normalized to match the model defined in Eq. 1. The total population .Ni for each federative unit i was taken from the same database, which infers into a constant .Ni for the entire analysis period. Both measurements are given as cumulative .zi,j (k) and incidence values .zi,j (k) = zi,j (k) − zi,j (k − 1) for each daily sampling time k; however, they are described in function of the cumulative measures to match the output of model .yi . The primary data preprocessing comprises Algorithms 1 and 2, which remove or smooth the outliers based on their incidences. The noise of the incidence is given by tol; a slack over the maximal outliers is defined by .δU , while a slack over the minimal outliers is defined by .δL . Unique values .tol = 10−7 , .δU = 1.3, and .δL = 0.7 were used in the preprocessing of all Brazilian federative units studied in this work, despite the expected uncertainty for tol since there are different magnitude orders among the total population between the case studies and among the cases and deceased notifications. Both algorithms are applied to the COVID-19 confirmed cases .zi,1 and confirmed deceased .zi,2 for each Brazilian federative unit i based on the data provided by the Ministry of Health of Brazil [33] on .t (Ndata ) = 11/October/2022. Finally, a posterior central average mean is   calculated according to Eq. 9 to define the measures . zi,j | j ∈ {1, 2} used in the model identification and the simulations  with the state and parameter estimation for a preprocessed data . zi,j | j ∈ {1, 2} : zi,j (k) =

.

zi,j (k + 3) − zi,j (k − 3) 7

(9)

Algorithm 1 identifies maximal outliers by comparing the incidences with their average values .dzjC and .dzjF calculated by central and forward differentiation, respectively. After that, the outliers identified have their distributed incidence

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Algorithm 1 Data preprocessing algorithm related to the maximal outliers of the confirmed cases and deceased notifications Require: δU , zi,j , tol for j ∈ {1, 2} do k ← Ndata − 6 while k > 30   do zi,j (k + 3) − zi,j (k − 3) dzjC ← 7   zi,j (k + 6) − zi,j (k) F dzj ← 7 2   2  2  2 if zi,j (k) ≥ tol 2 and zi,j (k) − δU dzjC ≥ tol 2 and zi,j (k) −  2 δU dzjF ≥ tol 2 then dzj ← zi,j (k) − δU dzjC if dzj ≥ 0 then dzj ← max(dzj , tol) else dzj ← min(dzj , −tol) end if m←1   zi,j (k) − zi,j (k − m) dzmj ← m 2 2 2  2    while k − m > 30 and dzmj ≥ δU dzjC and dzmj ≥ δU dzjF do m ← m +1  zi,j (k) − zi,j (k − m) dzmj ← m end while if k − m > 30 then for cont ∈ {1, . . . , m} do   cont zi,j (k − m − 1 + cont) ← zi,j (k − m − 1 + cont) + dzj m end for else zi,j (k − 1) ← zi,j (k − 1) + dzj end if k ←k+1 end if k ←k−1 end while end for

over the data in previous sampling times since they comprehend the review of a previously underreported time series. This outlier is frequent at the end of an epidemic wave following the unburden of the health system and subsequent decrease in hospital bed occupancy. The interval and amount of incidence distributed follow conditionals defined by tuning .δU ; thus, data preprocessing results depend on corresponding tuning parameters. The definition of .δU = 1 characterizes a constant incidence for preprocessed time series; thus, .δU ≥ 1 and the distance toward the lower bound defines the smoothness of the final result. The maximal outlier

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Algorithm 2 Data preprocessing algorithm related to the minimal outliers of the confirmed cases and deceased notifications. Require: δL , zi,j , tol for j ∈ {1, 2} do k ← 30 while k ≤ Nsim − 6 do  zi,j (k − 1) − zi,j (k − 7) dzjB ← 7 2   2 if δL dzjB − zi,j (k) ≥ tol 2 then dzj ← zi,j (k) − δL dzjB if dzj ≥ 0 then dzj ← max(dzj , tol) else dzj ← min(dzj , −tol) end if m←1   zi,j (k + m − 1) − zi,j (k − 1) dzmj ← m 2 2   while k + m − 1 < Ndata and dzmj ≤ δL dzjB do m ← m +1  zi,j (k + m − 1) − zi,j (k − 1) dzmj ← m end while if k + m − 1 = Ndata then zi,j (k − 1) ← zi,j (k − 1) + dzj else for cont ∈ {1, . . . , m} do zi,j (k + cont − 1) ← zi,j (k + cont − 1) − dzj end for end if k ←k−1 end 2  if if zi,j (k) ≤ 0 then if dzj ≥ 0 then dzj ← max(dzj , tol) else dzj ← min(dzj , −tol) end if m←1   zi,j (k) − zi,j (k − m) dzmj ← m 2 2   while k − m > 30 and dzmj ≤ δL dzjB do m ← m +1  zi,j (k) − zi,j (k − m) dzmj ← m end while

348

D. M. Silva and A. R. Secchi if k − m = 30 then m←1   zi,j (k + m − 1) − zi,j (k − 1) dzmj ← m 2 2   while k + m − 1 < Ndata and dzmj ≤ δL dzjB do m ← m +1  zi,j (k + m − 1) − zi,j (k − 1) dzmj ← m end while for cont ∈ {1, . . . , m} do zi,j (k + cont − 1) ← zi,j (k + cont − 1) + dzj end for else for cont ∈ {1, . . . , m} do zi,j (k − m − 1 + cont) ← zi,j (k − m − 1 + cont) + dzj



cont m



end for end if k ← k − cont − 1 end if k ←k+1 end while end for

identification and redistribution start at .k = Ndata − 6 and follow a backward search until an arbitrated lower bound .k = 30. The minimal outliers are defined by reviews of underreported or overreported notification, which were not identified in Algorithm 1. An underreporting review after non-working days is frequent among many case studies. A moving average reasonably smooths this effect for weekends; however, noncyclic events, such as holidays, are smoothed further by Algorithm 2. The proposed algorithm compares the measured incidences and a backward average incidence .dzjB weighted by .δL to define the beginning of the period with underreporting. After that, a posterior search is performed to find the end of the underreported period. Finally, a redistribution of the reviewed incidence is performed for the identified period. Reviews of underreported or overreported notifications also infer negative COVID-19 incidences, which are biologically impossible. Algorithm 2 identifies these negative incidences and searches for conditionals in the prior and posterior data. The incidence is redistributed to eliminate the negative values if a conditional is found. The negative outliers are pushed forward in the time series if their satisfactions are not found. This data outlier mostly happened at the end of epidemic waves during overall reviews of the notifications following the unburden of the health system. Algorithm 2 follows a forward search for the minimal outliers started at the same lower bound .k = 30. Graphical plots of some of the raw and preprocessed data in some case studies are shown in Figs. 1, 2, 3, and 4 to highlight the outcome of the previously mentioned algorithms. For example, Fig. 1 shows several meaningful outliers in the incidence of the cases from São Paulo (SP), while the same pattern is not followed for the

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Fig. 1 Raw and preprocessed data of the cases and deceased COVID-19 incidences for São Paulo (SP) between 26 February 2020 and 1 October 2022

Fig. 2 Raw and preprocessed data of the cases and deceased COVID-19 incidences for Roraima (RR) between 26 February 2020 and 1 October 2022

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Fig. 3 Raw and preprocessed data of the cases and deceased COVID-19 incidences for Piauí (PI) between 26 February 2020 and 1 October 2022

Fig. 4 Raw and preprocessed data of the cases and deceased COVID-19 incidences for Rio Grande do Norte (RN) between 26 February 2020 and 1 October 2022

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deceased notifications. These outliers comprehend mostly reviews of underreported periods under public holidays; however, overall notification reviews happened for both cases and deceased between the second and the third COVID-19 epidemic waves at different dates. A moving average would acceptably smooth the data for the reviews following public holidays; however, the epidemic wave reviews would still be subject to significant uncertainty. Figure 2, corresponding to Roraima data, shows a smoother deceased incidence over the analysis period; however, two relatively high outliers are presented for the cases between the second and third COVID-19 epidemic waves. Nonetheless, the positive overreporting followed by a notification review, characterized by the negative incidence, is only 3 days apart; thus, a moving average would still reasonably smooth the outlier. After that, however, there is approximately 1 month of underreported cases, followed by a review on 19 October 2021, which would correspond to significant uncertainty for a moving average. Figure 3, corresponding to Piauí data, presents an overreporting of cases followed by a review of high magnitude between the second and third COVID-19 epidemic waves, with an interval of 11 days. Both incidences would mostly be outside the same horizon for a moving average estimation; thus, a worst smooth of the outliers would be expected, with a period comprised of negative incidence of cases. In addition, there is a review of underreported deceased briefly past the maximal incidence in the second epidemic wave; thus, a different decease apex would be defined without the data preprocessing, which is a major error source for the analysis under state and parameter estimation. Finally, case studies, such as Rio Grande do Norte shown in Fig. 4, had significant outliers of different types during the analysis period. The data preprocessing improved the raw data; however, significant errors were observed for the simulations with state and parameter estimations, as further detailed in Sect. 5. Figure 4 showed frequent underreporting and overreporting reviews close to most non-working days in the analysis period for both cases and deceased incidences. Nonetheless, the cases underreporting review at the end of the second epidemic wave on 23 June 2021 stand out. The sudden incidence of almost 20% of the previous cumulative confirmed cases is not reasonably smoothed by the proposed data preprocessing. This maximal outlier was distributed around a prior maximal case incidence, defined over 2 months past the maximal incidence in the deceased notifications. The significant reporting dead time of the cases led to a failed case study for the proposed data preprocessing technique. Future development on integrating different available data might be required to expand further the methodology for case studies with poor data quality. Nonetheless, the proposed methodology was efficient in smoothing or removing outliers for all case studies, and the results were satisfactory for the remaining Brazilian federative units. Overall, the proposed data preprocessing is suitable for improving the data quality of epidemiological dynamics.

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3.2 Hospitalized and Healed with Treatment The average rates .σ˜ i and .τ˜i and the measurements .zi,3 and .zi,4 are taken from the Brazilian SARS database for each federative unit i [34, 35]. This database comprises clinical information from patients with the severe acute respiratory syndrome, such as age, gender, residence location, reporting location, hospital admission date, ICU admission data, discharge dates, clinical COVID-19 confirmation, and discharge type of the patient, among others. That information is used to estimate .σ˜ i and .τ˜i based on a hospital bed occupancy defined by the difference between the hospital admission and the discharge dates for the respective clinical outcome. The discharge date follows a priority order to filter filling errors, in which the initial priority is the evolution date, followed by the case closure date and the typing date. The analysis moves between the priority order if a node date is earlier than the hospital admission date or the filling is empty. The patient entry is neglected if there is no valid discharge date, the hospital admission date was not supplied, or the patient was not clinically confirmed for COVID-19. The hospital bed occupancy from each federative unit i is achieved by summing up all feasible entries from the respective reporting location. The measurement of the hospitalized or threatened individuals .zi,3 is defined by summing up all the feasible data entries clinically confirmed for COVID-19. The measurement of the healed with treatment .zi,4 follows the discharge dates of the feasible entries with discharge type different from deceased, thus, assuming feasible filling errors correspond to recovered patients. The measurement of the deceased individual .zi,2 is based on the Ministry of Health of Brazil data [33]; however, an analogous estimation of the deceased after treatment would be possible for the available data. Both measurements are almost equivalent to the oldest measures for all federative units; thus, the assumption that all individuals deceased for COVID-19 holds acceptably. However, there is a further increase as data gets closer to the data collection time. The Brazilian SARS database has a systematic dead time for data entry; however, the data from the Ministry of Health of Brazil are also subject to dead time uncertainties. Nonetheless, both databases were collected after the end of the analysis period .t (Nsim ) = 1/October/2022; thus, correlated uncertainty might be assumed negligible. The average rates .σ˜ i and .τ˜i are calculated by the recovery and death incidences among hospitalized individuals for an analysis defined between 1 April 2020 and 1 October 2022; thus, identical case studies presented slightly different estimations for their values compared to our previous work [14].

