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Computational and Mathematical Methods in Medicine
Volume 2017 (2017), Article ID 7294761, 20 pages
https://doi.org/10.1155/2017/7294761
Research Article

Threshold Dynamics in Stochastic SIRS Epidemic Models with Nonlinear Incidence and Vaccination

1Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, China
2College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Correspondence should be addressed to Zhidong Teng

Received 30 July 2016; Accepted 21 November 2016; Published 16 January 2017

Academic Editor: Giuseppe Pontrelli

Copyright © 2017 Lei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the dynamical behaviors for a stochastic SIRS epidemic model with nonlinear incidence and vaccination are investigated. In the models, the disease transmission coefficient and the removal rates are all affected by noise. Some new basic properties of the models are found. Applying these properties, we establish a series of new threshold conditions on the stochastically exponential extinction, stochastic persistence, and permanence in the mean of the disease with probability one for the models. Furthermore, we obtain a sufficient condition on the existence of unique stationary distribution for the model. Finally, a series of numerical examples are introduced to illustrate our main theoretical results and some conjectures are further proposed.

1. Introduction

As is well known, transmissions of many infectious diseases are inevitably affected by environment white noise, which is an important component in realism, because it can provide some additional degrees of realism compared to their deterministic counterparts. Therefore, in recent years, stochastic differential equation (SDE) has been used widely by many researchers to model the dynamics of spread of infectious disease (see [15] and the references cited therein). There are different possible approaches to include effects in the model. Here, we mainly introduce three approaches. The first one is through time Markov chain model to consider environment noise in SIS model (see, e.g., [6] and the references cited therein). The second is with parameters perturbation (see [2, 5, 7] and the references cited therein). The last issue to model stochastic epidemic system is to perturb around the positive equilibria of deterministic models (see, e.g., [1, 8, 9] and the references cited therein).

Now, we consider stochastic epidemic models with parameters perturbation. The incidence rate of a disease denotes the number of new cases per unit time, and this plays an important role in the study of mathematical epidemiology. In many epidemic models, the bilinear incidence rate is frequently used (see [2, 5, 7, 8, 1017]), and the saturated incidence rate is also frequently used (see, e.g., [1822]). Comparing with bilinear incidence rate and saturated incidence rate, Lahrouz and Omari [23] and Liu and Chen [24] introduced a nonlinear incidence rate into stochastic SIRS epidemic models. In [25], Tang et al. investigated a class of stochastic SIRS epidemic models with nonlinear incidence rate :

Lahrouz et al. [26] studied a deterministic SIRS epidemic model with nonlinear incidence rate and vaccination. If the transmission of the disease is changed by nonlinear incidence rate , and to make the model more realistic, let us suppose that the death rates of the three classes in the population are different, then a more general deterministic SIRS model is described by the following ordinary differential equation: where , , and denote the numbers of susceptible, infectious, and recovered individuals at time , respectively. denote a constant input of new members into the susceptible per unit time. is the rate of vaccination for the new members. is the rate of vaccination for the susceptible individuals. is the natural mortality rate or the removal rate of the . is the removal rate of the infectious and usually is the sum of natural mortality rate and disease-induced mortality rate. is the removal rate of the recovered individual. is the recovery rate of infective individual. is the rate at which the recovered individual loses immunity. represents the transmission coefficient between compartments and , and denotes the incidence rate of the disease. For biological reasons, we usually assume that functions and satisfy the following properties:(H1) is two-order continuously differentiable function; is monotonically nondecreasing with respect to ; and . (H2) is two-order continuously differentiable function; and for all , and .

It is well known that the basic reproduction number for model (2) is defined by , where . Applying the Lyapunov function method and the theory of persistence for dynamical systems, we can prove that, when , model (2) has a globally asymptotically stable disease-free equilibrium and, when , model (2) has a unique endemic equilibrium and disease is permanent.

In this paper, we extend model (1) to more general cases. As in [11], taking into account the effect of randomly fluctuating environment, we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in parameters , , , and in model (2) change to random variables , , , and such that Accordingly, model (2) becomesBy the central limit theorem, the error term () may be approximated by a normal distribution with zero mean and variance (), respectively. That is, . Since these may correlate with each other, we represent them by -dimensional Brownian motion as follows: where are all nonnegative real numbers. Therefore, model (4) is characterized by the following Itô stochastic differential equation:

Model (6) in the special case where , , and has been investigated by Yang and Mao in [11] and in the special case where and also has been discussed in [25]. It is well known that, in a stochastic epidemic model, the dynamical behaviors, like the extinction, persistence, stationary distribution, and stability of the model, are the most interesting topics. Therefore, in this paper, as an important extension and improvement of the results given in [11, 25], we aim to discuss the dynamical behaviors of model (6). Particularly, we will explore the stochastic extinction and persistence in the mean of disease with probability one and the existence of stationary distribution.

This paper is organized as follows. In Section 2, we introduce some preliminaries to be used in later sections. In Section 3, we establish the threshold condition for stochastic extinction of disease with probability one of model (6). In Section 4, we deduce the threshold conditions for the disease being stochastically persistent and permanent in the mean with probability one. In Section 5, we discuss the existence of the stationary distribution of model (6) under some sufficient conditions. In Section 6, the numerical simulations are presented to illustrate the main results obtained in this paper and some conjectures are further proposed. Finally, in Section 7, a brief conclusion is given.

