# Dynamics Analysis for a Stochastic SIRI Model Incorporating Media Coverage

Authors: Yanmei Wang; Guirong Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

We investigate a stochastic SIRI model incorporating media coverage. Firstly, existence and uniqueness of global positive solution of the model are established. By using the Hasminskii theory, we study stationary distribution of the model. Then, we prove extinction of the disease. Moreover, the asymptotic properties of solutions are studied by using Lyapunov method. Finally, numerical simulations are presented. In addition, it shows that media coverage can strain the spread of infectious diseases.


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## Body

## 1. Introduction

Although human society is constantly developing and progressing, various infectious diseases still plague humans (such as COVID-19 and mixed influenza A and B). Governments around the world have taken a series of measures to prevent the spread of infectious disease and protect human life and health. Furthermore, with the acceleration of globalization, the convenient transportation and the frequent movement of population have accelerated the spread of infectious diseases, so as to study the infectious law of disease, predict its development trend, and seek for prevention and treatment strategies, scholars have established various epidemic models and studied their dynamic behaviors (see [1–8]).Media coverage is an effective measure to prevent the spread of disease (see [9, 10]). Through media reports (such as TV, Internet, radio, advertising, Weibo, WeChat, etc.), the public can timely understand the latest information of diseases in order to take preventive measures and avoid bad behaviors caused by panic. Many mathematical models (see [11–16]) have been established to research the impact of media reports on communication dynamics with the different incidence rates, for example, βe−mISIm>0 in [13], β1−β2I/m+ISI/S+I in [11], β1−β2fISI/S+I in [15], and β1−β2It/m1+ItStIt/1+m2It in [16].As can be seen from the above discussion, in order to study the influence of media coverage on the spread of infectious disease, previous models only considered the influence of the number of infected individuals on media coverage. When infected individuals appear in an area, the public are conscious of the potential danger of infected individuals, they will voluntarily reduce their contact with the outside world for fear of infection. The more the infected individuals are, the less contact they have with the outside world. That is, media coverage can minimize the contact between susceptible and infected individuals. In addition, the number of recovered individuals also partly reflects the severity of the epidemic. Hence, the number of infected and recovered individuals is larger, and the impact of media coverage is larger on the exposure rate. Thus, the reduction of contact rate is related to the number of infected and recovered individuals. Additionally, due to the influence of psychological and social factors, susceptible individuals reduce their contact with infected individuals and strengthen protective measures, so that the effective contact rate between infected and susceptible individuals tends to saturation state. Based on this, a deterministic SIRI model with saturated incidence rate is established(1)S˙t=Λ−β1−β2It+αRtm1+It+αRtStIt1+m2It−μSt,I˙t=β1−β2It+αRtm1+It+αRtStIt1+m2It−γ+μIt+δRt,R˙t=γIt−μ+δRt.Here,St,It, Rt are the susceptible, infected, and recovered individuals, respectively; Λ is the birth rate or migration rate of susceptible individuals; γ is the recovery rate of infected individuals; μ is the natural mortality; δ denotes the recurrence rate of the recovered; β1 is the maximum effective contact rate; and β1−β2I+αR/m1+I+αR denotes the contact rate under media coverage. m1>0, m2>0, 0<α≤1. Since media coverage will not completely prevent the spread of diseases, then β1>β2. For model (1), the basic reproduction number is easily obtained as(2)R0=β1Λμμ+γ−γδ/μ+δ.In reality, infectious diseases are always influenced by environmental noise, which makes the related parameters (such as contact rate, mortality rate, and recovery rate) show random fluctuations. Thus, it is more realistic to research the dynamic behaviors of stochastic epidemic model (see [17–23]). At present, a lot of academics have studied stochastic epidemic models with media coverage and obtained some results (see [14, 16, 24]). In [24], a stochastic SIS model with media coverage on two patches was established, and almost sure exponential stability of disease-free equilibrium was obtained. In [14], asymptotic behaviors around the equilibria of a stochastic SIRI model were considered. In [16], ergodic stationary distribution of a stochastic SIRS model was investigated.Considering the influence of environmental random factors, we establish the stochastic SIRI epidemic model(3)dS=Λ−β1−β2I+αRm1+I+αRSI1+m2I−μSdt+σ1SdB1t,dI=β1−β2I+αRm1+I+αRSI1+m2I−γ+μI+δRdt+σ2IdB2t,dR=γI−δ+μRdt+σ3RdB3t,where Bit is the Brownian motion on Ω,ℱ,ℙ, which is a complete probability space; σi2 is the intensity of Bit, i=1,2,3.In the following, we shall study existence of the stationary distribution and sufficient conditions for extinction of the disease. The dynamic properties of solutions near equilibria are to be considered. The results reveal the influence of noise intensity on disease transmission. In addition, the numerical simulation shows that increasing the media coverage can reduce the number of infected individuals, so media coverage can reduce the spread of infectious diseases. Besides, the less impact the number of recovered individuals has on media coverage, the lower the transmission rate is, which in turn leads to fewer infected individuals. Thus, it provides a scientific theoretical basis for the prevention and control of infectious diseases.


