Advances in Epidemiological Modeling and Control of Viruses

By Hemen Dutta

Advances in Epidemiological Modeling and Control of Viruses covers recent and advanced research works in the field of epidemiological modeling, with special emphasis on new strategies to control the occurrence and reoccurrence of viruses. The models included in this book can be used to study the dynamics of different viruses, searching for control measures, and epidemic models under various effects and environments. This book covers different models and methods of modeling, including data-driven approaches. The authors and editors are experienced researchers, and each chapter has been designed to provide readers with leading-edge information on topics discussed.

• Includes models to describe global and local dynamics of various viruses
• Provides readers with control strategies for occurrence and reoccurrence of viruses
• Includes epidemic models under various effects and environments
• Provides readers with a robust set of mathematical tools and techniques for epidemiological modeling

• Language: English
• Publisher: Academic Press
• Release date: Jan 6, 2023
• ISBN: 9780323995580

Book preview

Preface

Hemen Dutta; Khalid Hattaf     Guwahati, India

Casablanca, Morocco

The book covers various aspects of epidemiological modeling, with a focus on virus control strategies. Readers can expect new and significant research information about the dynamics of various diseases under various effects and environments. The book should be a valuable resource for graduate students, researchers, and educators interested in epidemiological modeling, including data-driven models. The chapters are organized as follows.

Chapter "Global dynamics of a delayed reaction-diffusion viral infection in a cellular environment" discusses a reaction-diffusion equation with a general non-linear production term and a delayed inhibition term. Some methods for proving global convergence of solutions to the positive homogeneous in space equilibrium under certain conditions are presented. When these conditions are not met and the solution becomes unstable, numerical simulations have been used to study complex nonlinear dynamics and pattern formation.

Chapter "Hepatitis B virus transmission via epidemic model with treatment function" investigates an epidemiological mathematical model that demonstrates the dynamics of hepatitis B virus transmission under the influence of generalized incidence and treatment function. It is also emphasized to describe the transfer mechanism, i.e., preventive vaccination of susceptible populations; and various control mechanisms based on the size of the infective population, in which the control measure treatment can determine whether or not there is an epidemic outbreak, as well as the number of endemic equilibrium during endemic outbreaks. Numerical simulations were also performed to graphically represent the analytical findings.

Chapter "Global dynamics of an HCV model with full logistic terms and the host immune system" studies a mathematical model that describes the dynamics of the hepatitis C virus (HCV) while taking four populations into account: uninfected liver cells, infected liver cells, HCV, and T cells. It establishes the existence of two equilibrium states: the uninfected state and the endemically infected state, as well as the positivity and boundedness of the system solutions. It employs a geometrical approach to investigate the global stability of positive equilibrium. It also includes some numerical simulations to justify the analytical findings.

Chapter "On a Novel SVEIRS Markov chain epidemic model with multiple discrete delays and infection rates: modeling and sensitivity analysis to determine vaccination effects" investigates a novel discrete time general Markov chain SEIRS epidemic model with vaccination. The model includes finite delay times for disease incubation, natural and artificial immunity periods, and infected individuals' infectious period. The novel platform for representing the various disease states in the population employs two discrete time measures for the current time of a person's state, as well as how long a person has been in the current state. There are two sub-models based on whether the motivation to get vaccinated is motivated by close contact with infectious individuals or not. To determine how vaccination affects disease eradication, sensitivity analysis is performed on the two sub-models.

Chapter "Hopf bifurcation in an SIR epidemic model with psychological effect and distributed time delay" aims to replicate the recurrent epidemic wave of COVID-19 by investigating an SIR epidemic model with psychological effect and distributed time delay. It obtains the index values that determine the Hopf bifurcation characteristics for general distribution functions using the theory of center manifold and Poincaré normal form. As an example, it considers the truncated exponential distribution and demonstrates how the periodic solution can emerge via the Hopf bifurcation.

Chapter "Modeling of the effects of media in the course of rotavirus vaccination" investigates the influence of media cognizance and its impact at various points along the path of rotavirus vaccination. It investigates a model of the development of rotavirus diarrhoea based on time-based ordinary differential equations. It used numerical simulations to investigate the impact of vaccination as a preventive measure.

