Computational Methods for Biological Models (Studies in Computational Intelligence, 1109) [1st ed. 2023] 9789819950003, 9789819950010, 9819950007

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Studies in Computational Intelligence 1109

Harendra Singh Hemen Dutta   Editors

Computational Methods for Biological Models

Studies in Computational Intelligence Volume 1109

Series Editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Computational Intelligence” (SCI) publishes new developments and advances in the various areas of computational intelligence—quickly and with a high quality. The intent is to cover the theory, applications, and design methods of computational intelligence, as embedded in the fields of engineering, computer science, physics and life sciences, as well as the methodologies behind them. The series contains monographs, lecture notes and edited volumes in computational intelligence spanning the areas of neural networks, connectionist systems, genetic algorithms, evolutionary computation, artificial intelligence, cellular automata, selforganizing systems, soft computing, fuzzy systems, and hybrid intelligent systems. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution, which enable both wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Harendra Singh · Hemen Dutta Editors

Computational Methods for Biological Models

Editors Harendra Singh Department of Mathematics Post Graduate College Ghazipur Ghazipur, Uttar Pradesh, India

Hemen Dutta Department of Mathematics Gauhati University Guwahati, Assam, India

ISSN 1860-949X ISSN 1860-9503 (electronic) Studies in Computational Intelligence ISBN 978-981-99-5000-3 ISBN 978-981-99-5001-0 (eBook) https://doi.org/10.1007/978-981-99-5001-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book is designed to include several topics in the areas of biological models and their treatments using computational methods. Biological modeling is very useful to understand real-life problems. The chapters include useful computational methods for solving various types of biological models as well as their significance and relevance in other scientific areas of study and research. There are ten chapters in this book, and they are organized as follows. Chapter “Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model” deals with time-dependent partial differential equation (PDE) problems. The modeling and application of some epidemiological diseases have been the subject of activities. Most of these models mainly exist in the form of ordinary differential equations (ODE), due to some of the challenges encountered in solving systems with coupled partial differential equations (PDEs). At first, the problem is discretized with respect to space in such that, the arising system of ODEs is solved using the novel fourth-order ETD method of Runge-Kutta type. A typical example is given to the HIV model in the diffusive case which could rarely be found in the literature. Numerical results which reflect the physical behavior of the model under consideration are given in one and two dimensions. Chapter “An Effective Technique for Solving a Model Describing Biological Species Living Together” presents a reliable scheme to obtain an analytical approximation to the solution of a system of nonlinear integro-differential equations that arise in biology. The dynamics of the populations of two distinct species are characterized by this system. The suggested strategy involves two steps. Construct an integral operator first, and then apply Halpern’s iteration scheme to the constructed operator to develop the required iterative approach. This approach’s convergence is also addressed. Numerous numerical examples are taken into consideration to demonstrate the effectiveness of the strategy. The proposed approach has several key advantages over existing approaches, including the ability to handle nonlinearity without the use of Adomian polynomials, the ability to solve the problem without the use of the Lagrange multiplier or constrained variations, and the ability to take into account both interval endpoints. In contrast to existing semi-analytical

v

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Preface

methods, the suggested approach addresses the problems without requiring linearization, discretization, or perturbation assumptions. Three test problems are taken into account to confirm the method’s efficacy. Chapter “Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment” focuses to include a delay term to model key transformations taking place within the dynamics modeling the interaction among transformed epithelial cells (TECs), fibroblasts, myofibroblasts, transformed growth factor (TGF−β), and epithelial growth factor (EGF), in silico, in a setup mimicking experiments in a tumor chamber invasion assay. This consideration of such experiments resulted into a derivation of system of nonlinear quasi time-depended delay parabolic partial differential equations. The well-pose of the unique solution of the resulting system of nonlinear quasi time-depended delay parabolic partial differential equations presents that an extension of the existing Gronwall’s inequality for ordinary differential equations to Gronwall’s inequality for linear, delay ordinary differential equations is thus derived. The asymptotic stability and Hopf bifurcation analysis are established. Based on the asymptotic stability condition, a novel numerical method is derived, analyzed, and implemented, and improved results are presented for discussion. Hence, based on the obtained results, one strongly believes that the contribution in this chapter has a great deal toward the process of crafting a drug that can slow and/or confine tumor invasion. Chapter “A Mathematical Model to Study Regulatory Properties and Dynamical Behaviour of Glycolytic Pathway Using Bifurcation Analysis” deals with a dynamical system based on mass balance equations and S-system representation to study the regulatory properties, rate control distributions, and dynamical behavior of reactions in this glycolytic pathway. This representation involves several parameters. Few standard tools from the bifurcation analysis such as phase-portraits, time series plots, the Lyapunov exponents, and d∞ plots have been employed to investigate the stability of the pathway with respect to certain chosen parameters. The study is further extended by eliminating the glycogen branch from the original pathway and by adding an external perturbation into the system. Chapter “On Solutions of Biological Models Using Reproducing Kernel Hilbert Space Method” presents the reproducing kernel Hilbert space method (RKHSM) for some systems that have significant applications in biology. The proposed method’s error estimations and convergence analyses are discussed. The assessment of the RKHSM is made by testing some illustrative applications. The results suggest that the RKHSM is an efficient and highly convenient method to solve the fractional systems arising in biology. Chapter “A Study of the Fractional Tumour–Immune Unhealthy Diet Model Using the Pseudo-operational Matrix Method” deals with the fractional tumor-immune unhealthy diet (TIUNHD) model. The fractional-order Laguerre operational matrixbased method is used to solve it. The fractional derivative of Caputo is used in this chapter. In this method, first the pseudo-operational matrices of fractional, integer order integration are constructed, and then used these pseudo-operational matrices to approximate the unknown solutions of the given system of nonlinear fractional differential equations model. This results in an algebraic system of equations, which can be

Preface

vii

easily solved by Newton’s iterative method. A numerical experiment demonstrates the proposed algorithm’s applicability to solving the TIUNHD model. Chapter “Analysis of a Fractional Stage-Structured Model With Crowley–Martin Type Functional Response by Lagrange Polynomial Based Method” focuses on the dynamics of a stage-structured predator-prey system that replicates interactions between two densities of prey and predator populations. The adult predator population and the juvenile predator are the two compartments that make up the predator population in the model. The predator relies on both prey and juvenile predator, which is another element of the paradigm that can be termed cannibalism. CrowleyMartin type functional denotes the nature of the interaction between prey and adult predators, while Holling type-I functional response denotes the nature of contact between juvenile and adult predators. The concept of memory is introduced in the form of the Caputo fractional derivative to reflect the complicated dynamics of interaction among the species. As a result, the model is able to incorporate all relevant historical information about the occurrence, from its inception to the desired time, into its calculations. The boundedness and existence and uniqueness of solutions to the proposed model are also investigated. The numerical simulations are performed by using the Lagrange polynomial-based method which is novel in the field of mathematical biology. Simulations have been accomplished to examine the significance of parameters related to cannibalism, the conversion rate from prey to adult predator, harvesting of an adult predator, and growth rate of juvenile predators on the overall behavior of the system. The noteworthy performance of the fractional operator on the anticipated predator-prey model’s dynamical behavior is well demonstrated by numerical results. Chapter “Qualitative Theory and Approximate Solution to Norovirus Model Under Non Singular Kernel Type Derivatives” deals with a fractional-order nonlinear dynamical system of Norovirus disease. The considered mathematical model consists of susceptible, exposed, vaccinated, infected, and recovered classes. Firstly, the Caputo Fabrizio fractional derivative (CFFD) is used to examine the qualitative theory and approximate solutions. Some adequate results for the existence of approximate solutions are given using fixed point theory. The Laplace Transform (LT) and the Adomian decomposition (ADM) approach are used to get approximate results for each compartment. The graphical presentation that corresponds to available real facts for different fractional orders is given. Secondly, by using piecewise global fractional derivatives in sense of singular and non-singular kernels, the aforementioned system is also investigated. Sufficient conditions for the existence and uniqueness of the solution to the proposed model by using the fixed-point approach are established. Numerical simulation is performed by extending Newton’s interpolation formula for the considered model under the mentioned derivatives. Graphical presentations for various compartments of the model are given against the available real data for different fractional orders. The numerical findings are performed by using MATLAB 16. Chapter ”Study of the SIRI Model Utilizing the Caputo Derivative” presents qualitative studies of the SIRI model described by a fractional operator. In the investigations, the Caputo operator is utilized in the modeling of the epidemic.

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Using the reproduction number the stability of the model is analyzed. A qualitative property as the boundedness of the solutions of the SIRI model is focused. A numerical scheme including the numerical scheme of the Caputo operator has been utilized and explained rigorously. The results of the chapter have been illustrated by the graphics of the solutions obtained using the proposed numerical scheme. Chapter “Implementation of Vaccination in an Epidemic Model for COVID-19” aims to establish the Susceptible, Vaccination, Infectious, and Recovered (SVIR) pandemic paradigm. The proposed model’s general characteristics are first discussed. Particularly, both theoretical and numerical analysis is done on the basic reproduction numbers. Reproduction numbers are used to describe both the local and global stability of equilibrium points. Secondly, determine the stability of the equilibrium points using the Lypanouv function and the Painocare-Bandixson condition. The transmission of Covid-19 and its regulation are shown in simulation with the aid of MATLAB according to genuine data in India. In the suggested model, the effects of vaccination are investigated. This simulated result is consistent with the actual parameters and indicates the necessity for booster doses to eradicate the virus from the population. The editors sincerely acknowledge the cooperation of contributors while dealing with their works for possible publication in this book. Reviewers deserve deep gratitude for offering their kind help in selecting and finalizing the contents of the book. The editors also thankfully acknowledge the support of the editorial staff at Springer. The editors gratefully acknowledge the encouragement received from many colleagues and friends for this book project. Ghazipur, India Guwahati, India

Harendra Singh Hemen Dutta

Contents

Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kolade M. Owolabi, Edson Pindza, and Gulay Oguz

1

An Effective Technique for Solving a Model Describing Biological Species Living Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saurabh Tomar and Soniya Dhama

25

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kolade M. Owolabi, Albert Shikongo, and Edson Pindza

53

A Mathematical Model to Study Regulatory Properties and Dynamical Behaviour of Glycolytic Pathway Using Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shruti Tomar, Naresh M. Chadha, and Ankita Khanna

81

On Solutions of Biological Models Using Reproducing Kernel Hilbert Space Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Nourhane Attia and Ali Akgül A Study of the Fractional Tumour–Immune Unhealthy Diet Model Using the Pseudo-operational Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . 137 Saurabh Kumar and Vikas Gupta Analysis of a Fractional Stage-Structured Model With Crowley–Martin Type Functional Response by Lagrange Polynomial Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Chandrali Baishya and P. Veeresha Qualitative Theory and Approximate Solution to Norovirus Model Under Non Singular Kernel Type Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 181 Eiman, Waleed Ahmed, Kamal Shah, and Thabet Abdeljawad

ix

x

Contents

Study of the SIRI Model Utilizing the Caputo Derivative . . . . . . . . . . . . . . 211 Ndolane Sene Implementation of Vaccination in an Epidemic Model for COVID-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Yerra Shankar Rao

About the Editors

Dr. Harendra Singh is an assistant professor in the Department of Mathematics, Post-Graduate College, Ghazipur-233001, Uttar Pradesh, India. He has listed in top 2% scientist list published by Stanford University. His area of interest is mathematical modelling, fractional differential equations, integral equations, calculus of variations, analytical and numerical methods. His works have been published in applied mathematics and computations, applied mathematical modelling, applied numerical mathematics, Chaos Solitons & Fractals, numerical methods for partial differential equations, physica A, astrophysics and space science, Electronic Journal of Differential Equations, Few Body-system and several other peer-reviewed international journals. His 50 research papers have been published in various journals of repute with h-index of 22 and i10 index 31. He has delivered lectures in a number of national and international conferences and workshop. He has also attended short-terms programs and workshops. Dr. Hemen Dutta is serving Gauhati University, India, regularly as a faculty member since 2010. Prior to joining Gauhati University, he served three other academic institutions in different capacities. He is currently interested in applied analysis and mathematical modelling. He has over 170 publications in the forms of research papers, book chapters and proceedings papers and serving as an editor of several SCI/SCIE and Scopus indexed journals as well as book series editor of top-class publishers like Elsevier and Taylor & Francis. He has to his credit several authored and edited books and conference proceedings published by reputed publishers like Springer, Birkhauser, Wiley, Taylor & Francis, Elsevier, etc. He is a selected member as well as life member of some academies and societies. He is the recipient of several grants, research projects and holding some honorary positions at foreign and national university/institution.

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Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model Kolade M. Owolabi, Edson Pindza, and Gulay Oguz

Abstract In recent years, the modeling and application of some epidemiological diseases have been the subject of activities. Most of these models mainly exist in the form of ordinary differential equations (ODE), due to some the challenges encountered in solving systems with coupled partial differential equations (PDEs). This work is extended to PDE, by formulating a robust numerical technique based on the formulation of an exponential-time-differencing (ETD) scheme which could easily circumvent the issue of stiffness inherent in coupled PDE system. Spectral method is suitably applied to time-dependent PDE problems. At first, the problem is discretized with respect to space in such that, the arising system of ODEs is solved using the novel fourth-order ETD method of Runge-Kutta type. A typical example is given to the HIV model in the diffusive case which could rarely be found in the literature. Numerical results which reflect the physical behaviour of the model under consideration are given in one and two dimensions.

K. M. Owolabi (B) Department of Mathematical Sciences, Federal University of Technology Akure, PMB 704 Akure, Ondo State, Nigeria e-mail: [email protected] E. Pindza Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, South Africa Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria West, Pretoria 0183, South Africa G. Oguz Department of Mathematics, Harran Unıversity, Sanlıurfa, Turkey © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_1

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K. M. Owolabi et al.

1 Introduction This chapter considers higher order numerical method based on the novel exponentialtime-differencing (ETD) method for the solving of an evolution equation of the parabolic type (1.1) ut = DLu + N (t, u) where L stands for the linear operator representing diffusive term, and N represents a nonlinear operator which accounts for the totality of local reactions. D is diagonal matrix representing the diffusion constants of each of sub-population class or species. The linear terms of most time-dependent PDEs are mainly responsible for the issue of stiffness, with exponential decay property as in the case of dissipative PDE, or spatiotemporal oscillation as peculiar to dispersive cases. Therefore, it is desirable to use higher-order approximation techniques in both time and space for any timedependent parabolic PDE that adds a nonlinear low-order with a linear higher-order terms. When one discretize (1.1) in space, it leads to a large (or coupled) system of ODEs in the form (1.2) u t = DLu + N(t, u). Such diffusive systems often arise when wave equations such as the Navier-Stokes, Schrödinger, Kuramoto, Kortweg-de Vries (KdV), and nonlinear Burgers’ equations are discretized [1, 7]. The ETD methods are time integration solvers which when combined with the special methods can be used efficiently to obtain good numerical solutions for either stiff or moderately stiff PDE problems. The authors in [5, 7, 13, 15] have successfully applied the ETD methods as efficient time-solvers in conjunction with the spectral schemes to solve a range of linear and nonlinear time-dependent PDEs. Also, the stability and convergence properties for this class of schemes have been considered. Application of the ETD method for solving the diffusive HIV model in one-dimensional space will be considered in this work. A lot of time integrator algorithms are formulated to solve stiff problems have been proposed. For instance, some of these formulations are either based on the Runge-Kutta or Adams-Bashforth-Moulton methods. This class of numerical methods is most suitable to handle semi-linear equations which can be divided into a linear part, which harbors the stiffest part of the dynamical problem, and a non-linear-part, which often varies more slowly than the linear-part. Recall that many time-dependent reaction diffusion PDEs such as (1.1) mainly combined low-order (nonlinear terms) with higher-order (linear terms). Hence, the nature of such equations allows one to apply any method of choice in time and space directions [7, 13]. Exponential time-differencing schemes have been re-examined many times in different forms and under several authors. A case study is the exact linear part method which was suggested in [3] for random order. In [4], the explicit formulation of the linear part schemes of irregular order was reported, which was named the ETD schemes. According to Cox and Matthews, the idea of ETD methods was earlier introduced in the computational area of electrodynamics. In the year 2002, the duo

Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model

3

in [4] further designed a higher-order scheme, and they come up with the new RungeKutta (RK) types whose orders are ranging from first-order to fourth-order. Ever since the inception of these schemes, many authors have developed some interests in the application of the ETD time-solver schemes as a viable and effective approach to address the stiffness challenge posed in the computation of time-dependent PDE problems. This chapter presents and implements the novel Fourier spectral method in space with the fourth-order ETD Runge-Kutta (ETDRK4) method for the solution of HIV models. Hence, the remaining part of this work is arranged into different sections in the following manner. The Fourier spectral approximation technique and the novel exponential time differencing methods are introduced in Sect. 2. As a typical example, the multicomponent HIV model is introduced in Sect. 3, its stability analysis is also presented. Applicability and suitability of the numerical methods suggested is demonstrated on HIV model with detail numerical experiment in Sect. 4. Conclusion is given in the last part.

2 Numerical Techniques for Reaction-Diffusion Problems Spectral methods can be categorized into different types, and one clear difference often made is among the tau, Galerkin, and collocation (also known as the pseudospectral) spectral algorithms [28, 29]. The Galerkin and tau methods perform well using coefficients of the global expansion, while the collocation or pseudospectral method work well with its values at points. As a result of the general notion of the ansatz functions, spectral algorithms are commonly referred to as the general methods, which means the value of a derivative at a point with regards to space often depends on the solution at every other points in space, and off grid points. As a matter of fact, spectral discretization schemes have a high-order level of accuracy, talking in terms of spectral- convergence over any finite difference schemes. More importantly, diffusion and dispersion properties of the spatial Laplacian operator are of importance when one compares with the known finite difference (FD) schemes. It is not difficult to see that when the concept of a modified wave number is considered, spectral methods often results to the exact derivative of the function, the major concern should be the truncation error when a finite set of ansatz functions/coefficients is prematurely truncated. Though spectral algorithms are flexible geometrically and more cumbersome to compute than the FD methods, but spectral methods was regarded among the first sets of methods to be used in practical simulation processes [7, 21, 30]. Spectral method is most applicable when one desires high level of accuracy to compute the solution of the time-dependent PDEs. This class of methods is suitable for developing numerical schemes in most application areas of engineering, physics and mathematics. Spectral method has capacity to accurately differentiate the linear operator. Also, this method can be used to compute the numerical solution of ODEs and PDEs, based on the polynomial expansion of the solutions. The numerical integration is primarily based on the variational of the continuous problem in

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its formulation which is computed as quadrature scheme. For other numerical techniques which have been used to study models in mathematical biology, readers are referred to [6, 25, 26, 31].

2.1 Spectral Method for Spatial Discretization Define the Laplacian or linear operator L as a complete orthonormal set of eigenfunctions {ψ j } which satisfies the extreme conditions on D ⊂ Rd , d ≤ 3, and eigenvalues λ j , that is, (L)ψ j = λ j ψ j on D, it is reasonable to assume that u :=

⎧ ⎨ ⎩

u=

∞ 

u¯ j ψ j , u¯ j = u, ψ j ,

j=0

∞ 

|u¯ j |2 |λ|2 < ∞,

j=0

⎫ ⎬ ⎭

,

(2.3)

for some u ∈ u, it shows that Lu =

∞ 

u¯ j λ2j ψ j .

(2.4)

j=0

For notational convenience, one writes problem (1.1) in the form, taking u := (u, v, w)T , which implies ut = DLu + N (u), in ζ L , ∂u = 0 on ∂ζ × (0, L), u(x, 0) = u0 (x), x ∈ ζ, ∂ν with

and

(2.5) (2.6)

⎛

⎞ f 1 (u, v, w) N (u) ≡ ⎝ f 2 (u, v, w)⎠ f 3 (u, v, w)

(2.7)

⎛

⎞ η1 0 0 D = ⎝ 0 η2 0 ⎠ . 0 0 η3

(2.8)

With N = 0 in one dimension, (2.5) becomes a simple diffusion equation u t = DLu,

(2.9)

subject to the initial condition u(x, 0) = u 0 (x) and any boundary condition of choice with x ∈ [0, L]. With (2.3) and Eq. (2.4), the exact solution of (2.9) is given as

Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model

u(x, t) =

∞ 

u¯ j (t)ψ j (x) =

j=0

∞ 

u¯ j (0)e−Dλ j t ψ j (x), 2

5

(2.10)

j=0

with u¯ j (0) = u 0 (x), ψ j (x), for the case of homogeneous Dirichlet condition, we 



 ( j + 1)π 2 ( j + 1)π(x − 0) 2 have λ j = and ψ = sin . L L L To demonstrate this approach using multi-component reaction-diffusion system with the reaction kinetics, one uses the fast Fourier transform for the reactiondiffusion problem to have   Ut (ζx , ζ y , t) = η1 ζx2 + ζ y2 U (ζx , ζ y , t) + N [ f 1 (u(x, y, t), v(x, y, t), w(x, y, t))],   α/2 α/2 V (ζx , ζ y , t) + N [ f 2 (u(x, y, t), v(x, y, t), w(x, y, t))], Vt (ζx , ζ y , t) = η2 ζx + ζ y   α/2 α/2 W (ζx , ζ y , t) + N [ f 3 (u(x, y, t), v(x, y, t), w(x, y, t))] Wt (ζx , ζ y , t) = η2 ζx + ζ y

where U , V and W are the double Fourier transforms which stand for the population densities u(x, y, t), v(x, y, t) and w(x, y, t), respectively. In other words,  N [u(x, y, t)] = U (ζx , ζ y , t) = 

0

N [v(x, y, t)] = V (ζx , ζ y , t) =

∞



∞



∞





0

N [w(x, y, t)] = W (ζx , ζ y , t) = 0

∞ 0

∞

0

∞

u(x, y, t)e−i(ζx x+ζ y y) d xd y, v(x, y, t)e−i(ζx x+ζ y y) d xd y. w(x, y, t)e−i(ζx x+ζ y y) d xd y.

0

  With ζ 2 = ζx2 + ζ y2 , the issue of stiffness is explicitly removed in the linear piece via the integrating factors, next we set 2 U = eη1 ζ t U¯ ,

2 V = eη2 ζ t V¯ ,

2 W = eη3 ζ t W¯

in such that ∂t U¯ = eη1 ζ t N [ f 1 (u, v, w)], 2 ∂t V¯ = eη2 ζ t N [ f 2 (u, v, w)], 2 ∂t W¯ = eη3 ζ t N [ f 3 (u, v, w)], 2

(2.11)

Again, one requires to discretize the equal domain size (square) by via the equispaced points L x , L y along the directions x, y, respectively. We adopt the Matlab discrete fast Fourier transform to change Eq. (2.11) to ODEs

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∂t U¯ i, j = e K u ζi, j t N [ f 1 (u i, j , vi, j , wi, j )], 2 ∂t V¯i, j = e K v ζi, j t N [ f 2 (u i, j , vi, j , wi, j )], 2 ∂t W¯ i, j = e K W ζi, j t N [ f 3 (u i, j , vi, j , wi, j )], 2

(2.12)

 α/2 α/2 with u i, j = u(xi , y j ), vi, j = v(xi , y j ), wi, j = w(xi , y j ) and ζi, j = χx (i)+  α/2 χ y ( j) . At this point we utilize the zero-flux boundary conditions clamped at the two ends of the domain. Any explicit solver can be used to integrate in time at this point. In order to gain a better accuracy, we employ the novel ETDRK4 method as described below.

2.2 The ETDRK4 Method To derive the time integration solver, the notations used in [4] is followed. Consider the ordinary differential equation dυ(t) = Lυ(t) + N(υ(t), t), dt

(2.13)

where L denotes the stiffness parameter harbouring the diffusion, and N(υ(t), t) represents nonlinear coupled source or forcing term that accounts for the totality of local reactions. The formulation of the k-step ETD methods is followed directly in [2, 4]. To start with, one multiplies (2.13) with the integrating factor e−Lt , we then integrate it over a single time-step from t = tn and t = tn+1 = tn + ηt to have  υ(tn+1 ) = υ(tn )eLηt + eLηt

ηt

e−Lζ N(υ(tn + ζ), tn + τ )dτ .

(2.14)

0

Different ETD schemes arise from the approximation of integral function in (2.14). This scheme is called the exact method [4], and the next step is to derive approximations to the integral. By using the Newton backward-difference scheme, and source term N(υ(t), t) at the nth, and preceding time-steps, we approximate N(υ(tn + ζ), tn + ζ) as

 s−1  j −τ /ηt ∇ j An (tn ), N(υ(tn + τ ), tn + τ ) ≈ An (tn + τ ) = (−1) j j=0 where ∇ is the backward difference scheme expressed as

(2.15)

Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model

 j  j An−k (tn−k ), (−1)k k k=0

 j  j N(u(tn−k ), tn−k ), (−1)k ≈ k k=0

7

∇ j An (tn ) =

and j!

(2.16)

 − = (− − 1) · · · ( − j + 1), for j = 1, . . . , s − 1, j

where  = ζ/ηt. If one substitutes (2.15) (2.14), bear in mind that (0! υ(tn+1 ) − υ(tn )eLηt ≈ ηt

s−1 



1

(−1) j

eLηt (1−)

0

j=0

− 0

= 1), then

 − d∇ j An (tn ). j

(2.17)

At this point, the integral in (2.17) can be indicated as 

1

α j = (−1) j

eLηt (1−) d,

(2.18)

0

by obtaining in the generating function, one computes α j . For ω ∈ R, |ω| < 1, therefore, this can be executed by introducing the generating function as (ω) =

∞ 

αjω j,

(2.19)

j=0

which is readily found to be 

1

(ω) = 0



1

=

eLηt (1−)

 ∞  − (−ω) j d, j j=0

eLηt (1−) (1 − ω)− d,

0

= On rearranging (2.20) as

eLηt (1 − ω − e−Lηt ) . (1 − ω)(Lηt + log(1 − ω))

(2.20)

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(Lηt + log(1 − ω))(ω) = eLηt −

1 , 1−ω

and then obtain the expansion in powers of ω as

 ω2 ω3 Lηt − ω − − − · · · (α0 + α1 ω + α2 ω 2 + · · · ) = eL − 1 − ω − ω 2 − ω 2 − · · · , 2 3

for j ≥ 0, we have the recurrence relation for α j in the manner that Lηtα0 = eLηt − 1,

(2.21)

1 1 1 α0 = Lηtα j+1 + 1 = α j + α j−1 + α j−2 + · · · + 2 3 j +1

j  k=0

1 αk . j +1−k

On putting Eqs. (2.16), and (2.18) into (2.17), the general ETD method [4] of order s is presented as υn+1 = υn eLηt + ηt

s−1  j=0

αj

j  k=0

(−1)k

 j Nn−k , k

(2.22)

where u n and N are numerical approximation to u(tn ) and N(u(tn ), tn ), respectively. Note that the formulation and description of the ETD schemes are via the linear multi-step method (LMM) idea. Therefore, their usage require kth previous evaluations of the term N(υ(t), t) which make them difficult and more challenging to use due to the requirement of starting values. This issue was the reason the authors in [4] seeking for means to overcome the stiffness challenge, and finally opined that the problem can only be avoided by applying the Runge-Kutta methods which are self starter. More so, the importance of large area of stability (LAS) and small error constants than multi-step methods. It is of great interest to develop the Runge-Kutta based ETD schemes. Direct application of the classical Runge-Kutta method of order four and introduction of some necessary parameters results to the novel Cox and Matthews [4] method tagged the ETDRK4: υn+1 = υn eLηt + Nn [−4 − Lηt + eLηt (4 − 3Lηt + L2 ηt 2 )] +2(N (an , tn + ηt/2) + N (bn , tn + ηt/2))[2 + Lηt + eLηt (−2 + Lηt)] +N (cn , tn + ηt)[−4 − 3Lηt − L2 ηt 2 + eLηt (4 − Lηt)]/L3 ηt 2 , where an = υn eLηt/2 + (eLηt/2 − I)Nn /L, bn = υn eLηt/2 + (eLηt/2 − I)N (an , tn + ηt/2)/L, cn = υn eLηt/2 + (eLηt/2 − I)(2N (bn , tn + ηt/2) − Nn )/L.

(2.23)

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9

The terms an and bn are used for approximation of u at tn + ηt/2, while the term cn approximates u at tn + η [13, 22, 23, 27]. The method (2.23) serves as the quadrature formula for (2.14) which one derives via the quadratic interpolation points tn , tn + ηt/2 and tn + ηt, when an average value of N is applied at an and bn .

3 The HIV-I Model and Its Stability Analysis A lot of efforts has been made in the mathematical modeling of HIV-I infection model. Mathematical models of both fractional order and classical order have also been derived to discuss the HIV-I infection dynamics in the bloodstream where the cell-free-viral spread is the major route of viral infections [8–12]. The model which describe a cell-to-cell dynamic of HIV-I infection in the tissue is given by t   u + v + −∞ g(t−ξ)u(ξ)v(ξ) dξ du uv ϑ+u(ξ) = φu 1 − , −ϕ dt σ ϑ+u  t dv g(t − ξ)u(ξ)v(ξ) =ψ dξ − ηv, dt ϑ + u(ξ) −∞

(3.1)

where u and v denote the population densities at time t of healthy and infected cells, respectively, φ is the reproduction rate of class u, σ represents the carrying capacity, and the maximum rate of infection is given by ϕ, the death rate infected population η, the fraction of cells that survives the incubation period is denoted as ψ, while ρ is half saturation constant. By following the discussion in [8], we take the delay kernel g(ξ) as g(ξ) = ρe−ρξ , ξ ≥ 0, 

we let w(t) =

t −∞

ρe−ρ(t−ξ)

u(ξ)v(ξ) dξ, ϑ + u(ξ)

to finally arrive at equations

 u+v+w uv du = φu 1 − −ϕ , dt σ ϑ+u dv = ψw − ηv, dt dw uv =ρ − ρw. dt ϑ+u

(3.2)

In ecological context, Segel and Jackson [24] called the attention of many researchers to the fact that diffusion-driven or Turing instability can occur in an ecological context. They conducted their research based on a predator-prey dynam-

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ics where the inclusion of diffusion term (random dispersal) leads to an instability of the steady-state distribution to perturbation of a specific wavelength. In 2016, the same idea was extended to model the spread of Hepatitis B virus (HBV) in the presence of diffusion [15]. Further reports on diffusive instability in both ecological and epidemiological contexts can be found in [16–20]. Now assume that the cells u, v, w are population densities, that is population per unit volume, and depends on a spatial variable x and time t, that is u = u(x, t), v = v(x, t) and w = w(x, t). Then a conservation law can be formulated in the form 

uv u+v+w −ϕ , u t + x = f 1 (u, v, w) = φu 1 − σ ϑ+u vt + x = f 2 (u, v, w) = ψw − ηv, uv wt + x = f 3 (u, v, w) = ρ − ρw, ϑ+u

(3.3)

where  denotes the population flux. Assuming Fick’s law for the flux , we obtain the following reaction-diffusion HIV-I system u t − Du u x x vt − Dv vx x wt − D w w x x



uv u+v+w −ϕ , = f 1 (u, v, w) = φu 1 − σ ϑ+u = f 2 (u, v, w) = ψw − ηv, uv − ρw, = f 3 (u, v, w) = ρ ϑ+u

(3.4)

where Du , Dv and Dw are diffusion coefficients. System of the form (3.4) become one of the most essential classes of nonlinear phenomena due to their occurrence in many chemical (combustion) and biological (pattern formation) processes. In order to make the correct choice of parameters which will be used to simulate the full reaction-diffusion dynamics, it is necessary to examine the stability of HIV-I model. To achieve this, we set f i (u, v, w) = 0, i = 1, 2, 3 in the absence of diffusion (that is, Du = Dv = Dw = 0) as

 u+v+w uv 0 = φu 1 − −ϕ , σ ϑ+u 0 = ψw − ηv, uv − ρw. 0=ρ ϑ+u

(3.5)

It is obvious that the above algebraic system has three equilibrium states namely; the washout state corresponding to E 0 = (0,  0, 0),  the healthy cell state E 1 = (σ, 0, 0) 1 ψσ with basic reproduction number R0 = η ϑ+σ > 1. The infected equilibrium is calculated as E ∗ = (u ∗ , v ∗ , z ∗ ), where

Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model

11

ηϑ , ψ−η ψϑ(φηϑ − φσψ − φση) , y∗ = (ψ − η)(ψφϑ + φϑη + ϕϑψ − ϕση) ηϑ(φηϑ − φσψ − φση) w∗ = . (ψ − η)(ψφϑ + φϑη + ϕϑψ − ϕση) x∗ =

The community matrix corresponding to the autonomous system is given by ⎛ ⎜ ⎜ A(u,v,w) = ⎜ ⎜ ⎝

φ−

φ(2u+v+w) σ

−

ϕϑv (ϑ+u)2

− φu − σ

0

−η

ρϑv (ϑ+u)2

ρu ϑ+u

ϕu ϑ+u

− φu σ

⎞

⎟ ⎟ ψ ⎟ ⎟ ⎠ −ρ

(3.6)

Theorem 3.1 If R0 < 1, then healthy class E 1 = (σ, 0, 0) is locally asymptotically stable. Proof At healthy equilibrium, The characteristic polynomial at point E 1 = (σ, 0, 0) is given as   ϕσ  τ + ψ φ + ϑ+σ r        τ + η −ψ  Ψ (τ ) = τ I − A E1  =  0   σ  0 λ + ρ −ρ ϑ+σ = τ 3 + (τ + ρ + η) τ 2 + (φρ + φη + ρη (1 − R0 )) τ + φηρ (1 − R0 ) . For R0 < 1, it implies that all the coefficients of Ψ (τ ) are positive, and (τ + ρ + η) (φρ + φη + ρη (1 − R0 )) > φηρ (1 − R0 ) is satisfied. By using the Routh-Hurwitz criteria, since all the roots of Ψ (τ ) have negative real parts. This implies that the general condition | arg(τ )| > π2 holds, and as a result the equilibrium point E 1 = (σ, 0, 0) is locally asymptotically stable.  Next, the community matrix A at nontrivial point E ∗ = (u ∗ , v ∗ , w ∗ ) is calculated as

⎛

AE ∗

∗ ⎞ − p −q −φ uσ = ⎝ 0 ∗ −η ψ ⎠ ρϑv ρ ψη −ρ (ϑ+u ∗ )2 ∗

∗

ϕϑv η u where p = −φ + σφ v ∗ + 2 σφ u ∗ + σφ w ∗ + (ϑ+u ∗ )2 and q = φ σ + ϕ ψ . In attempt to gain insightful understanding, the following assumption was made on the parameters of the dynamical system

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− φ − 2φ

v∗ φ u∗ αϑv ∗ − φ − w∗ − < 0, σ σ σ (ϑ + u ∗ )2

(3.7)

we denote the discriminant by β and the trace by α as ∗

∗

∗

ϑψv ϑv α = 2 pη + p 2 + pφ uσ (ϑ+u − q (ϑ+u ∗ )2  ∗ )2∗    ∗ u ϑv 2 2 2 φ σ (ϑ+u ∗ )2 + p β = α − 4 p η + pη

Theorem 3.2 Assume that R0 > 1 and inequality in (3.7) is satisfied. (1) If either of β < 0 or β = 0 and α > 0, then E ∗ is locally asymptotically stable for all ρ > 0; (2) If β = 0 and α > 0, then there exists ρ0 > 0, such that E ∗ is locally asymptotically stable for ρ > 0 and ρ = ρ0 ; (3) If β > 0, then the following cases arise: (i) α > 0, then E ∗ is locally asymptotically stable for all ρ > 0; (ii) α < 0, then there exist 0 < ρ1 < ρ2 , such that E ∗ is locally asymptotically stable for ρ < ρ1 or ρ > ρ2 . Proof At interior equilibrium point E ∗ = (u ∗ , v ∗ , w ∗ ) , the characteristic polynomial takes the form  ∗   τ+p q φ uσ     0 τ + η −ψ  Ψ (τ ) = τ I − A(E ∗ )  =    ρϑv ∗ − −ρ βη τ + ρ  (ϑ+u ∗ )2 ∗ = τ 3 + ( p + ρ + η) τ 2 + pρ + pη + φuσ

ρϑv ∗ (ϑ+u ∗ )2



τ + η φuσ

∗

∗ ρϑv ∗ + q ρψϑv∗ 2 . (ϑ+u ∗ )2 (ϑ+u )

(3.8) Since ( 3.7) is satisfied, then for p > 0 and all the coefficients of Ψ (τ ) are positive. By adopting the Routh-Hurwitz Theorem to (3.8), bear in mind that E ∗ = (u ∗ , v ∗ , w ∗ ) is locally asymptotically stable if 

ρψϑv ∗ φu ∗ ρϑv ∗ φu ∗ ρϑv ∗ −q Λ (ρ) = ( p + ρ + η) pη + pρ + −η > 0. 2 2 ∗ ∗ σ (ϑ + u ) σ (ϑ + u ∗ )2 (ϑ + u )

(3.9)

We rewrite (3.9) into the following form 

∗ ϑv ∗ u + p ρ2 + αρ + pη 2 + p 2 η. Λ (ρ) = r σ (ϑ + u ∗ )2 Finally, we arrive at the following conclusion: If β < 0 or β = 0 and α > 0, then Λ (ρ) have no real-root or one negative realroot. In such a situation Λ (ρ) > 0 and E ∗ is locally asymptotically stable for all ρ > 0. If β = 0 and α > 0, then Λ (ρ) have one root ρ0 > 0. Thus, Λ (ρ) > 0 for all ρ > 0 and ρ = ρ0 . Then, E ∗ is locally asymptotically stable for all ρ > 0 and ρ = ρ0 . Now, let us assume that β > 0. Then, if α > 0, Λ (ρ) have two negative real roots, so Λ (ρ) > 0. the interior point E ∗ which correspond to the existence of all cells is locally asymptotically stable. In the case α < 0, Λ (ρ) have two positive real

Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model

13

Fig. 1 Chaotic attractors and time evolution results of different cells

roots 0 < ρ1 < ρ2 such that A (ρ) > 0 for ρ ∈ (0, ρ1 ) ∪ (ρ2 , +∞). Thus we have  that E ∗ is locally asymptotically stable for ρ ∈ (0, ρ1 ) ∪ (ρ2 , +∞). To illustrate the behavior of the dynamical system with respect to stability, the following parameters are used for the numerical simulations φ = 0.65/day, σ = 2 × 106 /ml, ϕ = 11/day, η = 0.3/day, ϑ = 3.7 × 106 /ml, ρ = (0, 1], ψ = 0.847, u 0 = 5 × 105 /ml, v0 = 5 × 102 /ml, w0 = 2 × 102 /ml. (3.10)

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Fig. 2 Chaotic attractors and time evolution results of different cells

With above parameters, it is obvious that R0 = 0.9906 < 1 which satisfies the condition on Theorem 3. The simulation results as shown in Figs. 1 and 2 show that the healthy cells predominate over the infected class. Clearly the point E 1 is asymptotically stable. The coexistence of both healthy and infected cells are also demonstrated. Each of the cells also enters into permanence, which in reality is true. No matter the level of sensitization which individual and government will put in place to wipe out infected or latent cells completely.

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15

4 Numerical Results and Discussions In this segment, we employ the numerical techniques in Sect. 2 to solve the diffusive HIV-I infection model as discussed in Sect. 3, to test the accuracy and efficiency of the proposed method. The experimental results are given in one and two spatial dimensions. All computations are carried out using the Matlab 2019a software package.

4.1 One Dimensional Case Consider the time-dependent reaction-diffusion HIV-I model (3.4), where u(x, t), v(x, t) and w(x, t) represent the species densities at time t and position x. The diffusion coefficients Du , Dv and Dw describe the intensity of spatial interaction due to self-motion of the cells within a given habitat or domain. In the computation, we experiment using Du = 0.07, Dv = 0.05, Dv = 0.02 and the homogenous (zeroflux) boundary conditions subject to non-negative random initial condition taken in the form

Fig. 3 One dimensional distribution of species with ϕ = 4 for t = 10

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Fig. 4 One dimensional distribution of species with ϕ = 4 for t = 10

u 0 = 0.07(ones(N , 1)),

v0 = 0.01(ones(N , 1)),

w0 = 0.02(ones(N , 1)), (4.1) where (u ∗ , v ∗ , w ∗ ) = (0.07, 0.01, 0.02) are the small perturbations. The effect of ϕ which symbolizes the maximum rate of infection is shown in Figs. 3 and 4 which correspond to ϕ = 4 and ϕ = 9, respectively at t = 10. It is obvious that all cells coexist. With increasing time (t), and varying value of ρ as indicated in the caption, we obtain different spatiotemporal distribution of dynamic system as displayed in Fig. 5.

4.2 Two Dimensional Case It is in higher dimensions that the numerical idea presented in Sect. 2 really become of serious value. We choose to illustrate the numerical technique using the result of nontrivial example which guarantee the existence of all cells as stated in Theorem 3. Here, we assume that about 30% of the infected class survive incubation which when calculated results to the value ϕ = 3.31, so that p = 0.0816, α = −0.0938 < 0, R0 = 3.8596, β = 0.00515 > 0. The second condition in Theorem 3 holds. Then the point E ∗ is locally asymptotically stable. Both infected and healthy cells do

Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model

17

Fig. 5 one dimensional evolution of (3.4) for t = 50. Columns 1–3 correspond to ρ = 0.6, 0.8, 1.0, respectively

coexist. For the simulation experiments in all 2D cases, the time step t = 0.05, the space step h is kept to 1/4, that is x =  y = 0.25, and the homogeneous (zero-flux) Neumann boundary conditions on equal domain size (L). The 2D experiments are repeated with different initial conditions and some perturbed parameters. In Figs. 6 and 7, we experiment with the initial condition set as u 0 = u ∗ − 0.5(sin(π(x − μ)/(2μ)))m (sin(π(y − μ)/(2μ)))m , v0 = v ∗ (sin(π(x − μ)/(2μ)))m (sin(π(y − μ)/(2μ)))m , w0 = w ∗ − 0.25(sin(π(x − ν)/(2μ)))m (sin(π(y − μ)/(2μ)))m

(4.2)

to mimic spatiotemporal evolution of the dynamic model with contour and surface plots, where (u ∗ , v ∗ , w ∗ ) = (1, 0.25, 3), m = 0.8 and μ = 0.5.

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Fig. 6 Two dimensional results for system (3.4) with initial condition (4.2) and ϕ = 5

Fig. 7 Two dimensional results for system (3.4) with initial condition (4.2) and ϕ = 3.5

Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model

19

Fig. 8 Two dimensional chaotic evolution of HIV-I system with random initial data (4.3) and ρ = 0.3. Simulation runs for t = 200

During the experiment, it was observed that variation of the initial conditions could also induce some strange and chaotic behavior of the system under consideration. To achieve this, we allow the initial condition to evolve naturally by using the computer random points computed as u 0 = randn(N , N )0.7, v0 = randn(N , N )0.5, w0 = randn(N , N )0.2,

(4.3)

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Fig. 9 Two dimensional chaotic evolution of HIV-I system with random initial data (4.3) and ρ = 0.6. Simulation runs for t = 200

with N = 200. This experiment is repeated for ρ = 0.3, 0.6, 1.0, to obtain the chaotic (spiral) results in Figs. 8, 9 and 10, respectively. With the experimental results in 2D, one could conclude that it is possible to obtain chaotic patterns in epidemiological models if such dynamics are extended to diffusive cases.

Exponential-Time-Differencing Method for the Solution of Diffusive HIV-I Model

21

Fig. 10 Two dimensional chaotic evolution of HIV-I system with random initial data (4.3) and ρ = 1.0. Simulation runs for t = 200

5 Conclusion This work is mainly concerned with the solution of the time-dependent HIV-I infection model in a diffusive form. To overcome the stiffness issue inherent with the diffusion term in the model, a Fourier spectral method is introduced for the spatial discretization, and the resulting system of ODEs is advanced in time with the modified ETDRK4 scheme. To explore the dynamic richness of the HIV-I model, numerical experiments are conducted in one and two dimensions to address the points and

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queries that may naturally occur. It was also observed that the model under certain parameter values and the choice of initial conditions can give rise to the evolution of chaotic pattern formation which has applications in applied physics and engineering. Extension of the numerical techniques to non-integer order models in sciences and technology is left for future research.

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An Effective Technique for Solving a Model Describing Biological Species Living Together Saurabh Tomar and Soniya Dhama

Abstract In this work, a reliable scheme is suggested to obtain an analytical approximation to the solution of a system of nonlinear integro-differential equations that arises in biology. The dynamics of the populations of two distinct species are characterized by this system. The suggested strategy involves two steps. Construct an integral operator first, and then apply Halpern’s iteration scheme to the constructed operator to develop the required iterative approach. This approach’s convergence is also addressed. Numerous numerical examples are taken into consideration to demonstrate the effectiveness of the strategy. The proposed approach has several key advantages over existing approaches, including the ability to handle nonlinearity without the use of Adomian polynomials, the ability to solve the problem without the use of the Lagrange multiplier or constrained variations, and the ability to take into account both interval endpoints. In contrast to existing semi-analytical methods, the suggested approach addresses the problems without requiring linearization, discretization, or perturbation assumptions. Three test problems are taken into account to confirm the method’s efficacy. Keywords Mathematical biology · Integro-differential equation · The equilibrium state of two species living together · Fixed point method

S. Tomar (B) Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, UP 208016, India e-mail: [email protected] S. Dhama Department of Mathematical Sciences, Rajiv Gandhi Institute of Petroleum Technology, Jais Amethi 229304, UP, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_2

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S. Tomar and S. Dhama

1 Introduction Differential equations, integral and delay integral equations, and their systems are used to simulate many problems in engineering and research [1]. In general, functional equations, such as integral and integro-differential equations, partial differential equations, stochastic equations, and others, are frequently the result of mathematical modelling of real-life problems[2–11]. Integro-differential equation has long been considered one of the most important tools in practical mathematics. Integral equations are found in a variety of scientific and technical domains. Integro-differential equations appear in many mathematical descriptions of physical processes. Fluid dynamics, biological models, and chemical kinetics are all examples of this [12–29]. Integro-differential equations appear in a variety of physical situations [30, 31]. As a result, computing the solutions to these equations is critical. This study is concerned with the following system of nonlinear delay integrodifferential equations, which are associated with the dynamics of two interacting species [32, 33]:    t ds1 = s1 (t) κ1 − λ1 s2 (t) − F1 (t − τ )s2 (τ )dτ + G 1 (t), κ1 , λ1 > 0, 0 ≤ t ≤ T, dt t−T0    t ds2 F2 (t − τ )s1 (τ )dτ + G 2 (t), κ2 , λ2 > 0, 0 ≤ t ≤ T, = s2 (t) −κ2 + λ2 s1 (t) + dt t−T0

(1.1) subject to s1 (0) = a1 , s2 (0) = a2 , where G 1 , G 2 , F1 and F2 are known functions while s1 and s2 are unknown. Here, s1 and s2 are two different species’ numbers at time t, respectively, with the profile being that the first species expands while the second contracts. If they are together and we suppose that the second species consumes the first, then the second species’ rate will increase at a rate of dsdt2 , which depends on both the current populations of the second species and all previous values of the first species. The replacement of a steady state condition between these two species is illustrated by the following system    t ds1 = s1 (t) κ1 − λ1 s2 (t) − F1 (t − τ )s2 (τ )dτ , dt t−T0    t ds2 = s2 (t) −κ2 + λ2 s1 (t) + F2 (t − τ )s1 (τ )dτ , dt t−T0

(1.2)

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where κ1 and −κ2 represent the first and second species’ rising and decreasing coefficients, respectively. Here, T0 is considered as the finite heredity duration for considered species s1 and s2 . Note that the above system (1.2) is a special case of (1.1). The evaluation of exact solutions for these problems is not an easy task, and to be more precise, the exact solutions are available only in a few special cases. Therefore, the evaluation of the solutions using numerical or semi-analytical schemes is critical. This prompted the development of numerical or semi-analytical techniques for finding solutions to problems. Over the past, some numerical as well as semianalytical techniques have been suggested to tackle the integro-differential equations. For example, in [34], Babolian and Biazar used the Adomian decomposition method to solve the problem. In [8], Shakeri and Dehghan applied the variational iteration method to the problem and suggested a numerical scheme which is based on the pseudospectral Legendre method to tackle this model. In [35], Yousefi developed a numerical scheme to obtain the approximate solutions of this biological model based on Legendre Multiwavelets. In [36], Sahu and Ray introduced the various schemes to handle this system of integro-differential equations based on Legendre spectral collocation. In [37], Shakourifar and Dehghan analysis the numerical issues of a piecewise polynomial collocation scheme for this system. In [38], Tari used the differential transform approach to solve this model. In [16], Yüzba¸sı and Sezer proposed a numerical methodology based on exponential polynomials to solve this model. In [39], Ramezani et al. suggested a complex spline collocation approach to handle such type of model. In [40], Hafez et al. have obtained the numerical solution to this model via Bernoulli collocation scheme. In [12], Sekar and Murugesan have suggested a single term Walsh series approach to solve this model. In [41], Thirumalai at el. have presented a spectral collocation methodology to solve this model. Fixed-point iterative methods have been studied for decades in order to better understand many applications in applied mathematics, and numerous fixed-point iterative strategies have been utilized to tackle a wide range of problems [42–52]. In this work, an efficient iterative methodology based on the Halpern fixed point approach is designed to solve the considered biological model. In the construction of the methodology, an equivalent integral operator form of the model is obtained, and then Halpern’s approach is combined with this integral representation. As a result, the methodology involves an unknown parameter that greatly accelerates the convergence of the obtained solutions. A method based on residual minimization is also provided to assess the unidentified parameter. The significance of this unknown parameter in the scheme is that it greatly accelerates the convergence of the obtained solutions. In the case of problems with strong nonlinearity or challenging integrable functions, the integration becomes very challenging to evaluate for further iterations or impossible to proceed with further iterations. To remedy this, Taylor’s series is utilized to simplify the iterative process and to improve the computational efficacy of the methodology. The convergence analysis of the methodology is also discussed. To verify the method’s effectiveness, some numerical examples are considered.

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This chapter is organized as follows. In Sect. 2, the proposed methodology to solve (1.1) is presented. In Sect. 3, numerical simulations are given and, the conclusion is summarized in Sect. 4.

2 Development of the Methodology In this section, we formulate a methodology by constructing different integral operators to obtain the approximate analytical solutions for the model (1.1). Let us consider the model    t ds1 = s1 (t) κ1 − λ1 s2 (t) − F1 (t − τ )s2 (τ )dτ + G 1 (t), dt t−T0    t ds2 = s2 (t) −κ2 + λ2 s1 (t) + F2 (t − τ )s1 (τ )dτ + G 2 (t). (2.1) dt t−T0

2.1 Type-I Re-writing (2.1) in the following manner    t ds1 − s1 (t) κ1 − λ1 s2 (t) − 1 = F1 (t − τ )s2 (τ )dτ − G 1 (t) = 0, dt t−T0    t ds2 − s2 (t) −κ2 + λ2 s1 (t) + F2 (t − τ )s1 (τ )dτ − G 2 (t) = 0, 2 = dt t−T0 (2.2) Now adding and subtracting dsdt1 and dsdt2 in the system (2.2), respectively as follows ds1 + 1 − dt ds2 + 2 − dt

ds1 = 0, dt ds2 = 0, dt

(2.3)

and re-write (2.3) as ds1 = −1 + dt ds2 = −2 + dt

ds1 , dt ds2 . dt

(2.4)

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Now, let us construct an equivalent integral form of the system (2.4) by integrating (2.4) from 0 to t as  t

 ds1 (u) − 1 (u) du, s1 (t) = s1 (0) + du 0   t ds2 (u) s2 (t) = s2 (0) + − 2 (u) du. du 0

(2.5)

Now using the following identities  t

 ds1 (u) du = s1 (t) − s1 (0), du 0   t ds2 (u) du = s2 (t) − s2 (0), du 0

(2.6)

the system (2.5) resulting in the following system 

t

s1 (t) = s1 (t) −

(1 (u)) du,

0



t

s2 (t) = s2 (t) −

(2 (u)) du.

