The Dynamical Behaviors in a Stochastic SIS Epidemic Model with Nonlinear Incidence

A stochastic SIS-type epidemic model with general nonlinear incidence and disease-induced mortality is investigated. It is proved that the dynamical behaviors of the model are determined by a certain threshold value R~0. That is, when R~0<1 and together with an additional condition, the disease is extinct with probability one, and when R~0>1, the disease is permanent in the mean in probability, and when there is not disease-related death, the disease oscillates stochastically about a positive number. Furthermore, when R~0>1, the model admits positive recurrence and a unique stationary distribution. Particularly, the effects of the intensities of stochastic perturbation for the dynamical behaviors of the model are discussed in detail, and the dynamical behaviors for the stochastic SIS epidemic model with standard incidence are established. Finally, the numerical simulations are presented to illustrate the proposed open problems.


Introduction
Our real life is full of randomness and stochasticity. Therefore, using stochastic dynamical models can gain more real benefits. Particularly, stochastic dynamical models can provide us with some additional degrees of realism in comparison to their deterministic counterparts. There are different possible approaches which result in different effects on the epidemic dynamical systems to include random perturbations in the models. In particular, the following three approaches are seen most often. The first one is parameters perturbation; the second one is the environmental noise that is proportional to the variables; and the last one is the robustness of the positive equilibrium of the deterministic models.
In recent years, various types of stochastic epidemic dynamical models are established and investigated widely. The main research subjects include the existence and uniqueness of positive solution with any positive initial value in probability mean, the persistence and extinction of the disease in probability mean, the asymptotical behaviors around the disease-free equilibrium and the endemic equilibrium of the deterministic models, and the existence of the stationary distribution as well as ergodicity. Many important results have been established in many literatures, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references cited therein. Particularly, for stochastic SI type epidemic models, in [6], Gray et al. constructed a stochastic SIS epidemic model with constant population size where the authors not only obtained the existence of the unique global positive solution with any positive initial value, but also established the threshold value conditions; that is, the disease dies out or persists. Furthermore, in the case of the persistence, the authors also showed the existence of a stationary distribution and finally computed the mean value and variance of the stationary distribution.
(1) The stochastic epidemic models with general nonlinear incidence are not investigated. Up to now, only some special cases of nonlinear incidence, for example, saturated incidence rate, are considered. But, we all know that the nonlinear incidence rate in the theory of mathematical epidemiology is very important.

Computational and Mathematical Methods in Medicine
(2) For the stochastic epidemic models with the standard incidence, up to now, we do not find any interesting researches.
(3) The conditions obtained on the existence of unique stationary distribution are very rigorous. Whether there is a unique stationary distribution only when the model is permanent in the mean with probability one is still an open problem.
Motivated by the above work, in this paper, we consider the following deterministic SIS epidemic model with nonlinear incidence rate and disease-induced mortality: (1) In model (1), and denote the susceptible and infectious individuals, Λ denotes the recruitment rate of the susceptible, is the natural death rate of and , is the disease-related death rate, the transmission of the infection is governed by a nonlinear incidence rate ( , ), where denotes the transmission coefficient between compartments and , ( , ) is a continuously differentiable function of and , and denotes the per capita disease contact rate. Now, we assume that the random effects of the environment make the transmission coefficient of disease in deterministic model (1) generate random disturbance. That is, → +( ), where ( ) is a one-dimensional standard Brownian motion defined on some probability space. Thus, model (1) will become into the following stochastic SIS epidemic model with nonlinear incidence rate: In this paper, we investigate the dynamical behaviors of model (2). By using the Lyapunov function method, Itô's formula, and the theory of stochastic analysis [17,18], we will establish a series of new interesting criteria on the extinction of the disease, permanence in the mean of the model with probability one. The stochastic oscillation of the disease about a positive number in the case where there is not diseaserelated death is also obtained. Further, we study the positive recurrence and the existence of stationary distribution for model (2), and a new criterion is established. Particularly, the effects of the intensities of stochastic perturbation for the dynamical behaviors of the model are discussed in detail. For some special cases of nonlinear incidence ( , ), for example, ( , ) = / (standard incidence) and ( , ) = ℎ( ) ( ), many idiographic criteria on the extinction, permanence, and stationary distribution are established. Lastly, some affirmative answers for the open problems which are proposed in this paper also are given by the numerical examples (the numerical simulation method can be found in [19]).
The organization of this paper is as follows. In Section 2, the preliminaries are given, and some useful lemmas are introduced. In Section 3, the sufficient conditions are established which ensure that the disease dies out with probability one. In Section 4, we establish the sufficient conditions which ensure that the disease in model (2) is permanent in the mean with probability one, and when there is not diseaserelated death the disease oscillates stochastically about a positive number. In Section 5, the existence on the unique stationary distribution of model (2) is proved. In Section 6, the numerical simulations are carried out to illustrate some open problems. Lastly, a brief discussion is given in the end to conclude this work.
In model (2), and denote the susceptible and infected fractions of the population, respectively, and = + is the total size of the population among whom the disease is spreading; the parameters Λ, , , and are given as in model (1); the transmission of the infection is governed by a nonlinear incidence rate ( ); ( ) denotes onedimensional standard Brownian motion defined on the above probability space; and represents the intensity of the Brownian motion ( ). Throughout this paper, we always assume the following.
This shows that ( , ) and ( , ) are continuous for ( , ) ∈ . Therefore, conclusion (4) also is true. Next, on the existence of global positive solutions and the ultimate boundedness of solutions for model (2) with probability one, we have the result as follows. Lemma 4 can be proved by using the method which is given in [6]. We hence omit it here.

