Survival and Stationary Distribution in a Stochastic SIS Model

The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: when R 0 ≤ 1, we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; when R 0 > 1, we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that randomeffectsmay lead the disease to extinction in scenarioswhere the deterministicmodel predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results.


Introduction
Mathematical epidemiology describing the population dynamics of infectious diseases has been made a significant progress in better understanding of disease transmissions and behavior of epidemics.Many epidemic models have been described by ordinary differential equations [1][2][3][4][5][6][7][8][9][10][11].These important and useful deterministic investigations offer a great insight into the effects of infectious disease, but in the real world, epidemic dynamics is inevitably affected by the environmental noise, which is an important component in the epidemic systems.As a matter of fact, the epidemic models are often subject to environmental noise; that is, due to environmental fluctuations, parameters involved in epidemic models are not absolute constants, and they may fluctuate around some average values.So, inclusion of random perturbations in such models makes them more realistic in comparison to their deterministic counterparts.In recent years, epidemic models under environmental noise described by stochastic different equations have been studied by many researchers.They introduce stochastic noises into deterministic models to reveal the effect of environmental variability on the epidemic dynamics in mathematical ecology [12][13][14][15][16][17][18][19][20].For example, Chen and Li [12] discussed the stability of the endemic equilibrium of a stochastic SIR model.Tornatore et al. [13] studied the stability of the disease-free equilibrium of a stochastic SIR model with and without distributed time delay.Ji et al. [14] discussed a multigroup SIR model with perturbation, where they showed that if the basic reproduction number  0 ≤ 1, then the solution of the model oscillates around the disease-free steady state, whereas, if  0 > 1, there is a stationary distribution.However, this field is still in its infancy.
As stated above, due to the existence of environmental noise, the parameters of model ( 1) are not absolute constants, and they exhibit continuous oscillation around some average values but do not attain fixed values with the advancement of time.Considering the biological significance and the effects of stochastic environmental noise perturbations, there are different possible approaches to include random effects in the model.The following four approaches are usually adopted in the literature (see [22]).The first one is through time Markov chain model to consider environment noise [23].The second is with parameters perturbation [24].The third one is the environmental noise that is proportional to the variables [25], and the last one is to robust the positive equilibria of deterministic models.In this paper, we will consider a stochastic counterpart of model ( 1) in a combination of the second and third approaches.That is, the stochastic perturbation is assumed to be of a white noise type which is directly proportional to (), (), influenced respectively on Ṡ() and İ() in model ( 1); meantime, the disease transmission coefficient  in model ( 1) is replaced by + 3 Ḃ 3 ().Our model takes the following form: where  1 (),  2 (), and  3 () are mutually independent Brownian motions and  1 ,  2 , and  3 are noise intensities.
The organization of this paper is as follows.In Section 2, we show that there is a unique positive solution of model (2) for any positive initial value.In Section 3, we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions.In Section 4, we prove that model (2) has a stationary distribution under certain parametric restrictions.In Section 5, an extinction result due to large noises is presented.Finally, numerical simulations are carried out to illustrate the main theoretical results.

Existence and Uniqueness of the Positive Solution
Throughout this paper, let (Ω, F, {F  } ≥0 , ) be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and F 0 contains all -null sets).Let   () ( = 1, 2, 3) denote the independent standard Brownian motions defined on this probability space.We also denote where throughout this paper we set inf  = ∞ (as usual  denotes the empty set).Obviously, we have  + ≤   .So, if we can prove that  + = ∞ a.s., then   = ∞ and ((), ()) ∈ R 2 + a.s.for all  ≥ 0.

Asymptotic Behavior around the Disease-Free Equilibrium 𝐸 0
As mentioned above, if  0 ≤ 1, then model (1) always has a globally asymptotically stable disease-free equilibrium  0 , which means the disease will die out with the advancement of time.Noting that  0 is not an equilibrium of stochastic model (2), it is natural to ask whether the disease will go to extinction in the population.In this section we mainly use the way of estimating the oscillation around  0 to reflect how the solution of model ( 2) spirals closely around  0 .We have the following theorem.
Proof.Define a function  : R 2 + → R + by Along the trajectories of system (2), we have where From ( 15), we have that where in the last step we have used the inequality ( + ) 2 ≤ 2 2 + 2 2 .Similarly, we can have from ( 16) that Noting that  0 = Λ/( +  + ) ≤ 1, it follows from the above that It follows from ( 17), (19), and (13) that Integrating from 0 to  on both sides of (20) and taking expectation yield Hence, we have that lim sup The proof of Theorem 2 is thus completed.