## 2. Existence and Uniqueness of Global Positive Solution

Theorem 1.
For everyS0,I0,R0∈ℝ+3, there is a unique global positive solution to (3) on 0,∞, namely, St,It,Rt∈ℝ+3, for any t≥0 almost surely.Proof.
Obviously, the coefficients of (3) are locally Lipschitz continuous. Hence, for every S0,I0,R0∈ℝ+3, there exists a unique maximal local solution St,It,Rt to model (3), where t∈0,τe and τe denotes the explosion time. In the following, it is proved that τe=∞ a.s. Let n0>0 such that S0,I0 and R0 all belong to 1/n0,n0. Define(4)τn=inft∈0,τe:maxSt,It,Rt≥norminSt,It,Rt≤1n,for any integer n≥n0. Obviously, τnn≥n0 is increasing. Set τ∞=limn⟶∞τn, then τ∞≤τe a.s. Let us show τ∞=∞ a.s. Or else there are ε∈0,1 and T>0 satisfying ℙτ∞≤T>ε. Then, for any n≥n1, there exists n1≥n0 satisfying ℙτn≤T>ε. Define a C2-function V:0,∞3⟶0,∞ by(5)V1S,I,R=S−1−lnS+R−1−lnR+I−1−lnI.
Forn≥n1,T>0 and 0≤t≤τn∧T, It o^′s formula shows that(6)dV1S,I,R=LV1S,I,Rdt+σ1S−1dB1t+σ2I−1dB2t+σ3R−1dB3t,where(7)LV1S,I,R=1−1SΛ−β1−β2I+αRm1+I+αRSI1+m2I−μS+1−1Iβ1−β2I+αRm1+I+αRSI1+m2I+δR−γ+μI+1−1R−δ+μR+γI+12σ12+σ22+σ32=Λ−μS−ΛS+β1−β2I+αRm1+I+αRI1+m2I−μI+γ−δRI−μR−γIR−β1−β2I+αRm1+I+αRS1+m2I+3μ+δ+12σ12+σ22+σ32≤Λ+β1m2+3μ+δ+γ+12σ12+σ22+σ32.
The remaining is similar to that of Theorem 2.1 in [23], so it is omitted.


