Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 5620301, 9 pages

https://doi.org/10.1155/2017/5620301

## The Threshold of a Stochastic SIRS Model with Vertical Transmission and Saturated Incidence

School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China

Correspondence should be addressed to Yufeng Sun

Received 27 January 2016; Accepted 15 November 2016; Published 28 February 2017

Academic Editor: Manuel De la Sen

Copyright © 2017 Chunjuan Zhu 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

The threshold of a stochastic SIRS model with vertical transmission and saturated incidence is investigated. If the noise is small, it is shown that the threshold of the stochastic system determines the extinction and persistence of the epidemic. In addition, we find that if the noise is large, the epidemic still prevails. Finally, numerical simulations are given to illustrate the results.

#### 1. Introduction

Mathematical models can describe the progress of a disease and predict the trend of the disease and they can provide the theoretical basis for people to undertake prevention strategies. At present, researches have constructed a series of mathematical models [1–10], including SIRS model [5, 9, 10]. They divided people into susceptible, , infective, , and removed, , categories and one of the most famous SIRS epidemic models is the following:Here, is the birth rate, is the transmission rate, and are the birth rate and natural death rate, respectively, of , , and individuals, is the death rate of the disease, and is the recovery rate of the infective individuals. Assume .

Medical research has shown that the herpes virus will be in the form of mother-to-child transmission (vertical transmission) to the baby. In addition, since susceptible individuals in contact with every infective individual are limited, we see that when the number of the susceptible individuals is large, the bilinear incidence is unreasonable to consider. In this case, saturated incidence is more suitable than bilinear incidence [11].

In this paper, the transmission rate is chosen as the saturated incidence rate , and the SIRS model is described as the following:

In system (2), is vertical transmission rate, is the threshold which determines whether the disease will die out or persist, and there always is a disease-free equilibrium ; see [5, 9, 10] and the references therein. When , the disease-free equilibrium is globally asymptotically stable; when , is unstable and there is an endemic equilibrium where , , , which is globally stable under a sufficient condition.

However, all parameters in system (2) are affected by environmental noise, so it is of benefit to use a stochastic model. Stochastic models are more realistic compared to deterministic models. Many stochastic models for epidemic populations have been studied [12–19]. Tornatore et al. [18] studied an stochastic SIR model. They showed that under the condition , the disease-free equilibrium is locally stable, but the authors do not discuss under which condition the disease will prevail. Concerning the transmission coefficient , Gray et al. [20] considered the stochastic SIS (susceptible-infective-susceptible) epidemic model with fluctuation. They proved threshold which determines the extinction and persistence of according to the fluctuation. Here, is the threshold of the deterministic model; however, it is more difficult to get the threshold of the stochastic model.

We consider certain stochastic environmental factors and assume that fluctuations in the environment will manifest themselves mainly as fluctuations in the parameter , as in [20],where is standard Brownian motion with and is the intensity. The stochastic version corresponding to the deterministic model (2) is the following:Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets) and let be the Brownian motion defined on the probability space.

For simplicity, define

#### 2. Existence and Uniqueness of the Nonnegative Solution

In this section, we will show that there is a unique positive solution of system (4).

Theorem 1. *For any initial value in , there is a unique solution of system (4) on , and the solution will remain in with probability 1.*

*Proof. *Since the coefficients of system (4) are locally Lipschitz continuous, for any initial value , there is a unique local solution on , where is the explosion time. To show that this solution is global, we need to have a.s. To show that this solution is global, we need to have a.s. Let be sufficiently large so that all lie in the interval . For each integer , define the stopping time where throughout this paper we set (as usual denotes the empty set). Clearly, is increasing as . Set , whence a.s. If we can show that a.s., then and a.s. for all . In other words, to complete the proof all we need to show is that a.s. If this statement is false, then there is a pair of constants and such that

Hence there is an integer such thatBesides, the total biomass of model (4) satisfies the following equation:It is easy to know that, for all , Define a -function byBy using Itô’s formula, we get where The remainder of the proof follows that in Li and Mao [21, Theorem ].

Since there is a positive solution of system (4) for any given initial value andis an invariant set [22], then from now on, we can assume that .

#### 3. Extinction

In this section, we discuss the conditions for the extinction of the disease.

Theorem 2. *Let be the solution of system (4) with initial value . If and thenNamely, tends to zero exponentially a.s. where*

*Proof. *Notice that ; that is, the quadratic function gets its maximum value on the interval at , where It follows from the monotonicity of the function on that when , we get Applying Itô’s formula to system (4) leads to Integrating this from 0 to and diving on both sides, we have where , which is local martingale and . Moreover, According to the large number theorem for martingales (see, e.g., [13]), we obtainThen The proof of Theorem 2 is completed.

*Remark 3. *We also notice that if , that is, , the quadratic function obtains the maximum value at The proof of Theorem 2 proceeded; we have if . It means that the disease will always die out if the noise is large enough such that .

*Remark 4. *From Theorem 2, we can get that the disease will die out if and the white noise is not large such that . Meanwhile if white noise is large enough such that is satisfied, then the disease will also die out. Moreover, we notice is smaller than the threshold of the corresponding deterministic model. The following examples illustrate this result more explicitly.

From Theorem 2, it is obvious that under some conditionswhich impliesNext, according to system (4), we get which implies that Together with (13), we haveAccording to (26) and the last equation of system (4), we obtainTherefore, by (29), we haveWe have the following theorem by combing these arguments.

Theorem 5. *Let be the solution of system (4) with initial value . If*(i)* and or*(ii)*then *

*Example 6. *We assume that the unit of time is one day and the population sizes are measured in units of 1 million. The parameters in system (4) are chosen as follows:Note thatand then by Theorem 5(i), the solution of system (4) obeys with any initial value . Then the disease tends to zero exponentially with probability one.

On the other hand, for the responding deterministic SIRS model, ; the disease will prevail. Using the method mentioned in [23], we provide the simulations shown in Figure 1 to support our results.