Abstract and Applied Analysis

Volume 2013 (2013), Article ID 127321, 8 pages

http://dx.doi.org/10.1155/2013/127321

Research Article

Extinction and Ergodic Property of Stochastic SIS Epidemic Model with Nonlinear Incidence Rate

Qixing Han,1,2 Daqing Jiang,¹ and Chengjun Yuan³

¹School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

²School of Mathematics, Changchun Normal University, Changchun 130032, China

³School of Mathematics and Computer, Harbin University, Harbin 150086, China

Received 28 September 2013; Accepted 11 November 2013

Academic Editor: Peixuan Weng

Copyright © 2013 Qixing Han 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.

We investigate a stochastic SIS model with nonlinear incidence rate. We show that there exists a unique nonnegative solution to the system, and condition for the infectious individuals to be extinct is given. Moreover, we prove that the system has ergodic property. Finally, computer simulations are carried out to verify our results.

1. Introduction

More attention has been paid to the epidemics models in order to monitor and curb the spread of some human diseases. A classical model is proposed by Kermack and McKendrick in 1927 [1]. They divided the population into three classes denoted by , , and , which expressed the number of susceptible individuals, infective individuals, and removed individuals at time , respectively. The model is called susceptible-infected-removed (SIR) model, and SIR models were investigated by many researchers [2–4].

However some diseases, such as some sexually transmitted and bacterial diseases, do not have permanent immunity. In [5], they introduced a SIS model to describe the spread of the disease, which takes the following form:

with initial values . is the total size of the population. Where and express the number of susceptible individuals and infective individuals at time , respectively, is the per capita birth (and death) rate, is the rate at which infected individuals become cured, and is the per capita contact rate. In model (1), they assumed that a rate of contacts by an infective individual with a susceptible individual is proportional to population size, and model (1) has been well studied [6].

In fact, population dynamics is inevitably affected by environmental white noise, which is always present. Since the parameters in the deterministic models are constant, they have some limitations when we describe the epidemics systems. Some researchers have paid their attention to the stochastic epidemics model [7–9]. Especially, in [10], Gray et al. consider that the parameter in (1) is perturbed with

where is Brownian motions and represents the intensities of the white noise. Corresponding to the deterministic model system (1), the stochastic system takes the following form:

Since , then (3) is reduced to

For model (4), they pointed out that

(i)if and , the disease will die out with probability one;(ii)if , then model (4) has a unique stationary distribution.

The incidence rate in (4) is bilinear, and several authors pointed out that the disease transmission process may have a nonlinear incidence rate [11, 12]. In [13], Xiao and Ruan propose an incidence rate

where is the parameter that measures the psychological or inhibitory effect, and describes the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals is very large. It can be used to explain some phenomena; for example, the outbreak of severe acute respiratory syndrome (SARS) had such psychological effects on the general public [14]: for a very large number of infective individuals, the infection force may decrease as the number of infective individuals increases. Some control measures and policies, such as border screening, mask wearing, quarantine, isolation, and so forth, can decrease the infection rate although the number of infective individuals was getting relatively larger. Equation (1) with nonlinear incidence rate (5) and the disturbed parameter () can be written as follows:

The parameters appearing in (6) have the same meaning as those above. Given that , it is sufficient to study the SDE for ,

with initial value . Notice that when , system (7) becomes (4). In this paper, we will analyze the dynamical behaviors of (7).

The organization of this paper is as follows. In the next section, we show that there exists a unique positive solution to (7). In Section 3, we carry out a qualitative analysis of the model (7) and extinction conditions for is derived. We prove that the system has ergodic property under some condition in Section 4. In Section 5, we present some numerical simulations to illustrate our mathematical findings. A brief conclusion is given in Section 6.

Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets) and let be a scalar Brownian motion defined on the probability space.

2. Existence and Uniqueness of the Global Positive Solution

In order for the model to make sense, we need to show the solution is global and nonnegative. However, theorem of existence and uniqueness (cf. Arnold [15] and Mao [16]) is not satisfied in (7). By using tools established by Mao et al. [17], we will show existence and uniqueness of the global positive solution of (7).

Theorem 1. For any given initial data , there exists a unique solution for all with probability 1.