## 3. Stationary Distribution and Ergodicity

LetXt be a Markov process in ℝd and satisfy(8)dXt=bXdt+∑k=1nσkXdBkt,where σkX=σk1X,…,σknXT. Then, the diffusion matrix is(9)A¯X=a¯ijXanda¯ijX=∑k=1nσkiXσkjX.Lemma 1 (see [25]).
For a bounded domainU⊂ℝd with regular boundary ∂U, if(i)
There existsω>0 such that ∑i,j=1da¯ijXξiξj≥ωξ2,X∈U,ξ∈ℝd;(ii)
There existC¯>0 and a C2-function V≥0 such that LVx<−C¯ for every x∈ℝd\U, then Xt has a unique ergodic stationary distribution μ⋅. Further, if f¯⋅ is an integrable function with respect to the measure μ, then for any x∈ℝd,(10)ℙlimQ⟶∞1Q∫0Qf¯Xtdt=∫ℝdf¯xμdx=1.
Denote(11)ℝs=β1−β2μ+2β1−β2/m2+σ12/2μ+γ+σ22/2.Theorem 2.
Ifℝs>1, then model (3) exists a unique ergodic stationary distribution.Proof.
For everyS0,I0,R0∈0,∞3, it follows from Theorem 1 that there exists a unique global positive solution St,It,Rt. Let ε>0 be sufficiently small. Denote(12)Uε=S,I,R∈ℝ+3:S,I∈ε,1ε,R∈ε2,1ε2,and AX=diagσ12S2,σ22I2,σ32R2. Let(13)M=minS,I,R∈Uεσ12S2,σ22I2,σ32R2.
For everyS,I,R∈U¯ε, ξ∈0,∞3,(14)∑i,j=13a¯ijξiξj=σ12S2ξ12+σ22I2ξ22+σ32R2ξ32≥Mξ2.
Thus,i in Lemma 1 holds. Define V˜S,I,R:0,∞3⟶−∞,∞ by(15)V˜S,I,R=NS+m2μ+γI+2R−c1lnS−c2lnI−lnR−lnS+1m+1R+S+Im+1≕NW1+W2+W3+W4,where c1=Λ/μ+2β1−β2/m2+σ12/2, c2=Λ/μ+γ+σ22/2. Positive constants m and N satisfy(16)μ−m2σ12∨σ22∨σ32>0,−3NΛℝs1/3−1+2Nγε+G<−1.
Here,(17)B=supS,I,R∈0,∞3ΛR+S+Im−142μ−mσ12∨σ22∨σ32S+I+Rm+1,G=supS,I,R∈0,∞3−12μ−m2σ12∨σ22∨σ32Sm+1+Im+1+Rm+1+N2β1−β2μ+γS−N2μ+2δ−m2δμ+γR+N+2μ+δ+2β1−β2m2+12σ12+σ32+B.
Obviously,(18)liminfl→∞,S,I,R∈0,∞3\UlV˜S,I,R=+∞,where Ul=1/l,l3. In addition, V˜S,I,R has a minimum point S¯0,I¯0,R¯0 in 0,∞3. Define V2:0,∞3⟶−∞,∞ by(19)V2S,I,R=−V˜S¯0,I¯0,R¯0+V˜S,I,R.
The Ito^ formula shows that(20)LW1=Λ−β1−β2I+αRm1+I+αRSI1+m2I−μS−c1ΛS+β1−β2I+αRm1+I+αRc1I1+m2I+m2μ+γβ1−β2I+αRm1+I+αRSI1+m2I−m2I−β1−β2I+αRm1+I+αRc2S1+m2I+c1μ+m2δμ+γR+c2μ+γ−c2δRI+2γI−2μ+δR+12c1σ12+12c2σ22≤−m2I+1−c1ΛS−c2β1−β2S1+m2I+1+Λ−μS−2μ+δR−β1−β2I+αRm1+I+αRSI1+m2I+β1−β2I+αRm1+I+αRc1I1+m2I+c1μ+m2δμ+γR+c2μ+γ−c2δRI+2γI+12c1σ12+12c2σ22+m22β1−β2SIμ+γ1+m2I≤−3Λβ1−β2c1c213+3Λ+1+2γI−μS−2μ+2δ−m2δμ+γR+m22β1−β2SIμ+γ1+m2I,LW2=−ΛS+β1−β2I+αRm1+I+αRI1+m2I+μ+σ122,LW3=−γIR+μ+δ+σ322,LW4=S+I+RmΛ−μS−μI−μR+12mS+I+Rm−1σ12S2+σ22I2+σ32R2≤ΛR+S+Im−μ−m2σ32∨σ22∨σ12R+S+Im+1≤B−142μ−mσ12∨σ22∨σ32Sm+1+Im+1+Rm+1.
