Threshold Dynamics in Stochastic SIRS Epidemic Models with Nonlinear Incidence and Vaccination

In this paper, the dynamical behaviors for a stochastic SIRS epidemic model with nonlinear incidence and vaccination are investigated. In the models, the disease transmission coefficient and the removal rates are all affected by noise. Some new basic properties of the models are found. Applying these properties, we establish a series of new threshold conditions on the stochastically exponential extinction, stochastic persistence, and permanence in the mean of the disease with probability one for the models. Furthermore, we obtain a sufficient condition on the existence of unique stationary distribution for the model. Finally, a series of numerical examples are introduced to illustrate our main theoretical results and some conjectures are further proposed.


Introduction
As is well known, transmissions of many infectious diseases are inevitably affected by environment white noise, which is an important component in realism, because it can provide some additional degrees of realism compared to their deterministic counterparts. Therefore, in recent years, stochastic differential equation (SDE) has been used widely by many researchers to model the dynamics of spread of infectious disease (see [1][2][3][4][5] and the references cited therein). There are different possible approaches to include effects in the model. Here, we mainly introduce three approaches. The first one is through time Markov chain model to consider environment noise in SIS model (see, e.g., [6] and the references cited therein). The second is with parameters perturbation (see [2,5,7] and the references cited therein). The last issue to model stochastic epidemic system is to perturb around the positive equilibria of deterministic models (see, e.g., [1,8,9] and the references cited therein). Now, we consider stochastic epidemic models with parameters perturbation. The incidence rate of a disease denotes the number of new cases per unit time, and this plays an important role in the study of mathematical epidemiology. In many epidemic models, the bilinear incidence rate is frequently used (see [2,5,7,8,[10][11][12][13][14][15][16][17]), and the saturated incidence rate /(1 + ) is also frequently used (see, e.g., [18][19][20][21][22]). Comparing with bilinear incidence rate and saturated incidence rate, Lahrouz and Omari [23] and Liu and Chen [24] introduced a nonlinear incidence rate / ( ) into stochastic SIRS epidemic models. In [25], Tang Lahrouz et al. [26] studied a deterministic SIRS epidemic model with nonlinear incidence rate / ( ) and vaccination. If the transmission of the disease is changed by nonlinear incidence rate ( ) ( ), and to make the model more realistic, let us suppose that the death rates of the three classes in the population are different, then a where ( ), ( ), and ( ) denote the numbers of susceptible, infectious, and recovered individuals at time , respectively. Λ denote a constant input of new members into the susceptible per unit time. is the rate of vaccination for the new members. is the rate of vaccination for the susceptible individuals. is the natural mortality rate or the removal rate of the . is the removal rate of the infectious and usually is the sum of natural mortality rate and disease-induced mortality rate.
is the removal rate of the recovered individual. is the recovery rate of infective individual. is the rate at which the recovered individual loses immunity. represents the transmission coefficient between compartments and , and ( ) ( ) denotes the incidence rate of the disease. For biological reasons, we usually assume that functions ( ) and ( ) satisfy the following properties: (H 1 ) ( ) is two-order continuously differentiable function; ( )/ is monotonically nondecreasing with respect to ; (0) = 0 and (0) > 0.
In this paper, we extend model (1) to more general cases. As in [11], taking into account the effect of randomly fluctuating environment, we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in parameters , , , and in model (2)   (4) By the central limit theorem, the error term error (0 ≤ ≤ 3) may be approximated by a normal distribution with zero mean and variance 2 (0 ≤ ≤ 3), respectively. That is, error =̃(0, 2 ). Since these error may correlate with each other, we represent them by -dimensional Brownian motion ( ) = ( 1 ( ), . . . , ( )) as follows: where are all nonnegative real numbers. Therefore, model (4) is characterized by the following Itô stochastic differential equation: Model (6) in the special case where ( ) = , ( ) = , and = = 0 has been investigated by Yang and Mao in [11] and in the special case where 1 = 2 = 3 = 0 (1 ≤ ≤ ) and = = 0 also has been discussed in [25]. It is well known that, in a stochastic epidemic model, the dynamical behaviors, like the extinction, persistence, stationary distribution, and stability of the model, are the most interesting topics. Therefore, in this paper, as an important extension and improvement of the results given in [11,25], we aim to discuss the dynamical behaviors of model (6). Particularly, we will explore the stochastic extinction and persistence in the mean of disease with probability one and the existence of stationary distribution.
This paper is organized as follows. In Section 2, we introduce some preliminaries to be used in later sections. In Section 3, we establish the threshold condition for stochastic extinction of disease with probability one of model (6). In Section 4, we deduce the threshold conditions for the disease being stochastically persistent and permanent in the mean with probability one. In Section 5, we discuss the existence of the stationary distribution of model (6) under some sufficient conditions. In Section 6, the numerical simulations Computational and Mathematical Methods in Medicine 3 are presented to illustrate the main results obtained in this paper and some conjectures are further proposed. Finally, in Section 7, a brief conclusion is given.