Existence of the Stationary Distribution
It is well known that, in the study of epidemic dynamical model, the endemic equilibrium, which means that disease will prevail and persist in a population, is one of the most important and interesting topics owing to its theoretical and practical significance.Since  * is not an epidemic equilibrium of stochastic model (2), in this section, we turn to prove the existence of its stationary distribution.Before proving the main theorem of this section we first cite a known result from Hasmiskii [27] which will be useful to prove the theorem.
Let () be a regular time-homogeneous Markov process in   (the  dimensional Euclidean space) described by stochastic equation The diffusion matrix is defined as follows: There exists a bounded domain  ⊂   with regular boundary Γ, having the following properties.
for all  ∈   .
Remark 5. We can find the proof of Lemma 4 in [27].
Hasmiskii [27] refers the existence of a stationary distribution with suitable density function.
To validate (B.1), it suffices to prove  is uniformly elliptical in , where  = () ⋅   + [tr ()  ]/2; that is, there is a positive number  such that (see [28] and Rayleigh's principle in [29]).To verify (B.2), it is sufficient to show that there exist some neighborhood  and a nonnegative  2 -function such that and for any   \ ,  is negative function [30].
On the other hand, we can write system (2) as the form of system (25): Here the diffusion matrix is There is an which shows that condition (B.1) is also satisfied.Therefore, we can conclude that stochastic model ( 2) has a stationary distribution (⋅).
Based on (30) in Theorem 6, we can further have the following persistence result of model (2).
Proof.Obviously, (30) holds.It follows that lim sup Noting also that we have that It follows from ( 40) and ( 43) that lim inf Similarly, we have lim inf Therefore, model ( 2) is persistent in the mean.

Extinction of the Disease
In this section, our goal is to find the conditions under which the disease will go to extinction.In the previous section we have showed that under certain conditions, deterministic model (1) and the associated stochastic model (2) behave similarly in the sense that both have positive solutions which will not explode to infinity in a finite time.In other words, we show that under certain condition the noise will not spoil these properties.For the deterministic epidemic model (1), the value of the basic reproduction number  0 determines the extinction or persistence of the disease: If  0 ≤ 1, the disease will go to extinction, and if  0 > 1, then the disease will be persistent in the population.However, we will show in this section that if the noise is sufficiently large, the disease will become extinct for stochastic model (2), although it may be persistent for its deterministic version (1).The following theorem gives a condition for the extinction of the disease expressed in terms of intensities of noise and system parameters.
Theorem 8. Consider stochastic model (2) with initial condition in R 2 + .We have Clearly, we have that 3 It follows from (47) that ln  () ≤ ln  (0 where () = ∫ Hence, we can divide both sides of (49) by  and then let  → ∞ to obtain that lim sup The proof is therefore completed.
Remark 9. When  0 > 1, the positive solution converges to the endemic equilibrium of deterministic model (1).However, the disease will die out exponentially regardless of the magnitude of  0 provided that  2 and  3 are big enough such that  2 /2 2 3 ≤ ( +  + ) +  2 2 /2.That is to say, large noises can lead to the extinction of disease.
where Δ is time increment and  1, ,  2, , and  3, ( = 1, 2, . . ., ) are independent Gaussian random variables (0, 1) which can be generated numerically by pseudorandom number generators.In order to understand their role on the dynamics, we use different values of  1 ,  2 , and  3 .In all the following figures, the blue lines and the red lines represent solutions of the deterministic system (1) and the stochastic system (2), respectively.
In Figure 1(a), we choose  1 = 0.1,  2 = 0.1, and  3 = 0.1, and in Figure 1(b), we choose  1 = 0.05,  1 = 0.05, and  3 = 0.05.We can see from Figure 1 that the solution of model (2) will oscillate around the disease-free equilibrium in time, and moreover, the larger the intensities of the white noises are, the larger the fluctuations of the solutions will be.
Furthermore, if we take  1 the same value as in Figure 1(b), but take  2 ,  3 smaller values  2 =  3 = 0.01.We can see from Figure 2(a) that the fluctuation of the solution will be very smaller.That is to say, the intensities of  2 and  3 have little effects on the solution.
Next we decrease intensities of environmental forcing to  1 = 0.05,  2 = 0.05, and  3 = 0.05 and again we observe that the population distribution fluctuates around the deterministic steady-state value but amplitude of fluctuation is less compared to earlier case (see Figure 4), which is also reflected in their stationary distributions.In Figure 4, the population of susceptible is distributed within (1.5, 2.6) and the population of infected remains within the range (1.4,2.8), while in Figure 3 they are, respectively, distributed within (1, 3.1) and (0.8, 3.4).Now, if we take   big enough, for example,  1 = 0.1,  2 = 0.1, and  3 = 0.33, other parameters take the same values as in (56).We can verify that the conditions in Theorem 8 are satisfied.Moreover, we have lim sup which means that () will go to zero exponentially with time (see Figure 5).That is to say, large noises can lead the disease to extinction, which is a phenomenon different from its corresponding deterministic model (1).
), where   is the explosion time.To show this solution is global, we need to show that   = +∞ a.s.Define the stopping time B.1) In the domain  and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix () is bounded away from zero.(B.2) If  ∈   \, the mean time  at which a path emerging from  reaches the set  is finite, and sup ∈    < ∞ for every compact subset  ⊂   .