# Survival and Stationary Distribution in a Stochastic SIS Model

Authors: Yanli Zhou; Weiguo Zhang; Sanling Yuan
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

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: whenR0≤1, we show how the solution spirals around the disease-free equilibrium of deterministic system under some
conditions; when R0>1, we show that the stochastic model has a stationary distribution under
certain parametric restrictions. In particular, we show that random effects may lead the disease
to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical
simulations are carried out to illustrate the theoretical results.


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## Body

## 1. 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 [1–11]. 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 [12–20]. 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 R0≤1, then the solution of the model oscillates around the disease-free steady state, whereas, if R0>1, there is a stationary distribution. However, this field is still in its infancy.The classical epidemic models proposed by Kermack and McKendrick in 1927 [21] have been extensively investigated in the literature. The deterministic SIS model takes the following form:
(1)S(t)˙=Λ-λS(t)I(t)-μS+γI(t),I(t)˙=λS(t)I(t)-(γ+μ+ϵ)I(t),
where S(t) represents the number of susceptible individuals and I(t) denotes the number of infected individuals at time t. The influx of individuals into the susceptible is given by a constant Λ. μ is the natural death rate; ϵ represents the death rate due to disease; λ and γ represent the disease transmission coefficient and the rate of recovery from infection, respectively. Model (1) always has a disease-free equilibrium E0=(S0,I0)=(Λ/μ,0) and it is globally asymptotical stable if the basic reproduction number R0=Λλ/μ(μ+ϵ+γ)≤1, which means the disease will disappear and the entire population will become susceptible. When R0>1, E0 becomes unstable and a globally asymptotical stable endemic equilibrium E*=(S*,I*)=((γ+μ+ϵ)/λ,(Λλ-μ(γ+μ+ϵ))/λ(μ+ϵ)) appears, which implies that the disease will always prevail and persist in the population.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 S(t), I(t), influenced respectively on S(t)˙ and I(t)˙ in model (1); meantime, the disease transmission coefficient λ in model (1) is replaced by λ+σ3B˙3(t). Our model takes the following form:
(2)dS(t)=(Λ-λS(t)I(t)-μS(t)+γI(t))dt+σ1S(t)dB1(t)-σ3S(t)I(t)dB3(t),dI(t)=[λS(t)I(t)-(γ+μ+ϵ)I(t)]dt+σ2I(t)dB2(t)+σ3S(t)I(t)dB3(t),
where B1(t), B2(t), and B3(t) are mutually independent Brownian motions and σ1, σ2, and σ3 are noise intensities.The organization of this paper is as follows. In Section2, 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.