4 State and Parameter Estimation The state estimation is a standard approach for real-time applications, which had been mentioned in our previous work [14] as a future goal for the research progress. In this and our previous work, however, the state estimator was used to mitigate model and measurement uncertainties present in the epidemiological system to

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achieve insightful information in past analyses. In this work, the epidemiological dynamics are defined as targets of an optimization problem to handle the lack of complete observation and to tune the state estimator. The model in Eq. 1 is based on the adjustment of the model over time by a state estimator to mitigate the model uncertainties. However, the system is only partially observable; thus, estimations of COVID-19 epidemiological dynamics based on observation studies are tracked to guarantee more realistic estimations. Several heuristic assumptions in Eq. 1 were added to describe real-world dynamics; however, tuning the state estimator by trial and error was a major drawback in [14] for generalizing the method. In this work, several assumptions and steps remained equivalent; however, some changes described further in this section were applied to improve the tuning of the state estimator. Most of the changes aimed at simplifying and generalizing this procedure, which is essential for expanding the method to further case studies. First, an augmented state .Xi is defined for the simultaneous estimation of state and parameter according to the formulation described in our previous work, thus, an augmented state  Xi =

.

xi ψi



 where .ψ i = α0,i xc,i xm,i are the estimated parameters. The state transition model of .Xi is given by   fi (xi , ui ) ˆ .Fi (Xi , ui ) = 0

(10)

The constrained extended Kalman filter and smoother (CEKF&S) presented similar performance to the moving horizon estimator for the prediction horizon .Np = 7 while taking almost 1% of the computational effort to run the same simulations [14]. The extension of the period of analysis for further case studies and the definition of the tuning of the state estimator through optimization problems led this work to follow the alternative with lower computational cost, which is the CEKF&S. The constrained extended Kalman filter (CEKF) implementation, which is necessary for the CEKF&S, remained equivalent to the proposal of Gesthuisen et al. [37], with a minor difference that the optimization problem was solved in this work at each sampling time by the interior point implementation of the quadprog routine from MATLAB. Furthermore, all integrations of Eq. 10 in CEKF and CEKF&S are calculated by the Runge-Kutta method implemented in the routine ode45 from MATLAB. In our previous work, the simulations with CEKF&s tuned with longer horizon .Np = 28 had worse estimations than those with .Np = 7; thus, a study into this issue was carried on. The cold start of the CEKF&S following .Np − 1 iterations of the CEKF increases the determinants of the error covariance matrix .Pk|k,i

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proportionally to the horizon .Np for the initial iterations of the estimator. Hence, the greater .Pk|k,i led the CEKF&S tuned with .Np = 28 to more aggressive parameter estimations, especially in the first iterations of the estimator. In addition, increasing the past horizon increases the violation of the constraints in the CEKF for the Rauch-Tung-Striebel (RTS) [36] smoothing equations since the CEKF&S is initially executed at greater .Pk|k,i . Hence, the process noises receive further estimations in the CEKF compared to the measurement noises to achieve feasible solutions, which led to further variance for the parameter estimations. Some additional simulations were performed with an optimization problem in the RTS equations to confirm these hypotheses; however, their results are not presented in this work as the implemented CEKF&S does not include it. The hypothetical algorithm significantly increases the computational time, which contradicts the criterion of lower computational effort defined for using the CEKF&S. Hence, two fundamental changes were applied to the CEKF&S concerning the formulation proposed by Salau et al.   [38]. The CEKF&S is started at .t (Nf ) assuming . Pk|k,i = 0 ∀k < Nf,i , k ∈ N and tuning violations. of .Np = 7, which presented asatisfactory balance for the constraint  The first assumption infers in . Qk−1,i = 0 ∀k < Nf,i , k ∈ N ; thus, an average covariance matrix of observation noise . Rk,i ∀k < Nf,i is given by the residual of the model identification as ⎛ Rk,i = ⎝

Nf,i 

.

k=N0,i

Nf,i

⎞ ⎞⎛ Nf,i  ˆ i (k) z zˆ i (k) ⎠ ⎠⎝ Nf,i − N0,i + 1 − N0,i + 1

(11)

k=N0,i

  where . zˆ i (k) = zi (k) − yi (k) ∀ k ∈ N . Finally, the definition of the endemic  point  in Eq. 1 by considering the loss of natural immunity .ϕ HD,i + HT ,i + HU,i allows the tuning of .PNf,i|Nf,i following the solution of the discrete-time algebraic Riccati equation according to the idare routine from MATLAB, which depends on the covariance matrix of process noise .Qk−1,i . Defining an adaptive estimation of the covariance matrix of observation noise .Rk,i leads to update the cumulative measures to their expected observation noise while correcting the assumptions uncertainties for eliminating the estimator tuning parameters over time. The RTS equations do not explicitly define matrix .Rk,i ; thus, the adaptive approach infers mainly in the CEKF estimations. Akhlaghi and Zhou [39] proposed an adaptive estimation of .Rk,i based on the average residual over a moving window, which is defined in this work by the prediction horizon .Np of the CEKF&S. In addition, only one update of .Rk,i is performed for each iteration of CEKF&S by the end of the last CEKF iteration. The average residual estimated in the prediction horizon .Np is used to update .Rk,i as Rk,i = βRk−1,i + ⎞ ⎛⎛ ⎞⎛ ⎞ Np Np .   zˆ i (k − Np + m) zˆ i (k − Np + m) ⎜ ⎠⎝ ⎠ + Hk,i Pk|k−1 Hk,i  ⎟ (1 − β) ⎝⎝ ⎠ Np Np m=1

m=1

(12)

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Algorithm 3 CEKF&S algorithm executed at each sampling time k for each Brazilian federative unit i Input: Xi , Pi , Qk−1,i , Rk,i , zi , β XSi (k − 1) ← Xi (k − 1|k − 1) PSk−1,i ← Pk−1|k−1,i for j ∈ {1, 2, . . . , Np } do  k−j ˆ i (t), ui (t))dt Xi (k − j |k − j − 1) ← Xi (k − j − 1|k − j − 1) + k−j −1 F(X  Pk−j|k−j−1,i ← φ k−j−1,i Pk−j−1|k−j−1,i φ k−j−1,i + Qk−1,i  −1 Ck−j−1,i ← Pk−j−1|k−j−1,i φ k−j−1,i  Pk−j|k−j−1,i   XSi (k − j − 1) ← Xi (k − j − 1|k − j − 1) + Ck−j−1,i XSi (k − j ) − Xi (k − j |k − j − 1)   PSk−j−1,i ← Pk−j−1|k−j−1,i + Ck−j−1,i PSk−j,i − Pk−j|k−j−1,i C k−j,i end for Pk−Np −1|k−Np −1,i ← PSk−Np −1,i Xi (k − Np − 1|k − Np − 1) ← XSi (k − Np − 1) residi ← 0 for j ∈ {0, 1, . . . , Np } do procedure CEKF Input: Xi (k − Np − 1 + j |k − Np − 1 + j ), Pk−Np −1+j|k−Np −1+j,i , Qk−1,i , Rk,i , zi (k − Np + j ) Output: Xi (k − Np + j |k − Np + j ), Pk−Np +j|−Np +j,i end procedure residi ← residi + zi (k − Np + j ) − Hi,j Xi (k − Np + j |k − Np + j ) end for residi ← residi /(Np + 1) Pk|k−1,i ← φ k−1,i Pk−1|k−1,i φ k−1,i  + Qk−1,i  Rk+1,i ← βRk,i + (1 − β) residi residi + Hi,j Pk|k−1,i Hi,j  Qk,i ← Qk−1,i Output: Xi , Pi , Qk,i , Rk+1,i

where .Pk |k − 1, i = φ k−1,i Pk−1|k−1,i φ k−1,i  + Qk−1,i , .Hk,i is the output matrix and .φ k−1,i is the state transition matrix. The adaptive strategy is based on the tuning parameter .β = 0.9 in this work. Further definitions of Eq. 12 are provided in [14]. The adaptive strategy proposed by Akhlaghi and Zhou [39] to estimate .Qk−1,i based on the innovation is unfeasible, since the system described by Eq. 1 is not observable. Hence, the tuning of the state estimator comprehends .QN , which is assumed as a constant diagonal matrix over time for f,i−1 

Qk−1,i = QNf,i−1 ∀ k > Nf,i , k ∈ N . This definition implies estimation uncertainties as epidemiological dynamics change following the predominant circulating variants [40, 41], which infers a time-varying covariance of the process noises. The implicit assumption of independent process noises inferred from the diagonal matrix definition is an additional uncertainty source. Algorithm 3 was written for each iteration of the CEKF&S at a sampling time k . The simulation with the CEKF&S is simplified by decreasing the tuning parameters to only the covariance .QNf−1,i , which did not infer unfeasible objectives for the tracked dynamics. The fine-tuning of the state estimator was designed as an optimization problem that tracks the COVID-19 epidemiological dynamics estimated

.

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based on observational studies. This approach provides realistic estimations for the system assuming the existence of feasible solutions despite the lack of observability in the system. In our previous work, we considered the relative variation of mortality and transmissibility as references for tuning the state estimator by trial and error; however, there is reasonable uncertainty in this approach related to the variable baseline. Several Brazilian federative units had different predominant circulating variants at the early stages of the COVID-19 pandemic, according to the data available from GISAID [6]. Hence, tracking an absolute target, such as the .R0,i , at different time series generalizes the tuning strategy concerning the formulation of an optimization problem for all case studies. Furthermore, the time series are bounded to intervals with a predominant circulation of a variant above 90% in the national territory, which minimize demographic uncertainties in the model and provide general targets. In this work, the values of .R0,i are constrained by .3.2 ≤ R0,i ≤ 8.0 for the Delta variant [30] and by .5.5 < R0,i < 24 for the Omicron variant [31]. These constraints are based on meta-analyses of estimations of the basic reproduction number in different countries under different sets of external transmissibility factors. Hence, the external transmissibility factors, such as requirements of mask usage indoors, are assumed to be comprised within the reference studies. The periods with predominant circulation were defined by the genomic sequence data available in GISAID [6], which defines the time interval of the Delta variant between 1 November 2021 and 1 December 2021 and the time interval of the Omicron variant between 1 February 2022 and 1 October 2022. Mortality constraints were also applied to the optimization problem based on the infection fatality rate .I F Ri , calculated according to Eq. 13. .I F Ri

= (1 − p)xc,i xm,i

(13)

The baseline of the .I F Ri for the ancestral lineages at each studied case i is given by = 1/December/2020, which is a sampling time before the predominant circulation of the Gamma variant that provides two months to the state and parameter estimations to adjust the model to the measures. A constraint .1.2 I F Ri (t = 1/12/2020) ≤ I F Ri (tγ ) ≤ 1.9 I F Ri (t = 1/12/2020) is defined for the estimated range of mortality increase observed by Faria et al. [42] for the Gamma epidemic wave in Manaus, Amazonas. In addition, a terminal constraint .I F Ri (tγ ) ≤ I F Ri (t = 1/10/2022) is added to constrain the mortality of the Omicron variant to be fewer than the mortality of the Gamma variant based on heuristic knowledge. The Gamma variant achieved predominant circulation in the studied cases during different vaccine coverage, which is a time-varying property that significantly affects the mortality of the overall population. Hence, the mortality of the Gamma was inferred by .1/12/2020 ≤ tγ = argmaxt I F Ri (t) ≤ 1/10/2021 since Faria et al. [42] estimated the mortality increase of the Gamma variant for an unvaccinated population which correlates to the maximal estimation of .I F Ri (t). Finally, some constraints are also .t

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applied to the tuning parameters in .Qid,i to match the model update to the expected state and parameter estimations. For example, . Q1,1,i ≥ Qj,j,i ∀j = {2, 3, . . . , 10} regarding to the transmissibility mitigation by the susceptible individuals .Si instead   of the infected compartments, and . Q12,12,i ≤ Qj,j,i , j = {11, 13} to follow the magnitude order of the estimated parameters .ψ i . Hence, the constraint set .gest,i is defined by Q1,1,i ≥ Qj,j,i , j = {2, 3, . . . , 10} Q12,12,i ≤ Qj,j,i , j = {11, 13} 1/12/2020 ≤ tγ = argmax I F Ri (t) ≤ 1/10/2021 t

.