Hence,(21)LV2S,I,R≤−3NΛℝs1/3−1+N+2NγI−NμS−N2μ+2δ−m2δμ+γR+δ+2μ+m2N2β1−β2μ+γSI1+m2I−ΛS+β1−β2I+αRm1+I+αRI1+m2I−γIR+12σ12+σ32+B−12μ−m2σ12∨σ22∨σ32Sm+1+Im+1+Rm+1.
Denote(22)F=supS,I,R∈ℝ+3−12μ−m2σ12∨σ22∨σ32Sm+1+Im+1+Rm+1+N2β1−β2μ+γS+2NγI−N2μ+2δ−m2δμ+γR+N+2μ+δ+2β1−β2m2+12σ12+σ32+B,H=supS,I,R∈ℝ+3−14μ−m2σ12∨σ22∨σ32Sm+1−142μ−mσ12∨σ22∨σ32Im+1+Rm+1+2β1−β2m2+N2β1−β2μ+γS−N2μ+2δ−m2δμ+γR+N+2μ+δ+2NγI+12σ12+σ32+B,J=supS,I,R∈ℝ+3−14μ−m2σ12∨σ22∨σ32Im+1−142μ−mσ12∨σ22∨σ32Sm+1+Rm+1+2β1−β2m2+N2β1−β2μ+γS−N2μ+2δ−m2δμ+γR+N+2μ+δ+2NγI+12σ12+σ32+B,L=supS,I,R∈ℝ+3−14μ−m2σ12∨σ22∨σ32Rm+1−142μ−mσ12∨σ22∨σ32Sm+1+Im+1+2β1−β2m2+N2β1−β2μ+γS−N2μ+2δ−m2δμ+γR+N+2μ+δ+2NγI+12σ12+σ32+B.
Letε>0 be sufficiently small for(23)−Λε+F<−1,−γε+F<−1,−14μ−m2σ12∨σ22∨σ321εm+1+H<−1,−14μ−m2σ12∨σ22∨σ321εm+1+J<−1,−14μ−m2σ12∨σ22∨σ321ε2m+2+L<−1.
Next,ℝ+3\Uε is divided into six areas:(24)U1=S,I,R∈0,∞3:S∈0,ε,U2=S,I,R∈0,∞3:S∈ε,1ε,I∈0,ε,U3=S,I,R∈0,∞3:S,I∈ε,1ε,R∈0,ε2,U4=S,I,R∈0,∞3:S∈1ε,∞,U5=S,I,R∈0,∞3:S∈ε,1ε,I∈1ε,∞,U6=S,I,R∈0,∞3:S,I∈ε,1ε,R∈1ε2,∞.
We shall proveLV2S,I,R<−1 in 0,∞3∖Uε.Case 1.
LetS,I,R∈U1. Then,(25)LV2≤−ΛS−142μ−mσ12∨σ22∨σ32Sm+1+Im+1+Rm+1+N2β1−β2γ+μS+2NγI−N2μ+2δ−m2δμ+γR+N+2μ+δ+2β1−β2m2+12σ12+σ32+B≤F−ΛS<F−Λε<−1.Case 2.
LetS,I,R∈U2. Then,(26)LV2≤−3NΛℝs1/3−1+2NγI−12μ−m2σ12∨σ22∨σ32Sm+1+Im+1+Rm+1+N+δ−N2μ+2δ−m2δμ+γR+2μ+N2β1−β2μ+γS+2β1−β2m2+12σ12+σ32+B≤−3NΛℝs1/3−1+2NγI+G≤−3NΛℝs1/3−1+2Nγε+G<1.Case 3.
LetS,I,R∈U3. Then,(27)LV2≤−γIR+F≤−γε+F<−1.Case 4.
LetS,I,R∈U4. Then,(28)LV2≤−182μ−mσ12∨σ22∨σ32Sm+1−182μ−mσ12∨σ22∨σ32Sm+1+N2β1−β2μ+γS−142μ−mσ12∨σ22∨σ32Im+1+Rm+1−N2μ+2δ−m2δμ+γR+2NγI+N+2μ+δ+2β1−β2m2+12σ12+σ32+B≤−182μ−mσ12∨σ22∨σ32Sm+1+H≤−182μ−mσ12∨σ22∨σ321εm+1+H<−1.Case 5.
LetS,I,R∈U5. Then,(29)LV2≤−182μ−mσ12∨σ22∨σ32Im+1−182μ−mσ12∨σ22∨σ32Im+1+N2β1−β2μ+γS−142μ−mσ12∨σ22∨σ32Sm+1+Rm+1−N2μ+2δ−m2δμ+γR+2NγI+N+2μ+δ+2β1−β2m2+12σ12+σ32+B≤−182μ−mσ12∨σ22∨σ32Im+1+J≤−182μ−mσ12∨σ22∨σ321εm+1+J<−1.Case 6.
LetS,I,R∈U6. Then,(30)LV2≤−182μ−mσ12∨σ22∨σ32Rm+1−182μ−mσ12∨σ22∨σ32Rm+1+N2β1−β2μ+γS−142μ−mσ12∨σ22∨σ32Sm+1+Im+1−N2μ+2δ−m2δμ+γR≤+2NγI+N+2μ+δ+2β1−β2m2+12σ12+σ32+B−182μ−mσ12∨σ22∨σ32Rm+1+L≤−182μ−mσ12∨σ22∨σ321ε2m+2+L<−1.
From (25)–(30), for every S,I,R∈0,∞3\Uε, LVS,I,R<−1. From Lemma 1, Theorem 2 holds.


