# Asymptotic Dynamics of a Stochastic SIR Epidemic System Affected by Mixed Nonlinear Incidence Rates

Authors: Ping Han; Zhengbo Chang; Xinzhu Meng
Journal: Complexity
 (2020)

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## Abstract

This paper considers a stochastic SIR epidemic system affected by mixed nonlinear incidence rates. Using Markov semigroup theory and the Fokker–Planck equation, we explore the asymptotic dynamics of the stochastic system. We first investigate the existence of a positive solution and its uniqueness. Furthermore, we prove that the stochastic system has an asymptotically stable stationary distribution. In addition, the sufficient conditions for disease extinction are also obtained, which imply that the white noise can suppress and control the spread of infectious diseases. Finally, in order to illustrate the analytical results, we give some numerical simulations.


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## Body

## 1. Introduction

In 1927, the threshold conditions for disease transmission were established by Kermack and McKendrick [1]. In recent decades, many studies have applied threshold theory to various epidemic systems, and there have been a large amount of results related to the dynamical behaviors for various models. In addition, nonlinear incidence rates are very significant and are used frequently in the dynamics of epidemic models [2–11]. Hethcote [12] introduced an SIR epidemic system affected by bilinear incidence rate λSI:(1)dS=u−uS−λSIdt,dI=λSI−uI−ηIdt,dR=ηI−uRdt,where the parameters u, λ, and η are positive constants. S, I, and R represent the density of susceptible, infectious, and recovered individuals, respectively. Suppose that the recruitment rate is the same as the natural death rate, denoted by u. λ is the transmission coefficient, η represents the recovery rate, and λ1SI represents the bilinear incidence rate. Furthermore, some researchers [13] proposed SIR epidemic models with saturated incidence λSI/1+mI similar to the following form:(2)dS=u−uS−λSI1+mIdt,dI=λSI1+mI−uI−ηIdt,dR=ηI−uRdt,where m is a positive constant. λ2SI/1+mI represents the saturated incidence rate. Particularly, Liu et al. [14] studied an SIS epidemic system with general nonlinear incidence rate effect by employing Markov semigroup theory. Inspired by previous works, we consider that the infected person may have immunity to return to recovery class after being cured or return to the susceptible class again. And the epidemic affected by mixed nonlinear incidence rates is more realistic compared to epidemic affected by a single nonlinear incidence rate. Thus, when two different incidence rates (bilinear and saturated incidence rates) are considered at the same time, the following deterministic SIR epidemic system affected by two different nonlinear incidence rates is studied:(3)dS=u−uS−pλ1SI−1−pλ2SI1+mIdt+1−αηIdt,dI=pλ1SI+1−pλ2SI1+mI−uI−ηIdt,dR=αηI−uRdt,where the parameters α, λ1, λ2, and p are positive constants. α denotes the probability that the infected returns to recovery class, 1−α denotes the probability that the infected returns to susceptible class correspondingly, λ1 and λ2 are the transmission coefficients, p represents the probability that bilinear incidence rate occurs, and 1−p represents the probability that saturated incidence rate occurs correspondingly. The basic reproduction number R0=pλ1+1−pλ2/u+η is the threshold of model (3).However, the population systems with stochastic effect present some complex dynamics, which attracts the attention of widespread researchers [15–18]. There are many kinds of noise in the environment, among which the white noise is a relatively stable noise in the process of propagation. It has been widely used and studied in physics and has a relatively complete system in mathematical analysis and application. In order to consider a more realistic disease model and make it more practical to study the influence of environmental noise on infectious diseases, the white noise is used. There is a long history of using the white noise to depict the influence of environmental randomness on the spread of disease; some researchers [19, 20] put forward that the environmental white noise disturbs the system parameters stochastically, and parameter λii=1,2 is an important parameter for the spread of disease. The researchers [21–29] investigated the effect of environment on the dynamic behaviors by introducing stochastic perturbation into deterministic models. Based on the discussion above, we assume that λi⟶λi+σiBi.t, where Bit represents a standard Brownian motion with intensity σi>0i=1,2. Now, the stochastic SIR epidemic system corresponding to system (3) is as follows:(4)dS=u−uS−pλ1SI−1−pλ2SI1+mIdt+1−αηIdt−pσ1SIdB1t−1−pσ2SI1+mIdB2t,dI=pλ1SI+1−pλ2SI1+mI−uI−ηIdt+pσ1SIdB1t+1−pσ2SI1+mIdB2t,dR=αηI−uRdt.SincedS+I+R=u−uS+I+Rdt, for any initial value S0,I0,R0∈ℝ+3 and S0+I0+R0=1, we always have St+It+Rt=1. Therefore, St,It,Rt∈ℝ+3St+It+Rt=1 is a positive invariant set of system (4). For simplicity, we let Rt=1−St−It; then, the dynamics of system (4) is equivalent to the following two-dimensional system:(5)dS=u−uS−pλ1SI−1−pλ2SI1+mIdt+1−αηIdt−pσ1SIdB1t−1−pσ2SI1+mIdB2t,dI=pλ1SI+1−pλ2SI1+mI−uI−ηIdt+pσ1SIdB1t+1−pσ2SI1+mIdB2t.Next, we study the dynamical behaviors of system (5) which are affected by mixed nonlinear incidence rates. Particularly, as the main purpose, we study the asymptotically stable stationary distribution and extinction of epidemic by establishing the corresponding sufficient conditions.In this paper,Ω,ℱ,ℱt≥0,ℙ represents a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions (i.e., it is increasing and right continuous, while ℱ0 contains all ℙ-null sets). Define Bt is on the probability space. Denote ℝ+2=xi>0,i=1,2.


## 2. Preliminary Knowledge

### 2.1. Markov Semigroups and Fokker–Planck Equation

In this section, we provide some definitions about Markov semigroups [14], asymptotic properties [30–34], and Fokker–Planck equation to verify our results.DenoteX, Σ, and m to be a metric space, σ-algebra of Borel sets, and the Lebesgue measure on X,Σ, respectively. D=sh∈L1:h≥0,h=1. Ptt≥0 represents an integral Markov semigroup with a continuous kernel Kt;x;y for t>0, which satisfies ∫XKx,ymdx=1 for ∀y∈X. For the diffusion process St,It, i.e., the transition probability function is Pt,x,y,A=ProbSt,It∈A. Define φ1x,y=x2y2, φ2x,y=x2y2/1+my2, and(6)f1x,y=u−ux−pλ1xy−1−pλ2xy1+my+1−αηy,f2x,y=pλ1xy+1−pλ2xy1+my−uy−ηy.Remark 1.
If for∀h∈D, we have(7)∫0∞Pthdt>0,a.e.
Then, this semigroup is asymptotically stable or sweeping with respect to compact sets [30, 31]; this property is called the Foguel alternative [34].Remark 2.
If the distribution ofSt,It is absolutely continuous with respect to the Lebesgue measure with the density Ut,x,y, then U satisfies the Fokker–Planck equation [32]:(8)∂U∂t=σ12p22∂2φ1U∂x2−2∂2φ1U∂x∂y+∂2φ1U∂y2+σ221−p22∂2φ2U∂x2−2∂2φ2U∂x∂y+∂2φ2U∂y2−∂f1U∂x−∂f2U∂y.Remark 3.
DenotePtVx,y=Ux,y,t for V∈D, and A represents the infinitesimal generator of semigroup Ptt≥0; then, we have(9)A∗V=1−p2σ22φ22∂2V∂x2−2∂2V∂x∂y+∂2V∂y2+p2σ12φ12∂2V∂x2−2∂2V∂x∂y+∂2V∂y2+∂f1V∂x+∂f2V∂y.


