Qualitative analysis of a nonautonomous stochastic S IS epidemic model with Lévy jumps

Abstract: In this paper, we study a nonautonomous stochastic S IS epidemic model with Lévy jumps. We first establish that this model has a unique global positive solution with the positive initial condition. Then, we investigate the condition for extinction of the disease. Moreover, by constructing suitable stochastic Lyapunov function, sufficient conditions for persistence and existence of Nontrivial T-periodic solution of system are obtained. Finally, numerical simulations are also presented to illustrate the main results.


Introduction
Infectious diseases are the public enemy of human population and seriously threatening the health and life. In many infectious disease research methods, epidemic models are widely used for analyzing the spread and control of infectious diseases. Here, the researchers construct an idea of compartmental model to study the dynamics of infectious diseases, initially proposed and investigated by Kermack and McKendrick. That is, S IS epidemic model [1]. Now, Most of models are descended from the classical S IS model. Some authors [2 − 4] analyzed the dynamic behavior of the S IS epidemic model based on variable population, delay, vaccination, pulse, etc. Papers [5 − 6] have investigated the influence of age structure and stage structure on the dynamic behavior of the S IS epidemic model, and studied the extinction and persistence of the disease. In addition, many results on the S IS epidemic models with network have been reported [7 − 9].
However, In the natural world, epidemic systems are inevitably infected by some stochastic environmental noise. Hence, many authors introduce stochastic interference into deterministic models to reveal the effect of environmental variability on the dynamic of infectious disease. In general the environmental fluctuations can be modeled by Gaussian white noises. The authors [10 − 12] have qualitatively analyzed stochastic S IS epidemic model with vaccination and vertical transmission. In [13], Zhang et al. investigated stochastic S IS epidemic model with vaccination under regime switching, and sufficient conditions for the existence of a unique ergodic stationary distribution has been established. The authors [14,15] studied stochastic models of different diseases, and completed dynamic analysis and optimal control research. But Gaussian white noise can only describe a class of continuous stationary random disturbances. Otherwise, the population may suffer from sudden environmental shocks, namely, some jump type stochastic perturbation, such as, earthquakes, hurricanes, floods, tsunami, epidemic diseases (SARS, avian influenza), commercial harvesting, and so on. These phenomena are discontinuous and jumping at random time t i , i ∈ N, and the waiting time of jumps is independent. Therefore, these phenomena cannot be described by Gaussian white noise alone. For this kind of sudden, The discontinuous random disturbance of Lévy jumps can more truly reflect the changing law of such sudden things. So, it's necessary to introduce Lévy jumps process into the dynamics of infectious disease. Bao et al. [16] considered stochastic Lotka-Volterra population systems with jumps for the first time. Zhang et al. [17] studied a stochastic S IQS epidemic model with Lévy jumps. Recently, T. Caraballo et al. [18] considered a stochastic S IS model with Lévy noise perturbation. M. Naim et al. [19] discussed a stochastic S IS epidemic model with vertical transmission, specific functional response and Lévy jumps. About the knowledge of jumps, the readers can further refer to [20,21].
For example, Zhou et al [22] proposed a stochastic S IS model with Lévy jumps. as follows: where dB i (t) , i = 1, 2 are independent Brownian motion defined on the complete probability space (Ω, F , P) with a filtration {F t } t≥0 . And it is increasing and right continuous with F 0 containing all P-null sets, σ 2 i , i = 1, 2 are the intensities of B i (t). Λ stands for recruitment rate of the population; µ is the natural death rate; β reprents the disease transmission rate; α is the mortality rate from disease; λ is the recovery rate. S (t−) and I (t−) are the left limit of S (t) and I (t). γ i (δ) > −1, i = 1, 2. Y is a measurable subset of (0, +∞) , ν is a Lévy measure on Y with ν (δ) < +∞, such that N (dt, dδ) = N (dt, dδ) − ν (dδ) dt. Therein, N is an independent Poisson counting measure with characteristic measure ν.
In (1.1), they obtain a unique positive and global solution of the system. Moreover, the sufficient conditions for extinction and persistence of the disease are established.
Furthermore, many disease are not only disturbed by various environmental noises, but also affected by time and seasonal variation. such as, rubella, measles, chickenpox etc. So, it is more realistic to assume that all of the coefficients in the models are positive T-periodic continuous functions [23 − 25]. In [26], Qi et al. investigated the dynamics of a nonautonomous stochastic S IS epidemic model with nonlinear incidence rate and double epidemic hypothesis. In [27], Zhang et al. formulated a new stochastic nonautonomous S IRI epidemic model, and the thresholds of the stochastic S IRI epidemic model have been obtained. To the best of our knowledge, there are few papers to deal with nonautonomous epidemic model with Lévy jumps. Motivated by the above analysis, we assume the following nonautonomous stochastic S IS model: (1. 2) The rest of this paper is organized as follows: In Section 2, by constructing Lyapunov function, we show the existence and uniqueness of the positive solution of the system (1.2); The conditions for the extinction of the system (1.2) is given In Section 3; In Section 4, we discuss the conditions for the existence of at least one nontrivial positive T-periodic solution of the system (1.2); In Section 5, we obtain the sufficient conditions of the system (1.2) for the persistence of the epidemic disease; The conclusion and numerical experiments of our theoretical results are given in Section 6.