(2.7)

0

Now using (2.2) and (2.7), we get  

 u ds1 − s1 (u) κ1 − λ1 s2 (u) − F1 (u − τ )s2 (τ )dτ − G 1 (u) du, du 0 u−T0  

 t  u ds2 − s2 (u) −κ2 + λ2 s1 (u) + F2 (u − τ )s1 (τ )dτ − G 2 (u) du. s2 (t) = s2 (t) − du 0 u−T0 s1 (t) = s1 (t) −

 t

(2.8) Now re-write (2.8) in the operator form s1 (t) = T1 [s1 , s2 ], s2 (t) = T2 [s1 , s2 ],

(2.9)

where T1 [s1 , s2 ] = s1 (t) − T2 [s1 , s2 ] = s2 (t) −

 t 0

 t 0

 

 u ds1 − s1 (u) κ1 − λ1 s2 (u) − F1 (u − τ )s2 (τ )dτ − G 1 (u) du, du u−T0  

 u ds2 − s2 (u) −κ2 + λ2 s1 (u) + F2 (u − τ )s1 (τ )dτ − G 2 (u) du. du u−T0

(2.10)

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Now, in view of (2.10), the Halpern fixed-point scheme will be implemented, and the Halpern’s approach [42–45] for system of equations is given as s1,n+1 (t) = αn v + (1 − αn )T1 [s1,n , s2,n ], v = s1,0 , s2,n+1 (t) = αn v + (1 − αn )T2 [s1,n , s2,n ], v = s2,0 , n ≥ 0,

(2.11)

From the combination of (2.10) and (2.11), we get the following iterative approach   t ds

1,n − s1,n (u) κ1 − λ1 s2,n (u) s1,n+1 (t) =αn s1,0 + (1 − αn )s1,n (t) − (1 − αn ) du 0  u    − F1 (u − τ )s2,n (τ )dτ − G 1 (u) du , n ≥ 0, (2.12) u−T0

  t ds

2,n s2,n+1 (t) =αn s2,0 + (1 − αn )s2,n (t) − (1 − αn ) − s2,n (u) κ2 − λ2 s1,n (u) du 0  u    − F2 (u − τ )s1,n (τ )dτ − G 2 (u) du , n ≥ 0. (2.13) u−T0

Begin with the initial guess s1,0 = a1 and s2,0 = a2 , which satisfies the given initial conditions for the system, and for the aforementioned problem, analytical iterative approximations can be obtained. Now, by establishing the residual norm, we outline the process for evaluating αn R1 (αn ) ≈

N 2 1  R1,r (t j ; αn ) , N i=1

(2.14)

R2 (αn ) ≈

N 2 1  R2,r (t j ; αn ) , N i=1

(2.15)

where ti , 1 ≤ i ≤ N are the discrete grids of [0, T ] and minimizing R1 (αn ) and R2 (αn ) for αn to evaluate αn , where R1,r (t; αn ) and R2,r (t; αn ) are the residual errors defined by R1,r (t; αn ) ≡

R2,r (t; αn ) ≡

   t ds1,n (t) − s1,n (t) κ1 − λ1 s2,n (t) − F1 (t − τ )s2,n (τ )dτ − G 1 (t), dt t−T0    t ds2,n (t) − s2,n (t) −κ2 + λ2 s1,n (t) + F2 (t − τ )s1,n (τ )dτ − G 2 (t). dt t−T0

In cases of strong nonlinearity or complicated integrands, the iterative process of the scheme requires more computational work and is challenging to compute further iterations. To minimize computational work, Taylor’s series is used to simplify the integrand in the iterative process.

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2.2 Type-II Now, let us construct a different integral operator to formulate the methodology. First, re-writing (2.1) in the following manner    t ds1 F1 (t − τ )s2 (τ )dτ − G 1 (t) = 0, 1 = − s1 (t)κ1 + s1 (t) −λ1 s2 (t) − dt t−T0    t ds2 + s2 (t)κ2 − s2 (t) λ2 s1 (t) + F2 (t − τ )s1 (τ )dτ − G 2 (t) = 0, 2 = dt t−T0 (2.16) Now adding and subtracting respectively as follows

ds1 dt

− s1 (t)κ1 and

ds2 dt

+ s2 (t)κ2 in system (2.16),

  ds1 ds1 − s1 (t)κ1 + 1 − − s1 (t)κ1 = 0, dt dt   ds2 ds2 + s2 (t)κ2 + 2 − + s2 (t)κ2 = 0, dt dt

(2.17)

and re-write (2.17) as ds1 − s1 (t)κ1 = −1 + dt ds2 + s2 (t)κ2 = −2 + dt

ds1 − s1 (t)κ1 , dt ds2 + s2 (t)κ2 . dt

(2.18)

Now, let us construct an equivalent integral form of the system (2.18) and for that treating (2.18) by means of the method of variation of parameters as 

 ds1 (u) s1 (t) = s1 (0) + − 1 (u) du, e du 0    t ds2 (u) − 2 (u) du. eκ2 (u−t) s2 (t) = s2 (0) + du 0 

t

−κ1 (u−t)

(2.19)

Now using the integrated by parts, (2.19) resulting in the following system 

t

s1 (t) = s1 (t) −

e−κ1 (u−t) (1 (u)) du,

0



t

s2 (t) = s2 (t) − 0

eκ2 (u−t) (2 (u)) du.

(2.20)

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In view of (2.16) and (2.20), we get

 

 u ds1 − s1 (u) κ1 − λ1 s2 (u) − F1 (u − τ )s2 (τ )dτ − G 1 (u) du, du 0 u−T0  

 t  u ds2 − s2 (u) −κ2 + λ2 s1 (u) + eκ2 (u−t) F2 (u − τ )s1 (τ )dτ − G 2 (u) du. s2 = s2 − du 0 u−T0 s1 = s1 −

 t

e−κ1 (u−t)

(2.21) Re-write (2.21) in the operator form s1 (t) = T1 [s1 , s2 ], s2 (t) = T2 [s1 , s2 ],

(2.22)

where  t ds

1 e−κ1 (u−t) T1 [s1 , s2 ] = s1 (t) − − s1 (u) κ1 − λ1 s2 (u) du 0  u   − F1 (u − τ )s2 (τ )dτ − G 1 (u) du, u−T0

 t ds

2 − s2 (u) − κ2 + λ2 s1 (u) eκ2 (u−t) T2 [s1 , s2 ] = s2 (t) − du 0  u   + F2 (u − τ )s1 (τ )dτ − G 2 (u) du. u−T0

Now, in view of (2.22) and the Halpern fixed-point scheme will be implemented, and from the combination of (2.22) and (2.11), we get the following iterative approach  t

ds 1,n s1,n+1 (t) =αn s1,0 + (1 − αn )s1,n (t) − (1 − αn ) e−κ1 (u−t) − s1,n (u) κ1 − λ1 s2,n (u) du 0  u    F1 (u − τ )s2,n (τ )dτ − G 1 (u) du , n ≥ 0, (2.23) − u−T0

 t

ds 2,n s2,n+1 (t) =αn s2,0 + (1 − αn )s2,n (t) − (1 − αn ) eκ2 (u−t) − s2,n (u) κ2 − λ2 s1,n (u) du 0  u    F2 (u − τ )s1,n (τ )dτ − G 2 (u) du , n ≥ 0. (2.24) − u−T0

Begin with the initial guess s1,0 = a1 and s2,0 = a2 , which satisfies the given initial conditions for the system, and for the aforementioned problem, analytical iterative approximations can be obtained. The evaluation of αn can be achieved via the residual norm (2.14) and (2.15).

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In cases of strong nonlinearity or complicated integrands, the iterative process of the scheme requires more computational work and is difficult to compute further iterates. To minimize computational work, multivariate Taylor’s series is used to simplify the integrand in the iterative process (2.23) and (2.24) as follows s1,n+1 (t) =αn s1,0 + (1 − αn )s1,n (t) − (1 − αn )



t

 Pr (t, u)du ,

(2.25)

 Q r (t, u)du ,

(2.26)

0

s2,n+1 (t) =αn s2,0 + (1 − αn )s2,n (t) − (1 − αn )



t 0

where

e−κ1 (u−t)

ds

1,n

du



− s1,n (u) κ1 − λ1 s2,n (u) −

u

  F1 (u − τ )s2,n (τ )dτ − G 1 (u)

u−T0

= Pr (t, u) + O(t r +1 ) + O(u r +1 ), and e

κ2 (u−t)

ds

2,n

du



− s2,n (u) κ2 − λ2 s1,n (u) −

u

  F2 (u − τ )s1,n (τ )dτ − G 2 (u)

u−T0

= Q r (t, u) + O(t r +1 ) + O(u r +1 ).

3 Implementation and Numerical Results This section presents the numerical simulation of the proposed methodology. To verify the applicability and efficacy of our approach, we consider some examples. Example 3.1 First, consider the following system [8]    t ds1 1 = s1 (t) 1 − s2 (t) − s2 (τ )dτ + G 1 (t), dt 3 t− 21    t ds2 = s2 (t) −2 + s1 (t) + (t − τ − 1)s1 (τ )dτ + G 1 (t), dt t− 21 where

23 5 49 17 G 1 (t) = − t 3 + t 2 + t − , 2 12 12 6 G 2 (t) =

15 3 1 2 3 t − t − t − 1, 8 4 8

(3.1)

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with initial data s1 (0) = 1 and s2 (0) = 0. The true solution is s1 (t) = 1 − 3t and s2 (t) = −t + t 2 . The proposed methodology type-I formula for (3.1) is   t ds

1 1,n s1,n+1 (t) =αn s1,0 + (1 − αn )s1,n (t) − (1 − αn ) − s1,n (u) 1 − s2 (u) du 3 0  u    − s2,n (τ )dτ − G 1 (u) du , n ≥ 0, (3.2) u−1/2

  t ds

2,n s2,n+1 (t) =αn s2,0 + (1 − αn )s2,n (t) − (1 − αn ) − s2,n (u) − 2 + s1,n (u) du 0  u    − (u − τ − 1)s1,n (τ )dτ − G 2 (u) du , n ≥ 0. (3.3) u−1/2

By selecting the initial approximation s1,0 (0) = 1 and s2,0 (0) = 0, we obtain the following results s1,1 (t) =1 − 2.833333333t − 0.6250000000t 4 + 1.361111111t 3 + 0.7083333333t 2 + 2.833333α0 t + 0.6250000000α0 t 4 − 1.361111111α0 t 3 − 0.7083333333α0 t 2 ,

s2,1 (t) = − t + 0.4687500000t 4 − 0.08333333333t 3 + 0.1875000000t 2 + α0 t − 0.4687500000α0 t 4 + 0.08333333333α0 t 3 − 0.1875000000α0 t 2 , and the corresponding values of α0 for s1,1 (t) and s2,1 (t) are identified via (2.14) and (2.15) with N = 10, which are 0.212566491334 and 0.5159049808, respectively. After substituting these values in the above approximations yield s1,1 (t) =1.0 − 2.23106160762256t − 0.492145942915818t 4 + 1.07178449781807t 3 + 0.557765401945012t 2 + 0.08333333333α0 t 3 − 0.1875000000α0 t 2 , s2,1 (t) = − 0.484095019159353t + 0.226919540230947t 4 − 0.0403412515949991t 3 + 0.0907678160923787t 2 .

In a similar way, we get s1,2 (t) =1 − 2.61929358205884t − 0.174965099041535t 4 + 1.04272781454564t 3 + 0.00934064327203696t 9 − 0.0310576388745547t 8 + 0.0160985245420873t 7 + 0.0190709989334935t 6

− 0.105091677912895t 5 − 0.0998384249658t 2 ,

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s2,2 (t) = − 0.805746743313668t + 0.292932489183872t 4 + 0.0377332701136180t 3 − 0.00499910038799772t 9 + 0.00762362808881598t 8 + 0.0170794661567121t 7 − 0.0195895303212047t 6

− 0.0686099838784783t 5 + 0.498677762195t 2 ,

and s1,3 (t) =1 − 2.92923543071108t + 0.00000202932247142605t 19 − 0.0000132807881680772t 18 + 0.0000260745045911674t 17 + 4.1633476063582510−7 t 16 − 0.0000519695905629862t 15 − 0.000205884840310831t 14 + 0.00128990424794279t 13 − 0.00223452861649316t 12 + 0.000363556855104520t 11 + 0.00510059589298002t 10 + 0.00369794905646681t 9 − 0.0333932807050378t 8 + 0.0141485215915262t 7 − 0.000803309330564085t 6 + 0.0229932835874042t 5 − 0.0659363332384586t 4 + 0.458240373533745t 3 − 0.0343263814770440t 2 , s2,3 (t) = − 0.956632866055380t − 0.00000117552087300173t 19 + 0.00000322615172814328t 18 + 0.0000127488126725305t 17 − 0.0000490257239538026t 16 + 0.00000826857412053931t 15 + 0.000253183655953299t 14 − 0.000374437492100575t 13 − 0.00100124698471227t 12 + 0.00351654888558956t 11 − 0.00313378894737590t 10 − 0.00606926377948803t 9 + 0.000606410328391509t 8 + 0.0358975526509382t 7 + 0.00888009012000952t 6 − 0.149865172735764t 5 + 0.185460620094t 4 + 0.011425647744927t 3 + 0.875460413974t 2 .

The achieved solutions along with the true solutions are represented in Figs. 1, 2, 3 and 4. From figures, it is clear that as the obtained solutions via the proposed methodology type-I is really well in line with the precise solutions as iterations progress. Now, let us implement the proposed methodology type-II formula for (3.1) which is given by ds  t

1 1,n − s1,n (u) 1 − s2 (u) s1,n+1 (t) =αn s1,0 + (1 − αn )s1,n (t) − (1 − αn ) e−(u−t) du 3 0  u    − s2,n (τ )dτ − G 1 (u) du , n ≥ 0, (3.4) u−1/2

ds  t

2,n s2,n+1 (t) =αn s2,0 + (1 − αn )s2,n (t) − (1 − αn ) e2(u−t) − s2,n (u) − 2 + s1,n (u) du 0  u    (u − τ − 1)s1,n (τ )dτ − G 2 (u) du , n ≥ 0. (3.5) − u−1/2

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(a) s1,2 (t) with exact

(b) s2,2 (t) with exact

Fig. 1 Graph of exact and approximate solutions for Example 3.1 for n = 1 (Type-I)

(a) s1,3 (t) with exact

(b) s2,3 (t) with exact

Fig. 2 Example 3.1 for n = 2 (Type-I), graph of exact and approximate solutions

(a) s1,4 (t) with exact

(b) s2,4 (t) with exact

Fig. 3 Example 3.1 for n = 3 (Type-I), graph of exact and approximate solutions

An Effective Technique for Solving a Model Describing Biological …

(a) s1,6 (t) with exact

37

(b) s2,6 (t) with exact

Fig. 4 Example 3.1 for n = 5 (Type-I), graph of exact and approximate solutions

By selecting the initial approximation s1,0 (0) = 1 and s2,0 (0) = 0, we obtain the following results s1,1 (t) =1.0 − 2.87547562844537t − 6.4029772411110610−12 t 15 − 9.6025091459746710−11 t 14 − 1.3445503821511210−9 t 13 − 1.7479498591924610−8 t 12 − 2.0975272192356710−7 t 11 − 0.00000230729129406353t 10 −

−0.000207656472353199t 8 − 0.00166125200690230t 7

− 0.0116287618645051t 6 − 0.0697725703916233t 5 − 0.348862852182235t 4 + 1.14173297037919t 3 − 0.718868906857623t 2 − 4.00151034981323 + ... s2,1 (t) = − 0.927879703429945t − 3.1606508862982310−8 t 15 + 2.3706248001542110−7 t 14 − 0.00000165935349750096t 13 + 0.0000107858071018801t 12 − 0.0000647148675836570t 11 + 0.000355932674395691t 10

− 00320339421135040t 7 + 0112118797587480t 6

− 0336356392361385t + 0840890981449893t 4 − 0811894740454808t 3 5

+ 110185714782306t 2 + 39513081604731010−9 t 16 + ...

and the corresponding values of α0 for s1,1 (t) and s2,1 (t) are identified via (2.14) and (2.15) with N = 10 which are −0.01487375133 and 0.072120296570, respectively. The obtained approximate solutions along with the exact solutions are represented in Figs. 5, 6, and 7. From figures, it is clear that as the obtained solutions via the proposed methodology type-II is really well in line with the precise solutions as iterations progress. As it is evident from figures that methodology type-II provides better results compare to methodology type-I with less number of iterations. The dynamics of the populations of two distinct species are depicted in Fig. 7, which demonstrates that as time increases, the population of the first species decays gradually throughout the domain, whereas the population of the second species

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(a) s1,2 (t) with exact

(b) s2,2 (t) with exact

Fig. 5 Example 3.1 for n = 1 (Type-II), graph of exact and approximate solutions

(a) s1,3 (t) with exact

(b) s2,3 (t) with exact

Fig. 6 Example 3.1 for n = 2 (Type-II), graph of exact and approximate solutions

decays gradually upto t = 0.5 and grows gradually. Also, the absolute errors are represented in Tables 1, 2, 3 and 4 to show the accuracy of the methodology type-I and type-II. Example 3.2 Next, consider the following system [8]    t ds1 (2(t − τ ) − 3)s2 (τ )dτ + G 1 (t), = s1 (t) 2 − s2 (t) − dt t− 13    t ds2 = s2 (t) −2 + s1 (t) + (t − τ )s1 (τ )dτ + G 1 (t), dt t− 13

(3.6)

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(b) s2,4 (t) with exact

(a) s1,4 (t) with exact

Fig. 7 Example 3.1 for n = 3 (Type-II), graph of exact and approximate solutions Table 1 Absolute errors for Example 3.1 of type-I t |s1,3 (t) − s1 (t)| |s1,4 (t) − s1 (t)| 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

where

0.0 0.00718507006898716 0.0163476879219852 0.0300346962119031 0.0506879676833916 0.0806577754170309 0.122187549122741 0.177350976276551 0.247924969052615 0.335189924569745 0.439662305629278

0.0 0.000891279284103597 0.00216035479162602 0.00454282171553791 0.00893980179413609 0.0164263431020639 0.0282479343832137 0.0457845059745663 0.0704658826323428 0.103630472896985 0.146329608018994

|s1,8 (t) − s1 (t)| 0.0 0.000889494472576802 0.00112653138860758 0.00115465578042079 0.00111615352292521 0.00100324038881405 0.000754709480734972 0.000308203207529667 0.000384502793038699 0.00137329894734139 0.00275032875682268

  7 −t 13 1 −t 22 1 −t −t 3 3 − 2t, + e G 1 (t) = t 2 − 3te − e + te 2 6 9 2

G 2 (t) =

 1 −t  e 342t 3 − 8t 2 + 325t + 324 , 648

with initial data s1 (0) = 0 and s2 (0) = 0. The true solution is s1 (t) = −t 2 and s2 (t) = 1 −t te . 2

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Table 2 Absolute errors for Example 3.1 of type-I t |s2,3 (t) − s2 (t)| |s2,4 (t) − s2 (t)| 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.003119803062820975 0.004033054674715708 0.00326236043635339 0.00145872782526993 0.000707024021335934 0.00260428631965243 0.00365090520793163 0.00327501981361736 0.000846028496138101 0.00439773375375940

0.0 0.000931011619598313 0.000698492761385477 0.0000859071352009855 0.000899328263183508 0.00130478506802512 0.000991188466249460 0.000149542530616359 0.00192046348971511 0.00371004855083354 0.00439574478085381

Table 3 Absolute errors for Example 3.1 of type-II t |s1,2 (t) − s1 (t)| |s1,3 (t) − s1 (t)| 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000891633770617073 0.00130775258405169 0.00157177926884652 0.00168513820878941 0.00159314894012341 0.00136337912438966 0.00128079560935612 0.00186567285102290 0.00381236263717399 0.00783021374842985

0.0 0.000600464801671752 0.000964971366172018 0.00125164153468421 0.00149510935742414 0.00166281063303275 0.00170516304411017 0.00158368883206395 0.00126767490137714 0.000696185124954063 0.000289768042434080

Table 4 Absolute errors for Example 3.1 of type-II t |s2,2 (t) − s2 (t)| |s2,3 (t) − s2 (t)| 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000593366270215279 0.00211686352074594 0.00369578102524581 0.00479926918788698 0.00526171571140666 0.00524668917517690 0.00517653869971149 0.00563692347044462 0.00725087080170486 0.0105017080812198

0.0 0.000541767646333416 0.000337971205008752 0.00154060099911887 0.00240675089359949 0.00263220868468192 0.00219593535811591 0.00129061657382865 0.000251270711093654 0.000516518877662742 0.000598803993307098

|s2,8 (t) − s2 (t)| 0.0 0.00107219324549537 0.000304756199101336 0.00105146411550314 0.00219590295873545 0.00270368146535593 0.00245228524538504 0.00154968498192945 0.000264227288312852 0.00104453386956990 0.00199883823548036

|s1,4 (t) − s1 (t)| 0.0 0.000798065720892693 0.000983812315787647 0.000987091330415968 0.000957606830722130 0.000915304446242882 0.000843768303506631 0.000739385128795567 0.000626173117829198 0.000544500688943739 0.000522043528108362

|s2,4 (t) − s2 (t)| 0.0 0.000396238727569748 0.000604898093448963 0.00193566961903682 0.00293405391055046 0.00329165641498311 0.00299218410610402 0.00224806226289570 0.00143951209357457 0.00106003706498801 00.00167292149102660

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41

The proposed methodology type-I formula for (3.6) is   t ds

1,n − s1,n (u) 2 − s2 (u) s1,n+1 (t) =αn s1,0 + (1 − αn )s1,n (t) − (1 − αn ) du 0  u    − 2(u − τ ) − 3)s2,n (τ )dτ − G 1 (u) du , n ≥ 0, (3.7) u−1/3

  t ds

2,n − s2,n (u) − 2 + s1,n (u) s2,n+1 (t) =αn s2,0 + (1 − αn )s2,n (t) − (1 − αn ) du 0  u    − (u − τ )s1,n (τ )dτ − G 2 (u) du , n ≥ 0. (3.8) u−1/3

By selecting the initial approximation s1,0 (0) = 0 and s2,0 (0) = 0, we obtain the following results s1,1 (t) = − 0.268431862536286t 2 + 0.171035570038697t 3 + 0.00753822992804326t 4 − 0.00365488244947080t 5 + 0.00119290910790607t 6 − 0.000293694245766914t 7 + 0.0000580588950322939t 8

− 0.00000958833992998826t 9 + 0.00000135968942726075t 10

− 1.6893094679747210−7 t 11 + 1.8674713141384310−8 t 12 − 1.8593996319187410−9 t 13 + 1.6840736977851610−10 t 14

− 1.3988522581440510−11 t 15 + 1.0729827747934810−12 t 16 ,

s2,1 (t) =0.220389070542336t + 0.000340106590343112t 2 − 0.0387721512991148t 3 + 0.0779694358361584t 4 − 0.0526031526397346t 5 + 0.0207663415620332t 6 − 0.00579071958938951t 7

+ 0.00125021225647604t 8 − 0.000220506938170207t 9

+ 0.0000329056500230278t 10 − 0.00000425825093366597t 11 + 4.8667173422000310−7 t 12 − 4.9819359161595810−8 t 13

+ 4.6192748435408510−9 t 14

− 3.9148526150224210−10 t 15 + 3.0556453036649410−11 t 16 ,

and the corresponding values of α0 for s1,1 (t) and s2,1 (t) are identified via (2.14) and (2.15) with N = 10 which are 0.7315681374637 and 0.55922185891, respectively. The achieved solutions along with the true solutions are represented in Figs. 8, 9, 10 and 11. From figures, it is clear that as the obtained solutions via the proposed methodology type-I is really well in line with the precise solutions as iterations progress. However, this approach requires more iterations to achieve a good agreement with the exact solutions. Now, let us implement the proposed methodology type-II formula for (3.6) which is given by ds  t

1,n − s1,n (u) 2 − s2 (u) s1,n+1 (t) =αn s1,0 + (1 − αn )s1,n (t) − (1 − αn ) e−2(u−t) du 0  u    − 2(u − τ ) − 3)s2,n (τ )dτ − G 1 (u) du , (3.9) u−1/3

42

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(a) s1,2 (t) with exact

(b) s2,2 (t) with exact

Fig. 8 Example 3.2 for n = 1 (Type-I), graph of exact and approximate solutions

(a) s1,3 (t) with exact

(b) s2,3 (t) with exact

Fig. 9 Example 3.2 for n = 2 (Type-I), graph of exact and approximate solutions

(a) s1,4 (t) with exact

(b) s2,4 (t) with exact

Fig. 10 Example 3.2 for n = 5 (Type-I), graph of exact and approximate solutions

An Effective Technique for Solving a Model Describing Biological …

43

(b) s2,6 (t) with exact

(a) s1,6 (t) with exact

Fig. 11 Example 3.2 for n = 7 (Type-I), graph of exact and approximate solutions ds  t

2,n s2,n+1 (t) =αn s2,0 + (1 − αn )s2,n (t) − (1 − αn ) e2(u−t) − s2,n (u) − 2 + s1,n (u) du 0  u    (u − τ )s1,n (τ )dτ − G 2 (u) du . (3.10) − u−1/3

By selecting the initial approximation s1,0 (0) = 0 and s2,0 (0) = 0, we obtain the following results 1

1

s1,1 (t) =0.785569668081790e 3 t 3 + 0.325832873693015e 3 t 4 1

1

− 0.0517761826690271e 3 t 5 + 0.0913522303413294e 3 t 6 1 3

1

− 0.00960707494515600e t 7 + 0.00602301960748580e 3 t 8 1

1

− 0.000232029280570617e 3 t 9 + 0.000196959206102791e 3 t 10 1

+ 0.00000344581998910735e11 + 0.00000435043719156179e 3 t 12 1

+ 2.7645142997530610−7 e 3 t 13

1

+ 7.6393392306879610−8 e 3 t 14

1

+ 7.0250300276984610−9 e 3 t 15 − 0.964108229009469t 2 − 1.12479293384438t 3 − 0.441882938296007t 4 + 0.0642738819339645t 5 − 0.125869685454014t 6 + 0.0128165181237371t 7 − 0.00834508362907489t 8 + 0.000302877915859399t 9 − 0.000274184218567433t 10 − 0.00000528949470284348t 11 − 0.00000608452910460047t 12 − 3.9449808286074610−7 t 13 − 1.0725057208143610−7 t 14 1

− 9.9391280887061210−9 t 15 − 1.5859124474361510−9 t 16 + 1.1270339535231910−9 t 16 e 3

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(a) s1,2 (t) with exact

(b) s2,2 (t) with exact

Fig. 12 Example 3.2 for n = 1 (Type-II), graph of exact and approximate solutions

s2,1 (t) =0.433780558559770t − 0.433111144117548t 2 + 0.212427516331739t 3 + 0.0472495027134935t 4 − 0.122435901482380t 5 + 0.0816852973121234t 6 − 0.0347362341756747t 7

+ 0.0111447875156318t 8 − 0.00291063199931374t 9

+ 0.000646892911249343t

10

− 0.000125998190919646t 11 + 0.0000219575896206182t 12

− 0.00000347614761264751t 13 + 8.5845335806808510

+ 5.0568439899227710−7 t 14 − 6.8195126990510710−8 t 15

−9 16

t ,

and the corresponding values of α0 for s1,1 (t) and s2,1 (t) are identified via (2.14) and (2.15) with N = 10 which are 0.0358917716425683 and 0.13243888254754, respectively. The obtained approximate solutions along with the exact solutions are represented in Figs. 12, and 13. From figures, it is clear that as the obtained solutions via the proposed methodology type-II is really well in line with the precise solutions as iterations progress. As it is evident from figures that methodology type-II provides better results compare to methodology type-I with less number of iterations. Even the obtained solutions via type-II using second iteration are in good agreement with true solutions. The dynamics of the populations of two distinct species are depicted in Fig. 13, which demonstrates that as time increases, the population of the first species decays gradually, whereas the population of the second species grows gradually throughout the domain. Also, the absolute errors are represented in Tables 5, 6, 7 and 8 to show the accuracy of the methodology type-I and type-II.

An Effective Technique for Solving a Model Describing Biological …

(b) s2,3 (t) with exact

(a) s1,3 (t) with exact

Fig. 13 Example 3.2 for n = 2 (Type-II), graph of exact and approximate solutions Table 5 Absolute errors for Example 3.2 of type-I t |s1,3 (t) − s1 (t)| |s1,4 (t) − s1 (t)| 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.00480136916120316 0.0200936701738036 0.0473264952538380 0.0881090249113091 0.144215549956031 0.217592173174226 0.310364193552304 0.42484391807577 0.5635388115645 0.72915999432191

0.0 0.00322069191865686 0.0137960584587350 0.0332561022468991 0.0633571248342677 0.106101246634957 0.163758817518988 0.238892728265536 0.334383942629798 0.453457727331305 0.599710105313309

Table 6 Absolute errors for Example 3.2 of type-I t |s2,3 (t) − s2 (t)| |s2,4 (t) − s2 (t)| 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.0036819673240259185 0.005258511510903147 0.005439147037049161 0.004631057495267987 0.003020017607675979 0.000627363524772967 0.00265059426164335 0.00701575289818374 0.0127499738355621 0.0202020329006536

0.0 0.00330246881953400 0.00536190186866586 0.00643120982770425 0.00656948321984713 0.00571427145600914 0.00373688127726224 0.000486260472108258 0.00417528910537326 0.0103430959730567 0.0180478121563033

|s1,8 (t) − s1 (t)| 0.0 0.0000174249722959229 0.0000306391477422549 0.000285078738717887 0.000846715759702549 0.00172996705824874 0.00279929801952972 0.00368516823133852 0.00367579102989757 0.00158076815295083 0.00443750096521933

|s2,8 (t) − s2 (t)| 0.0 0.000130732210885096 0.000333249984681358 0.0005193557798276464 0.000626890117261719 0.000617553144770078 0.000478472927137291 0.000225326285604482 0.0000948103089639662 0.000402306736056013 0.000586362971222137

45

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Table 7 Absolute errors for Example 3.2 of type-II t |s1,2 (t) − s1 (t)| |s1,3 (t) − s1 (t)| 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.0000970882166753675 0.000372857145899987 0.000786507026367282 0.00128044471014857 0.00179021027021825 0.00225415147462876 0.00262131623992051 0.00285690671206773 0.00294515883021340 0.00288993082563893

0.0 0.000120193078859533 0.000479683597553831 0.00105181080713520 0.00178164106987519 0.00259113472445216 0.00338198583480465 0.00403506324793529 0.00440531479661943 0.00431100804699747 0.00351629365242401

Table 8 Absolute errors for Example 3.2 of type-II t |s2,2 (t) − s2 (t)| |s2,3 (t) − s2 (t)| 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.000405054818801584 0.000711462381281988 0.000890379346648068 0.000900915503568023 0.000715845650909430 0.000343520163927885 0.000154136222035667 0.000648150240422335 0.000934087631095121 0.000728922919670083

0.0 0.000217970742560261 0.000391988256563461 0.000523349782605056 0.000603062752819683 0.000620852639215519 0.000571135014163976 0.000455237391732782 0.000278797425358718 0.0000431798322647892 0.000270075922915725

|s1,4 (t) − s1 (t)| 0.0 0.000123672826318245 0.000493708963232158 0.00108323941436941 0.00183612206692332 0.00267127296906178 0.00348519763166022 0.00415263796687698 0.00452541282204355 0.00442982283818383 0.00366364829838384

|s2,4 (t) − s2 (t)| 0.0 0.000250564757945076 0.000451164032148690 0.000603959806075624 0.000699613712112496 0.000725268421813668 0.000670025330357010 0.000528313970022870 0.000301602171264376 8.0888545300750210−7 0.000360588313551208

Example 3.3 Next, consider the following system [8]    t 1 ds1 = s1 (t) − 2s2 (t) − s2 (τ )dτ + G 1 (t), 3 dt 3 t− 10    t ds2 1 −(t−τ ) = s2 (t) − + s1 (t) + e s1 (τ )dτ + G 1 (t), 3 dt 2 t− 10

(3.11)

An Effective Technique for Solving a Model Describing Biological …

(a) s1,2 (t) with exact

47

(b) s2,2 (t) with exact

Fig. 14 Example 3.3 for n = 1 (Type-I), graph of exact and approximate solutions

where G 1 (t) =

1 1 cos t − sin t 4 4



  1 1 1 1 3 + sin t − cos t + cos t − , 3 2 4 4 10

      3 1 1 3 1 1 3 3 1 − sin t − , G 2 (t) = − cos t + sin t − + sin t − cos t + e− 10 cos t − 4 4 2 8 8 8 10 10

with initial data s1 (0) = 0 and s2 (0) = 0. The exact solution of this system is s1 (t) = 1 sin t and s2 (t) = − 41 sin t. 4 The proposed methodology type-I formula for (3.11) is   t ds

1 1,n s1,n+1 (t) =αn s1,0 + (1 − αn )s1,n (t) − (1 − αn ) − s1,n (u) − 2s2 (u) du 3 0  u    − s2,n (τ )dτ − G 1 (u) du , n ≥ 0, (3.12) u−3/10

  t ds

1 2,n − s2,n (u) − + s1,n (u) s2,n+1 (t) =αn s2,0 + (1 − αn )s2,n (t) − (1 − αn ) du 2 0  u    − e−(u−τ ) s1,n (τ )dτ − G 2 (u) du , n ≥ 0. (3.13) u−3/10

By selecting the initial approximation s1,0 (0) = 0 and s2,0 (0) = 0, we obtain the results and the obtained approximate solutions along with the exact solutions are represented in Figs. 14, 15, and 16. From figures, it is clear that as the obtained solutions via the proposed methodology type-I is really well in line with the precise solutions as iterations progress. However, this approach requires more iterations to achieve a good agreement with the exact solutions.

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(a) s1,3 (t) with exact

(b) s2,3 (t) with exact

Fig. 15 Example 3.3 for n = 2 (Type-I), graph of exact and approximate solutions

(a) s1,4 (t) with exact

(b) s2,4 (t) with exact

Fig. 16 Example 3.3 for n = 3 (Type-I), graph of exact and approximate solutions

Now, let us implement the proposed methodology type-II formula for (3.11) which is given by  t

1 ds 1,n s1,n+1 (t) =αn s1,0 + (1 − αn )s1,n (t) − (1 − αn ) e−1/3(u−t) − s1,n (u) − 2s2 (u) du 3 0  u    s2,n (τ )dτ − G 1 (u) du , n ≥ 0, (3.14) − u−3/10

 t

1 ds 2,n s2,n+1 (t) =αn s2,0 + (1 − αn )s2,n (t) − (1 − αn ) e1/2(u−t) − s2,n (u) − + s1,n (u) du 2 0  u    −(u−τ ) e s1,n (τ )dτ − G 2 (u) du , n ≥ 0. (3.15) − u−3/10

By selecting the initial approximation s1,0 (0) = 0 and s2,0 (0) = 0, we obtain the results and the obtained approximate solutions along with the exact solutions are

An Effective Technique for Solving a Model Describing Biological …

(a) s1,3 (t) with exact

49

(b) s2,3 (t) with exact

Fig. 17 Example 3.3 for n = 1 (Type-II), graph of exact and approximate solutions

(a) s1,4 (t) with exact

(b) s2,4 (t) with exact

Fig. 18 Example 3.3 for n = 2 (Type-II), graph of exact and approximate solutions

represented in Figs. 17, and 18. From figures, it is clear that as the obtained solutions via the proposed methodology type-II is really well in line with the precise solutions as iterations progress. As it is evident from figures that methodology type-II provides better results compare to methodology type-I with less number of iterations. Even the obtained solutions via type-II using second iteration are in good agreement with true solutions. The dynamics of the populations of two distinct species are depicted in Fig. 18, which demonstrates that as time increases, the population of the first species decays gradually, whereas the population of the second species grows gradually throughout the domain.

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4 Conclusion In this study, a practical methodology for solving a system of two nonlinear integro-differential equations is proposed. The considered nonlinear delay integrodifferential equations are associated with the dynamics of two interacting species. The methodology consists of two types of formulas, which are based on the different integral operators. The suggested methodology gives consistent approximate solutions to the problems. Some numerical test problems have been carried out to display the efficiency of the approach. It is clear from the computational simulations that as the iterations progress, it can be observed that the obtained solutions approach the exact solution. The dynamics of the populations of two distinct species are depicted in Figs. 7, 13 and 18 which demonstrate that as time increases, the population of distinct species decays or grows. It can be observed that the type-I approach is simple but requires more iterations to produce a good accurate solution. From the results, it is clear that type-II approach provides better results compared to type-I in less number of iterations. Also, the figures demonstrate that the proposed methodology is an effective tool to solve nonlinear delay integro-differential equations. In addition, this method can also be extended to other physical models in the future.

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Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment Kolade M. Owolabi, Albert Shikongo, and Edson Pindza

Abstract Many processes depend on recent history and recent history is mainly modeled by delay differential equations. This implies that delay differential equations introduce more realistic dynamics in modeling processes, due to their novel technique of capturing past and present processes simultaneously. Therefore, incorporating a delay term to model key transformations taking place within the dynamics modeling the interaction among transformed epithelial cells (TECs), fibroblasts, myofibroblasts, transformed growth factor (TGF−β), and epithelial growth factor (EGF), in silico, in a setup mimicking experiments in a tumor chamber invasion assay are thus, considered in this chapter. This consideration of such experiments resulted into a derivation of system of nonlinear quasi time-depended delay parabolic partial differential equations. The well-pose of the unique solution of the resulting system of nonlinear quasi time-depended delay parabolic partial differential equations, presents that an extension of the existing Gronwall’s inequality for ordinary differential equations to the Gronwall’s inequality for linear, delay ordinary differential equations is thus, derived. The asymptotic stability and Hopf bifurcation analysis are established. Based on the asymptotic stability condition, a novel numerical method is derived, analyzed, implemented and improved results are presented for discussion. Hence, based on the obtained results, one strongly believes that the contribution in this chapter has a great deal toward the process of crafting a drug that can slow and/or confine tumor invasion.

K. M. Owolabi (B) Department of Mathematical Sciences, Federal University of Technology Akure, PMB 704 Akure, Ondo State, Nigeria e-mail: [email protected] A. Shikongo Engineering Mathematics, School of Engineering and the Built Environment, JEDS Campus, University of Namibia, Private Bag 13301, Windhoek, Namibia E. Pindza Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, South Africa © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_3

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Keywords Tumor cells’ micro-environment · Time dependent quasi delay parabolic partial differential equations · Extended Gronwall’s inequality · Hopf bifurcation · Numerical solution

1 Introduction Following the fact that tumor micro-environment remained consistent and pertinent in a host due to active epithelial cells, fibroblasts, myofibroblasts, endothelial cells, and inflammatory cells [31], then the presence of such cells within the micro-environment implies that the surrounding extra-cellular matrix (ECM) plays a significant role in confining and/or confronting cancer within its micro-environment. This seems to be a fact because in [31, 32] it is deduced Matrix Metalloproteinase (MMP), human Ductal Carcinoma in Situ (DCIS) and Matrix Metalloproteinase (MMP) can be modulated as cell adhesion or blocking material. Consequently, presenting that several classes of MMPs can be expressed in the form of periductal fibroblasts and myofibroblasts. This presents an intense stromal involvement during early invasion. Therefore, it is vital that the dynamics corresponding to tumor cells degrading the basal membrane and invade into the stroma be considered, eventhough such invasion of transformed epithelial cells (TECs) into stroma is a complex step toward metastasis [2]. Therefore, as in [41] Dn , D f , Dm , D E , DG , D P are defined to denote constant diffusion coefficients for the density of transformed epithelial cells, density of fibroblasts, density of myofibroblasts, concentration of epidermal growth factor, concentration of transformed growth factor, and concentration of matrix metalloproteinase. Thus, from the dynamical system point of view, the process of simplifying the complex steps toward metastasis has led Kim and Friedman in [31] to derive the dynamics of confining and/or confronting cancer within its micro-environment as ⎫

∇ρ ∂n = ∇ · (D ∇n) − ∇ · χ n  ∇E ⎪ − ∇χ1n Is n  n n ⎪ ∂t ⎪ 1+(|∇ρ|/λρ )2 ⎪ 1+(|∇ E|/λ E )2 ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ E ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂f ⎪ ⎪ = ∇ · (D ∇ f ) − a G f + a f, 12 22 f ⎪ ∂t ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ⎬ ∂m = ∇ · (D ∇m) − ∇ · χ m  ∇G G f + a m, + a m m 21 31 ∂t 1+(|∇G|/λG )2 ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ∂ρ ρ ⎪ ⎪ = −a Pn + (a f + a m) 1 − , 41 42 43 ⎪ ρ∗ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ E = ∇ · (D ∇ E) + I ⎪ ⎪ − (a51 f + a52 m) − a53 E, E ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂G = ∇ · (D ∇G) + a I ⎪ n − a G, ⎪  61 62 G + ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ∂ P = ∇ · (D ∇ P) + a I m − a P,   +a11 4 n 1 − n −an ρI , ∗ 12 s k +E 4 E4

∂t

p

71 −

72

(1)

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment

55

where, ∀t > 0, xi ∈ [−L i , L i ], i := 1, 2, 3, L ∈ Z such that  = {x = (x1 , x2 , x3 )}, + =  ∩ {x1 > 0}, − =  ∩ {x1 < 0}, ∗ = + ∪ − , S = {−L 0 < x1 < L 0 , such that x1  = 0, −L i < xi < L i , ∀ i = 2, 3}.

When 0 < L 0 < L 1 , then Kim and Friedman in [31] set the characteristic function of a set A ∈ R to be denoted by I A . This implies that I A (x) = 1, if x ∈ A, A(x) = 0, if x∈ / A. Furthermore, Kim and Friedman in [31] defined n(x, t) ∈ + to denotes the density of transformed epithelial cells (TECs); f (x, t) ∈ − to denotes the density of fibroblasts (f); m(x, t) ∈ − to denotes the density of myofibroblasts (m); ρ(x, t) ∈ S to denotes the concentration of extracellular matrix (ECM); E(x, t) ∈ ∗ to denotes the concentration of epidermal growth factor (EGF); G(x, t) ∈ ∗ to denotes the concentration of transformed growth factor (TGF-β) and P(x, t) ∈ ∗ to denotes the concentration of matrix metalloproteinase (MMP). One can see that the experiment for the dynamics reported in Eq. (2) is carried out in such a way that the dependent variables are separated by semi-permeable membrane, which is defined for x1 = 0, ∀ i = 2, 3. But, the transmission conditions at the semi-permeable membrane are given in [31] as ∂ E+ ∂x 1

=

∂ E− ∂x 1 ,

− ∂∂xE 1 + γ(E + − E − ) = 0

+

∂G + ∂x 1

=

∂G − ∂x 1 ,

+ − − ∂G ∂x 1 + γ(G − G ) = 0

∂ P+ ∂x 1

=

∂ P− ∂x 1 ,

− ∂∂xP 1 + γ(P + − P − ) = 0

+

+

such that E(x, t) =

G(x, t) =

P(x, t) =

on on on

⎫ M, t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ M, t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ M, t > 0,

(2)

⎧ + ⎪ ⎨ E (x, t), if x1 > 0, ⎪ ⎩ E − (x, t) if x < 0, 1 ⎧ + ⎪ ⎨G (x, t), if x1 > 0, ⎪ ⎩G − (x, t), if x < 0, 1 ⎧ + ⎪ ⎨ P (x, t), if x1 > 0, ⎪ ⎩ P − (x, t), if x < 0, 1

where, γ denotes a positive parameter [17]. Since the dynamics reported in Eqs. (1)– (2) constitute a system of nonlinear time-dependent parabolic equations, then the boundary and initial conditions are required. But the γ presents that the membrane is not permeable to cells [31]. Hence, it necessitated the no flux boundary conditions to be imposed as

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 ∇E

Dn ∇n − χn n 

1+(|∇ E|/λ E )2

− χ1n I S n 

∇ρ

1+(|∇ρ|/λρ )2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

· ν = 0, on ∂+ , t > 0,

D f ∇ f · ν = 0, on ∂− , t > 0, 

 Dm ∇m − χm m 

∇G

1+(|∇G|/λG )2

· ν = 0, on ∂− , t > 0, Dρ ∇ρ · ν = 0,

D E ∇ E · ν = 0, DG ∇G · ν = 0, D P ∇ P · ν = 0, on ∂ − {x1 = 0, i = 2, 3}, t > 0,

in which ν denotes the outward normal vector. The initial conditions are ⎫    n 0 (x) = 21 1 + tanh − 1 (0.8 − x) ∈ + , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪    ⎪ 1 1 ⎪ f 0 (x) = 0, 143 2 1 + tanh − (x − 0.2 ∈ − , ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(3)

(4)

⎪ m 0 (x) = 0.0 ∈ − , ρ0 (x) = 1.0 ∈ S, E 0 (x) = 1.0 ∈ ∗ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ G 0 (x) = 1.0 ∈ ∗ , P0 (x) = 0.0 ∈ ∗ .

Due to the complexity of the dynamics within the micro-environment [31, 32], we strongly recommend that the incorporation of time (τ ∈ N) required for some transformations and/or productions to take place should be incorporated into the dynamics in Eq. (1). The incorporation of time (τ ) is the recommended to be in corporated into the following. • transformation of fibroblasts into myofibroblasts. • degradation of the extra-cellular matrix (ECM). • production of fibroblasts and myofibroblasts by the transformed epithelial cells (TEC). • production of transformed epithelial cells by the transformed growth factor (TGF−β). • the production of fibroblasts by the Matrix Metalloproteinase (MMP). Such incorporation of time (τ ) implies that the spatial domain for ρ should be increased from S to the entire domain ∗ . Thus ignoring the vertical variables, then the incorporation of time (τ ) implies that the dynamics in Eq. (1) becomes ∇E − ∇χ1n Is n √ ∇ρ 1+(|∇ρ|/λρ )2 1+(|∇ E|/λ E )2   nE4 n + ka411+E ], 1 − ∈ [0, L 1 4 n ∗ −a12 ρIs

∂n ∂t

− Dn n = −∇ · χn n √

∂f ∂t ∂m ∂t

− D f  f = (−a12 G(x,  t − τ ) + a22 ) f (x,  t − τ ) ∈ [−L 1 , 0),

E

− Dm m = −∇ · χm m √

∇G 1+(|∇G|/λG )2

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

⎪ +a21 G(x, t − τ ) f (x, t − τ ) + a31 m ∈ [−L1 , 0),  ⎪ ⎪ ⎪ ⎪ ∂ρ ρ ⎪ ⎪ ∂t − Dρ ρ = −a41 P(x, t − τ )n(x, t − τ ) + (a42 f + a43 m) 1 − ρ∗ ∈ [−L 1 , L 1 ], ⎪ ⎪ ⎪ ∂E ⎪ ⎪ − D E = I (a f (x, t − τ ) + a m(x, t − τ )) − a E ∈ [−L , L ], E  1 1 51 52 53 ⎪ − ∂t ⎪ ⎪ ∂G ⎪ − D G = a I n(x, t − τ ) − a G ∈ [−L , L ], ⎪ G 61 + 62 1 1 ∂t ⎪ ⎭ ∂P − D P = a I m(x, t − τ ) − a P ∈ [−L , L ], p 71  72 1 1 − ∂t

(5)

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment

57

for (x, t) ∈ [−L 1 , L 1 ] × [−τ , 0]. The dynamics in Eq. (5) is a system of discrete delay reaction-diffusion equations. Delay differential equations (DDEs) are widely used for analysis and predictions in various areas of life sciences see [41] and the references therein. Hence, the first aim in this chapter, is to carry out mathematical analysis, which leads to the investigation of how time delay τ affects the dynamics in Eq. (5). By applying the Poincare´ normal form and the center manifold theorem as in [22], one finds conditions for the functions and derives formulas which determine the properties of Hopf bifurcation. More specifically, the chapter presents that equilibrium point losses its stability and the dynamics exhibit Hopf bifurcation under certain conditions. The second aim is to present the improved numerical solution as compared to [41] and for more numerical method related literature, we refer our readers to [41] and the references therein. The rest of the chapter is structured as follow. Mathematical analysis of the extended model is presented in Sect. 2. A robust numerical scheme based on the fitted finite difference technique is formulated, implemented and analysed for convergence in Sect. 3. To justify the effectiveness of the novel numerical method numerical results are presented in Sects. 4 and 5 concludes the chapter.

2 Mathematical Analysis Let u = [n, f, m, ρ, E, G, P]T , D = [Dn , D f , Dm , Dρ , D E , DG , D P ]T , where T denote a transpose. Then the extended dynamics in Eq. (5) can be rewritten as ∂u − Du = F(u(x, t), ∇u(x, t), u(x, t − τ )), for t ∈ [t0 − τ , t0 ], ∂t

(6)

where, ∇E

F1 (u(x, t), ∇u(x, t), u(x, t − τ )) = −∇ · χn n 

1+(|∇ E|/λ E )2

− ∇χ1n Is n 

∇ρ

1+(|∇ρ|/λρ )2

  1 − n −an ρI ∈ [0, L 1 ], ∗ s 12 F2 (u(x, t), ∇u(x, t), u(x, t − τ )) = (−a12G(x, t − τ ) + a22 ) f (x,  t − τ ) ∈ [−L 1 , 0), +

a11 n E 4 k 4E +E 4

F3 (u(x, t), ∇u(x, t), u(x, t − τ )) = −∇ · χm m 

∇G

1+(|∇G|/λG )2

+a21 G(x, t − τ ) f (x, t − τ ) + a31 m ∈ [L 1 , 0), F4 (u(x, t), ∇u(x, t), u(x, t − τ )) = −a  41 P(x,  t − τ )n(x, t − τ )

+(a42 f + a43 m) 1 − ρρ∗ ∈ [−L 1 , L 1 ], F5 (u(x, t), ∇u(x, t), u(x, t − τ )) = I− (a51 f (x, t − τ ) +a52 m(x, t − τ )) − a53 E ∈ [−L 1 , L 1 ], F6 (u(x, t), ∇u(x, t), u(x, t − τ )) = a61 I+ n(x, t − τ ) − a62 G ∈ [−L 1 , L 1 ], F7 (u(x, t), ∇u(x, t), u(x, t − τ )) = a71 I− m(x, t − τ ) − a72 P ∈ [−L 1 , L 1 ],

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

with all other terminal conditions remained unchanged, as given in (2) through to (4).

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2.1 Solution Continuously Depending on the Data Let v(x, t, t − τ ), z(x, t, t − τ ) ∈ C21 [−L 1 , L 1 ] denote two solutions for the dynamics in Eq. (5), such that v(x, t, t − τ ) − z(x, t, t − τ ) =: u(x, t, t − τ ), (where −τ < t < 0), yields the following results. Theorem 2.1 Let ⎫

∂u(x,t,t−ø) −Du(x, t,t − τ ) = 0; x∈ [−L 1 , L 1 ], t > 0, ⎪ ⎪ ∂t ⎪ ⎪ 1 1 ⎪ ⎪ u 1 (x, 0) = 2 1 + tanh − (0.8 − x) ∈ [0, L 1 ], ⎬    1 1 u 2 (x, 0) = 0, 143 2 1 + tanh − (x − 0.2 ∈ [−L 1 , 0), u 3 (x, 0) = 0.0 ∈ [L 1 , 0), ⎪ ⎪

⎪ ⎪ ρ0 (x) = 1.0 ∈ [−L 1 , L 1 ], u 4 (x, 0) = 1.0 ∈ [−L 1 , L 1 ], u 5 (x, 0) = 1.0 ∈ [−L 1 , 0), ⎪ ⎪ ⎭ u 6 (x, 0)(x) = 0.0 ∈ [−L 1 , L 1 ].

Then u(x, t, t − τ ) is identically zero.

Proof Proceeding component-wise for elements of the vector u(x, t), one obtains, ⎫ ⎬

u(·, t, t − τ )ut (·, t, t − τ ) = 21 ∂t u2 (·, t, t − τ ), u(·, t, t − τ )ut−τ (·, t, t − τ ) = 21 ∂t−τ u2 (·, t, t − τ ).

⎭

Similarly, one also finds u(·, t, t − τ )ux x (·, t, t − τ ) = ∂x (u(·, t, t − τ )ux (·, t, t − τ )) − u2x (·, t, t − τ ), then in view of Theorem 2.1  L  L 1 1 1 1 d ∂t u2 (·, t, t − τ )d x u2 (·, t, t − τ )d x = 2 dt −L 1 −L 1 2  L  L 1 1 = u(·, t, t − τ )ut (·, t, t − τ )d x = D u(·, t)ux x (·, t, t − τ )d x, −L 1

=D

 L 1 −L 1 L

−L 1

∂x (u(·, t, t − τ )ux (·, t, t − τ ))d x −

1 − = Du(·, t, t − τ )ux (·, t, t − τ )|−L 1

 L 1 −L 1

u2x (·, t, t − τ )d x = −D

 L 1

 L 1

−L 1

−L 1

u2x (·, t, t − τ )d x,

u2x (·, t, t − τ )d x ≤ 0.

L These implies that the function t → −L1 1 u2 (·, t, t − τ )d x is a non-increasing function. Hence  L1  L1 u2 (·, t, t − τ )d x ≤ u2 (·, 0, 0)d x = 0, ∀x, t. 0≤ −L 1

A similar results can be established that,

−L 1

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment

d 1 2 d(t − τ )



L1

−L 1

 u (·, t, t − τ ) = −D 2

L1 −L 1

59

u2x (·, t, t − τ )d x ≤ 0.