1.2 I F Ri (t = 1/12/2020) ≤ I F Ri (tγ ) ≤ 1.9 I F Ri (t = 1/12/2020)

(14)

I F Ri (tγ ) ≤ I F Ri (t = 1/10/2022) 3.2 ≤ R0 (k) ≤ 8.0 , t (k) ∈ [1/10/2021, 1/11/2021] 5.5 ≤ R0 (k) ≤ 24.0 , t (k) ∈ [1/2/2022, 1/10/2022]

where .Qj,j,i corresponds to the component at the jth row and column of .Qid,i . A hybrid optimization was also carried out on the optimization problem for the tuning of the state estimator. A relaxed nondeterministic optimization was solved first to provide a better initial guess for a posterior deterministic optimization. The nondeterministic optimization minimizes the prediction error under several quadratic penalty functions weighted by .Wg,i = 1014 Ing,i to guarantee a minimal constraint violation. Similarly to Eq. 8, the nondeterministic optimization is solved by the particleswarm routine from MATLAB; however, a further relaxation is applied by defining the threshold .objectivelimit = 1 as a stopping criterion. Hence, the PSO searches for a realistic estimation that satisfies the set of constraints given in Eq. 14 for the defined analysis period. The feasible solution .Q∗id,i is defined as the initial guess of an SQP optimization afterward following the implementation comprised in the routine fmincon from MATLAB. The optimization problem solved by the PSO is defined by Nsim,i

min

Qid,i



zi (k) − yi (k)2Q

id,i

k=Nf,i

+

ng,i 

max {0, gi (xi )}2W

g,i

k=1

Subject to: . yi (k)

= Hi,j xi (k)

Qj,m,i = 0, ∀ {j = m | j = {1, 2, . . . , 13}, m = {1, 2, . . . , 13}} " ! Qj,j,i ∈ 10−12 , 10−2 , j = {1, 11, 13} " ! Qj,j,i ∈ 10−12 , 10−4 , j = {2, 3, · · · , 10, 12}

(15)

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Table 3 Optimized tuning parameters for the simulations with the state and parameter estimation ˆ j,i = − log10 Qj,j,i for each Brazilian federative unit i where .Q i RO AC AM RR PA AP TO MA PI CE RN PB PE AL SE BA MG ES RJ SP PR SC RS MS MT GO DF

ˆ 2,i ˆ 1,i .Q .Q

ˆ 3,i .Q

ˆ 4,i .Q

ˆ 5,i .Q

ˆ 6,i .Q

ˆ 7,i .Q

ˆ 8,i .Q

ˆ 9,i .Q

ˆ 10,i .Q

3.96 3.96 4.03 3.99 3.96 4.50 4.36 4.46 5.64 3.98 3.97 3.96 3.96 3.97 4.10 3.96 3.96 3.97 4.04 3.96 4.18 3.96 4.20 4.01 3.97 3.97 3.96

9.92 4.04 9.90 10.72 7.26 4.55 11.87 11.60 11.34 10.77 11.83 11.10 10.54 8.83 10.35 12.00 11.99 11.64 11.24 9.23 6.95 11.90 7.80 7.44 6.43 10.25 11.83

11.05 6.31 5.02 11.82 10.75 10.75 11.66 9.96 11.73 10.97 11.89 11.96 11.88 12.00 12.00 12.00 11.07 10.89 11.92 10.05 10.98 10.13 8.18 10.69 10.51 11.41 7.55

12.00 6.13 10.86 11.35 12.00 7.69 9.72 10.97 9.40 12.00 7.89 11.67 9.65 11.75 10.37 8.72 11.71 9.36 12.00 11.75 9.86 11.95 11.92 9.00 8.83 11.52 9.56

8.28 11.45 9.25 10.08 7.12 8.44 12.00 11.83 10.47 12.00 11.00 11.09 11.46 12.00 12.00 12.00 11.72 12.00 11.04 11.77 11.22 8.63 11.64 10.92 11.28 10.45 9.20

7.75 8.41 5.00 6.52 11.91 9.91 6.12 9.79 12.00 12.00 12.00 7.11 10.88 11.47 8.98 12.00 11.77 7.93 10.57 9.33 7.49 11.38 11.99 12.00 11.75 9.95 11.91

9.52 10.89 10.78 5.12 8.59 9.56 12.00 10.46 8.87 8.46 9.03 10.97 12.00 11.24 9.87 12.00 11.27 11.74 5.85 9.63 10.96 11.66 7.96 8.81 11.14 11.08 8.21

4.41 3.96 6.41 9.53 3.96 8.30 10.51 8.55 9.24 11.97 11.45 5.38 5.33 12.00 11.36 11.92 5.50 10.12 8.35 4.08 5.54 9.96 11.69 9.76 4.82 7.22 5.19

6.77 11.79 11.50 4.31 11.91 9.45 12.00 10.53 7.59 4.18 4.97 4.52 4.65 10.87 8.42 11.75 10.65 10.28 9.32 9.25 5.78 12.00 11.51 8.52 7.47 5.91 8.69

11.06 8.78 8.79 6.50 7.40 9.93 9.47 8.30 8.62 4.12 10.95 12.00 11.06 12.00 12.00 12.00 11.99 11.78 6.50 11.96 10.13 11.72 11.18 11.17 10.92 7.98 10.10

ˆ 11,i .Q ˆ 12,i .Q 5.01 2.65 2.15 2.64 4.62 4.98 6.69 4.23 5.08 4.04 5.16 6.13 5.64 5.17 5.50 5.55 6.13 5.73 4.09 4.45 5.71 5.04 4.27 4.62 4.55 5.86 5.01

11.25 7.90 6.01 5.85 11.73 10.98 9.67 7.37 9.85 10.28 11.73 11.93 9.02 9.83 12.00 7.44 11.72 7.20 12.00 12.00 10.71 11.59 9.42 9.59 9.78 7.71 11.32

ˆ 13,i .Q 2.56 5.59 2.00 5.06 3.75 2.72 2.44 4.50 7.72 2.03 4.23 2.06 5.73 9.00 4.77 5.90 7.81 2.10 3.13 4.88 3.94 5.29 5/96 6.32 5.12 4.95 5.85

where .gi is the .gest rewritten in the form .gi ≤ 0, and .xi (k) is calculated by the CEKF&S following the steps described in Algorithm 3. The deterministic optimization, however, was formulated by a cost function that minimizes the prediction error for a 14-day prediction horizon for each sampling time with the state and parameter optimization. The optimization problem is presented in Eq. 16. The data output of the simulations for each Brazilian federative unit i with state estimator tuned by the solution of Eq. 16 corresponds to the results analyzed in Sect. 5. The solutions of the optimized tuning parameters of all case studies are presented in Table 3. Most solutions therein are comprised in the upper or lower bounds of some of the heuristically defined limits; thus, improving further the tuning of the CEKF&S might be possible under the relaxation of the bound constraints:

Analysis of Covid-19 Dynamics in Brazil by Recursive State and Parameter Estimations

min

Qid,i

14 N sim   j =0 k=Nf,i

zi (k + j ) − yi (k + j )2Q + id,i

ng,i 

359

max {0, gi (xi )}2W

g,i

k=1

Subject to: .

yi (k) = Hi,j xi (k)

(16)

Qj,m,i = 0, ∀ {j = m | j = {1, 2, . . . , 13}, m = {1, 2, . . . , 13}} " ! Qj,j,i ∈ 10−12 , 10−2 , j = {1, 11, 13} " ! Qj,j,i ∈ 10−12 , 10−4 , j = {2, 3, · · · , 10, 12}

where .gi is the .gest,i rewritten in the form .gi ≤ 0 and .xi (k) is calculated by the CEKF&S following the steps described in Algorithm 3.

5 Results First, a check over the evolution of .R0,i and .I F Ri is performed to verify the satisfaction of the constraints defined in Eq. 14. The fine-tuning procedure guarantees these dynamics; however, studying their evolution over time provides insightful information about the implemented technique. Figures 5, 6, 7, 8, and 9 present graphical plots of these estimations according to the geographical region containing each case study. The approximate convergence of the .R0,i to a steady state at both bounded ranges is straightforward to observe. All case studies reach an approximately steady estimation for the Delta variant between 1 November 2021 and 1 December 2022 and for the Omicron variant between 1 February 2022 and 1 October 2022, which follows their respective predominant circulation according to GISAID [6]. A meta-analysis of the vaccine efficacies for the Delta variant showed moderate to high efficacy against infection, hospitalization, and death for the vaccine of several manufacturers [43], which is meaningful as over 60% of the population of each Brazilian federative unit had already been vaccinated by two doses against COVID-19 by 1 December 2021 [44]. The transition of the predominant circulating variants between Delta and Omicron followed the expected behavior; however, the same is not achieved between the Gamma and Delta predominant periods. This period matches a significant advance of the vaccination campaign for all the case studies; thus, the transmissibility increase of the emerging variant might be balanced by the decrease caused by the vaccination. In addition, the relaxation of the social distancing policies and the decreasing population compliance toward nonpharmaceutical interventions over the same period justify the average increase of the .R0,i . Finally, the frequent oscillations over the .R0,i estimation are mainly caused by the social distancing index .ui difference in nonworking days, such as weekends and public holidays.