## 4. Extinction

For convenience, denote(31)ℝ^s=2Λ2β1−β2γ+μ2m2μσ22γ2/2∧γ+μ2μ+σ32/2.Theorem 3.
For anyS0,I0,R0∈0,∞3, let St,It,Rt be the solution of (2). If ℝ^s<1, then(32)limt⟶∞It=0,limt⟶∞Rt=0,a.s.
Furthermore,(33)limsupt⟶∞1t∫0tSudu=Λμa.s.Proof.
DefineV3=lnγI+μ+γR. Then,(34)dV3=1γI+γ+μRβ1−β2I+αRm1+I+αRγSI1+m2I−μδ+γ+μR−σ22γ2I2+σ32γ+μ2R22γI+γ+μRdt+1γI+γ+μRγσ2IdB2+γ+μσ3RdB3≤2β1−β2Sm2dt−μγ+μ+δRγI+γ+μR+σ22γ2I2+σ32γ+μ2R22γI+γ+μR2dt+1γI+γ+μRγσ2IdB2+γ+μσ3RdB3≤2β1−β2Sm2dt−σ22γ22∧γ+μ2μ+σ322I2+R2γI+γ+μR2dt+1γI+γ+μRγσ2IdB2+γ+μσ3RdB3≤2β1−β2Sm2dt−σ22γ2/2∧γ+μ2μ+σ32/22γ+μ2dt+1γI+γ+μRγσ2IdB2+γ+μσ3RdB3.
From (3),(35)dS+R+I=−μS+R+I+Λdt+σ1SdB1+σ2IdB2+σ3RdB3.
Furthermore, it is similar to the proof in [26] and obtained that(36)limt⟶∞Itt=0,limt⟶∞Stt=0,limt⟶∞Rtt=0,a.s.limt⟶∞1t∫0tIvdB2v=0,limt⟶∞1t∫0tRvdB3v=0,limt⟶∞1t∫0tSvdB1v=0,a.s.
From (35) and (36),(37)limsupt⟶∞∫0tSu+Ru+Iudut≤Λμa.s.
From (34) and (37) and ℝ^s<1, we obtain that(38)limsupt⟶∞lnγI+μ+γRt≤2β1−β2Λm2μ−σ22γ2/2∧μ+σ32/2μ+γ22μ+γ2<0  a.s.
This implies(39)limt⟶∞Rt=0andlimt⟶∞It=0,a.s.
Further, it follows from (35) that(40)limsupt⟶∞∫0tSudut=Λμa.s.