This proves uniqueness of solution to the nonlinear quasi time-depended delay parabolic partial differential equations (DPPDEs) in Eq. (6). Corollary 2.2 Let v, z ∈ C21 [−L 1 , L 1 ] denote solutions to the dynamics in Eq. (6) with initial states v0 , z0 , such that v0 − z0 =: u0 and Fu , Fv denote the real-valued functions bounding v and z, respectively. Then 

t

v − z 2 ≤ exp(−ηt) v0 − z0 2 +

exp(−η(t − s)) Fu − Fv 2 ds,

0

and v − z 2 ≤ exp(−η(t − τ )) v0 − z0 2 + τ

 t t−τ

exp(−η(t − τ − s)) Fu − Fv 2 ds,

for some −τ < t < 0, and , τ , η ∈ R+ . In order to prove the Corollary 2.2, the following are preliminaries to the proof. Definition 2.3 Let v, z ∈ C21 [−L 1 , L 1 ] then  (v, z) :=

L1

v(·, t), z(·, t)d x, −L 1

defines an in inner product [14]. Lemma 2.4 If v, z ∈ C21 [−L 1 , L 1 ] then (vx x + σv, z) = σ(v, z) − (vx , zx ) = (v, zx x + σz), where, σ ∈ R. Proof Since ∂x [vx z] = vx x z + vx zx , then L1 = (vx x , z) + (vx , zx ), (∂x [vx , z]) = (vx x , z) + (vx , zx ), ⇒ (vx , z)|−L 1 ⇒ (vx x , z) = −(vx , zx ).

Similarly, ∂x [vzx ] = vzx x + vx zx ,

(7)

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then it follows from equation in (7) that (v, zx x ) = −(vx , zx ).

(8)

Combining equation in (7) with equation in (8) yields (vx x , z) = (v, zx x ). 

Hence, the results follows. Lemma 2.5 Let v ∈ C21 [−L 1 , L 1 ], then  1 2

v := (v, v) =

L1

−L 1

v (·, t)d x 2

 21

,

defines a norm. Proof See [14]. Corollary 2.6 Let v, z ∈ C21 [−L 1 , L 1 ] then |(v, v)| ≤ v v . Proof Let [14] P2 (σ) = v + σz 2 = v 2 + 2σ(v, z) + σ 2 z 2 , denotes a polynomial of degree two. Then P2 (σ) ≥ 0, ∀σ ∈ R. Thus,  P2 (σ) = 4(v, z)2 − 4 v 2 z 2 ≤ 0, ⇒ (v, z)2 ≤ v 2 z 2 , which concludes the proof of Corollary 2.6. Proof of Corollary 2.2: By means of Poincare´ inequality [14], Schwarz inequality in Lemma 2.5, Corollary 2.6 and in view of the prove to Theorem 2.1, d 1 d 1 2 u(·, t, t − τ ) 2 + u(·, t, t − τ ) 2 + D u(·, t) 2 2 dt 2 d(t − τ ) c ≤ 2(F(·, t, t − τ ), u(·, t, t − τ )) ≤ 2 F u(·, t, t − τ ) , where, c > 0, F := Fu − Fv . Thus, d u(·, t, t dt

− τ ) 2 +

d u(·, t, t d(t−τ )

2 c

 D − u(·, t, t − τ ) 2 ≤ 1 F 2 ,

− τ ) 2 +

2 c

⎫ ⎬

 ⎭ D − u(·, t, t − τ ) 2 ≤ 1 F 2 ,

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment

61

for some arbitrary sufficiently small > 0. Let η = 2c D −  > 0 and  = 1/. Then, Eq. (9) becomes d u(·, t, t dt

− τ ) 2 + η u(·, t, t − τ ) 2 ≤  F 2 ,

d u(·, t, t d(t−τ )

− τ ) 2 + η τ u(·, t, t − τ ) 2 ≤  τ F 2 .

⎫ ⎬ (9)

⎭

Applying the Gronwall’s inequality [14] to equation in (9), one finds the following results Theorem 2.7 (Extended Gronwall’s inequality) Let I ≡ [−L 1 , L 1 ]. Let  tau and u denote real-valued continuous functions defined on I. If u is differentiable in the interior I◦ of I and satisfies the differential inequality d u(·, t, t − τ ) 2 + η τ u(·, t, t − τ ) 2 ≤  τ F 2 , t − τ ∈ I◦ , d(t − τ ) then u is bounded by the solution of the corresponding differential equation v  (t) = β(t)v(t):  u(·, t, t − τ ) ≤ u(·, a0 , t − τ ) exp

t

a0

 η τ ds , ∀t ∈ I. 

2.2 Local Stability In [41] we have presented the following results. Theorem 2.8 The equilibrium point for system in Eq. (5) is given by ⎛

⎛



⎞ 21

⎞

L1 + 1 +  ⎜ a62 a22 ⎟ 2 ⎝ ⎠ , 0, a22 , 0⎟ . E =⎜ ⎝ a61 a12 , 0, 0, L 1 1 + (L 1 ) + ln  a12 ⎠ 1 + L2 − L L 21

1

1

To investigate the linearized stability of the system (5), we let (n, f, m, ρ, E, G, P) = (n¯ + n ∗ , f¯ + f ∗ , m¯ + m ∗ , ρ¯ + ρ∗ , E¯ + E ∗ , G¯ + G ∗ , P¯ + P ∗ ).

Substituting into (5), and retaining only the linear terms in n, f, m, ρ, E, G, P, one finds

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K. M. Owolabi et al. 

∂ n¯ − D ∂ n ∂x ∂t



∂ f¯ ∂ ∂t − D f ∂x ∂ m¯ − D ∂ m ∂x ∂t ∂ ρ¯ ∂ ∂t − Dρ ∂x

∂ ∂x n¯





∂ E¯ − D ∂ E ∂x ∂t ∂ G¯ − D ∂ G ∂x ∂t ∂ P¯ − D ∂ P ∂x ∂t



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

= 0, in ∈ [0, L 1 ], t > 0,



∂ ¯ ¯ ∂x f = a22 f (x, t − τ ) ∈ [−L 1 , 0], t > 0, ∂ ¯ ∂x m



= a31 m(x, ¯ t) ∈ [−L 1 , 0], t > 0,



∂ ¯ ¯ t) ∈ [−S, S], t > 0, ∂x ρ¯ = a42 f (x, t) + a43 m(x,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ E¯ = −a E(x, ⎪ ¯ ⎪ t) ∈ [−L , L ], t > 0, 1 1 53 ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ a62 a22 a62 a22 ∂ G ¯ = a61 n(x, ¯ ⎪ ¯ t − τ ) + a62 G(x, t) + a a − a ∈ [−L 1 , L 1 ], t > 0, ⎪ ⎪ ∂x ⎪ 61 12 12 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎭ ∂ P¯ = a I ¯ ¯ t − τ ) − a72 P(x, t) ∈ [−L 1 , L 1 ], 71 − m(x, ∂x





on (x, t) ∈ [−L 1¯, L 1 ] × [−τ , 0], where, the characteristic equation is (λ − Dn )(λ − D f − a22 e−λt )(λ − Dm − a31 )(λ − Dρ )(λ − D E + a53 ) (λ − DG − a62 )(λ − D P + a72 ) = 0. Hence, the following results. Theorem 2.9 The dynamics in Eq. (5) are asymptotically stable if λ < Dn , λ < D f + a22 e−λt , λ < Dm + a31 , λ < D E − a53 , λ < DG + a62 , λ < D P − a72 .

2.3 Hopf Bifurcation Analysis When τ = 0, we assume that λ = iω for ω > 0 and i = values, we have

√ −1. In view of the eigen-

iω − D f − a22 exp(iωτ ) = iω − D f − a22 (cos(ωτ ) + i sin(ωτ )) = 0. Separating real and imaginary parts we have ω + a22 sin(ωτ ) = 0 and − D f − a22 cos(ωτ ) = 0,

(10)

which yields 1 cos−1 τi = ω0



 Df + 2iπ , i = 0, 1, 2, 3, . . . . a22

Squaring on both sides of equations in (10), we find 2 sin2 (ωτ ) = 0, ω 2 + 2a22 ω sin(ωτ ) + a22 2 D 2f + 2D f a22 cos(ωτ ) + a22 cos2 (ωτ ) = 0.

(11)

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment

63

Adding the two equations in Eq. (11) one finds 2 = 0, ω 2 + D 2f + 2(D f cos(ωτ ) + ω sin(ωτ ))a22 + a22

which simplifies to 2 = 0, ω 2 + D 2f − 2(D 2f + ω 2 ) + a22  2 2 = 0, ⇒ ω0 = ± a22 − D 2f . ⇒ −ω 2 − D 2f + a22

Choosing τ0 = min{τi }, we need to show that  R

 dλ  = 0. dτ τ =τ0

From the eigenvalues we have  R

dλ dτ



 =R

d(D f + a22 (cos(ωτ ) + i sin(ωτ )) dτ

 = −a22 ω sin(ω0 τ0 ),

where ω0 τ0 = 0. By summarizing the above analysis, we arrive at the following theorem. Theorem 2.10 The equilibrium ⎛

⎛



⎞ 21

⎞

L1 + 1 +  ⎜ a62 a22 ⎟ 2 + ln ⎝  ⎠ , 0, a22 , 0⎟ , E =⎜ , 0, 0, L 1 + (L ) 1 1 ⎝ a61 a12 a12 ⎠ 1 + L2 − L L 21

1

1

of the system (5) is asymptotically stable for τ ∈ [0, τ0 ) and it undergoes Hopf bifurcation at τ = τ0 .

2.4 Global Stability Analysis Theorem 2.11 The dynamics in Eq. (5) is globally stable. Proof The prove to this theorem has been carried out already in [41].

3 Derivation and Analysis of the Numerical Method The derivation of the novel numerical method for discretising the dynamics (5) is as follows. We determine an approximation to the derivatives for the functions

64

K. M. Owolabi et al.

n(t, x), f (x, t), m(x, t), ρ(x, t), E(x, t), G(x, t), P(x, t), with respect to the spatial variable x. Let Sx be a positive integer. Discretize the interval [−L/2, L/2] through the points −L/2 = x0 < x1 < x2 < · · · < xs−1 < xs < xs+1 · · · < x Sx −2 < x Sx −1 < x Sx = L/2,

where, the step-size x = x j+1 − x j = 1/Sx , j = 0, 1, . . . , Sx . Let N j (t), F j (t), M j (t), R j (t), E j (t), G j (t), P j (t),

(12)

denote the numerical approximations for n(t, x), f (x, t), m(x, t), ρ(x, t), E(x, t), G(x, t), P(x, t). Then the spatial derivatives in the system in Eq. (5) are approximated as follows ⎛

⎞

∂ ⎜ ∂n  ∂x ⎝ Dn ∂x − χn n

∂ρ ∂E N j+1 −2N j +N j−1 ⎟ ∂x ∂x − χ1n Is n  ⎠ (ti , x j ) ≈ Dn φ2 ∂ρ 1+( ∂ E /λ E )2 n 1+( /λρ )2 ∂x ∂x − (Dx E j ) − −χn (Dx N j )  2   −  1+ Dx E j λE Dx+ (Dx− E j ) −χn N j ⎛  2 ⎞3/2 , Dx− E j ⎝1+ ⎠ λE (Dx− R j )  − 2  1+ Dx R j λρ Dx+ (Dx− R j ) −χ1n Is N j ⎛  2 ⎞3/2 , Dx− R j ⎝1+ ⎠ λρ

−χn (Dx− N j )  

∂ ∂x



D f ∂∂xf



(ti , x j ) ≈ D f

⎛

⎞

∂ ⎝ ∂m  ∂x Dm ∂x − χm m

∂ ∂x ∂ ∂x ∂ ∂x



F j+1 −2F j +F j−1 , φ2f

∂G M j+1 −2M j +M j−1 ∂x ⎠ (ti , x j ) ≈ Dm φ2 1+( ∂G /λG )2 m ∂x − (Dx G j ) − −χm (Dx M j )  2   −  1+ Dx G j λG Dx+ (Dx− G j ) −χm M j ⎛  2 ⎞3/2 , Dx− G j ⎝1+ ⎠ λG

 R −2R j +R j−1 ∂ρ , Dρ ∂x (ti , x j ) ≈ Dρ j+1 2 (t)





E −2E +E j−1 , D E ∂∂xE (ti , x j ) ≈ D E j+1 2 j φE



 G j+1 −2G j +G j−1 , DG ∂G 2 ∂x (ti , x j ) ≈ DG φG

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(13)

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment

65

where, D + (·) j :=

(·) j+1 − (·) j (·) j − (·) j−1 , D − (·) j := , x x

and the denominator functions "  !   f x 2 χn x 4 a22 2 exp( ) − 1 , φ f := 2 sin ,  f := , Dn 2 Df f  !   χm x a53 Dm x 4 e x 2 exp( φ2m := ) − 1 , φ2E := 2 sinh , e := , χm Dm 2 De e 2 2     g x  p x a62 2 a72 4 4 , g := ,  p := . φ2G := 2 sinh , φ P := 2 sinh 2 Dg 2 Dp g p Dn x φ2n := χn

Let St be a positive integer such that t = 1/St where 0 < t < St . Then discretizing the time interval [0, T ], through the points 0 = t0 < t1 < · · · < t St = T, where ti+1 − ti = t, i = 0, 1, . . . , (t St − 1). We approximate the time derivative at ti by ∂n ∂t (x j , ti )

≈

∂ρ ∂t (x j , ti )

≈

∂P ∂t (x j , ti )

≈

i+1 N j+1 −N ji

t

i Ri+1 j+1 −R j

t

i P i+1 j+1 −P j

ψP

∂f ∂t

, ,

(x j , ti ) ≈

∂E ∂t (x j , ti )

≈

,

i F i+1 j+1 −F j

ψf

,

∂m ∂t (x j , ti )

≈

ψE

,

∂G ∂t (x j , ti )

≈

i E i+1 j+1 −E j

i Mi+1 j+1 −M j

t

i G i+1 j+1 −G j

ψG

,

⎫ ,⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(14)

where, ψ f = ψw (t) = (1 − exp(−a22 t))/a22 , ψ E = (1 − exp(−a53 t))/a43 , ψG = (1 − exp(−a62 t))/a62 , ψ P = (1 − exp(−a72 t))/a72 , where, one can see that φn → x, φ f → x, ψ f → t, φm → x, φ E → x, ψ E → t, φG → x, ψG → t, φ P → x, ψ P → t as (t, x) → (0, 0).

The denominator functions in equations (13) and (14) are used explicitly to remove the inherent stiffness in the central finite derivatives parts and can be derived by using the theory of nonstandard finite difference methods, see, e.g., [35, 44, 45] and references therein. Combining the Eq. (13) for the spatial derivatives with Eq. (14) for time derivatives, we obtain

66

K. M. Owolabi et al. N ji+1 −N ji t

− Dn

i+1 i+1 N j+1 −2N ji+1 +N j−1 φ2n

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

(Dx− E ij )  − 2

= −χn (Dx− n ij )  

 1+

D x E ij λE

Dx+ (Dx− E ij )  − 2 ⎞3/2 D x E ij ⎝1+ ⎠

−χn N ji ⎛

λE

D + (D − Ri ) −χ1n Is N ji ⎛ x x j2 ⎞3/2 D x− Rij ⎝1+ ⎠ λρ

+

a11 (E 4 )ij

k 4E +(E 4 )ij N ji i+1 i+1 F i+1 j+1 −2F j +F j−1

i F i+1 j −F j ψf

(1 −

N ji n ∗ −a12 ρIs

), x ∈ [xs , L/2],

− Df = −a21 (HG )ij (H f )ij φ2f +a22 (H f )ij , x ∈ [− L2 , xs ], i+1 i+1 i Mi+1 Mi+1 (Dx− G ij ) j −M j j+1 −2M j +M j−1 − Dm = −χm (Dx− Mij )    − i 2 t φ2m  1+ Dx G j

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (D − G i ) ⎪ D ⎪ ⎪ −χm Mij ⎛ x x j2 ⎞3/2 ⎪ ⎪ ⎪ D x− G ij ⎪ ⎪ ⎝1+ ⎠ ⎪ λG ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +a21 (HG )ij (H f )ij + a31 Mij , x ∈ [− L2 , xs ], ⎪ ⎪ ⎪ i+1 i+1 i+1 i+1 i ⎪ R j −R j R j+1 −2R j +R j−1 ⎪ i i ⎪ ⎪ − D = −a ( H ) ( H ) ρ 41 p n 2 ⎪ j j t φρ ⎪ ⎪ ⎪ i ⎪ Rj ⎪ L L i i ⎪ +(a42 F j + a43 M j )(1 − ρ∗ ), x ∈ [− 2 , 2 ], ⎪ ⎪ ⎪ i+1 i+1 i+1 i ⎪ E i+1 − E E −2 E + E j j j+1 j j−1 L L ⎪ i i i ⎪ − DE = I a ( H ) + a ( H ) − a E , x ∈ [− , ], ⎪  f m 51 52 53 2 − j j j ⎪ ψE 2 2 φE ⎪ ⎪ i+1 i+1 i+1 i+1 i ⎪ G j −G j G j+1 −2G j +G j−1 ⎪ L L i i ⎪ ⎪ − D = I a ( H ) − a G , x ∈ [− , ], G  61 n 62 2 ⎪ + j j ψG 2 2 φG ⎪ ⎪ ⎪ i+1 i+1 i+1 i+1 i ⎪ P j −P j P j+1 −2P j +P j−1 ⎭ L L i i −D = I a (H ) − a P , x ∈ [− , ],

(15)

λG

P

ψP

− 71

φ2P

m j

72

j

2

2

with the terminal conditions ⎛

Nxis −1

⎜ ⎜ = Nxis +1 − 2xχn Nxis ⎜ ⎝

    Exi +1 −Exi s s 2x 1+ 2xλ

⎟ ⎟ 2 ⎟ ⎠ −1

E

⎛ ⎜ ⎜ −2xχ1n Nxis ⎜ ⎝

⎞ Exi s +1 −Exi s −1

⎞ ⎟

Rixs +1 −Rixs −1 ⎟   2 ⎟ ,  ⎠  Rix +1 −Rix −1 s s 2x 1+ 2xλρ

⎛

F i−L +1 = F i−L −1 , Mi−L −1 = Mi−L +1 − 2xχm Mi−L 2

2

2

2

2

Ri− L +1 = Ri− L −1 , E i−L −1 = (E − )i−L +1 (1 + 2xγ), 2

2

2

2

⎜ ⎜ ⎜ ⎜ ⎝

⎞ G i−L −G i−L 2 +1 2 −1   ⎛ i  −G i−L G −L  2 +1 2 −1 2x 1+⎝ 2xλG

G i−L −1 = (G − )i−L +1 (1 + 2xγ), P i−L −1 = (P − )i−L +1 (1 + 2xγ), 2 2  2     2  Nx0j = 21 1 + tanh − 1 (0.8 − x j ) , Fx0j = 0.143 21 1 + tanh − 1 (x − 0.2 , R0x j = 1.0, M0x j = 0.00, Ex0j = Gx0 j = Px0j = 1.00,

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

⎪ ⎟ ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎞2 ⎟ , ⎪ ⎪ ⎠ ⎪ ⎪ ⎪ ⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(16)

where, the no-flux boundary conditions are discretised by means of the central finite difference [10], j = −L/2, 2, . . . , L/2 − 1, i = 0, 1, . . . , T − 1 and

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment

67

(Hn )ij ≈ N (ti − τ , x j ), (H f )ij ≈ F(ti − τ , x j ), (HG )ij ≈ G(ti − τ , x j ), (Hm )ij ≈ M(ti − τ , x j ), (H P )ij ≈ P(ti − τ , x j ),

(17)

are denoting the history functions corresponding to n, f, m, G, P. The system in Eq. (15) can further be simplified as i+1 n −D N j−1 + φ2



n

1 t

=



Dn i+1 N j+1 φ2n i − (Dx E j ) −χn (Dx− n ij )    − i 2  1+ Dx E j

+

2Dn φ2n

N ji+1 −

λE

D + (D − E i ) −χn N ji ⎛ x x j2 ⎞3/2 D x− E ij ⎝1+ ⎠ λE

D + (D − Ri ) −χ1n Is N ji ⎛ x x j2 ⎞3/2 D x− Rij ⎝1+ ⎠ λρ

(E 4 )ij

Ni +a11 4 N ji (1 − n ∗ −a12j ρIs k E +(E 4 )ij



D

− φ2f F i+1 j−1 + f

1 ψf

+

2D f φ2f



F i+1 − j

Df φ2f

)+

N ji t

F i+1 j+1

= −a21 (HG )ij (H f )ij + a22 (H f )ij + m Mi+1 −D j−1 + φ2 m



1 t

+

2Dm φ2m



Mi+1 − j

= −χm (Dx− Mij )  

Dm

Mi+1 j+1

φ2m (Dx− G ij )

 1+



,

D x− G ij

2

F ij ψf

,

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

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λG ⎪ ⎪ ⎪ ⎪ i ⎪ M ⎪ ⎪ +a21 (HG )ij (H f )ij + a31 Mij + t j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ Dρ 2Dρ Dρ i+1 i+1 i+1 ⎪ 1 ⎪ − (x)2 R j−1 + (t) + (x)2 R j − (x)2 R j+1 ⎪ ⎪ ⎪ i ⎪ R ⎪ j R i i i i ⎪ = −a41 (H P ) j (Hn ) j + (a42 F j + a43 M j )(1 − ρ∗ ) + (t) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ D E i+1 2D E D E i+1 i+1 1 ⎪ ⎪ − φ2 E j−1 + ψ E + φ2 E j − φ2 E j+1 ⎪ ⎪ E E E ⎪ ⎪ i ⎪ E ⎪ ⎪ = I− (a51 (H f )ij + a52 (Hm )ij ) − a62 E ij + ψ Ej , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ DG i+1 2DG DG i+1 i+1 1 ⎪ − φ2 G j−1 + ψG + φ2 G j − φ2 G j+1 ⎪ ⎪ ⎪ G G G ⎪ ⎪ i ⎪ Gj ⎪ i i ⎪ = a61 I+ (Hn ) j − a62 G j + ψG , ⎪ ⎪   ⎪ ⎪ ⎪ D P i+1 2D P D P i+1 i+1 1 ⎪ − φ2 P j−1 + ψ P + φ2 P j − φ2 P j+1 ⎪ ⎪ ⎪ P P P ⎪ ⎪ i ⎪ Pj ⎭ i i = a71 I− (Hm ) j − a72 P j + ψ P , λG

D + (D − G i ) −χm Mij ⎛ x x j2 ⎞3/2 D x− G ij ⎝1+ ⎠

(18)

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which can be written as a tridiagonal system given by ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λE ⎪ ⎪ ⎪ ⎪ i i ⎪ + (D − Ri ) 4 )i E N N ( D x x j j j j ⎪ ⎪ i (1 − ⎪ −χ1n Is N ji ⎛  + a N ) + , ⎞ 11 k 4 +(E 4 )i ⎪ 3/2  j n −a ρI t 2 ∗ s 12 ⎪ j E ⎪ D x− Rij ⎪ ⎝1+ ⎠ ⎪ ⎪ λρ ⎪ ⎪ ⎪ ⎪ i ⎪ Fj ⎪ i+1 i i i ⎪ ⎪ A f F j = −a21 (HG ) j (H f ) j + a22 (H f ) j + ψ f , ⎪ ⎪ ⎪ i − ⎪ (Dx G j ) ⎬ i+1 i − Am M j = −χm (Dx M j )     2 − i  1+ Dx G j ⎪ ⎪ λG ⎪ ⎪ ⎪ ⎪ i + (D − G i ) ⎪ M D ⎪ ⎪ −χm Mij ⎛ x x j2 ⎞3/2 + a21 (HG )ij (H f )ij + a31 Mij + t j , ⎪ ⎪ ⎪ D x− G ij ⎪ ⎪ ⎝1+ ⎠ ⎪ ⎪ λG ⎪ ⎪ ⎪ ⎪ i ⎪ R ⎪ j i+1 R ⎪ Aρ R j = −a41 (H P )ij (Hn )ij + (a42 F ij + a43 Mij )(1 − ρ∗ ) + (t) , ⎪ ⎪ ⎪ i ⎪ Ej ⎪ i+1 i i i ⎪ ⎪ A E E j = I− (a51 (H f ) j + a52 (Hm ) j ) − a62 E j + ψ E , ⎪ ⎪ ⎪ i i G P ⎭ j i − a G i + j , A P i+1 = a I i − a Pi + A G G i+1 = a I ( H ) ( H ) , 61 + n j 62 j P j 71 − m j 72 j j ψG ψP

(19)

  ⎫   D 2D D n n n ⎪ An = Tri − D , 1 + 2D ,− D , A f = Tri − φ2f , ψ1f + φ2 f , − φ2f , ⎪ ⎪ φ2n t φ2n φ2n f f ⎪   f   ⎪ ⎪ ⎪ Dρ 2Dρ Dρ Dm 2Dm Dm 1 1 Am = Tri − φ2 , t + φ2 , − φ2 , Aρ = Tri − (x)2 , t + (x)2 , − (x)2 , ⎪ ⎬ m m   m   DG 2DG DG DE 2D E DE 1 1 ⎪ A E = Tri − φ2 , ψ E + φ2 , − φ2 , A G = Tri − φ2 , ψG + φ2 , − φ2 , ⎪ ⎪ ⎪ E E E  G G G ⎪  ⎪ ⎪ ⎪ DP 2D P DP 1 ⎭ A P = Tri − φ2 , ψ P + φ2 , − φ2 .

(20)

An N ji+1 = −χn (Dx− n ij )  

(Dx− E ij )  − 2

 1+

D x E ij λE

Dx+ (Dx− E ij )  − 2 ⎞3/2 D x E ij ⎝1+ ⎠

− χn N ji ⎛

where,

P

P

P

On the interval [0, τ ], the delayed arguments tn − τ belong to [−τ , 0], and therefore, the delayed variables in Eq. (15) are evaluated directly from the history functions n 0 (t, x), f 0 (t, x), m 0 (t, x), G 0 (t, x), P 0 (t, x), as (Hn )ij ≈ n 0 (ti − τ , x j ), (H f )ij ≈ f 0 (ti − τ , x j ), (Hm )ij ≈ m 0 (ti − τ , x j ), (HG )ij ≈ G 0 (ti − τ , x j ), (H P )ij ≈ P 0 (ti − τ , x j ), and Eq. (19) becomes

(21)

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment An N ji+1 = −χn (Dx− n ij )  

(Dx− E ij )  − 2

 1+

D x E ij λE

Dx+ (Dx− E ij )  − 2 ⎞3/2 D x E ij ⎝1+ ⎠

− χn N ji ⎛

λE

D + (D − Ri ) −χ1n Is N ji ⎛ x x j2 ⎞3/2 D x− Rij ⎝1+ ⎠ λρ

(E 4 )ij

Ni +a11 4 N ji (1 − n ∗ −a12j ρIs k E +(E 4 )ij

A f F i+1 j

)+

N ji t

,

= −a21 G 0 (ti − τ , x j ) f 0 (ti − τ , x j ) + a22 f 0 (ti − τ , x j ) +

Am Mi+1 j

=

(Dx− G ij ) −χm (Dx− Mij )    − i 2  1+ Dx G j

F ij ψf

,

D + (D − G i ) − χm Mij ⎛ x x j2 ⎞3/2 D x− G ij ⎝1+ ⎠

69 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λG λG ⎪ ⎪ ⎪ ⎪ ⎪ i ⎪ M ⎪ j i 0 0 ⎪ +a21 G (ti − τ , x j ) f (ti − τ , x j ) + a31 M j + t , ⎪ ⎪ ⎪ i ⎪ R ⎪ j 0 (t − τ , x )n 0 (t − τ , x ) + (a F i + a Mi )(1 − R ) + ⎪ Aρ Ri+1 = −a P , 41 i j i j 42 j 43 j j ⎪ ρ∗ (t) ⎪ ⎪ i ⎪ Ej ⎪ i+1 i 0 0 ⎪ ⎪ A E E j = I− (a51 f (ti − τ , x j ) + a52 m (ti − τ , x j )) − a62 E j + ψ E , ⎪ ⎪ ⎪ i ⎪ G j ⎪ i+1 i 0 ⎪ A G G j = a61 I+ n (ti − τ , x j ) − a62 G j + ψG , ⎪ ⎪ ⎪ i ⎪ Pj ⎭ i+1 i 0 A P P j = a71 I− m (ti − τ , x j ) − a72 P j + ψ P .

(22)

Let s denotes the largest integer such that τs ≤ τ . Then using the system in Eq. (22), one can compute N ji , F ij , Mij , Rij , E ij , G ij , P ij , for 1 ≤ i ≤ s. Up to this stage, one interpolates the data (t0 , N j0 ), (t1 , N j1 ), . . . , (ts , N js ), (t0 , F 0j ), (t1 , F 1j ), . . . , (ts , F sj ), (t0 , M0j ), (t1 , M1j ), . . . , (ts , Msj ), (t0 , R0j ), (t1 , R1j ), . . . , (ts , Rsj ), (t0 , E 0j ), (t1 , E 1j ), . . . , (ts , E sj ), (t0 , G 0j ), (t1 , G 1j ), . . . , (ts , G sj ), (t0 , P 0j ), (t1 , P 1j ), . . . , (ts , P sj ), using an interpolating cubic Hermite spline ϕ j (t) ([10]). Then N ji = ϕn (ti , x j ), F ij = ϕ f (ti , x j ), Mij = ϕm (ti , x j ), Rij = ϕρ (ti , x j ), E ij = ϕ E (ti , x j ), G ij = ϕG (ti , x j ), P ij = ϕ P (ti , x j ),

for all i = 0, 1, . . . , s and j = −L/2, 2, . . . , L/2 − 1. For i = s + 1, s + 2, . . . , T − 1, when we move from level i to level i + 1 we extend the definitions of the cubic Hermite spline ϕ j (t) to the point (ti + t, (Hn )ij , ti + t, (H f )ij , ti + t, (Hm )ij , ti + t, (HG )ij ), ti + t, (H P )ij ).

Then, the history terms (Hn )ij , (H f )ij , (Hm )ij , (HG )ij , (H P )ij can be approximated by the functions (ϕn ) j (ti − τ ), (ϕ f ) j (ti − τ ), (ϕm ) j (ti − τ ), (ϕG ) j (ti − τ ), (ϕ P ) j (ti − τ ) for i ≥ s. This implies that,

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(Hn )ij ≈ (ϕn ) j (ti − τ ), (H f )ij ≈ (ϕ f ) j (ti − τ ), (Hm )ij ≈ (ϕm ) j (ti − τ ), (HG )ij ≈ (ϕG ) j (ti − τ ), (H P )ij ≈ (ϕ P ) j (ti − τ ),

(23)

and Eq. (22) becomes An N ji+1 = −χn (Dx− n ij )  

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

(Dx− E ij ) i  − i 2 − χn N j ⎛

Dx+ (Dx− E ij )  − i 2 ⎞3/2 Dx E j ⎝1+ ⎠

 1+ Dx E j λE

λE

Dx+ (Dx− Rij ) −χ1n Is N ji ⎛  2 ⎞3/2 Dx− Rij ⎝1+ ⎠ λρ

(E 4 )ij +a11 4 k E +(E 4 )ij

Ni

Ni

N ji (1 − n −a j ρI ) + tj , ∗ 12 s Fi

A f F i+1 = −a21 (ϕG ) j (ti − τ )(ϕ f ) j (ti − τ ) + a22 (ϕ f ) j (ti − τ ) + ψ j , j f

(Dx− G ij ) Dx+ (Dx− G ij ) i  − i 2 − χm M j ⎛  − i 2 ⎞3/2  Dx G j 1+ Dx G j ⎝1+ ⎠ λ λ

Am Mi+1 = −χm (Dx− Mij )   j

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G G ⎪ ⎪ ⎪ ⎪ i ⎪ Mj ⎪ ⎪ i ⎪ +a21 (ϕG ) j (ti − τ )(ϕ f ) j (ti − τ ) + a31 M j + t , ⎪ ⎪ ⎪ i ⎪ R ⎪ j ⎪ i+1 R i i Aρ R j = −a41 (ϕ P ) j (ti − τ )(ϕn ) j (ti − τ ) + (a42 F j + a43 M j )(1 − ρ∗ ) + (t) , ⎪ ⎪ ⎪ ⎪ ⎪ i ⎪ E ⎪ j i+1 i ⎪ ⎪ A E E j = I− (a51 (ϕ f ) j (ti − τ ) + a52 (ϕm ) j (ti − τ )) − a62 E j + ψ , ⎪ ⎪ E ⎪ i ⎪ G ⎪ ⎪ i + j , ⎪ A G G i+1 = a I (ϕ ) (t − τ ) − a G ⎪ 61 + n j i 62 j j ψG ⎪ ⎪ ⎪ i ⎪ P ⎪ j i+1 i ⎭ A P P j = a71 I− (ϕm ) j (ti − τ ) − a72 P j + ψ ,

(24)

P

where, ϕn (ti − τ ) = [(Hn )i1 , (Hn )i2 . . . , (Hn )iL −1 ] , ϕ f (ti − τ ) = [(H f )i−L , (H f )i−L +1 . . . , (H f )ix0 −1 ] , 2

2

2

ϕm (ti − τ ) = [(Hm )i−L , (Hm )i−L +1 . . . , (Hm )ix0 −1 ] , ϕG (ti − τ ) = [G i−L , G i−L +1 . . . , G iL −1 ] , 2

2

2

2

2

ϕ P (ti − τ ) = [(H P )i−L , (H P )i−L +1 . . . , (H P )iL −1 ] . 2

2

2

The FOFDM is then consists of equations (18)–(23). Rewriting the FOFDM as a system of equations, we have ⎫ An N = Fn , A f F = F f , Am M = Fm , ⎬ Aρ R = Fρ , A E E = FE , A G G = FG , ⎭ A P P = FP . Let the functions n(x, t), f (x, t), m(x, t), E(x, t), G(x, t), P(x, t),

(25)

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment

71

and their partial derivatives with respect to both t and x be smooth such that they satisfy    i+ j  i+ j   ∂ n(t, x)  f (t, x)   ≤ ϒn ,  ∂   ∂t i x j  ≤ ϒ f ,  ∂t i x j     i+ j  i+ j    ∂ m(t, x)   ≤ ϒm ,  ∂ E(t, x)  ≤ ϒ E ,   ∂t i x j   ∂t i x j     i+ j  i+ j    ∂ G(t, x)   ≤ ϒG ,  ∂ P(t, x)  ≤ ϒ P , ∀i, j ≥ 0,    ∂t i x j   ∂t i x j

(26)

where, ϒn , ϒ f , ϒm ,ϒ E , ϒG , ϒ P , are constant that are independent of the time and space step-sizes. Then in view of the FOFDM one can see that the truncation errors ςn , ς f , ςm , ςρ , ς E , ςG , ς P , are given by ⎫ (ςn )ij = (An n)ij − (Fn )ij = (An (n − N ))ij , ⎪ ⎪ ⎪ ⎪ (ς f )ij = (A f f )ij − (F f )ij = (A f ( f − F))ij , ⎪ ⎪ ⎪ i i i i ⎪ (ςm ) j = (Am m) j − (Fm ) j = (Am (m − M)) j , ⎪ ⎬ (ςρ )ij = (Aρ ρ)ij − (Fρ )ij = (Aρ (ρ − R))ij , ⎪ (ς E )ij = (A E E)ij − (FE )ij = (A E (E − E))ij , ⎪ ⎪ ⎪ i i i i ⎪ ⎪ (ςG ) j = (A G Gn) j − (FG ) j = (A G (G − G)) j , ⎪ ⎪ ⎪ (ς P )ij = (A P P)ij − (FP )ij = (A P (P − P))ij . ⎭

(27)

Therefore, ⎫ i maxi, j |n ij − N ji | ≤ ||A−1 n || maxi, j |(ςn ) j |, ⎪ ⎪ ⎪ i ⎪ maxi, j | f ji − F ij | ≤ ||A−1 ⎪ f || maxi, j |(ς f ) j |, ⎪ i i i ⎪ −1 ⎪ maxi, j |m j − M j | ≤ ||Am || maxi, j |(ςm ) j |, ⎪ ⎬ i i i −1 maxi, j |ρ j − R j | ≤ ||Aρ || maxi, j |(ςρ ) j |, ⎪ i ⎪ maxi, j |E ij − E ij | ≤ ||A−1 E || maxi, j |(ς E ) j |, ⎪ ⎪ ⎪ −1 i i i ⎪ maxi, j |G j − G j | ≤ ||A G || maxi, j |(ςG ) j |, ⎪ ⎪ ⎪ ⎭ −1 i i i maxi, j |P j − P j | ≤ ||A P || maxi, j |(ς P ) j |, where,

(28)

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K. M. Owolabi et al.

⎫ 2 ⎪ (ςn )ij ≤ (t) |n tt (ξ)| − Dn (x) |n x x x x (ζ)|, x ∈ [xs , L/2], ⎪ 2 12 ⎪ ⎪ (x)2 (t) L ⎪ i ⎪ (ς f ) j ≤ 2 | f tt (ξ)| − D f 12 | f x x x x (ζ)|, x ∈ [− 2 , xs ], ⎪ ⎪ 2 ⎪ (t) (x) L i ⎪ ⎪ (ςm ) j ≤ 2 |m tt (ξ)| − Dm 12 |m x x x x (ζ)|, x ∈ [− 2 , xs ], ⎬ 2 (t) (x) L−x ∗ L−x ∗ i (ςρ ) j ≤ 2 |ρtt (ξ)| − Dρ 12 |ρx x x x (ζ)|, x ∈ [− 2 , 2 ], ⎪ 2 ⎪ ⎪ (ς E )ij ≤ (t) |E tt (ξ)| − D E (x) |E x x x x (ζ)|, x ∈ [− L2 , L2 ], ⎪ ⎪ 2 12 ⎪ ⎪ (t) (x)2 L L i ⎪ (ςG ) j ≤ 2 |G tt (ξ)| − DG 12 |G x x x x (ζ)|, x ∈ [− 2 , 2 ], ⎪ ⎪ ⎪ 2 ⎭ (t) (x) L L i (ς P ) j ≤ 2 |Ptt (ξ)| − D P 12 |Px x x x (ζ)|, x ∈ [− 2 , 2 ],

(29)

for ti−1 ≤ ξ ≤ ti+1 and x j−1 ≤ ζ ≤ x j+1 . In view of inequalities in (26) we see that the inequalities in (29) is equivalent to 



⎫

2 (t) ⎪ ϒ , x ∈ [xs , L/2], − Dn (x) ⎪ ⎪ 12  n ⎪  2 ⎪ 2 ⎪ (t) (x) L i ⎪ (ς f ) j ≤ 2 − D f 12 ϒ f , x ∈ [− 2 , xs ], ⎪ ⎪ ⎪   ⎪ 2 ⎪ (t) (x) L i ⎪ (ςm ) j ≤ 2 − Dm 12 ϒm , x ∈ [− 2 , xs ], ⎪ ⎪ ⎬   2 (t) (x) L−x ∗ L−x ∗ i (ςρ ) j ≤ 2 − Dρ 12 ϒρ , x ∈ [− 2 , 2 ], ⎪   ⎪ ⎪ (x)2 L L ⎪ (ς E )ij ≤ (t) ϒ − D , x ∈ [− , ], ⎪ E E ⎪ 12  2 2 ⎪  2 ⎪ 2 ⎪ (t) (x) L L i ⎪ (ςG ) j ≤ 2 − DG 12 ϒG , x ∈ [− 2 , 2 ], ⎪ ⎪ ⎪   ⎪ 2 ⎪ (t) (x) L L i ⎭ (ς P ) j ≤ 2 − D P 12 ϒ P , x ∈ [− 2 , 2 ],

(ςn )ij ≤

(30)

for ti−1 ≤ ξ ≤ ti+1 and x j−1 ≤ ζ ≤ x j+1 . Moreover, by a result in [51], we have −1 −1 −1 −1 ||A−1 n || ≤ n , ||A f || ≤  f , ||Am || ≤ m , ||Aρ || ≤ ρ , ||A E || ≤  E , −1 ||A−1 G || ≤ G , ||A P || ≤  P .

(31)

Using (30) and (31) in (28), we obtain the following results. Theorem 3.1 Let Fn (x, t), F f (x, t), Fm (x, t), Fρ (x, t), FE (x, t), FG (x, t), FP (x, t), be sufficiently smooth functions so that n(x, t), f (x, t), m(x, t), ρ(x, t), E(x, t), G(x, t), P(x, t) ∈ C ∞ ([−L , L] × [0, T ]). Let (N ji , F ij , Mij , Rij , E ij , G ij , P ij , ), j = 1, 2, . . . L , i = 1, 2, . . . T be the approximate solutions to (6), obtained using the FOFDM with N j0 = n 0j , F 0j = f j0 , M0j = m 0j , R0j = ρ0j , E 0j = E 0j , G 0j = G 0j , P 0j = P j0 ,. Then there exists n ,  f , m , ρ ,  E , G ,  P independent of the step sizes t and x such that

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 (x) (t) i i ⎪ maxi, j | f j − F j | ≤  f [ 2 − D f 12 (ζ)]ϒ f , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ (t) (x) i i ⎪ maxi, j |m j − M j | ≤ m [ 2 − Dm 12 ]ϒm , ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ 2 (t) (x) i i maxi, j |ρ j − R j | ≤ ρ [ 2 − Dρ 12 ]ϒρ , ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ (t) (x) i i maxi, j |E j − E j | ≤  E [ 2 − D E 12 ]ϒ E , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (t) (x)2 i i maxi, j |G j − G j | ≤ G [ 2 − DG 12 ]ϒG , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 (t) (x) i i maxi, j |P j − P j | ≤  P [ 2 − D P 12 ]ϒ P . ⎭

73

maxi, j |n ij − N ji | ≤ n [ (t) − Dn (x) ]ϒn , 2 12 2

(32)

4 Numerical Results and Discussions Let Sx = St = 60, times t = 20 and t = 30 and using the parameter values in Table 1, for L = 5 and T = 1, the numerical results for the system in Eq. (15) without a delay term (omitted) are the same as in [41]. Therefore, numerical results for discretised system in Eq. (15) with a delay term are presented for times t = 20, & 30 and delay terms τ = 5, & 10, in Figs. 1 and 2, respectively. The profiles for the dynamics without-delay (not shown) present that the density of the transformed epithelia cells are steadily rising to their steady state within their compartment. Similar phenomena can be seen on the behaviour of the density of fibroblasts, whereas, for the density of the myofobroblasts, a slight growth of the density of the cells is noted, which suddenly increases near the end of their prescribed compartment. Moreover, the concentration of the extracellular matrix,

Table 1 Dimensionless parameter values used for the invasion essay model [31] Dn = 3.6 × 10−4 D f = 6.12 × 10−5 Dm = 6.12 × 10−4 Dρ = 5.12 × 10−4 De = 5.98 × 10−1 λ E = λG = λρ ρ = a12 = 1.00 k E = 3.32 a62 = 2.89 × 10−2 a43 = 0.518 a31 = 4.53 × 10−3 χm = 3.96 × 10−6

Dg = 3.6 × 10−1 a11 = 0.69

D p = 3.6 × 10−1 a22 = 1.58 × 10−2

χn = 3.6 × 10−8 a51 = 2.03 × 10−1

κ = 2.88 × 103 a71 = 3732 n  = 2.88 × 103 r f = 100.0 a52 = 2.89 × 10−2

B = 5.00 a72 = 0.259 = 0.1 γ = 0.1 a61 = 2.03 × 10−1

a53 = 2.89 × 10−2 a41 = 3732 a21 = 2.61 × 10−2 χ1n = 1.8 × 10−4 a42 = 0.259

74

K. M. Owolabi et al. 1.2

0.15

(b)

(a)

1

n

0.8

0.1

f

0.6

0.4

0.05

0.2

0

0 -5

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x

0.035

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(c)

0.03

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0.025 -200

m

Rho

0.02

-300 -400

0.015

-500 0.01 -600 0.005

(d)

-700

0

-800 -5

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35

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1.06 1.04

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G

20 1

15 0.98 10 0.96 5

0.94 0.92

0 -5

-4

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x

x 2

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1.2 1 0.8 0.6 0.4 0.2 0 -5

-4

-3

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0

1

2

3

4

5

x

Fig. 1 Numerical solution of the system in (5) with delay τ = 5 and at time (t) = 20. Plots a–h correspond to the behaviors of the transformed epithelial cells (TECs), fibroblasts cells, myofibroblasts cells, concentration of extracellular matrix (ECM), concentration of epidermal growth factor molecules (EGF), concentration of transformed growth factor molecules (TGF-β), concentration of matrix metallo proteinase (MMP), respectively

Numerical Solution for a Tumor Cells Dynamics Within Their Micro-environment 1.2

75

0.16

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0.1 0.6

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0.03 0.025

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0.005 -1200

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x

Fig. 2 Numerical solution of the system in (5) with delay τ = 10 and at time (t) = 30. Plots a–h correspond to the behaviors of the transformed epithelial cells (TECs), fibroblasts cells, myofibroblasts cells, concentration of extracellular matrix (ECM), concentration of epidermal growth factor molecules (EGF), concentration of transformed growth factor molecules (TGF-β), concentration of matrix metallo proteinase (MMP), respectively

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possesses a slight growth of the concentration which is decreasing as the experiment progresses. It can be seen that the behaviour of fibroblasts gives rise to the behaviour of the density of the concentration of epidermal growth factor, which they secretes, whereas, the density of the transformed fibroblasts cells are influenced by the behaviour of the concentration of the matrix metalloproteinase to a certain extend, which once more enhanced by the concentration of the epidermal growth factor molecules. Another interesting phenomena is seen on the behaviour of the concentration of the transformed growth factor molecules, which is attributed by the density of the transformed epithelial cells. These phenomena are exactly the same as in Fig. 1. Interestingly, the sinusoidal behaviour for the density of transformed epithelial cells at an initial stage is notable, just before the density rises to its steady state. The effects of the delay term in the behaviour of the concentration of the transformed growth factor is notable, whereas the degradation of the extracellular matrix, behaviours of the concentration of epidermal growth factor and matrix metalloproteinase are presented in a manner, which one can deduce a relationship between the concentration of epidermal growth factor with that of the concentration of matrix metalloproteinase. Thus, the even though this experiment is in silico, the experimental results mimic that of real scenarios [2].

5 Conclusion In this chapter, we improved the results for the derived system of nonlinear quasi time-depended delay parabolic partial differential equations [31, 41]. Even though the main aim is to improve the results obtained in [31, 41], one can indeed see that the incorporation of a delay term for crucial transformations ought to take take place during the interaction of the experiment proposed in [31] is a significant contribution. The incorporation of some transformations, led the original model to be transformed to a system of non-linear quasi time-depended delay parabolic partial differential equations. The establishment of well-pose for existence of uniqueness of solution led to the extension of Gronwall’s inequality for linear delay differential equations. The analysis of the resulting system of non-linear quasi time-depended delay parabolic partial differential equations, enabled us to establish the asymptotic stability condition and the Hopf bifurcation. Consequently, the derived numerical method based on a fitted operator for solving the modified system in Eq. (5), is analyzed and implemented iterative in MATLAB R2021. Vividly, the delay term can be observed after some time of the transformations, which in turn enables one to see the sensitivity of the density of transformed epithelial cells. Thus, these findings are indeed essential toward the design of the drug that can slow and/or confine tumor invasion, particularly when the analysis present that the Hopf bifurcation affects the entire experiment through the density of fibroblasts. Hence, the future research direction is to extend the experiment to the higher dimensional space alond the recent developments reported in [3–5, 25, 37–40, 52–55].

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A Mathematical Model to Study Regulatory Properties and Dynamical Behaviour of Glycolytic Pathway Using Bifurcation Analysis Shruti Tomar, Naresh M. Chadha, and Ankita Khanna

Abstract The glycolytic pathway is an almost universal central pathway of glucose catabolism and is known to be highly conserved across various species. In some mammalian tissues and cell types (for example: erythrocytes, renal medulla, brain, and sperm), the glycolytic breakdown of glucose is the sole source of metabolic energy. In glycolysis, phosphofructokinase-1 (PFK-1) catalyzes the transfer of a phosphoryl group from ATP to fructose 6-phosphate to yield fructose 1, 6-bisphosphate. The PFK-1 reaction is essentially an irreversible reaction under cellular conditions. Additionally, it is the first “committed” step in the glycolytic pathway because glucose 6-phosphate and fructose 6-phosphate have other possible fates, but fructose 1, 6bisphosphate is targeted for glycolysis. In this chapter, a dynamical system based on mass balance equations and S-system representation has been formulated to study the regulatory properties, rate control distributions, and dynamical behaviour of reactions in this glycolytic pathway. This representation involves several parameters. Few standard tools from the bifurcation analysis such as phase-portraits, time series plots, the Lyapunov exponents and d∞ plots have been employed to investigate the stability of the pathway with respect to certain chosen parameters. The study is further extended by eliminating the glycogen branch from the original pathway and by adding an external perturbation into the system.

1 Introduction The beginning of the study of glycolysis started with Eduard Buchner’s discovery of fermentation in broken extracts of yeast cells in 1897. Throughout 1930s, the reactions of glycolysis in extracts of yeast and muscle were a major focus of biochemical research which led to whole pathway elucidation in yeast by Otto Warburg and Hans S. Tomar · N. M. Chadha (B) · A. Khanna School of Physical Sciences, DIT University, Uttarakhand, India e-mail: [email protected] A. Khanna e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_4

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von Euler-Chelpin [50]. The whole glycolytic pathway was elucidated in muscles by Gustav Embden and Otto Meyerhof [10, 25]. Glycolysis occurs in the cytosol of the cell. It is a metabolic pathway that creates ATP without the use of oxygen but can also occur in the presence of oxygen. In cells, during aerobic respiration, the pyruvate formed from the pathway enters the citric acid cycle and goes through oxidative phosphorylation to undergo oxidation into carbon dioxide and water. In highly oxidative tissues, such as the heart, pyruvate production is essential for acetyl-CoA synthesis and L-malate synthesis. It serves as a precursor to many molecules, such as lactate, alanine, and oxaloacetate. Glycolysis precedes lactic acid fermentation i.e., the pyruvate produced in glycolysis serves as pre-requisite for the lactate produced in the lactic acid fermentation. Skeletal muscles prefer to catalyze glucose into carbon dioxide and water during heavy exercise. But when oxygen is inadequate, these muscles simultaneously undergo anaerobic glycolysis and oxidative phosphorylation [19]. The chemistry of glycolysis has been completely conserved during evolution from vertebrates to yeast to spinach. Across different species, glycolysis differs only in its regulation and in subsequent metabolic fate of the pyruvate formed. The thermodynamic principles and types of mechanisms regulating glycolysis are common to all pathways of cell metabolism. Many authors have shown their interest in glycolysis pathway in different kind of cells namely E. coli cells, mammalian cells, pancreatic β-cells etc. [7, 8]. In a study by Maria et al. [22], some characteristics and cell components that caused stable glycolytic oscillations in E. coli cells have been identified. Mulukutla et al.[24] studied the diversity of steady states in glycolysis and shifts in metabolic states in cultured mammalian cells with the help of mathematical modelling. They reported bistable behaviour of glycolytic enzymes regulated by lactate inhibition, including the AKT pathway, and provided a new technique to control the metabolism of mammalian cells in fed-batch cultures. There are many studies available in the literature where the authors have attemped to study Glycolysis through experimental and mathematical means. Camacho et al. [6] have developed a model consisting a system of non-linear differential equations for those patients who were affected by retinitis pigmentosa, an inherited retinal disease which leads to irreversible blindness. This technique is used to model cellular and molecular interactions in a cone cell. Torres and Melendez-Hevia [42] studied the glycolysis pathway in a rat liver and determined some coefficients of glucokinase, glucose 6-phosphate isomers and phosphofructokinase with the help of shortening and enzyme titration method that can control the concentration of glycolysis in rat liver. Their study suggested that phosphofructokinase may play an important role in control of liver glycolysis. Bakker et al. [2] constructed and tested a kinetic model with experimental results that described the steady state behaviour of glycolysis in the blood-stream form of Trypanosoma brucei quantitatively. Verburg [45] constructed a dynamical system with the help of rate equations to control glycolysis in T. brucei and observed the effect of included rate constants. Refer to [12, 20, 28–31, 36–39, 47, 51], where the authors have used mathematical tools to study various related models.

