Dynamics for a stochastic delayed SIRS epidemic model

. In this paper, we consider a stochastic delayed SIRS epidemic model with seasonal variation. Firstly, we prove that the system is mathematically and biologically well-posed by showing the global existence, positivity and stochastically ultimate boundneness of the solution. Secondly, some sufﬁcient conditions on the permanence and extinction of the positive solutions with probability one are presented. Thirdly, we show that the solution of the system is asymptotical around of the disease-free periodic solution and the intensity of the oscillation depends of the intensity of the noise. Lastly, the existence of stochastic nontrivial periodic solution for the system is obtained.


Introduction
In the real world, epidemic spreads are always affected by demographic and environmental stochasticity. However, deterministic model cannot capture these fluctuating features. Hence, stochastic differential equation models, which can describe and predict the systems more accurately than that of their deterministic counterpart (see [6,7,10,19]), play an important role in many types of braches of applied sciences including epidemic dynamics. To make epidemic models even more realistic, combining the effects of disease latency or immunity, stochastic delay differential equations have attracted much attention in the last decade (see [1,2,18]). As is well known, due to the seasonal variation, individuals life cycle, hunting, food supplies, mating habits, harvesting and so on, the birth rate, the death rate of the population and other parameters will not remain unchanged, but exhibit a more or less periodicity. For humans, seasonal diseases are inherent in the organic growth of man from infancy to old age. Some examples of such disease are measles, cholera, rotaviruses, respiratory syncytial virus (RSV), etc. [8]. Therefore, some epidemic models were formulated as dynamical systems of nonautonomous differential equations (see [16,20]).
Recently, many authors have studied epidemic models with the effect of seasonal variation and stochastically. Jin et al. [13] considered an SIRS epidemic model with general seasonal variation in the constant rate. Bai et al. considered a nonautonomous SIR model with periodic transmission rate and a constant removal rate in [3]. Xing et al. investigated the stochastic nonautonomous logistic system with time delays and obtained the persistence and extinction of system [17]. The asymptotic behavior of a nonautonomous predator-prey model with Hassell-Varley-type functional response and random perturbation was discussed by the Zhang et al. [21]. Cai et al. investigated the stochastic dynamics of a simple epidemic model incorporating the mean-reverting Ornstein-Uhlenbeck process analytically and numerically [5].
To the best of our knowledge, the existence of nontrivial positive periodic solution for stochastic nonautonomous epidemic models have been studied not by theoretical methods, but only by numerical methods [4,9,12]. Especially, few studies have discussed the dynamic behaviors of nonautonomous stochastic differential equation systems for epidemic models with time delays. In this paper, we will consider a stochastic SIRS epidemic model with the seasonal parameters and time delay, which takes the following form: dS(t) = µ(t) − µ(t)S(t) − β(t)S(t)I(t) + γ(t)I(t − τ )e −µ(t)τ dt − σS(t)I(t) dB(t), where S(t), I(t) and R(t) denote the numbers of susceptible, infective and removed individuals at time t, respectively. The instantaneous birth rate µ(t) equals to the instantaneous death rate. For all the classes, it is assumed that the death rate µ(t) is the same because we assume that deaths associated with disease are small. β(t) denotes the transmissions coefficient between compartment S(t) and I(t), and γ(t) is the recovery rate of the infective individual. It is assumed that γ(t) < µ(t), for all t 0, i.e., the per capita rate of recovery is smaller than the per capita rate of leaving the infective. µ(t), β(t), γ(t) are positive, nonconstant and continuous functions of period ω. The time delay τ > 0 represents the duration of the immunity period, and the term I(t − τ )e −µ(t)τ reflects the fact that an individual has survived from natural death in a recovery pool before becoming susceptible again. Here, the perturbation to transmission coefficient β(t) is introduced in the following form: β(t) → β(t) + σξ(t), and ξ(t) = dB(t)/dt, B(t) is a standard Brownian motions defined on a complete probability space, σ is the intensity of the white noises. We also present some notations: if f (t) is an integrable Throughout this paper, unless otherwise specified, let (Ω, {F t } t 0 , P) be a complete probability space with a filtration {F t } t 0 satisfying the usual conditions. Let For biological reasons, we assume that the initial conditions of system (1) satisfy The aim of this work is to investigate the dynamic behavior of the solutions of the diseasefree state and the existence of positive periodic solutions of the proposed stochastic delayed epidemic model (1). For this purpose, we first prove that system (1) is mathematically and biologically well-posed by showing the global existence, positivity and stochastically ultimate boundneness of the solution in Section 2. The disease extinction and persistence of disease are investigated in Section 3. The sufficient conditions for the dynamic behavior of the solutions of the disease-free state are obtained in Section 4. The existence of positive periodic solutions is proved by using the Khasmimskii's boundary periodic Markov processes in Section 5. Simulation and conclusion in Section 6 complete the paper.

