Abstract and Applied Analysis

Volume 2014 (2014), Article ID 723825, 13 pages

http://dx.doi.org/10.1155/2014/723825

Research Article

Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

Can Chen and Yanmei Kang

Department of Applied Mathematics, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received 6 April 2014; Accepted 19 May 2014; Published 15 June 2014

Academic Editor: Debora Amadori

Copyright © 2014 Can Chen and Yanmei Kang. 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 introduce stochasticity into the SIS model with saturated incidence. The existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. Then, we carry out a detailed analysis on both its almost sure exponential stability and its pth moment exponential stability, which indicates that the pth moment exponential stability implies the almost sure exponential stability. Additionally, the results show that the conditions for the disease to become extinct are much weaker than those in the corresponding deterministic model. The conditions for the persistence in the mean and the existence of a stationary distribution are also established. Finally, we derive the expressions for the mean and variance of the stationary distribution. Compared with the corresponding deterministic model, the threshold value for the disease to die out is affected by the half saturation constant. That is, increasing the saturation effect can reduce the disease transmission. Computer simulations are presented to illustrate our theoretical results.

1. Introduction

Epidemiology has been investigated by mathematicians through establishing mathematical models for a long time; see, for instance, [1–4]. Mathematical models can help people to better understand the spread of the disease and thus take effective measures to reduce its transmission as much as possible. Particularly, the classical SIS epidemic model [5, 6] is often used to model the dynamics of the diseases with no protective immunity such as gonorrhea.

Usually, the incidence function describes the number of new infections per unit time and it plays a vital role in the deterministic models. As it is known, in the most existing literature, the bilinear incidence rate is frequently used. However, this kind of incidence rate has some limitations, since it does not consider the behavioral change of susceptible individuals or control measures taken by the government when the number of infective individuals gets large. For example, people would wash their hands frequently, wear masks, and limit their time of going out, and the government would also take measures such as quarantine and isolation when the A/H1N1 influenza became serious in 2009 [7]. Therefore, it is more reasonable to adopt saturated incidence rate [8] in the mathematical models. When the number of infective individuals gets large, tends to a saturation effect, where is the infection force of the disease and is the inhibition effect from the behavioral change or control measures. Here, represents the half saturation constant [9]. With the saturated incidence taken into consideration, the classical SIS epidemic model is transformed into

where and are the number of susceptible individuals and infective individuals at time , respectively. The total population is a constant. Consider the following parameters:   is the transmission rate, is the birth and death rate, and is the recovery rate. Assume all these parameters are positive; then, the classical stability analysis has shown that the system (1) always has a disease free equilibrium:

The basic reproduction number, , is a threshold value, which determines the extinction and persistence of the disease. When , the disease free equilibrium is globally stable. When , the disease free equilibrium is unstable, and model (1) has a unique endemic equilibrium:

which is globally stable.

Since the noise exists almost everywhere, epidemic models are inevitably affected by it. Hence, it is more reasonable to introduce random perturbations into mathematical models. At the same time, we note that taking the effect of environment noise on epidemic models into consideration has been a popular trend in disease spread modeling [10–13]. Gray et al. presented a stochastic SIS epidemic model in [14]. They established conditions for extinction and persistence of the disease and the existence of a stationary distribution. Stochastic SIR and SIRS models with and without distributed time delay were investigated in [15, 16], respectively, where the asymptotic behavior was discussed. In [17], stochastically perturbed SIR and SEIR epidemic models with saturated incidence were studied and the results of extinction and ergodicity were concluded. Ji et al. investigated a stochastic multigroup SIR model with fluctuations around the transmission coefficient and the death rate of each subpopulation in [18, 19] separately. Zhao et al. [20] discussed the extinction and persistence of the stochastic SIS epidemic model with vaccination. However, among all these researches, there are few literatures considering the parameter perturbation in SIS epidemic model with saturated incidence.