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Fig. 1 Pathway of glycolysis and glycogenolysis in terms of biological names and model variables [48]

In this study, we consider the Glycolytic-Glycogenolytic pathway in perfused rat-liver from Case study-4 of [48], shown in Fig. 1. Initial steps of glycolysis and glycogenolysis, in terms of biochemical names and, in terms of model variables are shown in Fig. 1. It is important to note that glycolysis and glycogenolysis are central steps of carbohydrate metabolism. They yield common intermediate, glucose-6-phosphate, which is subsequently used by the organism as a major source of energy. The concentration of glucose in both human and animal bodies varies according to nutritional requirements, just as fasting for a few days does not cause a noticeable decline in glucose levels. In these circumstances, the body must turn to other energy sources. In case of animals, if they starve for one day they loose almost all glycogen. Subsequent feeding of glycogenic substances lead to rapid replenishment in glucose level. The main route of glycogen utilization is glycogenolysis via glycogen phosphorolysis to glucose-1-phosphate and subsequent conversion to glucose-6-phosphate via the phosphoglucomutase reaction. Glucose6-phosphate can be used for energy production or formation of glucose. In the present study, the pathway depicted in Fig. 1 is represented as a dynamical system of non-linear ordinary differential equations, referred to as S-system representation, which is primarily based upon mass balance equations. As we shall further see that this dynamical system has many variables and parameters, and variation in any one of them may disturb the dynamics of the whole pathway. It is almost impossible to consider all the variables and the parameters involved in the pathway at one time, thus we restrict ourselves by choosing four parameters, namely h 33 , h 55 , h 5,10 and h 52 . These parameters are involved in the phosphofructokinase and the glucokinase reactions, respectively. Furthermore, these parameters may directly or indirectly influence the concentrations of glucose-6-P and fructose-6-P in the pathway.

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The main contributions of this study are highlighted here: we have studied the effect of range of the parameters h 33 , h 55 , h 5,10 and h 52 on the concentration of glucose in glycolytic pathway with the help of standard tools from the bifurcation analysis such as phase-portraits, time series plots, Lyapunov exponents and d∞ plots, [23]. In glycolytic pathway, glycolysis and glycogenolysis form central steps to produce glucose-6-P which can be further used to generate glucose. Since both the branches play an important role in this pathway, it was intriguing for us to study how this pathway behaves when we remove any one of these branches. We study the pathway by removing the right branch (glycogen branch) of the glycolytic pathway in rat liver. In our study, we have perturbed the glycolytic system by adding a forcing term of the form f 0 cos(ωt). The main observation of the study is that the change in the parameters h 33 , h 55 , and h 5,10 , have moderate effects on the dynamics of the pathway but a positive value of h 52 may convert a stable system into an unstable one. We have also observed that if we remove the right branch of the pathway then the converted system does not show any specific changes in the concentration of glucose but external perturbations such as f 0 cos(ωt) can reverse the process. The organization of chapter is as follows: In Sect. 2, the pathway is represented as a dynamical system using the mass balance equations and a S-system representation. In Sect. 3, we have discussed the interrelation between the parameters which are included in the dynamical system. The standard tools from bifurcation analysis have been employed to study the pathway in Sect. 4. In Sect. 5, one of the branches of pathway is removed and external perturbation is introduced in the dynamical system and the standard tools have been used to study the stability of the dynamics of the pathway. Results and future work have been discussed in Sects. 6 and 7, respectively. Few intermediate steps to calculate the rate constants, and certain important steps for the steady state analysis of the pathway are presented in Appendices 1 and 2.

2 Glycolytic Pathway, Mass Balance Equations, and S-System Representation To represent the glycolytic pathway as a system of differential equations, the first step is to identify the independent and dependent variables. The steady state concentrations of the dependent and independent variable have been taken from Case study-4 [48]. (1) Steady state concentrations and steady state fluxes: For the model, the primary data consist of steady-state concentrations and steadystate fluxes. V1+ = 1.068 µmol/min per gram fresh liver weight

V3− = 1.714 µmol/min per gram fresh liver weight V5− = 0.646 µmol/min per gram fresh liver weight

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(2) Dependent Variables: [G1P] = X 1s = 0.067 mM [G6P] = X 2s = 0.465 mM [F6P] = X 3s = 0.150 mM (3) Independent variables: [Pi ] = X 4 = 10.00 mM [Glucose] = X 5 = 5.00 mM [Phosphorylase a] = X 6 = 3.00 µmol/min/gm fresh liver weight [Phosphoglucomutase] = X 7 = 40.00 µmol/min/gm fresh liver weight [Phosphoglucose iso.] = X 8 = 136.00 µmol/min/gm fresh liver weight [Phosphofructokinase] = X 9 = 2.86 µmol/min/gm fresh liver weight [Glucokinase] = X 10 = 4.00 µmol/min/gm fresh liver weight [Glycogen] = X 11 = 50.00 µmol/min/gm fresh liver weight (4) Steady-state Fluxes: V1+ V2+ V3+ V5−

= V1− = 1.068 µmol/min/gm fresh liver weight = V2− = 1.714 µmol/min/gm fresh liver weight = V3− = 1.714 µmol/min/gm fresh liver weight = 0.646 µmol/min/gm fresh liver weight

Since X 1 , X 2 , X 3 are three dependent variables, so we have three equations with other variables. The mass-balance equation can represented as d X1 = V1+ (X 4 , X 6 , X 11 ) − V1− (X 1 , X 2 , X 7 ), dt d X2 = V2+ (X 1 , X 2 , X 5 , X 7 , X 10 ) − V2− (X 2 , X 3 , X 8 ), dt d X3 = V3+ (X 2 , X 3 , X 8 ) − V3− (X 3 , X 9 ). dt

(1)

The rate of change in concentration with respect to time is represented by ddtX 1 , ddtX 2 , and ddtX 3 , whereas the function Vi+,− s are functions of concentrations. This system is concerned with both the generation and depletion of X i ’s. A positive sign (+) indicates a change in product, while a negative sign (−) indicates a change in the substrate. These equations given by Eq. (1) have been converted into a system of non-linear ordinary differential equations, commonly referred to as S-system representation. Each flux term’s variable is represented by a product of power-law functions with a kinetic order parameter gi j or h i j , whose two indices correspond to the number of affected and effecting variables, respectively. Furthermore, each flux term is assigned

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Table 1 Kinetic order values and their representative reactions Kinetic orders Values Reaction g14 , g16

0.66, 1

h 11 , h 12 , h 17

1.53, −0.59, 1

h 55 , h 52 , h 5,10 h 22 , h 23 , h 28

0.84, −0.1, 1 3.97, −3.06, 1

h 33 , h 39

0.3, 1

Phosphorylase a reaction Phosphoglucomutase reaction Glucokinase reaction Phosphoglucose isomerase reaction Phosphofructokinase reaction

Reference [11, 34] [11, 32, 35] [3, 13, 26, 34, 44] [3, 26, 40] [13, 26, 33, 43, 44]

a rate constant αi or βi , whose index corresponds to the number of equation. The resulting S-system representation of the dynamical system is d X1 g g = α1 X 4 14 X 6 16 − β1 X 1h 11 X 2h 12 X 7h 17 , dt d X2 g g g g g = α2 X 1 21 X 2 22 X 5 25 X 7 27 X 102,10 − β2 X 2h 22 X 3h 23 X 8h 28 , dt d X3 g g g = α3 X 2 32 X 3 33 X 8 38 − β3 X 3h 33 X 9h 39 . dt

(2)

In this system Eq. (2), we have total 28 exponents and rate constant parameters, the values of some of them are taken from literature, shown in Table 1. The values of rest of the parameters viz. g21 , g22 , g25 , g27 , g32 , g33 , g38 and g2,10 are computed from the branch point constraints, refer to Appendix 1. These values are g21 = 0.95, g22 = −0.41, g25 = 0.32, g27 = 0.62 and g2,10 = 0.38 and g32 = h 22 = 3.97, g33 = h 23 = −3.06, g38 = h 28 = 1. At steady state, the rate constants are calculated by equating the measured fluxes with the relevant power-law terms [48]. The main emphasis in this study is on the parameters h 33 , h 55 , h 52 and h 5,10 which are involved in mathematical representation of the phosphofructokinase and the glucokinase reactions, respectively. Therefore, any change in the value of these parameters will affect the concentrations of X 2 i.e. Glucose-6-P and X 3 i.e. Fructose6-P. The dynamical system Eq. (2) consist many parameters which appear as exponents gi j , h i j , and the rate constants, αi ’s and βi ’s. In many practical applications, it may not be easy to control a parameter or a set of particular parameters in order to obtain a desired outcome. Thus, it is instructive to explore the inter-dependency of these parameters. For this, we formulate a set of transcendental equations obtained by considering the steady state of the dynamical system Eq. (2). The solution set of these transcendental equations will give the interrelations between certain important parameters. This is the subject matter of Sect. 3.

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3 Interrelation Between Certain Parameters By using certain precursor-product relations and the branch point restrictions (refer to Appendix 1), we have 1.068 0.646 1.608 , g25 = h 55 , g27 = h 17 , 1.741 1.741 1.714 0.646 1.068 0.646 , g22 = h 12 + h 52 . = h 5,10 1.714 1.741 1.741

g21 = h 11 g2,10

(3)

And by using the precursor-product relation, we have g1,11 = 0, g22 = h 22 , g33 = h 23 , g38 = h 28 , α3 = β2 .

(4)

From the dynamical system Eq. (2) at steady state, in conjunction with relations in Eqs. (3) and (4), we have the following set of transcendental equations α1 10g14 3g16 − β1 (0.067)h 11 (0.46)h 12 40h 17 = 0, α2 X 10.6314h 11 X 20.6314h 11 +0.3769h 52 50.3769h 55 400.6314h 17 40.3769h 5,10 − β2 X 2h 22 X 3h 23 136h 28 = 0, β2 X 2h 22 X 3h 23 136h 28 − β3 X 3h 33 2.86h 39 = 0.

(5)

By using symbolic calculation in MAPLE software, the solution set of the transcendental Eq. (5) is β1 0.067h 11 0.46h 12 40h 17 , 3g16 10g14 α2 = α2 , β1 = β1 , α1 =

α2 0.0670.6134405514h 11 0.465(0.6134405514h 12 +0.3768961494h 52 ) 50.3710511200h 55 0.465h 22 0.9381563594h 17 0.3768961494h 5,10 40 4 , 0.15h 23 136h 28 (1.591199772e − 18)α2 0.0670.6134405514h 11 0.4650.6134405514h 12 +0.3768961494h 52 β3 = 1 50.3710511200h 55 400.9381563594h 17 40.3768961494h 5,10 . (6) 0.15h 33 β2 =

As mentioned earlier, in the present study, we have considered the parameters h 52 , h 33 , h 5,10 and h 55 to analyse the behaviour of the dynamical system Eq. (2). By setting the values of other parameters from Tables 1 and 2, the solution set Eq. (6) reduces to the following form

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Table 2 Rate constants and their values Rate constant Values α1 α3 β2 β5

0.077884314 0.000793456 0.000793456 0.038706421

Rate constant

Values

α2 β1 β3

0.585012402 1.062708258 1.05880847

Fig. 2 Behaviour of the rate constant parameter β2 , and β3 for range of parameter h 5,10 ∈ (0.001, 1.5), h 55 ∈ (0.001, 1.5), h 33 ∈ (0.001, 1), and h 52 ∈ (−0.06, −0.001). The values of parameters are g14 = 0.66, g16 = 1, h 11 = 1.53, h 12 = −0.59, h 17 = 1, h 39 = 1, h 22 = 3.97, h 23 = −3.06, h 28 = 1, and rate constants values are α2 = 0.58501242, β1 = 1.062708258

α1 = 14.7388, β2 = (2.3451e + 06)0.465(0.3769h 52 −0.3619) 50.3711h 55 h 5,10 , β3 =

(2.3446e − 18)0.465(0.3769h 52 −0.3619) 40.3769h 5,10 50.3711h 55 . 0.15h 33

(7)

The relations between the parameters β2 , β3 and h 52 , h 33 , h 5,10 and h 55 given by Eq. (7) are depicted in Fig. 2. In Fig. 2a, b, the surface plots for β2 with respect to the parameters h 55 , h 52 and h 5,10 are plotted. For the range of parameters h 55 ∈ (0.001, 1.5) and h 5,10 ∈ (0.001, 1.5), β2 increases but for h 52 ∈ (−0.6, −0.001), a slight decrease in value of β2 is observed. For the same range of the parameters used to Fig. 2a, b, β3 show a slightly different trend as compared to β2 which is shown in Fig. 2c, d and e. It is important to note that the scale of the surface plots for β2 and β3 for the same range of the parameters h 55 , h 52 and h 5,10 is very different. Furthermore,

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a change in the values of these parameters can affect the concentration of X 2 and X 3 i.e., Glucose-6-P and Fructose-6-P in the Glycolytic pathway. In the next section Sect. 4, we study the dynamical system Eq. (2) to investigate the behaviour of the pathway with respect to these chosen parameters by using the standard tools from the Bifurcation Analysis. The commonly used tools are phaseportraits, Lyapunov exponents, d∞ parameters plots and time series plots [41].

4 Bifurcation Analysis of the Glycolytic Pathway Bifurcation analysis is a mathematical study of qualitative changes in system dynamics which is commonly used to investigate the unexpected appearance, disappearance, or change in the stability of equilibrium points with respect to certain parameters or perturbations. Bifurcation analysis has long been used to investigate dynamical systems emerging from varied real world problems, refer to [15, 41, 49] and references therein. For the bifurcation analysis also, we shall continue with our chosen parameters for the study, h 55 , h 33 ,h 52 and h 5,10 . We substitute the values of the independent variables X 4 to X 11 in the dynamical system Eq. (2) to obtain the following dynamical system d X1 = α1 10g14 3g16 − β1 X 1h 11 X 2h 12 40h 17 , dt d X2 = α2 X 10.6314h 11 X 20.6314h 11 +0.3769h 52 50.3769h 55 400.6314h 17 40.3769h 5,10 − β2 X 2h 22 X 3h 23 136h 28 , dt d X3 (8) = β2 X 2h 22 X 3h 23 136h 28 − β3 X 3h 33 2.86h 39 . dt

To investigate the stability of the dynamical system, we first calculate the Jacobian matrix [5, 21] of the dynamical system Eq. (8) which is given below ⎡

⎤ A B 0 J = ⎣C D E ⎦, 0 F G

(9)

where A=− D=

64.8752X 10.53 X 20.59 1



, B=

25.016X 11.53 X 21.59

, C=

17.4848(40.38h 5,10 )(50.37h 55 )X 2(0.38h 52 −0.3619) X 10.0614

(0.38h 52 −1.3619)

18.672(50.37h 55 )(4(0.38h 5,10 ) X 10.9386 X 2

X 33.06  1  − 3.06 0.3382(0.465)0.37h 52 −0.354 (40.37h 5,10 )X 22.97 , X3

 0.38h 52 − 0.3619

,

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E= G=

0.3382(0.465)0.37h 52 −0.354 40.37h 5,10 50.37h 55 X 23.97

, F=

1170.38(40.38h 5,10 )(50.37h 55 )X 22.97

X 34.06 −0.2607(0.465)0.37h 52 −0.354 40.37h 5,10 50.37h 55 X 23.97 X 34.06 (8.1112e − 18)0.4650.38h 52 −0.3599 40.376h 5,10 )50.375h 55 )h

−

X 33.06

h33−1 ) 33 X 3

0.15h 33

,

.

For obtaining the equilibrium point(s), we have solved the dynamical system Eq. (8) at steady state in MAPLE. Considering the number of parameters involved in the dynamical system, the expressions obtained for the equilibrium points are very complex. Thus, we consider the equilibrium point as (X 1 , X 2 , X 3 ) = (0.067, 0.465, 0.150) which is the concentration of dependent variables at the steady state reported in case study-4 [48]. For this equilibrium point, determinant of the Jacobian matrix Eq. (9) is |J | = (6.3627e − 115)((2.1808e + 117)0.4650.75h 52 40.75h 5,10 50.74h 55 h 52 − (4.1260e + 100)0.4650.75h 52 40.746h 5,10 50.755h 55 h 33 + (4.2913 + 99)0.4650.75h 52 40.756h 5,10 50.755h 55 h 33 h 52 ).

(10)

It is well known that the nature of an equilibrium point can be determined by the Jacobian matrix corresponding to a dynamical system. If the value of determinant of the Jacobian matrix is positive, i.e. |J | > 0, then, equilibrium point will be a stable point and if it is negative i.e. |J | < 0, then, the equilibrium point will be an unstable point or saddle point. For this dynamical system Eq. (8), determinant Eq. (10) for the range of the set value of (h 33 , h 52 ) ∈ (0.001, 1) × (−0.6, −0.001) and (h 55 , h 5,10 ) ∈ (0.001, 1.5) × (0.001, 1.5) is negative. Hence, the equilibrium point (X 1 , X 2 , X 3 ) = (0.067, 0.465, 0.150) is a saddle point which is shown in Fig. 3a, b. But when we choose positive value of inhibition parameter h 52 ∈ (0.001, 0.6), then the determinant value becomes positive. Hence, for this parameter range, the equilibrium point (X 1 , X 2 , X 3 ) = (0.067, 0.465, 0.150) will behave as a stable point, Fig. 4a. Characteristic polynomial of Jacobian matrix Eq. (9) is aλ3 + bλ2 + cλ + d = 0,

(11)

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Fig. 3 Negative value of determinant of Jacobian matrix Eq. (9) in range of parameters h 55 ∈ (0, 1.5), h 5,10 ∈ (0, 1.5), h 52 ∈ (−0.6, −0.001) and h 33 ∈ (0.001, 1) shown in (a), (b) Fig. 4 The positive value of determinant in the positive range of parameter h 52 ∈ (0.001, 0.6). Rest parameter values are h 33 = 0.3, h 5,10 = 1, h 55 = 0.84

where a = (1.2745e + 118)0.15h 33 , b = 0.15h 33 (0.465)0.37h 52 50.37h 55 ((6.5444e + 119)40.37h 5,10 − (2.0082e + 118)40.38h 5,10 h 52 + (9.0783e + 101)40.38h 5,10 h 33 + (1.9284e + 118)40.38h 5,10 ) + (3.1003e + 119)(3/20)h 33 ), c = (0.15)h 33 (0.465)0.37h 52 40.37h 510 50.37h 55 ((1.5920e + 121) + (2.2084e + 103)h 33 − (4.8852e + 119)h 52 ) + (0.15)h 33 (0.465)0.75h 52 40.76h 510 50.74h 55 ((1.3754e + 103)h 33 + (1.3737e + 102)h 33 ) − (7.2695e + 119)h 52 − (1.4305e + 102)h 33 h 52 + (6.9809e + 119)), d = (0.15)h 33 (0.465)0.75h 52 40.76h5,10 50.74h 55 ((3.3458e + 104)h 33 − (1.7684e + 121)h 52 − (3.4798e + 103)h 33 h 52 ).

Next, we investigate the nature of eigenvalues of the characteristic polynomial Eq. (11). According to lemma [9], if the coefficients of a polynomial are positive then, the roots of the polynomial will be negative. Here, in Fig. 5, we have shown

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Fig. 5 Mesh plots of the coefficients of characteristic polynomial Eq. (11) for range of h 33 ∈ (0.001, 1), h 55 ∈ (0.001, 1.5), h 52 ∈ (−0.6, −0.001) and h 5,10 ∈ (0.001, 1.5) shown in (a), (b), (d), (e), (g) and (h). For positive range of parameter h 52 ∈ (0.001, 0.6) shown in (c), (f), (i)

that for a chosen range of parameters h 33 , h 55 , h 52 and h 5,10 , the value of coefficients will be positive. Hence, roots of the characteristic polynomial will be negative. On this basis, for the range of these parameters, one can predict that the the dynamical system Eq. (8) will remain stable. The phase portrait of the dynamical system Eq. (8) for h 33 ∈ (0.001, 1) and related time series plots that illustrate the rate of rise in concentration of X 1 , X 2 , and X 3 with respect to these parameters are shown in Fig. 6. With increasing value of parameter h 33 , the concentration of X 3 increases faster than X 2 and X 1 ; for h 33 = 1, system reaches to a break point, refer to Fig. 6 (i). Figure 7 shows similar behaviour for other parameters h 55 , h 52 and h 5,10 with respect to time. In Fig. 8, we have noticed that the rate of change in X 1 is greater than X 2 and X 3 during short time periods (0 − 0.05). However, after a period of time, the rate of change in X 1 gradually decreases, and the concentration of X 3 gradually increases. This is due to the activation of feedback mechanism in a cell catalysed by Phosphoglucomutase and Phosphoglucoisomerase.

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Fig. 6 Phase portrait and time series plots of the dynamical system Eq. (8) for the range of parameter h 33 ∈ (0.001, 1), time step is [0, 50] and initial value are [0.067, 0.465, 0.150]

It is well known that a dynamical system can exhibit both regular and chaotic mode of evolution. The exponentially fast divergence or convergence of near orbits in the phase portrait can be defined by Lyapunov exponents which provide a qualitative and quantitative characterisation of the dynamical behaviour of a system [1, 21]. If any one value of the Lyapunov exponent becomes positive then the system is found to be chaotic. Figure 9 shows the dynamics of the Lyapunov exponent for two set of values for h 33 . Figure 9a shows all negative values of Lyapunov exponent

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Fig. 7 Time series plots of the dynamical system Eq. (8) for the range of parameter h 55 , h 5,10 ∈ (0.001, 1.5) and h 52 ∈ (−0.6, −0.001),. Time step is [0, 50] and initial value are [0.067, 0.465, 0.150]

(LE) for h 33 = 0.005, h 5,10 = 1, h 55 = 0.84, i.e. system may be stable but h 33 = 0.3 keeping all other parameters same as for in Fig. 9a, one L E ∗ (Lyapunov exponent for h 33 = 0.3) becomes positive with the progression in time, shown in Fig. 9b. Corresponding values of Lyapunov Exponents are shown in Table 3. The overall behaviour of dynamical system is decided by the largest Lyapunov exponent which is called the d∞ parameter. For this, we have considered the equation as (12) d˙z = λ1 dz − γ dz2 ,

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Fig. 8 Time series plots of the dynamical system Eq. (8) for the range of parameter h 33 ∈ (0.001, 1), h 55 , h 5,10 ∈ (0.001, 1.5) and h 52 ∈ (−0.6, −0.001), time step is [0, 0.05] and initial value are [0.067, 0.465, 0.150]

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Fig. 9 Lyapunov exponent graph of the dynamical system Eq. (8) for the parameter h 33 = 0.005 and h 33 = 0.3, h 55 = 0.84, h 5,10 = 1, h 52 and time interval is [0 : 50] and initial values are [0.067, 0.465, 0.150] Table 3 Lyapunov Exponents values with respect to time t = [0 : 50] when h 33 = 0.005 and 0.3, h 55 = 0.84, h 5,10 = 1. Here L E and L E ∗ represents the Lyapunov exponent for h 33 = 0.005 and h 33 = 0.3 respectively t L E1 L E2 L E3 L E 1∗ L E 2∗ L E 3∗ 5 10 15 20 25 30 35 40 45 50

−0.180282 −0.090572 −0.060607 −0.045592 −0.036566 −0.030538 −0.026226 −0.022988 −0.020465 −0.018445

−14.917085 −9.582886 −7.321299 −6.025724 −5.171144 −4.558621 −4.094747 −3.729381 −3.433004 −3.187015

−32.487695 −25.784965 −22.590820 −20.593075 −19.178657 −18.102693 −17.244812 −16.537723 −15.940341 −15.425905

−0.049546 −0.010262 −0.000617 0.003043 0.004698 0.005504 0.005898 0.006074 0.006127 0.006110

−11.507682 −6.976805 −5.141802 −4.120645 −3.461569 −2.997413 −2.651086 −2.381807 −2.165862 −1.988470

−29.511196 −22.750673 −19.570416 −17.595340 −16.204213 −15.150440 −14.313309 −13.625556 −13.046217 −12.548667

where distance between the trajectories is defined by dz from two different values of initial conditions, λ1 is the largest Lyapunov exponent and γ is a parameter accounting for the folding phenomena. For a chaotic system, the graph of dz increases with a slope λ1 until it reaches a stationary value around which it normally fluctuates. Figures 10 and 11 show that the trajectories starting from (X 1 , X 2 , X 3 ) and (X 1 + δ, X 2 + δ, X 3 + δ), (δ = 1e − 10) show fluctuating trend even for a very long time interval. Here initial value considered is (X 1 = 0.065, X 2 = 0.465, X 3 = 0.150)). This implies that the system is unpredictable which may lead to chaos. These figures correspond to range of h 33 ∈ (0.001, 1), h 55 ∈ (0.001, 1.5), refer to Fig. 10 and h 5,10 ∈ (0.001, 1.5) refer to Fig. 11 with time interval [0 : 500].

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Fig. 10 d∞ plots of the dynamical system Eq. (8) for the range of parameter h 33 ∈ (0.001, 1), h 55 ∈ (0.001, 1.5), time step is [0, 500] and initial value are [0.067, 0.465, 0.150]

5 Second Pathway In this section, we consider a pathway after removing the right branch of glycolysis pathway shown in Fig. 1. In addition to this, we introduce a perturbation term in the corresponding dynamical system. The perturbation has been considered in the form of a periodic function f 0 cos(ωt). In terms of biochemistry, a perturbation refers to addition or removal of any of the substrates/enzymes/metabolites in the pathway. Various studies have been carried out using perturbations in mathematical modelling to predict the behaviour of various biochemical systems. For instance, Vervekyo et al. [46] and Kar et al. [18] studied the nonlinear dynamics of a glycolysis pathway. Vervekyo et al. characterized the influence of periodic influx on the glycolytic oscillations within the forced Selkov system and its chaotic behaviour [4]. Brechmann et al. presented certain facts about the Selkov oscillator model by studying the properties of dynamics of various solutions of the model. Refer to [16, 17, 27] for similar studies. The dynamical system Eq. (2) takes the following form for this pathway.

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Fig. 11 d∞ plots of the dynamical system Eq. (8) for the range of parameter h 5,10 ∈ (0.001, 1.5), and h 52 ∈ (−0.6, −0.001), time step is [0, 500] and initial value are [0.067, 0.465, 0.150]

0.646h 5,10 0.646h 52 0.646h 55 d X2 = α2 X 2 1.714 X 5 1.741 X 10 1.714 − β2 X 2h 22 X 3h 23 X 8h 28 + f 0 cos(ωt), dt d X3 = α3 X 2h 22 X 3h 23 X 8h 28 − β3 X 3h 33 X 9h 39 . dt

(13)

By substituting the values of the desired parameters given in Tables 1 and 2, and the independent variables given in Sect. 2, we have the following dynamical system from Eq. (13). d X2 = 0.585012402X 20.38h 52 50.37h 55 40.38h 5,10 − 0.107910016X 23.97 X 3−3.06 + f 0 cos(ωt), dt d X3 (14) = 136X 23.97 X 3−3.06 − 3.0281922242X 3h 33 . dt

We further divide our investigation into two cases, namely when (a) f 0 = 0, and (b) f 0 = 0. These two cases are separately investigated in Sects. 5.1 and 5.2, respectively.

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5.1 When Forcing Coefficient is Equal to Zero i.e. f0 = 0 Let us consider f 0 = 0, then, reduced dynamical system will be d X2 = 0.585012402X 20.38h 52 50.37h 55 40.38h 5,10 − 0.107910016X 23.97 X 3−3.06 , dt d X3 (15) = 136X 23.97 X 3−3.06 − 3.0281922242X 3h 33 . dt At steady state, we get two sets of equilibrium points (0, 0) and (E 1 , E 2 ) where 0.38h 1 −3.97 52 136β2 ∗ E1 = α2 50.37h 55 40.38h 52 3.06

 3.97  0.38h33 h52 −3.97h 33 +1.1628h 52 2.86β3 α2 50.37h 55 40.38h 5,10 ) 0.38h52 −3.97 , 136β2 136β2

 0.38h3.97−3.97  h +3.06+3.061 3.97 33 52 136β2 136β2 0.38h 52 −3.97 . E2 = 0.37h 0.38h 55 52 2.86β3 α2 5 4 

(16)

(17)

The second equilibrium point is further dependent on the four parameters h 55 , h 33 , h 5,10 and h 52 . At the first equilibrium point, no noticeable feature worth reporting was observed in the corresponding phase portrait. Hence, we consider (0.001, 0.001) which is very close to (0, 0) for further investigation. According to the theory due to bifurcation analysis, if two trajectories are moving towards each other then, equilibrium point will be stable. But in case, if they show repulsive behaviour towards each other then, equilibrium point will be unstable or a saddle point. Here, at point (0.001, 0.001), trajectories are moving towards this point which means this point will be stable and system is stable in the neighbourhood of this point. Similar behaviour is observed for the second equilibrium point, shown in Fig. 12. The Jacobian matrix corresponding to the dynamical system Eq. (15) is J=

A B , C D

(18)

where A = 0.2223(40.38h 5,10 )50.37h 55 h 52 X 20.38h 52 −1 − C=

0.431936X 22.97 0.332928X 23.97 , B= , 3.06 X3 X 34.06

0.332928X 23.97 0.431936X 22.97 , D = − − 3.0316h 33 X 3h 33 −1 . X 32.06 X 34.06

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Fig. 12 Phase-portrait of Dynamical system Eq. (13) corresponding parameter values are h 33 = 0.3, h 55 = 0.84, h 52 = −0.1 and h 5,10 = 1

Determinant of Jacobian matrix Eq. (18) is |J | = −

1 ((2.08e − 08)(3558168(40.38h 5,10 )(50.37h 55 )h 52 X 20.38h 52 X 23.97 X 2 X 34.06

− 62954672h 33 X 23.97 X 3h 33 + 32400225(40.38h 5,10 )(50.37h 55 )h 33 h 52 X 20.38h 52 X 3h 33 X 33.06 )). (19)

General form of the characteristic polynomial is λ2 + (3.0037X 23.97 X 3−3.06 + 2.3152X 22.97 X 3−3.06 + 0.3299h 33 X 3h33−1 − 4.4984(40.38h 5,10 ) (0.38h 52 −1)

(0.38h 52 −1)

X 3h33−1 0.38h + 0.7637h 33 X 22.97 X 3h33−4.06 − 13.5117(40.38h 5,10 )(50.37h 55 )h 52 X 2 52 X 22.97 = 0,

(50.37h 55 )h 52 X 2

λ − 1.4838(40.38h 5,10 )(50.37h 55 )h 33 h 52 X 2

(20)

Stability of the system at the point (0.001, 0.001). With respect to point i.e. (0.001, 0.001), the determinant Eq. (19) is |J | = (2.4383e + 03)(0.001)h 33 h 33 − 137.8129(0.001)0.38h 52 40.38h 5,10 50.37h 55 h 52 − 6.7392 + 05(0.001)0.38h 52 (0.001)h 33 4.0.38h 5,10 50.37h 55 h 33 h 52 .

(21)

In Fig. 13a, b, c, for range of parameters h 55 ∈ (0.001, 1.5), h 5,10 ∈ (0.001, 1.5), h 33 ∈ (0.001, 1) and h 52 ∈ (−0.6, −0.001), it is shown that the determinant Eq. (21) is positive i.e. |J | > 0. Therefore, the point (0.001, 0.001) is a stable point. In system Eq. (13), if we take positive values of this parameter i.e. h 52 ∈ (0.001, 0.6), then determinant will become negative implicating that the point (0.001, 0.001) will be a saddle point for positive values of h 52 , refer to Fig. 13d. Next, we investigate the behaviour of the dynamical system by using the eigenvalues of the Jacobian matrix Eq. (18). By putting the point (0.001, 0.001) in the general

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Fig. 13 Determinant of the Jacobian matrix Eq. (21) for range of h 33 ∈ (0.001, 1), h 55 ∈ (0.001, 1.5), h 52 ∈ (−0.6, −0.001) and h 5,10 ∈ (0.001, 1.5) with respect to initial value (X 2 , X 3 ) = (0.001, 0.001) and for positive range of h 52 ∈ (0.001, 0.6) shows negative determinant

form of characteristic polynomial Eq. (20), we get the characteristic polynomial as follows λ2 + (1.4242 + (3.0316e + 03)(0.001)h 33 h 33 − 222.3000(0.001)0.38h 52 40.38h 5,10 50.37h 55 ) h 52 )λ − ((6.7392e + 05)(0.001)0.38h 52 (0.001)h 33 40.38h 5,10 5).37h 55 h 33 h 52 + (2.4383e + 03) (0.001)h 33 h 33 − 137.8129(0.001)0.38h 52 40.38h 5,10 50.37h 55 h 52 ) = 0,

(22)

and corresponding eigenvalues are 1 (111.1500(0.001)0.38h 52 40.38h 5,10 50.37h 55 )h 52 − 1.5158e + 03(0.001)h 33 h 33 − 2 (9.1906e + 06(0.001)2h 33 h 233 − (1.1178e + 03(0.01)h 33 h 33 − 81.9671(0.001)0.38h 52 40.38h 5,10

λ1 =

50.37h 55 h 52 + 4.9417e + 04(0.001)0.76h 52 40.76h 5,10 50.74h 55 h 252 + 1.3478e + 06(0.001)0.38h 52 (0.001)h 33 40.38h 5,10 50.37h 55 h 33 h 52 + 2.0285)1/2 − 0.7121, (23) 1 2h 33 2 h 33 0.38h 52 λ2 = (9.1906e + 06(0.001) h 33 − 1.1178e + 03(0.001) h 33 − 81.9671(0.001) 2 40.38h 5,10 50.37h 55 h 52 + 4.9417e + 04(0.001)0.76h 52 40.76h 5,10 50.74h 55 h 252 + 1.3478e + 06

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(0.001)0.38h 52 (0.001)h 33 40.38h 5,10 50.37h 55 h 33 h 52 + 2.0285)1/2 − (1.5158e + 03(0.001)h 33 h 33 + 111.1500(0.001)0.38h 52 40.38h 5,10 50.37h 55 h 52 − 0.7121.

(24)

These two eigenvalues are further dependent upon the three major parameters h 33 , h 55 and h 5,10 and inhibition parameter h 52 . If (λ1 , λ2 ) = (−, −) then, system will be stable and if (λ1 , λ2 ) = (+, −) i.e. if any one eigenvalue is positive then system will become unstable. For the range of h 33 ∈ (0.001, 1), h 55 ∈ (0.001, 1.5), h 52 ∈ (−0.6, −0.001) and h 5,10 ∈ (0.001, 1.5), both eigenvalues are negative which is shown in Fig. 14a–g in three dimensional graphs. But for h 52 ∈ (0.001, 0.6), one eigenvalue is positive. Hence, system will become unstable for positive value of h 52 as shown in Fig. 14h. Stability of the system at the point (E 1 , E 2 ). Next, we investigate the behaviour of the dynamical system Eq. (15) at the second point (E 1 , E 2 ). Recall that the equilibrium point (E 1 , E 2 ) is given by Eq. (16). 

0.38h 1 −3.97 52 136β2 E1 = 0.37h 0.38h 55 52 α2 5 4 3.06

 3.97  0.38h33 h52 −3.97h 33 +1.1628h 52 2.86β3 α2 50.37h 55 40.38h 5,10 ) 0.38h52 −3.97 , 136β2 136β2

 0.38h3.97−3.97  h +3.06+3.061 3.97 33 52 136β2 136β2 0.38h 52 −3.97 . E2 = 0.37h 0.38h 55 4 52 2.86β3 α2 5 Figure 15a–d show the surface plots for E 1 and E 2 for h 33 ∈ (0.001, 1), h 55 ∈ (0.001, 1.5), h 52 ∈ (−0.6, −0.001) and h 5,10 ∈ (0.001, 1.5). The equilibrium points E 1 and E 2 for these range of these parameters remain positive. With respect to this equilibrium point (E 1 , E 2 ), the Jacobian matrix for the dynamical system Eq. (15) is evaluated as |J | = −(46256184(40.38h 5,10 )50.37h 55 h 52 T E R M1 ∗ (T E R M2)0.38h 52 3.97

T E R M1 ∗ (T E R M2)3.97 − 818410736h 33 (((272(68/625) 0.38h 52 −3.97 ) 1

397

/(7579((117(40.38h 52 )(50.37h 55 )/200) 38h 52 −397 ))) (h 33 +60741/(5000((0.38h 52 )−3.97))+153/50) ))h 33 T E R M1 (T E R M2)3.97 + 421202925(40.38h 5,10 )50.37h 55 h 33 h 52 397

(((272(68/625)397/100(0.38h 52 −3.97) )/(7579((117(40.38h 52 )50.37h 55 )/200) 38h 52 −397 ))) 1 h 33 +60741/5000(0.38h 52 −3.97)+153/50

397

)153/50 (((272(68/625) 100(0.38h 52 )−3.97 )/(7579((117(40.38h 52 )50.37h 55 )

397

1

/200) (38h 52 −397) ))) (h 33 +60741/(5000((0.38h 52 )−3.97))+153./50) )h 33 T E R M1 ∗ (T E R M2)((0.38h 52 )) )/ 397

(625000000(((272(68/625) 38h 52 −397 )/(7579((117(40.38h 52

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Fig. 14 Behaviour of eigenvalues for range of h 33 ∈ (0.001, 1), h 55 ∈ (0.001, 1.5), h 52 ∈ (−0.6, −0.001) and h 5,10 ∈ (0.001, 1.5) with respect to initial value (X 2 , X 3 ) = (0.001, 0.001) and for positive range of h 52 ∈ (0.001, 0.6)

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Fig. 15 Second equilibrium point (E 1 , E 2 ) always remains in positive quadrant for h 33 ∈ (0.001, 1), h 55 ∈ (0.001, 1.5), h 52 ∈ (−0.6, −0.001) and h 5,10 ∈ (0.001, 1.5)

397

1

50.37h 55 )/200) 38h 52 −397 )) (h 33 +60741/(1900h 52 −19850)+153./50) )(203/50) T E R M1(544/(2925 (40.38h 52 )50.37h 55 ))(1/((0.38h 52 )−3.97)) ),

(25)

where 3.06    3.97  1.1628h52 −3.97h 33 +0.38h 33 h 52 T E R M1 = 27.864 5.377(40.38h 5,10 )(50.37h 55 ) (0.38h52 )−3.97 , 1   T E R M2 = 0.1850/40.38h 52 50.37h 55 (0.38h52 )−3.97 .

In this case, we found that for the range of h 55 ∈ (0.001, 1.5) and h 33 ∈ (0.001, 1), determinant of Jacobian matrix Eq. (18) remains positive. Hence, equilibrium point is stable point which is also shown in Fig. 16a, b. For the range of h 5,10 and h 52 , the determinant is positive. Hence, for these parameters equilibrium point is stable. But, for sufficiently big positive values of the inhibition parameter, h 52 and the equilibrium point becomes saddle, which is shown in Fig. 16c, d. As shown in Fig. 16d, the positive value of h 52 leads to instability in the system. It means that the pathway becomes activating instead of inhibiting. This further implies that, when G-6-P concentration is high, it will positively regulate (increase) its concentration instead of

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Fig. 16 Behaviour of determinant of Jacobian matrix Eq. (18) with respect to second equilibrium point (E 1 , E 2 ), parameters range are h 55 ∈ (0.001, 1.5), h 33 ∈ (0.001, 1), h 52 ∈ (−0.6, −0.001) and h 5,10 ∈ (0.001, 1.5). Figure c, d shows the behaviour for the negative range of h 52 and positive range of inhibition parameter h 52 , (0.001, 0.6)

negatively regulating (decrease) itself. This is a deviation of the cell or system from the homoeostatic behaviour and will lead to instability. We further investigate the behaviour of dynamical system Eq. (15) by using the eigenvalues of the Jacobian matrix Eq. 18. For the defined range of parameters, Tables 4 and 5, show negative eigenvalues which implies that this system will be stable. These eigenvalues are plotted in Fig. 17a–d.

5.2 When Forcing Coefficient is Not Equal to Zero i.e. f0  = 0 In this section, we consider the dynamical system Eq. (14), when f 0 = 0 i.e. perturbation f 0 cos(ωt) is introduced in this system. Our motivation here is to investigate the effect of this external perturbation onto the dynamical system, separately for the forcing coefficient f 0 and the frequency ω. Hence, we consider only one point (0.001, 0.001) for further investigation in this section.

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Table 4 Eigenvalues of Jacobian matrix Eq. (18) for second equilibrium point (E 1 , E 2 ), with respect to h 33 and h 55 parameters h 33 λ1 λ2 h 55 λ1 λ2 0.1

−5.9148

−0.8237

−3.2721e + 10 −1.3425e + 06 −2.6565e + 04 −1.6614e + 03 −399.9017

0.2

−1.6451

0.3

−1.0944

0.45

−0.8788

0.6

0.001

−2.5297

0.1

−2.2920

0.5

−1.5376

0.8

−1.1391

1

−0.9324

0.8

−0.8207

−136.202

1.2

−0.7631

1

−0.8470

−71.4937

1.5

−0.5647

−7.4918e + 04 −6.6279e + 04 −4.0420e + 04 −2.7908e + 04 −2.1808e + 04 −1.7044e + 04 −1.1782e + 04

Table 5 Eigenvalues of Jacobian matrix Eq. (18) for second equilibrium point (E 1 , E 2 ) with respect to h 52 and h 5,10 parameters h 52 λ1 λ2 h 5,10 λ1 λ2 −0.6 −0.5

−1.6905 −1.5647

−0.4

−1.4427

−0.3

−1.3241

−0.2

−1.2083

−0.1

−1.0944

−0.001

−0.9825

−1.497e + 04 −1.6434e + 04 −1.8193e + 04 −2.0350e + 04 −2.3059e + 04 −2.6565e + 04 −3.1216e + 04

0.001 0.1

−2.8466 −2.6290

−267.3779 −414.3657

0.5

−1.8208

0.8

−1.3469

1

−1.0944

1.2

−0.8863

1.5

−0.6433

−2.5656e + 03 −1.0379e + 04 −2.6565e + 04 −6.8302e + 04 −2.8342e + 05

For the range of forcing coefficient f 0 ∈ (0.001, 0.8), the system shows the periodic behaviour for a fixed value of other parameters ω = 0.4, h 33 = 0.3, h 5,10 = 1, h 52 = −0.1 and h 55 = 0.84. The periodic behaviour is shown in Fig. 18a–d. Corresponding time series plots are shown in Fig. 18e–h. For ω, the system shows oscillating behaviour for ω ∈ (0.001, 0.8) and f 0 = 0.4 (rest of the parameters are same as in Fig. 18), behaviour and time series plots are shown in Fig. 19. Periodic behaviour (scaled and zoomed) with respect to both parameters f 0 , and ω has been shown as Fig. 20. The impact of the presence of the perturbation term in a dynamical system

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Fig. 17 Eigenvalues of Jacobian matrix Eq. (18) with respect to second equilibrium point (E 1 , E 2 ), parameters range are h 55 ∈ (0.001, 1.5), h 33 ∈ (0.001, 1), h 52 ∈ (−0.6, −0.001) and h 5,10 ∈ (0.001, 1.5)

is shown in Fig. 21a and b with the help of Lyapunov exponents for a chosen values of the parameters. By perturbing the system through mathematical modelling, we have observed that the concentrations of X 2 (Glucose-6-P) and X 3 (Fructose-6-P) show oscillations with respect to time, i.e the concentration of both the metabolites first increases and then decreases. It has already been shown that Phosphofructokinase (PFK) is the control enzyme responsible for the oscillations. In [14], the authors have demonstrated that oscillations can only be achieved by infusing one of those metabolites pre-cursing PFK.

6 Conclusion In the present study, few initial committed steps of the glycolytic pathway were formulated into a dynamical system with the help of mass balance equations and S-system formulation. This dynamical system consists various parameters which involve rate constants, exponents and dependent and independent variables. We stud-

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Fig. 18 Effect of forcing term f 0 on dynamical system Eq. (15) with f 0 = [0.001, 0.1, 0.4, 0.8] and rest parameters values are h 33 = 0.3, h 55 = 0.84, h 5,10 = 1, ω = 0.4 and h 52 = −0.1. Time interval is 0 : 300 and initial values are (0.001, 0.001)

ied the interrelation between few of the important parameters by setting the system at steady state which eventually lead to solving a set of transcendental equations. We identified four parameters, namely, h 33 , h 55 , h 5,10 , and h 52 which relate the dynamical system to the phosphofructokinase and glucokinase reactions, respectively. We investigated the stability of this dynamical system with the help of the bifurcation tools such as the phase portraits, the Jacobian stability, Lyapunov exponents, d∞ plots and the time series plots for the parameters h 33 , h 55 , h 5,10 , and h 52 . The equilibrium point (0.067, 0.465, 0.150) was considered for this investigation. These values are the steady state concentration of the dependent variable X 1 , X 2 , and X 3 . We found that h 33 is the only parameter which may disturb the stability of the system with respect to time and rest three parameters can contribute to an increase in the concentration of the fructose 1,6-bi-phosphate. The second pathway considered in this study is by deleting the right-side branch of the original pathway, or by shrugging off the contribution of glucose-1-P from the pathway. We further added an external perturbation term f 0 cos(ωt) in it to study the effect of forcing coefficient f 0 , and the frequency ω. We have considered two cases when f 0 = 0, and f 0 = 0.

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Fig. 19 Effect of frequency ω on the dynamical system Eq. (15) ω = [0.001, 0.01, 0.2, 0.8] and rest parameters are h 55 = 0.84, h 33 = 0.3, h 5,10 = 1, f 0 = 0.4 and h 52 = −0.1, time interval is [0 : 300] and initial values are (0.001, 0.001)

Fig. 20 Scaled and Zoomed view of effect of frequency ω = 0.2, forcing coefficient f 0 = 0.4 and rest parameters are h 55 = 0.84, h 33 = 0.3, h 5,10 = 1 and h 52 = −0.1, time interval is [0 : 300] and initial values are (0.001, 0.001)

• For the case of f 0 = 0, we investigated the stability of the system around two points (0.001, 0.001), and (E 1 , E 2 ). For both the points, the system is found to be stable for the defined range of the parameters. However, we found that the parameter h 52 can significantly affect the stability of the system. In fact, beyond a critical value of h 52 system became unstable and rather than inhibiting the production of glucose-6-P, it triggers activation.

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Fig. 21 Comparison of Lyapunov exponents when f 0 = 0 and f 0 = 0, rest parameters are h 55 = 0.84, h 33 = 0.3, h 5,10 = 1, f 0 = 0.4, ω = 0.8 and h 52 = −0.1, time interval is 0 : 0.5 : 100 and initial values are (0.001, 0.001)

• For the second case, when f 0 = 0, the behaviour corresponding to rest of the parameters h 33 , h 55 , h 5,10 , and h 52 was observed similar to the first case. But in presence of the forcing term with forcing coefficient f 0 , and the frequency ω. The system showed periodic behaviour with oscillations. The results for both cases were also verified with the help of the Lyapunov exponents.

7 Future Work • This glycolytic pathway has various independent variables, and parameters, therefore, it is impossible to see the effect of range of all these parameters. In future, another set of parameters can be chosen based on the experimental justification and interest, and the effect of these parameters can be investigated on the glycolytic pathway. • The values of the parameters and independent variables can be perturbed and/or additional perturbation term can be added in this system to study the behaviour of system with respect to these perturbation terms. • In the second pathway, we have removed only right branch of the glycolytic pathway. The glycolytic pathway can be investigated by adding or removing more branches based on theory in future studies. • In this study, we have studied first few committed of the complete glycolytic pathway. This study can be expanded for a complete pathway or one can study the dynamical behaviour of other metabolic pathways by using standard tools from bifurcation tools as demonstrated in this study.

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Appendix 1: S-System Study of Pathway The mass-balance equation can be translated into S-system, d X1 g g g = α1 X 4 14 X 6 16 X 111,11 − β1 X 1h 11 X 2h 12 X 7h 17 , dt d X2 g g g g g = α2 X 1 21 X 2 22 X 5 25 X 7 27 X 102,10 − β2 X 2h 22 X 3h 23 X 8h 28 , dt d X3 g g g = α3 X 2 32 X 3 33 X 8 38 − β3 X 3h 33 X 9h 39 . dt

(26)

There is one rate constant for each pool’s production and one rate constant for its degradation in this S-system. These are α1 , α2 , α3 and β1 , β2 , β3 , respectively. For each X i s, one subscript is used for each power. Power is represented by g in the production term and h in the degradation term. The parameters αi and gi j always refer to production or gain, whereas the parameters βi and h i j always refer to degradation or loss, i.e. the greater αi s indicate more production and the larger βi s imply a loss in production. With the help of the precursor-product relationship between X 2 & X 3 and branch point constraint at X 2 , we can reduce the number of unknowns. Because V2− and V3+ reflect the same process, the precursor-product relationship implies that the values of the constraints’ parameters be equal. α3 = β2 , g32 = h 22 , g33 = h 23 , g38 = h 28 .

(27)

According to the branch point restriction at X 2 , the total of the two fluxes entering X 2 from X 1 and X 5 , namely V1− and V5− , must equal the influx V2+ . The constraint on branch points can be written as V2+ = V1− + V5− , i.e. g

g

g

g

g

h

α2 X 1 21 X 2 22 X 5 25 X 7 27 X 102,10 = β1 X 1h 11 X 2h 12 X 7h 17 + β5 X 2h 52 X 5h 55 X 105,10 .

(28)

These expressions can be understood as a power-law representation in this case. We may use the partial differentiation of the left-hand side of the equation to get the power terms g21 , g22 , g25 , g27 , and g2,10 .

g27 =

∂ V2+ X 7 ∂ X 7 V2+ h

∂(β1 X 1h 11 X 2h 12 X 7h 17 + β5 X 2h 52 X 5h 55 X 105,10 ) X 7 = . ∂ X7 V2+

(29)

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By doing partial differentiation, the first term is V− g27 = β1 h 17 X 1h 11 X 2h 12 X 7h 17 −1 VX+7 , which is equivalent to h 17 V1+ and the second term of 2 2 the equation will become zero. Thus, V− g27 = h 17 V1+ . Similarly, we can compute the other ones i.e. 2

V−

V−

V−

2

2

2

g21 = h 11 V1+ , g25 = h 55 V5+ , g2,10 = h 5,10 V5+ . The last one is g22 whose derivation

may be difficult because of both the terms V5− and V1− depend upon X 2 .

g22 =

∂ V2+ X 2 , ∂ X 2 V2+ h

=

∂(β1 X 1h 11 X 2h 12 X 7h 17 + β5 X 2h 52 X 5h 55 X 105,10 ) X 2 , ∂ X2 V2+

= β1 h 12 X 1h 11 X 2h 12 −1 X 7h 17 + β5 h 52 X 2h 52 −1 X 5h 55 X 105,10 h

X7 , V2+

h

= h 12 = h 12

β5 X 2h 52 X 5h 55 X 105,10 β1 X 1h 11 X 2h 12 X 7h 17 + h , 52 V2+ V2+ V5− V1− + h 52 + . V2+ V2

(30)

Appendix 2: Steady State Analysis of the Dynamical System Eq. (8) We require equations that characterise the steady state in an explicit algebraic form for more theoretical investigations, such as a general analysis of logarithmic sensitivities. The structure and characteristics of these equations can be investigated. To find the steady state, divide each equation by its rate constant αi and set the three symbolic equations to zero. The result is β1 h 11 h 12 h 17 X X X = 0, α1 1 2 7 β2 h22 h23 h28 g g g g X 1 21 X 2 22 X 5 25 X 7 27 X 1 0g2,10 − X X 3 X 8 = 0, α2 2 β3 h33 h39 g g g X 2 32 X 3 33 X 8 38 − X X 9 = 0. (31) α3 3   Next, we define yi = log X i for i = 1, 2, . . . , 10 and bi = log αβii for i = 1, 2, 3. g14

g16

X4 X6

−

g

Power term can be expressed as X i i j = exp(gi j log X i ), and taking log to all terms,

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we obtain b1 + h 11 y1 + h 12 y2 + h 17 y7 − g14 y4 + g16 y6 = 0, b2 + h 22 y2 + h 23 y3 + h 28 y8 − g21 y4 − g22 y2 − g25 y5 − g27 y7 − g2,10 y10 , b3 + h 33 y3 + h 39 y9 − g32 y2 − g33 y3 − g38 y8 = 0.

(32)

By solving the system of linear equations (32), we will get two solution sets. One is with respect to the parameters h 55 , h 52 , h 33 , h 5,10 , αi and βi for i = 1, 2, 3. While putting the value of the parameters from Sect. 2, Table 1. The value of the variables yi will be log(X i ), all X i are from Sect. 2, Table 2. Corresponding solutions set for this system Eq. (32) is h 33 = h 33 , h 52 = h 52 , h 55 = h 55 , h 5,10 = h 5,10 , α1 = 0.8333333333e − 2 exp(−1.530000000y1 + 0.5900000000y2 + 1.519706161)β1 , α2 = exp(0.5129289136h 5,10 + 0.5954920276h 55 − 0.3700000000y2 h 52 + 0.9486000000y1 − 3.604200000y2 + 3.060000000y3 − 2.625549624)β2 , α3 = α3 , β1 = β1 , β2 = β2 , β3 = 0.7352941176e − 2 exp(h 33 y3 − 3.970000000y2 + 3.060000000y3 + 1.050821625)α3 .

