The Dynamics Study Of Stochastic Epidemic Model

It is well known that there are various infectious diseases in the real world,which might do great harm to human health and the society.Mathematical epidemic models have become important tools in investigating the spread and the control of infectious diseases.In the real world,environmental noises are everywhere that reflect the actual phenomenon more accurately and reveal the effect of stochastic perturbations on epidemic model by introducing environmental noises into deterministic model.The aim of this paper is to investigate how environmental noises and the intensity of the noises influence the dynamic behaviors of epidemic model.Firstly,we formulate a stochastic SIV epidemic model.The sufficient conditions for the existence and uniqueness of the solution of the system are obtained.By using the (?)'s formula,Lyapunov function and Chebyshev's inequality,we establish the sufficient conditions for stochastically permanent and extinct of the model.A series of numerical simulations are used to illustrate the reliability of these theoretical results.Secondly,a stochastic epidemic model with two different kinds of infectious diseases that spread through horizontal and vertical transmission is proposed.We obtain the sufficient conditions for existence,stochastically ultimately bounded and permanent of the global positive solution by using the (?)'s formula and suitable Lyapunov function.We also make some numerical simulations by Matlab to confirm the above theoretical analysis and the rich dynamics phenomenons,which are causes by stochastic perturbations.Finally,we summarize the work of this paper and prospect the future research work.