Motivated by the above consideration, in this paper, we introduce random effect into the SIS model (1) by replacing the transmission rate with , where is standard Brownian motion and is the intensity of white noise. The stochastic version corresponding to the deterministic model (1) can be represented as

Given that , it is sufficient to study the following equation:

with initial value . In the following sections, we will restrict ourselves to (5).

The paper is organized as follows. In Section 2, the existence of unique positive solution is shown. Both its almost sure exponential stability and its th moment exponential stability are then investigated in Section 3. The conditions for persistence in the mean of the disease are established in Section 4. The existence of a stationary distribution and the expressions for the mean and variance are presented in Section 5. In Section 6, we give a brief conclusion. Besides, the computer simulations which support our results are given in each section. Finally, we give an appendix containing some theory used in the previous sections.

2. Existence of Unique Positive Solution

To investigate the dynamical behavior of the epidemic model, we need to show that the model has a unique global solution and the solution will remain within whenever it starts there. Hence, in this section, employing the Lyapunov analysis method (mentioned in [18, 21]), we establish Theorem 1.

Theorem 1. There is a unique global solution of system (5) on for any given initial value with probability 1; namely,

Proof. Since the coefficients of (5) are locally Lipschitz continuous for any given initial value , there is a unique local solution on , where is the explosion time [22]. To show that the solution is global, we need to show that a.s. Let be sufficiently large such that . 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 a.s. and a.s. for all . In other words, to complete the proof, all what we need to show is that a.s. If this statement is not true, then there exists a pair of constants and such that

Therefore, there is an integer such that

Define a function by

Using the Itô formula, we get

where is defined by

where . Therefore,

which implies that

The Gronwall inequality yields that

Set for all . Then, by (9), . For every , equals either or , and therefore

It then follows from (9) and (15) that

where is the indicator function of . Letting leads to the contradiction that

So, we must have a.s., whence the proof is complete.

3. Extinction

In discussing the extinction of system (5), we focus on the two kinds of exponential stabilities, almost sure exponential stability and th moment exponential stability.

3.1. Almost Sure Exponential Stability

Theorem 2. Let be the solution of system (5) with initial value . If

(a) and ,(b).

then,

     . if (a) holds;    . if (b) holds;

that is, tends to zero exponentially .; namely, the disease will die out with probability one.

Proof. Applying Itô formula to system (5) leads to

where . Integrating both sides of (19) from 0 to , we have

where . Obviously,

where , for .

If condition (a) is satisfied, it then follows from (20) that

This implies that

Clearly, is a local martingale and vanishes at . Moreover,

By strong law of large numbers [22], we get

Then,

If condition (b) is satisfied, the function takes its maximum value at ; that is, . It then follows from (20) that

Then,

This finishes the proof.

From Theorem 2, we know that when and the perturbation is small, the disease will die out a.s., whereas the disease may still persist for the corresponding deterministic model (1). If the noise intensity is larger than , then the disease will also die out a.s. For , system (5) becomes the model discussed in [14] and the conditions for the disease to become extinct we obtain here are in agreement with those derived in [14]. If there exists no environment noise, that is, , then is the basic reproduction number of the deterministic model (1). In addition, the half saturation constant also influences , which shows that increasing the saturation effect can reduce the value of and thus can reduce the disease spread. The following simulations indicate these results.

Example 3. We assume that the unit time is one day. The parameters are given by

Note that

That is, condition (a) is satisfied and the solution satisfies

Hence, tends to zero exponentially a.s.; that is, the disease will die out a.s.

On the other hand, for the deterministic model (1), the basic reproduction number is

The solution obeys

Employing the Euler-Maruyama (EM) method in [23], the computer simulations are given in Figure 1, which support our results.

(a)

(b)

Figure 1: Computer simulations of for model (5) and the deterministic model (1), using the EM method with step size 0.001 and with initial values (a) and (b) .

