The Threshold Of Several Stochastic Epidemic Model

Infectious diseases have been endangering human health and life,mathematical modeling has been an important approach in analyzing the spread and control of infec-tious diseases.In recent years,many scholars discuss stochastic noises into deterministic models to reveal the effect of environmental noise,they introduce the persistence,ex-tinction,existence and uniqueness of the positive solution and stationary distribution of the model Its main contents can be summarized as follows:In the first part,we introduce the biological background and significance of stochas-tic epidemic model,and then introduce the present investigation of stochastic epidemic model,finally,we give the main work of our paper.In the second part,some definitions,symbols,lemmas and theorem which will be used in this paper are introduced exhaustively.In the third part,we consider a stochastic SIVS epidemic model with general nonlinear incidence rate.It is shown that the extinction and permanence in the mean of the disease are determined by a threshold value (?).If (?)<1,then disease dies out with probability one,and if (?)>1,then the disease is permanent in the means with probability one.In the fourth part,we discussed persistence and extinction for a stochastic SIS epidemic model with isolation.we assume that the fluctuations in the environment will manifest themselves mainly as fluctuation in the transmission coefficient ? and the disease-related constant ?2 and ?3,the proportional coefficient of isolated of disease.In the previously known literature,the extinction has been discussed,but the persistence has never been discussed.It is shown that the extinction and permanence in the mean of the disease are determined by a threshold value R0s.If.R0s<1,then disease dies out with probability one,and if-R0s>1,then the disease is stochastic persistent in the means with probability one.There exists a stationary distribution when the noise is not large.Finally,some numerical simulations are carried out to confirm the analytical results.In the fifth part,we discuss a class of stochastic SIR epidemic model with general nonlinear incidence rate and temporary immunity.The existence of the unique global positive solution any positive initial value is established.Sufficient condition for extinc=tion and permanence in the mean of the disease are established.That is,if (?)<1,then disease dies out with probability one,and if (?)>1,then the disease is permanent in the means with probability one.In the sixth part,we do some discussions and conclusions about the research results on this paper.