(33)

The second solution set is generated only for Y1 , Y2 , Y3 , in which only αi , βi and h 33 , h 55 , h 52 and h 5,10 parameters are unknown. Solution set is Y1 = −

1 ((2.042483660e + 06) (46250.h 52 h 33 + (4.04800e + 05)h 33 + (1.41525e + 05)h 52 − (2.79837e + 05))

((−1.405361185e + 10)h 33 h 55 − (1.210512236e + 10)h 33 h 5,10 + (1.441680000e + 11) α log 1 h 33 + (4.836322660e + 10)h 52 h 33 − (2.355910293e + 11) + (4.528800000e + 10)h 52 log β1 α α + (1.480000000e + 10) log 1 h 33 h 52 − (4.477392000e + 10) log 1 + (7.221600000e + 10) log β1 β1 α α (7.221600000e + 10) log 2 + (5.330730828e + 11)h 33 + (2.360000000e + 10)h 33 log 2 β2 β2 + (1.479914734e + 11)h 52 − (4.300405227e + 10)h 55 − (3.704167443e + 10)h 5,10 )),

Y2 = −

α1 β1 α3 + β3

(34)

1 ((3.125000000e + 04) (46250h 52 h 33 + (4.04800e + 05)h 33 + (1.41525e + 05)h 52 − (2.79837e + 05))

((−2.381968110e + 08)h 33 h 55 − (2.051715654e + 08)h 33 h 5,10 + (1.860630674e + 09)h 33 + (α2 ) − (7.288822418e + 08)h 55 − (6.2782499033 + 08)h 5,10 + β2 α2 α1 h 33 + (1.224000000e + 09) log + (9.666459509e + 08)+ (2.48000000e + 08) log β1 β2 α3 α1 + (1.224000000e + 09) log )), (7.58880000e + 08) log β1 β3

(4.00000000e + 08)h 33 log

(35)

114 Y3 =

S. Tomar et al. 1 (0.6250000000e − 5 (46250h 52 h 33 + (4.04800e + 05)h 33 + (1.41525e + 05)h 52 − (2.79837e + 05))

((2.857756613e + 10)h 52 + (4.728206699e + 10)h 55 + (4.072655574e + 10)h 5,10 − (7.400000000e + 09) (α3 α2 − (1.192119721e + 11) − (7.940000000e + 10) log − (4.922800000e + 10) β3 β2 α3 α1 − (6.476800000e + 10) log )). log β1 β3

h 52 log

(36)

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45. L.W.M. Verburg, Modeling and control, of glycolysis in trypanosoma brucei. Dissertation Master Thesis, Department of Mathematics, Vrije Universiteit, Amsterdam, 2006 46. D.V. Verveyko, A.Y. Verisokin, E.B. Postnikov, Mathematical model of chaotic oscillations and oscillatory entrainment in glycolysis originated from periodic substrate supply. Chaos: Interdiscip. J. Nonlinear Sci. 27(8), 083104 (2017) 47. E.O. Voit, The best models of metabolism. Wiley Interdiscip. Rev.: Syst. Biol. Med. 9(6), e1391 (2017) 48. E.O. Voit, (Case Study-4) Computational Analysis of Biochemical Systems: A Practical Guide for Biochemists and Molecular Biologists (Cambridge University Press, 2000) 49. X. Wang, Bifurcation analysis of two biological systems: a tritrophic food chain model and an oscillating networks model. Doctoral dissertation, The University of Western Ontario, Canada 2018 50. O. Warburg, W. Christian, Uber ein neues Oxyda-tionsferment und sein Absorptionsspektrum. Biochem. Z. 254, 438–458 (1932) 51. P.O. Westermark, A. Lansner, A model of phosphofructokinase and glycolytic oscillations in the pancreatic β-cell. Biophys. J. 85(1), 126–139 (2003)

On Solutions of Biological Models Using Reproducing Kernel Hilbert Space Method Nourhane Attia and Ali Akgül

Abstract Differential equations (DEs, for short) are becoming more and more indispensable for modeling real-life problems. Modeling and then analyzing these DEs help scientists to understand and make predictions about the system that they want to analyze. And this is possible only in one case when their solutions are available. However, the majority of fractional differential equations lack exact solutions. This chapter’s goal is to introduce the reproducing kernel Hilbert space method (RKHSM) for some systems that have significant applications in biology. The proposed method’s error estimations and convergence analyses are discussed. The assessment of the RKHSM is made by testing some illustrative applications. The results suggest that the RKHSM is an efficient and highly convenient method to solve the fractional systems arising in biology. Keywords Computational biology · Reproducing kernel Hilbert space method · Numerical solution AMS (2010) Subject Classifications 46E22 · 34A08

N. Attia National High School for Marine Sciences and Coastal (ENSSMAL), Dely Ibrahim University Campus, Bois des Cars, B.P. 19, 16320 Algiers, Algeria e-mail: [email protected] A. Akgül (B) Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon e-mail: [email protected] Department of Mathematics, Arts and Science Faculty, Siirt University, 56100 Siirt, Turkey Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, PC: 99138 Nicosia/Mersin 10, Turkey © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_5

117

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1 Introduction PDEs serve as a crucial link between mathematics and its applications. Numerous biological phenomena have been described using PDEs, and their solutions are essential to comprehending nature. However, especially for the non-linear equation, it is quite challenging to find exact solutions. One of the main tasks for researchers now is to create analytical and numerical approaches to address challenging nonlinear problems [15–18]. For nonlinear PDEs, only a few basic numerical methods are employed. This offers us the inspiration to propose a method for these problems. The primary contribution of this research is the development of the RKHSM in order to numerically solve the nonlinear Newell–Whitehead–Segel equations (NWSE) [20] that appear in biology. The RKHSM is employed in this chapter to create numerical solutions for NWSEs. The dynamical behavior near the bifurcation point for the Reyleigh–Benard convection of binary fluid mixtures is described by these equations that can be found in mathematical biology. An efficient numerical approach for difficult nonlinear problems without discretization is the RKHSM, which was first presented in 1908 [2]. The RKHSM, which was proposed in 1908 [2] has been used to resolve a variety of equation forms, including Riccati differential equations [7], forced Duffing equations [10], generalized Schamel equation [3], fuzzy fractional IVPs [13], nonlocal fractional boundary value problems [9], second-order PDEs [5], and many other interesting studies [4, 8, 14, 19, 21]. The main benefits of the RKHSM are: 1. how simple it is to apply, especially since it is mesh-free. 2. its capacity to solve a variety of difficult differential equations. The section that follows contains some fundamental RK theory definitions and theorems. In our general section (Sect. 3), The RKHSM’s explanation and application to the suggested problem are presented. Error estimates are provided in the fourth section. In the fifth section, two applications are used to validate the RKHSM’s efficacy. The sixth section contains a discussion, graphics, and tables. The conclusion is presented in the end.

2 Useful Mathematical Concepts The RK theory contains the following basic definitions and theorems. In order to start, we list a few RKHSs that are crucial for our study and are found in numerous articles (see, e.g. [1, 12]). The reproducing kernel function (RKF) expressions are then derived. For “completely continuous” and “absolutely continuous function(s),” we used the acronyms “CC” and “ACF(s)” throughout the text.

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119

Definition 2.1 If 1. (·, ϑ) ∈ J , ∀ ϑ ∈ ℵ, 2. f, (·, ϑ) = f(ϑ), ∀ ∈ J and ∀ ϑ ∈ ℵ, a function  : ℵ × ℵ → C is said to be a reproducing kernel of the space H given with J over ℵ = ∅ is a Hilbert space. Remarks • A Hilbert space that has a RK is a RKHS. • The second assertion in the previous definition is a reproducing property (for short, RP). Definition 2.2 ([11]) The space of functions f where f(j ) , j = 0, 1, . . . , m − 1 are ACFs on [a, L] and f(m) ∈ L 2 [a, L] is known as the function space G m 2 [a, L]. Definition 2.3 ([11]) The inner product and norm of G m 2 [a, L] are    L m m dı f  dı g  d f d g f, gG m = +   m dκ m dx, 2 dκ ı κ=a dκ ı κ=a dκ a ı=0 m−1 

f, g ∈ G m 2 [a, L],

and f G m2 =



f, fG m2 ,

f ∈ Gm 2 [a, L].

As examples, 1. The space N32 [a, L] : N32 [a, L] = {f| f, f and f

are ACFs on [a, L], f(3) ∈ L 2 , and f(a) = f(L) = 0}. 2. The space W22 [a, L] : W22 [a, L] = {f| f and f are ACFs on [a, L], f

∈ L 2 , and f(a) = 0}. 3. The space W12 [a, L] : W12 [a, L] = {f| f is ACF on [a, L] and f ∈ L 2 }. We use I = [a, L] × [0, T] ⊂ R × R+ throughout the entire research. Definition 2.4 ([11]) The space of functions Z(κ, ϑ) where ∂κ∂m−1 ∂ϑn−1 Z(κ, ϑ) is m+n CC in I and ∂κ∂ m ∂ϑn Z(κ, ϑ) ∈ L 2 (I) is known as the space W(3,2) (I) . 2 m+n−2

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Definition 2.5 ([11]) The inner product and norm of W(3,2) (I) are 2 Z, Q

(3,2) W2

=

 m−1 n−1   T  ∂n ∂ı   ∂ı ∂n ∂ı ∂ı Z(a, ϑ) Q(a, ϑ) dϑ + Z(κ, 0), Q(κ, 0) ∂ϑn ∂κ ı ∂ϑn ∂κ ı ∂ϑı ∂ϑı Gm ı=0 0 ı=0  T L m ∂ ∂n ∂m ∂n + m ∂ϑn Z(κ, ϑ) ∂κ m ∂ϑn Q(κ, ϑ)dκdϑ, ∂x 0 a

2

(3,2)

Z, Q ∈ W2

(I) ,

and Z W(3,2) = 2

 Z, ZW(3,2) ,

Z ∈ W(3,2) (I) . 2

2

As examples, 1. The space W(3,2) (I) : 2 ∂3 ∂5 Z(κ, ϑ) ∈ L 2 (I), Z(κ, ϑ) is CC in I, ∂κ 2 ∂ϑ ∂κ 3 ∂ϑ2 and Z(a, ϑ) = Z(L, ϑ) = Z(κ, 0) = 0}.

W(3,2) (I) = {Z(κ, ϑ)| 2

2. The space W(1,1) (I) : 2 W(1,1) (I) = {Z(κ, ϑ)| Z(κ, ϑ) is CC in I and 2

∂2 Z(κ, ϑ) ∈ L 2 (I)}. ∂κ∂ϑ

Theorem 2.1 From the space W12 [a, L], we derive the RK function Mμ (κ) as [11]:

Mμ (κ) =

−a + 1 + κ , κ ≤ μ, μ − a + 1 , κ > μ.

Proof We must prove  f, Mμ W1 = f(μ). 2

We have 

f, Mμ

W12

 = f(a)Mμ (a) +

L a

We have to calculate Mμ (a) : Mμ (a) = 1.

f (κ)M μ (κ) dκ.

(1)

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121

By applying the aforementioned equation, we get:  f, Mμ W1 = f(a) +



2

L a

f (κ)M μ (κ) dκ.

We have M μ (κ)

=

1 , κ ≤ μ, 0 , κ > μ.

Thus 

f, Mμ



W12

= f(a) +

μ

a

f (κ)M μ (κ) dκ +



L μ

f (κ)M μ (κ) dκ

so, we deduce 

f, Mμ

W12

= f(μ). 

Theorem 2.2 From the space W22 [a, L], we derive the RK function Rη (ϑ) as [11]:

Rη (ϑ) =

−1/3 a3 + 1/2 a2 η + a2 − η a + 1/2 ϑ a2 − ϑ η a − ϑ a + ϑ η + 1/2 η ϑ2 − 1/6 ϑ3 , ϑ ≤ η, −1/3 a3 + 1/2 a2 η − 1/6 η 3 + a2 − η a + 1/2 ϑ a2 − ϑ η a + 1/2 ϑ η 2 − ϑ a + ϑ η , ϑ > η.

(2) Proof We must demonstrate  f, Rη W2 = f(η). 2

We have 

f, Rη

W22

= f(a)Rη (a) + f (a)R η (a) +



L a

f

(ϑ)R

η (ϑ) dϑ.

Integrating by parts yields the following results:

 f, Rη W2 = f(a)Rη (a) + f (a) R η (a) − R

η (a) + f (L)R

η (L) − 2



 L a

(3)

f (t)Rη (t)dt.

Since f(t) ∈ W22 [a, L], f(a) = 0 follows. Then,    f, Rη W2 = f (a) R η (a) − R

η (a) + f (L)R

η (L) − 2

 a

L

f (ϑ)R(3) η (ϑ)dϑ.

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We have to calculate the following: R η (a) = R

η (a) = −a + η, R

η (L) = 0. By applying the aforementioned equations, we get: 

f, Rη



W22

L

=− a

f (ϑ)R(3) η (ϑ)dϑ.

(3)

We have R(3) η (ϑ) =



−1 , ϑ ≤ η, 0 , ϑ > η.

As a result, 

f, Rη

W22



η



(ϑ)R(3) η (ϑ)dϑ

=− f a  η f (ϑ) dϑ =

 −

L η

f (ϑ)R(3) η (ϑ)dϑ

a

and, since f(ϑ) ∈ W22 [a, L], we infer 

f, Rη

W22

= f(η). 

Remarks The reproduction kernel function Zμ (κ) of N32 [a, L] is expressed here, but due to its length, we have chosen not to include it. For additional information on this point, the interested reader is directed to [12]. The reproducing kernel function of the space W(3,2) (I) 2 1. The RK function O(μ,η) (κ, ϑ) of the space W(3,2) (I) is produced by multiplying 2 the RKFs of the spaces N32 [a, L] and W22 [0, T]—i.e.: O(μ,η) (κ, ϑ) = Zμ (κ) Rη (ϑ). 2. The RK function V(μ,η) (κ, ϑ) of W(1,1) (I) is produced by multiplying two RKFs 2 of the same space W12 [a, L]—i.e.: V(μ,η) (κ, ϑ) = Mμ (κ) Mη (ϑ).

On Solutions of Biological Models Using Reproducing …

123

3 The RKHS Approach Our primary objective in this part is to outline the steps to apply the suggested approach for solving the NWSE, which has the following form [20]: 2 [Z (κ, ϑ)] + A Z(κ, ϑ) + A Zq (κ, ϑ), 0 ≤ κ ≤ L, 0 ≤ ϑ ≤ T, Lϑ [Z(κ, ϑ)] = A1 Lκ 2 3

(4)

accompanied by the following boundary conditions (BCs) Z (0, ϑ) = φ1 (t), and Z (L, ϑ) = φ2 (ϑ),

(5)

and initial conditions (IC) Z (κ, 0) = Z0 (κ),

(6)

∂ ∂ 2 = ∂κ where Lκ 2 and Lϑ = ∂ϑ . A1 , A2 , and A3 are real constants. q is a positive integer. The function Z(κ, ϑ) will be determined and it could be seen as the temperature distribution of temperature in an infinitely thin and long rod or as the flow velocity of a fluid in an infinitely long pipe with small diameter. The functions φ1 (ϑ), φ2 (ϑ), and Z0 (κ) are known. One clever way to investigate problem (4) is to homogenize (5) and (6) using an appropriate transformation. Thus, let’s take: 2

Q(κ, ϑ) = Z(κ, ϑ) + P(κ, ϑ), with   1 1 P(κ, ϑ) = (φ1 (0) − φ1 (ϑ)) 1 − κ + (φ2 (0) − φ2 (t))κ − Z0 (κ). L L Then, (4)–(6) become 2 [Q(κ, ϑ)] + R [Q(κ, ϑ)] = (κ, ϑ), 0 ≤ κ ≤ L, 0 ≤ ϑ ≤ T, (7) Lϑ [Q(κ, ϑ)] − A1 Lκ

with BCs Q(0, ϑ) = Q(L, ϑ) = 0,

(8)

Q(κ, 0) = 0,

(9)

and IC the remaining linear terms are denoted by the term R [Q(κ, ϑ)] , and the nonlinear terms are expressed by the term (κ, ϑ). Here, we present the linear operator : W(3,2) (I) → W(1,1) (I) , as 2 2 2

Q(κ, ϑ) = Lϑ [Q(κ, ϑ)] − A1 Lκ [Q(κ, ϑ)] + R [Q(κ, ϑ)] .

(10)

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Lemma 3.1 The operator : W(3,2) (I) → W(1,1) (I) is bounded and linear. 2 2 Proof The linearity of can be easily verified. We may demonstrate the boundedness of directly. And, showing that Q W(1,1) ≤ C Q W(3,2) , with C > 0, 2

2

is necessary to prove this. From Definition 2.5, we get Q(κ, ϑ) 2 (1,1) =  Q(κ, ϑ), Q(κ, ϑ) W2

(1,1)

W2

2  T ∂

Q(a, ϑ) dϑ +  Q(κ, 0), Q(κ, 0)W1 2 ∂ϑ 0 2  T L ∂ ∂

Q(κ, ϑ) dκdϑ + ∂κ ∂ϑ 0 0 2 2  L  d ∂ ∂ = [ Q(0, 0)]2 +

Q(κ, 0) dκ +

Q(a, ϑ) dϑ ∂κ ∂ϑ 0 0 2  T L ∂ ∂

Q(κ, ϑ) dκdϑ. + ∂κ ∂ϑ 0 0 =

The RP has allowed us to have  Q(κ, ϑ) = Q(•, ∗), O(κ,ϑ) (•, ∗) W(3,2) . 2

In the same way,  ∂ı ∂j ∂ı ∂j

Q(κ, ϑ) = Q(•, ∗),

O(κ,ϑ) (•, ∗) , ı, j = 0, 1. ∂κ ı ∂ϑj ∂κ ı ∂ϑj W(3,2) 2 The Schwarz inequality yields     ı     ∂ ∂j ∂ı ∂j    Q(•, ∗), =

Q(κ, ϑ)

O (•, ∗)   (κ,ϑ)    ∂κ ı ∂ϑj  ∂κ ı ∂ϑj W(3,2) 2  ı  j  ∂ ∂   ≤ Q W(3,2)   ∂κ ı ∂ϑj O(κ,ϑ) (•, ∗) (3,2) . 2 W2 Thanks to the continuity of O(κ,ϑ) (•, ∗), we deduce   ı   ∂ ∂j  ≤ Cı,j Q (3,2) , j, ı = 0, 1. 

Q(κ, ϑ) W2   ∂κ ı ∂ϑj Hence

On Solutions of Biological Models Using Reproducing …

 Q(κ, ϑ) 2W(1,1) 2

≤C20,0 

Q 2W(3,2) 2 T



+ 0

L

125

 C21,0

Q 2W(3,2) 2

T

dκ + 0

C20,1 Q 2W(3,2) dϑ 2

L

+ C21,1 Q 2W(3,2) dκdϑ 2 0 0   ≤ C20,0 + C21,0 L + C20,1 T + C21,1 L T Q 2W(3,2) . 2

By taking C = C20,0 + C21,0 L + C20,1 T + C21,1 L T, we get the desired result. Applying (10) allows us to rewrite (7) and (9) as follows ⎧ 0 ≤ κ ≤ L, 0 ≤ ϑ ≤ T, ⎨ Q(κ, ϑ) =  (κ, ϑ) , Q (0, ϑ) = Q (L, ϑ) = 0, ⎩ Q (κ, 0) = 0.



(11)

(I) is necessary now. Let’s start Building the orthogonal function system of W(3,2) 2 with some helpful functions: ρı (κ, ϑ) = V(κı ,ϑı ) (κ, ϑ) and ψı (κ, ϑ) = ∗ ρı (κ, ϑ), where ∞ • {(κı , ϑı )}ı=1 is a dense countable set in I. (I) is represented by V(κı ,ϑı ) (κ, ϑ). • The RKF connected to W(1,1) 2 • The formal neighbor of is ∗ . ∞ Using the Gram–Schmidt procedure, we obtain the orthonormal system {ψ¯ı }ı=1 in (3,2) W2 (I) shown below:

ψ¯ı (κ, ϑ) =

ı 

ςık ψk (κ, ϑ), ςıı > 0, ı = 1, 2, . . . .

k=1 ∞ Here, {ψı }ı=1 designates a function system in W(3,2) whose expression can be found 2 using the formula:

 ψı (κ, ϑ) = ∗ ρı (κ, ϑ) = ∗ ρı (μ, η), O(κ,ϑ) (μ, η) W(3,2) 2  = V(μı ,ηı ) (μ, η), (μ,η) O(κ,ϑ) (μ, η) W(1,1) 2

= (μ,η) O(κ,ϑ) (μ, η)|(μ,η)=(κı ,ϑı ) = (μ,η) O(μ,η) (κ, ϑ)|(μ,η)=(κı ,ϑı ) . Applying to (μ, η) variables is shown by the operator (μ,η) . However, the orthogonalization coefficients ςık are provided by:

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⎧ 1 ⎪ for ı = j = 1, ⎨ ψ1 , 1 , for ı = j = 1, ςıj = eı ⎪ ⎩ − 1 ı−1 C ς , for ı > j, k=j ık kj eı where eı =



ψı 2 −

ı−1 k=1

 2 Cık , Cık = ψı , ψ¯k W(3,2) .

∞ {(κı , ϑı )}ı=1

Theorem 3.1 If (I). system of W(3,2) 2

(12)

2

∞ is dense on I, then {ψı }ı=1 represents the complete

Proof It is evident that ψı ∈ W(3,2) (I). Consequently, for each Q ∈ W(3,2) , we write 2 2 Q(κ, ϑ), ψı (κ, ϑ)W(3,2) = 0, ı = 1, 2, . . . . 2

Since Q(κ, ϑ), ψı (κ, ϑ)W(3,2) =  Q(κ, ϑ), ρı (κ, ϑ)W(1,1) = Q(κı , ϑı ) = 0. 2

2

∞ As a result of the density of {(κı , ϑı )}ı=1 in I,

Q(κ, ϑ) = 0.

(13)

And the fact that −1 exists implies that Q(κ, ϑ) = 0.  ∞ Theorem 3.2 Assuming that {(κı , ϑı )}ı=1 is dense on I and (11) has a unique (3,2) solution in W2 (I), then this solution takes the form:

Q(κ, ϑ) =

∞  ı 

ςık (κk , ϑk )ψ¯ı (κ, ϑ),

(14)

ςık (κk , ϑk )ψ¯ı (κ, ϑ) − P(κ, ϑ).

(15)

ı=1 k=1

and the solution of (4)–(6) is Z(κ, ϑ) =

∞  ı  ı=1 k=1

∞  (I) allows us to compute: Proof The completeness of ψ¯ı (κ, ϑ) ı=1 in W(3,2) 2

On Solutions of Biological Models Using Reproducing …

Q(κ, ϑ) = = = = =

127

∞  

Q(κ, ϑ), ψ¯ı (κ, ϑ) W(3,2) ψ¯ı (κ, ϑ) 2

ı=1 ∞  ı 

 ςık Q(κ, ϑ), ∗ ρk (κ, ϑ) W(3,2) ψ¯ı (κ, ϑ) 2

ı=1 k=1 ∞  ı 

ςık  Q(κ, ϑ), ρk (κ, ϑ)W(1,1) ψ¯ı (κ, ϑ) 2

ı=1 k=1 ∞  ı 

 ςık Q(κ, ϑ), V(κk ,ϑk ) (κ, ϑ) W(1,1) ψ¯ı (κ, ϑ) 2

ı=1 k=1 ∞  ı 

ςık (κk , ϑk )ψ¯ı (κ, ϑ),

ı=1 k=1

with (κk , ϑk ) = Q(κk , ϑk ). Additionally, we mention that it is simple to obtain (15) from Z(κ, ϑ) = Q(κ, ϑ) − P(κ, ϑ) and (15).  Remarks In this case, the approximate solution is obtained if we take a finite many terms in (14): p ı   Q p (κ, ϑ) = ςık (κk , ϑk )ψ¯ı (κ, ϑ). ı=1 k=1

W(3,2) (I) is a Hilbert space, hence it is obvious that 2 ∞  ı 

ςık (κk , ϑk )ψ¯ı (κ, ϑ) < ∞.

ı=1 k=1

We come to the conclusion that Q p (κ, ϑ) is convergent in the norm.

4 Analysis of Convergence The following is the approximate answer for problem (14) provided by the RKHSM: Q p (κ, ϑ) =

p 

ı ψ¯ı (κ, ϑ),

(16)

ı=1

in which ı =

ı  k=1

ςık (κk , ϑk ).

(17)

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Here, by letting (κ1 , ϑ1 ) = (0, L), the values of Q(κ1 , ϑ1 ) will be known from the IC and BCs. And, Q0 (κ1 , ϑ1 ) = Q(κ1 , ϑ1 ). ∞ Theorem   4.1 Assume that {(κı , ϑı )}ı=1 is dense on I, (16) has a unique solution, and Q p W(3,2) is bounded. Then, 2

1. Q p (κ, ϑ) converges to Q(κ, ϑ). 2. Q p (κ, ϑ) =

p 

ı ψ¯ı (κ, ϑ),

ı=1

where ı is given by (17). Proof 1. From (16), we write Q p+1 (κ, ϑ) = Q p (κ, ϑ) +  p+1 ψ¯ p+1 (κ, ϑ),  ∞ since ψ¯ı (κ, ϑ) ı=1 is orthogonal,     Q p+1 2 (3,2) = Q p 2 (3,2) + 2 p+1 W2 W2  2 = Q p−1 W(3,2) + 2p + 2p+1 2

.. . = Q0 2W(3,2) + 2

and so

p+1 

ı2 ,

(18)

ı=1

    Q p  (3,2) ≤ Q p+1  (3,2) . W W 2

2

  The fact that Q p 

is bounded makes its convergence obvious. Consequently, there is a positive constant  in which W(3,2) 2

∞ 

ı2 = .

ı=1

Consequently

 2 ∞ ı ı=1 ∈ 2 .

    Since Qq (κ, ϑ) − Qq−1 (κ, ϑ) ⊥ · · · ⊥ Q p+1 (κ, ϑ) − Q p (κ, ϑ) , for q > p we have

On Solutions of Biological Models Using Reproducing …

129

    Qq − Q p 2 (3,2) = Qq − Qq−1 + Qq−1 − · · · + Q p+1 − Q p 2 (3,2) W W 2

2

   2 2 2 = Qq − Qq−1  (3,2) + Qq−1 − Qq−2  (3,2) + · · · + Q p+1 − Q p  (3,2) . W W W 2

Furthermore,

2

2

(19)

  Qq − Qq−1 2 (3,2) = 2 . q W 2

Consequently, we have q    Qq − Q p 2 (3,2) = 2 → 0, as n → ∞. W 2

= p+1

Since W(3,2) (I) is complete, we arrive at: Qn → Qˆ as n → ∞. 2 2. We begin by taking the limits in (16) ˜ Q(κ, ϑ) =

∞ 

ı ψ¯ı (κ, ϑ).

(20)

ı=1

When we apply the linear operator to (20), we discover ˜

Q(κ, ϑ) =

∞ 

ı ψ¯ı (κ, ϑ),

ı=1

it follows that ˜  , t ) =

Q(x = =

∞ 

 ı ψ¯ı (κ, ϑ), ρ (κ, ϑ) W(1,1) 2

ı=1 ∞ 

 ı ψ¯ı (κ, ϑ), ∗ ρ (κ, ϑ) W(3,2) 2

ı=1 ∞ 

 ı ψ¯ı (κ, ϑ), ψ (κ, ϑ) W(3,2) .

(21)

2

ı=1

Multiplying (21) by ςj  and taking j 

˜  , t ) = ςj  Q(x

=1

∞  ı=1

=

∞ 

j

=1

to find

 ı ψ¯ı (κ, ϑ),

j 

 ςj  ψ (κ, ϑ)

=1

W(3,2) 2

 ı ψ¯ı (κ, ϑ), ψ¯j (κ, ϑ) W(3,2) 2

ı=1

= j .

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Using (17), we discover ˜  , ϑ ) = (κ , ϑ ).

Q(κ ∀(μ, η) ∈ I, ∃{(κ pj , ϑ pj )}∞ j =1 such that (κ pj , ϑ pj ) → (μ, η), as j → ∞. ∞ is dense on I. That is a result of the fact that {(κı , ϑı )}ı=1 Also we are aware of the following:

˜ pj , ϑ pj ) = (κ pj , ϑ pj ).

Q(κ Using j → ∞ and the continuity of , we can determine the desired result.



5 Numerical Experiments The theoretical foundation is tested using two applications of biological problems. Now, the following algorithm can be used to apply the RKHSM: 1st Step : Fix p =  × q; 2nd Step : Set ψı (κı , ϑj ) = (μ,η) O(κ,ϑ) (μ, η)|(μ,η)=(κı ,ϑı ) ; 3rd Step : Calculate ςıj using  (12); 4th Step : Set ψ¯ı (κ, ϑ) = ık=1 ςık ψk (κ, ϑ), ςıı > 0, ı ∈ {1, . . . , p}; 5th Step : Choose an initial guess Q0 (κ1 , ϑ1 ); 6th Step : Set ı = 1; 7th Step : Set ı = ık=1  ςpık (κk , tk ); 8th Step : Qı (κı , ϑı ) = k=1 k ψ¯ k (κı , ϑı ); 9th Step : If n > ı, set ı = ı + 1. Go to step 7. Else stop. Example 5.1 Considering the following NWSE: ⎧ ∂ϑ Z(κ, ϑ) = ∂ 2 2 Z(κ, ϑ) + 3Z(κ, ϑ) − 4Z3 (κ, ϑ), ⎪ ⎪ κ  ⎪ √ ⎪  ⎪ ⎪ 3 e 6κ√ ⎪ ⎨ Z(κ, 0) = , 4

√

0 ≤ κ, ϑ ≤ 1,

6

e 6κ +e 2 κ ! " ⎪ √ ⎪ √ ⎪ 3  , and Z 1 3 1− 1 ⎪  ⎪ ϑ) = Z(0, ϑ) = . (1, ⎪ 9ϑ 2 9ϑ + 3 ⎪ ⎩ 2 +1 2 e− 2 +1 e 2

(22)

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The aforementioned problem has the following exact solution: √ #  e 6κ 3 √ Z(κ, ϑ) = . √ 4 e 6κ + e 26 κ− 29 ϑ

Example 5.2 Considering the NWSE as: ⎧ 2 2 ⎨ ∂t Z(κ, ϑ) = ∂κ 2 Z(κ, ϑ) + 2Z(κ, ϑ) − 3Z (κ, ϑ), Z(κ, 0) = λ, ⎩ λe2ϑ λe2ϑ Z(0, ϑ) = 3λeϑ sinh(ϑ)+1 , and Z (1, ϑ) = 3λeϑ sinh(ϑ)+1 .

0 ≤ κ, ϑ ≤ 1, (23)

The aforementioned problem has the following exact solution: 2λ exp(2ϑ) . Z(κ, ϑ) = −  3 λ(− exp(2ϑ)) + λ − 23

6 Computing Outcomes and Analysis As we previously explained, the RKHSM is used for all cases. Considering p collocation points with κı = ı , ı = 1, 2 . . . ,  and ϑj = qj , j = 1, 2 . . . , q with  × q = p. By computing the absolute error values defined as |Z − Z p | with (κ, ϑ) ∈ [0, 1]2 , the numerical solution to each problem is determined and compared with the exact solution. In Example 5.1, we used 10 collocation points ( p = 10). Figures 1 and 2 show the approximate and exact solutions for Example 5.1. These graphs demonstrate how similar the behavior of the graphs is. To illustrate more contrasts between the RKHSM’s and exact solutions, as well as other approaches already in use [6] (Extended cubic uniform B-spline, Trigonometric cubic B-spline, and Uniform Cubic B-spline; for short ECBS, TCBS, and UCBS, respectively), We calculated and displayed the absolute errors in Table 1. The accuracy of the suggested method is confirmed by the good agreement between the results from the RKHSM and true solutions. Furthermore, we may assert that the RKHSM is more accurate than the alternative approaches. In Example 5.2, we used 10 collocation points( p = 10). Figures 3 and 4 show the approximate and exact solutions for Example 5.2. These graphs demonstrate how similar the behavior of the graphs is. To illustrate more contrasts between the RKHSM’s and exact solutions, as well as other approaches already in use [6] (UCBS, TCBS, and ECBS), We calculated and displayed the absolute errors in Table 2. The accuracy of the suggested method is confirmed by the good agreement between the results from the RKHSM and true solutions. Furthermore, we may assert that the RKHSM is more accurate than the alternative approaches.

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Fig. 1 Comparison of the RKHSM solution to Example 5.1 at ϑ = 0.3 and the exact solution

Fig. 2 Comparison of the RKHSM solution to Example 5.2 at κ = 0.6 and the exact solution

On Solutions of Biological Models Using Reproducing … Table 1 Results’ comparison for Example 5.1 κ ϑ UCBS [6] 0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

133

TCBS [6]

ECBS [6]

RKHSM

6.129 × 10−2

6.134 × 10−2

5.166 × 10−2

× 10−2

7.129 × 10−2

× 10−2

8.206 × 10−4 4.215 × 10−4 2.344 × 10−4 3.560 × 10−5 3.300 × 10−4 7.496 × 10−4 4.584 × 10−5 3.047 × 10−5 8.582 × 10−6 6.559 × 10−4 4.713 × 10−4 2.153 × 10−4 5.270 × 10−7 2.042 × 10−4 1.020 × 10−3 2.060 × 10−5 3.690 × 10−4 3.494 × 10−6 3.464 × 10−4 1.067 × 10−3

7.120 4.312 × 10−2 4.661 × 10−2 1.560 × 10−2 8.711 × 10−2 1.005 × 10−1 8.771 × 10−2 6.001 × 10−2 1.640 × 10−2 8.030 × 10−2 9.072 × 10−2 7.637 × 10−2 4.899 × 10−2 9.878 × 10−3 4.857 × 10−2 5.319 × 10−2 6.420 × 10−2 2.582 × 10−2 2.862 × 10−3

6.430 × 10−2 4.670 × 10−2 1.563 × 10−2 8.718 × 10−2 1.006 × 10−1 8.786 × 10−2 6.015 × 10−2 1.650 × 10−2 8.036 × 10−2 9.085 × 10−2 7.651 × 10−2 4.911 × 10−2 9.969 × 10−3 4.862 × 10−2 5.328 × 10−2 4.321 × 10−2 2.590 × 10−2 2.915 × 10−3

5.402 4.511 × 10−2 2.860 × 10−2 1.518 × 10−3 7.224 × 10−2 7.421 × 10−2 5.888 × 10−2 3.321 × 10−2 4.323 × 10−3 6.461 × 10−2 6.456 × 10−2 4.822 × 10−2 2.328 × 10−2 9.679 × 10−3 3.721 × 10−2 3.588 × 10−2 2.500 × 10−2 9.644 × 10−3 9.118 × 10−3

Fig. 3 Comparison of the RKHSM solution to Example 5.2 at κ = 0.6 and the exact solution

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Fig. 4 Absolute error of the RKHSM for Example 5.2 at ϑ = 0.6 Table 2 Results’ comparison for Example 5.2 κ ϑ UCBS [6] 0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

TCBS [6]

ECBS [6]

RKHSM

1.011 × 10−3

8.295 × 10−4

1.007 × 10−3

6.068 × 10−4

8.581 × 10−4 4.745 × 10−4 1.991 × 10−5 1.226 × 10−3 1.520 × 10−3 1.302 × 10−3 7.334 × 10−4 4.932 × 10−6 1.226 × 10−3 1.520 × 10−3 1.302 × 10−3 7.334 × 10−4 4.932 × 10−6 8.323 × 10−4 1.011 × 10−3 8.581 × 10−4 4.745 × 10−4 1.991 × 10−5

8.514 × 10−4 4.651 × 10−4 3.239 × 10−5 9.013 × 10−4 1.023 × 10−3 8.289 × 10−4 4.221 × 10−4 8.366 × 10−5 1.222 × 10−3 1.514 × 10−3 1.292 × 10−3 7.194 × 10−4 2.342 × 10−5 8.295 × 10−4 1.007 × 10−3 8.514 × 10−4 4.651 × 10−4 3.239 × 10−5

6.800 5.476 × 10−4 2.746 × 10−4 6.339 × 10−5 1.222 × 10−3 1.514 × 10−3 1.292 × 10−3 7.194 × 10−4 2.342 × 10−5 9.013 × 10−4 1.023 × 10−3 8.289 × 10−4 4.221 × 10−4 8.366 × 10−5 6.068 × 10−4 6.800 × 10−4 5.476 × 10−4 2.746 × 10−4 6.339 × 10−5

1 × 10−10 0 1 × 10−10 1 × 10−10 1 × 10−10 1 × 10−10 0 1 × 10−10 1 × 10−10 1 × 10−10 1 × 10−10 0 1 × 10−10 1 × 10−10 1 × 10−10 1 × 10−10 0 1 × 10−10 1 × 10−10 1 × 10−10

8.323 × 10−4

× 10−4

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7 Conclusion The NWSEs, which describe the dynamical behavior near the bifurcation point, were effectively solved numerically in the current work. The two primary steps in using the RKHSM are creating an orthonormal function system and defining a suitable bounded linear operator. It is demonstrated that the suggested approach has good convergence. To demonstrate the RKHSM’s capability and reliability, two examples were used. When exact results are compared to our acquired results, it is seen that they are in strong agreement. Figure 1 displays the variations of Z(κ, ϑ) in Example 5.1 over a fixed time ϑ with respect to position κ, demonstrating that Z(κ, ϑ) increases as x increases. The variations of Z(κ, ϑ) in Example 5.2 are shown in Figs. 2 and 3. This graph demonstrates that Z(κ, ϑ) increases as t increases. The absolute error of RKHSM in Example 5.2 is shown in Fig. 4. The accuracy of the RKHSM is truly demonstrated by this figure. Finally, it can be said that the suggested approach is very convincing and effective for solving a wide range of nonlinear equations that describe real biological systems. As part of our goal, we intend to make a novel contribution to the literature by applying a novel method to partial differential equations with multiple variables.

References 1. A. Akgül, A novel method for a fractional derivative with non-local and non-singular kernel. Chaos Solitons Fractals 114, 478–482 (2018) 2. S. Zaremba, Sur le calcul numérique des fonctions demandées dans le problème de Dirichlet et le problème hydrodynamique. Bulletin International de l’Académie des Sciences de Cracovie 68, 125–195 (1908) 3. B. Ghanbari, A. Akgül, Abundant new analytical and approximate solutions to the generalized Schamel equation. Phys. Scr. 95, 075201 (2020) 4. E. Babolian, S. Javadi, E. Moradi, RKM for solving Bratu-type differential equations of fractional order. Math. Methods Appl. Sci. 39, 1548–1557 (2016) 5. M. Modanli M, A. Akgül, On solutions to the second-order partial differential equations by two accurate methods. Numer. Methods Part. Diff. Equ. 34, 1678–1692 (2018) 6. W.K. Zahra, W.A. Ouf, M.S. El-Azab, Cubic B-spline collocation algorithm for the numerical solution of Newell Whitehead Segel type equations. Electr. J. Math. Anal. Appl. 2, 81–100 (2014) 7. M.G. Sakar, Iterative reproducing kernel Hilbert spaces method for Riccati differential equation. J. Comput. Appl. Math. 309, 163–174 (2017) 8. W. Jiang, T. Tian, Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method. Appl. Math. Model. 39, 4871–4876 (2015) 9. F. Geng, M. Cui, A reproducing kernel method for solving nonlocal fractional boundary value problems. Appl. Math. Lett. 25, 818–823 (2012) 10. F. Geng, M. Cui, New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions. J. Comput. Appl. Math. 233, 165–172 (2009) 11. M. Cui, Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space (Nova Science Publishers, Inc, New York, 2009) 12. N. Attia et al., On solutions of time-fractional advection-diffusion equation. Numer. Methods Part. Diff. Equ. (2020). https://doi.org/10.1002/num.22621

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13. O. Abu Arqub, J. Singh, B. Maayah, M. Alhodaly, Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag-Leffler kernel differential operator. Math. Methods Appl. Sci. 1–22 (2021). https://doi.org/10.1002/mma.7305 14. S. Momani, N. Djeddi, M. Al-Smadi, S. Al-Omari, Numerical investigation for Caputo-Fabrizio fractional Riccati and Bernoulli equations using iterative reproducing kernel method. Appl. Numer. Math. 170, 418–434 (2021) 15. H. Singh, Analysis for fractional dynamics of Ebola virus model. Chaos Solitons & Fractals 138, 109992 (2020) 16. H. Singh, Analysis of drug treatment of the fractional HIV infection model of CD4+ T-cells. Chaos Solitons & Fractals 146, 110868 (2021) 17. H. Singh, D. Baleanu, J. Singh, H. Dutta, Computational study of fractional order smoking model. Chaos Solitons & Fractals 142, 110440 (2021) 18. H. Singh, H.M. Srivastava, D. Baleanu (Eds.), Methods of Mathematical Modeling: Infectious Diseases (Elsevier Science, 2022). https://doi.org/10.1016/C2021-0-00445-4 19. T. Allahviranloo, H. Sahihi, Reproducing kernel method to solve fractional delay differential equations. Appl. Math. Comput. 400, 126095 (2021) 20. D. Kumar, R.P. Sharma, Numerical approximation of Newell-Whitehead-Segel equation of fractional order. Nonlinear Eng. 5(2), 81–86 (2016) 21. Y. Chellouf, B. Maayah, S. Momani, A. Alawneh, S. Alnabulsi, Numerical solution of fractional differential equations with temporal two-point BVPs using reproducing kernal Hilbert space method. AIMS Math. 6(4), 3465–3485 (2021)

A Study of the Fractional Tumour–Immune Unhealthy Diet Model Using the Pseudo-operational Matrix Method Saurabh Kumar and Vikas Gupta

Abstract One of the main causes of death in the world is cancer. Cancer is a collection of illnesses characterized by the formation of tumours, malignant cells, or cancer cells capable of becoming cancerous. Several studies have been conducted on cancer and related disorders in recent years. In this chapter, we study the fractional tumour-immune unhealthy diet (TIUNHD) model, and to solve it; we use the fractional-order Laguerre operational matrix-based method. The fractional derivative of Caputo is used in this chapter. In this method, first, we constructed the pseudooperational matrices of fractional, integer order integration and then used these pseudo-operational matrices to approximate the unknown solutions of the given system of non-linear fractional differential equations model. This results in an algebraic system of equations, which can be easily solved by Newton’s iterative method. A numerical experiment demonstrates the proposed algorithm’s applicability to solving the TIUNHD model. Keywords Caputo fractional derivative · Fractional-order Laguerre polynomials · Pseudo-operational matrix of integration · Fractional TIUNHD model

1 Introduction An abnormal cell that divides uncontrollably anywhere on the body is defined as cancer. It is typically used in epidemiology studies to determine the extent of infection and what might be causing it. Several studies show that new dietary habits, climatic conditions, and other factors have increased cancer survivorship rates [1]. During the Leadership for the Twenty-First Century conference, Venter said, “human S. Kumar · V. Gupta (B) Department of Mathematics, Centre for Mathematical and Financial Computing, The LNM Institute of Information Technology, Jaipur 302031, India e-mail: [email protected]; [email protected] S. Kumar e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_6

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biology is considerably more complex than we thought.” Biological issues are not only influenced by genetic factors, but also by the proteins and cells affected by environmental factors [2]. Everyone is indeed born with some genetic characteristics; however, these genes have minimal impact on how some people will turn out in life. Although genes communicate essential information about high infection risk, they rarely play a direct role in determining a disease’s actual occurrence or cause. The relationship between diet (such as vitamins E, A, D, C, and B) and immunity has gained importance in recent years. According to studies [3–5], these vitamins are essential in maintaining immune system activity to protect tissues from injury. For cancer studies, researchers have proposed various mathematical models using ordinary differential equations (ODEs), delay differential equations, and partial differential equations to interrogate the effects of tumour evolution on the dynamics of other cells [6–10]. It has been found that obesity, hormones, and diet are among the primary causes of cancer. However, most of the components remain unknown. The Michaelis–Menten function has been used to study tumour-immune interactions [11, 12]. Rambely and Alharbi proposed a healthy-immune model to illustrate how the immune system resists tumour development dynamically [13]. According to the analysis, their model compares results with those of a weakened immune system. A model explaining how obesity and cancer interact to boost tumour response to chemotherapy has been developed by Roberto et al. [14]. As a means of exploring the effects of vitamin interventions on tumours and immune cells, Alharbi and Rambely developed the tumour-immune-vitamin system via ODEs in [15]. A mathematical model based on ODEs was developed in 2012 by Mufudza and others [16] on the influence of oestrogen on breast cancer dynamics. A mathematical model of pancreatic cancer is considered by [17]. Arciero and others [18] presented a mathematical model that examined tumour growth, immune escape, and siRNA treatment of tumours. In this study, an unhealthy diet has been identified as one of the main risks for increased cancer cases. According to a numerical simulation, vitamins can boost immunity when taken regularly. The immune system is boosted by 16% per day, with abnormal cells appearing in the tissues. The TIUNHD model dynamics might assist in understanding how vitamin intake affects immune and tumour cell dynamics. The TIUNHD model consists of a system of non-linear ODEs. Numerous fractional operators have been described in the literature, including Riemann–Liouville, Hadamard, Grunwald–Letnikov, Caputo derivatives, and other adaptive derivatives [19]. Here are some references on how to solve fractional models that govern real-life applications such as [20–24]. Because we seek a highly accurate, fully discrete numerical scheme for solving the fractional TIUNHD model, we wish to evaluate the applicability of the operational matrix approach based on fractional order Laguerre polynomials. According to the proposed technique, we first develop the fractional, integer order pseudo-operational integration matrices for fractionalorder Laguerre polynomials. The fractional Taylor’s functions are used to define the fractional-order Laguerre polynomials. These pseudo-operational matrices are used to approximate the integer and fractional order derivatives contained in the TIUNHD model, leading to an algebraic system of equations. After collocating this system of equations at the Newton–Cotes nodal points, we used Newton’s iterative

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approach to solve the resulting system of algebraic equations in order to determine the unknown coefficients. The fractional order Laguerre polynomials are easy to work with and offer a highly accurate approximation solution, which is an advantage of the suggested operational matrix approach. Contrary to previous discretization methods [25–27] the proposed method converts the non-linear system into algebraic equations without requiring any linearization. Some collocation methods based on Laguerre and other polynomials to solve fractional differential equations are given in [28–31]. The proposed pseudo-operational matrix (POM) technique can be considered as an alternative approach to the adaptive finite difference, spline collocation techniques [32–35]. No article has yet discussed the numerical method based on fractional order Laguerre polynomials for solving the fractional TIUNHD model. So we were motivated to solve the fractional TIUNHD model using the fractional order Laguerre operational matrix. The rest of the chapter is structured as follows. Preliminaries will be discussed in the next section. The fractional TIUNHD model is introduced in Sect. 3. In Sect. 5, we discuss the function approximation. Section 6 presents the pseudo-operational integration matrix for integer and fractional orders. In Sects. 7 and 8, we discussed the numerical method and discussion of results, respectively. In Sect. 9, the conclusion is discussed.

2 Preliminaries Several definitions of fractional calculus are presented in this section, which is essential to the rest of the chapter. Definition 2.1 The Riemann–Liouville integral of the function f (x) ∈ L 1 [a, b] in the interval (a, b) with fractional order μ ≥ 0 is defined as follows [36]:  μ

J f (x) =

1 (μ)

x

f (x),

a

f (t)(x − t)μ−1 dt, μ > 0, μ = 0,

(2.1)

where, (μ) is widely known as the gamma function. For μ, η > 0, f (x) ∈ L 1 [a, b] and ρ > −1, the Riemann–Liouville integral exhibits the following properties. (i) J μ J η f (x) = J η J μ f (x). (ρ+1) x (μ+ρ) . (ii) J μ x ρ = (ρ+1+μ) μ η μ+η f (x). (iii) J J f (x) = J Definition 2.2 The Riemann–Liouville fractional-order derivative of the function f (x) with order μ ∈ (0, 1), is defined as follows [36]: d 1 D f (x) = (1 − μ) d x μ

 a

x

f (t)(x − t)−μ dt,

(2.2)

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where, D μ denotes the fractional derivative with order μ. Definition 2.3 The Caputo fractional derivative of f (x) of order μ ∈ (r − 1, r ) can be defined as follows [37, 38]: D μ f (x) =

1 (r − μ)



x

(x − t)r −μ−1

0

dr f (t) dt, dt r

(2.3)

where r is the smallest positive integer such that r − 1 < μ < r. When μ ∈ N, the differential operator treat as same as the integer order derivative operator. For r − 1 < μ ≤ r , and f (x) ∈ L 1 [a, b], following properties can be determined by Caputo’s derivative definition: (i) In case of constant functions, the Caputo derivative is zero when its order μ is greater than zero. )(x) = f (x). (ii) (D μ J μ f  (ρ+1) ρ−μ x , if ρ > μ − 1, r − 1 < μ < r, ρ ∈ R (iii) D μ x ρ = 1+ρ−μ . 0, if ρ ≤ μ − 1, r − 1 < μ < r, ρ ∈ {−1, −2, . . .}  −1 xm (iv) (J μ D μ f )(x) = f (x) − rm=0 f m (0+ ) (m+1) . Definition 2.4 (The Kronecker delta function [39]) Mathematically, the Kroneckerdelta function is a function of two variables; when the two variables are equal, the function returns 1; otherwise, it returns 0, i.e.,  1, if i = j, . (2.4) δi j = 0, if i = j. Definition 2.5 (Fractional-order Laguerre functions) The analytic form of the fractional order Laguerre functions in the interval (0 ≤ x < ∞) is defined as follows [40]: L μm (t) =

m  s=0

(−1)s (m + 1) t μs . (s + 1)(s + 1)(m − s + 1)

(2.5)

In the form of recurrence relation, the fractional order Laguerre polynomials are defined as follows: L μm+1 (t) = (2m + 1 − t μ )L μm (t) − m 2 L μm−1 (t), μ

m = 1, 2, . . . ,

(2.6)

μ

where L 0 (t) = 1 and L 1 = 1 − t μ . μ

Theorem 2.1 In terms of the weight function wμ (t) = t μ−1 e−t in interval [0, ∞), the fractional order Laguerre polynomials are orthogonal with the following orthogonal property:  ∞ δi j μ μ , (2.7) wμ (t)L i (t)L j (t)dt = μ 0

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where δi j is the Kronecker delta function defined in Eq. (2.4). μ

1

Proof By substituting y = t μ and using the property L m (z μ ) = L m (z) of the fractional order Laguerre polynomials, we obtained, 

∞



∞



∞

1 μ μ L i (z)L j (z)dz. μ 0 0 0 (2.8) By orthogonal property of fractional order Laguerre polynomials, we obtained the desired result.  μ

μ

wμ (t)L i (t)L j (t)dt=

μ

t μ−1 e−t L i (t μ )L j (t μ )dt =

3 Fractional TIUNHD Model In this section, we explain how the TIUNHD model works. This model is based on a nonlinear system of ODEs. Further, we hypothesized that tumour cells emerge in tissue when the immune system cannot eliminate or suppress abnormal cells. The primary role of the immune system is to protect the body from cancer. There are two main populations in the TIUNHD model that is based on non-linear ODEs. First is the tumour cells population and it is denoted by the symbol T. The cells of this type may divide rapidly in the presence of a weak immune system while invading other tissues or countering them [41]. Following is an ordinary differential equation that represents tumour cell growth: dT = c1 T (1 − c2 T ) − c3 T I, dt

(3.1)

where the parameter c2 denotes the tumour reduction due to dietary metabolization of the garbled tumour, and c1 represents the growth limit of tumour cells. Thirdly, parameter c3 determines the rate at which tumour cells are eliminated or suppressed by the immune system under the assumption that a poor diet compromises the immune system’s response. The immune system plays a crucial role in preventing the development of cancer. When tumours are present in the tissue, several immune cells are produced. In some cases, these cells are ineffective at delaying or eliminating tumour cells. I represents the immune response behaviour governed by the following non-linear ordinary differential equation: δ3 I T dI = δ1 − δ2 I + − δ5 T I. dt δ4 + T

(3.2)

Parameters are described below in details (i) δ1 represents the constant production of immune cells by the body on a regular basis. (ii) δ2 represents natural death percentage of the immune cells.