Existence and uniqueness of the positive solution
In this section, based on the theorem of Mao [28], the coefficients of system (1.2) are locally Lipschitz continuous [29]. In what follows, we shall prove that the solution of system (1.2) is positive and global.(i.e. the solution of system (1.2) will not explode in a finite time with probability one).

For a bounded integrable function
There is a unique and positive solution (S (t) , I (t)) ∈ R 2 + of system (1.2) with initial condition (S (0) , I (0)) ∈ R 2 + with probability one, assumptions A 1 and A 2 hold. We suppose the jump diffusion coefficient that for each h > 0 there exists L h > 0. i.e.
where κ i is positive constant, which biologically interpret intensities of Lévy jumps are not too large.
Proof. According to the theorem of Mao [28] and (A 1 ), the system (1.2) exist a unique local solution (S (0) , I (0)) on t ∈ [0, τ e ), where τ e is the explosion time. To prove that this solution is global, i.e. τ e = +∞. Let the solution defined on the 1 h 0 , h 0 , where h 0 is a sufficiently large positive number. Define the stop-time where a (t) is a positive function on [0, 1].
Using the Itô's formula, we have Denote and D is a positive constant which is independent of S , I and t. we can get that Taking the expectations of the above inequality leads to Let Therefore, we obtain τ ∞ = ∞. The proof is completed.
Then, their quadratic variations The proof is therefore completed.
Theorem 3.1. Let (S (t) , I (t)) be a solution of system (1.2) with initial value (S (0) , Then lim t→∞ sup ln Proof. By the system (1.2), one has Note that Clearly, we have where

From Lemma 3.1, it follows that lim
Applying Itô's formula to system (1.2) leads to Intergrating (3.6), we obtain According to the large number of theorem [28], we have Moreover, Deriving from (3.4) and (3.5) that This completes the proof.
Remark 3.1. Theorem 3.1 shows that the disease will die out under the influence of random perturbation. Note that even if σ 2 (t) = 0 only Lévy process is large, the disease also will die out. That is, Lévy process can suppress the spread of the disease.

Existence of Nontrivial T-periodic Solution
In this section, we discuss the existence of the nontrivial positive T-periodic solution. Lemma 4.1 [30]. Assume that (A 1 ) System has a unique global solution; (A 2 ) There is a function V (t, x) ∈ C 2 which is T-periodic in t, and satisfies the following conditions: and LV −1 outside some compact set, where the operator L is given by x) .
Then the system has a T-periodic solution. Theorem 4.1. Assume that then there exists a nontrivial positive T-periodic solution of system (1.2).
Proof. Define where V 1 = −C ln S (t) − ln I (t) − I (t) and C is a positive constant. By the Itô's formula, we have Define the T-periodic function ω (t) satisfying (4.1) Then Define Z (t) = S (t) + I (t) and V 2 = Z θ (t) , θ > 1, by Itô's formula, we obtain where Then,

Now, define a compact subset
where ε is a sufficiently small positive number. Set A C such that A ∪ A C = R 2 + and A ∩ A C = φ. Then In the set A C , we choose appropriate positive constant M and a sufficiently small positive constant ε satisfying the following inequalities

Permanence
In this section, we verify the conditions for the permanence of disease I (t). Theorem 5.1. If R 1 > 1 and lim t→∞ inf I t Q (t) T (R 1 − 1), then the disease of system (1.2) is permanent in mean.

Proof. Define
Then integrating both side of (5.1), we have By Lemma 3.1 and the large number of theorem [28], one can get that lim t→∞ ξ (t) = 0.
Remark 5.1. Theorem 5.1 shows that if the random perturbation is not large and R 1 > 1, the disease is persistent.

Numerical simulations and conclusion
Now, we will introduce some numerical simulation to support our main theoretical results.
Note that R 1 > 1 and lim t→∞ inf I t Q (t) T (R 1 − 1), That is, I (t) will lead the disease to permanent. From Fig 1, the disease will go persistent when the noises are sufficiently small.
From the above conclusion, we know that the effect of the noise cannot be ignored in model process. In addition, considering the sudden environmental shocks and the influence of seasonal changes, nonautonomous stochastic model with jumps is better than stochastic model be described by Gaussian white noise alone. In this paper, Firstly, By constructing suitable stochastic Lyapunov function, we show that the system (1.2) has a unique positive solution.
(3) If R 1 > 1, there is at least one positive T-periodic solution of system (1.2). This means the occurrence of disease may be periodic in the ecosystem.
On the basis of this article, some issues deserve further investigation. For instance, We can consider the effects of impulsive or delay perturbations in system (1.2). Furthermore, we can also explore some complex nonautonoumous stochastic epidemic models , such as S IR and S EIR model with non-Gaussian white noise. We leave these investigations for the future work.