2. Preliminaries

Through this paper, we let be a complete probability space with a filtration satisfying the usual conditions (that is, it is right continuous and increasing while contains all -null sets). In this paper, we always assume that stochastic model (6) is defined on probability space . Furthermore, we denote , , , and

Firstly, on the existence and uniqueness of global positive solution for model (4) we have the following result.

Lemma 1. Assume that and hold; then, for any initial value , model (6) has a unique solution defined for all , and the solution will remain in with probability one.

This lemma can be proved by using a similar argument as in the proof of Theorem given in [11]. We hence omit it here.

Lemma 2. Assume that and hold and let be the solution of model (6) with initial value . Then Moreover, let be any continuous function defined on ; then for each we have

Proof. Let ; then we have from model (6) where and By the comparison theorem of stochastic differential equations, we further have where Define , where and It is clear that from Lemma 1 and (9) is nonnegative for , and and are continuous adapted increasing processes for and . Therefore, by Theorem given in [27], we obtain that exists. From (9), we further have DenoteBy (11), we haveHence, the strong law of large number (see [27, 28]) implies This completes the proof.

For any function defined on , we denote the average value on by .

Lemma 3. Assume that and hold. Let be any positive solution of model (6); then where function is defined for all satisfying .

Proof. Taking integration from to for model (6), we get Hence, we haveWith a simple calculation from (16) we can easily obtain formula (14) with which is defined byBy Lemma 2, we further have

Lemma 4. Assume that and hold and . Then, for any solution of system (6) with , one has where . Furthermore, the region is positive invariant with probability one for model (6), where .

In fact, for , from model (6) we have This implies that (18) holds, and set is positive invariant with probability one for model (6).

Lemma 5. Assume that and hold, , , and with constant . Then, for any solution of model (6) with , one has where

Proof. Since then where . From the third equation of model (6) we have Therefore, Let ; then Solving , we obtain Substituting (29) into (27), we immediately obtain (21)–(23). This completes the proof.

Remark 6. When in model (6), whether we can also establish a similar result as in Lemma 5 still is an interesting open problem.

Consider the following -dimensional stochastic differential equation:where , , and are standard Brownian motions defined on the above probability space. The diffusion matrix is defined byFor any second-order continuously differentiable function , we define The following lemma gives a criterion for the existence of stationary distribution in terms of Lyapunov function.

Lemma 7 (see [27]). Assume that there is a bounded open subset in with a regular (i.e., smooth) boundary such that(i)there exist some and positive constant such that for all ;(ii)there exists a nonnegative function such that is second-order continuously differentiable function and that, for some , for all , where .Then (30) has a unique stationary distribution . That is, if function is integrable with respect to the measure , then for all

To study the permanence in mean with probability one of model (6) we need the following result on the stochastic integrable inequality.

Lemma 8 (see [13]). Assume that functions and satisfy a.s. If there is such thatfor all , then

3. Extinction of Disease

For the convenience of following statements, we denote We further define a threshold value

Theorem 9. Assume that and hold. If one of the following conditions holds: (a)  and  ,(b)  and  ,then, for any initial value , one has That is, disease is stochastically extinct exponentially with probability one. Moreover,

Proof. Applying Itô’s formula to model (6) leads to where and Assume that condition (b) holds. Sincethen from (40)By Lemma 2, we haveTherefore,Assume that condition (a) holds. Choose constant such that We compute thatWhen , which implies , we have from (46)Since where , from , we can obtain . Hence, we haveAccording to (14), (40), and (49), we haveHence, from (44) and Lemma 3, we finally haveWhen , from (40) and (46) we have Define a function Clearly, is a monotone increasing for and monotone decreasing for . With condition , that is, , we have Hence, by (14) and (49), we have Choose ; from (44) and Lemma 3, we finally have From (45), (51), and (56), it follows that (38) holds.
Since , by (14) of Lemma 3 and the last equation of (15), we further obtain This completes the proof.

Remark 10. Condition (b) in Theorem 9 can be rewritten in the following form:It is clear thatTherefore, when condition (b) holds, from (58) we also have

Remark 11. From Remark 10 above, we see that in Theorem 9 if condition (a) holds, then we directly have , and if condition (b) holds, then we also have . Therefore, an interesting open problem is whether we can establish the extinction of disease with probability one for model (6) only when .

4. Stochastic Persistence in the Mean

In this section, we discuss the stochastic persistence and permanence in the mean with probability one for model (6) only for the following two special cases: (1) and (2) . Furthermore, we also assume that in model (6) function .

4.1. Case

When and in model (6), we have

Theorem 12. Assume that holds, , and . If ; then disease in model (6) is stochastically persistent in the mean; that is,

Proof. Let be any positive solution of model (6). Lemma 2 implies that there is a constant such that for all . Define a Lyapunov function Using Itô’s formula to model (6) leads to From , which implies that , we have where . Since , then . Substituting (65) into (64) and then integrating from to , we get where . From Lemma 2, we have Define a function as follows. When , , and when , . Then is continuous for and differentiable for . Applying Lagrange’s mean value theorem to , we have from (66) Substituting (14) into (68), it follows that Since we have where From (67) and Lemma 3 we have Finally, by Lemma 8, we obtain