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Fig. 5 Time evolution of the basic reproduction number R0,i and the infection fatality rate I F Ri for the federative units i in the Mid-west region

Fig. 6 Time evolution of the basic reproduction number R0,i and the infection fatality rate I F Ri for the federative units i in the North region

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Fig. 7 Time evolution of the basic reproduction number R0,i and the infection fatality rate I F Ri for the federative units i in the Northeast region

Fig. 8 Time evolution of the basic reproduction number R0,i and the infection fatality rate I F Ri for the federative units i in the South region

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Fig. 9 Time evolution of the basic reproduction number .R0,i and the infection fatality rate .I F Ri for the federative units i in the Southeast region

The evolution of .I F Ri observed in Figs. 5, 6, 7, 8, and 9 partly follow the expected trends as the constraints applied in Eq. 14 are satisfied, while the mortality decreases following the Omicron emergence for all case studies. However, the analysis period between those events is subject to an effective vaccination campaign [44], which should have decreased .I F Ri estimation before the Omicron emergence. Some federative units, such as Santa Catarina (SC) and Alagoas (AL), failed in this estimation; however, others, such as São Paulo (SP) and Maranhão (MA), achieved the expected behavior. Hence, there might be a different tuning set for the state estimators that guarantees a more realistic parameter; however, adding further constraints in Eq. 14 might constrain the nondeterministic optimization to an unfeasible set. Furthermore, the mortality is correlated to the infected compartments for the state estimator; thus, a different tuning could achieve more realistic estimations if none or minimal model assumptions are violated. Finally, the numerical update of the .I F Ri or .R0,i at the beginning of the simulation with the state estimator CEKF&S corresponds to the adjustment of the model to the observed data. The drawback of the technique for considering dynamics with a smaller magnitude over time as noise in the state and parameter estimation also has some advantages. The state estimator filters effective noise in the measurements and provides insightful estimations of epidemiological properties that would be less accurate for a standard moving average analysis. For example, including vaccination dynamics in the model from Eq. 1 temporarily provides a more accurate estimation;

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however, several studies point out the waning of immunity related to the last immunization event and the COVID-19 variants involved [43, 45]. The state and parameter estimation limitation, however, highlight the epidemiological dynamics from the relaxation of social distancing and the emergence of circulating variants with transmissibility advantages. At the same time, it mitigates model uncertainties, such as testing and vaccination. The estimated parameter related to the transmission dynamics corresponds to .α0,i , which was the relative baseline for the transmissibility analysis in our previous work [14]. An analysis of the .α0,i estimations over time highlights further the events boosting the COVID-19 transmissibility for the local population, such as the reopening of crowded indoor events and the emergence of variants with evolutionary advantages. Hence, the interval between the emergence of a variant and its subsequent predominant circulation might be detected strictly by the available data since other events boosting the virus spread are expected to be acknowledged. In addition, a comparative analysis of the start and end of a transition among predominant circulating variants for each of the Brazilian federative units provides additional information about the connectivity at a national and international level. Furthermore, the smoother increases of the estimations of .α0,i highlight the effectiveness of decentralized restrictive policies at a federative unit level, the existence of overlapping predominant circulating variants over time, and the increasing disrespect of the population to nonpharmaceutical interventions or suboptimal tuning of the state estimator. Hence, heuristic knowledge is essential for a conclusion over the parameter estimated. The definition of the contagion rate .νi by a linear relation to the social distancing index .ui provides the trend of the parameter as function of additional data. However, the model uncertainty caused by transmission nonlinearities is expected to be mitigated by the state estimator for each sample time. Figures 10, 11, 12, 13, and 14 show that the overall strategy of clustering minor territories to define the restrictive policies based on local epidemiological indicators was efficient in smoothing the viral transmission at a relatively small economic cost. Most Brazilian federative units followed similar strategies during the COVID-19 pandemic, characterized by the .α0,i estimation by a steady increase of its value over time. In addition, the full reopening of the economic activities under the restriction or orientation of nonpharmaceutical intervention is identifiable for most case studies by reaching an approximately steady estimation of .α0,i afterward. Nonetheless, some exceptions are mentioned further in the section. An analysis of the .α0,i estimation helps to identify sudden events that boost the virus spread in the system. This property allows an assessment of the connectivity of each federative unit based on the emergence date of a new advantageous circulating variant and the interval required for its spread in the case study until achieving predominant circulation. The connectivity analysis does not infer binary connections but provides connectivity trends helpful in understanding past epidemic waves and predicting better government policies for future epidemic waves. An analysis of the emergence of the Omicron variant and the duration for it to

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Fig. 10 Time evolution of the .α0,i estimations for each federative units i in the Mid-west region

Fig. 11 Time evolution of the .α0,i estimations for each federative units i in the North region

outcompete previously circulating variants provides essential information about its spread countrywide. This work evaluates these properties from the .α0,i estimations. Figure 15 shows estimations of the initial Omicron variant emergence and the period required to achieve predominant circulation for each federative unit i bases on manual dates taken from .α0,i estimations. It shows that this COVID-19 lineage spread countrywide mostly by three initial outbreaks. One cluster is defined in the South region comprised of the three federative units: Paraná (PR), Santa Catarina (SC), and Rio Grande do Sul (RS). Another cluster is defined in the Southeast region by Minas Gerais (MG) and Espírito Santo (ES), and the final cluster is defined by Alagoas (AL), Bahia (BA), Ceará (CE), and Pernambuco (PE). The early

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Fig. 12 Time evolution of the .α0,i estimations for each federative units i in the Northeast region

Fig. 13 Time evolution of the .α0,i estimations for each federative units i in the South region

Omicron identification in the Mid-west region is not considered a separate cluster due to the slow transition identified for Mato Grosso do Sul and the geographical proximity between Goiás (GO) and Minas Gerais (MG). The aforementioned clusters comprehend an unexpected behavior for the Omicron epidemic wave compared to the first epidemic wave [10]. Candido et al. [10] identified São Paulo (SP) as the major international importer and national exporter of the virus during the first epidemic wave. However, São Paulo was the last federative unit in the Southeast region to identify the emergence of the Omicron variant. The same pattern is observed for the most relevant COVID-19 national importer in the first epidemic wave: Rio de Janeiro, which had an intermediate emergence date and an intermediate outcompeting period for the Omicron variant among the Southeast

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Fig. 14 Time evolution of the .α0,i estimations for each federative units i in the Southeast region

Fig. 15 Spatial patterns of the emergence of the Omicron variant for each Brazilian federative unit i. (a) Intervals in days until the initial circulation of the Omicron variant identified by the state estimator concerning the identified emergence of 24 December 2021 from Minas Gerais (MG). (b) Required time for the Omicron variant to achieve predominant circulation according to the state estimator

region. The estimations of Fig. 13 show a smoother increase for both studied cases, with a step pattern before reaching a steady estimation. The step pattern highlights the end of mask requirements indoors for both federative units. The efficiency of this specific nonpharmaceutical intervention is remarkable from the estimation of

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standalone since the same step pattern is observed for several other studied cases. The more strictly social distancing policies would explain why two of the most densely populated federative units with further international connectivity and economic index were not among the first and faster virus spreaders for the Omicron epidemic wave, by following a single restrictive policy. Figure 14 also highlights another remarkable event in São Paulo: the reopening of the economic activities following the unburdening of the health system in June/2021, which is characterized by a sudden transmissibility increase. While the two previously mentioned restrictive policies inferred changes in the .α0,i estimations, the end of the mask requirement outdoors showed minimal effects for the RJ and SP federative units. The same conclusion is taken from Fig. 10 for the Distrito Federal (DF), another densely populated Brazilian federative unit with significant international connectivity. Distrito Federal partially reopened economic activities in March 2021 and completely reopened them in October 2021. Both events are inferred in the .α0,i estimations, which shows robustness for the methodology to identify transmission boosting events in densely populated case studies. Figure 15 shows that the federative units in the South region had the earliest emergence and most intense transitions between Delta and Omicron variants. It is the complete opposite pattern from the one observed by Candido et al. [10] for the first epidemic wave. The delay for reaching the predominant circulation of Omicron might be explained by the restrictive policies and the geographical area of the case study; however, an analysis of the genome sequences of the neighboring countries Argentina and Paraguay from GISAID [6] provides a feasible explanation. These neighboring countries had a prior circulation of the Omicron variant; thus, they were likely the source of the virus spread to the federative units in the South region. In addition, the heterogeneity between the study of Candido et al. [10] for the first epidemic wave and this study highlights great connectivity between Santa Catarina, Rio Grande do Sul, and Paraná among themselves and their neighboring countries, which explains the slower spread for the first epidemic wave. The federative units in the Mid-west region showed considerably similar behavior concerning the South region. Figure 15 shows a similar emergence of the Omicron variant for Mato Grosso (MT), Mato Grosso do Sul, Goiás, and Distrito Federal, which a slightly earlier emergence for the federative units neighboring Bolivia and Paraguay. However, a slower interior virus spread happens for the bigger and less densely populated federative units of the Mid-west region. The same slower spread is seen for the biggest federative units in the North region. The COVID19 spread in this region also follows an opposite pattern from the one observed by Candido et al. [10] for the first epidemic wave. The federative units closer to areas infected by the Omicron variant, such as neighboring countries, had an earlier emergence, which was later propagated to more populated and nationally connected states. The Omicron spread is characterized further by the geographical distance than economic connectivity, and this property had already been observed by Candido et al. [10] from the virus spread reduction in air travels following the NPI compliance. Figure 11 also remarks the NPI requirements in Acre (AC), Pará (PA), Amazonas (AM), and Roraima (RR) by the step response during the Omicron

.α0,i

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epidemic wave. Overall, the federative units in the North region showed the smallest spatial connectivity since neighboring countries were already under an Omicron circulating transition according to GISAID data [6]. The federative units in the Northeast showed intermediate emergence and average transition among circulating variants for the Omicron variant. Similarly to the North region, Fig. 15 remarks two clusters for the virus spread. Bahia, Alagoas, Pernambuco, and Ceará had earlier identifications for the Omicron variant, which took a further week to be identified in the remaining federative units. Hence, these four aforementioned federative units have higher national or international connectivity, leading to a local COVID-19 spread afterward. Minas Gerais was the second biggest international importer identified in the study of Candido et al. [10], which agrees with the first emergence of Omicron in this case study. COVID19 probably spread from Minas Gerais and Espirito Santo to the Northeast region following the spatial trend observed for the other regions. Both states were under restrictions over mask usage indoors; however, their initial infection by the Omicron variant led to a secondary spread in the Northeast region earlier than at the nearest federative units São Paulo and Rio de Janeiro, which was an unexpected pattern. The smooth increase of the .α0,i estimations for ES and MG in Fig. 14 suggests a progressive disrespect of the population for the social distancing policies. The same pattern is observed for Rio de Janeiro, Alagoas, Bahia, and Pernambuco, however, at a smaller slope. Hence, the national spread pattern for the Omicron variant might have been defined by the populational compliance to restrictive policies; however, further study on the issue is required for a definitive conclusion as tuning errors would generate the previously mentioned behavior in the .α0,i estimation. The tuning error, however, is unlikely for several federative units in the Northeast, since the estimation described the transmission increases following the reopening of the economic activities between May and July 2021 following the unburden of the hospital bed occupancy for the Gamma epidemic wave. While the methodology was efficient in identifying several epidemiological dynamics for most case studies, including the emergence of the sublineage B.A.5 for some federative units around July 2022, there were three case studies with poor results. Goias (GO) has a steady and progressive increase in the .α0,i estimations between the simulation start and the Omicron emergence; thus, it has a likely tuning error. Additional constraints in Eq. 14 might improve the tuning procedure; however, it might create an unfeasible problem. In addition, model uncertainties with a severe assumption violation are another source for the observed behavior. Rio Grande do Norte and Paraná presented underreporting reviews in June 2021 for the case and deceased notifications, respectively. The resulting data outliers are not reasonably redistributed for the proposed data preprocessing; thus, the state estimator identifies transmission-boosting events in Figs. 12 and 13. The failure of the proposed methodology due to data outliers configures the current main drawback of the method. The proposed data preprocessing improves the data quality; however, further developments are required to generalize the method further. Nonetheless, this work presents a generalization of the method that achieved realistic estimations for most of the studied cases. Model and tuning uncertainties are two unavoidable

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drawbacks of the method; which did not affect the major analysis. The study on the transmissibility dynamics was satisfactory; however, further studies might also improve the .I F Ri estimations for analyzing the mortality.