Chapter "Mathematical models of early stage COVID-19 transmission in Sri Lanka" discusses several mathematical models developed to investigate the COVID-19 transmission dynamic in Sri Lanka. An SIER compartmental model is used, and the optimal initial parameters of the early stage of the outbreak are estimated by calibrating reported cases using an optimization algorithm. Scenario-based control measures are also introduced into the model at the parameter level, and their impact is evaluated using numerical simulations. Then, an optimal control model incorporating a SIER type of model is developed, taking into account the heterogeneity of cases such as asymptomatic, symptomatic with mild indications, and cases requiring intensive care treatments. All of the measures and interventions are being implemented at a significant social and economic cost; therefore, optimal control techniques have been used to identify the most appropriate strategies to reduce this cost.

Chapter "Global stability of a diffusive HTLV-I infection model with mitosis and CTL immune response" proposes a spatially dependent HTLV-I infection model. The model describes the interactions between uninfected CD4+T cells, latent HTLV infected cells, Tax-expressing HTLV-infected cells, and HTLV-specific CTLs within the host. The solutions' well-posedness, including the existence of global solutions and boundedness, is justified. It determines the existence and stability of the model's three steady states by calculating two threshold parameters: the basic infection reproduction number and the HTLV-specific CTL mediated immunity reproduction number. It also investigates the global stability of all steady states using suitable Lyapunov functions and the Lyapunov-LaSalle asymptotic stability theorem. The validity of theoretical results is justified using numerical simulations.

Chapter "Mathematical tools and their applications in dengue epidemic data analytics" aims to introduce mathematical tools that are powerful in analyzing time-dependent disease transmission data in the context of external forces such as climate variability. It considers the Fast Fourier Transformation (FFT) method, which converts time or space domain data series to frequency domain data series by decomposing a sequence of data into components of different frequencies. The wavelet theory is then applied to noisy epidemiological time series data. The FFT analysis makes use of reported dengue and climate data. Certain dengue weekly incident data are used for the wavelet analysis. Certain temperature and rainfall data were obtained in order to investigate the patterns of dengue transmission in relation to climate. FFT and wavelet analysis are both carried out using structure computer algorithms to investigate periodic patterns of dengue incidents in relation to external variables and to obtain spectral properties of time series data.

Chapter "COVID-19 pandemic model: a graph theoretical perspective" emphasises that the COVID-19 virus spread (pandemic) pattern can be analyzed using graph theory. Each vertex in the network represents an individual at any stage of infection (asymptomatic, pre-symptomatic, or symptomatic), and edges represent transmissions from person to person. It looks into the spread of COVID-19 among those who are susceptible, exposed, infected, recovered, and dead. It takes into account each individual's neighborhood prevalence, i.e., the proportion of each individual's contacts who are either exposed or infected, and introduces certain parameters. Furthermore, it proposes a threshold value and describes the effects of this value on pandemic spread.

Chapter "Towards nonmanifest chaos and order in biological structures using the multifractal paradigm" investigates nonlinear behaviors of biological structures (viruses systems) in Schrödinger type regimes at various scale resolutions in a multifractal motion paradigm. The functionality of a hidden symmetry of SL(2R) type then implies, via a Riccati type gauge, different synchronization modes among these virus systems in the stationary case of these regimes. Furthermore, assuming that nonmanifest chaos is not present, specific patterns corresponding to virus system dynamics can be highlighted. In such a framework, using artificial intelligence methods, it is demonstrated that, based on the dynamics of certain patterns, changes in the acoustic field can constitute a method of COVID-19 detection.

Chapter "Global stability of epidemic models under discontinuous treatment strategy" takes a look at the SIR, SIS, and SEIR epidemic models under discontinuous treatment. In each compact interval, the treatment rate has a finite number of jump discontinities. The basic reproductive number is demonstrated to be a sharp threshold value that completely determines the dynamics of the model using Lyapunov theory for discontinuous differential equations and other techniques on non-smooth analysis. It discusses how the disease will die out in a finite amount of time, which the corresponding model with continuous treatment cannot do. Furthermore, the numerical simulations show that increasing treatment after infective individuals reach a certain level is beneficial to disease control.