## 2. Existence and Uniqueness of the Positive Solution

Throughout this paper, let(Ω,ℱ,{ℱt}t≥0,P) be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all P-null sets). Let Bi(t)(i=1,2,3) denote the independent standard Brownian motions defined on this probability space. We also denote
(3)ℝ+2={(x,y)∈ℝ2:x>0,y>0}.
Since the S(t) and I(t) in model (2) are, respectively, the sizes of the susceptible individuals and the infected individuals at time t, they should be nonnegative. Therefore, we are only interested in positive solutions. In order to get a unique global solution (i.e., no explosion in a finite time) for any given initial value, the coefficients of the stochastic differential equation are generally required to satisfy the linear growth condition and local Lipschitz condition [26]. However, the coefficients of model (2) do not satisfy the linear growth condition, though they are locally Lipschitz continuous; hence the solution of model (2) may explode at a finite time. In what follows, we shall prove that the solution of model (2) is positive and global.Theorem 1.
For any given initial condition(S(0),I(0))∈ℝ+2, there is a unique positive solution (S(t),I(t)) of model (2) on t≥0, and the solution will remain in ℝ+2 with probability one, namely, (S(t),I(t))∈ℝ+2 for all t≥0 a.s.Proof.
Since the coefficients of the equation are locally Lipschitz continuous, for any initial value(S(0),I(0))∈ℝ+2, there is a unique local solution (S(t),I(t)) on t∈[0,τe), where τe is the explosion time. To show this solution is global, we need to show that τe=+∞ a.s. Define the stopping time
(4)τ+=inf{t∈[0,τe):S(t)≤0,I(t)≤0},
where throughout this paper we set inf ϕ=∞ (as usual ϕ denotes the empty set). Obviously, we have τ+≤τe. So, if we can prove that τ+=∞ a.s., then τe=∞ and (S(t),I(t))∈ℝ+2 a.s. for all t≥0.
Assume thatτ+<∞; then there exists a T>0 such that ℙ(τ+<T)>0. Define a C2-function V:ℝ+2→ℝ+ by
(5)V(S(t),I(t))=lnS(t)I(t).
Applying Itô’s formula, we can get
(6)dV(S(t),I(t))=1S(t)dS(t)-12S2(t)(dS(t))2+1I(t)dI(t)-12I2(t)(dI(t))2.
That is,
(7)dV(S(t),I(t))=[ΛS(t)-(2μ+ϵ+γ)+γI(t)S(t)+λS(t)===-λI(t)-σ12+σ222-σ32I2(t)+σ32S2(t)2]dt==+σ1dB1(t)+σ2dB2(t)==-σ3I(t)dB3(t)+σ3S(t)dB3(t).
Due to the positivity of S(t) and I(t), it follows from the above equality that
(8)dV(S(t),I(t))≥k(S(t),I(t))dt+σ1dB1(t)+σ2dB2(t)-σ3I(t)dB3(t)+σ3S(t)dB3(t),
where k(S(t),I(t))=-(2μ+ϵ+γ+λI(t))-(σ12+σ22)/2-(σ32I2(t)+σ32S2(t))/2.
Integrate both sides of (8) from 0 to t; we yield
(9)V(S(t),I(t))≥V(S(0),I(0))+∫0t‍k(S(u),I(u))du+σ1B1(t)+σ2B2(t)-∫0t‍σ3I(u)dB3(u)+∫0t‍σ3S(u)dB3(u).
It is easy to see that, for almost all ω in τ+<τe, S(t),I(t) are positive on [0,τ+) and S(τ+)I(τ+)=0. Accordingly,
(10)limt→τ+[lnS(t)+lnI(t)]=-∞.
Letting t→τ+, it follows from (9) that
(11)-∞≥V(S(0),I(0))+∫0τ+‍k(S(t),I(t))du+σ1B1(τ+)+σ2B2(τ+)-∫0τ+‍σ3I(t)dB3(u)+∫0τ+‍σ3S(t)dB3(u)>-∞
which is a contradiction; therefore we have τ+=∞ a.s. The proof is therefore completed.