Extinction of the Disease
Define the constants 0 = ( ( 0 , 0) / ) + + , We have that 0 is the basic reproduction number of deterministic model (1). On the extinction of the disease in probability for model (2) we have the following result.
When 0 ≤ 1, then, for any > 0,̃0 < 1, and it is easy to prove that one of the conditions (a) and (b) of Theorem 5 holds. Therefore, for any > 0, the conclusions of Theorem 5 hold. Let 1 < 0 ≤ 2. From̃0 = 1 we have Denote Since 1 ≤ 2 , we easily prove that when > one of the conditions (a) and (b) of Theorem 5 holds. Therefore, for any > , the conclusions of Theorem 5 hold. When 0 > 2, we have 1 > 2 and 1 ≥ ≥ 2 . Hence, condition (a) in Theorem 5 does not hold. We only can obtain that for any > 1 the conclusions of Theorem 5 hold. Summarizing the above discussions we have the following result as a corollary of Theorem 5.

Then disease in model
Computational and Mathematical Methods in Medicine 5 Corollary 8. Let ( , ) = ℎ( ) ( ). Assume that (H * ) holds and one of the following conditions holds: Then disease in model (2) is extinct with probability one.
Remark 9. It is easy to see that in Theorem 5 the conditions 0 > 2 and ≤ ≤ 1 are not included. Therefore, an interesting conjecture for model (2) is proposed; that is, if the above condition holds, then the disease still dies out with probability one. In Section 6, we will give an affirmative answer by using the numerical simulations; see Example 1.

Remark 10.
In the above discussions, we see that casẽ0 = 1 has not been considered. An interesting open problem is whether wheñ0 = 1 the disease in model (2) also is extinct with probability one. A numerical example is given in Section 6; see Example 2.

Permanence of the Disease
On the permanence of the disease in the mean with probability one for model (2), we establish the following results.
Remark 12. From (20), we have that̃0 > 1 is equivalent to < . Therefore, Theorem 11 also can be rewritten by using intensity of stochastic perturbation in the following form: if < , then disease in model (2) is permanent in the mean with probability one.
Remark 13. Combining Corollary 6 and Remark 12 we can obtain that when 1 < 0 ≤ 2, number is a threshold value. When 0 < < , the disease in model (2) is permanent in the mean and when > , the disease is extinct with probability one. However, when 0 > 2, then the alike results are not established. Therefore, it yet is an interesting open problem. (2) also is permanent in the mean with probability one. That is, there is a constant > 0 such that, for any initial value ( (0), (0)) ∈ 2 + , solution ( ( ), ( )) of model (2) satisfies

Theorem 14. Susceptible in model
Proof. By Lemma 4 we easily see that, for any initial value ( (0), (0)) ∈ 2 + , solution ( ( ), ( )) of model (2) Computational and Mathematical Methods in Medicine 7 Therefore, with the large number theorem for martingales, we finally have This completes the proof.
As consequences of Theorems 11 and 14, we have the following corollaries.  (2) is permanent in the mean with probability one.
We further have the result on the weak permanence of model (2) in probability.