## 5. Asymptotic Property around the Disease-Free Equilibrium

Theorem 4.
For everyS0,I0,R0∈0,∞3, let St,It,Rt be the solution to (3). If ℝ0<1, σ12<μ,σ22<2μ, and σ32<2μ, then(41)limsupt⟶∞1tE∫0tSv−Λμ2+I2v+R2vdv≤K1σ12l1,where K1=21+2μ/δΛ/μ2, l1=min21+2μ/δμ−σ12,1+2μ/δ2μ−σ22,−σ32−2μ.Proof.
Define(42)V4S,I,R=v1+2μδv2+2μβ1δ2δ+2μ+γv3,where(43)v1=S+R+I−Λμ2,v2=I+S−Λμ2,v3=kR+I,where k>0 is a constant to be chosen. Then,(44)Lv1=−2μS−Λμ2−2μI2+R2−4μR+IS−Λμ−4μIR+σ12S2+σ22I2+σ32R2≤−2μ−σ12S−Λμ2−4μR+IS−Λμ−2μ−σ22I2−2μ−σ32R2−4μIR+2Λσ1μ2,Lv2=−2μ−Λμ+S2−2μ+γ−σ22I2+2δS−ΛμR−22μ+γ−Λμ+SI+2δIR+σ12S2≤−2−σ12+μ−Λμ+S2−2γ+2μS−ΛμI−2γ+μ−σ22I2+2δS−ΛμR+2δRI+2Λσ1μ2,Lv3=β1−β2I+αRm1+I+αRIS1+m2I+−γ1−k−μI+δ1−k−kμR≤β1SI+−μ−γ+kγI+δ1−k−kμR=β1−Λμ+SI+Λβ1μ−μ−γ1−kI+δ1−k−kμR.
Takeδγ/μ+δ≤kγ≤γ+μ−Λβ1/μ. From ℝ0<1,(45)Lv3≤β1IS−Λμ.
It follows from (44) and (45) that(46)LV4≤−2μ−σ122μδ+1−Λμ+S2−2μδ+1−σ22+2μI2+22μδ+1σ1Λμ2−−σ32+2μR2.
Hence,(47)dV4=LV4dt+2σ12S−Λμ+I+R+2μδS−Λμ+IS2dB1t+2σ221+2μδS−Λμ+I+RI2+2μσ2β1δ2δ+2μ+γIdB2t+2σ32S−Λμ+I+RR2+2μσ3kβ1δ2δ+2μ+γRdB3t≤−l1R2+I2+−Λμ+S2+22μδ+1σ1Λμ2dt+2σ12S−Λμ+I+R+2μδS−Λμ+IS2dB1t+2σ221+2μδS−Λμ+I+RI2+2μσ2β1δ2δ+2μ+γIdB2t+2σ32S−Λμ+I+RR2+2μσ3kβ1δ2δ+2μ+γRdB3t.
Then,(48)0≤V4t≤V40−l1∫0tSv−Λμ2+I2v+R2vdv+21+2μδΛσ1μ2t+M^t,where(49)M^t=2σ12∫0tS−Λμ+I+R+2μδS−Λμ+IS2dB1v+∫0t2σ221+2μδS−Λμ+I+RI2+2μσ2β1δ2δ+2μ+γIdB2v+∫0t2σ32S−Λμ+I+RR2+2μσ3kβ1δ2δ+2μ+γRdB3v.
Hence,(50)limsupt⟶∞1tE∫0tSv−Λμ2+I2v+R2vdv≤K1σ12l1.Remark 1.
By Theorem4, if ℝ0<1, σ12<μ,σ22<2μ,σ32<2μ, then the solution of (3) fluctuates around the disease-free equilibrium. In addition, if σ1=0, then E0=Λ/μ,0,0 is the disease-free equilibrium of (3). Then, (46) becomes(51)LV4≤−2μ2μδ+1−Λμ+S2−2μδ+1−σ22+2μI2−2μ−σ32R2.
Thus,E0 is globally asymptotically stable if ℝ0<1, σ22<2μ,σ32<2μ.