6 Conclusions The generalization of the tuning strategy of the method removed a significant disadvantage of our previous work [14]. In addition, the data preprocessing allows a more aggressive tuning, which can be bounded further in Eq. 14 for improving the realistic estimations. The changes in the tuning strategy successfully provided state and parameter realistic estimations for most of the studied cases while removing the number of tuning parameters. The data preprocessing allows more aggressive tuning for the state estimators; however, the performance and accuracy of the estimation are functions of the data quality. The proposed algorithms succeeded for all case studies, except Rio Grande do Norte and Paraná; thus, further development in this issue must be performed to further generalize the method. The methodology had shown robustness in modeling and tuning uncertainties; however, more epidemiological information, such as vaccination coverage, might be included to improve the estimation accuracy. The COVID-19 epidemiological dynamics changes following the predominant circulating variant [40, 41]; thus, the tuning of .Qk−1,i would also change for a rigorous formulation; however, the system lacks observability and an adaptive approach for this tuning parameter is unfeasible. Therefore, the method performance is also bounded to reasonable model uncertainties. An analysis of the transmissibility evolution showed that Delta variant mostly did not increase the transmissibility in the system. In addition, the estimations of the infection rate .α0,i allow the identification and characterization of past events which boosted the virus spread. The .α0,i estimations provided spatial information of the COVID-19 spread trends for the Omicron epidemic wave, while successfully matching several acknowledged boosting events in the analysis period. Despite the robustness shown in the methodology for most federative units, specific errors in Goiás, Rio Grande do Norte, and Paraná occurred. Further improvements are required, but the methodology is already considered suited for defining a model predictive in the research progress.

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Computational Modeling of Membrane Blockage via Precipitation: A 2D Extended Poisson-Nernst-Planck Model H. Lefraich

1 Introduction Ionic transport has been noticed in a broad variety of technological applications and biological processes, such as membrane ion channels, electrochemical devices, and electrokinetics. Based on the mean field approximation, the Poisson-NernstPlanck (PNP) equations can be derived to describe the transport of ions in thin membrane under an electric field. The model is developed by coupling the NernstPlanck equation (which describes the diffusion of ions under the effect of an electric potential) with the Poisson equation (which relates charge density with electric potential). The conventional (PNP) equations for ion transport in thin membranes have been the subject of much study and numerical simulation for both steady-state and time-dependent ion dynamics [1]. Despite their success in various applications [2–5], the (PNP) equations fail in predicting the dynamics of ionic concentrations in confined environments, because of the ignorance of the phenomenon of precipitation and its impact on the mobility of ions. In fact, due to these chemical reactions of precipitation, precipitates form and effectively block the pore. In the present work, we consider a highly generalized 2D (PNP) equations where the reactions between ions form an insoluble product that physically interferes with species transport in the membrane. The system modeled here is similar to plant transpiration experiments performed by Schreiber et al. [6] and resembles less directly to those conducted by Ranathunge et al. [7]. In those experiments, after having suspended plant membranes between two salt reservoirs, they realized that the membranes were blocked by precipitation. The Nernst-Planck equation can describe electrochemical systems involving diffusion, migration, and chemical reactions [8]:

H. Lefraich () Laboratory (MISI), Faculty of Sciences and Technology, Department of Mathematics and Computer Science, University Hassan First, Settat, Morocco © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6_21

373

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H. Lefraich

  ∂Ci F + ∇ −Di ∇Ci − zi Di Ci ∇φ = Fi . ∂t RT

(1)

For each i, .Ci is the concentration of the i species which has mobility .Di and valency .zi . F is the Faraday constant, R is the molar gas constant, T is the absolute temperature, and .φ is the electrical potential. .Fi is the reaction term which describes the consumption or generation of species i due to chemical reactions with one or more other species j . We suppose that .Fi depends continuously on the .Cj ’s. In order to describe the electrostatic behavior of the system, an additional equation is required. A natural choice is Poisson’s equation: .

− εφ = F

 zi Ci ,

(2)

i

where .ε is the absolute permittivity of the medium. Consequently, the complete formulation of the (PNP) is given by (1) and (2). The effective diffusivity [9] in a porous membrane can be given by De =

.

Daq t δ τf

(3)

Daq is the diffusivity in water, and .t , .δ, and .τf are the dimensionless parameters accounting for the porosity, constrictivity, and tortuosity of the membrane, respectively. Equation (3) supposes that diffusion occurs only in the pore space and not in the solid phase of the membrane. The porosity is defined as the volume fraction of the pore space in the membrane .(0 ≤ t ≤ 1). Typical porosities are located between .0.3 and .0.7 [10]. In our model we consider a space- and timedependent diffusivity, since pore can become locally constricted by the precipitation reaction. The tortuosity .(τf ≥ 1) of a porous membrane ranges from .1.5 to .2.5 [10], meantime the constrictivity .(0 ≤ δ ≤ 1) equals 1 when the membrane pores are supposed too large compared to the diffusing species [9].

.

2 Mathematical Model The classical (PNP) equations serve as a basis for our extended mathematical model including precipitation, where the diffusivity values not only vary in space and time but also depend on the time integrated local precipitation rate. To demonstrate the behavior of the reaction diffusion system including insoluble reaction products, the model was initially computed for a generalized system involving a reaction of the form − A+ (aq) + B(aq) → AB(s)

.

Computational Modeling of Membrane Blockage via Precipitation: A 2D. . .

375

Fig. 1 Computational domain constructed. The computational domain . is a rectangle of length L and thickness l representing the porous membrane. The boundary of . consists of four parts, .∂ =

bottom ∪ left ∪ top ∪ right , where . left and . right are the interfaces between the membrane and the left and right bathing solutions, respectively

We consider the system to hold on a bounded spatial domain . ⊂ R2 , representing a hydrated porous membrane located between two reservoirs containing solutions of AM and BN, respectively, where .M − and .N + are nonreactive ions. We assume that the computational domain . is a rectangle of length L and thickness l (see Fig. 1). All species .A+ , .B − , .M − , and .N + can diffuse and migrate within the water-filled pores of the membrane. The ions .H + and .OH − are also present in the pore fluid; however, these species were neglected because their concentrations are negligible compared to those of other ions. The .A+ and .B − ions react to form the insoluble product AB, which precipitates out of solution to create a solid obstruction in the membrane pores. The rate of chemical reaction can be quantified by the precipitation rate .kp and the solubility product .Ksp . If we denote the concentrations of the reactants by   C1 = A+ ,

.

  C2 = B − ,

C3 = [AB] ,

  C4 = M − ,

We have, F1 (C1 , C2 ) = F2 (C1 , C2 ) = −kp C1 C2 .

.

Moreover, the rate equation for AB formation can be written as

  C5 = N + .

376

H. Lefraich

F3 (C1 , C2 ) = kp (C1 C2 − Ksp ).

.

For computational simplicity, the product AB is assumed to be insoluble, i.e., .Ksp = 0. As the other ions are nonreactive, we get F4 = F5 = 0.

.

Let’s note that the accumulation of solid precipitates causes a local reduction in porosity and, consequently, in the effective diffusivity of all moving species in the membrane. The relative amount of pore volume occupied by the precipitate equals the product of its concentration and molar volume. By incorporating these considerations to (3), we get a new expression for the effective diffusivity:  De (C3 ) =

.

δ τf

 (t0 − VAB C3 (t, x)) Daq

(4)

where .t0 is a globally defined initial porosity and .VAB is the molar volume of the insoluble product AB. Instead of studying the behavior of the system in a single pore as in [11], the two-dimensional model just described aims to estimate the average behavior across the entire cross-sectional space of a porous medium. Especially, (4) insinuate that blockage in one pore does not stop diffusion in a neighboring pore and can thus provide a better description of complete membrane behavior than single-pore models. In our case, we take the initial porosity of the membrane as .0.5 (which means a pore content of .50% by volume)[10]. Furthermore, we neglect the tortuosity and constrictivity effects by setting .δ and .τf to unity, because they are not important for the present study and would only serve to ensure a uniform decrease of the effective diffusivity values. We assume that the relative charge of the solid equals zero, i.e., z3 = 0, and then the computational model is of the form ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ .

  ∂C1 F + ∇. −De (C3 )∇C1 − z1 De (C3 )C1 ∇φ = −kp C1 C2 ∂t RT   F ∂C2 + ∇. −De (C3 )∇C2 − z2 De (C3 )C2 ∇φ = −kp C1 C2 ∂t RT ∂C3 + ∇. [−De (C3 )∇C3 ] = kp C1 C2 ∂t   ∂C4 F ∇φ = 0 + ∇. −De (C3 )∇C4 − z4 De (C3 )C4 RT ∂t   ∂C5 F + ∇. −De (C3 )∇C5 − z5 De (C3 )C5 ∇φ = 0 ∂t RT

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 5 ⎪  ⎪ ⎪ ⎪ −εφ = F ( zi Ci ) ⎪ ⎪ ⎪ ⎪ ⎩ i=1 i =3

in QT 

in QT  in QT 

in QT  in QT  in QT 

Computational Modeling of Membrane Blockage via Precipitation: A 2D. . .

377

  where .QT  = 0, T  × , and .T  > 0. As can be seen in Fig. 1, the boundary of . consists of four parts: ∂ = bottom ∪ left ∪ top ∪ right ,

.

where . left and . right are the interfaces between the membrane and the left and right bathing solutions, respectively. We assume that the reservoirs are large enough that any change in the bathing solution concentrations from the initial values is negligible. Initially, the membrane contained zero concentrations of all species. Furthermore, the electrical potential boundary conditions were specified with the left boundary electrically grounded and a zero-charge condition at .∂  left . Based on the aforementioned assumptions, the system is closed with the following homogeneous boundary and initial conditions: ⎧ C1 (t, x) = Cleft,1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C1 (t, x) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ F ∂C1 ⎪ ⎪ ⎪ ⎪ De (C3 ) ∂υ + z1 De (C3 )C1 RT ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C2 (t, x) = Cright,2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C2 (t, x) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ F ∂C2 ⎪ ⎪ + z2 De (C3 )C2 De (C3 ) ⎪ ⎪ ⎪ RT ∂υ ⎪ ⎪ ⎪ ⎪ ∂C3 ⎪ ⎪ ⎪ D (C ) =0 ⎪ ⎪ e 3 ∂υ ⎪ ⎨ .