### 2.2. Some Lemmas about the Asymptotically Stable Stationary Distribution

In this section, five lemmas are given to study the asymptotically stable stationary distribution of system (5).Lemma 1.
The transition probability functionPt,x0,y0,A has a continuous density Kt,x,y;x0,y0 with respect to the Lebesgue measure.Proof.
Denote(10)a⟶x,y=a1a2,b1⟶x,y=−pσ1xypσ1xy,b2⟶x,y=−1−pλ2xy1+my1−pλ2xy1+my,where a1=u−ux−pλ1xy−1−pλ2xy/1+my+1−αηy, a2=pλ1xy+1−pλ2xy/1+my−uy−ηy, and x,y∈ℝ+2. a⟶,bi⟶i=1,2 is defined as(11)a⟶,bi⟶jx=∑k=12ak∂bji∂xk−bki∂aj∂xkx,j=1,2.
Then,(12)a⟶,bi⟶=a1∂b1i∂x−b1ia1∂x+a2∂b1i∂y−b2ia1∂ya1∂b2i∂x−b1ia2∂x+a2∂b2i∂y−b2ia2∂y.
Denoteb1i=−b2i, and by direct calculation, we have(13)b1⟶a⟶,b1⟶=b11a1∂b11∂x−b11a1∂x+a2∂b11∂y−b21a1∂yb21a1∂b21∂x−b11a2∂x+a2∂b21∂y−b21a2∂y=αηp2σ12x2y2>0,b2⟶a⟶,b2⟶=b12a1∂b12∂x−b12a1∂x+a2∂b12∂y−b22a1∂yb22a1∂b22∂x−b12a2∂x+a2∂b22∂y−b22a2∂y=αη1−p2σ22x2y21+my2>0,which shows that bi⟶ and a⟶,bi⟶ are independent on ℝ+2. Thus, according to Ho..rmander theorem (see [35]), Pt,x0,y0,A has a continuous density Kt,x,y;x0,y0 and K∈C∞0,∞×ℝ+2×ℝ+2. The proof is completed. In addition, we can obtain that the semigroup Ptt≥0 is an integral Markov semigroup.Lemma 2.
For eachx0,y0∈E and x,y∈E, where suppU∗=x,y∈ℝ+2:u/u+αη<x+y<1≔E, there exists T>0 such that KT,x,y;x0,y0>0.Proof.
Fix a pointx0,y0∈ℝ+2 and a continuous function ϕ∈L20,T;ℝ; the following system is first considered:(14)xϕt=x0+∫0tf1xϕs,yϕs−pσ1ϕxϕsyϕsds−∫0t1−pσ2ϕxϕsyϕs1+myϕsds,yϕt=y0+∫0tf2xϕs,yϕs+pσ1ϕxϕsyϕsds+∫0t1−pσ2ϕxϕsyϕs1+myϕsds.Step 1.
FindDx0,y0;ϕi by the perturbation method. Denote f′, g1′, and g2′ are the Jacobians of(15)f=f1x,yf2x,y,g1=−pσ1xypσ1xy,g2=−1−pσ2xy1+my1−pσ2xy1+my,respectively; then,(16)Ϝi=f′xϕt,yϕt+gi′xϕt,yϕtϕ.
For0≤t0≤t≤T, let Qit0,t0=I and ∂Qit,t0/∂t=ϜitQit,t0; we get(17)Dx0,y0;ϕih=∫0TQiT,sgishsds.
Assume that the rank ofDx0,y0;ϕi is 2 for some ϕ∈L20,T;ℝ; then, for x,y=xϕT,yϕT, we have KT,x,y;x0,y0>0.Step 2.
Prove that the derivativeDx0,y0;ϕi has rank 2. Replace system (14) with(18)xϕ′t=yϕ′t=f1xϕt,yϕt−pσ1ϕxϕtyϕt−1−pσ2ϕxϕtyϕt1+myϕt,f2xϕt,yϕt+pσ1ϕxϕtyϕt+1−pσ2ϕxϕtyϕt1+myϕt.
Defineϵ∈0,T and ht=1T−ϵ,T/xϕtyϕtfxϕt,yϕt for t∈0,T; here, 1T−ϵ,T denotes the characteristic function of interval T−ϵ,T. Using Taylor expansion, we obtain(19)QiT,s=I+ϜiTT−s+oT−s.
Then, we get(20)Dx0,y0;ϕih=∫0TI+ϜiTT−s+oT−sgishsds=ϵv+ϵ22ϜiTv+oϵ2,and v1=−σ1,σ1T,v2=−σ2,σ2T.
Then,(21)Ϝ1Tv1=σ1AB,v1Ϝ1Tv1=−σ1σ1Aσ1σ1B=σ12αη>0,Ϝ2Tv2=σ2CD,v2Ϝ2Tv2−σ2σ2Cσ2σ2D=σ22αη>0,where(22)A=pλ1+σ1ϕy−x+1−pλ2−x+y+my21+my2+1−αη+u,B=pλ1+σ1ϕx−y+1−pλ2x−y−my21+my2−u+η,C=u+pλ1x+1−pλ2x1+my2−1−pσ2x1+my2+1−αη−pλ1y−1−pλ2y1+my+1−pσ2y1+my,D=−u−pλ1x−1−pλ2x1+my2+1−pσ2x1+my2−η+pλ1y+1−pλ2y1+my−1−pσ2y1+my.
Therefore, the rank ofDx0,y0;ϕi is 2.Step 3.
Verify that, for anyx0,y0∈E and x,y∈E, there exist control function ϕ and T>0 such that xϕ0,yϕ0=x0,y0 and xϕT,yϕT=x,y. Let zϕ=xϕ+yϕ, and(23)g1x,z=u−ux−pλ1xz−x−1−pλ2xz−x1+mz−x,g2x,z=u+αηx−u+αηz.
System (18) becomes(24)xϕ′t=g1xϕt,zϕt−pσ1ϕxϕtzϕt−xϕt−1−pσ2ϕxϕtzϕt−xϕt1+mzϕt−xϕt,zϕ′t=g2xϕt,zϕt.
Denote(25)E0=x,z∈ℝ+2:0<x<1,uu+αη<z<1,x<z.
Using the same proof method of Lemma 3.2 in [36], we can obtain that, for any x0,z0∈E0 and x,z∈E0, there exist control function ϕ and T>0 such that xϕ0,zϕ0=x0,z0 and xϕT,zϕT=x,z. Now, the positivity of K is proved by support theorems (see [37–39]).Lemma 3.
IfR0−p2σ12+1−p2σ22/2u+η>1, for every density f and the semigroup Ptt≥0, the following conclusion holds:(26)limt⟶∞∬EPtfx,ydxdy=1.Proof.
SubstituteZt=St+It; we can obtain that(27)u−u+αηZt<dZtdt<u−uZt,t∈0,∞a.s.
Next, in the following three cases, (i)Z0∈0,u/u+αη, (ii) Z0∈u/u+αη,1, and (iii) Z0∈1,∞, we verify that there is t0=t0ω such that(28)uu+αη<Zt,ω<1,t>t0,for almost every ω∈Ω; this proof is completed.(i)
If inequality (28) is not true, then there exists Ω′⊂Ω with ProbΩ′>0 which satisfies(29)Zt,ω∈0,uu+αη.On the one hand, we can get thatdZt/dt>u−u+αηZt>0 from (27). Thus, Zt,ω is strictly increasing on 0,∞ for any ω∈Ω′, and then limt⟶∞Zt,ω=u/u+αη. In view of the expression of g2, we have(30)limt⟶∞St,ω=0,limt⟶∞It,ω=uu+αη.Namely,(31)limt⟶∞lnIt−lnI0t=0,ω∈Ω′.On the other hand, according to Itô’s formula, we get(32)dlnIt=pλ1S+1−pλ2S1+mI−u−ηdt−12p2σ12S2+1−p2σ22S21+mI2dt+pσ1SdB1t+1−pσ2S1+mIdB2t.Therefore,(33)lnIt−lnI0t=1t∫0tpλ1Ss+1−pλ2Ss1+mIsds−1t∫0t12p2σ12S2sds−1t∫0t121−p2σ22S2s1+mIs2ds+1t∫0t−u−ηds+M1tt+M2tt,where(34)M1t=pσ1∫0tSsdB1s,M2t=1−pσ2∫0tSs1+mIsdB2s,M1,M1t=p2σ12∫0tS2sds,M2,M2t=1−p2σ22∫0tSs1+mIs2ds.Clearly,(35)limsupt⟶∞M1,M1tt=limsupt⟶∞p2σ12∫0tS2sdst<p2σ12∫0t12dst<p2σ12<∞,a.s.,limsupt⟶∞M2,M2tt=limsupt⟶∞1−p2σ22∫0tSs/1+mIs2dst<1−p2σ22∫0t12dst<1−p2σ22<∞,a.s.By the strong law of large numbers (see [40]), we get(36)limt⟶∞M1tt=limt⟶∞M2tt=0,a.s.Namely,(37)limt⟶∞lnIt−lnI0t=limt⟶∞−u−η+pλ1∫0tSsdst+limt⟶∞1−pλ2∫0tSs/1+mIsdst−limt⟶∞p2σ12∫0tS2sds2t−limt⟶∞1−p2σ22∫0tSs/1+mIsds2t=−u+η,a.s.onΩ′,this contradiction arises. Hence, the result follows forZ0∈0,u/u+αη.(ii)
From (27), we obtain that(38)Zt>uu+α+Z0−uu+αeu+αt0e−u+αt,Zt<1+Z0−1eut0e−ut<Zt,Clearly, conclusion (28) holds in the case of Z0∈u/u+αη,1.(iii)
Suppose that inequality (28) does not hold. On the one hand, using the same way to (ii), we have limt⟶∞Zt,ω=1, ω∈Ω′, and(39)limt⟶∞St,ω=1,limt⟶∞It,ω=0.Clearly,(40)limt⟶∞lnIt−lnI0t=0,ω∈Ω′.On the other hand, ifR0−1>p2σ12+1−p2σ22/2u+η, we obtain(41)lnIt−lnI0t=pλ1+1−pλ2−u−η−p2σ12+1−p2σ222=u+ηR0−1−p2σ12+1−p2σ222u+η>0,a.s.onΩ′,so there is a contradiction. Thus, the result holds forZ0∈1,∞. This proof is completed.Lemma 4.
IfR0−p2σ12+1−p2σ22/2u+η>1, the semigroup Ptt≥0 is asymptotically stable or sweeping with respect to compact sets.Proof.
Using the same proof method of Lemma 4.5 in [14], this means that our result holds. We omit it here.Lemma 5.
The semigroupPtt≥0 is asymptotically stable if R0−p2σ12+1−p2σ22/2u+η>1.Proof.
By system (5), we obtain that(42)L−lnI=−pλ1S−1−pλ2S1+mI+u+η+p2σ12S2+1−p2σ22S/1+mI22≤−pλ1S−1−pλ2S1+mI+u+η+p2σ12+1−p2σ222=−pλ1S−1−pλ2S+u+η+p2σ12+1−p2σ222+1−pλ2S−1−pλ2S1+mI=−pλ1S−1−pλ2S+u+η+p2σ12+1−p2σ222+mI1+mI1−pλ2S.
Denote(43)f1S,I=u−uS−pλ1SI−1−pλ2SI1+mI+1−αηI,f2S,I=pλ1SI+1−pλ2SI1+mI−uI−ηI,(44)A∗V=F∂2V∂S2−2∂2V∂S∂I+∂2V∂I2+∂f1V∂S+∂f2V∂I,where F=S2I2p2σ12+1−p2σ221/1+mI2/2, and we call V a Khasminskii˘ function [41].
P
t
t
≥
0 satisfies the Foguel alternative according to Lemma 4. We can exclude sweeping by constructing a nonnegative C2-function V and a closed set U∈Σ satisfying(45)supS,I∈X\UA∗VS,I<−1,Step 1.
Construct a nonnegativeC2-function V. For S,I∈E, define(46)gS,I=G−lnI+pλ1+1−pλ2u−S−I−lnS−ln1−S−I−lnS+I−uu+αη,where G>0 such that(47)−Gu+ηR0−1−p2σ12+1−p2σ222u+η+3u+αη+pλ1+1−pλ2+p2σ12+1−p2σ222≤−2.
Let(48)VS,I=gS,I−gS¯,I¯=G−lnI+pλ1+1−pλ2u−S−I−lnS−ln1−S−I−lnS+I−uu+αη−gS¯,I¯.
Denote(49)V1=−lnI+pλ1+1−pλ2u−S−I,V2=−lnS,V3=−ln1−S−I,V4=−lnS+I−uu+αη−gS¯,I¯.Step 2.
Denote a closed setU∈Σ. For a sufficiently small positive number ε, we denote(50)U=S,I∈E:ε≤S,ε≤I,K1≤S+I≤K2,where K1=u/u+αη+ε2,K2=1−ε2. In the set E\U, let ε be sufficiently small such that(51)Gmε1−pλ2+Gpλ1+1−pλ2uαη+uε<1,(52)−uε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1,(53)−αηε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1.Step 3.
Prove(54)supS,I∈X\UA∗VS,I<−1,A∗V1≤−pλ1+1−pλ2S+u+η+p2σ12+1−p2σ222+mI1+mI1−pλ2S−pλ1+1−pλ2+pλ1+1−pλ2S+pλ1+1−pλ2uαη+uI=−pλ1+1−pλ2+u+η+p2σ12+1−p2σ222+mI1+mI1−pλ2S+pλ1+1−pλ2uαη+uI≤−pλ1+1−pλ2+u+η+p2σ12+1−p2σ222+mI1+mI1−pλ2+pλ1+1−pλ2uαη+uI≔mI1+mI1−pλ2+pλ1+1−pλ2uαη+uI−λ,where λ=pλ1+1−pλ2−u−η−p2σ12+1−p2σ22/2=u+ηR0−1−p2σ12+1−p2σ22/2u+η>0.(55)A∗V2=−uS+u+pλ1+1−pλ2I1+mI−1−αηIS+p2σ12I2+1−p2σ22I2/1+mI22≤−uS+u+pλ1+1−pλ2+p2σ12+1−p2σ222=−uS+pλ1+1−pλ2+u+p2σ12+1−p2σ222,(56)A∗V3=u−uS+I−αηI1−S−I=u−αηI1−S−I,(57)A∗V4=−u−uS−uI−αηIS+I−u/u+αη=u+αη−αηSS+I−u/u+αη.
Thus, according to (54)–(57), we have(58)A∗V≤−Gλ+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI−uS−αηI1−S−I−αηSS+I−u/u+αη+H,where H=3u+αη+pλ1+1−pλ2+p2σ12+1−p2σ22/2.
Now, we prove thatA∗VS,I<−1 on the following four domains which are equivalent to A∗VS,I<−1 on the E\U.(i)
U1=S,I∈E:0<I<ε. If S,I∈U1, by (47) and (51), we have(59)A∗V≤−Gλ+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI+H<−Gλ+Gmε1−pλ2+Gpλ1+1−pλ2uαη+uε+H<1+−2=−1.(ii)
U2=S,I∈E:0<S<ε. If S,I∈U2, using (52), we get(60)A∗V≤−uS+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI+H<−uε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1.(iii)
U3=S,I∈E:ε≤I,1−ε2<S+I. If S,I∈U3, in view of (53), we derive(61)A∗V≤−αηI1−S−I+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI+H<−αηε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1.(iv)
U4=S,I∈E:ε≤S,S+I<u/u+αη+ε2. If S,I∈U4, according to (53), we obtain(62)A∗V≤−αηSS+I−u/u+αη+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI+H<−αηε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1.
In conclusion, for a sufficiently small positive numberε,(63)supS,I∈E\UA∗VS,I<−1.
We apply similar arguments to those in [41]; this means that the semigroup is not sweeping from the set U. Thus, the semigroup Ptt≥0 is asymptotically stable by Remark 1.