## 6. Asymptotic Property around the Endemic Equilibrium

Assumeℝ0>1. E∗S∗,I∗,R∗ is the endemic equilibrium of (1). From(52)Λ−β1−β2I∗+αR∗m1+I∗+αR∗S∗I∗1+m2I∗−μS∗=0,β1−β2I∗+αR∗m1+I∗+αR∗S∗I∗1+m2I∗−μ+γI∗+δR∗=0,γI∗−μ+δR∗=0,we can easily show(53)S∗=μ+γ−δγ/μ+δm1μ+δ+μ+δ+αγI∗1+m2I∗β2μ+δ+αγI∗,R∗=γI∗δ+μ.SubstitutingS∗,R∗ into (52) yields(54)A1I∗2+A2I∗+A3=0,where(55)A1=−δ+μ+αγγ+μ−δγμ+δβ1−β2+μm2,A2=μ+αγ+δ−μμ+γ−δγδ+μ+Λ−β2+β1−m1μ+δβ1−μm2−δγδ+μ+γ+μ,A3=m1μ+δΛβ1−μ−δγδ+μ+γ+μ.Fromβ1≥β2 and ℝ0>1, we have A1<0 and A3>0. So, E∗S∗,I∗,R∗ is a unique endemic equilibrium of (1). Denote(56)p=2μ+γ1+m2I∗β1−β2I∗+αR∗/m1+I∗+αR∗,q=δ2μ+γ1+m2I∗R∗γI∗β1−β2I∗+αR∗/m1+I∗+αR∗.Theorem 5.
For anyS0,I0,R0∈ℝ+3, let St,It,Rt be the solution of (3). If ℝ0>1, σ12<μ/2−pβ2α,σ22<μ/2+γ−γ2/2δ,σ32<μ+δ/2−δ2/μ, then(57)limsupt⟶∞1tE∫0tSv−S∗2+Rv−R∗2+Iv−I∗2dv≤K2l2,where(58)K2=σ12+pβ2αS∗2+σ22I∗2+σ32R∗2+12pσ22I∗+12qσ32R∗+12pβ2αR∗+I∗α2,l2=minμ2−σ12−pβ2α,μ2+γ−σ22−γ22δ,μ+δ2−δ2μ−σ32.Proof.
DefineV5S,I,R:ℝ+3⟶ℝ by(59)V5S,I,R=w1S,I+w2R+pw3I+qw4R,where(60)w1S,I=I−I∗+S−S∗22,w2R=−R∗+R22,w3I=−I∗lnII∗+I−I∗,w4R=−R∗lnRR∗+R−R∗.
Using the Ito^ formula and (52),(61)Lw1=I−I∗+S−S∗−μI−I∗+S−S∗+δR−R∗−γI−I∗+12σ12S2+12σ22I2≤−μ−σ12S−S∗2−I−I∗2γ+μ−σ22+μS−S∗δμR−R∗−γ+2μI−I∗S−S∗+μI−I∗δμR−R∗+σ12S∗2+σ22I∗2≤δ2μ−R∗+R2−μ2−σ12−S∗+S2−γ+μ2−σ22−I∗+I2−2μ+γ−I∗+I−S∗+S+σ12S∗2+σ22I∗2,(62)Lw2=−δ+μR−R∗+γI−I∗R−R∗+σ32R22≤γR−R∗I−I∗−δ+μ−σ32R−R∗2+σ32R∗2≤γ22δI−I∗2+σ32R∗2−μ+δ2−σ32R−R∗2,(63)Lw3=I−I∗β1−β2I+αRm1+I+αRS1+m2I−β1−β2I∗+αR∗m1+I∗+αR∗S∗1+m2I∗−δR∗I∗+δRI+12σ22I∗=−β2SI−I∗1+m2IαR+Im1+I+αR−I∗+αR∗m1+αR∗+I∗−αR∗+I∗m1+αR∗+I∗−β2SI−I∗1+m2I+β1SI−I∗1+m2I−β1−β2αR∗+I∗m1+αR∗+I∗I−I∗S∗1+m2I∗+δI−I∗−R∗I∗+RI+12σ22I∗=−β2m1I−I∗2S1+m2Im1+αR+Im1+I∗+αR∗+δR∗RR∗−II∗−I∗RIR∗+1−β2αm1SI−I∗R−R∗m1+I∗+αR∗1+m2Im1+I∗+αR+12σ22I∗+I−I∗S1−m2I−S∗1+m2I∗β1−β2I∗+αR∗m1+I∗+αR∗≤β2αS−S∗2+β2αS∗2+S−S∗I−I∗1+m2I∗β1−β2I∗+αR∗m1+I∗+αR∗+12β2αR∗+I∗α2+δR∗RR∗−II∗−RI∗IR∗+1+12σ22I∗,(64)Lw4=γ−R∗R+1−RI∗R∗+I+12σ32R∗=γI∗1−IR∗I∗R−RR∗+II∗+12σ32R∗.
It follows from (61)–(64) that(65)LV5≤−μ2−σ12−pβ2α−S∗+S2−μ2+γ−σ22−γ22δ−I∗+I2−δ2+μ−δ2μ−σ32−R∗+R2+σ12S∗2+σ22I∗2+σ32R∗2+pβ2αS∗2+12pβ2αR∗+I∗α2+pσ22I∗2+qσ32R∗2≤−l2−S∗+S2+−I∗+I2+−R∗+R2+K2.
Hence,(66)dV5≤−l2−S∗+S2+−R∗+R2+−I∗+I2+K2+pσ2−I∗+IdB2t+−S∗+S−I∗+Iσ1SdB1t+σ2IdB2t+R+qσ3R−R∗dB3t.
Then,(67)EV5St,It,Rt≤V5S0,I0,R0+K2t−l2E∫0tSv−S∗2+Rv−R∗2+Iv−I∗2dv.
Therefore,(68)limsupt⟶∞1tE∫0tSv−S∗2+Rv−R∗2+Iv−I∗2dv≤K2l2.
The proof is complete.Remark 2.
By Theorem5, if the noise intensity is small and ℝ0>1, the solution of (3) fluctuates around the endemic equilibrium. That is, if the intensity of random disturbance is small enough, the disease will persist.