## 3. Asymptotic Behavior around the Disease-Free EquilibriumE0

As mentioned above, ifR0≤1, then model (1) always has a globally asymptotically stable disease-free equilibrium E0, which means the disease will die out with the advancement of time. Noting that E0 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 E0 to reflect how the solution of model (2) spirals closely around E0. We have the following theorem.Theorem 2.
AssumeR0≤1. Then for any solution (S(t),I(t)) of model (2) with initial value (S(0),I(0))∈ℝ+2, if σ12<μ and σ22/2<μ+ϵ, we have
(12)limsupt→∞1tE∫0t‍[(S(u)-Λμ)2+I2(u)]du≤σ12Λ2mμ2,
where m=min{μ-σ12,μ+ϵ-σ22/2}.Proof.
Define a functionV:ℝ+2→ℝ+ by
(13)V(t)=V1(t)+2μ+ϵλV2(t)=(S(t)-(Λ/μ)+I(t))22+2μ+ϵλI(t).
Along the trajectories of system (2), we have
(14)dV1(t)=(S(t)-Λμ+I(t))(dS(t)+dI(t))+12(dS(t)+dI(t))2=LV1(t)dt+(S(t)-Λμ+I(t))×(σ1S(t)dB1(t)+σ2I(t)dB2(t)),dV2(t)=LV2(t)dt+σ2I(t)dB2(t)+σ3S(t)I(t)dB3(t),
where
(15)LV1(t)=(S(t)-Λμ+I(t))(Λ-μS(t)-(μ+ϵ)I(t))+σ12S2(t)+σ22I2(t)2,(16)LV2(t)=λS(t)I(t)-(μ+γ+ϵ)I(t).
From (15), we have that
(17)LV1(t)=(S(t)-Λμ+I(t))[-μ(S(t)-Λμ)-(μ+ϵ)I(t)]=+σ12(S(t)-Λ/μ+Λ/μ)2+σ22I2(t)2≤-(μ-σ12)(S(t)-Λμ)2-(2μ+ϵ)(S(t)-Λμ)I(t)=-(μ+ϵ-σ222)I2(t)+σ12(Λμ)2,
where in the last step we have used the inequality (a+b)2≤2a2+2b2. Similarly, we can have from (16) that
(18)LV2(t)=λ(S(t)-Λμ+Λμ)I(t)-(μ+γ+ϵ)I(t)=λ(S(t)-Λμ)I(t)+[λΛμ-(μ+γ+ϵ)]I(t).
Noting that R0=Λλ/μ(μ+ϵ+γ)≤1, it follows from the above that
(19)LV2(t)≤λ(S(t)-Λμ)I(t).
It follows from (17), (19), and (13) that
(20)LV(t)≤-(μ-σ12)(S(t)-Λμ)2-(μ+ϵ-σ222)I2(t)+σ12(Λμ)2.
Integrating from 0 to t on both sides of (20) and taking expectation yield
(21)EV(t)-V(0)≤-(μ-σ12)E∫0t‍(S(u)-Λμ)2du-(μ+ϵ-σ222)E∫0t‍I2(u)du+σ12(Λμ)2t.
Hence, we have that
(22)limsupt→∞1tE∫0t‍[(μ-σ12)(S(u)-Λμ)200000000000i+(μ+ϵ-σ222)I2(u)]du≤σ12(Λμ)2.
Let m=min{μ-σ12,μ+ϵ-σ22/2}; it follows from (22) that
(23)limsupt→∞1tE∫0t‍[(S(u)-Λμ)2+I2(u)]du≤σ12Λ2mμ2.
The proof of Theorem 2 is thus completed.Notice that whenσ1=0, E0 is also the disease-free equilibrium of system (2), and (20) is reduced to the following form:
(24)LV(t)≤-μ(S(t)-Λμ)2-(μ+ϵ-σ222)I2(t),
which means that LV(t) is negative definite provided that σ22/2<μ+ϵ. Therefore, we can have the following result (see [26]).Corollary 3.
Assume thatR0≤1 and σ1=0. Then the disease-free equilibrium E0 of system (2) is stochastically asymptotically stable in the large provided σ22/2<μ+ϵ.