## 2.1. Markov Semigroups and Fokker–Planck Equation

In this section, we provide some definitions about Markov semigroups [14], asymptotic properties [30–34], and Fokker–Planck equation to verify our results.DenoteX, Σ, and m to be a metric space, σ-algebra of Borel sets, and the Lebesgue measure on X,Σ, respectively. D=sh∈L1:h≥0,h=1. Ptt≥0 represents an integral Markov semigroup with a continuous kernel Kt;x;y for t>0, which satisfies ∫XKx,ymdx=1 for ∀y∈X. For the diffusion process St,It, i.e., the transition probability function is Pt,x,y,A=ProbSt,It∈A. Define φ1x,y=x2y2, φ2x,y=x2y2/1+my2, and(6)f1x,y=u−ux−pλ1xy−1−pλ2xy1+my+1−αηy,f2x,y=pλ1xy+1−pλ2xy1+my−uy−ηy.Remark 1.
If for∀h∈D, we have(7)∫0∞Pthdt>0,a.e.
Then, this semigroup is asymptotically stable or sweeping with respect to compact sets [30, 31]; this property is called the Foguel alternative [34].Remark 2.
If the distribution ofSt,It is absolutely continuous with respect to the Lebesgue measure with the density Ut,x,y, then U satisfies the Fokker–Planck equation [32]:(8)∂U∂t=σ12p22∂2φ1U∂x2−2∂2φ1U∂x∂y+∂2φ1U∂y2+σ221−p22∂2φ2U∂x2−2∂2φ2U∂x∂y+∂2φ2U∂y2−∂f1U∂x−∂f2U∂y.Remark 3.
DenotePtVx,y=Ux,y,t for V∈D, and A represents the infinitesimal generator of semigroup Ptt≥0; then, we have(9)A∗V=1−p2σ22φ22∂2V∂x2−2∂2V∂x∂y+∂2V∂y2+p2σ12φ12∂2V∂x2−2∂2V∂x∂y+∂2V∂y2+∂f1V∂x+∂f2V∂y.