Global positive solution
Model (1) describes the dynamics of a biological population. Hence, the population sizes should be nonnegative and bounded. For this reasons, we first establish the global existence, positivity, and boundedness of solutions. Let Z(t) = S(t) + I(t) + R(t). It is easy to see that dZ(t) = (µ(t) − µ(t)Z(t)) dt. So, Then one can obtain that It is clear that if Z(0) ∈ Γ , then Z(t) 1, i.e., S(t), I(t), R(t) ∈ (0, 1] for t ∈ [0, T ] a.s. Theorem 1. Assume that µ u , γ u , σ are positive real numbers. Let X(0) = (S(0), I(0), R(0)) T ∈ Γ be any initial condition. Then, there is a unique solution X(t) = (S(t), I(t), R(t)) of system (1) for t 0 and the solution will remain in R 3 + with probability one. Proof. It is clear that the coefficients of system (1) are locally Lipschitz continuous. It follows from [15] that for any initial value (2), system (1) admits a unique maximal local solution (S(t), where τ e is the explosion time. To verify this solution is global, we only need to show that τ e = ∞ a.s. For this, we consider the following stopping time: where we set inf φ = ∞ (φ denote the empty set). Clearly, τ + τ e , so if we prove that τ + = ∞ a.s., then τ e = ∞, which means that (S(t), I(t), R(t)) ∈ R 3 + a.s. for all t 0. Assume the τ + < ∞, then there exists a T > 0 such that P(τ + < T ) > 0.
Theorem 1 shows that the solutions of system (1) with positive initial value in Γ will remain positive in Γ . The properties of positivity and nonexplosion are essential for a population system. Once they have been established, we can discuss some other properties of the solutions of the system. From a biological point of view, due to the limitation of resources, the property of stochastically ultimate boundedness is more desirable than the nonexplosion property. In the following, we will show system (1) is stochastically ultimate bounded. Now, we give the following definition. (1) is said to be stochastically ultimately bounded if for any ε ∈ (0, 1), there exists a positive constant H = H(ε) such that for any initial value (S(0), In order to prove the stochastically ultimate boundness of system (1), we show that system (1) is asymptotically bounded firstly.
Theorem 2. If θ ∈ [1, ∞), and there exists a positive constant H 0 , which is independent of the initial value X(0) = (S(0), I(0), R(0)) ∈ Γ , the solution X(t) = (S(t), I(t), R(t)) of system (1) has the following property: Proof. From Theorem 1 the solutions will remain in Γ for all t 0 with probability 1 a.s. Let us choose Applying the Itô's formula, we get where For each integerk k 0 , define the stoping time by Clearly, τ k → ∞ a.s. as k → ∞. It follows from (5) that Since then Thus, Therefore, we can obtain Theorem 3. Under the assumption of Theorem 2, system (1) is stochastically ultimately bounded.
Proof. By Theorem 2, for θ = 3/2, there is a H 2 > 0 such that Let be given, we can choose H( ) = (H 2 / ) 2/3 . Therefore, by the application of the Chebyshev's inequality, one gets that This completes the proof of the theorem.

Extinction and persistence of the disease
In this section, we shall investigate the persistence and extinction of system (1) and obtain a threshold, which determines whether the disease dies out or persists. Now, we present two auxiliary lemmas, which will be used in the proof of the main results of this section.

The dynamic behaviors of the solution in the disease-free state
In this section, we analyze the dynamic behavior of the solutions of the disease-free state.
The following theorem shows that the solution of system (1) is asymptotical around of the bounding periodic solution (1, 0, 0) if β u − γ l < 0. Furthermore, the intensity of the oscillations depends of σ.
Theorem 6. If β u − γ l < 0, then system (1) is globally asymptotically stable, i.e., for any initial value X 0 ∈ Γ , the solution X(t) will tend to the disease-free periodic solution (1, 0, 0) asymptotically with probability 1. Simplifying and dividing the above inequality by t > 0, it follows that Using the fact that lim t→∞ t 0 σS(s) dB(s)/t = 0 a.s., we get Thus, for all t > 0. Since I(t) > 0 for all t > 0 a.s., we get lim t→∞ I(t) = 0 a.s. Now, from the third equation of system (1) it follows that Thus, Since Z(t) = S(t) + I(t) + R(t), it is easy to know that lim t→∞ S(t) = 1.
Then we complete the proof.
Next, we show the solution of (1) is oscillatory around of the disease-free periodic solution (1, 0, 0), and the intensity of this oscillation depends on the parameter σ (which is the intensity of the noise).
Integrating both sides of (14) from 0 to t and taking expectation, we obtain that Let K = min(2µ l , c 2 µ l ), we get that We complete the proof. http://www.journals.vu.lt/nonlinear-analysis

