Project topics and materials
Get Complete Project Material File(s) Now! »
CHAPTER 3 SIS MODEL
Introduction
The aim of this chapter is to construct and analyze a reliable numerical method for solving an SIS model (where infection with the disease does not confer permanent immunity against re-infection so that those who survived the infection revert to the class of wholly-susceptible individuals [55]) with discrete time delay. Although the SIS model with time delay has been studied in the literature (see, for instance [25, 59, 60, 91, 104] and some of the refer-ences therein), some of the pertinent aspects of the analyses are unreported. As discussed in Chapter 1, disease transmission models are usually designed by splitting the total population of interest at time t, denoted by N(t), into mutually-exclusive epidemiological compartments based on the infection sta-tus of the members of the population. The simplest of such models take the form of an SIS model where N(t) is split into compartments of susceptible individuals (S(t)) and infected individuals (I(t)), so that N(t) = S(t) + I(t). Diseases, such as Meningitis, Gonorrhea, Influenza, Chagas, Rocky moun-tain, Malaria and Sleeping sickness, can be modeled using an SIS framework [54, 58]. Numerous classical SIS models, using different assumptions on de-mographic and incidence parameters,are widely studied in the literature in [58, 108, 119] and the references therein.
Infection occurs following effective contact between an infectious individ-ual and a susceptible individual. There is often a time lag (delay) before the newly-infected individual become infectious (typically at the end of the incu-bation period, when the infected individual displays clinical symptoms of the disease). In other words, time delay is used to represent a latent period (in-cubation period), maturation time or wearing time of immunity [25, 59, 104]. Compartmental models with time delay are known to generally exhibit com-plex dynamic of behaviour (including sustained oscillations associated with the (Hopf) bifurcation of an EE into a limit cycle [47, 57, 69, 113]). More-over, the explicit solutions of such models are formulated using systems of non-linear differential equations, whose exact solution, if at all exists, are very difficult (or impossible) to compute in close form. This necessitate the use of numerical methods, preferably, those that best approximate and repli-cate the basic properties of the governing continuous-time systems. Time delays have also been used in other biological and non biological studies, such as respiratory system [109] tumor growth [110], chemostat models [118] and neural networks [19].
The main objective of this chapter is two-fold. The first is to quali-tatively analyze the SIS delay differential equation system ((3.3.1)-(3.3.2)). The second is to construct a robust numerical method for approximating its solution. For the later objective, a reliable numerical method (NSFD scheme) for a linear delay differential equation is designed for solving some classes of epidemiological models (including an SIS delay model). It is shown, theoret-ically and computationally, that the NSFD scheme is dynamically consistent with respect to the asymptotic stability of the trivial equilibrium solution of the continuous time model. The NSFD scheme has been extended to a logistic epidemic model and finally to SIS delay model in a reliable manner.
Before studying the DDE system (3.3.1)-(3.3.2), it is instructive to study the non-delayed model (3.2.1), for comparison purposes. This is done in Section 3.2. Afterwards, in Section 3.3, we consider the complete qualitative and quantitative analysis of the SIS model with discrete delay. Section 3.4 is based on the construction of dynamically consistent NSFD scheme for SIS model with delay (starting, first, with setting down the foundation by analyzing a linear delay differential equation, followed by nonlinear logistic epidemiological delay logistic model). Numerical simulations are provided for each of these cases to illustrate the theoretical results derived.
SIS model without time dela
As stated earlier, the focus of this chapter is to study the dynamics of an SIS model with discrete time delay. However, to make the presentation self-contained, it is outlined, in this section, some of the key facts and notations that will later be used in future (regarding the SIS model without delay). The non-delayed SIS model is given by the following deterministic system of nonlinear differential equations [18, 55]:
where I Π is the constant recruitment rate of individuals by birth and immi-gration; β is the effective contact rate; µ is the natural death rate; δ is the disease-induced death rate; γ is the transition rate from the infected class to susceptible class. It can also stand for assumed rate of loss of infection-acquired immunity.