Preliminaries
Through this paper, we let (Ω, F, {F } ≥0 , ) be a complete probability space with a filtration {F } ≥0 satisfying the usual conditions (that is, it is right continuous and increasing while F 0 contains all -null sets). In this paper, we always assume that stochastic model (6) is defined on probability space Firstly, on the existence and uniqueness of global positive solution for model (4) we have the following result.
This lemma can be proved by using a similar argument as in the proof of Theorem 3.1 given in [11]. We hence omit it here.
Proof. Taking integration from 0 to for model (6), we get Computational and Mathematical Methods in Medicine Hence, we have With a simple calculation from (16) we can easily obtain formula (14) with which ( ) is defined by By Lemma 2, we further have lim →∞ ( ) = 0 a.s. is positive invariant with probability one for model (6), where = min{ , , }.

Remark 6.
When ̸ = in model (6), whether we can also establish a similar result as in Lemma 5 still is an interesting open problem.

Lemma 7 (see [27]). Assume that there is a bounded open subset in with a regular (i.e., smooth) boundary such that
(i) there exist some = 1, 2, . . . , and positive constant > 0 such that ( ) ≥ for all ∈ ; (ii) there exists a nonnegative function ( ) : → such that ( ) is second-order continuously differentiable function and that, for some > 0, Then (30) has a unique stationary distribution . That is, if function is integrable with respect to the measure , then for To study the permanence in mean with probability one of model (6) we need the following result on the stochastic integrable inequality.
Remark 10. Condition (b) in Theorem 9 can be rewritten in the following form: It is clear that Therefore, when condition (b) holds, from (58) we also havẽ Remark 11. From Remark 10 above, we see that in Theorem 9 if condition (a) holds, then we directly havẽ0 < 1, and if condition (b) holds, then we also havẽ0 < 1. Therefore, an interesting open problem is whether we can establish the extinction of disease with probability one for model (6) only wheñ0 < 1.

Remark 13.
In the proof of Theorem 12, we easily see that three constants 0 , = sup 0≤ ≤ 0 { / ( )}, and * given in (74) are dependent on every solution ( ( ), ( ), ( )) of model (6). This shows that in Theorem 12 we only obtain the stochastic persistence in the mean of the disease.