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(iii) Symbol δ3 represents the percentage of immune cells that naturally die. (iv) δ4 represents the threshold percentage for the immune system. (v) δ5 represents the percentage of immune cells that are suppressed. A TIUNHD model is defined as a system of ODEs as follows: 

dT dt dI dt

= c1 T (1 − c2 T ) − c3 T I, IT = δ1 − δ2 I + δδ43+T − δ5 T I.

(3.3)

In this chapter our aim to consider the fractional order TIUNHD model which is defined as follows:  β d T = c1 T (1 − c2 T ) − c3 T I, dt β (3.4) IT dβ I = δ1 − δ2 I + δδ43+T − δ5 T I, dt β where 0 < β ≤ 1. The initial conditions are choose as T (0) = 1, I (0) = 1.22 [13].

4 Stability For the stability analysis of the TIUNHD model, we refer to [15].

5 Function Approximation A function g(t) in the interval [0, ∞) can be expanded in terms of fractional order Laguerre polynomials as the following formula: g(t) =

∞ 

am L μm (t),

(5.1)

m=0

where {a0 , a1 , a2 , . . .} are the unknowns coefficients, and derived by the following formula:  ∞ am = μ

0

wμ (t)g(t)L μm dt, m = 0, 1, 2, 3, . . . .

(5.2)

Function approximation in terms of M number of fractional order Laguerre polynomials are considered in particular applications as follows g(t) = AL μ (t),

(5.3)

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where A is the row vector and L μ (t) is the column vector defined as A = [a0 a1 μ μ μ a2 · · · a M ] and L μ (t) = [L 0 (t) L 1 (t) · · · L M (t)]T respectively, here T denotes the transpose of the vector.

6 Pseudo-operational Matrix The fractional Taylor functions are used to calculate the POM of integer and fractional order, and they are defined as follows [42]: μ

Ti = t iμ .

(6.1)

Using Eq. (2.5), we may express L μ (t) in the form of Eq. (6.2) as follows: L μ (t) = D1 Tμ (t),

(6.2)

where D1 = [di j ]((M+1)×(M+1)) , and entries di j are defined as follows:  di j =

(−1) j (i+1) , (i− j+1)(( j+1))2

0,

otherwise,

Now, for finding the value of 

t

if i ≥ j,

t 0

(6.3)

L μ ( p)dp, we follow the following steps: 

t

L μ ( p)dp =

0

i, j = 0, 1, 2, . . . , M.



t

D1 Tμ ( p)dp = D1

0

Tμ ( p)dp

0

(6.4)

= t D1 H1 Tμ (t) = t D1 H1 D1−1 L μ (t). If D1 H1 D −1 = Q 1 , then Eq. (6.4) implies 

t

L μ ( p)dp = t Q 1 L μ (t).

0

Here Q 1 is called the POM of integration with integer order, and entries h i j of the matrix H1 = [h i j ] is defined as follows:  hi j =

1 , iμ+1

0,

if i = j, i, j = 0, 1, 2, . . . , M. otherwise,

(6.5)

Similarly, we can find the POM of integration with fractional order, which is determined by using the following procedure:

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J η (L μ (t)) = J η (D1 Tμ (t)) = D1 J η (Tμ (t)) = t η D1 R1 Tμ (t) = t η D1 R1 D1−1 L μ (t),

(6.6)

suppose D1 R1 D −1 = Q 2 . So, Eq. (6.6) implies J η (L μ (t)) = t η Q 2 L μ (t). In this case, Q 2 is called the POM of integration with fractional order, and entries ri j of the matrix R1 = [ri j ] is defined as follows:  ri j =

(iμ+1) , (iμ+1+η)

0,

if i = j, i, j = 0, 1, 2, . . . , M. otherwise,

(6.7)

7 Numerical Method to Solve the Fractional TIUNHD Model with Caputo’s Derivative This section focuses on the solution of the fractional order TIUNHD model defined in Eq. (3.4) by using the fractional order Laguerre polynomials. First, we approximate the highest-order derivatives of the tumour cell and immune system in terms of the fractional order Laguerre polynomials. So, we obtained the following dI dT = AL μ (t), and = B L μ (t), (7.1) dt dt where, ⎡

⎤T a0 ⎢ a1 ⎥ ⎢ ⎥ A=⎢ . ⎥ , ⎣ .. ⎦

⎡

⎤T b0 ⎢ b1 ⎥ ⎢ ⎥ B=⎢ . ⎥ , ⎣ .. ⎦

am

⎡

⎤ L 0 (t) ⎢ L 1 (t) ⎥ ⎢ ⎥ L μ (t) = ⎢ . ⎥ . ⎣ .. ⎦

bm

(7.2)

L m (t)

Based on the initial conditions and integration with respect to variable t of Eq. (7.1), we arrive at the following approximations: T (t) = t AQ 1 Lμ(t) + 1,

(7.3)

I (t) = t B Q 1 Lμ(t) + 1.22.

(7.4)

and

To find the values of

dβ T dt

, and

dβ I dt

, we have

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 dβ T 1−β dT 1−β = Jt = Jt AL μ (t) dt dt 1−β = A Jt L μ (t) = t 1−β Q 2 L μ (t).

(7.5)

β

Similarly, we can find the value of ddtI . β β Substituting the values of ddtT , ddtI , T (t), and I (t) into the Eq. (3.4), we obtained the system of equations; then, we collocate this system of equations at the Newton– 2i−1 . And, finally we obtained the unknowns Cotes nodal points defined as the ti = (M+1)2 A and B, by using Newton’s iterative method. So, we obtained the solutions T (t) and I (t).

8 Discussion of Numerical Results Numerical results derived using the proposed numerical technique are discussed in this section. Using the symbolic and algebraic software package Mathematica, we demonstrate the numerical solution of the fractional TIUNHD model. This analysis assumes some arbitrarily chosen orders, so the system’s complexity under consideration can be illustrated. The derivatives of arbitrary orders are more accessible in terms of degree, allowing for a wider range of geometry. Table 1 contains the parameter values, and primary conditions used in numerical simulations are given as T (0) = 1 and I (0) = 1.22. Figure 1 represents the number of tumour and immune cell populations with different time sequences. The behavior of the tumour and immune cells are shown for the various fractional order derivatives in Fig. 2a–b. Our graphs demonstrate that enhancing immune cells can reduce tumour cell production. If the fractional order is smaller, the decay or growth process is faster. So, the fractional order operators provide a better depiction of the dynamics of the relationship between the growth and decay of Immune cells and the growth or decay of tumours. Table 1 The values of parameters

Parameters

Cost

c1 c2 c3 δ1 δ2 δ3 δ4 δ5

0.4426 0.4 0.1469 0.7 0.57 0.7829 0.8620 0.3634

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Population

1.25 1.20 1.15 1.10

Tumour Cells

1.05

Immune Cells

1.00 0

1

2

3

4

5

6

t

Fig. 1 The behaviour of the Tumour and Immune cells during different timing in Days and the given values of μ = 21 , M = 12, β = 1, c1 = 0.4426, c2 = 0.4, c3 = 0.1469, δ1 = 0.7, δ2 = 0.57, δ3 = 0.7829, δ4 = 0.8620 and δ5 = 0.3634 Tumour Cells

Immune Cells 1.29

1.14

1.28

1.12 1.10

1.27

1.08

1.26

1.06

1.25

1.04

1.24 1.23

1.02 0.5

1.0

(a)

1.5

2.0

t

0.5

1.0

1.5

2.0

t

(b)

Fig. 2 For β = {0.75, 0.90, 1}, μ = 21 , and M = 12 a The behaviour of Tumour cells concerning with the time t (Days), b The behaviour of Immune cells concerning with the time t (Days)

9 Conclusion This chapter generalizes the fractional TIUNHD model, which is governed by the system of non-linear differential equations. We solved the fractional TIUNHD model using the fractional order Laguerre polynomials POM technique. Additionally, the fractional-order curves approximate integer-order when β → 1. The proposed method is simple to implement and efficiently solve the mathematical non-linear TIUNHD model converted to non-linear algebraic equations. The proposed method may be extended to approximate new fractional and adaptive derivatives used to model complicated events in nutrition, food science, and biological science. We will likely achieve better accuracy in future communications by using other classes of polynomials also.

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Analysis of a Fractional Stage-Structured Model With Crowley–Martin Type Functional Response by Lagrange Polynomial Based Method Chandrali Baishya and P. Veeresha

Abstract The dynamics of a stage-structured predator-prey system that replicates interactions between two densities of prey and predator populations were investigated in this work. The adult predator population and the juvenile predator are the two compartments that make up the predator population in the model. The predator relies on both prey and juvenile predator, which is another element of the paradigm that can be termed cannibalism. Crowley–Martin type functional denotes the nature of the interaction between prey and adult predators, while Holling type-I functional response denotes the nature of contact between juvenile and adult predators. The concept of memory is introduced in the form of the Caputo fractional derivative to reflect the complicated dynamics of interaction among the species. As a result, the model is able to incorporate all relevant historical information about the occurrence, from its inception to the desired time, into its calculations. We have also investigated the boundedness and existence and uniqueness of solutions to the proposed model. The condition of existence and stability of various points of equilibrium are investigated. The numerical simulations are performed by using the Lagrange polynomial-based method which is novel in the field of mathematical biology. Simulations have been accomplished to examine the significance of parameters related to cannibalism, the conversion rate from prey to adult predator, harvesting of an adult predator, and growth rate of juvenile predators on the overall behavior of the system. The noteworthy performance of the fractional operator on the anticipated predator-prey model’s dynamical behavior is well demonstrated by numerical results. Keywords Stage-structured model · Caputo fractional derivative · Crowley–Martin type functional response · Holling type · Lagrange polynomial based method

C. Baishya Department of Studies and Research in Mathematics, Tumkur University, Tumkur 572103, India P. Veeresha (B) Department of Mathematics, Center for Mathematical Needs, CHRIST (Deemed to be University), Bengaluru 560029, India e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_7

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2020 Subject Classification 34A34 · 70K20 · 92D40 · 26A33

1 Introduction The study of the dynamics of species in the form of a dynamical system is one of the most captivating issues in all of academia. It has gone beyond the realm of species, and these systems have been used to represent a variety of natural phenomena such as hydrodynamics [1], chemical reactions [2], plasma physics; social science and economics issues such as business, finance, defense, criminology, epidemiology, and others [3, 4]. These are the addition to novel applications on population dynamics. In the literature of species dynamics stage-structured models have significant importance because every species naturally have physical growth in stages. So, many mathematical biologists have contributed to the evaluation of stage-structured predator-prey models [5–10]. The factors like predator harvesting, prey refuge, disease in prey/predator, delay, etc. have an impact on stage-structured models just like they do on other predator-prey models. [11, 12]. The practice of eating another member of the same species as food is known as cannibalism. Cannibalism comes in a variety of forms, including ritualistic, sacrificial, and survival cannibalism. Cannibalism is a widely prevalent ecological interface that has been seen in more than 1,500 species of animals. In ecology, cannibalism acts as a controller of the structure of a population. In [13], authors included forty experimental research on cannibalism in crows, copepods, beetles, and fish species, several scorpions, and others, taking cannibalism as a density control parameter. In the field of mathematical biology, cannibalism has gotten a lot of attention. When population densities are high and resources are few, cannibalism appears to occur in many animals [14]. Depending on the system under examination, it might have destabilizing or stabilizing impacts on the species. In [15], cannibalism is depicted as a tool of stabilization in a prey-predator model. For growth, in a system of stage-structured species, young species cohabit alongside adult species while neither reproducing nor predating. Only adult predators can attack prey and breed, and almost all species in the environment have an adult and juvenile stage structure. The pressure of predation on prey lessens as giant con-specifics consume small predators, which has a substantial impact on organism dynamics. Cannibalism occurs within a species, and its systems differ structurally from those of other populations [13, 16]. In [17], the conditions that led to the evolutionary emergence of cannibalism are created, as well as the criteria for cannibalism’s persistence in organisms are studied. Including cannibalism in predator species, author in [18] investigates a prey-predator system. A cannibalistic predator-prey model has been proposed and examined in [19] where the predator population is infected with a transmissible disease. In their research, Deng et al. [20] have demonstrated that cannibalism affects the stability of the system in both favorable and unfavorable ways. In [21], authors have analyzed that cannibalism has significant impact on the evolution of predator species. On the other hand, the

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dynamic growth of a population is significantly impacted by harvesting. Population harvesting is a natural part of wildlife management, fishing, and other industries. Constant harvesting [22–24], proportional harvesting [25] and nonlinear harvesting [26] are the most common harvesting components included in mathematical models. An excellent harvesting strategy incorporating stage structure in predator species was investigated in [27]. Including cannibalism in the predator, authors in [28] constructed a stage-structured predator-prey system which was later analyzed further by Zhang et al. [29]. The functional response (FR), on the other hand, is a unique and noteworthy component that plays a key role in all predator-prey interactions [30]. Some of the notable FRs are Holling type I, II, III, and IV FR [31], Bedington-DeAngelis FR [32–34], Leslie-Gower FR[35], square root FR [36], etc. When both prey and predator populations are large, the Crowley–Martin FR (CMFR) is used as a predator interference function. The predator-prey model involving CMFR was originally put forward by Crowley and Martin [37]. This FR is comparable to the Beddington–Deangelis functional reaction. This varies only by one extra term, which describes how predators interfere with one another when their prey is densely packed. Assuming that the larger density of the predator population is what causes the depletion in the feeding rate of predators [38], in this formulation the feeding rate per capita is: g(u, v) =

γ uv , (1 + au)(1 + bv)

where u and v are the population density of the prey and predator respectively [38]. γ is the rate of capture effect, a denotes the holding time and b indicates the magnitude of interference among predators. In [39, 40], authors studied the global stability and persistence in a model with CMFR where the prey population has two growth stages. More research related to the CMFR can be found in [27, 37, 38, 41, 42]. Other than the predator-prey dynamics, involvement of CMFR can also be noticed in the works related to Human Immunodeficiency Virus (HIV), viral dynamics models, and other infectious [43–46]. In [47], authors have used CMFR to observe a phytoplanktonzooplankton system with delay. In [48, 49], for a prey-predator model with CMFR, non-autonomous stochastic analysis is performed. Further to analyze bifurcation and chaos control involving two delay times, discrete-time CMFR is used in [50]. Although fractional differentiation is as old as the classical order derivative, the concept has gotten a lot of attention and has been studied extensively since 1967, when it was first modified as part of the research in Caputo’s work [51]. Recently, for modeling numerous real-world phenomena, rigorous applications of fractional calculus can be noticed in several fields such as chemistry, image and signal processing, control theory, electricity, biology, mechanics, and economics [52–54] by using fractional derivatives (FDs), involving Caputo, Riemann–Liouville, Riesz, Weyl, Grünwald–Letnikov, Marchaud and Hifler, Atangana–Baleanu, and Caputo–Fabrizio operators. The most crucial subjects include history-dependent process, power law, Levy statistics, anomalous diffusion, vibration and control, random walk, porous media, special functions, chaos and groundwater problems, Riesz potential, singu-

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larities analysis and integral representations, fractional Brownian motion, computational fractional differential equations, non-Fourier heat conduction, FD and fractals, nonlocal phenomena, biomedical engineering, fractional variational principles, relaxation, fractional predator-prey system, rheology, soft matter mechanics, fractional transforms, fractional wavelet, fractional signal and image processing, acoustic dissipation, geophysics, fluid dynamics, viscoelasticity. Supporting documentation of the above-mentioned works can be found in [55–58]. Riemann–Liouville and Caputo FD received special attention from the research community over the decades. Continuity and differentiability of an arbitrary function are not mandatory with Riemann–Liouville FD at the origin. Benefits of the Caputo FD include the ability to formulate models using standard initial and boundary conditions [51]. Furthermore, under the Caputo FD, the derivative of a constant is zero. Although these FDs have a lot of advantages, they cannot be used in every case. The fact that a constant’s Riemann–Liouville derivative is not zero gives rise to various switches to the modeling presumptions, and thus necessitates the generalization of the initial conditions with fractional order. Additionally, when any arbitrary function is a constant at the origin, the FD results in the singularity at it. Mittag–Leffler and exponential functions are examples of this. Only differentiable functions are defined for the Caputo derivative. In the Riemann–Liouville sense, a function may nevertheless have FDs of all orders lower than one even when they lack the first-order derivative. Due to the singularity of the power-law-based FDs, the frequently employed fractional differential operators (FDOs) have considerable limits when it comes to reproducing real-world issues. Caputo and Fabrizio brought in a novel fractional operator to address this issue. They emphasized the fact that many physical phenomena are non-singular and that utilizing singular operators to model such events could produce incorrect outcomes. An FDO with an exponential function as its kernel, known as the Caputo–Fabrizio FD, has been developed to tackle this issue. With classical derivative, exponential functions generate an evolution partial differential equation and they are eigenfunctions of the decay differential ordinary equations [59]. Atangana and Baleanu in 2016, to incorporate time non-locality into the system with a non-singular kernel, introduced the generalized Mittag–Leffler function. Numerous academics have investigated stage structure predator-prey models incorporating FDs [60–70]. Recent literature contains some investigations on the fractional cannibalistic model [71, 72]. In the present study, with Holling type I and CMFR with the Caputo FD, we have examined a stage structure predator-prey model. Even though some works on the fractional-order stage structure model can be found in the literature on predator-prey dynamics, however, incorporation of cannibalism with CMFR under the influence of FD is a novel contribution in this regard. Another important aspect that we have integrated into the proposed model is that juvenile predators do not reproduce even though they capture prey. The structure of the remaining part of this chapter is presented here. In Sect. 2, some definitions concerning the Caputo fractional-order systems are given. The proposed fractional-order stage structure cannibalistic model is presented in Sect. 3. In Sect. 5, the existence and uniqueness of the solution of the proposed system are analyzed. Section 6 concerns the boundedness of solutions. The dynamics of the system

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are presented in Sect. 6. Numerical simulations are performed in Sect. 7 to verify the obtained analytical results. Section 8 contains the conclusion of the present work.

2 Preliminaries In the present work, we have used the Caputo FDs because it supports the integer order initial condition. In this section, we have presented certain theorems that have been applied to determine the theoretical results corresponding to the solution of the projected model. Definition 2.1 ([56] Caputo FD) Suppose g(t) is k times continuously differentiable function and g (k) (t) is integrable in [t0 , T ]. The FD of the order α for g(t) in the Caputo sense is  t g (k) (τ ) 1 α dτ, (1) Dt g(t) = (k − α) t0 (t − τ )α+1−k where k is a positive integer with the property k − 1 < α < k and t > a. Here,  (·) refers to Gamma function. Lemma 2.2 ([73]) Consider the system Dtα v(t) = g(t, v), t > t0 ,

(2)

choosing the initial condition as v(t0 ), where 0 < α ≤ 1 and g : [t0 , ∞) ×  → Rn ,  ∈ Rn . Equation (2) has a unique solution on [t0 , ∞) × , when g(t, v) holds the locally Lipchitz conditions concerning to v. Lemma 2.3 ([74]) g(t) is a continuous function on [t0 , +∞) satisfying Dtα g(t) ≤ −g(t) + ξ, g(t0 ) = g0 , where t0 ≥ 0 is the initial time, 0 < α ≤ 1,  = 0, (, ξ ) ∈ R2 . Then, 

 ξ ξ g(t) ≤ g(t0 ) − E α [−(t − t0 )α ] + .  

(3)

3 Mathematical Model Formulation In this chapter, a stage-structured cannibalism model has been analyzed interconnecting three species. We have considered two stages of the predator population viz. juvenile and adult predators. Here u denotes the prey population and v and w denote juvenile and adult predator populations respectively. Preys are the victim of an attack

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by both juvenile and adult predators. However, juvenile predators do not reproduce. Predator-prey interaction is represented by CMFR whereas cannibalism is denoted by Holling type I FR. The proposed work is observed within the frame of the Caputo FD. To the best of our knowledge incorporation of FDs in a cannibalistic model with CMFR is novel in the literature.  γ uw u − βuv − , Dtα0 u = r u 1 − k (au + 1)(bw + 1) Dtα0 v = w − d1 v − μv − σ vw, (4) qγ uw α − ξ w. Dt0 w = μv − d2 w + pσ vw + (au + 1)(bw + 1) Here, k denotes the carrying capacity and r denotes the specific growth rate of the prey. positive parameters a, b, and γ are used to analyze the impact of handling time, the level of predator interference, and the proportion of prey captured by adult predators respectively. The parameter ξ is the harvesting effort of the adult predator to protect the prey population, q is the conversion factor of the adult predator, and d1 and d2 are the natural death rate of juvenile and adult predators respectively. The parameter μ represents the proportionality constant of the conversion of juvenile to adult predators,  represents the birth rate of the juvenile predator, and β is the capture rate of prey by juvenile predators. The catchability rate of young predators by adults is represented by sigma, and the associated conversion factor is denoted by p.

4 Boundedness Theorem 4.1 All the non-negative solutions of (4) that started in  are uniformly bounded. Proof Let (t) = u(t) + v(t) + q1 w(t). Then   1 Dtα (t) = Dtα u(t) + Dtα v(t) + Dtα w(t) q   u μ d2 pσ ξ = ru 1 − − βuv + w − μv − d1 v − σ vw + − w + vw − w. k q q q q

We have      u μ 1 Dtα (t) + L (t) =u r 1 − + L − μ + d1 − − L v − (ξ + d2 − q − L) k q q − βuv − σ ( p − q)).

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  Let L be a positive constant. If L = min (μ + d1 − μq ), (d2 + ξ − q) , then Dtα (t)

  k(r + L) 2 k(r + L)2 1 u− , + L (t) ≤ − + k 2 4

which implies that, Dtα (t) + L (t) ≤

k(r + L)2 . 4

By Lemma 2.3,   

k(r + L)2 k(r + L)2 + (t0 ) − E α −L(t − t0 )α , t ≥ t0 . 0 < (t) ≤ 4L 4L We have,  

k(r + L)2 k(r + L)2 k(r + L)2 + (t0 ) − E α −L(t − t0 )α → , t → ∞. 4L 4L 4L

This implies that (t) ≤

k(r + L)2 , t → ∞. 4L

Hence, all the solution associated for the system (4) that starting in  limited in 

k(r + L)2 + η, η > 0 ,

= (u, v, w) ∈ + : u(t) + v(t) + w(t) ≤ 4L for η being sufficiently small.

5 Existence and Uniqueness of Solutions of the Model Here, we examine the uniqueness and existence of the solution of the fractional differential Eq. (4) by employing the Banach fixed-point theorem. Dtα [u(t)] = ϕ1 (t, u), Dtα [v(t)] = ϕ2 (t, v), Dtα [w(t)] = ϕ3 (t, w).

(5)

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Then, Eq. (5) can be presented in Volterra type integral equation  t 1 u(t) − u(0) = ϕ1 (τ, u(τ ))(t − τ )α−1 dτ, (α) 0  t 1 ϕ2 (τ, v(τ ))(t − τ )α−1 dτ, v(t) − v(0) = (α) 0  t 1 ϕ3 (τ, w(τ ))(t − τ )α−1 dτ. w(t) − w(0) = (α) 0

(6)

Theorem 5.1 In the region  × [t0 , T ], where    = (u, v, w) ∈ R3 : max {|u|, |v|, |w|} ≤ M , and T < +∞, the Lipschitz condition holds and contraction occurs by the kernel ϕ1 if   +L)2 . 0 ≤ r + 2rkM + β M + γ M < 1, Where M = k(r4L Proof Consider the function u(t) and u(t) ¯ such as:      u γ uw u¯ γ uw ¯  ||ϕ1 (t, u) − ϕ1 (t, u)|| ¯ = ¯ − ) − β uv ¯ − r u(1 − k ) − βuv − ((1 + au)(1 + bw)) − r u(1 k ((1 + a u)(1 ¯ + bw))    2r M γw ≤ r+ + β M ||u − u|| ¯ + ||u − u|| ¯ k (1 + bw)(1 + au)(1 + a u) ¯   2r M + β M ||u − u|| ¯ + γ w||u − u|| ¯ ≤ r+ k   2r M ≤ r + βM + + γ M ||u − u|| ¯ k ¯ = ψ1 ||u − u||,

(7)

  where ψ1 = r + β M + 2rkM + γ M . The Lipschitz condition true for ϕ1 and if 0 ≤ ψ1 < 1, then ϕ1 follows contraction. Identically, it can be proved for ϕ2 and ϕ3 ||ϕ2 (t, v) − ϕ2 (t, v)|| ¯ ≤ ψ2 ||v − v||, ¯ ||ϕ3 (t, w) − ϕ3 (t, w)|| ¯ ≤ ψ3 ||w − w||, ¯

(8)

where ψ2 = (μ + d1 + σ M) and ψ3 = (d2 + 2M + pσ M + ξ + qγ M). Contractions occurs if 0 < ψi < 1, i = 2, 3. Theorem 5.2 The solution of the system (4) exists and will be unique, if we attains some tα such that 1 ψi tα < 1, for i = 1, 2, 3. (α)

Analysis of a Fractional Stage-Structured Model …

157

Proof This proof is shown in three steps: 1. Using the system (4), the recursive form can be written as  t 1 (ϕ1 (τ, u n−1 ) − ϕ1 (τ, u n−2 ))(t − τ )α−1 dτ, (α) 0  t 1 (ϕ2 (τ, vn−1 ) − ϕ2 (τ, vn−2 ))(t − τ )α−1 dτ, ζ2,n (t) = vn (t) − vn−1 (t) = (α) 0  t 1 (ϕ3 (τ, wn−1 ) − ϕ3 (τ, wn−2 ))(t − τ )α−1 dτ. (9) ζ3,n (t) = wn (t) − wn−1 (t) = (α) 0 ζ1,n (t) = u n (t) − u n−1 (t) =

The initial conditions are u 0 (t) = u(0), v0 (t)v(0), w0 (t) = w(0). When the first equation is subjected to the norm, we get ||ζ1,n (t)|| = ||u n (t) − u n−1 (t)||    t   1 α−1  (ϕ1 (τ, u n−1 ) − ϕ1 (τ, u n−2 ))(t − τ ) dτ  =  (α) 0  t 1 ≤ ||(ϕ1 (τ, u n−1 ) − ϕ1 (τ, u n−2 ))(t − τ )α−1 dτ ||. (α) 0

(10)

Utilizing the Lipschitz condition Eq. (7), we get: 1 ψ1 ||ζ1,n (t)|| ≤ (α)



t

||ζ1,n−1 (τ )dτ ||.

(11)

0

Correspondingly,  t 1 ψ2 ||ζ2,n−1 (τ )dτ ||, (α) 0  t 1 ψ3 ||ζ3,n−1 (τ )dτ ||. ||ζ3,n || ≤ (α) 0 ||ζ1,n || ≤

(12)

From the result, we can write: u n (t) =

n  i=1

ζ1,i , vn (t) =

n 

ζ2,i , wn (t) =

i=1

n 

ζ3,i .

i=1

Employing Eqs. (11) and (12) recursively, one can get 

n 1 ψ1 t , ||ζ1,i || ≤ ||u n (0)|| (α)  n 1 ||ζ2,i || ≤ ||vn (0)|| ψ2 t , (α)  n 1 ||ζ3,i || ≤ ||wn (0)|| ψ3 t . (α)

(13)

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As a result, the existence and continuity are illustrated. 2. To examine that the correlation (13) composed is the solution for the system (4), we have imagine the following: u(t) − u(0) = u n (t) − ϑ1n , v(t) − v(0) = vn (t) − ϑ2n , w(t) − w(0) = wn (t) − ϑ3n .

(14)

We set object to prove required results, as    t  1   (ϕ1 (τ, u) − ϕ1 (τ, u n−1 ))dτ  ||ϑ1n (t)|| =  . (α)

(15)

0

This yields, ||ϑ1n (t)|| ≤

1 ψ1 ||u − u n−1 ||. (α)

(16)

By recursively repeating the same technique, we get  ||ϑ2n (t)|| ≤

1 ψ1 t (α)

n+1 M.

(17)

M.

(18)

At some tα , we have 

1 ψ1 tα (α)

||ϑ2n (t)|| ≤

n+1

1 Equation (18) shows that, as n → ∞, ||ϑ1n (t)|| → 0 if (α) tα < 1. In the same manner, it can be shown that ||ϑ2n (t)|| and ||ϑ3n (t)|| trend to zero.

3. We shall now exhibit the system’s uniqueness in terms of solution (4). Assume that the system (4) has a new set of solutions, namely u, ˆ v, ˆ and w. ˆ Then, using the first equation in Eq. (5) as a starting point, we write: 1 u(t) − u(t) ˆ = (α)



t

(ϕ1 (τ, u) − ϕ1 (τ, u))dτ. ˆ

(19)

0

The following equation emerges after employing the norm ||u(t) − u(t)|| ˆ =

1 (α)



t

||(ϕ1 (τ, u) − ϕ1 (τ, u))dτ ˆ ||.

0

We acquire the Lipschitz condition by applying it.

(20)

Analysis of a Fractional Stage-Structured Model …

||u(t) − u(t)|| ˆ ≤

159

1 ψ1 t||u − u||. ˆ (α)

(21)

This results yields, at some tα  ||u(t) − u(t)|| ˆ 1−  Hence, 1 − u(t). ˆ

1 ψt (α) 1 α



1 ψ1 tα (α)

 ≤ 0.

(22)

> 0, which implies ||u(t) − u(t)|| ˆ = 0. Such that u(t) =

6 Existence and Stability of the Points of Equilibrium of the System (4) Here, we shall evaluate the existence and stability of the points of equilibrium of the (4).

6.1 Existence of Points of Equilibrium Now, by the help of the system, we have (4).  γ uw u − βuv − = 0, ru 1 − k (au + 1)(bw + 1) w − d1 v − μv − σ vw = 0, qγ uw − ξ w = 0. μv − d2 w + pσ vw + (au + 1)(bw + 1)

(23)

The significant non-negative equilibrium points of the system (4) are the trivial equilibrium Q 0 (0, 0, 0), Prey free equilibrium Q 2 (0, v2 , w2 ), Predator free equilibrium Q 1 (u 1 , 0, 0), and co-existence equilibrium Q 3 (u ∗ , v ∗ , w ∗ ). 1. Predator free equilibrium point Q1 (u 1 , 0, 0) = (k, 0, 0) always exists.  2 +ξ )−μ (d1 +μ)(d2 +ξ )−μ . , 2. Prey free equilibrium point is Q 2 0, (dσ1 +μ)(d σ ( p−(d2 +ξ )) ( p(d1 +μ)−μ) Q 2 exists if (d1 + μ) (d2 + ξ ) < μ, p (d1 + μ) < μ, and p < d2 + ξ. Second and third conditions imply the first condition. Therefore, Q 2 exists if R1 < 1 μ . The quantity R1 means the mutual growth of juvenile where R1 = (d1 +μ)(d 2 +ξ ) and adult predators. 3. Coexistence equilibrium point Q 3 (u ∗ , v ∗ , w ∗ ) shall be obtained from the following isocline:

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 γw u − βv − = 0, r 1− k (au + 1)(bw + 1) w − d1 v − μv − σ vw = 0, qγ uw − ξ w = 0. μv − d2 w + pσ vw + (au + 1)(bw + 1)

(24) (25) (26)

We shall verify the existence of u ∗ and w ∗ from isoclines (24) and (26). This will lead to the stability of v ∗ from isocline (24). From Eq. (25), we obtain v = w . From Eq. (24), we obtain the following: d1 +μ+σ w (a) For u = 0, we get w =

bd1 r +bμr −β−γ μ−γ d1 +r σ +

√

(d1 (γ −br )−bμr +β+γ μ−r σ )2 +4r (d1 +μ)(bβ−br σ +γ σ ) 2(bβ−br σ +γ σ )

=

w1 .

w1 > 0 if

bβ − br σ + γ σ > 0.

(b) For w =0, we get u = k >0. γ = 1 rk − (1+au)awγ (c) dw 2 (1+bw) , where  = − (1+au)2 (1+bw) − du dw du

< 0, if

r awγ > . k (1 + au)2 (1 + bw)

(27)

β(d1 +μ) (μ+σ w+d1 )2

< 0. (28)

Then, the isocline (24) passes through the points (0, w1 ) and (k, 0) and w decreases as u increases under the conditions (27) and (28). From the Eq. (26), we obtain the following 2 +ξ )−μ (a) For u = 0, we get w = (dσ1 +μ)(d = w2 . w2 > 0 due to the condition of (−d2 −ξ + p) existence of prey free equilibrium point. qγ u (μ+d1 )(μ+2 pσ w)+ pσ 2 w 2 ¯ = 1¯ (1+au)qwr (b) dw 2 (1+bw) > 0., where  = (1+au)2 (1+bw) + du (μ+σ w+d1 )2 − (ξ − d2 ).

Clearly, the isocline (25) passes through the points (0, w2 ) and w increases as u increases under the conditions. The analysis performed above interpret that the two isoclines (24) and (26) intersect at a unique point (u ∗ , v ∗ ) if w2 < w1 .

(29)

Theorem 6.1 Coexistence equilibrium point Q 3 (u ∗ , v ∗ , w ∗ ) exists and is unique if the conditions (27), (28), and (29) are satisfied.

Analysis of a Fractional Stage-Structured Model …

161

6.2 Stability Analysis Jacobian matrix  corresponding to the system (4) is J (u, v, w) =

− (au+1)γ2w(bw+1) − 0

2r u k

+ r − βv

γ qw (au+1)2 (bw+1)

β(−u) −d1 − μ − σ w μ + pσ w

γu − (au+1)(bw+1) 2  − σv γ qu − d2 − ξ + pσ v (au+1)(bw+1)2

.

If all the eigenvalues λi , i = 1, 2, . . . , n, of the Jacobian matrix J (E), E being the point of equilibrium, satisfy the condition |arg(eig(J (E)))| = |arg(λi )| >

απ , i = 1, 2, . . . , n. 2

(30)

Then the E is a stable point of equilibrium. To evaluate the eigenvalues we solve the characteristic equation |J (E) − λi I | = 0. Lemma 6.1 ([75]) Define the following characteristic equation P(λ) = λn + A1 λn−1 + A2 λn−2 + · · · + An = 0.

(31)

All the roots of the characteristic Eq. (31) satisfy the Eq. (30) if we have 1. For n = 1, the condition is A1 > 0.

√ 4 A2 −A21 2. For n = 2, the conditions are A1 > 0, 4 A2 > A21 , |tan −1 | > απ . A1 2 3. For n = 3, if the discriminant of the polynomial P(λ) is positive then necessary and sufficient conditions to satisfy the Eq. (30) are A1 > 0, A2 > 0, A1 A2 > A3 . If the discriminant of the polynomial P( ) is negative then necessary and sufficient conditions to satisfy the Eq. (30) are A1 > 0, A2 > 0, A1 A2 = A3 . 4. For general n, An > 0 is the necessary condition for Eq. (30) to be admit.

• At the trivial equilibrium Q 0 (0, 0, 0) the eigenvalues of the Jacobian matrix are λ0 = r,   1 λ1 = 2 −d1 − d2 − μ − ξ − (d1 + d2 + μ + ξ ) 2 − 4 ((d1 + μ) (d2 + ξ ) − μ) ,    λ2 = 21 −d1 − d2 − μ − ξ + (d1 + d2 + μ + ξ ) 2 − 4 ((d1 + μ) (d2 + ξ ) − μ) .

The eigenvalue corresponding to the u-direction is the positive number r. Since product of the eigen value is a negative quantity (d1 + μ) (d2 + ξ ) − μ, eigenvalues corresponding to the v and w direction are of opposite signs. Trivial equilibrium Q 0 (0, 0, 0) is a saddle point. • At predator free equilibrium Q 1 (k, 0, 0), the eigenvalue corresponding to udirection is the negative number −r. The product of the eigenvalues correspond1 +μ) + (μ + d1 ) (d2 + ξ ) − μ. Since ing to the v and w direction is P = − γ kq(d ak+1

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(d1 + μ) (d2 + ξ ) < μ, clearly P is a negative quantity. Therefore, Q 1 is a saddle point. • At the prey-free equilibrium point Q 2 , the characteristic equation of the Jacobian matrix is (32) P1 (λ) = λ3 + A1 λ2 + A2 λ + A3 , wher e γw + d1 + d2 + μ + ξ − pσ v − r + σ w, bw + 1     γw γw A2 = r − (−d2 − ξ + pσ v) + r − (−d1 − μ − σ w) bw + 1 bw + 1 γ qw( − σ v) − + (−d1 − μ − σ w) (−d2 − ξ + pσ v) , bw + 1     γw γw A3 = ( − σ v) r − (μ + pσ w) − r − (−d1 − μ − σ w) (−d2 − ξ + pσ v) . bw + 1 bw + 1

A1 =

The equilibrium point Q 2 is stable if the third condition of the Lemma 6.1 is satisfied. • The characteristic equation corresponding to the Jacobian matrix at Q 3 (u ∗ , v ∗ , w ∗ ) is (33) P2 (λ) = λ3 + B1 λ2 + B2 λ + B3 , where B1 = −(11 + 22 + 33 ), B2 = 11 22 + 22 33 + 11 33 − 12 21 − 23 32 − 13 31 , B3 = 12 21 33 + 23 32 11 + 13 31 22 − 11 22 33 − 12 23 31 − 13 21 32 .

Here, i j , i, j = 1, 2, 3 are entries of the Jacobian matrix. The equilibrium point Q 3 is stable if the third condition of the Lemma 6.1 is satisfied.

7 Numerical Simulation The Lagrange interpolation polynomial has been used as the basis for a large number of effective numerical techniques for the solution of linear and nonlinear DEs. In the setting of classical nonlinear systems, the Adams–Bashforth method, which makes use of the Lagrange interpolation polynomial, has been recognized as an effective numerical technique for solving complex system. The Adams–Bashforth has been expanded due to its accuracy and effectiveness within the context of fractional integration and differentiation. But the majority of these studies employ polynomials with linear Lagrange interpolation. In this section, we have presented the Lagranges polynomial-based numerical method described in [76, 77] to solve the system (4),

Analysis of a Fractional Stage-Structured Model …

163

which is a modified Adams–Bashforth and take into account the approximation at two successive places. We consider a general nonlinear Cauchy’s problem C

Dtα x(t) = φ(t, x(t)), 0 ≤ t ≤ T, x(0) = x0 .

(34)

Corresponding to (34), the Volterra integral equation is as follows: 1 x(t) = x0 + (α)



t

(t − s)α−1 φ(s, x(s))ds, x(0) = x0 .

(35)

0

Considering a time interval divided into (n − 1) subintervals such that at a given time t = tn , Eq. (35) becomes x(tn ) = x0 +

1 (α)



tn

φ(s, x(s))(tn − s)α−1 ds.

(36)

0

Equation (36) can be transformed into a sum as 1  x(tn ) = x0 + (α) k=0 n−1



tn+1

φ(s, x(s))(tn − s)α−1 ds.

(37)

tn

The function f (s, x(s)) is approximated by linear Langrange polynomial. On integration from Eq. (37) we get x(tn ) = x0 +

n−1 n−1   hα hα φ(sk , x(sk ))ck,n − φ(sk−1 , x(sk−1 ))τk,n , α(α + 2) α(α + 2) k=1

k=1

(38) where ck,n = (n − k)α (2α + n − k − 1) − (n − k − 1)α (n − k + 1 + 2α), α+1

τk,n = (n − k)

α

− (n − k − 1) (n − k + α).

(39) (40)

Again, subdivide the time interval into n subintervals such that at a given time t = tn+1 , Eq. (38) becomes 1 x(tn+1 ) = x0 + (α)



tn+1 0

φ(s, x(s))(tn+1 − s)α−1 ds.

(41)

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From Eq. (41) we get, 1  (α) k=0 n

x(tn+1 ) = x0 +



tn+1

φ(s, x(s))(tn+1 − s)α−1 ds.

(42)

tn

Equation (42) yields x(tn+1 ) = x0 +

n n   hα hα φ(sk , x(sk ))δk,n − φ(sk−1 , x(sk−1 ))μk,n , α(α + 2) α(α + 2) k=1

k=1

(43) where δk,n = (n − k)α (2α + n − k − 1) − (n − k − 1)α (n − k + 2 + 2α), α+1

μk,n = (n − k − 1)

α

− (n − k) (n − k + 1 + α).

(44) (45)

Subtracting (38) from (43) we obtain the formula to solve the Eq. (34) as follows:  n n−1   hα x(tn+1 ) = x(tn ) + φ(sk , x(sk ))δk,n − φ(sk , x(sk ))ck,n α(α + 2) k=1 k=1  n n−1   hα − φ(sk−1 , x(sk−1 ))μk,n − φ(sk−1 , x(sk−1 ))τk,n . α(α + 2) k=1 k=1 (46) Using the method described in Eq. (46), the formula corresponding to the system (4) becomes u n+1 = u n +

     n hα γ u i wi ui δi,n r u i 1 − − β − βu i vi − α(α + 2) k (au i + 1)(bwi + 1) i=1

    ui γ u i wi ci,n r u i 1 − − β − βu i vi − − k (au i + 1)(bwi + 1) i=1     n  γ u i−1 wi−1 u i−1 δi,n r u i−1 1 − − βu i−1 vi−1 − − k (au i−1 + 1)(bwi−1 + 1) n−1 

i=1

+

n−1  1=1

    γ u i−1 wi−1 u i−1 c1,n r u i−1 1 − − βu i−1 vi−1 − , k (au i−1 + 1)(bwi−1 + 1)

(47)

Analysis of a Fractional Stage-Structured Model …

vn+1

165

 n hα = vn + δi,n (wi − d1 vi − μvi − σ vi wi ) α(α + 2) i=1 − −

n−1  i=1 n  i=1

+

n−1 

ci,n (wi − d1 vi − μvi − σ vi wi ) (48) δi,n (wi−1 − d1 vi−1 − μvi−1 − σ vi−1 wi−1 )  c1,n (wi−1 − d1 vi−1 − μvi−1 − σ vi−1 wi−1 ) ,

1=1

wn+1 = wn +

   n hα qγ u i wi − ξ wi δi,n μvi − d2 wi + pσ vi wi + α(α + 2) (au i + 1)(bwi + 1) i=1

−

n−1  i=1

−

n 

 ci,n μvi − d2 wi + pσ vi wi +

qγ u i wi − ξ wi (au i + 1)(bwi + 1)

 δi,n μvi−1 − d2 wi−1 + pσ vi−1 wi−1 +

i=1

+

n−1  1=1

 c1,n μvi−1 − d2 wi−1 + pσ vi−1 wi−1 +



qγ u i−1 wi−1 − ξ wi−1 (au i−1 + 1)(bwi−1 + 1)



qγ u i−1 wi−1 − ξ wi−1 (au i−1 + 1)(bwi−1 + 1)

 .

(49) For numerical simulation, we considered r = 3.5, k = 70, β = 0.01, γ = 5,  = 2.5, p = 0.5, q = 0.2, μ = 0.25, d1 = 1.2, d2 = 0.2, σ = 0.2, ξ = 1, a = 0.01, b = 1.2. Initial populations are taken as u 0 = 1, v0 = 1, w0 = 1. For these values, the points of equilibrium are Q 0 (0, 0, 0), Q 1 (70, 0, 0), Q 2 (0, 11.7368, 111.5), Q 3 (2.77162, 4.26873, 3.75984). That means the complex dynamic of the system features the existence of two positive equilibria. Eigenvalues of the Jacobian matrix are respectively {3.5, −2.12539, −0.524609}, {−3.5, 3.05634, −1.58869}, {−23.8231, −0.635757, 0.0468033}, {−2.68984, −0.101439 + 0.47488i, −0.101439 − 0.47488i}. This indicate that coexistance point of equilibrium Q 3 is stable spiral and Q 2 is unstable. In Fig. 1, we can view the convergence to the coexistence points of equilibrium. Clearly, as the value of the FD α decreases, the system attains stability faster than before. In this section, we also looked at the impact of one of the parameters on the dynamics of the system while maintaining the other parameters constant. In Figs. 2, 3, 4, we have presented the effect of the rate of cannibalism on the solution profile for α = 1, α = 0.95, α = 0.9 respectively. It is observed that a stable system in Fig. 1 converted to unstable limit cycle in Fig. 2 with the increase in the rate if cannibalism. This means an increase in the cannibalism rate destabilizes the system. Figure 2 also indicates that loss of stability by a Hopf bifurcation will take place as the level of cannibalism increases. But the presence of FD stabilizes the system which can be observed in Figs. 3 and 4.

166

Fig. 1 Nature of the system (4) for distinct α

C. Baishya and P. Veeresha

Analysis of a Fractional Stage-Structured Model …

Fig. 2 Impact of cannibalism rate on the population of the system (4) for α = 1

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Fig. 3 Impact of cannibalism rate on the population of the system (4) for α = 0.95

Analysis of a Fractional Stage-Structured Model …

Fig. 4 Impact of cannibalism rate on the population of the system (4) for α = 0.9

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Figures 5, 6, and 7 represent the effect of change in the conversion rate of prey to adult predators. It is noticed that an increase in this conversion rate leads to a decrease in the prey population and an increase in the juvenile population. This indicates that the adult predators’ need for food is satisfied with catching prey and their attentions are diverted from juvenile predators. However, juvenile predator increase is comparatively slow due to the cannibalism factor. Again FD has been observed as a stabilizing tool here. In Figs. 8 and 9 we have seen the influence of the harvesting rate on the population. An increase in the harvesting rate of adult predator influence in increasing the prey population and decrease the juvenile predator population. Fractional derivative helps in achieving stability faster. Figures 10 and 11 help us understand the impact of growth rate of juvenile predators on the population. It is observed that this growth rate affects the prey population negatively.

8 Conclusion The analysis of a stage structure predator-prey model presented in this chapter in the context of the Caputo FD. Combination of the Caputo FD, CMFR, Cannibalism, and non-reproductivity of the juvenile predator after consuming prey make the proposed model unique from the available stage structure models in the literature. The existence and stability of trivial, predator-free, prey-free, and co-existence points of equilibrium points are examined. We have observed that, the solution of the system shows a switch from stable spiral behavior to stable focus in some cases and unstable to stable in other cases as the order of the FD reduces. One of the significantly noticeable points is that the increase in the rate of cannibalism destabilizes the system. Moreover, as the level of cannibalism increases, loss of stability takes place by a Hopf bifurcation. But the presence of FDs reduces the destabilization. It can also be noticed that the conversion factor is directly proportional to the growth of juvenile predator. The effects of capturing these observations are effectively validated by numerical simulations in this chapter using the Lagrange polynomial-based method. The simulated results show the efficiency of the numerical technique. As a future direction of studies, one can always look for chaotic dynamics which may occur in a system as a result of cannibalism and then find a way to control them.

Analysis of a Fractional Stage-Structured Model …

Fig. 5 Impact of conversion rate of prey to adult predator on the population for α = 1

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Fig. 6 Impact of conversion rate of prey to adult predator on the population for α = 0.95

Analysis of a Fractional Stage-Structured Model …

Fig. 7 Impact of conversion rate of prey to adult predator on the population for α = 0.9

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Fig. 8 Effect of harvesting rate of adult predator on the population for α = 0.95

Fig. 9 Effect of harvesting rate of adult predator on the population for α = 0.9

Analysis of a Fractional Stage-Structured Model …

Fig. 10 Effect of growth rate of juvenile predator on the population for α = 0.95

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Fig. 11 Effect of growth rate of juvenile predator on the population for α = 0.95

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Qualitative Theory and Approximate Solution to Norovirus Model Under Non Singular Kernel Type Derivatives Eiman, Waleed Ahmed, Kamal Shah, and Thabet Abdeljawad

Abstract We study a fractional order nonlinear dynamical system of Norovirus disease. The considered mathematical model consists of susceptible, exposed, vaccinated, infected, and recovered classes. Firstly, the Caputo-Fabrizio fractional derivative (CFFD) is used to examine the qualitative theory and approximate solutions. Some adequate results for the existence of approximate solutions are given using fixed point theory. We use the Laplace Transform (LT) and the Adomian decomposition (ADM) approach to get approximate results for each compartment. The graphical presentation that corresponds to available real facts for different fractional orders is given. Secondly, by using piecewise global fractional derivatives in sense of singular and non-singular kernels, the aforementioned system is also investigated. We establish sufficient conditions for the existence and uniqueness of the solution to the proposed model by using the fixed-point approach. Numerical simulation is performed by extending Newton’s interpolation formula for the considered model under the mentioned derivatives. Graphical presentations for various compartments of the model are given against the available real data for different fractional orders. The numerical findings are performed by using Matlab 16. Keywords Nonlinear dynamical system · Norovirus · Approximate solution · ADM · CFFD · Crossover behavior · Mathematical biology · Piecewise global fractional derivatives · Newton interpolation formula

Eiman · W. Ahmed · K. Shah Department of Mathematics, University of Malakand, Chakdara Dir(L) 18000 Khyber Pakhtunkhwa, Pakistan K. Shah · T. Abdeljawad (B) Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia e-mail: [email protected] T. Abdeljawad Department of Medical Research, China Medical University, Taichung 40402, Taiwan © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_8

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1 Introduction Nonlinear dynamical systems are useful for modeling a variety of real-world phenomena and processes. Algebraic, difference and differential/integral equations are widely used to represent dynamic systems. Nonlinear equations have been used to model the majority of real-world processes and phenomena. Engineering, physics, chemistry, control theory, and mathematical biology are among most of the fields where the indicated area has applicability. Many diseases have affected human population throughout the history but the most dangerous of which are viral diseases. Diseases like Measles, TB, malaria, HBV, HCV, Dengue fever, malignant malignancies, Spanish flu etc that primarily impact human populations have resulted in millions of deaths. As a result, they have developed a variety of tactics for limiting and decreasing the rate of infections in their communities. Most epidemics have been limited by the discovery of treatments and cures but still, there are some diseases that have no proper medications or having no treatment i.e incurable. Among these diseases, one of the infectious diseases is Norovirus disease. For more than 75 years, outbreaks of non-bacterial gastroenteritis have been linked to viruses. Because of the increased in incidence during the winter months, Zahorsky [1] during 1929, suggested the name “winter vomiting disease” to investigate outbreak. It should be kept in mind that the initial outbreak was discovered in a village Norwalk of USA in 1968. The virus was first reported in children and workers at a local elementary school. It was named Norwalk virus at that time [2]. Later on the name Norovirus was approved by an international committee of taxonomy in 2002. Norovirus is considered the main reason for a fifth of all acute gastroenteritis (AGE) cases each year worldwide [3–5]. And is linked to a majority of morbidity, mortality, and healthcare costs. The transmission dynamics of this relatively prevalent virus at the community level are still poorly known, in part due to underreporting of sickness. According to serological studies, the first Norovirus infection occurs early in infancy [6, 7]. AGE is responsible for over a quarter of all fatalities in children under the age of five in Africa and Southeast Asia [8]. Norovirus is responsible for the bulk of non-bacterial outbreaks and an estimated 18 percent of all endemic AGE [9, 10]. In the United Kingdom alone, Norovirus is predicted to cost 81 million pounds per year to healthcare systems and people [11]. Norovirus is spread mostly through fecal, oral transmission, which might be aided by vomiting episodes [1]. Other factors, on the other hand, are more frequently linked to outbreaks spread by the environment [12, 13]. This can happen through the contamination of food, fluid, and surfaces [14, 15]. Vomiting, diarrhea, muscle aches, low-grade fever or chills, headache, and stomach cramping are some of the symptoms of an infected person. Symptoms normally appear 12 to 48 hours after the virus is consumed and last 1 to 3 days [16, 17]. But vomiting and diarrhoea can seriously dehydrate people, especially elderly people, young children, and people who already have medical conditions. Therefore, necessary medical attention is required. There is a dose-response association, meaning that people who are exposed to more virus particles have a

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greater attack rate [18–20]. However, the infectious dose is modest, with only a 49 percent chance of infection from a single particle, implying that a dosage response is improbable. Many variables can aggravate Norovirus transmission and the resulting sickness. According to researchers , the virus has a high seasonality, with over half of infections occurring during the cold season. Seasonal variation is attributed to both environmental factors and population behavior. For example, Norovirus thrives in lower temperatures and may be aided by increased rainfall. Other factors in the population can also influence the intensity of Norovirus outbreaks. The virus affects people of all ages, but it is most common in children under the age of five. Because Norovirus is a virus, it cannot be treated with antibiotics. To prevent the problem one has to manage dehydration induced by vomiting and diarrhea. Also, hand washing with soap and water is an excellent way to reduce Norovirus infection transmission. Here, we refer some more details on the mentioned viral disease as [21–24, 26–29, 34]. It is remarkable that researchers are progressively using new tools and analysis to investigate epidemiological diseases. The mathematical formulation of the aforementioned phenomena is a strong tool. Mathematical models are becoming a powerful technique for making predictions and analyzing the transmission dynamics of infectious illnesses. Usually daily problems, such as infectious diseases are nonlinear in nature. This has led to a long standing interest in nonlinear mathematical models that can be used to describe many real-world problems. As a result, multiple mathematical models have been developed to describe various diseases. Similarly, nonlinear differential equations have been used by numerous authors to formulate the Norovirus infection. Since the majority of research regarding Norovirus infection has been based on mathematical models using classical derivatives. Classical derivatives are local in nature and lack a higher degree of freedom. As a result, most dynamic problems cannot be stated accurately and efficiently. As a result, mathematicians have determined that utilizing non-integer order derivatives is superior to traditional ones for describing a variety of real-world phenomena and processes. It should be noted that fractional order integral and differential operators are nonlocal in nature and have the ability to provide global dynamics which include the traditional order dynamics as a special case (see some detail in [30, 31]). As a result, the researchers were drawn to examine numerous dynamical issues in context of fractional calculus as can be seen in [32–34]. The said differential operators can be divided into two kinds of classes on the basis of their kernels. Riemann-Liouville and Caputo operators have singular kernel-type operators so placed in one of the classes. On the other hand, Caputo-Fabrizio and Atangana-Baleanu, etc are placed in the non-singular kernel type class. Both kinds have a variety of applications in the mathematical modeling of various real-world problems (see details in [35–39]). For more recent contribution, we refer some work as [40–43].

