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DYNAMICS OF AN IMPULSIVE STOCHASTIC SIR EPIDEMIC MODEL WITH SATURATED INCIDENCE RATE

In this paper, the dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate are analyzed. The existence and uniqueness of the global positive solution is proved by constructing the equivalent system without pulses. The threshold which determines the extinction and persistence of the disease is obtained. The global attraction of disease-free periodic solution is addressed. Sufficient condition for the existence of a positive periodic solution is established. These results are supported by computer simulations.


Introduction
From UK, South America to India, pulse vaccination strategy (PVS) has been widely used as a powerful way to eliminate infectious diseases [1,18]. PVS is a series of periodic vaccinations applied to susceptible group in a very short period of time [18]. Comparing with routine constant vaccination strategy, theoretical studies suggest that PVS can largely reduce the incidence of disease at lower vaccination rates because it keeps the average number of the susceptible during vaccination interval below the epidemic threshold [33]. Mathematical analysis of PVS begins with Agur et al [1], further investigations can be refered in [6,7,11,14,20,28,30,33].
Epidemic models in aforementioned papers are all described by the ordinary differential equations. However, environmental noises are ubiquitous in real world and can induce different dynamics in real system [2, 3, 5, 8-10, 34, 35]. Therefore researchers have shown great interest in stochastic epidemic models incorporated with white noises, colore noises or Lévy noises [4,12,[21][22][23]39,40]. The investigations demonstrate that environmental noises can help to suppress the disease and change the basic reproduction number of the disease [12,39]. Although we can find intensive studies in impulsive stochastic population models [24,25,31,32,41], there are few papers about impulsive stochastic epidemic models [13,36,37]. And none of them gives the threshold which determines the extinction and persistence of the disease because the hybrid of stochastic perturbation and impulsive effects adds an extra level of complexity to deal with. To explore the effect of white noises and inspired by above works, we will study a stochastic SIR epidemic model with saturated incidence rate and pulse vaccinations.
Our model is derived from the deterministic impulsive system in Jin's research [17]. The deterministic SIR epidemic model with pulses is as follows: (1.1) where S(t), I(t) and R(t) stand for the population number of the susceptible, infectious and recovery at time t respectively. The parameter µ represents the birth rate (and the natural death rate is assumed to be identical), K is total population size, β denotes the transmission rate, α reflects the disease-related death rate and λ is the recovery rate of the infective individuals. In model (1.1) the period of pulse vaccination is 1, k is the time at which we applied the pulse, and k − is the time just before applying the pulse. p is the fraction of all the susceptible to whom the vaccine is inoculated at discrete time t = k, k ∈ N . All the parameters are positive constants. For deterministic system (1.1), there exists a periodic infection-free solution (S * (t), 0, R * (t)), where Then there is the basic reproduction number R 0 = β⟨S * ⟩1 µ+α+λ . If R 0 < 1, the periodic infection-free solution (S * (t), 0, R * (t)) is globally stable; if R 0 > 1, the disease will uniformly persist and system (1.1) has a positive periodic solution [17].
One of the approaches to introduce white noises into biological models is proposed by Imhof and Walcher [16]. They give a detailed and rigorous derivation of a stochastic model by considering a discrete time Markov chain in which the random amount is supposed to be linear to the microbe population. In this paper, our approach to include random perturbation is analogous to that of Imhof and Walcher [16,21]. Here we assume that the white noises are proportional to S(t), I(t), R(t), directly influencing on the S ′ (t), I ′ (t), R ′ (t) in the model (1.1). By this way, our stochastic model takes the form of where B 1 (t) and B 2 (t) are independent standard Brownian motions with B 1 (0) = 0, B 2 (0) = 0 and σ i > 0 represents the intensity of B i (t), i = 1, 2. For the dynamic of group R has no effects on the transmission dynamics, so we omit it in system (1.2).
Corresponding to the results of deterministic model, the novelties and contributions of our paper are: • we give the threshold which determines the extinction and persistence of the disease.
• we verify the global attraction of disease-free periodic solution.
• we demonstrate the existence of positive periodic solution.
The rest of this paper is organized as follows. In Section 2, we demonstrate the existence and uniqueness of global positive solution. In Section 3, we establish the threshold which determines disease to die out or prevail. In Section 4, we show that there is a globally attractive boundary periodic solution for system (1.2). In Section 5, we prove the existence of nontrivial positive periodic solution of the system (1.2). Finally, we summarize the main results in this paper and provide a brief discussion.
Throughout this paper, unless otherwise specified, let (Ω, F , {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is right continuous and F 0 contains all P-null sets). Let B 1 (t) and B 2 (t)be the Brownian motions defined on this probability space.
For convenience, we always use the following notations.