## 2.2. Some Lemmas about the Asymptotically Stable Stationary Distribution

In this section, five lemmas are given to study the asymptotically stable stationary distribution of system (5).Lemma 1.
The transition probability functionPt,x0,y0,A has a continuous density Kt,x,y;x0,y0 with respect to the Lebesgue measure.Proof.
Denote(10)a⟶x,y=a1a2,b1⟶x,y=−pσ1xypσ1xy,b2⟶x,y=−1−pλ2xy1+my1−pλ2xy1+my,where a1=u−ux−pλ1xy−1−pλ2xy/1+my+1−αηy, a2=pλ1xy+1−pλ2xy/1+my−uy−ηy, and x,y∈ℝ+2. a⟶,bi⟶i=1,2 is defined as(11)a⟶,bi⟶jx=∑k=12ak∂bji∂xk−bki∂aj∂xkx,j=1,2.
Then,(12)a⟶,bi⟶=a1∂b1i∂x−b1ia1∂x+a2∂b1i∂y−b2ia1∂ya1∂b2i∂x−b1ia2∂x+a2∂b2i∂y−b2ia2∂y.
Denoteb1i=−b2i, and by direct calculation, we have(13)b1⟶a⟶,b1⟶=b11a1∂b11∂x−b11a1∂x+a2∂b11∂y−b21a1∂yb21a1∂b21∂x−b11a2∂x+a2∂b21∂y−b21a2∂y=αηp2σ12x2y2>0,b2⟶a⟶,b2⟶=b12a1∂b12∂x−b12a1∂x+a2∂b12∂y−b22a1∂yb22a1∂b22∂x−b12a2∂x+a2∂b22∂y−b22a2∂y=αη1−p2σ22x2y21+my2>0,which shows that bi⟶ and a⟶,bi⟶ are independent on ℝ+2. Thus, according to Ho..rmander theorem (see [35]), Pt,x0,y0,A has a continuous density Kt,x,y;x0,y0 and K∈C∞0,∞×ℝ+2×ℝ+2. The proof is completed. In addition, we can obtain that the semigroup Ptt≥0 is an integral Markov semigroup.Lemma 2.
For eachx0,y0∈E and x,y∈E, where suppU∗=x,y∈ℝ+2:u/u+αη<x+y<1≔E, there exists T>0 such that KT,x,y;x0,y0>0.Proof.
Fix a pointx0,y0∈ℝ+2 and a continuous function ϕ∈L20,T;ℝ; the following system is first considered:(14)xϕt=x0+∫0tf1xϕs,yϕs−pσ1ϕxϕsyϕsds−∫0t1−pσ2ϕxϕsyϕs1+myϕsds,yϕt=y0+∫0tf2xϕs,yϕs+pσ1ϕxϕsyϕsds+∫0t1−pσ2ϕxϕsyϕs1+myϕsds.Step 1.
FindDx0,y0;ϕi by the perturbation method. Denote f′, g1′, and g2′ are the Jacobians of(15)f=f1x,yf2x,y,g1=−pσ1xypσ1xy,g2=−1−pσ2xy1+my1−pσ2xy1+my,respectively; then,(16)Ϝi=f′xϕt,yϕt+gi′xϕt,yϕtϕ.
For0≤t0≤t≤T, let Qit0,t0=I and ∂Qit,t0/∂t=ϜitQit,t0; we get(17)Dx0,y0;ϕih=∫0TQiT,sgishsds.
Assume that the rank ofDx0,y0;ϕi is 2 for some ϕ∈L20,T;ℝ; then, for x,y=xϕT,yϕT, we have KT,x,y;x0,y0>0.Step 2.
Prove that the derivativeDx0,y0;ϕi has rank 2. Replace system (14) with(18)xϕ′t=yϕ′t=f1xϕt,yϕt−pσ1ϕxϕtyϕt−1−pσ2ϕxϕtyϕt1+myϕt,f2xϕt,yϕt+pσ1ϕxϕtyϕt+1−pσ2ϕxϕtyϕt1+myϕt.
Defineϵ∈0,T and ht=1T−ϵ,T/xϕtyϕtfxϕt,yϕt for t∈0,T; here, 1T−ϵ,T denotes the characteristic function of interval T−ϵ,T. Using Taylor expansion, we obtain(19)QiT,s=I+ϜiTT−s+oT−s.
Then, we get(20)Dx0,y0;ϕih=∫0TI+ϜiTT−s+oT−sgishsds=ϵv+ϵ22ϜiTv+oϵ2,and v1=−σ1,σ1T,v2=−σ2,σ2T.
Then,(21)Ϝ1Tv1=σ1AB,v1Ϝ1Tv1=−σ1σ1Aσ1σ1B=σ12αη>0,Ϝ2Tv2=σ2CD,v2Ϝ2Tv2−σ2σ2Cσ2σ2D=σ22αη>0,where(22)A=pλ1+σ1ϕy−x+1−pλ2−x+y+my21+my2+1−αη+u,B=pλ1+σ1ϕx−y+1−pλ2x−y−my21+my2−u+η,C=u+pλ1x+1−pλ2x1+my2−1−pσ2x1+my2+1−αη−pλ1y−1−pλ2y1+my+1−pσ2y1+my,D=−u−pλ1x−1−pλ2x1+my2+1−pσ2x1+my2−η+pλ1y+1−pλ2y1+my−1−pσ2y1+my.
Therefore, the rank ofDx0,y0;ϕi is 2.Step 3.
Verify that, for anyx0,y0∈E and x,y∈E, there exist control function ϕ and T>0 such that xϕ0,yϕ0=x0,y0 and xϕT,yϕT=x,y. Let zϕ=xϕ+yϕ, and(23)g1x,z=u−ux−pλ1xz−x−1−pλ2xz−x1+mz−x,g2x,z=u+αηx−u+αηz.
System (18) becomes(24)xϕ′t=g1xϕt,zϕt−pσ1ϕxϕtzϕt−xϕt−1−pσ2ϕxϕtzϕt−xϕt1+mzϕt−xϕt,zϕ′t=g2xϕt,zϕt.
Denote(25)E0=x,z∈ℝ+2:0<x<1,uu+αη<z<1,x<z.
Using the same proof method of Lemma 3.2 in [36], we can obtain that, for any x0,z0∈E0 and x,z∈E0, there exist control function ϕ and T>0 such that xϕ0,zϕ0=x0,z0 and xϕT,zϕT=x,z. Now, the positivity of K is proved by support theorems (see [37–39]).Lemma 3.
IfR0−p2σ12+1−p2σ22/2u+η>1, for every density f and the semigroup Ptt≥0, the following conclusion holds:(26)limt⟶∞∬EPtfx,ydxdy=1.Proof.
SubstituteZt=St+It; we can obtain that(27)u−u+αηZt<dZtdt<u−uZt,t∈0,∞a.s.
Next, in the following three cases, (i)Z0∈0,u/u+αη, (ii) Z0∈u/u+αη,1, and (iii) Z0∈1,∞, we verify that there is t0=t0ω such that(28)uu+αη<Zt,ω<1,t>t0,for almost every ω∈Ω; this proof is completed.(i)
If inequality (28) is not true, then there exists Ω′⊂Ω with ProbΩ′>0 which satisfies(29)Zt,ω∈0,uu+αη.On the one hand, we can get thatdZt/dt>u−u+αηZt>0 from (27). Thus, Zt,ω is strictly increasing on 0,∞ for any ω∈Ω′, and then limt⟶∞Zt,ω=u/u+αη. In view of the expression of g2, we have(30)limt⟶∞St,ω=0,limt⟶∞It,ω=uu+αη.Namely,(31)limt⟶∞lnIt−lnI0t=0,ω∈Ω′.On the other hand, according to Itô’s formula, we get(32)dlnIt=pλ1S+1−pλ2S1+mI−u−ηdt−12p2σ12S2+1−p2σ22S21+mI2dt+pσ1SdB1t+1−pσ2S1+mIdB2t.Therefore,(33)lnIt−lnI0t=1t∫0tpλ1Ss+1−pλ2Ss1+mIsds−1t∫0t12p2σ12S2sds−1t∫0t121−p2σ22S2s1+mIs2ds+1t∫0t−u−ηds+M1tt+M2tt,where(34)M1t=pσ1∫0tSsdB1s,M2t=1−pσ2∫0tSs1+mIsdB2s,M1,M1t=p2σ12∫0tS2sds,M2,M2t=1−p2σ22∫0tSs1+mIs2ds.Clearly,(35)limsupt⟶∞M1,M1tt=limsupt⟶∞p2σ12∫0tS2sdst<p2σ12∫0t12dst<p2σ12<∞,a.s.,limsupt⟶∞M2,M2tt=limsupt⟶∞1−p2σ22∫0tSs/1+mIs2dst<1−p2σ22∫0t12dst<1−p2σ22<∞,a.s.By the strong law of large numbers (see [40]), we get(36)limt⟶∞M1tt=limt⟶∞M2tt=0,a.s.Namely,(37)limt⟶∞lnIt−lnI0t=limt⟶∞−u−η+pλ1∫0tSsdst+limt⟶∞1−pλ2∫0tSs/1+mIsdst−limt⟶∞p2σ12∫0tS2sds2t−limt⟶∞1−p2σ22∫0tSs/1+mIsds2t=−u+η,a.s.onΩ′,this contradiction arises. Hence, the result follows forZ0∈0,u/u+αη.(ii)