We are grateful to the contributors for their cooperation throughout the book's development process. The reviewers deserve a much appreciation for their voluntary service in making this book a success. We would like to express their gratitude to their family members, friends, and well-wishers for their ongoing support in developing such books. Elsevier editors and staff also deserve our appreciation for their timely assistance and support.

10 August, 2022

1: Global dynamics of a delayed reaction-diffusion viral infection in a cellular environment

Mohammed Nor Friouia,b; Tarik Mohammed Touaoulaa    aLaboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées, Dépt. de Mathématiques, Université Aboubekr Belkaïd, Tlemcen, Algeria

bFaculty of Mathematics, University of Sciences and Technology Houari Boumedienne, Algiers, Algeria

Abstract

Reaction-diffusion equation with a general nonlinear production term and a delayed inhibition term is studied. Global stability of the homogeneous in space equilibrium is proved under some conditions on the delay term. In the case where these conditions are not satisfied, this solution can become unstable, resulting in the emergence of spatiotemporal pattern formation studied in numerical simulations.

Keywords

Delay reaction-diffusion equation; Global convergence; Spatiotemporal pattern formation

Acknowledgements

The authors are partially supported by DGRSDT, ALGERIA, project PRFU, code C00L03UN130120200004. They are grateful to the anonymous reviewer for his∖her suggestions.

1.1 Introduction

Infections are usually initiated by a few virions that are individual viral particles. They enter healthy cells and modify the genetic structure of their hosts. Then they replicate and produce a swarm of progeny viruses. During the process of viral infection, the immune response is critical in controlling the transmission of disease. The interaction of the immune cells with a target population of viruses must be considered as a dynamic process. The mathematical modeling of such a process plays an important role in understanding the population dynamics of viruses (see, e.g., the recent papers [1], [2], [3], [7] [8], [9] and the references therein).

Another interesting aspect of the dynamics of the spread of a viral infection is related to the spatial structure of the system and the effects of its processes, such as random dispersal of virions. It was showed, in many works, that the spatial structure of the system may influence the dynamics of the population under consideration.

Mathematical models have been extensively used to study the dynamics of viral infections, taking into account the immune cells mostly under the simplifying assumptions of spatial homogeneity, with a few models considering the spatial spread of viruses in infected hosts.

When a spatial process is taken into account, reaction-diffusion equations represent an appropriate framework to describe the evolution of the dynamic of viruses and immune cells. Many reaction-diffusion equations are formulated to investigate the roles of diffusion on the transmission of disease.

Motivated by the description made, in this chapter we aim to study the following reaction-diffusion equation with delay:

(1.1)

The first term in the right-hand side of Eq. (1.1) describes virus diffusion, the second one represents virus production, and the last term models virus elimination by immune cells. The parameter D is the diffusion coefficient. The function stands for the number of immune cells generated after some time-delay τ; this means that the production of immune cells depends on the concentration of viruses some time before. In our context, we consider the concentration of immune cells as a function of the virus concentration a τ time ago; that is, , where is the concentration of the immune cells at space x and at time t. Eq. (1.1) was suggested in [8], [9] for the particular case as a model of viral infection spreading in tissues, but not only; it can also be considered as a generic model of population growth with delayed inhibition (see also [7], [8], [9]).

This simple equation in appearance can have solutions with very complex behavior: even when , chaos can appear as established in [6].

When , under distributed delay, Eq. (1.1) can have many very known forms [4], [5], [12], [15], [17], [19], [34].

In [5] a unimodal function f (function having exactly one maximum) is considered for the study of

(1.2)

The authors succeed in proving the global convergence of solution to the unique positive equilibrium of (1.2).

In [34] the authors consider the equation

(1.3)

and they obtain a complete analysis of the global behavior of the solutions.

In addition to global asymptotic stability of the unique positive equilibrium, the author in [19] obtains an exponential stability result for the equation

(1.4)

When , so for a delay reaction-diffusion, there is a huge interest, as this kind of equations can appear in models describing predator-prey interaction, population dynamics, and more precisely in very well-known Nicholson's blowflies and Mackey-Glass models. Some solutions of such problems can take the form of waves connecting different equilibria.