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2 Formulation of Proposed Model A model is designed that further categories all of the population members as: H denotes susceptible, V vaccinated, exposed people are denoted by U, infected people are represented by A,and recovered class C. A schematic dia gram of the model is given in Fig. 1. The classical form of the model we need to study is recalled from [44] as ⎧ ηH(t)A(t) ⎪ ˙ ⎪ H(t) =∧− − (ρ + d)H(t), ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ (1 − τ )ηA(t)V(t) ⎪ ˙ ⎪ − dV(t), ⎪ ⎪ V(t) = ρH(t) − N ⎪ ⎨ ηH(t)A(t) (1 − τ )ηA(t)V(t) ˙ U(t) = + − (β + d)U(t), ⎪ N N ⎪ ⎪ ⎪ ⎪ ˙ A(t) = βU(t) − (δ + d)A(t), ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ⎪ C(t) = δA(t) − dC(t), ⎪ ⎪ ⎩ H(0) = H0 , V(0) = V0 , U(0) = U0 , A(0) = A0 , C(0) = C0 ,

(1)

where H0 , V0 , U0 , A0 , C0 ≥ 0. Here, ∧ is the recruitment rate into the susceptible class, η is the effective control rate while ρ is the vaccination coverage rate, and d is the natural morality rate. The rate of recovery is given by δ, β denotes developing clinical symptoms and τ is vaccine effectiveness. Where N is total population in t. N = H + V + U + A + C.

Fig. 1 A flow chart of the proposed model (1)

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Fractional calculus is nowadays the most popular branch of calculus among researchers. But dealing with FODEs, in particular, has received a lot more attention. Researchers have examined fractional differential operators from a variety of perspectives, including existence theory, stability, and numerical methods (see details [45, 46]). They developed a number of strategies, methodologies, and theories for computing precise or numerical FODE solutions. For the desired results, researchers employ well-known techniques such as HPM, ADM and a variety of other numerical methods see [47–49]. Motivated by this work, in our project we handle the model (1) under CFFD as ⎧ ηHA ⎪ CF σ ⎪ − (ρ + d)H, ⎪ Dt H(t) = ∧ − ⎪ N ⎪ ⎪ ⎪ ⎪ (1 − τ )ηVA ⎪ CF σ ⎪ − dV, Dt V(t) = ρH − ⎪ ⎪ N ⎪ ⎨ ηHA (1 − τ )ηVA CF σ Dt U(t) = + − (β + d)U, ⎪ N N ⎪ ⎪ ⎪ CF σ ⎪ Dt A(t) = βU − (δ + d)A, ⎪ ⎪ ⎪ ⎪ CF σ ⎪ Dt C(t) = δA − dC, ⎪ ⎪ ⎪ ⎩ H(0) = H0 , V(0) = V0 , U(0) = U0 , A(0) = A0 , C(0) = C0 ,

(2)

where H0 , V0 , U0 , A0 , C0 ≥ 0 and σ ∈ (0, 1]. Firstly some qualitative results, such as the existence and uniqueness of the solution that corresponds to the given model, are proven. The fixed point theory of Krassnoselskii and Banach are employed to obtain these results. To analyze some approximate analytical results, the LT and ADM tools are also applied. The said method already has been used in many papers. For instance see [50, 51]. Finally, graphical representations of approximation results are produced using Matlab. For some more frequent results, we refer readers to see [52–59]. Here we remark that we have studied the model (1) from different aspects. Our analysis is devoted to the existence theory through a fixed point approach and analytical as well as the numerical study of the proposed model. The model has been studied under the different concepts of fractional calculus. Some updated form of fractional differential operators has been considered here in the form of piecewise derivative. Both analytical and numerical results have been presented graphically by using Matlab 16.

3 Preliminaries Some basic definitions and results of FODEs are needed to obtain all these results. Which are given as Definition 1 ([60]) Let m ∈ H1 (0,  ),  > 0, σ ∈ (0, 1), then the CFFD is given by

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Dσt (m(t))



(M)(σ ) = 1−σ

0

t

 t− d. m ( ) exp − σ 1−σ 



However if m does not belong to H1 (0,  ), the derivative is given as CF

Dσt (m(t))

(M)(σ ) = 1−σ



t 0

 t− d. (m(t) − m( )) exp − σ 1−σ 

Definition 2 ([60]) For m ∈ H1 (0,  ),  > 0, the CF integral is given as CF

Iσt [m(t)] =

σ (1 − σ ) m(t) + (M)(σ ) (M)(σ )



t

m( )d, σ ∈ (0, 1].

0

Definition 3 ([60]) If M(σ ) = 1, then the transform due to Laplace of C F Dσt m(t) can be defined as L [C F Dσt m(t)] =

sL [m(t)] − m(0) , s ≥ 0, σ ∈ (0, 1]. s + σ (1 − s)

If J = [0,  ],  < ∞, then Z = C ([0,  ] × R2 , R) is Banach space under the norm (x, y) = supt∈J [|x(t)| + |y(t)|]. Theorem 1 ([61]) Y ⊂ Z and there exist operators P1 , P2 such that 1. P1 v + P2 v ∈ Y for every v ∈ Y; 2. P1 is contraction; 3. P2 is completely continuous, then P1 v + P2 v = v has at least one solution. Definition 4 ([62, 63]) Let U ∈ H 1 (0, τ ), then ABC

σ 0 D t (U (t))

ABC(σ ) = 1−σ



t 0



σ   −σ d U (α)E σ t −α dα. dα 1−σ

(3)

Definition 5 ([64]) We define ABC

σ 0 I t U (t) =

(1 − σ )U (t) + ABC(σ )

 0

t

σ (t − α)σ −1 U (α)dα. ABC(σ ) (σ )

(4)

Definition 6 ([65]) Let U , then the arbitrary order derivative in Caputo sense is defined by C σ 0 Dt U (t)

 = 0

t



(t − α)n−σ −1 [U (α)] dα.

(1 − σ )

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Definition 7 For U , fractional integral is defined as C σ 0 It U (t)

1 =

(σ )



t

(t − α)σ −1 dα, σ > 0.

0

Definition 8 ([66]) Consider U and g as differentiable and increasing functions respectively, one has ⎧ t ⎪ ⎪ U (τ )dτ, 0 < t ≤ t1 , ⎨ 0 PG  t 0 It U (t) = ⎪ ⎪ ⎩ U (τ )g  (τ )d(τ ), t1 < t ≤ T. t1

Definition 9 If U is differentiable, one can define   U (t), 0 < t ≤ t1 , PC σ 0 Dt U (t) = C σ 0 Dt U (t), t1 < t ≤ T. Definition 10 Let U be usual differentiable, we define ⎧ t ⎪ ⎪ U (τ )dτ, 0 < t ≤ t1 , ⎨ 0 PC I U (t) =  t t 0 ⎪ 1 ⎪ ⎩ (t − α)σ −1 U (α)d(α), t1 < t ≤ T.

σ t1 Definition 11 For differentiable function U in ABC sense, we have C σ 0 Dt U (t), 0 < t ≤ t1 , PC ABC σ Dt U (t) = ABC σ 0 Dt U (t), t1 < t ≤ T. 0 Definition 12 For differentiable function U , we have ⎧  t 1 ⎪ ⎪ (t − α)σ −1 U (α)d(α), 0 < t ≤ t1 , ⎪ ⎨ σ t1 PC ABC I U (t) = ,  t t 0 ⎪ σ 1−σ ⎪ ⎪ U (t) + (t − α)σ −1 U (α)d(α) t1 < t ≤ T. ⎩ ABCσ ABCσ σ t1

[65] Lemma 3.13.1 The solution of piecewise problem PC ABC σ Dt U 0

(t) = G(t, U (t)), 0 < σ ≤ 1

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such that right sides vanish at t = 0 is given by

U (t) =

⎧ ⎪ ⎪ ⎪ ⎨ U0 +

1

(σ )



t

G(α, U (α))(t − α)σ −1 dα, 0 < t ≤ t1

0

⎪ 1−σ ⎪ ⎪ G(t, U (t)) + ⎩ U (t1 ) + ABC(σ )



t t1

σ (t − α)σ −1 G(α U (α))d(α) t1 < t ≤ T. ABC(σ ) (σ )

4 Theoretical Results by Using CF Concept To elaborate the existence result, applying the integral operator C F Iσt on both sides of  CF σ Dt U (t) = ϕ(t, U (t)), (5) U (0) = U0 , we get U (t) = U0 +

(1 − σ )ϕ(t, U (t)) + (M)(σ )



t

0

σ ϕ(, U ( )) d, (M)(σ )

(6)

where ⎧ ⎧ ⎧ H0 ϕ1 (t, H, V, U, A, C), ⎪ ⎪ ⎪ ⎪ H(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V(t) V ⎪ ⎪ ⎪ ⎨ ⎨ 0 ⎨ ϕ2 (t, H, V, U, A, C), U (t) = U(t) , U0 = U0 , ϕ(t, U (t)) = ϕ3 (t, H, V, U, A, C), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A(t) ϕ4 (t, H, V, U, A, C), A0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎩ C(t) ϕ5 (t, H, V, U, A, C). C0

(7)

For further analysis, the following assumptions are taken into consideration. (D1)

If Lϕ > 0, and U, U¯ ∈ Z , one have |ϕ(t, U(t)) − ϕ(t, U¯ (t))| ≤ Lϕ [|U − U¯ |].

(D2)

Let Cϕ , Cϕ > 0 and Mϕ > 0, such that |ϕ(t, U (t))| ≤ Cϕ |U | + Mϕ .

Using (6) and (7), the two operators are defined as: (1 − σ )ϕ(t, U (t)) , P1 (U ) = U0 (t) + (M)(σ )  t σ ϕ(, U ( )) P2 (U ) = d. (M)(σ ) 0

(8)

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189

Theorem 2 Problem (6) has at least one solution if

Lϕ (M)(σ )

< 1.

Proof Consider Y = {U ∈ Z : U  ≤ ρ, ρ > 0} be closed convex set of Z . If U − U¯ ∈ Y, one has

(1 − σ ) ¯ ¯ P1 U − P1 U  = sup ϕ(t, U (t)) − (ϕ(t, U (t)) (M)(σ ) t∈J (1 − σ ) ≤ Lϕ sup |U (t) − U¯ (t)| (M)(σ ) t∈J Lϕ U − U¯ . ≤ (M)(σ ) Hence P1 is contraction. Now to show P2 is completely continuous, let for any U ∈ Y, one has  t σ d ϕ(, U ( )) P2 (U ) = sup (M)(σ ) t∈J

0

 ≤ [Cϕ ρ + Mϕ ] := . (M)(σ )

(9)

From (9) we conclude that P2 is bounded. Continuity of σ implies continuity of P2 .  Similarly, it can be proved that P2 is equi-continues by setting t1 < t2 ∈ J. Theorem 3 The problem (6) has a unique solution if

(1+ )L (M)(σ )

< 1.

Proof Let define P : A → A by P(U ) = U0 +

(1 − σ )ϕ(t, U (t)) + (M)(σ )



t 0

σ ϕ(, U ( )) d. (M)(σ )

If U , U¯ ∈ A, one has (1 − σ ) ¯ ¯ ϕ(t, U (t)) − ϕ(t, U (t)) P(U ) − P(U ) ≤ sup t∈J (M)(σ )  t σ + sup |ϕ(, U ( )) − ϕ(, U¯ ( ))|d (M)(σ ) t∈J 0 (1 +  )L U − U¯ . (10) ≤ (M)(σ ) Hence the result.



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5 Computational Results of (2) Setting (M)(σ ) = 1, and applying LT on both sides of (2), and using (σ, s) = 1 [s + σ (1 − s)], one has s ⎧ ηHA ⎪ ⎪ L [H(t)] = s −1 H0 + (σ, s)L [∧ − − (ρ + d)H], ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ (1 − τ )ηVA ⎪ −1 ⎪ − dV], ⎪ ⎨ L [V(t)] = s V0 + (σ, s)L [ρH − N ηHA (1 − τ )ηVA ⎪ ⎪ L [U(t)] = s −1 U0 + (σ, s)L [ + − (β + d)U], ⎪ ⎪ N N ⎪ ⎪ ⎪ ⎪ L [A(t)] = s −1 A0 + (σ, s)L [βU − (δ + d)A], ⎪ ⎪ ⎪ ⎩ L [C(t)] = s −1 C0 + (σ, s)L [δA − dC]. Assume H(t) =

∞

Hq (t), V(t) =

q=0

U(t) =

∞

∞

Vq (t),

q=0

Uq (t), A(t) =

q=0

C(t) =

∞

(11)

∞

Aq (t),

(12)

q=0

Cq (t),

q=0

and H(t)A(t) =

∞

Aq (H, A),

q=0

V(t)A(t) =

∞

(13) Bq (V, A),

q=0

such that the “Adomian polynomial” Aq (H, A) can be defined as  q  q

1 dq j j Aq (H, A) = λ H j (t) λ A j (t) . q q! dλ j=0 λ=0 j=0 and the “Adomian polynomial” Bq (V, A) can be defined as  q  q

1 dq j j Bq (V, A) = λ V j (t) λ A j (t) . q! dλq j=0 λ=0 j=0

Qualitative Theory and Approximate Solution …

191

Reference to (12) and (13), the system (5) becomes ⎧  ∞ 

⎪ ⎪ ⎪ L Hq (t) = ⎪ ⎪ ⎪ ⎪ q=0 ⎪ ⎪ ⎪ ⎪   ∞ ⎪ ⎪ ⎪ ⎪ L Vq (t) = ⎪ ⎪ ⎪ ⎪ q=0 ⎪ ⎪ ⎪ ⎪  ⎪  ∞ ⎨ L Uq (t) = ⎪ ⎪ q=0 ⎪ ⎪ ⎪ ⎪   ∞ ⎪ ⎪ ⎪ ⎪ Aq (t) = ⎪ ⎪L ⎪ ⎪ ⎪ q=0 ⎪ ⎪ ⎪   ⎪ ∞ ⎪ ⎪ ⎪ ⎪ Cq (t) = ⎪ ⎩L q=0

H(0) + (σ, s)L s

 ∧ −η

 ∞ ∞

Aq (H, A) − (ρ + d) Hq (t) , N

q=0

q=0

  ∞ ∞ ∞

Bq (V, A) V(0) Hq (t) − (1 − τ )η Vq (t) , + (σ, s)L ρ −d s N q=0

q=0

q=0

  ∞ ∞ ∞

Aq (H, A) Bq (V, A) U(0) Uq (t) , + (σ, s)L η + (1 − τ )η − (β + d) s N N q=0

q=0

q=0

  ∞ ∞

A(0) + (σ, s)L β Uq (t) − (δ + d) Aq (t) , s q=0

q=0

  ∞ ∞

C(0) Aq (t) − d Cq (t) . + (σ, s)L δ s q=0

q=0

(14) From (14), using initial conditions, we equate terms as ⎧ ⎪ ⎪ ⎪ L [H0 (t)] = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L [V0 (t)] = ⎪ ⎪ ⎪ ⎪ ⎨ L [U0 (t)] = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L [A0 (t)] = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ L [C0 (t)] =

H0 + (σ, s)L (∧), s V0 , s U0 , s A0 , s C0 . s

(15)

  ⎧ ηA0 (H, A) ⎪ ⎪ L [H1 (t)] = (σ, s)L − − (ρ + d)H0 (t) , ⎪ ⎪ N ⎪ ⎪   ⎪ ⎪ ⎪ (1 − τ )ηB0 (V, A) ⎪ ⎪ − dV0 (t) , L [V1 (t)] = (σ, s)L ρH0 (t) − ⎪ ⎪ N ⎪ ⎪ ⎪   ⎨ ηA0 (H, A) (1 − τ )ηB0 (V, A) + − (β + d)U(t) , L [U1 (t)] = (σ, s)L ⎪ N N ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ L [A1 (t)] = (σ, s)L βU0 (t) − (δ + d)A0 (t) , ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ⎩ L [C1 (t)] = (σ, s)L δA0 (t) − dC0 (t) . (16)

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⎧   ηA1 (H, A) ⎪ ⎪ L [H − (ρ + d)H (t)] = (σ, s)L − (t) , ⎪ 2 1 ⎪ ⎪ N ⎪ ⎪   ⎪ ⎪ (1 − τ )ηB1 (V, A) ⎪ ⎪ ⎪ (t)] = (σ, s)L ρH (t) − (t) , L [V − dV 2 1 1 ⎪ ⎪ N ⎪ ⎪   ⎪ ⎪ ⎪ ηA1 (S, I ) (1 − τ )ηB1 (V, A) ⎪ ⎪ + − (β + d)U (t)] = (σ, s)L (t) , L [U ⎪ 2 1 ⎪ N N ⎨   ⎪ (t)] = (σ, s)L βU (t) − (δ + d)A (t) , L [A ⎪ 2 1 1 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ (t)] = (σ, s)L δA (t) − dC (t) . L [C 2 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ (17)   ⎧ ηAq (S, I ) ⎪ ⎪ L [Hq+1 (t)] = (σ, s)L − − (ρ + d)Hq (t) , ⎪ ⎪ N ⎪ ⎪   ⎪ ⎪ ⎪ (1 − τ )ηBq (V, A) ⎪ ⎪ − dV (t)] = (σ, s)L ρH (t) − (t) , L [V ⎪ q+1 q q ⎪ N ⎪ ⎪ ⎪   ⎨ ηAq (S, I ) (1 − τ )ηBq (V, A) L [Uq+1 (t)] = (σ, s)L + − (β + d)Uq (t) , ⎪ N N ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ (t)] = (σ, s)L βU (t) − (δ + d)A (t) , L [A q+1 q q ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ⎩ L [Cq+1 (t)] = (σ, s)L δAq (t) − dCq (t) . (18) After performing simplification, we get

⎧ ⎪ ⎪ H0 (t) = H0 + ∧ 1 + (t − 1)σ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V0 (t) = V0 , U0 (t) = U0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A0 (t) = A0 , ⎪ ⎪ ⎩ C0 (t) = C0 ,

(19)

Qualitative Theory and Approximate Solution …





⎧ η ⎪ ⎪ H − A (t) = H − (ρ + d) 1 + (t − 1)σ ⎪ 1 0 0 ⎪ N ⎪ ⎪ ⎪



⎪ ⎪ t2 η ⎪ 2 ⎪ ⎪ 1 + 2σ (t − 1) + σ 1 − 2t + A , + ∧ 1 − 0 ⎨ N 2!



⎪ η(1 − τ ) ⎪ ⎪ 1 + (t − 1)σ V (t) = ρH − V − dV A ⎪ 1 0 0 0 0 ⎪ N ⎪ ⎪ ⎪



⎪ ⎪ t2 ⎪ ⎪ ⎩ + ρ ∧ 1 + 2(t − 1)σ + 1 − 2t + σ2 , 2! ⎧



η η(1 − τ ) ⎪ ⎪ U1 (t) = A0 H0 + V0 A0 − (β + d)U0 1 + (t − 1)σ ⎪ ⎪ ⎪ N N ⎪ ⎪



⎪ ⎪ t2 η ∧ A0 ⎪ ⎪ ⎪ 1 + 2(t − 1)σ + 1 − 2t + σ2 , + ⎨ N 2!



⎪ ⎪ ⎪ A1 (t) = βU0 − (δ + d)A0 1 + (t − 1)σ , ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎩ C1 (t) = (δA0 − dC0 ) 1 + (t − 1)σ .

193

(20)

(21)

⎧ 



−η −η ⎪ ⎪ H H A βU + A (t) = − (δ + d)A H − (ρ + d)H ⎪ 2 0 0 0 0 0 0 0 ⎪ ⎪ N N ⎪ ⎪





⎪ ⎪ ⎪ t2 −η ⎪ ⎪ 1 + 2(t − 1)σ + 1 − 2t + A σ2 H − (ρ + d)H − (ρ + d) 0 0 0 ⎪ ⎪ N 2! ⎪ ⎪ ⎪

⎨ η η + ∧ βU0 − (δ + d)A0 + A0 (1 − A0 ) − (ρ + d) 1 − A0 ⎪ N N ⎪ ⎪ ⎪ 2

⎪ t ⎪ ⎪ ⎪ σ2 1 + 3(t − 1)σ + 3 1 − 2t + ⎪ ⎪ 2! ⎪ ⎪ ⎪

 ⎪ ⎪ t3 t2 ⎪ ⎪ σ3 , ⎩ − 1 − 3t + 3 − 2! 3!





(22)  ⎧ η(1 − τ ) −η η(1 − τ ) ⎪ V2 (t) = ρ A0 H0 − (ρ + d)H0 − d + A0 ρH0 − V0 A0 − dV0 ⎪ ⎪ ⎪ N N N ⎪ ⎪ ⎪





 ⎪ ⎪ t2 ⎪ ⎪ σ2 1 + 2(t − 1)σ + 1 − 2t + ⎪ ⎨ + V0 βU0 − (δ + d)A0 ) 2! 

 ⎪ η η(1 − τ ) ⎪ ⎪ A0 − d + ρ ∧ 1 − A0 − ⎪ ⎪ ⎪ N N ⎪ ⎪ ⎪ ⎪ 2

2 3

⎪ ⎪ ⎩ 1 + 3(t − 1)σ + 3 1 − 2t + t σ 2 − 1 − 3t + 3 t − t σ 3 , 2! 2! 3!

(23)

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Eiman et al.

⎧

 η −η ⎪ ⎪ U βU H A (t) = − (δ + d)A ) + A ( H − (ρ + d)H ⎪ 2 0 0 0 0 0 0 0 ⎪ ⎪ N N ⎪ ⎪



 ⎪ ⎪ η(1 − τ ) η(1 − τ ) ⎪ ⎪ ⎪ ρH + V βU A V + − A − dV − (δ + d)A 0 0 0 0 0 0 0 0 ⎪ ⎪ N N ⎪ ⎪ ⎪



⎨ t2 σ2 1 + 2(t − 1)σ + 1 − 2t + ⎪ 2! ⎪ ⎪ 

 ⎪ ⎪ η η ⎪ ⎪ ⎪ ∧ βU A + − (δ + d)A + A (1 − ) + (1 − τ )ρA − (β + d)A 0 0 0 0 0 0 ⎪ ⎪ N N ⎪ ⎪ ⎪



⎪ ⎪ t2 t3 t2 ⎪ ⎪ σ 2 − 1 − 3t + 3 − σ3 , ⎩ 1 + 3(t − 1)σ + 3 1 − 2t + 2! 2! 3! (24)



  ⎧ η η(1 − τ ) ⎪ A2 (t) = β A0 H0 + A0 V0 − (β + d)U0 − (δ + d) βU0 − (δ + d)A0 ⎪ ⎪ ⎪ N N ⎪ ⎪ ⎪



⎪ ⎪ t2 ⎪ 2 ⎪ ⎪ σ 1 + 2(t − 1)σ + 1 − 2t + ⎪ ⎪ 2! ⎪ ⎪ ⎪

⎪ ⎨ t2 βη∧ A0 1 + 3(t − 1)σ + 3 1 − 2t + σ2 + N 2! ⎪ ⎪ ⎪



⎪ ⎪ ⎪ t2 t3 ⎪ ⎪ σ3 , − 1 − 3t + 3 − ⎪ ⎪ 2! 3! ⎪ ⎪ ⎪ ⎪ ⎪





.  ⎪ ⎪ t2 ⎪ . ⎩ C2 (t) = δ βU0 − (δ + d)A0 − d δA0 − dC0 1 + 2(t − 1)σ + 1 − 2t + σ 2 ., 2!

(25)

and so on. In this way, the other terms are computed, hence one has ⎧ H(t) = H0 + H1 (t) + H2 (t) + H3 (t) + . . . . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V(t) = V0 + V1 (t) + V2 (t) + V3 (t) + . . . . U(t) = U0 + U1 (t) + U2 (t) + U3 (t) + . . . . ⎪ ⎪ ⎪ A(t) = A0 + A1 (t) + A2 (t) + A3 (t) + . . . . ⎪ ⎪ ⎪ ⎩ C(t) = C0 + C1 (t) + C2 (t) + C3 (t) + . . . .

(26)

6 Results and Discussion The solutions, up to few terms, of (26) is plotted by using MATLAB. The numerical values for parameters are used from Table 1. Against various fractional orders, the solutions are shown as Figs. 2, 3, 4, 5, and 6 respectively.

Qualitative Theory and Approximate Solution …

195

Table 1 Nomenclature and their values. Nomenclature Stands for ∧ η ρ d δ β τ H0 V0 U0 A0 C0

Rate of birth Rate of effective contact Rate of vaccination coverage Rate of mortality due to nature Rate of recovery Rate of appearing of clinical symptoms Rate of vaccine effectiveness

Value A

Value B

0.90 0.020 0.030 0.020 0.050 0.020 1.0 75.0 20.0 55.0 30.0 20.0

5.0 0.020 0.020 0.020 0.50 0.20 1.0 75.0 20.0 55.0 30.0 20.0

80 0.90 0.92 0.94 0.96 0.98 1.00

70

H

60 50 40 30 20 0

10

20

30

40

50

60

Time (Days)

Fig. 2 Dynamical behavior of approximate values of H at several fractional order 45

40

35

V

0.90 0.92 0.94 0.96 0.98 1.00

30

25

20 0

10

20

30

40

50

Time (Days)

Fig. 3 Dynamical behavior of approximate values of V at several fractional order

60

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Eiman et al.

60 0.90 0.92 0.94 0.96 0.98 1.00

50

U

40 30 20 10 0 0

10

20

30

40

50

60

Time (Days)

Fig. 4 Dynamical behavior of approximate values of U at several fractional order 0.90 0.92 0.94 0.96 0.98 1.00

25 20 15 10 5 0 0

10

20

30

40

50

60

Time (Days)

Fig. 5 Dynamical behavior of approximate values of V at several fractional order 32 30

C

28 0.90 0.92 0.94 0.96 0.98 1.00

26 24 22 20 0

10

20

30

40

50

Time (Days)

Fig. 6 Dynamical behavior of approximate values of C at several fractional order

60

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197

7 Setting of Model (2) in Caputo and Atangana-Baleanu Piecewise Equations Fractional Derivative Now in this section, we investigate the mathematical model given in (1) by using the ABC piecewise differential operators. For recent work on using the said operators, we refer some frequent work like [67, 68]. We formulated the proposed model in the aforementioned operators form with 0 < σ ≤ 1, t ∈ [0, T ], as ηHA − (ρ + d)H, N (1 − τ )ηVA C ABC σ − dV, Dt (V(t)) = ρH − 0 N ηHA (1 − τ )ηVA C ABC σ + − (β + d)U, Dt (U(t)) = 0 N N C ABC σ Dt (A(t)) = βU − (δ + d)A, 0 C ABC σ Dt (H(t)) 0

=∧−

(27)

C ABC σ Dt (C(t)) 0

= δA − dC, H(0) = H0 ≥ 0, V(0) = V0 ≥ 0,

U(0) = U0 ≥ 0, A(0) = A0 ≥ 0 C(0) = C0 ≥ 0, In more explicit form, the model (27) can also be written as C C ABC σ Dt (H(t)) 0

=

σ C 0 Dt (H(t)) = G 1 (H, V, U, A, C, t), 0 < ABC σ Dt (H(t)) = ABC G 1 (H, V, U, A, C, t), 0

t ≤ t1 ,

σ C 0 Dt (V(t)) = G 2 (H, V, U, A, C, t), 0 < ABC σ Dt (V(t)) = ABC G 2 (H, V, U, A, C, t), 0

t ≤ t1 ,

σ C 0 Dt (U(t)) = G 3 (H, V, U, A, C, t), 0 < ABC σ Dt (U(t)) = ABC G 3 (H, V, U, A, C, t), 0

t ≤ t1 ,

σ C 0 Dt (A(t)) = G 4 (H, V, U, A, C, t), 0 < ABC σ Dt (A(t)) = ABC G 4 (H, V, U, A, C, t), 0

t ≤ t1 ,

C C ABC σ Dt (V(t)) 0

=

C C ABC σ Dt (U(t)) 0

=

C C ABC σ Dt (A(t)) 0

=

C C ABC σ Dt (C(t)) 0

=

σ C 0 Dt (C(t)) = G 5 (H, V, U, A, C, t), 0 < ABC σ Dt (C(t)) = ABC G 5 (H, V, U, A, C, t), 0

t1 < t ≤ T, t1 < t ≤ T, t1 < t ≤ T, t1 < t ≤ T, t ≤ t1 ,

(28) t1 < t ≤ T.

where C0 Dtσ and 0ABC Dtσ are Caputo and ABC derivative respectively. We investigate the existence and uniqueness of at least one approximate solution. It should be noted that fixed point theory plays important role in investigating existence theory for dynamical problems. Further, Newton’s interpolation formula will be used for numerical investigation. In the end, all the numerical findings are displayed graphically by using some real data.

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8 Qualitative Analysis of (2) Using Piecewise Derivative We can re-write the system (28) as in the Lemma 3.13.1 to develop some results about analysis as PC ABC σ Dt U 0

(t) = G(t, U (t)), σ ∈ (0, 1],

is ⎧  t G(ζ, U (ζ ))(t − ζ )σ −1 ⎪ ⎪ dζ, 0 < t ≤ t1 , + U ⎪ 0 ⎨

(σ ) 0 U (t) =  t σ (t − ζ )σ −1 ⎪ (1 − σ )G(t, U (t)) ⎪ ⎪ + G(ζ U (ζ ))d(ζ ), t1 < t ≤ T, ⎩ U (t1 ) + ABC(σ ) t1 ABC(σ ) σ

(29)

where

⎧ H(t) ⎪ ⎪ ⎪ ⎪ ⎪ V(t) ⎪ ⎪ ⎨ U (t) = U(t) ⎪ ⎪ ⎪ ⎪ A(t) ⎪ ⎪ ⎪ ⎩ C(t),

⎧ H0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V ⎪ ⎨ 0 U 0 = U0 ⎪ ⎪ ⎪ ⎪ A0 ⎪ ⎪ ⎪ ⎩ C0 ,

⎧ Ht1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V ⎪ ⎪ t1 ⎨ U t1 = Ut1 ⎪ ⎪ ⎪ ⎪ ⎪ At ⎪ ⎪ ⎪ 1 ⎩ Ct1 ,

⎧ ⎧ ⎨ C G 1 (H, V, U, A, C, t) ⎪ ⎪ ⎪ ⎪ G1 = ⎪ ⎪ ⎩ ABC G (H, V, U, A, C, t), ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ C G (H, V, U, A, C, t) ⎪ ⎨ ⎪ 2 ⎪ ⎪ G2 = ⎪ ⎪ ⎪ ⎩ ABC G (H, V, U, A, C, t), ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎨ ⎨ C G 3 (H, V, U, A, C, t) G(t, W(t)) = G 3 = ⎪ ⎩ ABC G (H, V, U, A, C, t), ⎪ ⎪ 3 ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ C G (H, V, U, A, C, t) ⎨ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ G 4 = ⎩ ABC ⎪ ⎪ G 4 (H, V, U, A, C, t), ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ C G (H, V, U, A, C, t) ⎨ ⎪ ⎪ 5 ⎪ ⎪ ⎪ ⎩ G 5 = ⎩ ABC G 5 (H, V, U, A, C, t).

(30)

Here, we described some assumptions as (C1)

There exists constants L U > 0; for every G, U¯ ∈ Z , as |G(t, U ) − G(t, U¯ )| ≤ L G |U − U¯ |.

(C2)

Let one has constants C G > 0 and MG > 0, such that |G(t, U (t))| ≤ C G |U | + MG

holds. Theorem 4 If G be a piecewise continuous operator on the sub interval 0 < t ≤ t1 , and t1 < t ≤ T on [0, T ] together with hypothesis (C2), then problem (28) has at least one solution on each sub interval.

Qualitative Theory and Approximate Solution …

199

Proof Let define a closed subset by B = {U ∈ Z : U  ≤ R1,2 , R1,2 > 0}, and define T : B → B as ⎧  t 1 G(ζ, U (ζ ))(t − ζ )σ −1 ⎪ ⎪ dζ, 0 < t ≤ t1 + U ⎪ ⎨ 0

(σ ) 0 T (U ) = (31)  t ⎪ (t − ζ )σ −1 σ (1 − σ )G(t, U (t)) ⎪ ⎪ + G(ζ U (ζ ))d(ζ ), t1 < t ≤ T. ⎩ U (t1 ) + ABC(σ )

(σ ) t1 ABC(σ )

On any U ∈ B, we get

|T (U )(t)| ≤

⎧  t 1 (t − ζ )σ −1 ⎪ ⎪ |G(ζ, U (ζ ))|dζ, ⎪ ⎨ |U 0 | + 0

(σ )

 t σ (t − ζ )σ −1 1−σ 1 |G(t, U (t))| + |G(ζ U (ζ ))|d(ζ ), ABC(σ ) ABC(σ ) t1

(σ )  ⎧ t1 1 σ −1 [C |U | + M ]dζ, ⎪ ⎪ ⎪ G G ⎨ |U 0 | + (σ ) 0 (t − ζ ) ≤  t ⎪ σ (t − ζ )σ −1 1−σ ⎪ ⎪ [C |U| + MG ] + [C |U | + MG ]d(ζ ), ⎩ |U (t1 ) | + ABC(σ ) G ABC(σ ) (σ ) G t1 ⎧ Tσ ⎪ ⎪ ⎨ |U 0 | + (σ + 1) [C G |U | + MG ] = R1 , 0 < t ≤ t1 , ≤ σ ⎪ ⎪ ⎩ |U (t ) | + 1 − σ [C G |U | + MG ] + σ (T − T) [C |U | + MG ]d(ζ ) = R2 , t1 < t ≤ T, 1 ABC(σ ) ABC(σ ) σ + 1 G  R1 , 0 < t ≤ t1 , T (U ) ≤ R2 , t1 < t ≤ T. ⎪ ⎪ ⎪ ⎩ |U (t ) | + 1

Fixing R1,2 = max{R1 , R2 }, we see that T (U ) ≤ R1,2 which demonstrates that T is bounded and also T (U ) ∈ B. Thus T (B) ⊂ B. Further, for completely continuity, we take ti < t j ∈ [0, t1 ] in the Caputo sense, such that  t  j (t j − ζ )σ −1 |T (U )(t j ) − T (U )(ti )| =  G(ζ, U (ζ ))dζ, 

(σ ) 0  ti 1 (ti − ζ )σ −1 G(ζ, U (ζ ))dζ −

(σ ) 0  ti 1 ≤ [(ti − ζ )σ −1 − (t j − ζ )σ −1 ]|G(ζ, U (ζ )) dζ

(σ ) 0  tj 1 + (t j − ζ )σ −1 |G(ζ, U (ζ ))|dζ

(σ ) ti  t i 1 [(ti − ζ )σ −1 − (t j − ζ )σ −1 ]dζ ≤

(σ ) 0   tj + (t j − ζ )σ −1 dζ (C G |U | + MG ) ti

(C G U + MG ) σ [t j − tiσ + 2(t j − ti )σ ]. ≤

(σ + 1)

(32)

200

Eiman et al.

From (32), we see that as ti → t j , then |T (U )(t j ) − T (U )(ti )| → 0, as i → j. Since T is bounded on [0, t1 ], so is uniformly continuous also over [0, t1 ]. Hence, we concluded that T over the said interval is equi-continuous also. 1−σ In addition let ti , t j ∈ [t1 , T ], here use 1 (σ ) = ABC(σ , and 2 (σ ) = ABC(σσ ) (σ ) )  |T (U )(t j ) − T (U )(ti )| = 1 (σ )G(t, U (t)) + 2 (σ )

tj

(t j − ζ )σ −1 G(ζ, U (ζ ))dζ,

t1

− 1 (σ )G(t, U (t)) +  ≤ 2 (σ )

ti

[(ti − ζ )

t1 tj

 + 2 (σ )

(σ ) ABC(σ ) (σ )

σ −1

− (t j − ζ )



ti t1

σ −1

(ti − ζ )σ −1 G(ζ, U (ζ ))dζ

]|G(ζ, U (ζ ))|dζ

(t j − ζ )σ −1 |G(ζ, U (ζ ))|dζ

ti

 ≤ 2 (σ )  +

ti

[(ti − ζ )σ −1 − (t j − ζ )σ −1 ]dζ

t1

tj

 (t j − ζ )σ −1 dζ (C G |U | + MG )

ti

≤

σ (C G U + MG ) σ [t − tiσ + 2(t j − ti )σ ]. ABC(σ ) (σ + 1) j

(33)

Also from (33), we see that |T (U )(t j ) − T (U )(ti )| → 0, as ti → t j . Due to the uniform continuity of T over the said interval, we conclude that T is equi- continuous in [t1 , T ] interval. Therefore, T is equi- continuous mapping. Using Arzelá-Ascoli Theorem, operator T is completely continuous and also uniform continuous and has bounds. Hence, problem (28) has at least one solution on each subinterval.  Theorem 5 In view of hypothesis (C1) together with the condition that   Tσ σ Tσ L G , L G 1 (σ ) + < 1, max

(σ + 1) ABC(σ ) (σ + 1) 

the proposed model has a unique solution. Proof We define here operator T : E → E and take U and U¯ ∈ E on [0, t1 ] in Caputo sense as  t  t ¯ )) (t − ζ )σ −1 (t − ζ )σ −1 G(ζ, W(ζ G(ζ, U (ζ ))dζ − dζ T (U ) − T (U¯ ) = max

(σ )

(σ ) t∈[0,t1 ] 0 0 Tσ ≤ L G U − U¯ . (34)

(σ + 1)

Qualitative Theory and Approximate Solution …

201

Equation (34) yields T (U ) − T (U¯ ) ≤

Tσ L G U − U¯ .

(σ + 1)

(35)

So T is contraction. Therefore in the given sub-interval, in sense of the Banach contraction theorem, considered problem has a unique solution. Next, in the sense of ABC derivative, for the other interval t ∈ [t1 , T ], we have T (U ) − T (U¯ ) ≤ 1 (σ )L G U − U¯  +

σ (T − T σ )L G U − U¯ . (36) ABC(σ ) (σ + 1)

or  T (U ) − T (U¯ ) ≤ L G 1 (σ ) +

 σ (T − T)σ U − U¯ . ABC(σ ) (σ + 1)

(37) 

Hence, we get the required result.

9 Approximate Results Using Piecewise Derivative Concept Here following the procedure [66, 67], we recall the scheme as ⎧ ⎪ ⎪ ⎪ ⎨ H0 +

 t 1 1 (t − α)σ −1C G 1 (α)dα, 0 < t ≤ t1 ,

(σ ) 0 , H(t)) =  t ⎪ ⎪ ABC G (t) +  (σ ) ⎪ (t − α)σ −1 ABC G 1 (α)dα, t1 < t ≤ T, ⎩ H(t1 ) + 1 (σ ) 1 2 ⎧ ⎪ ⎪ ⎪ ⎨ V0 +

t1

⎧ ⎪ ⎪ ⎪ U0 + ⎨

t1

⎧ ⎪ ⎪ ⎪ A0 + ⎨

t1

⎧ ⎪ ⎪ ⎪ C0 + ⎨

t1

 t 1 1 (t − α)σ −1C G 2 (α)dα, 0 < t ≤ t1 ,

(σ ) 0 V(t)) = ,  t ⎪ ⎪ ABC G (t) +  (σ ) ⎪ (t − α)σ −1 ABC G 2 (α)dα, t1 < t ≤ T, ⎩ V(t1 ) + 1 (σ ) 2 2  t 1 1 (t − α)σ −1C G 3 (α)dα, 0 < t ≤ t1 ,

(σ ) 0 U(t)) = ,  t ⎪ ⎪ ABC G (t) +  (σ ) σ −1 ABC G (α)dα, t < t ≤ T, ⎪ ) +  (σ ) (t − α) U(t ⎩ 1 1 3 2 3 1  t 1 1 (t − α)σ −1C G 4 (α)dα, 0 < t ≤ t1 ,

(σ ) 0 A(t)) = ,  t ⎪ ⎪ ABC G (t) +  (σ ) σ −1 ABC G (α)dα, t < t ≤ T, ⎪ ) +  (σ ) (t − α) A(t ⎩ 1 1 4 2 4 1  t 1 1 (t − α)σ −1C G 5 (α)dα, 0 < t ≤ t1 ,

(σ ) 0 , C(t)) =  t ⎪ ⎪ ABC G (t) +  (σ ) σ −1 ABC G (α)dα, t < t ≤ T, ⎪ (t − α) ⎩ C(t1 ) + 1 (σ ) 2 1 5 5 t1

(38)

202

Eiman et al.

where C G i (t) =C G i (H, V, U, A, C, t) and ABC G i (t) =iABC G(H, V, U, A, C, t) are the left hand side of equation (27) for i = 1, 2, 3, 4, also given in equation (28). We derive the scheme for the first equation of system (38) and the same procedure can be repeated for other compartments as well. At t = tn+1 , one has  t1 ⎧ 1 ⎪ ⎪ H0 + (t − α)σ −1C G 1 (H, V, U, A, C, α)dα, ⎪ ⎪

(σ ) 0 ⎪ ⎨ H(tn+1 )) = H(t1 ) + 1 (σ ) ABC G 1 (H, V, U, A, C, tn ) ⎪  tn+1 ⎪ ⎪ ⎪ ⎪ ⎩ + 2 (σ ) (t − α)σ −1 ABC G 1 (α)dα, t1 < t ≤ T.

(39)

t1

Following [37], (39) implies ⎧ ⎧ i  

C ⎪ ⎪ ( t)σ −1  ⎪ ⎪ ⎪ ⎪ ⎪ G 1 (Hk−2 , Vk−2 , Uk−2 , Ak−2 , Ck−2 , t) ⎪ ⎪ ⎪ ⎪

(σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ i ⎪ C ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G 1 (Hk−1 , Vk−1 , Uk−1 , Ak−1 , Ck−1 , t) + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ∇ ⎪ ⎪ ⎨ C ⎪ k−2 , Vk−2 , Uk−2 , Ak−2 , Ck−2 , t) ( t) ⎪ ⎪ ⎪ H0 + − G 1 (H

(σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C ⎪ i ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G 1 (Hk , Vk , Uk , Ak , Ck , tk ) + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − 2C G 1 (Hk−1 , Vk−1 , Uk−1 , Ak−1 , Ck−1 , t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ +C G 1 (Hk−2 , Vk−2 , Uk−2 , Ak−2 , Ck−2 , t) σ ( t) ⎪ ⎪ ⎪ 2 (σ + 3) ⎪ ⎨ ⎧ ⎫ ABC G ((Hn , Vn , Un , An , Cn , t ) H(tn+1 ) = (40)  (σ ) ⎪ ⎪ n 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   n ⎪ ⎪

ABC ⎪ ⎪ ⎪ (δt)σ −1 ⎪ σ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G 1 ((Hk−2 , Vk−2 , Uk−2 , Ak−2 , Ck−2 , tk−2 ) ⎪+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  n ⎪ ⎪ ⎪ ABC

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ G (H , V , U , A , C , t) + ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪  ⎪ σ −1 ⎪ (δt) ∇σ ⎪ H(t1 ) + + ABC G ((Hk−2 , Vk−2 , Uk−2 , Ak−2 , Ck−2 , t ⎪ ⎪ ) ⎪ ⎪ 1 k−2 ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n  ABC ⎪ ⎪ ⎪

⎪ ⎪ ⎪ k k k k k ⎪ ⎪ ⎪ ⎪+ ⎪ ⎪ G ((H , V , U , A , C , t ) ⎪ ⎪ 1 ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC G (Hk−1 , Vk−1 , Uk−1 , Ak−1 , Ck−1 , t) ⎪ ⎪ ⎪ − 2 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ (δt) σ ⎪ ⎪ ABC k−2 k−2 k−2 k−2 k−2 ⎪ ⎩+ ⎭ ⎩ G 1 ((H ,V ,U ,A ,C , tk−2 ) . ABC(σ ) (σ + 3)

Other compartments can also be expressed as

Qualitative Theory and Approximate Solution …

203

⎧ ⎧ i C ⎫ 

⎪ ⎪ ( t)σ −1 ⎪ ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎪ ⎪ ⎪ ⎪ G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪

(σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i C ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ + G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ∇( t) ⎨ ⎬ ⎪ C k−2 k−2 k−2 k−2 k−2 ⎪ − G 1 (H , V , U , A , C , t) ⎪ ⎪ V0 + .

(σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i C ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + G 1 (H k , V k , U k , Ak , C k , tk ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ − 2 G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ σ ( t) ⎪ ⎪ ⎪ C k−2 k−2 k−2 k−2 k−2 ⎪ ⎩ ⎭ ⎪ + G (H , V , U , A , C , t) 1 ⎪ ⎪ 2 (σ + 3) ⎪ ⎨ ⎧ ⎫ ABC V(tn+1 )) = G 1 ((Hn , Vn , Un , An , Cn , tn ) ⎪ ⎪ ⎪ 1 (σ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ n ABC ⎪ ⎪ σ −1 ⎪ ⎪

⎪ ⎪ ⎪ σ (δt) ⎪ ⎪ ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎪ ⎪ + G 1 ((H , V , U , A , C , tk−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  n ⎪ ⎪ ⎪ ABC

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + G 1 (H , V , U , A , C , t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪  ⎪ σ −1 ⎪ (δt) ∇σ ⎪ . ⎪ V(t1 ) + + ABC G 1 ((Hk−2 , Vk−2 , Uk−2 , Ak−2 , Ck−2 , tk−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n  ABC ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + G 1 ((Hk , Vk , Uk , Ak , Ck , tk ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ G 1 (H , V , U , A , C , t) ⎪ −2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ABC ⎪ ⎪ σ (δt) σ ⎪ ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎩+ ⎭ ⎩ . G 1 ((H , V , U , A , C , tk−2 ) ABC(σ ) (σ + 3)

(41)

⎧ ⎧ i C ⎫ 

⎪ ⎪ ⎪ ( t)σ −1 ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎪ ⎪ ⎪ ⎪ G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

(σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ i C ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ + G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ∇( t) ⎨ ⎬ ⎪ C k−2 k−2 k−2 k−2 k−2 ⎪ − G (H , V , U , A , C , t) 1 ⎪ ⎪ U0 + .

(σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i C ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + G 1 ((Hk , Vk , Uk , Ak , Ck , tk ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ G (H , V , U , A , C , t) − 2 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ σ ( t) ⎪ ⎪ ⎪ C k−2 k−2 k−2 k−2 k−2 ⎪ ⎩ ⎭ ⎪ + G ((H , V , U , A , C , t ) 1 k−2 ⎪ ⎪ 2 (σ + 3) ⎪ ⎨ ⎧ ⎫ ABC U(tn+1 )) = H1 ((Hn , Vn , Un , An , Cn , tn ) ⎪ ⎪ ⎪ 1 (σ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ n ABC ⎪ ⎪ σ −1 ⎪

⎪ ⎪ ⎪ (δt) σ ⎪ ⎪ ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎪ ⎪ + G 1 (H , V , U , A , C , tk−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  n ⎪ ⎪ ⎪ ABC

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G 1 (H , V , U , A , C , t) + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪  ⎪ σ −1 ⎪ (δt) ∇σ ⎪ k−2 k−2 k−2 k−2 k−2 ) + U(t . ⎪ 1 + ABC G 1 (H , V , U , A , C , tk−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n  ABC ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G 1 ((Hk , Vk , Uk , Ak , Ck , tk ) ⎪+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ G 1 (H , V , U , A , C , t) ⎪ −2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ σ (δt) ⎪ ⎪ ABC k−2 k−2 k−2 k−2 k−2 ⎪ ⎩+ ⎭ ⎩ G 1 (H , V , U , A , C , tk−2 ) . ABC(σ ) (σ + 3)

(42)

204

Eiman et al. ⎧ ⎧ i C ⎫ 

⎪ ⎪ ( t)σ −1 ⎪ ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎪ ⎪ ⎪ ⎪ G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪

(σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i C ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ + G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ∇( t) ⎨ ⎬ ⎪ C k−2 k−2 k−2 k−2 k−2 ⎪ − G 1 (H , V , U , A , C , t) ⎪ ⎪ A0 + .

(σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i C ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + G 1 ((Hk , Vk , Uk , Ak , Ck , tk ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ − 2 G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ σ ( t) ⎪ ⎪ ⎪ C k−2 k−2 k−2 k−2 k−2 ⎪ ⎩ ⎭ ⎪ + G (H , V , U , A , C , t) 1 ⎪ ⎪ 2 (σ + 3) ⎪ ⎨ ⎧ ⎫ ABC A(tn+1 )) = G 1 (Hn , Vn , Un , An , Cn , tn ) ⎪ ⎪ ⎪ 1 (σ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ n ABC ⎪ ⎪ σ −1 ⎪ ⎪

⎪ ⎪ ⎪ σ (δt) ⎪ ⎪ ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎪ ⎪ + G 1 (H , V , U , A , C , tk−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  n ⎪ ⎪ ⎪ ABC

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + G 1 (H , V , U , A , C , t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪  ⎪ σ −1 ⎪ (δt) ∇σ ⎪ . ⎪ A(t1 ) + + ABC G 1 (Hk−2 , Vk−2 , Uk−2 , Ak−2 , Ck−2 , tk−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n  ABC ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + G 1 ((Hk , Vk , Uk , Ak , Ck , tk ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ G 1 (H , V , U , A , C , t) ⎪ −2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ABC ⎪ ⎪ σ (δt) σ ⎪ ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎩+ ⎭ ⎩ . G 1 ((H , V , U , A , C , tk−2 ) ABC(σ ) (σ + 3)

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⎧ ⎧ i C ⎫ 

⎪ ⎪ ( t)σ −1 ⎪ ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎪ ⎪ ⎪ ⎪ G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪

(σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ i C ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ + G (H , V , U , A , C , t) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ∇( t) ⎨ ⎬ ⎪ C k−2 k−2 k−2 k−2 k−2 ⎪ − G (H , V , U , A , C , t) 1 ⎪ ⎪ C0 + .

(σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i C ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + G 1 ((Hk , Vk , Uk , Ak , Ck , tk ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ G (H , V , U , A , C , t) − 2 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ σ ( t) ⎪ ⎪ ⎪ C k−2 k−2 k−2 k−2 k−2 ⎪ ⎩ ⎭ ⎪ + G (H , V , U , A , C , t) 1 ⎪ ⎪ 2 (σ + 3) ⎪ ⎨ ⎧ ⎫ ABC C(tn+1 )) = G 1 (Hn , Vn , Un , An , Cn , tn ) ⎪ ⎪ ⎪ 1 (σ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ n ABC ⎪ ⎪ σ −1 ⎪

⎪ ⎪ ⎪ (δt) σ ⎪ ⎪ ⎪ k−2 k−2 k−2 k−2 k−2 ⎪ ⎪ ⎪ + G 1 (H , V , U , A , C , tk−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  n ⎪ ⎪ ⎪ ABC

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G 1 (H , V , U , A , C , t) + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪  ⎪ σ −1 ⎪ (δt) ∇σ ⎪ k−2 k−2 k−2 k−2 k−2 ) + C(t , ⎪ 1 + ABC G 1 (H , V , U , A , C , tk−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC(σ ) (σ + 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n  ABC ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ G 1 ((Hk , Vk , Uk , Ak , Ck , tk ) ⎪+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=i+3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ABC k−1 k−1 k−1 k−1 k−1 ⎪ ⎪ ⎪ G 1 (H , V , U , A , C , t) ⎪ −2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ σ −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ σ (δt) ⎪ ⎪ ABC k−2 k−2 k−2 k−2 k−2 ⎪ ⎩+ ⎭ ⎩ G 1 ((H , V , U , A , C , tk−2 ) , ABC(σ ) (σ + 3)

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where ⎡

2 σ σ σ 2 σ σ ⎢ 2(n − k) (1 + n − k) + (1 + n − k) (3σ + 10)(n − k) + 2(1 + n − k) σ + 9(1 + n − k) σ + 12(1 + n − k) ⎢ =⎢

⎣ − 2(n − k)(n − k)2 + (n − k)(5σ + 10)(−k + n) + 6(n − k)σ 2 + 18(n − k)σ + 12(n − k)

⎤ ⎥ ⎥ ⎥, ⎦

⎡

⎤ 3 + 2σ − k + n (1 + n − k)σ ⎥ ⎢ ⎥,

∇=⎢ ⎦ ⎣ − n − k + 3σ + 3 (n − k)   = (1 + n − k)σ − (n − k)σ .