From (27), we obtain that(38)Zt>uu+α+Z0−uu+αeu+αt0e−u+αt,Zt<1+Z0−1eut0e−ut<Zt,Clearly, conclusion (28) holds in the case of Z0∈u/u+αη,1.(iii)
Suppose that inequality (28) does not hold. On the one hand, using the same way to (ii), we have limt⟶∞Zt,ω=1, ω∈Ω′, and(39)limt⟶∞St,ω=1,limt⟶∞It,ω=0.Clearly,(40)limt⟶∞lnIt−lnI0t=0,ω∈Ω′.On the other hand, ifR0−1>p2σ12+1−p2σ22/2u+η, we obtain(41)lnIt−lnI0t=pλ1+1−pλ2−u−η−p2σ12+1−p2σ222=u+ηR0−1−p2σ12+1−p2σ222u+η>0,a.s.onΩ′,so there is a contradiction. Thus, the result holds forZ0∈1,∞. This proof is completed.Lemma 4.
IfR0−p2σ12+1−p2σ22/2u+η>1, the semigroup Ptt≥0 is asymptotically stable or sweeping with respect to compact sets.Proof.
Using the same proof method of Lemma 4.5 in [14], this means that our result holds. We omit it here.Lemma 5.
The semigroupPtt≥0 is asymptotically stable if R0−p2σ12+1−p2σ22/2u+η>1.Proof.
By system (5), we obtain that(42)L−lnI=−pλ1S−1−pλ2S1+mI+u+η+p2σ12S2+1−p2σ22S/1+mI22≤−pλ1S−1−pλ2S1+mI+u+η+p2σ12+1−p2σ222=−pλ1S−1−pλ2S+u+η+p2σ12+1−p2σ222+1−pλ2S−1−pλ2S1+mI=−pλ1S−1−pλ2S+u+η+p2σ12+1−p2σ222+mI1+mI1−pλ2S.
Denote(43)f1S,I=u−uS−pλ1SI−1−pλ2SI1+mI+1−αηI,f2S,I=pλ1SI+1−pλ2SI1+mI−uI−ηI,(44)A∗V=F∂2V∂S2−2∂2V∂S∂I+∂2V∂I2+∂f1V∂S+∂f2V∂I,where F=S2I2p2σ12+1−p2σ221/1+mI2/2, and we call V a Khasminskii˘ function [41].
P
t
t
≥
0 satisfies the Foguel alternative according to Lemma 4. We can exclude sweeping by constructing a nonnegative C2-function V and a closed set U∈Σ satisfying(45)supS,I∈X\UA∗VS,I<−1,Step 1.
Construct a nonnegativeC2-function V. For S,I∈E, define(46)gS,I=G−lnI+pλ1+1−pλ2u−S−I−lnS−ln1−S−I−lnS+I−uu+αη,where G>0 such that(47)−Gu+ηR0−1−p2σ12+1−p2σ222u+η+3u+αη+pλ1+1−pλ2+p2σ12+1−p2σ222≤−2.
Let(48)VS,I=gS,I−gS¯,I¯=G−lnI+pλ1+1−pλ2u−S−I−lnS−ln1−S−I−lnS+I−uu+αη−gS¯,I¯.
Denote(49)V1=−lnI+pλ1+1−pλ2u−S−I,V2=−lnS,V3=−ln1−S−I,V4=−lnS+I−uu+αη−gS¯,I¯.Step 2.
Denote a closed setU∈Σ. For a sufficiently small positive number ε, we denote(50)U=S,I∈E:ε≤S,ε≤I,K1≤S+I≤K2,where K1=u/u+αη+ε2,K2=1−ε2. In the set E\U, let ε be sufficiently small such that(51)Gmε1−pλ2+Gpλ1+1−pλ2uαη+uε<1,(52)−uε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1,(53)−αηε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1.Step 3.
Prove(54)supS,I∈X\UA∗VS,I<−1,A∗V1≤−pλ1+1−pλ2S+u+η+p2σ12+1−p2σ222+mI1+mI1−pλ2S−pλ1+1−pλ2+pλ1+1−pλ2S+pλ1+1−pλ2uαη+uI=−pλ1+1−pλ2+u+η+p2σ12+1−p2σ222+mI1+mI1−pλ2S+pλ1+1−pλ2uαη+uI≤−pλ1+1−pλ2+u+η+p2σ12+1−p2σ222+mI1+mI1−pλ2+pλ1+1−pλ2uαη+uI≔mI1+mI1−pλ2+pλ1+1−pλ2uαη+uI−λ,where λ=pλ1+1−pλ2−u−η−p2σ12+1−p2σ22/2=u+ηR0−1−p2σ12+1−p2σ22/2u+η>0.(55)A∗V2=−uS+u+pλ1+1−pλ2I1+mI−1−αηIS+p2σ12I2+1−p2σ22I2/1+mI22≤−uS+u+pλ1+1−pλ2+p2σ12+1−p2σ222=−uS+pλ1+1−pλ2+u+p2σ12+1−p2σ222,(56)A∗V3=u−uS+I−αηI1−S−I=u−αηI1−S−I,(57)A∗V4=−u−uS−uI−αηIS+I−u/u+αη=u+αη−αηSS+I−u/u+αη.
Thus, according to (54)–(57), we have(58)A∗V≤−Gλ+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI−uS−αηI1−S−I−αηSS+I−u/u+αη+H,where H=3u+αη+pλ1+1−pλ2+p2σ12+1−p2σ22/2.
Now, we prove thatA∗VS,I<−1 on the following four domains which are equivalent to A∗VS,I<−1 on the E\U.(i)
U1=S,I∈E:0<I<ε. If S,I∈U1, by (47) and (51), we have(59)A∗V≤−Gλ+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI+H<−Gλ+Gmε1−pλ2+Gpλ1+1−pλ2uαη+uε+H<1+−2=−1.(ii)
U2=S,I∈E:0<S<ε. If S,I∈U2, using (52), we get(60)A∗V≤−uS+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI+H<−uε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1.(iii)
U3=S,I∈E:ε≤I,1−ε2<S+I. If S,I∈U3, in view of (53), we derive(61)A∗V≤−αηI1−S−I+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI+H<−αηε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1.(iv)
U4=S,I∈E:ε≤S,S+I<u/u+αη+ε2. If S,I∈U4, according to (53), we obtain(62)A∗V≤−αηSS+I−u/u+αη+GmI1+mI1−pλ2+Gpλ1+1−pλ2uαη+uI+H<−αηε+Gm1−pλ2+Gpλ1+1−pλ2uαη+u+H<−1.
In conclusion, for a sufficiently small positive numberε,(63)supS,I∈E\UA∗VS,I<−1.
We apply similar arguments to those in [41]; this means that the semigroup is not sweeping from the set U. Thus, the semigroup Ptt≥0 is asymptotically stable by Remark 1.