Existence and uniqueness of global positive solution
In this section, we will prove the existence and uniqueness of global positive solution.
Before the proof, we first need the definition of the solution of stochastic differential equation with impulses (ISDE), (see [26] for details).

Definition 2.1 ( [26]
). Consider the following ISDE: with initial condition X(0). A stochastic process X(t) = (X 1 (t), . . . , X n (t)) T , t ∈ R, is said to be a solution of with probability one; (iii) for almost all t ∈ [0, t 1 ), X(t) obeys the integral equation And for almost all [t k , t k+1 ), k ∈ N , X(t) obeys the integral equation Moreover, X(t) satisfies the impulsive conditions at each t = t k , k ∈ N with probability one.
With the existence and uniqueness of solution (x(t), y(t)) to system (2.2) and (S(t), I(t)) = (W (t)x(t), y(t)), hence we can hold the existence and uniqueness of solution (S(t), I(t)) to system (1.2) for t ≥ 0 with any given initial value (S(0), I(0)) ∈ R 2 + . The proof is completed.

Extinction and persistence of the disease
In this section, based on the global existence of the solution, we shall explore the threshold which determines the disease to die out or persist. Let and R s 0 < 1, then for any given initial value (S(0), I(0)) ∈ R 2 + , the solution (S(t), I(t)) of system (1.2) has the following property: In other words, the disease will go extinct exponentially with probability one.
Proof. First, we define a 1-periodic auxiliary function h(t) which will be used later. we shall give its explicit form and calculate lim t→+∞ In the latter part of the proof, we will establish the threshold which determines the disease to go extinct or prevail. Define .
Thus we obtain the explicit expression of h(t), Because W (t) and h(t) are 1-periodic functions, it's easy to prove that By using the similar arguments as in Zhao et al. [42], we can get that if µ ≥ Applying Itô formula, we have This combined with (3.2) implies Similarly we can get that Integrating this from 0 to t and dividing t on the both sides, and combining with (3.4) we have In view of (3.3) and (3.4), it follows that lim t→+∞ 1 t Φ(t) = 0, a.s. Taking the limit superior of both sides of (3.5), and if R s 0 < 1, then it follows that lim sup For Therefore (3.6) and (3.7) means the disease I(t) will go extinct exponentially with probability one. The proof is completed.
and R s 0 > 1, then for any given initial value (S(0), I(0)) ∈ R 2 + , the disease I will persist in the sense that: Proof. From the first inequality of (3.5) and Lemma A.2 in [42], it is easy to get lim sup It is easy to verify that h(t) ≤ 1 µ W (t) from Eq. (3.1), then from the third equality of (3.5) and (3.4) we have (3.8) By Lemma 17 in [38] it can obtain .
For I(t) = y(t), the claim is proved.