Existence of ω-periodic solution
In this section, we shall investigate the existence of nontrivial positive periodic solution of system (1). First, we introduce some results concerning the periodic Markov process.
Definition 2. (See [14].) A Markov process x(t) is ω-periodic if and only if its transition probability function is ω-periodic and the function P 0 (t, A) = P{X(t) ∈ A} satisfies the equation P 0 (s, A) = R n P 0 (s, dx)P(s, x, s + ω, A) ≡ P 0 (s + ω, A). Consider the following integral equation: Lemma 3. (See [22].) Suppose that the coefficient of (15) is ω-periodic in t and satisfies the conditions in every cylinder I × U , where B is a constant. And suppose further that there exists a function V (t, x) ∈ C 2 in R n , which is ω-periodic in t and satisfies the following conditions: and LV −1 out side some compact set, where the operator L is given by Then there exists a solution of (15), which is an ω-periodic Markov process.

Remark 1.
In view of the proof of Theorem 1, we can see that the local Lipschitz condition and linear growth condition are only used to guarantee the existence and uniqueness of the solution of system (1). Now, we prove the existence of nontrivial positive periodic solution to stochastic system (1). Theorem 8. If R 0 ω − σ 2 /2 > 0, then there exists an ω-periodic solution of system (1).

Simulation and conclusion
In this section, we will illustrate our analytical results by some examples with the help of numerical simulations firstly.   (1), we choose µ(t) = 0.06 + 0.012 cos πt, γ(t) = 0.05 + 0.14 sin πt, β(t) = 0.15 + 0.05 sin πt, τ = 0.1, and the initial values are taken as S(0) = 0.65, I(0) = 0.15, R(0) = 0.05. We carry out the numerical simulation with noise intensities σ = 0.1 for the stochastic differential equation model (SDE), compare to its deterministic model with σ = 0 (i.e., the SDE model (1) without noise leads to ordinary differential equation (ODE) model). By computing, R 0 ω − σ 2 /2 = 0.044 > 0, and  (1), we choose µ(t) = 0.4 + 0.55 cos t, γ(t) = 0.15 + 0.05 sin t, β(t) = 0.05 + 0.4 sin t, τ = 0.12, and the initial values are taken as S(0) = 0.55, I(0) = 0.15, R(0) = 0.15. We start our numerical simulation with noise intensities σ = 0 and σ = 0.076, respectively. By computing, we obtain R 0 ω = −0.954 < 0 and ω 0 (µ(t) − σ 2 /2) dt = 0.79 > 0, then the conditions in Theorem 5 hold, and the disease of SDE model (1) will become extinct exponentially almost surely; see Fig. 1(b). The green curve is the path of disease I(t) of SDE model, and the red curve is the path of disease I(t) of the corresponding deterministic model. In this paper, we consider a stochastic seasonal epidemic model with time delay in a population, where the stochasticity is introduced on the baseline transmission rate. We investigate the existence and uniqueness of the solution of the stochastic model and prove positivity and boundedness. In addition, we show the system is stochastically ultimately bounded. We obtain the threshold of stochastic system, which determines whether the disease occurs or not, i.e., when R 0 ω − σ 2 /2 > 0, the disease will persist, and the disease will become extinct exponentially almost surely when R 0 ω < 0. Especially, the dynamic behavior of the solution in the disease-free state is discussed, we prove that the solution of system (1) is oscillatory around of the disease-free state (1, 0, 0), and the intensity of this oscillation depends on the intensity of the noise. Furthermore, by using the Khasmimskii's boundary periodic Markov processes, the existence of stochastic nontrivial periodic solutions for the model is also obtained.
During the process of proving the persistence and extinction and the existence of positive periodic solutions of the stochastic system, we extend the method of proving nonautonomous stochastic systems to nonautonomous stochastic differential systems with time delays. It has been illustrated that this method is suitable for models with a stationnary population. We believe that this approach can also be applied to some nonautonomous and deleyed models in different areas.