## 3. Dynamics of System (5)

### 3.1. The Property of the Positive Solution

The first result is the existence and uniqueness of the positive solution of system (5).Theorem 1.
System (5) has a unique positive solution St,It on t≥0 with any initial value S0,I0∈ℝ+2 almost surely.Proof.
By system (5), we obtain that(64)St+It≤1+e−utS0+I0−1≤K,where K=maxS0+I0,1.
Define(65)VS,I=S−1−lnS+I−1−lnI.
Clearly,VS,I is nonnegative. Using Itô’s formula and the fact St+It≤K, we get(66)dVS,I=LVS,Idt+pσ1I−SdB1t+1−pσ21+mII−SdB2t,(67)LVS,I=u−uS+1−αηI−uS+u−1−αηIS+pλ1I+1−pλ2I1+mI−uI−ηI+u+η+I22p2σ12+1−p2σ221+mI2−pλ1S−1−pλ2S1+mI+S22p2σ12+1−p2σ221+mI2≤3u+pλ1I+1−pλ2I1+mI+η+I22p2σ12+1−p2σ221+mI2+S22p2σ12+1−p2σ221+mI2≤3u+pλ1K+1−pλ2K1+mI+η+K22p2σ12+1−p2σ221+mI2+K22p2σ12+1−p2σ221+mI2≤3u+pλ1K+1−pλ2K+η+K22p2σ12+1−p2σ22+K22p2σ12+1−p2σ22=3u+pλ1K+1−pλ2K+η+K2p2σ12+K21−p2σ22≔K0.
Here,K0 is a positive constant. Then, refer to [42], and we complete the proof.