Globally attractive boundary periodic solution
In this section and next section, we will prove the existence and global attraction of the disease-free periodic solution and the existence of the positive periodic solution respectively. However,the periodic solution of SDE is in the sense of distribution. For the convenience of readers, we first present the definition of the periodic solution of SDE and cite a result of the periodic solution of stochastic differential equations without impulses.
Consider the following periodic stochastic differential equation without impulse: where the vectors b(s, X), σ 1 (s, X), …,σ k (s, X) (X ∈ R l ) are continuous functions of (s, X) and satisfy the conditions: where B is a constant. Let I = {t : 0 ≤ t < +∞}, U be a given open set in R l and E = I × R l . Let C 2 denote the class of functions on E which are twice continuously differentiable with respect to x 1 , …, x l and continuously differentiable with respect to t.

Lemma 4.1 ( [19]).
Suppose that the coefficients of system (4.1) are T -periodic in t and satisfy the conditions (4.2) in every cylinder I × U , and suppose further that there exists a function V (t, x) ∈ C 2 in E which is T -periodic in t, and satisfies the following conditions: where the operator L is given by Then there exists a solution of Eq. (4.1) which is a T -periodic process.

Lemma 4.2.
Consider the following linear stochastic differential equation with initial value X(t) = x(0). Then Eq. Proof. First we construct a Lyapunov function to prove the existence of X p (t).
Next we will prove that X p (t) is globally attractive. Now X p (t) satisfies Eq.
Thus we have Then it follows that where M (t) = σ 1 B 1 (t). By the property of Brownian motion [29], we can get that lim t→+∞ M (t)/t = 0. Take limits in above equation, one can see that, This implies that X(t) → X p (t), a.s., so the periodic solution X p (t) of Eq. (4.3) is globally attractive.
Proof. Because (S(t), I(t)) = (W (t)x(t), y(t)), we just need to prove that the equivalent system (2.2) has a boundary periodic solution (x p (t), 0) which is globally attractive.
To prove the global attraction of boundary periodic solution (x p (t), 0), we should prove that y(t) tends to 0 and x(t) tends to x p (t) respectively for any solution (x(t), y(t)) under assumed conditions.
If µ ≥ 1 2 (σ 2 1 ∨ σ 2 2 ) and R s 0 < 1 are satisfied, then from theorem 3.1 we know that lim t→+∞ y(t) = 0 a.s. Combining with Eq. (3.2) that lim t→+∞ x(t)/t = 0 a.s., for any arbitrary small τ > 0, there exists a t 0 = t 0 (ω) and a set Ω τ ∈ Ω such that P Let X(t) be the solution of the equation with initial value X(0) = x(0). Then it follows from the stochastic comparison theorem that for almost all ω ∈ Ω τ , where X(t) is the solution of Eq. Obviously, x p (t) = X p (t). Then the boundary periodic solution (x p (t), 0) of System (2.2) is globally attractive. For (S(t), I(t)) = (W (t)x(t), y(t)), the boundary periodic solution (S p (t), 0) = (W (t)x p (t), 0) of System (1.2) is also globally attractive. The proof is completed. Proof. We just need to prove the existence of a periodic solution of the equivalent system (2.2) without impulses.

Existence of the nontrivial positive periodic solution
Since any (x(0), y(0)) ∈ R 2 + system (2.2) has a unique global positive solution, we take R 2 + as the whole space. It is clear that the coefficients of system (2.2) satisfy the local Lipschitz condition. Next we will testify the conditions (1), (2) of Lemma 4.1.
In order to confirm the condition (2) of Lemma 4.1, it is obvious that we only need to prove that Therefore it is easy to see that V (t, x, y) satisfies the condition (2) of Lemma 4.1.
Next we will find a closed set
The proof is complete.

