Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 5620301, 9 pages
https://doi.org/10.1155/2017/5620301

Research Article

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

Chunjuan Zhu, Guangzhao Zeng, and Yufeng Sun

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

Correspondence should be addressed to Yufeng Sun; suny2001@sina.com

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 that

Besides, the total biomass of model (4) satisfies the following equation:

It is easy to know that, for all ,

Define a -function by

By 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 and

is 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 then

Namely, 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 obtain

Then

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 conditions

which implies

Next, according to system (4), we get

which implies that

Together with (13), we have

According to (26) and the last equation of system (4), we obtain

Therefore, by (29), we have

We 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 that

and 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.

Figure 1: Simulations of the path for the corresponding deterministic system (2) and stochastic system (4) with initial value , .

Example 7. We choose the same values of the parameters of (33) except for the value of . Because , , we let ; then by Theorem 5(ii), the solution of system (4) obeys a.s., in which case the disease will also die out. Note that , so that the disease for the corresponding system will prevail too (see Figure 2).

Figure 2: Simulations of the path , , for the corresponding deterministic system (2) and stochastic system (4) with initial value , .

4. Persistence

Definition 8. System (4) is said to be persistent in the mean if

Theorem 9. If , then for any initial value , the solution of system (4) has the following property:

Proof. An integration of system (4) yields

According to (39), we have

After simple computing

where and .
Let .
By Itô’s formula, we obtain

and then

Then the inequality can be rewritten as

By (29), then taking the limit inferior of both sides (44) leads to

This completes the proof of Theorem 9.

Example 10. Choose the parameters in system (4) as follows:

Note that

and then by Theorems 2, with any initial value , we get that the solution of system (4) obeys

Using the method mentioned in [23], we give the simulations shown in Figure 3, and we find the disease is persistent.

Figure 3: Simulations of the path , , for the corresponding deterministic system (2) and stochastic system (4) with initial value , .

Example 11. We consider the same values of the parameters of (46) except . Change to 0.06 to further illustrate the effect of the noise intensity on this SIRS model. Notice that and Using the method mentioned in [14], we give the simulations shown in Figure 4 to support our results. And we find the fluctuation of the solution of system (4) is becoming weaker compared with the picture in Figure 3.

Figure 4: Simulations of the path , , for the corresponding deterministic system (2) and stochastic system (4) with initial value , .

5. Conclusion

In this paper, we study a stochastic SIRS model with vertical transmission and saturated incidence. We find that the extinction or persistence of the disease is mainly determined by the value of , and it is obvious that when . We find when the noise is so small (in fact ), the extinction and persistence are completely determined by : if , the disease dies out; if , the disease prevails. So the noise is adverse to the survival of the disease. We also find that if the noise is so large that , the disease prevails too.

To understand how the noise effects the survival of the disease better, the conditions in Theorem 2 can be written as

and rewrite the condition in Theorem 5 as follows:

We obtain the main results as follows:

(a)If , the disease will always die out without considering the existence of the noise according to (15).(b)If , the disease will persist for the deterministic model (2) while for the stochastic model (4), the disease will persist when and die out when The existence of noise determines the extinction of the disease.(c)If , the disease will always die out.

From (b) we get that when , is the threshold which determines the extinction and persistence of the disease for stochastic system (4). Meanwhile, we can obtain from cases (b) and (c) that if

then disease will die out when ; meanwhile if

the disease will die out only for

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The research was supported by the Project for Science and Technology in Shaoguan (2014CX/K231) and the High-Level Talents Project of Guangdong Province Colleges and Universities (2013-178).

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