Case
. When ( ) = and 1 = 2 = 3 = 0 (1 ≤ ≤ ) in model (6), we havẽ In order to obtain the stochastic permanence in the mean with probability one for model (6), we need to introduce a new threshold value Obviously, we have 0 ≤̃0.
Proof. We here use the Lyapunov function method to prove this theorem. The proof given here is similar to Theorem 5.1 Computational and Mathematical Methods in Medicine 13 in [11]. But, due to nonlinear function ( ), the Lyapunov function structured in the following is different from that given in [11]. By Lemma 7, it suffices to find a nonnegative Lyapunov function ( ) and compact set ⊂ 3 + such that ( ) ≤ − for some > 0 and ∈ 3 + / . Denote = ( , , ) ∈ 3 + . Define the function Calculating 1 ( ), we have Define the function Calculating 2 ( ), we have Define the function Calculating 3 ( ), we get Define the function Calculating 4 ( ), we get Define the Lyapunov function for model (6) as follows: If condition (96) holds, then the surface lies in the interior of 3 + . Hence, we can easily obtain that there exists a constant > 0 and a compact set of 3 + such that, for any ∈ 3 + / , Therefore, model (6) has a unique stationary distribution. This completes the proof.
Remark 17. In fact, the variances of errors usually should be small enough to justify their validity of real data; otherwise, the data may not be considered as a good one. It is clear that when are very small, condition (96) is always satisfied.

Numerical Examples
To verify the theoretical results in this paper, we next give numerical simulations of model (6). Throughout the following numerical simulations, we choose = 2 and ( ) = /(1 + 2 ), where is a positive constant. It is easy to verify that assumption ( 1 ) holds. By Milstein's higher-order method [29,30], we drive the corresponding discretization equations of model (6): Here, ( = 1, 2, . . . , = 1, . . . , ) are (0, 1)-distributed independent Gaussian random variables and Δ > 0 is time increment.  (6) is still stochastically extinct with probability one. Therefore, as an improvement of Theorem 9, we have the following interesting conjecture. also stochastically persistent with probability one. Therefore, as an improvement of Theorem 12, we have the following interesting conjecture.  By computing, we obtain 0 = 0.8687 < 1 and̃0 = 1.2931 > 1. The numerical simulations given in Figure 3 show that disease ( ) of model (6) is still stochastically permanent in the mean. Therefore, combining Theorem 12 and Theorem 14, we can obtain the following interesting conjecture about the stochastic permanence in the mean of disease ( ).
That is, all conditions in Theorem 16 are satisfied. The stationary distributions about the susceptible, infected, and removed individuals obtained through the numerical simulations are reported in Figure 4, which shows that after some initial transients the population densities fluctuate around the deterministic steady-state values * = 1.4230, * = 0.3845, and * = 0.1372.

R(t)
Relative frequency density That is, the conditions in Theorem 16 are not satisfied. However, we obtain that threshold valuẽ0 = 2.7192 > 1. The numerical simulations given in Figure 5 show the stationary distributions about the susceptible, infected, and removed individuals. Therefore, we can obtain the following interesting conjecture about the stationary distribution for model (6), as described in the conclusion part.

Conclusion
In this paper, as an extension of the results given in [11,25], we investigated the dynamical behaviors for a stochastic SIRS epidemic model (6) with nonlinear incidence and vaccination. In model (6), the disease transmission coefficient and the removal rates , , and are affected by noise. Some new basic properties of model (6) are found in Lemmas 2, 3, and 5. Applying these lemmas, we established a series of new threshold value criteria on the stochastic extinction, persistence in the mean, and permanence in the mean of the disease with probability one. Furthermore, by using the Lyapunov function method, a sufficient condition on the existence of unique stationary distribution for model (6) is also obtained.
In fact, under the above case, from the proofs of Theorems 12 and 14, we can see that an important question is to deal with terms ( ( )) and 2 ( ( )) ( ( )). If we may get ( ( )) ≥ ( 0 ) + V 1 ( ( ) − 0 ) a.s., 2 ( ( )) ( ( )) ≤ 2 ( 0 ) (0) where V 1 and V 2 are two positive constants; then the following perfect result may be established. Assume that ( 1 ) holds. If̃0 > 1, then disease in model (6) is stochastically persistent in the mean; that is, Another important open problem is about the existence of stationary distribution of model (6), that is, whether we can establish a similar result as in Theorem 16 when ( ) is a nonlinear function. The best perfect result on the stationary distribution is to prove that model (6) possesses a unique stationary distribution only when threshold valuẽ0 > 1. But this is a very difficult open problem.
However, the numerical examples given in Section 6 propose some affirmative answer for above open problems.