Example 4. We keep the parameters the same as Example 3 but increase to 0.09. Note that

That is, condition (b) is satisfied and

Therefore, tends to zero exponentially a.s.; namely, the disease will die out a.s. The simulations are shown in Figure 2 to support our results completely.

(a)

(b)

Figure 2: Computer simulations of for model (5) and the deterministic model (1), using the EM method with step size 0.001 and with initial values (a) and (b) .

3.2. th Moment Exponential Stability

Following (20), we obtain

As , then [24]

Using the fact that

we find

Since

then

Theorem 5. Assume that ; the solution with initial value is th moment exponentially stable.

Next, we will make a comparison between the almost sure exponential stability and the th moment exponential stability, since

From Theorem 5, we know that ; that is,

Moreover,

Let be positive; then

Therefore, the condition of Lemma A.4 (in the appendix) [22] is satisfied, which suggests that the th moment exponential stability of system (5) implies the almost sure exponential stability.

4. Persistence

We will investigate persistence in the mean for system (5) in this section. The definition of the persistence in the mean was initially proposed for the deterministic system [25] and was also used for the stationary stochastic system with ergodicity [19, 20]. According to the ergodicity for the stationary process, the time average in the long time limit is equal to the ensemble expectation over phase space, so we introduce the notation

Definition 6. System (5) is said to be persistent in the mean, if

Theorem 7. If

then, for any initial value , the solution of system (5) has the following property:

where

Proof. By (20), we obtain

The inequality can be written as

Since , if condition (48) is satisfied, then

Together with Lemma . in [20], we have

On the other hand,

That is,

Since , then

Moreover,

Therefore, we complete the proof.

Example 8. The parameters are given by

Note that

Then the solution of system (5) satisfies

For the corresponding deterministic model (1),

The simulation results are shown in Figure 3.

(a)

(b)

Figure 3: Computer simulations of for model (5) (red line) and the deterministic model (1) (blue line), using the EM method with step size 0.001 and with initial values (a) and (b) . and .

5. Stationary Distribution

Applying Lemma A.5 (in the appendix) [26], if we can show that conditions and are satisfied, then system (5) has a stationary distribution. Obviously, the square of the diffusion coefficient of system (5), that is, , is bounded away from zero for ; then, condition is satisfied. Next, we will prove that is also valid.

Theorem 9. Assume that and ; then, there is a unique stationary distribution of system (5). Here, is the unique endemic equilibrium of system (1); that is, . Moreover,

Proof. If , it is clear that ; then, model (1) has a unique endemic equilibrium ; that is,

Define

Then is positive definite. By Itô formula, we yield

where

Note that if , then the neighborhood lies entirely in . Hence, for , ( is a positive constant), which implies that condition is satisfied; for more details, we refer to [27]. Therefore, system (5) has a stationary distribution.

Moreover,

Then

Next, we will give the expressions for the mean and variance of the stationary distribution.

Theorem 10. If and . Let and denote the mean and variance of the stationary distribution of model (5). Then,

Proof. Multiply both sides of (5) by ; that is,

Integrating (71) from 0 to and dividing by on both sides, then

Since

letting , we have

Then, applying the ergodic property of the stationary distribution and strong law of large numbers, we get

Here, denotes the second moment of the stationary distribution [14].

In a similar way, multiplying both sides of (19) by , we obtain that

Then, integrating (76) from 0 to and dividing by on both sides,

Since

then

Namely,

Therefore,

From (75) and (81), we can get

Example 11. We keep the parameters the same as Example 8 but reduce to 0.005, 0.001, and 0.0005. Note that is 3.2083, 3.3283, and 3.3321, respectively, and all of them are bigger than 1. Furthermore,