## 7. Numerical Simulations

Example 1.
Letβ1=0.5,β2=0.01,μ=0.1,Λ=0.6,γ=0.2,δ=0.1,m1=m2=1,α=0.5,σ12=0.02,σ22=0.01,S0,I0,R0=0.7,0.12,0.18. Then ℝs≈1.4605>1, and model (3) has a stationary distribution by Theorem 2. As can be seen from Figure 1, when S0,I0,R0=0.5,0.3,0.2 or 0.4,0.4,0.2, there is a stable distribution in model (3).Figure 1
The probability density function trajectories of susceptible, infected, and recovered individuals of (3) with S0,I0,R0=0.5,0.3,0.2 or 0.4,0.4,0.2.
(a)(b)(c)Example 2.
Letβ1=0.2,β2=0.18,μ=0.15,Λ=0.2,γ=0.8,δ=0.1,m1=1,m2=2.7,α=0.5,σ22=0.4,σ32=0.2, S0,I0,R0=0.5,0.3,0.2. Then, ℝ^s≈0.6566<1. By Theorem 3, the disease of (3) becomes extinct (see Figure 2).Figure 2
The trajectories of (1) and sample path of (3).
(a)(b)(c)Example 3.
Letβ1=0.2,β2=0.1,μ=0.2,Λ=0.3,γ=0.2,δ=0.1,m1=m2=1,α=0.5,σ12=0.1,σ22=0.1,σ32=0.2, S0,I0,R0=0.7,0.12,0.18. We obtain ℝ0≈0.8996<1, 0.1=σ12<μ=0.2,0.1=σ22<2μ=0.4,0.2=σ32<2μ=0.4. It is known from Theorem 4 that the solution of (3) fluctuates around the curve of (1). Figure 3 is consistent with this result.Figure 3
The trajectories of (1) and sample path of (3).
(a)(b)(c)Example 4.
Letβ1=0.65,β2=0.05,μ=0.5,Λ=0.8,γ=0.3,δ=0.3,m1=1,m2=0.1,α=0.01,σ12=0.02,σ22=0.03,σ32=0.04,S0,I0,R0=4.5,3.8,2.6. Then, ℝ0≈1.5125>1,p≈2.1242,0.02=σ12<μ/2−pβ2α≈0.2489,0.03=σ22<μ/2+γ−γ2/2δ=0.4,0.04=σ32<μ+δ/2−δ2/μ=0.47. Solution of (3) fluctuates around the endemic equilibrium of (1) by Theorem 5. Figure 4 is consistent with this result.Figure 4
The trajectories of (1) and sample path of (3).
(a)(b)(c)Example 5.
To show the impact of media coverage on the spread of diseases, different media coverage parameters are taken asβ2=0.01,0.1,0.2. Let β1=0.18,μ=0.25,Λ=0.8,γ=0.2,δ=0.2,m1=1,m2=0.1,α=0.01,σ12=0.01,σ22=0.04,σ32=0.03,S0,I0,R0=0.5,0.4,0.1. As can be seen from Figure 5(a), the larger the media coverage parameter β2 is, the smaller the time mean 1/t∫0tIsds of the infected individuals is. This suggests that media coverage can decrease the spread of disease.Figure 5
The sample path of the time mean of infected individuals in (3).
(a)(b)Example 6.
To illustrate the influence of the number of recovered individuals on the spread of infectious diseases, we take respectivelyα=0.01,0.1,0.15. β1=0.18,β2=0.15,μ=0.25,Λ=0.8,γ=0.2,δ=0.2,m1=1,m2=0.1,σ12=0.01,σ22=0.03,σ32=0.02,S0,I0,R0=0.5,0.4,0.1. From Figure 5(b), the time mean of the infected individuals decreases with the decrease of the parameter α. This shows that the less impact the number of recovered individuals has on media coverage, the lower the transmission rate is, which in turn leads to fewer infected individuals.


## 8. Conclusions

We analyze a stochastic SIRI model with media coverage. To begin with, existence and uniqueness of global positive solution of the model are established. Using the Hasminskii theory, we prove existence of a stationary distribution. Then, conditions of disease extinction are proved. By constructing the suitable Lyapunov functions, the dynamic properties of solutions near equilibria are studied. Ifℝ0<1, when the intensity of random disturbance is sufficiently small, infectious diseases will be extinct; if ℝ0>1 and the random disturbance is small, infectious diseases will persist. In addition, the less impact the number of recovered individual has on media coverage, the lower the transmission rate is, which in turn leads to fewer infected individuals.



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