### 3.2. Asymptotically Stable Stationary Distribution and Extinction of System (5)

Theorem 2.
DenoteSt,It a solution of system (5) with any initial value S0,I0∈ℝ+2; the distribution of St,It has a density Ut,x,y for every t>0. If R0−p2σ12+1−p2σ22/2u+η>1, then there is a unique density U∗x,y satisfying(68)limt⟶∞∬ℝ+2Ut,x,y−U∗x,ydxdy=0,suppU∗=x,y∈ℝ+2:uu+αη<x+y<1≔E.Proof.
The proof of Theorem2 is the following steps:Step 1. According to Ho..rmander theorem [35], we prove that the transition function of the process St,It is absolutely continuous (see Lemma 1)Step 2. We verify the positivity of the density of the transition function on E by support theorems [37–39] (refer to Lemma 2)Step 3. We prove that the Markov semigroup satisfies the Foguel alternative (see Lemmas 3 and 4)Step 4. Excluding sweeping by verifying there is a Khasminskiiˇ function (refer to Lemma 5)
Notice that the above strategies can be proved by Lemmas1–5; thus, this proof is complete.Theorem 3.
For any initial valueS0,I0∈ℝ+2, St,It is a solution of system (5). If one of the following conditions holds, the epidemic It becomes extinct with probability one.(i)
(69)σ12≤λ1p,σ22≤λ21−p,R0−p2σ12+1−p2σ222u+η<1.(ii)
(70)λ122σ12+λ222σ22<u+η.Particularly, if (i) holds,(71)limsupt⟶∞lnItt≤u+ηR0−1−p2σ12+1−p2σ222u+η<0,a.s.If (ii) holds,(72)limsupt⟶∞lnItt≤λ122σ12+λ222σ22−u−η<0,a.s.Proof.
Ifσ12≤λ1/p,σ22≤λ2/1−p, and R0−p2σ12+1−p2σ22/2u+η<1, we obtain(73)limsupt⟶∞lnItt≤−p2σ1221−λ1pσ122+λ122σ12−1−p2σ2221−λ21−pσ222+λ222σ22−u−η=−p2σ122+pλ1−1−p2σ222+1−pλ2−u−η=u+ηR0−1−p2σ12+1−p2σ222u+η,and notice that R0−p2σ12+1−p2σ22/2u+η<1; then,(74)u+ηR0−1−p2σ12+1−p2σ222u+η<0.
Then,(75)limsupt⟶∞lnItt<0,a.s.
Ifλ12/2σ12+λ22/2σ22<u+η, we easily get that(76)limsupt⟶∞lnItt=limsupt⟶∞pλ1∫0tSsdst−p2σ12∫0tS2sds2t+limsupt⟶∞1−pλ2∫0tSs/1+mIsdst−u−η−limsupt⟶∞1−p2σ22∫0tSs/1+mIs2ds2t+limsupt⟶∞M1tt+M2tt≤limsupt⟶∞pλ1∫0tSsdst−limsupt⟶∞p2σ122∫0tS2sdst2+limsupt⟶∞1−pλ2∫0tSs/1+mIsdst−u−η−limsupt⟶∞1−p2σ222∫0tSs/1+mIsdst2=−p2σ122liminft⟶∞∫0tSsdst−λ1pσ122+λ122σ12−1−p2σ222liminft⟶∞J2+λ222σ22−u−η≤λ122σ12+λ222σ22−u−η<0,a.s.,where J=∫0tSs/1+mIsds/t−λ2/1−pσ22. This proof is completed.Remark 4.
R
0
−
p
2
σ
1
2
+
1
−
p
2
σ
2
2
/
2
u
+
η is the threshold of system (5) when σ12≤λ1/p and σ22≤λ2/1−p.


## 3.1. The Property of the Positive Solution

The first result is the existence and uniqueness of the positive solution of system (5).Theorem 1.
System (5) has a unique positive solution St,It on t≥0 with any initial value S0,I0∈ℝ+2 almost surely.Proof.
By system (5), we obtain that(64)St+It≤1+e−utS0+I0−1≤K,where K=maxS0+I0,1.
Define(65)VS,I=S−1−lnS+I−1−lnI.
Clearly,VS,I is nonnegative. Using Itô’s formula and the fact St+It≤K, we get(66)dVS,I=LVS,Idt+pσ1I−SdB1t+1−pσ21+mII−SdB2t,(67)LVS,I=u−uS+1−αηI−uS+u−1−αηIS+pλ1I+1−pλ2I1+mI−uI−ηI+u+η+I22p2σ12+1−p2σ221+mI2−pλ1S−1−pλ2S1+mI+S22p2σ12+1−p2σ221+mI2≤3u+pλ1I+1−pλ2I1+mI+η+I22p2σ12+1−p2σ221+mI2+S22p2σ12+1−p2σ221+mI2≤3u+pλ1K+1−pλ2K1+mI+η+K22p2σ12+1−p2σ221+mI2+K22p2σ12+1−p2σ221+mI2≤3u+pλ1K+1−pλ2K+η+K22p2σ12+1−p2σ22+K22p2σ12+1−p2σ22=3u+pλ1K+1−pλ2K+η+K2p2σ12+K21−p2σ22≔K0.
Here,K0 is a positive constant. Then, refer to [42], and we complete the proof.


## 3.2. Asymptotically Stable Stationary Distribution and Extinction of System (5)

Theorem 2.
DenoteSt,It a solution of system (5) with any initial value S0,I0∈ℝ+2; the distribution of St,It has a density Ut,x,y for every t>0. If R0−p2σ12+1−p2σ22/2u+η>1, then there is a unique density U∗x,y satisfying(68)limt⟶∞∬ℝ+2Ut,x,y−U∗x,ydxdy=0,suppU∗=x,y∈ℝ+2:uu+αη<x+y<1≔E.Proof.
The proof of Theorem2 is the following steps:Step 1. According to Ho..rmander theorem [35], we prove that the transition function of the process St,It is absolutely continuous (see Lemma 1)Step 2. We verify the positivity of the density of the transition function on E by support theorems [37–39] (refer to Lemma 2)Step 3. We prove that the Markov semigroup satisfies the Foguel alternative (see Lemmas 3 and 4)Step 4. Excluding sweeping by verifying there is a Khasminskiiˇ function (refer to Lemma 5)
Notice that the above strategies can be proved by Lemmas1–5; thus, this proof is complete.Theorem 3.
For any initial valueS0,I0∈ℝ+2, St,It is a solution of system (5). If one of the following conditions holds, the epidemic It becomes extinct with probability one.(i)
(69)σ12≤λ1p,σ22≤λ21−p,R0−p2σ12+1−p2σ222u+η<1.(ii)
(70)λ122σ12+λ222σ22<u+η.Particularly, if (i) holds,(71)limsupt⟶∞lnItt≤u+ηR0−1−p2σ12+1−p2σ222u+η<0,a.s.If (ii) holds,(72)limsupt⟶∞lnItt≤λ122σ12+λ222σ22−u−η<0,a.s.Proof.
Ifσ12≤λ1/p,σ22≤λ2/1−p, and R0−p2σ12+1−p2σ22/2u+η<1, we obtain(73)limsupt⟶∞lnItt≤−p2σ1221−λ1pσ122+λ122σ12−1−p2σ2221−λ21−pσ222+λ222σ22−u−η=−p2σ122+pλ1−1−p2σ222+1−pλ2−u−η=u+ηR0−1−p2σ12+1−p2σ222u+η,and notice that R0−p2σ12+1−p2σ22/2u+η<1; then,(74)u+ηR0−1−p2σ12+1−p2σ222u+η<0.
Then,(75)limsupt⟶∞lnItt<0,a.s.
Ifλ12/2σ12+λ22/2σ22<u+η, we easily get that(76)limsupt⟶∞lnItt=limsupt⟶∞pλ1∫0tSsdst−p2σ12∫0tS2sds2t+limsupt⟶∞1−pλ2∫0tSs/1+mIsdst−u−η−limsupt⟶∞1−p2σ22∫0tSs/1+mIs2ds2t+limsupt⟶∞M1tt+M2tt≤limsupt⟶∞pλ1∫0tSsdst−limsupt⟶∞p2σ122∫0tS2sdst2+limsupt⟶∞1−pλ2∫0tSs/1+mIsdst−u−η−limsupt⟶∞1−p2σ222∫0tSs/1+mIsdst2=−p2σ122liminft⟶∞∫0tSsdst−λ1pσ122+λ122σ12−1−p2σ222liminft⟶∞J2+λ222σ22−u−η≤λ122σ12+λ222σ22−u−η<0,a.s.,where J=∫0tSs/1+mIsds/t−λ2/1−pσ22. This proof is completed.Remark 4.
R
0
−
p
2
σ
1
2
+
1
−
p
2
σ
2
2
/
2
u
+
η is the threshold of system (5) when σ12≤λ1/p and σ22≤λ2/1−p.