## 4. 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. SinceE* 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.LetX(t) be a regular time-homogeneous Markov process in El (the l dimensional Euclidean space) described by stochastic equation
(25)dX(t)=b(X)dt+∑r=1k‍frdBr(t).
The diffusion matrix is defined as follows:
(26)A(x)=(aij(x)),aij(x)=∑r=1k1‍fri(x)frj(x).
There exists a bounded domain U⊂El with regular boundary Γ, having the following properties.(B.1)
In the domainU and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero.(B.2)
Ifx∈El∖U, the mean time τ at which a path emerging from x reaches the set U is finite, and supx∈KExτ<∞ for every compact subset K⊂El.Lemma 4.
If above assumptions(B.1) and (B.2) hold, then the Markov process X(t) has a stationary distribution μ(·). Let f(·) be a function integrable with respect to the measure μ. Then
(27)Px{limT⇀∞∫0T‍f(X(t))dt=∫El‍f(x)μ(dx)}
for all x∈El.Remark 5.
We can find the proof of Lemma4 in [27]. Hasmiskii [27] refers the existence of a stationary distribution with suitable density function.To validate(B.1), it suffices to prove F is uniformly elliptical in U, where Fu=b(x)·ux+[trA(x)uxx]/2; that is, there is a positive number M such that
(28)∑i,j=1k‍ai,jξiξj≥M|ξ|2,x∈U,ξ∈Rk
(see [28] and Rayleigh’s principle in [29]). To verify (B.2), it is sufficient to show that there exist some neighborhood U and a nonnegative C2-function such that and for any El∖U, LV is negative function [30].Theorem 6.
Consider stochastic model (2) with initial condition in (S(0),I(0))∈ℝ+2. Assume that R0>1 and 0<δ<min(m1S*2,m2I*2) then there exists a stationary distribution μ(·) for model (2). Here (S*,I*) is the unique endemic equilibrium of (1) and
(29)m1=μ-σ12-σ32(2μ+ϵ)I*λ,m2=μ+ϵ-σ22,δ=σ12S*2+(I*2+2μ+ϵ2λI*)σ22+σ32I*(2μ+ϵ)λS*2.
Especially, we have
(30)limsupt→∞1t∫0t‍[m1(S(u)-S*)2+m2(I(u)-I*)2]du≤δ.Proof.
Our approach is inspired by the work of Ji and Jiang [31]. Since R0>1, then there exists a unique positive equilibrium point E*=(S*,I*) of model (1). Furthermore, we have that
(31)Λ+γI*=λS*I*+μS*,λS*I*=(γ+μ+ϵ)I*.
Define a positive definite function V:E2→ℝ+:
(32)V(t)=12(S(t)+I(t)-S*-I*)2+2μ+ϵλ(I(t)-I*-I*lnI(t)I*)=V1(t)+2μ+ϵλV2(t),
where E2=Int(ℝ2+). Applying Itô’s formula and using (31), we can calculate
(33)dV1(t)=(S(t)+I(t)-S*-I*)(dS(t)+dI(t))+12(dS(t)+dI(t))2=LV1(t)+(S(t)+I(t)-S*-I*)×(σ1S(t)dB1(t)+σ2I(t)dB2(t)),dV2(t)=(1-I*I(t))dI(t)+I*2I2(t)(dI(t))2=LV2(t)+(1-I*I(t))×(σ2I(t)dB2(t)+σ3S(t)I(t)dB3(t)),
where
(34)LV1(t)=(S(t)-S*+I(t)-I*)[Λ-μS(t)-(μ+ϵ)I(t)]+12(σ12S2(t)+σ22I2(t))=(S(t)-S*+I(t)-I*)×[μS*+(μ+ϵ)I*-μS(t)-(μ+ϵ)I(t)]+12σ12(S(t)-S*+S*)2+12σ22(I(t)-I*+I*)2≤(S(t)-S*+I(t)-I*)×[-μ(S(t)-S*)-(μ+ϵ)(I(t)-I*)]+σ12(S(t)-S*)2+σ12S*2+σ22(I(t)-I*)2+σ22I*2=-(μ-σ12)(S(t)-S*)2-(2μ+ϵ)(S(t)-S*)(I(t)-I*)-(μ+ϵ-σ22)(I(t)-I*)2+σ12S*2+σ22I*2,LV2(t)=(I(t)-I*)[λS(t)-(ϵ+γ+μ)]+I*2σ22+σ32I*2S2(t)=(I(t)-I*)(λS(t)-λS*)+I*2σ22+σ32I*2(S(t)-S*+S*)2≤λ(S(t)-S*)(I(t)-I*)+I*2σ22+σ32I*(S(t)-S*)2+σ32I*S*2.
It follows from (32)-(33) that
(35)LV(t)=LV1(t)+2μ+ϵλLV2(t)≤-(μ-σ12)(S(t)-S*)2-(μ+ϵ-σ22)(I(t)-I*)2+σ12S*2+σ22I*2+I*(2μ+ϵ)2λσ22+2μ+ϵλσ32I*(S-S*)2+2μ+ϵλσ32I*S*2=(μ-σ12-2μ+ϵλσ32I*)(S(t)-S*)2-(μ+ϵ-σ22)(I(t)-I*)2+σ12S*2+σ22I*2+I*(2μ+ϵ)2λσ22+2μ+ϵλσ32I*S*2≜-m1(S(t)-S*)2-m2(I(t)-I*)2+δ,
where m1,m2, and δ are defined, respectively, in (29).
Notice that if0<δ<min(m1S*2,m2I*2), then the ellipsoid
(36)-m1(S(t)-S*)2-m2(I(t)-I*)2+δ=0
lies entirely in ℝ+2. We can take U to be any neighborhood of the ellipsoid such that U¯⊆El=ℝ+2, so for x∈U∖El, LV(t)≤-C (C is a positive constant), which implies condition (B.2) is satisfied.
On the other hand, we can write system (2) as the form of system (25):
(37)d(S(t)I(t))=(Λ-λSI-μS+γIλSI-(μ+ϵ+γ)I)dt+(σ1S0)dB1+(0σ2I)dB2+(-σ3SIσ3SI)dB3.
Here the diffusion matrix is
(38)A=(σ12S*2+σ32S*2I*2-σ32S*2I*2-σ32S*2I*2σ22I*2+σ32S*2I*2).
There is an M=min{σ12S2,σ22I2}>0, such that for all (x1,x2)∈U¯ and ξ∈ℝ2,
(39)∑i,j=12‍aijξiξj=(σ12S2+σ32S2I2)ξ12+(σ22I2+σ32S2I2)ξ22-2σ32S2I2ξ1ξ2=σ12S2ξ12+σ22I2ξ22+σ32S2I2(ξ1-ξ2)2≥σ12S2ξ12+σ22I2ξ22≥min{σ12S2,σ22I2}|ξ|2=M|ξ|2,
which shows that condition (B.1) is also satisfied. Therefore, we can conclude that stochastic model (2) has a stationary distribution μ(·).Based on (30) in Theorem 6, we can further have the following persistence result of model (2).Corollary 7.
IfR0>1 and δ<min{m1(S*)2,m2(I*)2}, then model (2) is persistent in the mean.Proof.
Obviously, (30) holds. It follows that
(40)limsupt→∞1t∫0t‍(S(u)-S*)2du≤δm1a.s.,(41)limsupt→∞1t∫0t‍(I(u)-I*)2du≤δm2a.s.
Noting also that
(42)2(S*)2-2S*S=2S*(S*-S)≤(S*)2+(S-S*)2,
we have that
(43)S≥S*2-(S-S*)22S*.
It follows from (40) and (43) that
(44)liminft→∞1t∫0t‍S(u)du≥S*2-limsupt→∞∫0t‍(S(u)-S*)22S*du≥S*2-δ2S*m1a.s.
Similarly, we have
(45)liminft→∞1t∫0t‍I(u)du≥I*2-limsupt→∞∫0t‍(I(u)-I*)22I*du≥I*2-δ2I*m2a.s.
Therefore, model (2) is persistent in the mean.