10 Numerical Simulation and Discussion In this section, we present our proposed numerical scheme to simulate the considered model numerically and graphically under the piecewise equation using ABC type derivative. We use numerical values given in Table 1 and simulate the model in Figs. 7, 8, 9, 10 and 11 respectively. Further, the domain [0, 60] is divided into two sub intervals [0, 20], [20, 60] respectively.

80

Suscceptible class

Fig. 7 Approximate results presentation using piecewise concept of H at several fractional order in the model (27)

0.90 0.94 0.96 1.00

60

40

20 0

10

20

30

40

50

60

Time (Days) 45

Vacinnated class

Fig. 8 Approximate results presentation using piecewise concept of V at several fractional order in the model (27)

40 35

0.90 0.94 0.96 1.00

30 25 20 0

10

20

30

Time (Days)

40

50

60

206 60

0.90 0.94 0.96 1.00

50

Exposed class

Fig. 9 Approximate results presentation using piecewise concept of U at several fractional order in the model (27)

Eiman et al.

40 30 20 10 0 0

10

20

30

40

50

60

Time (Days) 30

0.90 0.94 0.96 1.00

25

Infected class

Fig. 10 Approximate results presentation using piecewise concept of A at several fractional order in the model (27)

20 15 10 5

0

10

20

30

40

50

60

Time (Days) 500

Recovered class

Fig. 11 Approximate results presentation using piecewise concept of C at several fractional order in the model (27)

400 0.90 0.94 0.96 1.00

300 200 100 0 0

10

20

30

40

50

60

Time (Days)

11 Suberization of This Work We have investigated a five-compartmental fractional order mathematical model of Norovirus. The five compartments are of susceptible, vaccinated, exposed, infected, and recovered individuals. Also by Laplace transform coupled with ADM some approximate analytical results have been determined. From the graphs, it has been concluded that the solution tending to the classical (integer) order solution when σ → 1. It has concluded that by taking a few terms of the series solutions, the system under investigation can be demonstrated properly. We also check the investigated proposed model under piecewise equations by using ABC fractional order derivative. Theoretical results by using fixed point theory and numerical analysis via the Adam

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Bash-forth method have been used to obtain the required results. Several graphical results have been presented to understand the crossover behavior of the proposed model. The obtained analysis can be extended in the future for a more general model of epidemiology. Acknowledgements “Authors are thankful to Prince Sultan University for support through TAS research lab.” Conflict of Interest “The authors declare that they have no competing interests.”

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Study of the SIRI Model Utilizing the Caputo Derivative Ndolane Sene

Abstract In this article, we make qualitative studies of the SIRI model described by a fractional operator. In the investigations, the Caputo operator is utilized in the modeling of the epidemic. We have used the reproduction number and we have analyzed the stability of the model. A qualitative property as the boundedness of the solutions of the SIRI model is focused. A numerical scheme including the numerical scheme of the Caputo operator has been utilized and explained rigorously. The results of the paper have been illustrated by the graphics of the solutions obtained using the proposed numerical scheme in the paper. Keywords Fractional SIRI model · Frcational derivative · Stability analysis · Reproduction number

1 Introduction Modeling world problems via fractional operators has attracted many authors these last times. The researchers find many applications of fractional operators in fractional calculus fields. The operators have applications in physics [11, 15], they have applications in biological models [3, 10, 18, 20, 27–29], they have applications in mathematical physics [13, 16, 26], they have applications in chaos theory [21] notably findings new types of attractors with the variations of the orders of the fractional derivatives. The pandemic started in 2019 and had many consequences on our life and generated interest in the epidemical models. The novel problems with fractional operators consist to model the disease models and finding optimal control to stop the spread of the different diseases. And it is stipulated that fractional operators can bring advancements in modeling problems. Many operators are introduced in fractional calculus, we can cite the old fractional operators: the Caputo type and the N. Sene (B) Section mathematics and statistics, Institut des Politiques Publiques, Cheikh Anta Diop University, Dakar Fann, Senegal e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_9

211

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Riemann-Liouville type of derivatives [17, 19]. These two derivatives are the most used in the fractional operators and recently the new fractional operators with new types of kernels began to emerge, we have operators from Atangana and Baleanu [12] and the operator from Caputo and Fabrizio [14]. These new operators grow many interests this decade with many papers appearing to justify their applicability in modeling problems. We review the literature on the epidemic models which have used fractional operators. In [3], the authors propose the SIR model via the Atangana-Baleanu derivative. The qualitative properties such as the stability and the existence of the solutions and the graphical representations of the dynamics have been proposed in this work. In [4], the authors proposed biological modeling via stochastic perturbation, the boundedness, the permanency, the reproduction number of the model and the graphics of the stochastic dynamics have been provided in this paper. In [5], for a model against the novel coronavirus, the present paper gives exciting findings which can serve to reduce the evolution of the coronavirus. In [6], the authors have proposed the analytical solutions of the biological model using the Natural Transform. In [2], the authors have proposed the ergodicity property, the extinction of disease, and the density function of a SIRI model with nonlinear incidence rate and high-order stochastic perturbations. In [1], the authors have described the SIRI model via stochastic perturbation. In [7], the authors have investigated on SIRS reaction-diffusion epidemic model. In [8], the problem of the stability study of the equilibriums in epidemic modeling has attracted many researchers this decade, in their paper, the authors have presented investigations on the stability of the equilibrium points of some epidemic models. In [9], the authors investigated the HIV infection model via the fractional operators, and they have used it in their modeling of the drug therapy effect. In [10], the authors proposed investigations related to the modeling of the SIR model using the fractional model. The model presented in this paper has a particularity to be described by the nonlinear incidence rate. See also in [22, 25]. In the present paper, we address the SIRI model. The objective is to model using the fractional operator and to study the global stability and the local stability using the reproduction number. Another important finding of this paper is that we prove via the mean value theorem that there exists an endemic equilibrium point for the SIRI model, in other words, we do not need to get its form, we have just to prove this point exists. The numerical scheme will be proposed and applied to the Caputo derivative in the way to get the form of the behaviors of the solutions. We will use the numerical scheme to propose the graphics of the solutions of the considered SIRI model.

2 Fractional Calculus Operators Let’s recall the operators used in this chapter. We will use the fractional integral named the Riemann-Liouville operator integral. We have utilized the Caputo operator in the investigations. We will recall it too. The function and transformation necessary for

Study of the SIRI Model Utilizing the Caputo Derivative

213

our investigations as the Simon Laplace transformation and the function known as the Mittag-Leffler function will be recalled as well. Definition 1 ([17, 19, 23]) Assume that the function w : [0, +∞[−→ R. Let’s the Liouville-Riemann integral of the function w as the following form I α w (t) =

1 (α)



t

(t − s)α−1 w(s)ds,

(1)

0

where t > 0, α ∈ (0, 1) represents the order of the integral operator, and the function gamma denotes (...). Definition 2 ([17, 19, 24]) Assume that the function w : [0, +∞[−→ R. Let’s the Liouville-Caputo operator of the function w as the form 

Dcα w



1 (t) = (1 − α)



t

(t − s)−α w (s)ds,

(2)

0

where t > 0, α ∈ (0, 1) represents the order of the integral operator, and the function gamma denotes (...). The Laplace transform and the Mittag-Leffler function will be used in the solutions procedures concerning the fractional equations and then we recall them as well. For the Laplace transform of the Caputo operator, we obtain the following form[17, 19] L

 α   Dc w (t) = sα L {w(t)} − sα−1 w(0).

(3)

with the condition α ∈ (0, 1). The Mittag-Leffler function is used to describe the form of the solutions of the differential equations. Here recall this important function for our investigations ∞  zn , (4) Eα,β (z) = (αn + β) n=0 with α > 0, β ∈ R and the variable t is into the set C. We have many other fractional operators, we have the CF derivative proposed in the literature in [14] in 2015. We have another fractional operator known as the fractional derivative with the ML function proposed in the investigation [12]. These two new operators have had many successes in these recent years, notably in biological modeling.

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3 Fractional SIRI Model Under Caputo Operator In this part, we will focus on the SIRI model under the Caputo derivative. Before introducing the model we first recall the original model described by the integerorder operator. We have many classes of SIRI models presented in the research, here referring to the investigation in the literature, we have the following differential equations ϑSI dS =  − μS − , dt 1 + νS ϑSI dI = − (μ + c + γ) I + δR, dt 1 + νS dR = γI − (μ + δ) R, dt

(5) (6) (7)

The initial condition considered for our present problem can be described in the following form, that is in the form that S (0) = S0 ,

I (0) = I0 ,

R (0) = R0 ,

(8)

The signification of the parameters utilized in the SIRI model is assigned in the following Table, for more clarity, we have that (Table 1). In the present SIRI model, the incidence function is represented by the function ϑSI which is considered to be Lipchitz continuous function. The secthat f (S, I ) = 1+νS ond step of the modeling will be to introduce the memory effect in the differential equations (5)–(7), to arrive at our end we use the Caputo fractional operator which is known to be an operator with singularity. Therefore, under Caputo fractional derivative, the fractional differential under consideration will be defined in the following equations Table 1 Parameters of the model

Parameters

Descriptions

 μ γ ϑ δ c

The recruitment rate The natural death rate The rate of recovery from infection The transmission rate The relapse rate The death rate generated by infection

Study of the SIRI Model Utilizing the Caputo Derivative

Dcα S =  − μS −

215

ϑSI , 1 + νS

(9)

ϑSI − (μ + c + γ) I + δR, 1 + νS α Dc R = γI − (μ + δ) R, Dcα I =

(10) (11)

The initial conditions of the model (5)–(7) are maintained, in other words, we conserve the same initial conditions, they are defined by the following equation S (0) = S0 ,

I (0) = I0 ,

R (0) = R0 ,

(12)

We first prove that the total population variable is well-bounded. To arrives at this end we proceed in the following form by calculating the sum of the three variables and then we have that P = S + I + R, thus Dcα P(t) =  − μP(t) − cI (t).

(13)

We notice that Eq. (13) is a simple linear differential equation and then we can determine its exact analytical solution by applying the Laplace transformation, and then we get that  ¯ − μP(s) − cI¯ (s), s  sα−1 sα−1 P(0) cI¯ (s)  1 ¯ − α + α − α . P(s) = μ s s +μ s +μ s +μ

¯ − sα−1 P(0) = sα P(s)

(14)

The inversion of the Laplace transformation applied to Eq. (14) gives the exact solution which in particular includes the Mittag-Leffler function, we obtain the form P(t) ≤

 

1 − Eα (−μ (t)α ) + P(0)Eα (−μ (t)α ) . μ

(15)

We conclude that the solutions of our present model are well-bounded because the total population is bounded. And then, our SIRI model is biologically well definite. An important remark is it follows from the convergence of the time to infinity, we get the bound represented in the following relation P(t) ≤

 . μ

(16)

Equation (16) will play a role because it will permit us sometimes to write our model with two variables and to study the stability analysis. Let’s analyze the reproduction number which has an important role in the analysis of the epidemics model, in general. The method to obtain is not new and can be found

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in the literature, the method utilized in this paper is similar to the technique used in. The first step is that Eqs. (5)–(7) can be written in the form described by Dcα x = V − W,

(17)

in the previous equation (17) the variable x = (S, I , R), and furthermore the matrix V and W are described by the form that ⎛

ϑSI 1+νS

⎞

⎞ (μ + c + γ) I − δR W = ⎝ −γI + (μ + δ) R ⎠ . ϑSI −  + μS 1+νS ⎛

V =⎝ 0 ⎠ 0

and

(18)

  The next step consists to apply the Jacobian at the equilibrium point P0 = μ , 0, 0 to the previous matrices V and W , the method to get the free equilibrium is illustrated in the next section, we get the following ⎛

ϑ μ+ν

JV (E0 ) = ⎝ 0 0  F=

0 0 0

ϑ μ+ν

0

⎞ 0 0⎠ 0 0 0

and

⎞ μ + c + γ − 0 −γ μ + δ 0⎠. J W (E0 ) = ⎝ ϑ 0 γ μ+ν

and

V =

⎛





μ + c + γ − −γ μ+δ

 .

(19)

To get the reproduction number for our original model in Eqs. (5)–(7), we apply the definition of the spectral radius on the matrix defined by FV −1 , we have that   R0 = ρ FV −1 =

ϑ (μ + δ) . (μ + β) [(μ + c + γ) (μ + δ) − γδ]

(20)

The study in this paper will be based on the reproduction number found in Eq. (20). Notably, the local and global stability will be analyzed according to the reproduction number R0 .

4 Stability Investigations In this sub-section, we investigate the stability local of the equilibriums for our model described in Eqs. (5)–(7). The first part before will be to establish the points where the equations of our present model are zero. They are obtained by letting that

Study of the SIRI Model Utilizing the Caputo Derivative

 − μS −

217

ϑSI = 0, 1 + νS

ϑSI − (μ + c + γ) I + δR = 0, 1 + νI γI − (μ + δ) R = 0,

(21) (22) (23)

After calculation, we get as a trivial equilibrium point the point given by the relation that    , 0, 0 . (24) P0 = μ The nontrivial equilibrium point known in the literature as the endemic point can be represented by the following relationship that 0 =  − μS ∗ − ∗ ∗

ϑS ∗ I ∗ , 1 + νS ∗

(25)

ϑS I − (μ + c + γ) I ∗ + δR∗ , 1 + νS ∗ 0 = γI ∗ − (μ + δ) R∗ .

0=

(26) (27)

The endemic point P1 = (S ∗ , I ∗ , R∗ ) required to prove this point really exist because it values at this stage is not trivial. To prove the existence of this point we proceed as follows. We consider that the function I ∗ exists and we have to write the other comportment depending on this variable I ∗ and we combine it with the use of the classical mean value Theorem. We have the following for the last variable R∗ =

γ ∗ I . μ+δ

(28)

With the sum between the first and the second equation of our model (5)–(7), we get the following representation  − μS ∗ − (μ + c + γ) I ∗ + δR∗ = 0.

(29)

Solving Eq. (29) previously mentioned function, we finally arrive at the end that S∗ =

 μ+c+γ− − μ μ

δγ μ+δ

I ∗.

(30)

It proves the existence and the unicity of the variable S ∗ because it depends on the variable I ∗ . We now continue with the last by proving the variable I ∗ exists and is unique, to this end we use the mean value Theorem, the sketch is to consider the function that

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N. Sene

 ϑ

 μ



f (i) = 1+ν

−  μ

δγ μ+c+γ− μ+δ μ

−

 i

δγ μ+c+γ− μ+δ μ

 − (μ + c + γ) + i

γ . μ+δ

(31)

Let us calculate the following values including the reproduction number in its expression. We suppose that a = R0 (μ+ν) and b = 0, we have the forms defined by ϑ f (a) = − (μ + c + γ) + f (b) =

γ < 0. μ+δ

(32)

  γ ϑ 1 ϑ > 0. − (μ + c + γ) + = 1− μ + ν μ+δ μ + ν F0

(33)

Using the fact that the function f is a bijective function, we arrive at that there exists a unique I ∗ which is the solution of equation f (i) = 0 and which verifies the expression defined by the form that R0 (μ + ν) . (34) 0 ≤ I∗ ≤ ϑ We conclude that the endemic point Pe = (S ∗ , I ∗ , R∗ ) exists and is unique. We consider the endemic point, and due to the complexity, of the calculations, we try to establish the global stability of the endemic equilibrium point using the Lyapunov direct method. To finish, we take the Lyapunov function which is represented by (35) L (S, I , R) = L1 (S, I , R) + L2 (S, I , R) + L3 (S, I , R) . We calculate step by step the Caputo derivative operator applied to the previous function represented in the previous Eq. (35). Note that we suppose that the function L1 is represented by 

S L1 (S, I , R) = S − S − S ln S∗ ∗

∗





I + I − I − I ln ∗ I ∗

∗

 ,

(36)

Let’s calculate the Caputo operator to the previous function respect to the trajectories of the system represented in Eqs. (5)–(7), we obtain the following relationships  Dcα V1 ≤ 1 −  ≤ 1−  + 1−

   S∗ I∗ Dcα S + 1 − Dcα I , S I   ϑSI S∗  − μS − , S 1 + νS   ϑSI I∗ − (μ + c + γ) I + δR . I 1 + νS

(37)

Study of the SIRI Model Utilizing the Caputo Derivative

219

Using the fact the endemic point satisfied the following conditions, in other words applying the conditions that  = μS ∗ +

ϑS ∗ I ∗ , 1 + νS ∗

(38)

ϑS ∗ I ∗ = (μ + c + γ) I ∗ − δR∗ , 1 + νS ∗ γI ∗ = (μ + δ) R∗ .

(39) (40)

Applying Eqs. (38)–(41), the calculations can be expressed in the following forms, they are given by the form    S∗ αSI ϑS ∗ I ∗ (S − S ∗ )2 , − 1− − Dα L1 ≤ −μ S S 1 + νS 1 + νS ∗    ϑSI ϑS ∗ I ∗ I∗ , − + 1− I 1 + βS 1 + νS ∗       I∗

I∗

I − I∗ + δ 1 − R − R∗ . − (μ + c + γ) 1 − I I

(41)

We suppose the second Lyapunov function described by the function given in the next line and we calculate it Caputo derivatization α,ρ

L2 (S, I , R) = κIt

 I (t − τ ) − I ∗ ,

(42)

We apply the Caputo derivative to the previous function and we consider the trajectories of our model we get the following forms, which are described by the forms

2 Dα L2 = κDα Itα I − I ∗ ,

2 = κ I − I∗ .

(43)

We end by considering the last Lyapunov function and we calculate its Caputo operator derivation by considering our model (5)–(7). The Lyapunov function considered in this last section is represented by the form that 

R L3 (S, I , R) = R − R − R ln R∗ ∗

∗

 .

(44)

As a previous procedure, we consider the trajectories of the Eqs. (5)–(7) and we apply the Caputo derivative on the previous function described in the previous Eq. (44), we have that

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  R∗ Dα,ρ R, Dα,ρ L3 ≤ 1 − R   R∗ ≤ 1− [γI − (μ + δ) R] . R

(45)

Here, we consider the endemic equilibrium point, for the simplification of the calculations, and then we obtain the following relations       R∗

R∗

∗ I − I − (μ + δ) 1 − R − R∗ . D L3 ≤ γ 1 − R R α

(46)

Note that the condition that R0 ≤ 1 ensures the existence of the endemic equilibrium point as previously proved in this section. Summing Eqs. (41), (43), and (47), we get the following condition Dα L = Dα L1 + Dα L1 + Dα L3 ≤ 0.

(47)

And then using the Lyapunov direct method characterization, and considering the condition Eq. (47), we summarize that the endemic point is global asymptotic stable when the condition that R0 > 1 is held.

5 Results and Discussions This section, we present the numerical procedure and analyze the behaviors of the dynamics of the SIRI epidemic model described by the Caputo operator. We notably focus on the impact of the order of the derivative operator in the dynamics of the model. The first sub-section addresses the numerical procedure used to get the graphics.

5.1 Solution Procedure This section describes the numerical procedure adopted in the present paper. This section is significant in the computational epidemic model because it will permit drawing the dynamics of the considered epidemic model. We begin by determining the form of the solution by the utilization of the fractional integral, we get the forms S(t) = S (0) + I α u (t, x) , α

I (t) = I (0) + I v (t, x) , R(t) = R (0) + I α w (t, x) ,

(48) (49) (50)

Study of the SIRI Model Utilizing the Caputo Derivative

221

where x = (S, I , R) and the functions u, v, and w are represented in our context by the elementary functions defined by the forms that u (t, x) =  − μS −

ϑSI , 1 + νS

(51)

ϑSI − (μ + c + γ) I + δR, 1 + νI w (t, x) = γI − (μ + δ) R, v (t, x) =

(52) (53)

The next part consists to find the numerical scheme of the integral used in the previous solutions (51)–(53). And then at points (tn , xn ), we get the following form in the procedure of the discretization α

α

I u (tn , xn ) = h

s 

n     (α)  α n,j u tj , xj + h n−j u tj , xj ,

j=0

I α v (tn , xn ) = hα

s 

n     (α)  n,j v tj , xj + hα n−j v tj , xj ,

j=0

I α w (tn , xn ) = hα

(54)

j=0

(55)

j=0

s 

n     (α)  n,j w tj , xj + hα n−j w tj , xj .

j=0

(56)

j=0

here we consider that h is the step size and the starting weight is denoted in our previous equation by the symbol that n, j. Furthermore, the starting weight satisfies the condition expressed in the following relation, for more explanation about this formula see in [30] s  j=0

γ

n,j j = −

s  j=0

n−j j γ +

 (γ + 1) nγ+α  (γ + 1 + α)

(57)

where the assumption on the parameter γ can be found in [30], note that for n = 0, 1, 3, ... we obtain the following relationships n(α) = (−1)n

 (1 − α)  (n + 1)  (1 − α − n)

(58)

Utilizing the numerical discretization described in Eqs. (54)–(56) for the Riemann Liouville integral we get as numerical discretization for our model the relationships defined by the form that

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N. Sene

x(tn ) = x (0) + hα

s 

n     (α)  n,j u tj , xj + hα n−j u tj , xj ,

j=0

y(tn ) = y (0) + hα

s 

n     (α)  n,j v tj , xj + hα n−j v tj , xj ,

j=0 α

z(tn ) = z (0) + h

s 

(59)

j=0

(60)

j=0 n     (α)  α n,j w tj , xj + h n−j w tj , xj .

j=0

(61)

j=0

The numerical procedure of the functions utilized in Eqs. (5)–(7), in other words, used in our epidemic model are given in the forms given as   ϑSj Ij u tj , xj =  − μSj − , 1 + νSj   ϑSj Ij v tj , xj = − (μ + c + γ) Ij + δRj , 1 + νSj   w tj , xj = γIj − (μ + δ) Rj ,

(62) (63) (64)

In this new part, we give the graphics and analyze the influence of the fractional operator used in our modeling. In this section, we suppose that S(0) = 5, I (0) = 10 and R(0) = 0. let that  = 20, μ = 0.3, ϑ = 0.375, ν = 10, c = 0.4, γ = 0.2, and δ = 0.02. In the first graphic, we utilize that the order of the fractional operator is considered as α = 0.95. We obtain the following graphics (Figs. 1 and 2). To see, the impact or the influence of Caputo derivative operator, we now consider the order fixed at α = 0.85 and we try to observe the behaviors of the dynamics of the considered SIRI model. We have the following graphics (Figs. 3 and 4). We notice that the impact or the influence of the order of the derivative operator exists, we can notice that the graphics of the susceptible increase more rapidly

60

10 9

50

8 7

40

6 30

5 4

20

3 2

10

1 0

0 0

10

20

30

40

(a)

50

60

70

0

10

20

30

40

(b)

Fig. 1 Susceptible individuals versus time (a). Infective individuals versus time (b)

50

60

70

Study of the SIRI Model Utilizing the Caputo Derivative

223

1.4 1.5 1.2 1 1 0.5

0.8 0.6

0 0

0.4 20 0.2 40 0 0

10

20

30

40

50

60

70 60

(a)

2

4

6

8

10

0

(b)

Fig. 2 Recovered individuals versus time (a). SIRI in three dimensional (b) 10

70

9 60 8 50

7 6

40

5 30

4 3

20

2 10 1 0

0 0

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

(b)

(a)

Fig. 3 Susceptible individuals versus time (a). Infective individuals versus time (b) 1.4 1.5 1.2 1 1 0.5

0.8 0.6

0 0

0.4 20 0.2

40 60

0 0

10

20

30

40

50

60

70 80

(a)

10

6

8

(b)

Fig. 4 Recovered individuals versus time (a). SIRI in three dimensional (b)

4

2

0

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N. Sene

when the order α = 0.85 than when the order of the operator is α = 0.95. The same behavior can be noticed with the recovery individuals, where the pic of the recovered is attended more rapidly by the order α = 0.85 than when the order of the derivative operator is α = 0.95. The conclusion is that the order of the operator plays a deceleration effect in the case of modeling the SIRI model. Thus the order of the fractional operator can be used to control the considered disease.

6 Conclusion This paper has discussed the SIRI model via the Caputo operator. We have introduced the fractional operator in modeling the biological model. The equilibrium points of the model have been provided and their stability analysis using the reproduction number has been studied. The proof of the existence of the endemic equilibrium via the mean value theorem is an interesting result in this paper. The second part of this proposition is the numerical discretization used to obtain the solutions of the proposed epidemic model via the Caputo derivative. The optimal control studies are not proposed in this chapter and it can be interesting to focus in this direction in future investigations.

Conflict of Interest The authors declare that they have no conflict of interest.

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Implementation of Vaccination in an Epidemic Model for COVID-19 Yerra Shankar Rao

Abstract The vaccination campaign against the pandemic diseases started in India in 1st week of January 2021, and by 10th December 2021, there are 13, 123, 75,088 people have already been vaccinated. It is observed that people who were vaccinated but again infected with the disease COVID-19. This article establishes the SVIR (Susceptible, Vaccination, Infectious and Recovered) pandemic paradigm. The proposed model’s general characteristics are first discussed. Particularly, both theoretical and numerical analysis is done on the basic reproduction numbers. Reproduction numbers are used to describe both the local and global stability of equilibrium points. Secondly, determine the stability of the equilibrium points using the Lypanouv function and the Painocare-Bandixson condition. The transmission of Covid-19 and its regulation are shown in simulation with the aid of MATLAB according to genuine data in India. In the suggested model, the effects of vaccination are investigated. This simulated result is consistent with the actual parameters and indicates the necessity for booster doses to eradicate the virus from the population. Keywords Vaccinations · Epidemic model · Lyponouv function · Stability

Nomenclature Notations N S I V R A

Descriptions Total number of populations Number of the susceptible population at the time t Number of the infected population at the time of t Number of vaccinated populations Number of recovered population after infected at the time t Influx rate that represented the migrant population

Y. S. Rao (B) Department of Mathematics, NIST Institute of Science and Technology (Autonomous), Institute Park, Pallur Hills, Berhampur, Odisha, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Singh and H. Dutta (eds.), Computational Methods for Biological Models, Studies in Computational Intelligence 1109, https://doi.org/10.1007/978-981-99-5001-0_10

227

228

β γ α ε d1 d2

Y. S. Rao

The infected rate at time t in the Corena transmission model Reassembly rate of vaccination at time t Recover rate of the infected population at time t Vaccinated population getting the infected rate at time t Natural death i.e. death rate other than COVID-19 The death rate due to COVID-19

1 Introduction The ongoing pandemic disease is Coronavirus (COVID-19) caused by severe acute respiratory syndrome coronavirus-2(SARS-COV-2). This disease spreads rapidly all over the world and its transmission rate is very high. The outbreak of SARSCoV-19 first appears in Wuhan city of China, in the last of December 2019[1]. Then it was spread worldwide across the globe. Therefore due to the imbalance situation in the world on March 11th, 2020, the world health organization (WHO) declared it a pandemic. There are more than 269,468,456 people infected, having death 53, 11,895, across 169 countries as on 10th December 2021. But in India, there were 34, 68, 2736 people infected and the death due to covid-19 was 47, 4479 people as on 10th December 2021. So the impact of covid-19 in various fields like socio-economic imbalance, educational organizations, small-scale and largescale industries, unemployment, etc. Initially, the main symptoms of this disease are mild cold with cough, high fever, aches, tiredness, loss of taste, headache, diarrhea, chest pain, and shortness of breath whereas in complicated cases may include severe pneumonia in lungs. Primarily this disease spreads with close contact with infected persons, during coughing, sneezing, and talking. [2] Studied the non-integer order smoking model using the memory principle. This method is used to that model to minimize smoking in society. Infectious disease modeling in real-world problems like COVID-19, Cancer, diabetes, and other diseases was discussed. The author mainly discussed the basic concept of modeling and how it was implemented in a real-world problem. The fixed point theory used for [3] analysis of COVID-19 transmission and its mitigations in the sense of fractional order Caputo operator. This operator provides deeper grasp of the epidemic diseases. At an early stage, the government had taken initiatives like complete lock-down, contact tracing and implementation of strict social distancing measures, and quarantine processes either home or hospitalization [4] studied the SIQR model for preventing the covid-19. They mainly emphasize that people remain isolated to prevent or control the disease. [5] adopted three quarantine processes according to their stages. They were mainly to see the transmission behavior of the virus and its preventive measures stages. [6] proposed a mathematical model and suggested that the hospitalized quarantine is the best policy for curb the Covid-19. The study conducted by [[7] designed an epidemic model that the double quarantine comes close to preventing diseases. [8] carried out to investigation on the movement of

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migrant People in Orissa during the lockdown period. The government of Odisha adopted the quarantine method in schools or local areas for suppressing COVID-19. In this paper, the authors [9] perform a trend analysis of the fatality rate, impact of lockdown, and infection rate with valid data. They take the example of Bhilwara, a city in Rajasthan, India which successfully controls diseases by use of tracing, testing, and treatment technique. Due to the implementation of the above preventative measures the rate of active cases somehow decreased. The infected population trajectory for India may have shown positive signs of slowing down. [10] In this paper, the authors proposed an improved SEIR model to predict the covid-19 trends in Italy, the United State of America, and New York. However, continuing this process leads to chaos in society. Vaccination is the main weapon to control or prevent the spread of the disease. Implementation of blocking measures, admitting mild patients into the hospital, and tracking of the covid-19 cases controlled by Singapore, and other countries studied in this paper [11]. So all over the globe desperately started vaccination campaigns as a marathon journey to eradicate or reduce the COVID-19 Pandemic. In 2020 there were more than 50 companies developed a vaccine to fight against SARSCOV-2. The U.K. government first approved the vaccine on 2nd Dec 2020 named as Pfizer-BioNTech BNT162b26. Then followed by the Oxford-AstraZeneca vaccine7 on 20th December 2020, and subsequently, the pharmaceutical company Moderna TX, Inc named Moderna COVID-19 vaccine started on 8th Jan 2021, whereas only the two companies vaccines Pfizer-BioNTech and Oxford-AstraZeneca were widely provided to the population of U.K. by February 2021. Now there are different types of vaccines available in the globe like a whole virus, protein subunit, viral vector, and nucleic acid (mRNA, DNA). But the Indian government approved three vaccines, out of which Covishield manufactured by Serum Institute of India, Pune in collaboration with Oxford University-AstraZeneca, and Covaxin manufactured by Bharat Biotech International Limited Hyderabad in collaboration with ICMR India are being used extensively for the vaccination program. In the first phase, the vaccination program was started on January 16th, 2021, covering the health care workers like doctors and nurses, then it was extended to frontline workers In early February, and the vaccination program was expanded to include frontline workers on 2nd February 2021. On March 1st, 2021 the government of India started 2nd phase of vaccination campaigns for people of senior citizens above 60 years and people above 45 years of age with comorbidity. The country launched vaccination for all people aged more than 45 years on April 1st , 2021. Subsequently, the government then decided to expand its vaccination drive by allowing everyone above 18 years to be vaccinated from 1st May 2021. As a result, more than 13, 123, 75,088 doses are given to the people of India from 16th Jan 2021 to 10th December 2021. Till now there is no vaccine approved by the government of India for the age group below 18 years. Governments plan to implement the vaccination program for below 18 years people and kids not to increase the infection rate of the above age groups. An epidemic model is a tool used to study the mechanism through which the virus transmits, predict the future of the epidemic, and identify possible preventive measures for the virus. Some of the researchers studied the different mathematical modeling of different diseases like the Ebola virus, Cancer

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treatment, HIV, etc. [12] analyzing the Ebola virus model in fractional modeling. In that paper, he was shown the mathematical stability of the virus with the help of CPU time computations. Analysis of drug treatment of HIV infection model CD4+ T cell was discussed by [2]. He was used to show the efficiency of the treatment of HIV infected population. He was using the memory principle and decentralization of the said diseases [13]. In this book, the authors describe the diseases modeling and computational modeling of different diseases in real-world problems and their remedies by different approaches. Many researchers have studied a different mathematical model for covid19 and related issues. Reference [14] mainly focusing on the fundamental solution for preventing COVID-19 and efficient vaccines through a mathematical model. The herd immunity concept and its implications for Covid-19 were conducted by [15]. Reference [16] discussed the effectiveness and complicatedness of the vaccine in real-world conditions for Pfizer-BioNTech’s and Moderna’s mRNA vaccines. [17] proposed a mathematical model that explains vaccination strategy is a major role in reducing social distance and minimize the covid-19 infections. Reference [18] In this paper, the authors have discussed the age structure model and suggested that by providing the vaccine to the elder age group, the mortality rate can be decreased. Also, they suggest an effective vaccine given to all people of Korea to reduce the incidence and mortality rate. Reference [19] focus on the optimal duration of the vaccine campaign properly for eliminating the infection at Marocco. They were taking a statistical method named Bootstrap for estimating the parameter and characterized the optimal function of the Pontryagins maximal principle for optimal control of the diseases. This paper [17] suggested the high rate of vaccination given to people covid-19 cases should be resurgence. Reference [20] studied the process of preparing vaccines after dealing with multiple morbidities and fatalities and also said that personal health education is required for the destruction of the virus. Reference [21] the 1st year covid-19 vaccine program carried out by Oliver et al. They studied the global impact of vaccination programs to determine lives lost if no vaccine was distributed. The above-mentioned work on mathematical modeling of vaccination supply and the infection rate is constant. Comparing the vaccination policy in different counties like Italy, South Africa, and India that leads a positive impact and reduces the mortality rate The vaccination policy and their implementation effect in India, Italy, and the UK comparison that a positive impact. Receiving of vaccine doses of [22] Delta variant and alpha variant virus and the effectiveness of the doses supporting maximize vaccine and minimize the infection discussed in the country UK. The existing work has been done on the variable of infection rate. The sack of that work leads to human congestion and trouble for some people infected. Thus we present the transmission model with vaccinated people getting infected and susceptible, motivated by the work and reports. Recently there is a new mutant or variant of COVID-19 was detected first in South Africa on 24th November 2021. It is observed by scientists and medical researchers that the virus spreads very rapidly across all countries including Australia and European countries. On 26th November 2021, WHO declared the name of the new Covid-19 virus as Omicron. In India, on 2nd December the first case of Omicron was confirmed in Karnataka. As of 12th December 2021, there were 38 people were

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detected with the omicron virus in India. The U.K. declared the first death due to the omicron variant of covid -19 on 13th December 2021. Fully vaccinated people should require the booster dose for preventing Covid19 virus transmission. The contribution of the study is that stability analysis is performed in both analytical and graphical ways by using the data available in the world meter. The behavior of the SVIR model is explained in mathematically using a nonlinear system of ordinary differential equations and its solutions are performed by MATLAB simulations. The rate of transmission, fatality, rate of vaccinations, and rate of recovery from the susceptible people are considered in this model. Our results were compared with other countries’ results like the USA, and the U.K. the model constructed in this chapter is the extension of standard SIR models including a new compartment for vaccination. That is essential for SARS COVID-19 transmission dynamics. Our model uses continuous vaccination. In this paper, we try to understand the transmission behavior of COVID-19 in the populations with a briefing assessment of the transmission rate fatality rate vaccination rate, and the recovery rate in the bad effect of vaccination in India as well as the world. This study determines the potential anti corena virus vaccine. This shows eradication success depends on the vaccine and its coverage used. The remainder of this paper is organized as follows in Sect. 2 a mathematical approach is described briefly. In Sect. 3 we analyze the stability of equilibriums and their behaviors. Mathematical results are explained by numerical simulations and control techniques are given in Sect. 4. Lastly, we summarise the work and propose the future scope.

2 Mathematical Hypothesis To explain the mathematical model for the transmission of diseases we consider a deterministic compartmental modeling approach. We consider the total number of population N can be divided into four sub-populations or groups susceptible, infected, vaccinated, and recovered populations. Because of COVID-19 Vaccine is available in the hospital so the transition of the vaccinated and susceptible population and the infected population is given at the rate γ and ε respectable. Since the efficacy of the vaccine is not a hundred percent so the population was infected after taking the vaccine. The influxes of the population are at the rate A getting susceptible. The susceptible population was in contact with the infected population get infectious at the rate β. We assume that the rate of infection is very less after getting the vaccination. Those people who were infected were treated with hospitalization or isolation getting recovery at the rate α. The death rate considers as d2 due to COVID19. The people were suffering from severe diseases like sugar, heart, blood pressure, and low immunity power their death/ natural death rate consider as d1. We consider the recovery population died at the rate of d1 and the susceptible at the rate δ. The assumption of the model we represented a system of nonlinear systems of the ordinary differential as in Fig. 1 and its equation.

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Y. S. Rao δR

Fig. 1 Diagrammatic representation of the model A

S

βSI

d1S

αI

I (d1+d2)I

γV

V

R d1R

εV

(d1+d2)V

3 Mathematical Model for Vaccination Epidemic Model 3.1 The Mathematical Equation of the Model dS = A − d1 S − β S I + γ V + δ R dt dI = β S I + εV − (α + d1 + d2 )I dt dV = −(γ + ε + d1 + d2 )V dt dR = α I + (δ + d1 )R dt

(1)

3.2 Mathematical Analysis Throughout the paper we assume that the initial conditions of the system (1) are nonnegative: S(0) = S0 ≥ 0, I (0) = I0 ≥ 0, V (0) = V0 ≥ 0, R(0) = R0 ≥ 0

3.3 Positivity and Boundedness of the Solutions In this section, we shall define a region within which the solution of the model is uniformly bounded as the set  ∈ 4+ for the populations. The total population is defined as

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N (t) = S(t) + I (t) + V (t) + R(t) Hence

dN dt

=

dS dt

+

dI dt

+

dV dt

+

dR dt

dN = A − d1 N − d2 (I + V ) dt dN ≤ A − d1 N ⇒ dt dN ⇒ ≤ dt A − d1 N Integrating both sides we have 

 dN ≤ dt A − d1 N ln(A − d1 N ) ≤t +C ⇒− d1

where C is a constant of Integrations ln(A − d1 N ) ≥ −(d1 t + C) Thus . ⇒ (A − d1 N ) ≥ C1 e−d1 t Where C1 is a constant of integrations. Since N (0) = N0 then (A − d1 N0 ) ≥ C1 . Accordingly, (A − d1 N ) ≥ (A − d1 N0 )e−d1 t A (A − d1 N0 )e−d1 t − d1 d1 A ⇒ N (t) = ; t → ∞ d1 ⇒ N (t) ≤

  Thus N (t) ∈ 0, dA1 . Hence the feasible set of the  enter and remains  solution of the model equations A 4 in the invariant regions:  = (S, I, V, R) ∈ + : N (t) ≤ d1 .

4 Basic Reproduction Number In an emerging epidemic, the basic reproduction number is usually useful as a measure of the strength of the control measure needed to break the epidemic. This number is the spectral radius of the next-generation matrix. According to the nextgeneration matrix, one first identifies the infected subsystem of the model. These are

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the set of equations that describe the new infection and change in the population of the system. The infected population is linearised about the infection-free equilibrium forming the Jacobian Matrix. This Jacobian matrix is decomposed as J = T − K. Where T is the rate of appearance of new infection in the population and K represents the rate of transformation of the population into the population. The matrix T and K are obtained as ⎛ ⎞ β S0 0 0 T = ⎝ 0 0 0⎠ 0 00 ⎛ ⎞ −ε 0 (α + d1 + d2 ) ⎠ K =⎝ 0 0 (ε + d1 + d2 ) −α 0 (d1 + δ)) ⎛ Lastly K −1 =

−ε 0 (α+d1 +d2 )(ε+d1 +d2 ) ⎜ 1 0 0 ⎝ (ε+d1 +d2 ) −α αε 1 −α (α+d1 +d 2 )(d1 +δ) (α+d1 +d2 )(ε+d1 +d2 )(d1 +δ) (d1 +δ) 1 (α+d1 +d2 )

⎞ ⎟ ⎠.

Thus the next-generation matrix is ⎛ ⎜ T K −1 = ⎝

−β S0 ε β S0 (α+d1 +d2 ) (α+d1 +d2 )(ε+d1 +d2 )

0 0

0 0

⎞ 0 ⎟ 0⎠ 0

The eigenvalues of the next-generation matrix are ρ=

β S0 (α+d1 +d2 )

 00

Hence the spectral radius is ρ(T K −1 ) = R0 =

βA β S0 = . (α + d1 + d2 ) d1 (α + d1 + d2 )

5 Stability Analysis In this section, we can discuss two equilibrium points of the epidemic model. i.e. Corena free equilibrium point and endemic equilibrium point. These two equilibrium points are based on the existence of the Corena virus in the population over continuous time. The Corena-free equilibrium point is the point at which the corena virus does

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not spread in an area because the infected population equal to zero during the time. While the endemic equilibrium point is the point at which the virus must spread in a defined area. The Corena Free equilibrium (CFE) of the model is given by E 0 = (S0 , I0 , V0 , R0 ) = (S0 , 0, 0, 0)

The endemic equilibrium of the model is given by Aε(d1 + δ) + γ (d1 + δ)(d1 + d2 + α)(1 − R0 )I ∗ + εδα I ∗ ε(d1 + δ)(β I ∗ + d1 ) (d1 + d2 + α)(1 − R0 )I ∗ V∗ = ε ∗ α I R∗ = (d1 + δ) S∗ =

Theorem 1 The CFE is locally asymptotical stable if the reproduction number, R0 < 1, and endemic equilibrium if it exists, is unstable. Proof According to the Next generation Matrix Principle ⎛

JC F E

⎞ −β S0 γ −d1 ⎠ = J (S0 , 0, 0) = ⎝ 0 β S0 − (d1 + d2 + α) ε 0 0 −(ε + d1 + d2 )

The eigenvalues of the Jacobian matrix J are λ1 = −d1 λ2 = −(ε + d1 + d2 ) λ3 = β S0 − (d1 + d2 + α) < 0 β S0 1. Proof In order proof the endemic equilibrium of the system we can linearization the Eq. (1) and the Jacobian matrix is given by E E∗ E = E ∗ (S ∗ , I ∗ , V ∗ , R ∗ ) ⎞ ⎛ −d1 − β S ∗ −β S ∗ γ δ ⎟ ⎜ β S ∗ − (d1 + d2 + α) ε 0 βI∗ ⎟ =⎜ ⎠ ⎝ 0 0 0 −(d1 + d2 + ε) 0 α 0 −(d1 + δ) The eigenvalues of E* are obtained by solving the equation det(E*-λI) = 0. Thus we have that χ1 = −(d1 + d2 + ε) which has negative real parts and the other three can be obtained by solving the cubic equation χ 3 + aχ 2 + bχ + c = 0. Where a = (d1 + β S ∗ − β S ∗ + (d1 + d2 + α) + (d1 + δ))   (d1 + β S ∗ )(d1 + δ) − (d1 + δ)(β S ∗ − (d1 + d2 + α)) b= −(d1 + β S ∗ )(β S ∗ − (d1 + d2 + α) + αε − β 2 I ∗ S ∗ .   −(d1 + β S ∗ )(d1 + δ)(β S ∗ − (d1 + d2 + α) c= +(d1 + β S ∗ )αε + β 2 I ∗ S ∗ (d1 + δ) − β I ∗ αδ Clearly a.b > c. So by the Raurth-Hurtwz condition, endemic equilibrium is locally asymptotic stable.

6 Global Stability for Infection-Free Equilibrium 4 Theorem 3 The disease-free equilibrium is global asymptotical stable in R+ , if R0 < 1.

Proof Consider the Lyapunov function of the form

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237

1 I (α + d1 + d) dM 1 dI = ( ) dt (α + d1 + d2 ) dt dM 1 = (β S I − (α + d1 + d2 )I + εV ) dt (α + d1 + d2 ) βSI (α + d1 + d2 )I εV dM = − + dt (α + d1 + d2 ) (α + d1 + d2 ) (α + d1 + d2 ) βS εV dM =( − 1)I + dt (α + d1 + d2 ) (α + d1 + d2 ) εV dM = (R0 − 1)I + dt (α + d1 + d2 ) εV dM ≤ (R0 − 1)I + dt (α + d1 + d2 ) dM ≤0 dt

M= ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒

If R0 < 1 then ddtM ≤ 0. Hence by [23, 24] Maximum invariant principle, the E0 is globally asymptotically stable in the region.

7 Convergence Theorem for Lyapunov Stability Theequilibriumpoint xeq ∈ Rn is Lyapunov stability if for all xeq ∈ ε > 0,∃δ >  0 : x(t0 ) − xeq  ≤ δ⇒ x(t) − xeq  ≤ ε, ∀t ≥ t0 ≥ 0. If the solutions start close to xeq it, they will remain close to it forever and ε can make arbitrarily small by choosing δ sufficiently small. This theorem established the global asymptotically stable if it is Lyapunov stable and for every initial state the solution exist [0, ∞) s on and x(t) → xeq as t → ∞.

8 Global Stability for Endemic Equilibrium In this part, we will prove the global stability of the endemic equilibrium using the [25] Pinocare Bendixson principle. For proofing the Bendixson principle we can assume some assumptions. Consider a function f (x) :  ⊆ R n → R n , where  is an open set and its differential ddtx = f (x) determines the solution for every value of x(t) with its initial conditions x(t) = × 0. Therefore the equilibrium points satisfy some conditions:

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1.  is simply connected. 2. This open set  exists a compact absorbing sub-set K of . 3. This equilibrium point in the open set  is globally stable when it satisfies the t − condition given by q2 = lim sup q < 0, where q = (Z (x(s, x0 )))ds, such t→∞ x ∈k 0

0

that the Bandixson matrix Z = (C f C −1 + C J [2] C −1 ) and (Z ) ≤ 0. Here represent Loziniskii Measure and J[2] denotes the second additive compound matrix J. Theorem 4 The system of Eq. (1) is globally stable in a given region when R0 > 1. Proof For finding the global stability at the unique endemic equilibrium points (S*, I*, V*) of the system (1) leaving the Recovery class the jacobian matrix can be represented as ⎛

⎞ −(d1 + β I ) −β S γ ⎠ J =⎝ ε βI β S − (α + d1 + d2 ) 0 0 −(ε + γ + d1 + d2 ) Since J ∈ R3X 3 so its second additive matrix J [2] becomes J [2] = ⎛ ⎞ β S − (α + 2d1 + d2 + β I ) ε −γ ⎜ ⎟ −β S 0 −(2d1 + d2 + γ + ε + β I ) ⎝ ⎠ 0 −β I β S − (2d1 + d2 + γ + ε + α)

For obtaining the Bandixson matrix Z we can represent a diagonal matrix C = ⎛ ⎞ 1 0 0 C(S, I, V ) = ⎝ 0 VI 0 ⎠ = Diag(1, VI , VI ) 0 0 VI In the system the vector field f then 

CfC

−1



I = Diaog 0, V

  f

     V V I , I V f I

where 

⎛

Then C f C −1

I V

0 0 / ⎜ 0 II − =⎜ ⎝ 0 0

  f

V I

 = 0

 /

V V



0 I/ I

−

I/ V/ − I V ⎞

⎟ ⎟  ⎠. /

V V



Implementation of Vaccination in an Epidemic Model for COVID-19

239

So the Bandixson matrix is defined as Z = (C f C −1 + C J [2] C −1 ) = ⎛ ⎜ Z =⎜ ⎝

β S − (α + 2d1 + d2 + β I )



Z 11 Z 12 Z 21 Z 22

ε VI

0

/ / β S − (α + d1 + d2 ) + II − VV

0

−β I

 ⎞

γ VI −β S

/ / β S − (2d1 + d2 + γ + ε + α) + II − VV

⎟ ⎟ ⎠

where

 Z 22 =

Z 11 = (β S − (α + 2d1 + d2 + β I ))   Z 12 = ε VI γ VI   0 Z 21 = 0 / / β S − (α + d1 + d2 ) + II − VV −β S / / −β I β S − (2d1 + d2 + γ + ε + α) + II − VV

The matrix Z can be calculated as (Z ) ≤ sup{y1 , y2 }. Again above two values are V I I/ V/ y2 = |Z 21 | + (Z 22 ) = β S − (2d1 + d2 + γ + ε + α) + − I V y1 = (Z 11 ) + |Z 12 | = β S − (α + 2d1 + d2 + β I ) + γ

Hence I/ V ≤ − d1 I I V/ I/ I/ − ≤ − d1 y2 = β S − (2d1 + d2 + γ + ε + α) + I V I y1 = β S − (α + 2d1 + d2 + β I ) + γ

Thus. (Z ) ≤ sup{y1 , y2 } ≤ Therefore

I/ I

− d1 .

t (Z )dt ≤ ln I (t) − d1 t 0

Finally, we obtain as



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Y. S. Rao

t − q2

=

(Z )dt

0

t

≤

ln I (t) − d1 < 0. t

This satisfies the Benxdison principle. These results show the global stability of the endemic equilibrium.

9 Simulation and Discussion of Results This figure indicates the susceptible populations recovered after giving the vaccination doses. The rate of susceptibility decreases and the recovered rate increases simultaneously concerning time (Figs. 2 and 3). It explains that susceptible populations get infectious with low immunity in the body. The infected population increases due to not taking the vaccine and also having other diseases. Figure 4 explains the vaccination program during the initial stage of the epidemic goes steadily. Initially, the infected population was very high during the 1st phase of vaccination programs. It indicates the effective vaccination given to the people to control the epidemic diseases (Fig. 5). This figure shows the dynamic behavior of the population. It is clear that when the reproduction number is less than unity, it is locally stable and shows the asymptotical behavior of the population observed (Fig. 6). The optimal control has been shown in the above figure. The more vaccine is less infected concerning time is observed in the real parameter. It reveals the level of infection the population reaches after a certain time the vaccination used as a control measures (Fig. 7).

Fig. 2 Susceptible population versus recovered population in the CFE point

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Fig. 3 Susceptible population versus infected population

Fig. 4 Infected population versus vaccinated population

This figure indicates at the endemic equilibrium point the vaccination program has been considered at the early time the infected population would have been significantly lower and the outbreak of covid-19 would be extinguished. That leads to the death rate would decrease (Fig. 8). The recovery of the population is very high and the vaccination program should implement during this period. When people get the vaccination and are free from other diseases the infected population getting more recovery from COVID-19. This

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Fig. 5 Behaviour of the populations

Fig. 6 Infected population versus vaccination concerning the time

figure shows the recovered population increases when more people get the vaccine in two doses (Tables 1 and 2).

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Fig. 7 Susceptible population versus recovered population

Fig. 8 Vaccination population versus recovered populations concerning the time Table 1 CPU time for different values t and n

Table 2 CPU time for different values t and n

t

n

0.1

1000

0.123

0.01

10,000

0.912

0.001

100,000

17.132

Times (Days)

t

n

0.1

2000

0.1

20,000

0.1

200,000

27.132

0.1

20,000,000

34.234

Times (Days) 0.923 16.612

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10 Limitations of the Model There were some drawbacks to this investigation. Unreported cases were not taken into account during the vaccination period. A booster dose is not mentioned in the vaccination. The reality of today’s seniors does not show any evidence of disease control. Due to some confounding variables, the amount of covid-19 vaccination coverage cannot be calculated. The uncertainty could not be taken into account when estimating the outcomes. inherent effects of vaccination. As a result, the model creates the best current method for available doses and its consideration of controlling infectious diseases.

11 Conclusion and Future Scope Here explore the SVIR model with variable infection rates for investigating the action of virus propagation. Based on this model a control parameter basic reproduction number R0 that determines the local and global dynamic behaviour of the disease propagation has been obtained by mathematical analysis. The virus may die out in the population when basic reproduction is below unity and they will be prevalent otherwise. Also using the Bendixson’s principle and Lyapunov function to obtain global stability for infection-free and endemic equilibrium. Further numerical simulations verify the correctness of the theoretical analysis. More importantly, investigate the effect of infection rate after the vaccinated population gets prevalence the peoples. When providing some booster dose or a powerful effective vaccine according to simulating results. This analysis can provide some insight into covid-19 countermeasures. In the real world, the result can help vaccine companies or related organizations to make a powerful effective vaccine or booster dose countermeasure to work well. In the future time delay, the existence of an invariant distribution and erotic property problem will study for preventing the diseases.

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