There is a vast literature devoted to delay reaction-diffusion equations (see [6], [10], [20], [21], [23], [24], [25], [27], [28], [29], [30] [31], [32], [35] and the references therein). Most of them consider models in population dynamics. Let us mention the work [30] where the authors studied the following delay reaction-diffusion equation:

(1.5)

with Neumann boundary conditions and unimodal function f. It is proved that the convergence to the unique positive steady state of (1.5) is strongly linked to the convergence of the sequence determined by the difference equation . We also mention the work [33] where the authors investigated a nonlocal reaction-diffusion equation in a semiinfinite interval and obtained the global attractivity of the nontrivial equilibrium.

Investigation of delay reaction-diffusion equations is often personalized in the sense that the methods of their analysis should be adapted to the particular form of the equation. In this work, we develop some methods to prove global convergence of solutions to the positive homogeneous in space equilibrium under certain conditions on the function f (Section 1.2). In the case where these conditions are not satisfied and this solution becomes unstable, we use numerical simulations to study complex nonlinear dynamics and pattern formation (Sections 1.3 and 1.4).

1.2 Global convergence to the homogeneous solution

In this section, we study the following class of functional differential equations:

(1.6)

Throughout this chapter, we will make the following assumptions:

(T1) f and g are with g is a bounded function on and .

(T2) There exists such that , and for all .

Let and be equipped with the usual supremum norm . Also, let and . For any , we write if ; if .

A function v is Hölder continuous with exponent on if there exists a constant such that

We write in this case .

We define the ordered intervals

and for any , we write for the element of X satisfying for all . The segment of a solution is defined by the relation for and . The family of maps

such that

defines a continuous semiflow on [26]. The map is defined from to , which is the semiflow denoted by

The set of equilibria of the semiflow generated by (1.6) is given by

Let ( ) be a strongly continuous semigroup of bounded linear operators on C generated by the Laplace operator Δ under periodic conditions. It is well known that ( ) is an analytic, compact, and strongly positive semigroup (see example 1.13, page 26 in [26]). Define by

(1.7)

We consider the following integral equation with the given initial data:

(1.8)

For each , with values in C on its maximum interval is called a mild solution of (1.6) (for the existence and uniqueness of this solution, see for instance [11], [13], [14], [26]), and it is called classical if it is in with respect to x and in with respect to t.

Definition 1.2.1

We call a pair of smooth functions sub- and supersolutions of (1.6) if it satisfies the following properties:

i)  

for all . Denote by the set of all functions that are once continuously differentiable in t and twice continuously differentiable in x for all .

ii)   for .

iii)  The functions verify the following problems:

(1.9)

and

(1.10)

Lemma 1.2.2

Suppose that a smooth function w satisfies the following problem:

(1.11)

for some bounded function . Then in . Moreover, in provided is not identically null.

The proof of this result is well known (see for instance [18]). Set and the following problems:

(1.12)

and

(1.13)

Lemma 1.2.3

Suppose that f is nondecreasing and a smooth pair exists. Then we have the following inequalities:

Proof

We set . Then w satisfies the following problem:

From Lemma 1.2.2, we get . Similarly, we can show that . Next, we set . Then, we have

Hence,

where θ is a function that lies between and . Since f is a nondecreasing function and , we get

Choosing and using Lemma 1.2.2, we obtain . The assertion of the lemma can now be obtained by induction. □

Theorem 1.2.4

Suppose that f is a nondecreasing function, and let , be coupled super- and subsolutions of problem (1.6). Then the sequences , given by (1.12)–(1.13) converge monotonically to a unique solution u of (1.8) and in .

Proof

From Lemma 1.2.3 we conclude that the sequences , converge to some limits and , respectively, and . By a classical regularity theorem for parabolic equation (see for instance [16]), these limits satisfy the equations

(1.14)

and the boundary and initial conditions in (1.6). Hence the limits of , are solutions of (1.6) if in . We begin by choosing . Subtracting the equations in (1.14) and using the mean value theorem, we see that the function satisfies the relation

and the same boundary conditions as in (1.6). The function satisfies

and the same boundary conditions as in (1.6) with

(after a suitable choice of ).