## 5. 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 R0 determines the extinction or persistence of the disease: If R0≤1, the disease will go to extinction, and if R0>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
(46)limsupt→∞lnI(t)t≤λ22σ32-(γ+μ+ϵ)-σ222a.s.
If λ2/2σ32≤(γ+μ+ϵ)+σ22/2 holds, then I(t) will go to zero exponentially with probability one.Proof.
DefineV(I(t))=lnI(t); by Itô’s formula we have
(47)lnI(t)=lnI(0)+∫0t‍[λS(u)-(γ+μ+ϵ)-σ222-σ32S2(u)2]du+σ2B2(t)+∫0t‍σ3S(u)dB3(t)=lnI(0)+∫0t‍[-σ322(S2(u)-2λσ32S(u))00000-(γ+μ+ϵ)-σ222(S2(u)-2λσ32S(u))]du+σ2B2(t)+M(t).
Clearly, we have that
(48)-σ322(S2(t)-2λσ32S(t))=-σ322(S(t)-λσ32)2+λ22σ32≤λ22σ32.
It follows from (47) that
(49)lnI(t)≤lnI(0)+∫0t‍[λ22σ32-(γ+μ+ϵ)-σ222]du+σ2B2(t)+M(t),
where M(t)=∫0t‍σ3S(u)dB3(u) is a continuous local martingale whose quadratic variation is
(50)〈M(t),M(t)〉=σ32∫0t‍S2(u)du,
which implies that
(51)limsupt→∞〈M,M〉tt<∞.
Hence, by the strong law of large numbers of martingales,
(52)limt→∞M(t)t=0a.s.
Hence, we can divide both sides of (49) by t and then let t→∞ to obtain that
(53)limsupt→∞lnI(t)t≤λ22σ32-(γ+μ+ϵ)-σ222.
The proof is therefore completed.Remark 9.
WhenR0>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 R0 provided that σ2 and σ3 are big enough such that λ2/2σ32≤(γ+μ+ϵ)+σ22/2. That is to say, large noises can lead to the extinction of disease.