C4 (t, x) = Cleft,4 ⎪ ⎪ ⎪ ⎪ ⎪ C4 (t, x) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ F ∂C4 ⎪ ⎪ ⎪ + z4 De (C3 )C4 De (C3 ) ⎪ ⎪ RT ∂υ ⎪ ⎪ ⎪ ⎪ ⎪ (t, x) = C C ⎪ 5 right,5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C (t, x) = 0 ⎪ ⎪ 5 ⎪ ⎪ ⎪ ⎪ F ∂C5 ⎪ ⎪ + z5 De (C3 )C5 De (C3 ) ⎪ ⎪ ∂υ RT ⎪ ⎪ ⎪ ⎪ ⎪ φ(t, x) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂φ = 0 ∂υ

  on 0, T  × left   on 0, T  × right ∂φ =0 ∂υ

  on 0, T  × ( top ∪ bottom )   on 0, T  × right   on 0, T  × left

∂φ =0 ∂υ

  on 0, T  × ( top ∪ bottom )   on 0, T  × ∂   on 0, T  × left   on 0, T  × right

∂φ =0 ∂υ

  on 0, T  × ( top ∪ bottom )   on 0, T  × right   on 0, T  × left

∂φ =0 ∂υ

  on 0, T  × ( top ∪ bottom )   on 0, T  × left   on 0, T  × (∂  lef t )

where .Cleft,i and .Cright,i are the concentrations of the ith species in the left and right bathing solutions. Moreover, we have the following initial conditions

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H. Lefraich

⎧ ⎪ C1 (0, x) = C10 = 0 ⎪ ⎪ ⎪ ⎪ C2 (0, x) = C20 = 0 ⎪ ⎪ ⎨ C3 (0, x) = C30 = 0 . ⎪ C4 (0, x) = C40 = 0 ⎪ ⎪ ⎪ ⎪ C5 (0, x) = C50 = 0 ⎪ ⎪ ⎩ φ(0, x) = φ 0 (x)

3 Numerical Method In this section, we present the numerical scheme for solving the problem. We used a finite element discretization for all equations of the (PNP) system, and we made a modification of the Poisson equation. In fact, the Poisson equation is rewritten 5  F2 slightly, by adding the term . RT ( zi2 Ci )φ from both sides. The Poisson equation i=1 i =3

becomes nonlinear in the potential and would be solved in a fixed-point loop, where the .Ci ’s are taken from the last iteration. This correction prevents the fixedpoint iteration from exploding. This type of iteration is known in semiconductor device modeling as Gummel’s method [14, 15]. However, instead of treating all occurrences of .φ equally, the .φ on the right side of the Poisson equation is treated as a known quantity. We arrive at a corrected Poisson equation which features the previous potential .φ0 on the right-hand side: 5 5 5  F2  2 F2  2 ( ( . − εφ + zi Ci )φ = F ( zi Ci ) + zi Ci )φ0 RT RT i=1 i =3

i=1 i =3

i=1 i =3

3.1 Time Marching Scheme We used the following modified Gummel procedure to solve the system at every time step .t = tn+1 . To this end, let us denote by .(C1n+1 , C2n+1 , . . . , C5n+1 , φ n+1 ) and .(C1n , C2n , . . . , C5n , φ n ) the approximate value at time .t = t n+1 and .t = t n , respectively, and by .δt = tn+1 − tn the time step size. Then by using the following algorithm, we determine the unknown fields. 3.1.1

Algorithm of Resolution

1. Initialization for time marching scheme: set time step n = 0, and take the initial

value C10 , C20 , C30 , C40 , C50 , φ 0 .

Computational Modeling of Membrane Blockage via Precipitation: A 2D. . .

379

2. Initialization for nonlinear iteration for Poisson equation: Let φ n+1,0 = φ n when n  0. 3. Finite element computation on each nonlinear iteration: For l  0 compute φ n+1,l+1 such that

.

−εφ n+1,l+1 +

5 5 5  F 2  2 n n+1,l F 2  2 n n+1,l+1 ( ( zi Ci )φ = F( zi Cin )+ zi Ci )φ RT RT i=1 i =3

i=1 i =3

i=1 i =3

4. Checking the stopping iteration: For a fixed tolerance δ1 ,  criteria for nonlinear  stop the iteration if φ n+1,l+1 − φ n+1,l L2 ( ) ≤ δ1 , and set φ n+1 = φ n+1,l+1 . Otherwise, set l ← l + 1 and go to Step 3 and continue the nonlinear iteration for Poisson equation. 5. Initialization for nonlinear iteration for Nernst-Planck equation: Let Cin+1,0 = Cin , for i = 1, 2, . . . , 5, when n  0. 6. Finite element computation on each nonlinear iteration: For k  0 compute Cin+1,k+1 , for i = 1, 2, . . . , 5, such that

.

.

.

 C1n+1,k+1 − C1n + ∇. −De (C3n+1,k )∇C1n+1,k+1 δt  n+1,k n+1,k+1 F n+1 = −kp C1n+1,k+1 C2n+1,k ∇φ −z1 De (C3 )C1 RT

 C2n+1,k+1 − C2n + ∇. −De (C3n+1,k )∇C2n+1,k+1 δt  F ∇φ n+1 = −kp C1n+1,k+1 C2n+1,k+1 −z2 De (C3n+1,k )C2n+1,k+1 RT

  C3n+1,k+1 − C3n + ∇. −De (C3n+1,k )∇C3n+1,k+1 = kp C1n+1,k+1 C2n+1,k+1 δt

.

 C4n+1,k+1 − C4n + ∇. −De (C3n+1,k+1 )∇C4n+1,k+1 δt  n+1,k+1 n+1,k+1 F n+1 =0 ∇φ −z4 De (C3 )C4 RT

380

H. Lefraich

.

 C5n+1,k+1 − C5n + ∇. −De (C3n+1,k+1 )∇C5n+1,k+1 δt  n+1,k+1 n+1,k+1 F n+1 =0 ∇φ −z5 De (C3 )C5 RT

7. Checking the stopping criteria for nonlinear iteration: For a fixed tolerance δ2 , stop the iteration if

.

5     n+1,k+1  − Cin+1,k  C i i=1

L2 ( )

≤ δ2 ,

and set Cin+1 = Cin+1,k+1 for i = 1, 2, . . . , 5. Otherwise, set k ← k + 1 and go to Step 6 and continue the nonlinear iteration for Nernst-Planck equations. 8. Time marching: Stop if n + 1 = N . Otherwise set n ← n + 1 and go to Step 2.

4 Results and Discussion In this section we present two numerical applications of the model introduced previously. In the first application, we consider the generalized case where all mobile ions are supposed to have the same initial (unconstricted) diffusivity. Although, in the second application, the effect of varying the ion mobilities in the system was examined by replacing the generalized .A+ and .B − species with silver and chloride ions, respectively. The parameter values used in the numerical simulation are summarized in Table 1.

4.1 Result 1 In the case of .10−3 mol/l AM solution at the left boundary and .10−3 mol/l BN solution at the right boundary, the reservoir concentrations are .Clef t,1 = Clef t,4 = Cright,2 = Cright,5 = 1 mol/m3 . For simplicity, we take the initial (unconstricted) diffusivities of all five mobile species as .1.5 × 10−9 m2 /s. Figure 2 illustrates the formation of the precipitate AB, which begins at the middle area of the membrane. The time-dependent concentration profiles (along an horizontal line at the center of the membrane) for the reactive ions (.A+ and .B − ) are presented in Figs. 3 and 4, and the corresponding profiles for the unreactive ions (.M − and .N + ) are presented in Figs. 5 and 6. The time-dependent behavior of the system consists of an initial transient period followed by a metastable period where concentrations change slowly over time, a second transient period and a final steady state. At .t = 0 s, ions

Computational Modeling of Membrane Blockage via Precipitation: A 2D. . .

381

Table 1 Baseline parameters values for the model Parameter Daq (unconstricted diffusivity for A+ , B − , M −, N +) Daq (unconstricted diffusivity for Ag + ) Daq (unconstricted diffusivity for Cl − ) Daq (unconstricted diffusivity for N a + ) Daq (unconstricted diffusivity for N O3− ) L (membrane length) l (membrane thickness) VAB (precipitate molar volume) kp (reaction rate constant) ε (absolute permittivity of water) δ (membrane constrictivity) t0 (initial membrane porosity) τf (membrane tortuosity) δ1 , δ2 (stopping criteria)

Dimensional value 1.5 × 10−9

Unit m2 /s

Reference [12]

1.65 × 10−9 2.03 × 10−9 1.33 × 10−9 1.90 × 10−9 0.2 0.1 2.85 × 10−5 4.2 7.08 × 10−10 1 0.5 1 10−6

m2 /s m2 /s m2 /s m2 /s mm mm m3 /mol m3 /(mol s) F/m – – – –

[12] [12] [12] [12] – – [12] [13] [12] – – – –

Fig. 2 Formation of the precipitate AB at several instants of time indicated in seconds after initiation ion flux at t = 0 s

382

H. Lefraich 1 "C1_(t=0).dat" "C1_(t=0.1).dat" "C1_(t=0.2).dat" "C1_(t=0.3).dat" "C1_(t=0.5).dat" "C1_(t=0.6).dat" "C1_(t=2).dat" "C1_(t=5).dat"

Concentration (mol/m^3)

0.8

0.6

0.4

0.2

0

-0.2 0

10

20

30

40

50

60

70

80

90

100

Position (in micrometer)

Fig. 3 Concentration profiles for reactive ion .A+ at several instants of time (from .t = 0 s to = 5 s) indicated in seconds after initiating ion flux at .t = 0 s

.t

1 "C2_(t=0).dat" "C2_(t=0.1).dat" "C2_(t=0.2).dat" "C2_(t=0.3).dat" "C2_(t=0.5).dat" "C2_(t=0.6).dat" "C2_(t=2).dat" "C2_(t=5).dat"

Concentration (mol/m^3)

0.8

0.6

0.4

0.2

0

-0.2 0

10

20

30

40

50

60

70

80

90

100

Position (in micrometer)

Fig. 4 Concentration profiles for reactive ion .B − at several instants of time (from .t = 0 s to = 5 s) indicated in seconds after initiating ion flux at .t = 0 s

.t

start to penetrate the porous membrane, and a reaction front is established at the mid-area of the membrane as the ions .A+ and .B − begin to meet and eliminate each other. After approximately .0.2 s, an initial near steady state is established. In this state the concentration profiles of the unreactive ions are nearly linear, as would be expected for a simple diffusion between two reservoirs of constant concentrations. However, the concentration profiles of the reactive ions are sublinear because of the depletion of ions at the reaction front. This could have been the final state of the system, if there was no precipitation reaction.

Computational Modeling of Membrane Blockage via Precipitation: A 2D. . .

383

1 "C4_(t=0).dat" "C4_(t=0.1).dat" "C4_(t=0.2).dat" "C4_(t=0.3).dat" "C4_(t=0.5).dat" "C4_(t=0.6).dat" "C4_(t=2).dat"

Concentration (mol/m^3)

0.8

0.6

0.4

0.2

0 0

10

20

30

40

50

60

70

80

90

100

Position (in micrometer)

Fig. 5 Concentration profiles for unreactive ion .M − at several instants of time (from .t = 0 s to = 2 s) indicated in seconds after initiating ion flux at .t = 0 s

.t

1 "C5_(t=0).dat" "C5_(t=0.1).dat" "C5_(t=0.2).dat" "C5_(t=0.3).dat" "C5_(t=0.5).dat" "C5_(t=0.6).dat" "C5_(t=2).dat"

0.9 0.8

Concentration (mol/m^3)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

10

20

30

40

50

60

70

80

90

100

Position (in micrometer)

Fig. 6 Concentration profiles for reactive ion .N + at several instants of time (from .t = 0 s to = 2 s) indicated in seconds after initiating ion flux at .t = 0

.t

Between approximately .t = 0.2 s and .0.45 s, the ion concentration profiles in the membrane change gradually as precipitate builds up in the membrane pores. This reaction product buildup affects the effective membrane porosity as shown in Fig. 7. We can see that the more the effective porosity within the reaction zone decreases, the more the ion concentration gradients around this region gradually increase. At approximately .t = 0.5 s, the effective porosity at the center of the reaction zone reaches the near zero minimum value allowed by the model, and then the local diffusivities of all ions become negligible. At this stage, the new .A+

384

H. Lefraich 0.5 "P_(t=0).dat" "P_(t=0.1).dat" "P_(t=0.2).dat" "P_(t=0.3).dat" "P_(t=0.4).dat" "P_(t=0.5).dat" "P_(t=1.5).dat"

0.45 0.4

Effective porosity

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

10

20

30

40

50

60

70

80

90

100

Position (in micrometer)

Fig. 7 Local effective porosity in the membrane at several instants of time (from .t = 0 s to .t = 1.5 s) indicated in seconds after initiating ion flux at .t = 0

and .B − reactants are prevented from diffusing to the reaction front in considerable quantities, and their concentrations quickly reach a constant value on either side of the blockage as ions diffuse inward from the boundaries. The concentrations of .N + ions on the left side of the obstruction and .M − ions on the right side reach zero as the counterions diffuse back out of the membrane into the reservoirs. In the final steady state, seen at .t = 5 s in Figs. 3 and 4, the concentration profiles of reactive ions resemble step functions with uniform concentrations nearly equal to the reservoir concentrations on either side of the blockage.