The values of are satisfied for the conditions of Theorem 10. The histograms of are shown in Figure 4, respectively, for different values of , , and . The simulations were run for 100,000 iterations with step size 0.001, and the first 90,000 iterations were discarded to allow to reach its recurrent level. The distribution is slightly negative skewed for and positive skewed for and . Additionally, the corresponding sample skewness coefficients are −0.00067, 0.0016, and 0.0017, separately. These simulation results illustrate that the distribution of has reached the stationary distribution. When , from (70), the mean and variance of the stationary distribution are 75.9710 and 2.2016, respectively, while the sample mean and variance are 76.3754 and 0.3209. For , the mean and variance of the stationary distribution are 75.9988 and 0.0876, compared to the sample mean and variance which are 75.9285 and 0.0157, separately. When reduces to 0.0005, the mean and variance of the stationary distribution are 75.9997 and 0.0219 and the corresponding sample mean and variance are 76.0248 and 0.0050. As decreases, becomes more symmetric about and the perturbations become much weaker. What is more, the normal quantile-quantile plots in Figure 5 suggest that these data are not far from being normally distributed.

Figure 4: Histograms of the solution for model (5) for and differing values of , , and .

Figure 5: Normal quantile-quantile plots of the solution for model (5) for and differing values of , , and .

From (70), we know that when , the mean value equals the value of the endemic equilibrium and the variance is zero. Furthermore, if increases, the mean value decreases and the variance increases as shown in Figure 6. Obviously, the greater the intensity of the noise is, the larger the variance is, that is, the stronger the fluctuations of the distribution are. And this is in accordance with the actual case.

(a)

(b)

Figure 6: Plot the mean and variance against . The endemic equilibrium of the deterministic model (1) is 76.

6. Conclusion

In this paper, we established a stochastic SIS epidemic model with saturated incidence to investigate the effect of environment noise. After proving the existence and uniqueness of the positive solution, we considered two kinds of stabilities: almost sure exponential stability and th moment exponential stability. Our deduction shows that the th moment exponential stability implies the almost sure exponential stability. As for the extinction of the disease, our results indicate that when the noise is larger than , the disease will die out a.s. If the noise is small, the disease will also die out a.s. when , instead, the disease will be persistent in the mean when . Our investigation shows that the conditions for the disease to become extinct are much weaker than those in the corresponding deterministic model; that is, even if the disease dies out for the stochastic model, it may still persist for the corresponding deterministic model. Additionally, the half saturation constant influences the threshold value , which is not obtained in the deterministic model. Increasing the saturation effect by changing the behavior of the susceptible individuals and taking effective control measures of the government can reduce and thus can reduce the spread of the disease. Finally, we proved the existence of a stationary distribution and derived the expressions for the mean and variance when the noise is small and . Our theoretical results were further verified by computer simulations.

Appendix

First, we give some theory in stochastic differential equations.

Let be a complete probability space with satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). We use to denote , to denote , and a.s. to denote almost surely.

Definition A.1 (see [22]). Consider the -dimensional stochastic differential equation

with initial value and being the -dimensional standard Brownian motion. The solution is represented as . Assume that, for any , there is

So (A.1) has the solution . This solution is called the trivial solution.

Definition A.2 (see [22]). The trivial solution of (A.1) is said to be almost surely exponentially stable if

for all .

Definition A.3 (see [22]). The trivial solution of (A.1) is said to be th moment exponentially stable if there is a pair of positive constants and such that

for all . When , it is said to be exponentially stable in mean square.

Lemma A.4 (see [22]). Assume that there is a positive constant such that

Then, the th moment exponential stability of the trivial solution of (A.1) implies the almost sure exponential stability.

Next, we give some theory about stationary distributions [26].

Let be a homogeneous Markov process in ( denotes Euclidean -space) described by

The diffusion matrix is , .

Assumption B. There exists a bounded domain with regular boundary , having the following properties.

In the domain and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix is bounded away from zero.If , the mean time at which a path issuing from reaches the set is finite and for every compact subset .

Lemma A.5 (see [26]). If and hold, then the Markov process has a stationary distribution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The work is financially supported by the National Natural Science Foundation of China (nos. 11072182 and 11372233).

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