## 4. Simulations and Conclusion

Now, in order to illustrate the analytical results, we make some numerical simulations. For systems (3) and (4), the parameter values are u=0.2,α=0.4,p=0.6,λ1=0.5,λ2=0.6,η=0.2,andm=0.2; then, R0=pλ1+1−pλ2/u+η=1.35>1.We employ the following discrete equations:(77)Sk+1=u−uSk−pλ1SkIk−1−pλ2SkIk1+mIkΔt+1−αηIkΔt−1−pσ2SkIk1+mIkΔtξk−1−pσ22SkIk21+mIkξk2−1Δt+Sk−pσ1SkIkΔtξk−pσ12SkIk2ξk2−1Δt,Ik+1=pλ1SkIk+1−pλ2SkIk1+mIk−uIk−ηIkΔt+pσ1SkIkΔtξk+pσ12SkIk2ξk2−1Δt+Ik+1−pσ2SkIk1+mIkΔtξk+1−pσ22SkIk21+mIkξk2−1Δt,Rk+1=Rk+αηIk−uRkΔt,where ξkk=1,2,…,n are N0,1-distributed independent Gaussian random variables.Example 1.
Letσ1=0.2andσ2=0.3; then,(78)R0−1=0.35>0.0065=p2σ12+1−p2σ222u+η,which satisfies the condition in Theorem 2; this means that epidemic It is persistent for long time. Figures 1(a) and 1(b) are the paths of individuals in deterministic model (3) and stochastic model (4), respectively. Figure 1(c) is the phase portrait of Figures 1(a) and 1(b). Figures 2(a)–2(c) are the distribution diagrams of S,I and R, respectively.Figure 1
(a) Deterministic system, (b) corresponding stochastic systems of (a), and (c) the phase portrait of (b). (a)σ1=0,σ2=0; (b) σ1=0.1,σ2=0.1.
(a)
(b)
(c)Figure 2
(a, b, and c) The distribution diagram ofS,I, and R, respectively. σ1=0.1,σ2=0.1.
(a)
(b)
(c)Example 2.
Case 1: letσ1=0.95andσ2=0.95; then,
(79)
λ
1
2
2
σ
1
2
+
λ
2
2
2
σ
2
2
=
0.33795
<
0.4
=
u
+
η
,
which satisfies condition (ii) in Theorem3; this means that epidemic It becomes extinct with probability one. Figures 3(a) and 3(b) show the paths of individuals in deterministic model (3) and stochastic model (4), respectively. Figure 3(c) shows the phase portrait of Figures 3(a) and 3(b).
Case 2: letσ1=0.85andσ2=0.85; then,
(80)
σ
1
2
=
0.7225
≤
0.8333
=
λ
1
p
,
σ
2
2
=
0.7225
≤
1.5
=
λ
2
1
−
p
,
R
0
−
1
=
0.35
<
0.46962
=
p
2
σ
1
2
+
1
−
p
2
σ
2
2
2
u
+
η
,
which satisfies condition (i) in Theorem3; this means that epidemic It becomes extinct with probability one. Figures 4(a) and 4(b) show the paths of individuals in deterministic model (3) and stochastic model (4), respectively. Figure 4(c) is the phase portrait of Figures 4(a) and 4(b).
We have studied a stochastic SIR epidemic system affected by mixed nonlinear incidence rates in this paper. Using Markov semigroup theory and Fokker–Planck equation, we proved that the value ofR0−p2σ12+1−p2σ22/2u+η is the threshold of system (5) when σ12≤λ1/p and σ22≤λ2/1−p. If R0−p2σ12+1−p2σ22/2u+η>1, the epidemic is persistent, and there is a unique asymptotically stable stationary distribution. If R0−p2σ12+1−p2σ22/2u+η<1, the sufficient conditions for disease extinction are established, that is, if σ12≤λ1/p and σ22≤λ2/p, then limsupt⟶∞lnIt/t<0; this means that the epidemic becomes extinct with probability one. Moreover, if λ12/2σ12+λ22/2σ22<u+η, the epidemic is also extinct.
By the conclusion of Theorem1, it is worthy to point out that two different incidence rates which are considered at the same time will not destroy a great property that existence and uniqueness of the positive solution. From Theorems 2 and 3, we know that when two infection rates are considered at the same time, in addition to the values of σ1 and σ2, the value of p also affects the threshold value R0−p2σ12+1−p2σ22/2u+η. In addition, from the sufficient conditions for disease extinction, parameter p is an important variable related to the extinction of the population. Especially, when the parameters p=1 and α=1, system (3) becomes (1) in [12], and system (3) is similar to (2) in [13] when the parameters p=0 and α=1. To some extent, our model is more realistic than considering the epidemic affected by a single nonlinear incidence rate.
Since the proposed system is degenerate, we use Markov semigroup theory to study the stationary distribution and ergodicity of the system, and we can also study color noise and other noises in the future. For other noises, this method is feasible as long as it conforms to the relevant properties of Gaussian white noise. In addition, two aspects can be used as a guide for further research. First of all, we can investigate some other systems, for example, one can consider the systems with the impulsive perturbation effects [43, 44]. In addition, it is interesting to study the chemostat as well as population dynamics systems [45–51]. Notice that it is a meaningful question to investigate whether the way used in this article is applied to other epidemic systems. These questions are worthy of further study.Figure 3
(a) Deterministic system, (b) corresponding stochastic systems of (a), and (c) the phase portrait of (b). (a)σ1=0,σ2=0; (b) σ1=0.95,σ2=0.95.
(a)
(b)
(c)Figure 4
(a) Deterministic system, (b) corresponding stochastic systems of (a), and (c) the phase portrait of (b). (a)σ1=0,σ2=0; (b) σ1=0.85,σ2=0.85.
(a)
(b)
(c)



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