4.2 Result 2 The previous modeling gives a good description of ideal precipitation process in porous membrane. However, the generalized system is unrealistic because the mobilities of all ionic species are not equal. In order to verify the last results on a more realistic footing, the model was modified by replacing the generalized .A+ and − ions with silver and chloride ions, respectively, which react according to .B + − Ag(aq) + Cl(aq) → AgCl(s) ,

.

and we considered the counterion species .Na + and .N O3− . Schreiber et al [6], in their experimental work, formed AgCl precipitates in order to block aqueous pores in an isolated plant cuticle by suspending it between reservoirs of .0.001M N aCl and .0.001M .AgNO3 . After computing the model under those experimental conditions, the formation of the precipitate AgCl is illustrated in Fig. 8. We can see

Computational Modeling of Membrane Blockage via Precipitation: A 2D. . .

385

Fig. 8 Formation of the precipitate AB at several instants of time indicated in seconds after initiation ion flux at .t = 0 s

that the initial reaction zone location where the ions begin to form precipitate depend on the identity of ions. While these modification have a negligible effect over large timescale, the initial transient behavior is quite different from the generalized case.

5 Conclusion A computational model for a 2D diffusion-migration-reaction system with insoluble reaction products had been developed. The system is based on the Poisson-NernstPlanck equations along with a nonlinear diffusivity expression to describe the physical obstruction caused by a precipitation reaction. The model is investigated for the generalized case and then extended to cover plant transpiration experiments. It’s believed that generalized PNP system of this sort can improve our understanding of other situations where ion reaction or phase transformation affects ion motion.

386

H. Lefraich

References 1. H. Cohen and J.W. Cooley, Biophys. J. 5, 145 (1965). 2. M. G. Kurnikova, R. D. Coalson, P. Graf and A. Nitzan Biophys J. 76, 642-656 (1999). 3. B. Lu, Y. Zhou, G. A. Huber, S. D. Bond, M. J. Holst and J. A. McCammon, J. Chem. Phys 127, 135102 (2007). 4. V. Barcilon, D. P. Chen and R. S. Eisenberg, SIAM J. Appl. Math 52, 1405-25 (1992). 5. D. Gillespie, W. Nooner and R. S. Eisenberg, J. Phys: Condens. Matter 14, 12129-45 (2002). 6. L. Schreiber, S. Elshatshat, K. Koch, J. Lin and J. Santrucek, Planta 223, 283 (2006). 7. K. Ranathunge, E. Steudle and R. Lafitte, Plant Cell Environ 28, 121 (2005). 8. B. E. McNealy and J. L. Hertz, Int. J. Hydrogen Energy 38, 5357 (2013). 9. P. Grathwohl, Kluwer Academic Publishers, Norwell, (1998). 10. R. Baker,Wiley, Hoboken, (2012). 11. M.-T. Wolfram, M. Burger and Z. S. Siwy, J. Phys. Condens. Matter22, 454101 (2010). 12. D. R. Lide (ed.), CRC Press, Boca Raton, (1993). 13. C. W. Davies, G. H. Nancollas, Trans. Faraday Soc. 51, 818 (1955). 14. H. K. Gummel, IIIE Trans. Electron Devices 11 10, 455-465 (1964). 15. T. Kerkhoven, SIAM J. Sci. Stat. Comput. 9 1, 48-60 (1988).

Index

A Agent-based model (ABM), 18, 19, 21, 23–26, 254 Age-structured epidemic model, 254 Age-structured model, 253, 256, 257 Allee effect, 74 Alpha variant, 216, 230 Alphaviruses, 125 Altruistic offspring, 176 Altruistic population, 173 Anderson and May’s model, 17, 197 Andronov-Witt theorem, 13 Antibody immune cells, 106 Aorta segmentation, 308, 309 Aortic root, 307 Asymptotically mean square stable, 86 Atangana-Baleanu fractional derivative, 126, 127

B Bacillus subtilis, 168, 35 Banach space, 102, 202 Basic reproduction number, 3, 60, 104, 120, 154, 158, 168, 236, 250, 257, 269, 341, 356, 360 Basic reproduction ratio, 5 Batch pharmaceutical manufacturing process, 294 Batch processes, 291, 294 Beddington-DeAngelis incidence function, 234 Beddington-DeAngelis response function, 74 Behavioural and cultural traits, 171 Beta variant, 216 Bifurcation analysis, 75

Bifurcation diagram, 91, 160, 166, 209, 212, 322 Bifurcation parameter, 8, 68, 80, 82, 92, 280 Biological feasible region, 60, 65 Bi-stability, 58, 70 Blood viscosity, 315 Bogdavov-Takens (BT) bifurcation, 92 Bogdanov-Takens (BT) point, 92 Both strains endemic equilibrium, 237, 238, 244, 245, 249 Bottom trophic level, 275, 277 Boundary equilibria, 77 Boundary equilibrium, 8, 11 Boundary wall, 21, 26 Butler’s lemma, 205

C Caenorhabditis elegans, 35 Cancer cells, 33, 36, 125, 126, 136 Cancer therapy, 126 Cancer virotherapy, 126 Caputo-Fabrizio fractional derivative, 126 Chaotic oscillation, 198, 209 Characteristic polynomial, 10 Chronically infected cells, 100–101 Clustering, 253–255, 258–264, 363 Clustering algorithm, 259, 261 Coexistence equilibrium, 77, 78, 84, 87, 94, 96, 128, 129, 134, 136, 197, 200, 208–213, 276, 277, 287, 288 Coexistence equilibrium of the stochastic differential system, 84 Coexistence of multiple strains, 233 Color noise, 74, 75

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. P. Mondaini (ed.), Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, https://doi.org/10.1007/978-3-031-33050-6

387

388 Compartmental model, 1, 18, 21, 217, 256, 335 Competition-free equilibrium, 128–129, 135 Contact matrices, 253–261, 265, 266, 269 Containment zones, 23 Continuous manufacturing, 291, 295, 298, 299 Coronary hemodynamics model, 306 Coronary stenosis, 305, 315 Cost of fear, 73 COVID-19, 153, 215–217, 229, 230, 233, 253, 254, 333–340, 345, 348, 349, 351, 352, 363, 365, 367, 368 COVID-19 epidemiological dynamics, 334, 338, 353, 355, 369 COVID-19 infection, 18 COVID-19 pandemic, 18, 215, 333, 334, 341–343, 356 CRISPR, 33 CRISPR/Cas9, 31–33 CRISPR knockout screens, 34, 39 CRISPR loss-of-function screens, 33 CRISPR score, 42, 51 Critical quality attributes (CQA), 292, 293, 295, 399 Cross-immunity between novel strains, 231 Cross-immunity matrix, 218–220, 225, 229 Crowded cities, 25 Crowded environments, 23 Crowding of infected population, 57 Crowley-Martin incidence function, 234 Crystallization, 297, 298 Crystallization processes, 297 Crystal size distribution (CSD), 297, 298 CTL cells, 125–127, 135 CTL immunity, 129, 136 Curse of dimensionality, 255, 258, 260

D Database of Essential Genes (DEG), 33, 41 Delayed and diffusive viral infection model, 100 Delayed two-strain epidemic model, 233 Delayed two-strain infection model, 233 Delay mathematical model, 100 Delta variant, 216, 356, 359, 369 Deterministic compartmental model, 335 Diffusion-migration-reaction system, 385 Diffusion process of Ornstein-Uhlenbeck type, 138 Dirac delta function, 88 Discrete Lyapunov functional, 116 Discrete-time algebraic Riccati equation, 354 Discrete two-variable host-parasite system, 318

Index Disease-free equilibrium, 8, 13, 154, 158, 159, 168, 198, 237, 239, 240, 247, 347 Disease-free steady state, 60, 63–65 Disease transmission, 2, 57, 253, 341 Dominance period, 221 Dominant strain, 221 Drosophila melanogaster, 31, 36, 37 Dynamics of phytoplankton-zooplankton interaction, 198

E Eco-epidemiological system, 197, 198 Effectiveness of current COVID-19 vaccines, 230 Effective reproduction number, 216, 217, 222, 223, 225, 227–229 Emergence of new variants, 333 Endemic equilibrium, 3, 8–12, 70, 158, 160, 161, 163–165, 237, 238, 240, 242, 244, 247, 347 Endemic steady state, 62–64, 66, 69, 70 Environmental fluctuation, 74, 87, 93, 94 Environmental noise, 74, 75, 97 Epicenter of the virus spread, 334 Epigraph region, 190 Escherichia coli, 35, 37, 329 Escort probability, 185, 189 Essential genes, 29, 33, 41, 51 Essential Genes on Genome Scale (EGGS), 33 Essentiality of a gene, 35 Essentialome, 29–33 Euler-Maruyama and Milstein methods, 93 Euler-Maruyama method, 145 Evolutionary dynamics, 172–177 Evolutionary dynamics of influenza viruses, 18 eXpression-based Gene Essentiality Prediction (XGEP), 36

F Fear of predation, 73, 75 Final pharmaceutical product, 293 First Lyapunov coefficient, 82, 90, 91, 165 First passage time (FPT), 138, 140, 143–149 First strain endemic equilibrium, 247, 249 Fokker-Planck equations, 142, 194 Food chain model, 74 Force of infection, 58, 197 Forward bifurcation, 3 Fractional derivative, 125, 126, 135 Fractional flow reserve (FFR), 305–309, 311–315 Frangi vesselness filter, 308

Index G Gamma variant, 216, 335, 336, 356 Gaussian bridge distribution, 145 Gaussian-diffusion process, 143, 144, 146 Gaussian white noise, 74, 84, 88 Gene essentiality, 29–33, 42, 51 General incidence function, 101, 105, 110, 111 Generalist predator, 274 General nonlinear incidence function, 100 Gibbs-Shannon entropy measure, 186, 187 GISAID database, 333–334, 356, 359, 367, 368 Globally asymptotically stable, 65, 66, 68, 107, 110, 116, 117, 119–121, 154, 239, 240, 242, 244, 246 Globally asymptotically stable equilibria, 13 Global stability, 3, 64, 66, 70, 107, 233, 235, 238–240, 242, 244, 249 Global stability of disease-free steady state, 64 Global stability of equilibria, 13 Granulation, 298, 299 Graph embedding, 41 Graph neural networks (GNNs), 37 Greatest lower bound (g.l.b.), 190 Green engineering, 293 Grid-characteristic method, 307 Growths in predator populations, 210 Gummel procedure, 378 Gummel’s method, 378