Next, we multiply the above equation by v and integrate over . We obtain

Thus, in view of the equality , we conclude that

Therefore, by the continuity of w, we get for all . We iterate on ,..., , and we show the existence of a solution. We proceed similarly to prove the uniqueness of the solution . □

Lemma 1.2.5

Assume (T1)–(T2) hold. Then, for , problem (1.8) admits a unique solution u. In addition, we have the following results:

(i)  There exists such that for all and . Moreover, for all with B defined in (T2).

(ii)   for all . Furthermore, if is not identically zero in , then

where is the interior of the set X.

(iii)  The semiflow admits a compact attractor and is a classical solution of(1.6)for and .

Proof

We first prove that each solution of (1.8) is bounded. Indeed, the function u solution of (1.6) is a subsolution of the following problem:

(1.15)

The function v satisfies

(1.16)

with c chosen such that is a nondecreasing function over . Hence, by the comparison principle (see Theorem 8.1.10 in [26]), we have for all and . Next, (B being defined in (T2)) is a supersolution of (1.16), then for all . Therefore for all and . We claim now that . Indeed, suppose that the contrary holds, then . From the fluctuation method (see for instance Proposition A.22 in [22]), there exists a sequence , as such that as and . Substituting in (1.15) and passing to the limit, we get , which is a contradiction to (T2). The claim is proved. This completes the proof of statement (i). Now, note that for all and for all small . This implies that . Furthermore, maps into . By Theorem 8.3.1 and Remark 8.3.2 in [26], we know that . Moreover, for each , problem (1.6) may be rewritten as

with , where θ is a function lying between 0 and u. Since ϕ is nonnegative and not identically null, in view of Lemma 1.2.2, we have for all and , (ii) is proved. Finally, statement (iii) follows from (i), (ii), and Theorem 2.2.6 in [26]. □

We consider the following problem:

(1.17)

Suppose that G satisfies the one-sided Lipschitz condition

(1.18)

where c is a nonnegative function and is any pair of ordered lower and upper solutions of (1.17).

Lemma 1.2.6

Let , be ordered bounded positive sub- and supersolutions of (1.17), respectively, and suppose that (1.18) holds and that is either a decreasing or increasing function for . Then problem (1.17) admits a unique positive solution .

Proof

The existence of the minimal solution and the maximal solution of problem (1.17) such that can be deduced by Theorem 3.2.2 in [16]. (The same proof can be applied when Neumann conditions are replaced by periodic ones.) Now we claim that . Indeed, we have

and

Subtracting these two equations and integrating the result over , we get

By integration by parts and from the periodic conditions, we obtain

so,

It follows from and the monotone property of that

and thus . □

Let be a solution of the stationary problem associated to (1.6); namely,

(1.19)

The following result is a direct consequence of Lemma 1.2.6.

Lemma 1.2.7

Suppose that f is an increasing function and is a decreasing function. Then problem (1.19) admits at most one positive solution.

In the subsequent part of this section, we suppose that there exists such that

(1.20)

Corollary 1.2.8

Under the hypotheses of Lemma 1.2.7 and (1.20), problem (1.19) admits a unique positive solution.

Next, we study the persistence of solutions of problem (1.6). First observe, from (T2), that there exists such that .

Definition 1.2.9

We say that u solution of (1.6) is strongly persistent in if there exists such that

for all and for all .

Lemma 1.2.10

Suppose that f is an increasing function and either for all or is a nonincreasing function. Assume also that , then the solution of (1.6) is strongly persistent provided .

Proof

Since , there exists such that . First, suppose that . Then, is a couple of sub- and supersolutions of (1.6). Indeed, it is sufficient to note that relations (1.9)–(1.10) are verified for the couple . For the general case, where without loss of generality is nonidentically zero, we conclude from Lemma 1.2.5 that for all . Then for all . Hence, if we consider u in as the initial condition, then, as before, this last implies that for all . To finish the proof, we will show that each positive solution of problem (1.6) enters the interval and stays there. For , we introduce the following