The impact of nonlinear relapse and reinfection to derive a stochastic threshold for SIRI epidemic model
Introduction
The 21st century has already been marked by major epidemics. Old diseases cholera, plague and yellow fever have returned, and new ones have emerged SARS, pandemic influenza, MERS, Ebola and Zika. Therefore, there is a lot of interest in both understanding diseases and mathematically modelling the spread of epidemics. Mathematical modeling of infectious diseases is a tool which has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak. Epidemics are commonly modeled by using compartmental models where the population is divided into several classes, such as susceptible, infected and removed individuals. The first complete mathematical model was introduced by Kermack and Mckendrick in 1927 [12]. Based on this work, many researches have studied the spread of infectious diseases in a population by compartmental models such as SIS, SIR, SIRS, SEIR or SVIS, see for instance [5], [9], [14], [22]. For some diseases such as tuberculosis and herpes, recovered individuals may relapse with reactivation of latent infection and revert back to the infective class [4]. These types of diseases can be modeled by SIRI models. Tudor (1990) was the first to construct and study a compartmental epidemic model with relapse, which incorporates bilinear incidence rate and constant total population. Sanchez et al. [19] introduced a SIRI epidemic model with nonlinear relapse to describe the dynamics of drinking behaviors generated from contacts between individuals in shared drinking environments. The model is given by the following set of nonlinear differential equations: where the population is divided in three classes: S(t), moderate and occasional drinkers, I(t), problem or heavy drinkers, and temporarily recovered, R(t). The parameter β denotes the effective contact rate (infection rate). That is, βS(t)I(t) denotes the rate of transitions from S to I, the result of the frequency dependent interactions between individuals in the classes S and I, μ denotes the recruitment rate of susceptible corresponding to births and immigration, δ denotes the relapse rate. That is, δR(t)I(t) denotes the rate of transitions from R to I, the result of the frequency-dependent interactions between R and I, this nonlinear process assumes that R and I individuals (as well as S individuals) share the same environments, and α denotes the recovery (treatment or education) rate. The basic reproduction number of system (1.1) is given by
Rα is a dimensionless quantity (ratio or number) that can be interpreted as the number of I-individuals “generated” in a population of primarily S-individuals sharing a common environment. Sanchez et al. [19] have shown that the dynamics of system (1.1) depends also on the size of the initial population and the ratios
If Rα > 1, then the I-class becomes established. That is, the persistence of a regular drinking class over time is guaranteed.
Whenever Rc < Rα < 1 and Rp < 1 or whenever Rα < Rc < 1 the I-class becomes extinct.
Whenever Rc < Rα < 1 and Rp > 1 whether or not the I-class becomes established is a function of the initial size of the class of I-individuals.
However, in real life, there is a lot of randomness and hazardous events. This is why stochastic calculus is considered as a solid approach to describe this real random events seen in nature. Exciting results were found thanks to the consideration of stochasticity in epidemic models and population dynamics. The technique of parameter perturbation has been used by a number of researchers [5], [9], [15], [16]. The case of a color noise was investigated by Gray et al. and Settati and Lahrouz [10], [20]. Both of them have accomplished a detailed analysis on asymptotic behavior of an epidemic model under a finite regimes-switching. Motivated by those previous works, in this paper, we consider fluctuations in the environment (see [1], [2], [3], [6], [7], [17]), which are assumed to manifest themselves as fluctuations in parameters β and δ involved in the previous deterministic model (1.1), so that where B1(t) and B2(t) are two independent brownian motions with intensities σ1 > 0, σ2 > 0 respectively. Incorporating the above perturbations in the deterministic system (1.1), we get the following stochastic model
From system (1.3), it is obvious that
With initial condition S(0) > 0, I(0) > 0, R(0) > 0, and one can easily show that
Denote the meaningful domain for system (1.3) by
This paper is organized as follows. In Section 2, we introduce some needed results and lemmas which will be used in the following analysis. In Section 3, we investigate the disease extinction of our model (1.3). In Section 4, we explore the persistence in mean. In Section 5, we obtain the existence of a stationary distribution and the ergodicity of solutions for system (1.3). In Section 6, we introduce some examples and numerical simulations to illustrate the analytical results. In the last section, we provide a brief discussion and summary of main results.
Section snippets
Preliminaries
Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and increasing while contains all -null sets). Let X(t) be a regular time-homogeneous Markov process in described by the stochastic differential equation and the diffusion matrix is defined as follows
We introduce the operator associated with (1.3) as
Stochastic Disease-Free Dynamics
When studying the behavior of a stochastic epidemic model, the special case when the threshold equal to one still always untreated. Here, we propose an approach based on stopping times to overcome this difficulty. Let us now introduce the two positive numbers. Theorem 3.1 For any initial values (S(0), I(0), R(0)) ∈ Δ, if one of the followings assumptions and holds, then we get Proof Let 0 < ε < I(0). Define the stopping times
Persistence in mean of the disease
The most interesting topics in studying epidemic modeling concern the extinction and the persistence. The first one has been treated in section 3. In this section, we will show that the disease is persistent in mean. Theorem 4.1 If then for any given initial value (S(0), I(0), R(0)) ∈ Δ, the solution of system (1.3) has the property that .
Proof
(i) From the first
Stationary Distribution and Positive Recurrence
In the study of population models, we are also interested in when the disease will persists and prevails in a population. In the deterministic models, the problem can be resolved by showing that the endemic equilibrium is a global attractor or is globally asymptotically stable. But for stochastic models such system (1.3), there is no endemic equilibrium. In this section, we will show the positive recurrence and the existence of stationary distribution for our system (1.3), provided that
Simulations
In this section, we will introduce some numerical simulations to illustrate our main results. Using the Milstein method developed in [11], we get the following discretization equation of system (1.3): where τk, ξk are N(0, 1)-distributed independent random variables. Example 1 In system (1.3),
Conclusion
This paper is concerned with the dynamics of a stochastic SIRI epidemic model with nonlinear relapse, by perturbing consecutively β and δ. We show sufficient conditions on extinction and persistence of the disease. In addition, we prove the existence of an ergodic stationary distribution to the model (1.3). As consequence of Theorem 3.1, Theorem 3.2, and 4.1, the quantity is a threshold of system (1.3), if and which is not the case for for the reason that, we cannot prove
Declaration of Competing Interest
All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript.
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