Study On Asymptotic Behaviors Of The Solutions To Stochastic Epidemic Models

This paper consists of four parts.In the first chapter,we describe the development of stochastic epidemic models and the related definitions and theorems.In the second chapter,a type of stochastic SIQS model is investigated.Firstly,the unique global solution of the model is checked by using of Lyapunov functions.Then the asymptotic behavior of the global solution of the stochastic model near the equilibrium point of the deterministic model is studied.In the third chapter,a kind of stochastic SIRS model with time delay is investigated in which the contact rate is disturbed by the white noise.Firstly,the existence of positive solution to stochastic model is proved by using of Lyapunov functions.Secondly,the ext:inction and existence of the diseases are investigated.The study indicates that if the intensity of the white noise is small and stochastic indicator R0<1,the disease will disappear;and if R0>1,then the disease is prevalent.But when the intensity of the white noise is large,together with the basic reproduction number R0>1,the disease will disappear.Furthermore,the main results show that extinction of the diseases does not depend on time delay r,persistence of the diseases does depend on time delay r.In the last chapter,a two-disease driven stochastic epidemic model is studied with system perturbation.Firstly,we prove a unique global solution to this stochastic model.Then extinction and existence of the diseases are put into investigation.The sufficient conditions for prevalence of the diseases are derived.If the stochastic indicator Ri<1(i=1,2),then the diseases Ii(i=1,2)will disappear;and if Ri>1(i=1,2),then the diseases Ii(i=1,2)will be prevalent.Finally,numerical simulations are figured out to support the main results.