The Asymptotic Behavior Of SIRS And SEIR Stochastic Epidemic Models

In this paper,we study the asymptotic behavior of stochastic SIRS and SEIR epi-demic model dS(t)=(Λ-βS(t)I(t)/1+αI(t)-dsS(t)+μR(T))dt+σ1S(t)dB1(t), dI(t)=[βS(t)I(t)/1+αI(t)-(dI+δ+γ)I(t)]dt+σ2I(t)dB2(t), dR(t)=[λI(t)-(dR+λ)R(t)]df+σ3R(t)dB3(t) and dS(t)=(λ-βS(t)I(t)/1+αI(t)-dsS(t))dt+σ1S(t)dB1(t), dE(t)=[βS(t)I(t)/1+αI(t)-(dE+θ)E]dt+σ2E(t)dB2(t), dI(t)=[θE(t)-(dI+δ+γ)I(t)]dt+σ3I(t)dB3(t), dR(f)=(γI(t)-dRR(t))dt+σ4R(t)dB4(t).In chapter 2,for the convenience of the readers,we give the preliminary knowledges needed in this paper:In chapter 3,we present a sufficient condition for the extinction of stochastic SIRS epidemic model,that is,when R0S<1,the disease becomes extinct almost surely:In chapter 4,firstly,we study the ergodicity of stochastic SIRS epidemic model,that is,when R0S>1,there is a unique stationary distribution which is ergodic; Finally we study the ergodicity of stochastic SEIR epidemic model.We show that when R0S>1 there is a unique stationary distribution which is ergodic.