## 6. Numerical Simulations

In order to make the readers understand our results more better, we will perform some numerical simulations to illustrate our theoretical results. The numerical simulation is given by the following Milstein scheme [32]. Consider the discretization of model (2) for t=0,Δt,2Δt,…,nΔt:
(54)Sk+1=Sk+(Λ-λSkIk-μSk+γIk)Δt+Sk[σ1ξ1,kΔt+12σ12(ξ1,k2-1)Δt]+SkIk[σ3ξ3,kΔt+12σ32(ξ3,k2-1)Δt],Ik+1=Ik+[λSkIk-(γ+μ+ϵ)Ik]Δt+Ik[σ2ξ2,kΔt+12σ22(ξ21,k2-1)]Δt+SkIk[σ3ξ3,kΔt+12σ32(ξ3,k2-1)Δt],
where Δt is time increment and ξ1,k, ξ2,k, and ξ3,k(k=1,2,…,n) are independent Gaussian random variables N(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 σ1, σ2, and σ3. 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 Figures1 and 2, we always choose initial value S(0)=0.8,I(0)=0.8, and parameters Λ=1, λ=0.4, μ=0.4, ϵ=0.4, and γ=0.4 with different intensities of white noise which satisfy the conditions in Theorem 2. We can easily compute that R0=0.8333<1, E0=(Λ/μ,0)=(2.5,0). By Matlab software, we simulate the solution of model (2) with different values of σi,i=1,2,3 and the solution of model (1).The trajectories of deterministic system (1) and stochastic system (2) with initial value (S(0),I(0))=(0.8,0.8). Here Λ=1,λ=0.4,μ=0.4,ϵ=0.4, and γ=0.4; (a) σ1=0.1, σ2=0.1, and σ3=0.1; (b) σ1=0.05, σ2=0.05, and σ3=0.05.
(a)
(b)The trajectories of deterministic system (1) and stochastic system (2) with initial value (S(0),I(0))=(0.8,0.8). Here Λ=1, λ=0.4, μ=0.4, ϵ=0.4, and γ=0.4; (a) σ1=0.05, σ2=0.01, and σ3=0.01; (b) σ1=0, σ2=0.01, and σ3=0.01.
(a)
(b)In Figure1(a), we choose σ1=0.1, σ2=0.1, and σ3=0.1, and in Figure 1(b), we choose σ1=0.05, σ1=0.05, and σ3=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 takeσ1 the same value as in Figure 1(b), but take σ2,σ3 smaller values σ2=σ3=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 σ2 and σ3 have little effects on the solution.Whenσ1=0, then E0 becomes the disease-free equilibrium of model (2). We can see from (20) that
(55)LV(t)≤-0.4(S(t)-2.5)2-0.79995I2(t)≤0;
that is, the solution of model (2) is stochastically asymptotically stable in the large.In the following, we consider the the long behavior of model (2) in the case of R0>1. Choose the parameters
(56)Λ=2,λ=0.4,μ=0.4,ϵ=0.2,γ=0.2.
We can compute that R0=2.5, E*=(S*,I*)=(2,2). We start our numerical simulation with environmental forcing strengths σ1=0.1, σ2=0.1, and σ3=0.1 and starting from the initial point (1.9,1.9). Figure 3 reports the result of one simulation run and stationary distribution about the populations of susceptibles and infectives. After some initial transients the population densities fluctuate around the deterministic steady-state values S*=2 and I*=2, respectively. It is to compute that m1=0.34, m2=0.59, m1S*2=1.46, m2I*2=2.36, and δ=0.305. The conditions in Theorem 6 are satisfied. Accordingly, there exists a stationary distribution for model (2). From the histograms in Figure 3 we can see that the values of S(t) and I(t) are distributed normally around the mean values 2 and 2, respectively.The solution of the stochastic system (2) and its histogram with initial value (S(0),I(0))=(1.9,1.9). Here Λ=2, λ=0.4, μ=0.4, ϵ=0.2, and γ=0.2; σ1=0.1, σ2=0.1, and σ3=0.1.
(a)
(b)Next we decrease intensities of environmental forcing toσ1=0.05, σ2=0.05, and σ3=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).The solution of the stochastic system (2) and its histogram with initial value (S(0),I(0))=(1.9,1.9). Here Λ=2, λ=0.4, μ=0.4, ϵ=0.2, and γ=0.2; σ1=0.05, σ2=0.05, and σ3=0.05.
(a)
(b)Now, if we takeσi big enough, for example, σ1=0.1, σ2=0.1, and σ3=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
(57)limsupt→∞lnI(t)t≤-0.0704,
which means that I(t) 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).Figure 5
Computer simulation of a single path ofI(t) for the SDE model (2) with initial value (S(0),I(0))=(1.9,1.9) and its corresponding deterministic model (1) for parameter values Λ=2, λ=0.4, μ=0.4, ϵ=0.2, γ=0.2, σ1=0.1, σ2=0.1, and σ3=0.33.



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*Source: 592821-2013-12-04.xml*