Corollary 17. Assume that̃0 > 1. Then there is a constant
> 0 such that, for any initial value ( (0), (0)) ∈ 2 + , solution ( ( ), ( )) of model (2) Now, we discuss special case: = 0 for model (2); that is, there is not disease-related death in model (2). We can establish the following more precise results on the weak permanence of the disease in probability compared to the conclusion given in Corollary 17.
From Theorem 18, we easily see that number will arise from the change when the noise intensity changes. Therefore, it is very interesting and important to discuss how number changes along with the change of . We have the following result.
Computing the derivative of with respect to , we have Computational and Mathematical Methods in Medicine 9 Since we have / > 0. From the definition of , we easily see that is monotone decreasing for . From (49) and (H), we obtain that / exists and is continuous for . Since Therefore, we have lim →̂= 2 , and 2 satisfies This completes the proof.
Remark 22. When ( , ) = , we easily validate that Theorems 20 and 24 degenerate into Theorems 5.1 and 5.4 which are given in [19], respectively. Therefore, Theorems 18 and 20 are the considerable extension of Theorems 5.1 and 5.4 in general nonlinear incidence cases, respectively.
Remark 23. For the case > 0 in model (2), an interesting and important open problem is wheñ0 > 1 whether we also can establish similar results as Theorems 18 and 20. Furthermore, as an improvement of the results obtained in Corollary 17 we also propose another open problem: only wheñ0 > 1 we also can establish the permanence of the disease with probability one; that is, there is a constant > 0 such that, for any solution ( ( ), ( )) of model (2) with initial value ( (0), (0)) ∈ 2 + , one has lim →∞ ( ) ≥ , a.s. In Section 6, we will give an affirmative answer by using the numerical simulations; see Example 3.

Stationary Distribution
From Theorems 11 and 14 we obtain that wheñ0 > 1 model (2) is permanent in the mean with probability one. However, wheñ0 > 1 model (2) also has a stationary distribution. We have an affirmative answer as follows.
Corollary 29. (a) Let 0 ≤ 1. Then for any > 0 the disease in model (2) is extinct with probability one.
(b) Let 1 < 0 ≤ 2. Then for any 0 < < model (2) is permanent in the mean with probability one and has a unique stationary distribution, and for any > the disease in model (2) is extinct with probability one. (c) Let 0 > 2. Then for any 0 < < model (2) is permanent in the mean with probability one and has a unique stationary distribution, and for any > 1 , where 1 is given in (20), the disease in model (2) is extinct with probability one.

Numerical Simulations
In this section we analyze the stochastic behavior of model (2) by means of the numerical simulations in order to make readers understand our results more better. The numerical simulation method can be found in [19]. Throughout the following numerical simulations, we choose ( , ) = /(1 + ), where > 0 is a constant. The corresponding discretization system of model (2) is given as follows: where ( = 1, 2, . . .) are the Gaussian random variables which follow standard normal distribution (0, 1).
By computing we have 0 = 4.195 > 2,̃0 = 0.6715 < 1, / 0 − 2 = −0.0023 < 0, and 2 − 2 /2( + ) = −0.0019 < 0 which is the case of Remark 9. From the numerical simulations, we see that the disease will die out (see Figure 1). An affirmative answer is given for the open problem proposed in Remark 9. By computing we havẽ0 = 1. From the numerical simulations given in Figure 2 we know that the disease will die out. Therefore, an affirmative answer is given for the open problem proposed in Remark 10. We havẽ0 = 1.200, 0 = 1.2500, and = 0.1037. The numerical simulations are found in Figure 3. We can see that solution ( ) of model (2) oscillates up and down at , which further show that the conclusions of Theorems 14 and 18 are true. At the same time, this example also shows that the disease in model (2) is permanent with probability one. Therefore, an affirmative answer is given for the open problems proposed in Remarks 19 and 23.

Discussion
In this paper we investigated a class of stochastic SIS epidemic models with nonlinear incidence rate, which include the standard incidence, Beddington-DeAngelis incidence, and nonlinear incidence ℎ( ) ( ). A series of criteria in the probability mean on the extinction of the disease, the persistence and permanence in the mean of the disease, and the existence of the stationary distribution are established. It is easily seen that the research given in [6] for the stochastic SIS epidemic model with bilinear incidence is extended to the model with general nonlinear incidence and disease-induced mortality. Particularly, we see that stochastic SIS epidemic model with standard incidence is investigated for the first time.
The researches given in this paper show that stochastic model (2) has more rich dynamical properties than the corresponding deterministic model (1). Particularly, stochastic model (2) has no endemic equilibrium. Thus, this can bring more difficulty for us to investigate model (2), but, on the other hand, this also makes model (2) have more rich researchful subjects than deterministic model (1). We can discuss not only the extinction, persistence, and permanence in the mean of disease in probability, but also the existence and uniqueness of stationary distribution, the asymptotical behaviors of solutions of stochastic model (2) around the equilibrium of deterministic model (1), and so forth.
In addition, we easily see that when intensity > 0 of the stochastic perturbation, then 0 >̃0. This shows that when 0 > 1 we still can havẽ0 < 1. Therefore, there is a very interesting and important phenomenon; that is, for