The term βSI/N is the incidence rate. It measures the average number of susceptible individuals infected by infectious person per contact per unit of time. The system (3.2.1) is to be solved subject to the following initial conditions:
S(0) = S0 ≥ 0, I(0) = I0 ≥ 0.
Nonstandard finite difference method
Although the SIS model is simple, it cannot be solved explicitly (in terms of elementary functions of S(t) and I(t)) owing to its nonlinearity. Thus, its solution has to be obtained numerically. In this section, we present an NSFD scheme which captures the essential qualitative properties of the SIS model (3.2.1). The equations of the model (3.2.1) are descritized in the following way (see [73]):
SIS model with discrete time delay
To account for effect of latency in disease transmission, the non-delayed SIS model (3.2.1) is extended to include time delay, denoted by τ > 0, resulting in the following delay differential equation (DDE) system [59, 91, 119]:
The parameters (noting the addition of τ > 0, the discrete time delay) and variables of the delayed SIS model (3.3.1) have the same meaning as those for the non-delayed SIS model (3.2.1). However, it is worthwhile noting henceforth that:
- I(t−τ) is the population of individuals who were infected at time (t−τ) and become infectious after τ units of time have elapsed;
- e−µτ is the probability that a newly-infected individual survives natural death (µ) during the time τ period, and become infectious.
Moreover, the initial conditions (3.2.2) must be replaced by S(t) = ψ1(t), I(t) = ψ2(t), t ∈ [−τ, 0].
Basic properties of continuous-time delayed system
As a consequence to the way the analysis of delay differential equations sys-tem are scattered, and in view of our future quest of constructing a dynami-cally consistent discrete delay scheme, it is of paramount importance to study the qualitative properties of the continuous model in full detail. We start by proving the well-posedness of the model (3.3.1)-(3.3.2), as below.
For the choice of q > LB, the operator T is a contraction. Hence, by Banach contraction principle, there exists a unique fixed point of T (and, thus, a unique local solution of the DDE system (3.3.1)).
Stage 2: The second stage of the proof is based on using Theorem 3.1 in [51]. We show that whenever the solution exists at some time t, it satisfies some apriori estimate, namely 0 ≤ S(t), 0 ≤ I(t) and S(t) + I(t) ≤ Πµ . This is specifically done in Theorem 3.3.2 and Theorem 3.3.3 below.
Stage 3: The final (third) stage is to establish the global existence result and is covered by Theorem 3.3.4 below.
The following result, which is needed in the proof of Theorem 3.3.1, re-flects a qualitative property of the solution.
Theorem 3.3.2 Whenever it exists, the solution S(t), I(t) of the DDE sys-tem (3.3.1) corresponding to the non-negative initial data (3.3.2), remains non negative for all t > 0.
Declaration
Acknowledgements
Abstract
1 Introduction
2 Mathematical preliminaries
2.1 Continuous-time dynamical systems
2.1.1 Stability of dynamical systems
2.1.2 Lyapunov function
2.2 Reproduction threshold
2.2.1 Next generation operator method
2.3 Bifurcations
2.3.1 Backward bifurcation
2.3.2 Hopf bifurcation
2.4 Discrete-time dynamical systems
2.4.1 Jury stability criterion
2.5 Nonstandard finite difference method
3 SIS Model
3.1 Introduction
3.2 SIS model without time delay
3.3 SIS model with discrete time delay
3.4 Towards the construction of NSFD scheme for delayed SIS model
4 Modeling transmission dynamics of BTB-MTB in humanbuffalo population
4.1 Introduction
4.2 Basic SEIR model for TB
4.3 A model for TB with exogenous reinfection
4.4 A model for TB with exogenous reinfection and two stage exposed classes
4.5 The model of tuberculosis in human-African buffalo population
4.6 Analysis of the BTB-MTB model
5 Conclusion and Future Work
5.2 Future work
Bibliography
GET THE COMPLETE PROJECT