Numerical simulations
In this section we give numerical simulations by Milstein's Higher Order Method [15]. We assume that the unit of time is one year and the population sizes are measured in unit of 1 million. The examples are just numerical experiments to confirm our results. Example 6.1. To illustrate the threshold of disease and the effects of the environment white noises, we choose the parameters in deterministic system and stochastic system as follows: µ = 0.08, K = 1, β = 0.87, α = 0.05, λ = 0.22, p = 0.1.
With σ 2 in denominator, the white noise σ 2 decreases the basic reproduction number of disease. From case (a), (b) and (c) in Fig. 1, we can know that in the deterministic impulsive model (1.1), I(t) tends to 0 if and only if R 0 = β⟨S * ⟩1 µ+α+λ < 1, while in the ISDE SIR model (1.2), In other words, the conditions for I(t) to become extinct in the ISDE SIR model are weaker than that in the corresponding deterministic impulsive model. Furthermore, from Theorem 3.1 one can see that I(t) tends to 0 exponentially in a speed e (µ+α+λ+ 1 2 σ 2 2 )(R s 0 −1) when R s 0 < 1. In theory, the bigger σ 2 is, the faster the As illustrated in Fig 1, above parameters can make sure the disease will die out in stochastic system. In addition, to simulate the disease-free periodic solution in the deterministic system (as showed in case (a) and (b) of Fig 2.), we decrease the transmission rate β = 0.5 for deterministic system (therefore the basic reproduction number R 0 = 0.6171 < 1). Although Example 6.2 and Example 6.3 are just pathway simulations, they can also confirm our results from another aspect.
It shows that when the disease becomes extinct, the disease-free solution S * (t) of the deterministic model will display periodic behavior after some time. The stochastic solution S(t) of stochastic model (1.2) will fluctuate in a very small neighborhood around the deterministic periodic solution when the white noise σ 1 is not so big, which indicates the existence of the positive stochastic boundary periodic solution. It also shows that the amplitude of the oscillation around the trajectory of the deterministic periodic solution depends on the intensity of white noise (σ 1 = 0.020 or σ 1 = 0.015).
We note that the pathways in case (c) and (d) of Figure 2 overlap each other very well which implies that wherever S(t) start from, the density functions of S(t) converge to the disease-free periodic solution respectively.
In summary, the simulations in Figure 2 confirm our conclusion that the diseasefree periodic solution is global attractive under assumed conditions. Moreover, the global attraction of stochastic periodic solution is analogue to the convergence of density functions to a a stationary distribution. But it is hard to demonstrate because it is a spectrum of density functions. So here we use pathway simulation to substitute it. However, if readers are interested in the convergence of density functions to a stationary distribution, it could be found in Figure 1 and 2 in Lin's paper [27]. Therefore, the condition of Theorem 5.1 holds. We can see that, after a while, the trajectory of the deterministic solution goes into periodic orbit and the pathways of stochastic solution also show some measure of periodic behavior but with oscillations. Same as Example 2, the fluctuation of the stochastic pathways also depend on the intensities of white noises (σ 1 = 0.02, σ 2 = 0.02 or σ 1 = 0.01, σ 1 = 0.02).

Conclusion
In this paper, we study a stochastic SIR model with pulse vaccinations in which we assume random effects directly influence the susceptible, infective and indirectly the recovered group. First, we transform the impulsive stochastic model into equivalent stochastic system without pulses. Then we establish the threshold R s 0 : under extra mild condition µ ≥ 1 2 (σ 2 1 ∨ σ 2 2 ), if R s 0 < 1 then the disease will go extinct; if R s 0 > 1, then the disease will prevail. We also prove that: if µ ≥ 1 2 (σ 2 1 ∨ σ 2 2 ) and R s 0 < 1 are satisfied, then there exists a disease-free periodic solution which is globally attractive; if R s 0 > 1, then there exists at least one positive periodic solution which means the disease will persist.
The environmental noises play an important role in determining the epidemic dynamics. It follows from Theorem 3.1 and Figure 1. that white noises reduce the basic reproduction number R 0 and the disease will die out if R s 0 < 1 even as R 0 > 1. Therefore, white noises help to suppress the spread of disease. According to Theorem 4.1 and Theorem 5.1, the existences of disease-free periodic solution and positive periodic solution are governed by R s 0 which indicates that the noises can influence the long time behavior of the disease.
Pulse vaccinations also have effects on the dynamic behavior of disease. From Equation