Proof. It is obvious that the coefficients of the SDE (7) are locally continuous. For any given initial data , there exists a unique maximal local solution on , where is the explosion time. In order to show that the solution is global, it is sufficient to show a.s. Let be sufficiently large so that lies within the interval . For each integer , we define the stopping time

where, throughout this paper, we set (as usual denotes the empty set). It is clear that is increasing as . Let , whence . It is easy to show that a.s. implies a.s. and a.s. for all . Therefore, to complete this proof, it is enough to show that a.s.

Define a function as follows:

By Itô’s formula, we get

where

By almost the same method in the proof of [10], the desired result will be obtained.

3. Extinction

In this section, we will point out the condition for to be extinct. We firstly do some preparation work.

Consider the following stochastic equation:

and assume that the coefficients , satisfy

(1), ,(2), such that ,

where ; .

Lemma 2 (see [18]). Assume that (1) and (2) hold, and let be a weak solution of (12) in , with nonrandom initial condition . Let be given by

If , , then

Theorem 3. If , then for any initial data , the solution of SDE (7) has the following property:

that is, the disease dies out with probability one.

Proof. Applying Lemma 2 with , and , we can compute

Clearly, conditions (1) and (2) are satisfied. By calculation, we have

We see that

Let ; we obtain that

Clearly,

When , we have

that is,

It can be straightly shown by Lemma 2 that

The proof is therefore completed.

4. Ergodic Property

In this section, we show that the small perturbation forces the infective individuals to be ergodic.

Theorem 4. If , then, for any initial data , the solution of SDE (7) is ergodic.

Proof. If , we get

It follows from (20) that

Besides,

Let ; then

When , that is, , we have

The conditions of Theorem 1.16 in Kutoyants [19] follow from (24), (25), and (28). Therefore is ergodic, and the invariant density is given by

where is a constant.

Remark 5. If , then

By the property of density function, we have

Let ; then

which implies

Next, we compute :

This together with (33) shows that

Similarly, we have

Consequently,

When , model (7) becomes (4), and the mean and variance of the stationary distribution of model (4) are the same as the results in [10].

5. Simulations

We utilize the method developed in [20] to illustrate our findings. We have the corresponding discretization equation:

where are the Gaussian random variables .

Using the discretized equation and with the help of Matlab software, choosing the appropriate parameters , , , , and , we get simulations of system (7) and the corresponding deterministic system.

In Figure 1, we choose initial values and 0.4, respectively, , noting that , as the case in Theorem 3 expected, for any initial value , the large white noise leads to the extinction of and the solution of system (7) tends to zero; that is, the disease dies out. While the solution of corresponding deterministic system does not tend to zero.

(a)

(b)

Figure 1: The solution of SIS model (7) compared to the deterministic system with different initial value and , . The blue line represents the solution of stochastic system, and the black line represents the solution of the corresponding deterministic system.

In Figure 2, we choose and , respectively, with initial values , which satisfy the cases in Theorem 4; that is, . From the left picture in Figure 2, we can see that the solution of system (7) is fluctuating in a small neighborhood, and there is a stationary distribution (see the histogram in Figures 2(b) and 2(d)); the disease becomes endemic.

(a)

(b)

(c)

(d)

Figure 2: The solution of SIS model (7) and its histogram with differing value of .

6. Conclusion

In this paper, we analyze the dynamic behaviors of a stochastic SIS model with nonlinear incidence rate, under the assumption that the population lives in an environment subjected to random fluctuations which mainly affect the disease transmission term. First of all, we show that there exists a unique positive solution in system (7). Moreover, we obtain the threshold between prevalence and extinction of the disease; that is, if , the disease will die out with probability one; if , is ergodic, which means the disease will become endemic. Finally, we illustrate our results with computer simulations.

We see that if , (7) becomes (4), which is studied in [10]; they show that if and , the disease will die out with probability one. Obviously, and are equivalent. The condition , however, is not necessary in our investigation. In addition, if , the invariant density is obtained, which is not mentioned in [10].

An extension of our work is to consider a stochastic SIS model with the general incidence rate , and it is currently a work in progress.

Acknowledgments

The work was supported by the Program for Changjiang Scholars and Innovative Research Team in University, NSF of China (nos. 11371085 and 11101072); the Ph.D. Programs Foundation of Ministry of China (no. 200918); and the Nature Science Foundation of Changchun Normal University (no. 2010007).

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