H Hessian matrix, 186, 188, 191 Heterogeneous spread of COVID-19, 333 Hierarchical clustering, 254, 259, 262 Higher trophic level, 288 High parasite burden, 197 HIV/AIDS epidemiological rates, 1 HIV-1 dynamics, 100 HIV-1 infection, 101, 102 HIV-1 infection dynamics, 99 Holling I functional response, 73 Holling II response function, 75, 95 Holling type II, 58, 74, 274 Holling type III treatment, 58, 70 Hopf bifurcation, 80, 82, 96, 154, 162, 205, 206, 210, 288 Hopf point, 90, 91, 165 Host-parasite (HP) system, 318, 319, 328 Hough circleness transform, 308 Human Metabolic Atlas (HMA), 39 Human Protein Atlas (HPA), 40, 41 Humoral immunity, 102, 106, 110, 111, 117, 119–122, 216

389 Hyperemia, 305, 307, 313 Hysteresis, 58, 70

I Immune evasive, 230 Immune-free equilibrium, 128, 129, 131, 133, 134, 136 Incidence rate function, 57–59, 70, 234 Infection equilibrium with humoral immunity, 106, 118, 120, 122 Infection equilibrium without humoral immunity, 105, 110, 120, 121 Infection-free equilibrium, 104, 106–108, 116, 119, 120, 200, 207, 210, 213 Infectious disease, 2, 17, 19, 25, 58, 154, 253 Infectious disease modelling, 17 Infective population, 57, 155 Insertional mutagenesis, 30 Instability of disease-free steady state, 66 Intermediate predator, 274, 275, 277, 280, 288 Inter-spike intervals (ISIs), 140, 141, 143, 145 Intraspecies competition, 199, 209–211, 213 Intrinsic growth rate, 75, 92, 198, 319 Inverse Gaussian law, 141 Isoperimetric distance trees (IDT) method, 308 It¯o stochastic differential system, 84

J Jacobian, 6–8, 10, 280–283 Jacobian matrix, 8, 63, 64, 68, 78, 82, 88, 158, 159, 161, 162

K Kalman filter, 355 Kermack and McKendrick model, 17 Kernel singularity, 144 Kolmogorov equation, 138, 142

L Landsberg-Vedral entropy measure, 186, 187 Langevin equations, 87 LaSalle invariance principle, 108, 110 Leaky-Integrate-and-Fire (LIF) models, 137, 139, 140, 142–144, 146, 147, 149 Left coronary artery (LCA), 307 Light gradient boosting machine (LGBM), 42 Limit cycle, 13, 90–92, 165, 177, 179, 198 Limit cycle oscillations, 207, 208, 210, 213 Local center manifold, 8

390 Locally asymptotically stable, 5, 7, 9, 11, 12, 64, 79, 88, 162, 164, 201, 205, 206 Locally Lipschitz, 103 Local stability, 3, 5, 7, 63, 95, 201, 202, 204 Logistic delay, 198, 213 Logistic map, 319, 321 Lozinskii measure, 67 Lyapunov function, 77, 85, 93, 107, 130, 238–240, 245 Lyapunov functional, 107, 108, 110, 116, 130, 132, 134, 238, 249

M Markovian processes, 88 Markov process, 138 Matthews correlation coefficient (MCC), 43, 45 Maximal cluster, 259 Maximal extension of the parameter space, 187 Maximum negative correlation, 61 Maximum positive correlation, 61 Mean fitness, 176–178 Media awareness programmes, 154 Metzler matrix, 65, 66 Microreactors, 294 Minimal gene-set, 29 Minimal genome, 29, 34 Mittag-Leffler function, 126 Mixed Euler method, 100, 111 Modification of the Poisson equation, 378 Monod-Haldane function, 58 Moore neighbourhood, 20 Multiple endemic steady states, 70 Multiple stochastic models, 18 Multiplicative noise, 84 Multiscale Convolutional Neural Network, 37 Multivascular stenosis, 315 Murray’s law, 307 M1 virotherapy, 125, 136 M1 virus, 125–127, 135 Mycoplasma mycoides derivative, 30

N Nernst-Planck equation, 373, 379, 380 Network dynamics, 138 Neumann boundary conditions, 102, 128 Neural networks, 36, 37, 296, 297, 300, 315 Neuronal dynamics, 137–139, 149 Neuronal network, 138, 147 Neuronal signals, 139 Newtonian viscous incompressible fluid, 306 Newton-Leibniz theorem, 4

Index Next-generation matrix (NGM), 6, 13, 159, 222, 224, 225, 228, 257, 341, 342 Next-generation matrix (NGM) method, 60, 257, 268 Node2vec, 37–39, 42, 44 Noise intensity, 94 Non-altruistic organisms, 176 “Non-extensive” Gaussian entropy measure, 186, 187 Nonlinear incidence function, 100, 101 Nonlinear iteration for Nernst-Planck equation, 379, 380 Nonlinear iteration for Poisson equation, 379 Non-monotone incidence function, 234 Non-monotonic incidence rates, 248, 250 Non-pharmaceutical interventions (NPI), 18, 26, 254, 266, 334, 359, 363, 366 Non-pharmaceutical measures, 215 Novel mutations, 229

O ODD protocol, 19 Omicron-era Netherlands data, 231 Omicron subvariants, 230 Omicron variant, 216, 341, 356, 359, 365, 367, 368 Oncolytic M1 virus, 125 Online GEne Essentiality (OGEE) Database, 33, 43, 44, 51 Ornstein-Uhlenbeck (OU) colored noise perturbation, 87 Ornstein-Uhlenbeck (OU) process, 74, 89, 138, 142, 143, 145

P Partial least squares (PLS) regression, 299 Partial success immune-free equilibrium, 128, 134, 136 Particle size distribution, 300 Particle swarm optimization (PSO) algorithm, 343 Pfam database, 193, 194 Pipeline, 255, 260, 261, 266 Poincaré criterion, 12 Poisson-Nernst-Planck (PNP) equations, 373, 374, 378, 385 Poisson’s equation, 373, 378 Population Balance Models, 297, 300 Population-free equilibrium, 200 Positive coexistence equilibrium, 77 Postsynaptic neuron, 139 Predator-free equilibrium, 77, 200, 207, 209

Index Predator-prey-parasite phenomenon, 197 Predator-prey system, 73, 212 Predator’s functional response, 74 Presynaptic neuron, 139 Prey behaviour, 73 Prey-only equilibrium, 276 Prey refuge, 74, 75 Prey species’ logistic growth function, 75 Principal component analysis (PCA), 258, 261, 262 Principal minors, 186, 188, 189, 191 Probabilities of occurrence, 181, 182, 184 Probabilities of occurrence of t-sets, 182 Process Analytical Technology (PAT), 291–293, 295–296 Process of filtration, 298 Process Systems Engineering (PSE), 295, 296, 298, 300, 301 Protein domain families, 194 Protein family PF01926, 193 Protein-protein interaction (PPI) network, 32–40, 44, 51

Q Quality by Design (QbD), 292, 301 Quasi-periodic dynamics, 318, 321, 326, 327

R Random Poisson excitatory and inhibitory inputs, 141 Rauch-Tung-Striebel (RTS) smoothing equations, 354 Reaction-diffusion fractional model, 126, 135 Reactome, 36, 40 Recovered individuals, 1, 18, 154, 217, 218, 220, 222, 225, 227 Reduced model system, 60 Reef fish, 171 Refractoriness, 137, 139, 143 Reinfection cycles, 333 Renyi entropy measure, 186, 187 Reproduction number, 106, 128, 222, 247 Right coronary artery (RCA), 307 Rouche’s Theorem, 204 Routh-Hurwitz condition, 282, 283 Routh-Hurwitz criterion, 79, 161, 162 Runge-Kutta method, 343, 353

S Saccharomyces cerevisiae, 30, 37 Saddle-node bifurcation, 276, 279, 281

391 SARS-CoV-2 variants, 230, 333 SARS-CoV-2 virus, 333, 334 Second strain endemic equilibrium, 249 SEIR model, 154, 233, 336 SEIR two-strain epidemic model, 233 Selective feeding behaviour of zooplankton on phytoplankton in the presence of free-viruses, 197 Selective predation of zooplankton, 198 Semilinear parabolic systems, 104 Sensitive indices of parameters, 61 Severe acute respiratory syndrome, 333, 352 Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), 215, 216 Sharma-Mittal class of entropy measures, 185, 186 Significant negative correlation, 61 SI model, 20, 21 SIS model, 1, 256 Skeletonization, 308, 310 Social contact matrices, 254, 255, 268, 269, 281 Social contact patterns, 254, 262 Social distance parameter, 222 Social distancing, 217, 219, 221, 254, 367 Social distancing index, 336, 338, 359, 363 Social distancing measure, 215, 218 Social distancing policies, 340, 342, 359, 367, 368 Social isolation, 155 Social media, 153 Sotomayor’s theorem, 83, 280 Spike-frequency adaptation, 143 Spike Response Model, 143, 149 Spikes, 138–141, 143, 149 Spiral waves in cardiac tissue, 318 Spreading pattern of infection, 21 Spread of infectious diseases, 1, 17, 19, 25, 58, 154 Stability analysis, 58, 126, 135, 154, 202, 329 Stein’s model, 142 Stenosis severity, 305, 315 Stochastically stable, 94, 97 Stochastic asymptotic stable, 97 Stochastic model, 18, 74, 84, 138, 139 Stochastic modeling of a single neuron, 138 Stochastic stability, 75, 94 Stochastic stability of the coexistence equilibrium, 84, 87 Strain reproduction number, 247–249 Strict concavity, 185, 186, 188, 190, 193 Strictly diagonally dominant matrix, 113, 114

392 Superpredator, 274, 276, 277, 280 Super-threshold endemic equilibrium, 8 Survival strategy, 73 Susceptible individuals, 1, 18, 57, 217, 219, 220, 226, 234, 247, 359 Susceptible prey only equilibrium, 200, 208 Symbiosis, 273 Syn3.0, 30 Synapses, 139–141, 147 T Temporal gene expression profiles, 37 Time series, 206, 334, 346, 348, 356 Time varying effective reproduction number, 217 Transcritical bifurcation, 9, 13, 58, 68–70, 280, 282 Transmission of disease, 1, 156, 198, 202 Transversality condition, 80, 163 Treatment failure equilibrium, 128, 129, 134, 136 Treatment failure immune-free equilibrium, 128, 131, 133, 136 Trophic chain, 273 Trophic interactions, 197 Tryptophan, 317 T-set of ordered columns, 181 Tumor cells, 31, 32, 125, 127, 128, 135

Index Tumor-free equilibrium, 128, 130, 131, 135 Two variable Fitzhugh-Nagumo model, 317 U Unique endemic equilibrium, 10 V Vaccination of the infected hosts, 8, 13 Vaccine-resistant mutant virus (MT), 2161 Variable heart rate, 315 Variant dynamics, 231 Vertical transmission, 1, 2, 13, 171, 179 Viral mutations, 333 Viral replication process, 100 Virotherapy, 125, 126 Virtual FFR, 305, 308, 311, 312 Virus dynamics, 100 Volterra integral equation, 144, 146

W Wiener processes, 74, 84, 141, 144, 145 Wild-type virus (WT), 216 World Health Organization (WHO), 1, 99, 215

Z Zeta variant, 335, 336, 340