Stochastic Epidemic Model With Nonlinear Incidence

With the development of human society, epidemic disease also is in constant de-velopment, for a long time, humans struggles with all kinds of epidemic disease. So in the field of mathematical biology, epidemic model research is still very important. But in fact, epidemic model inevitably affected by other environmental factors around, so compared with deterministic system, stochastic differential equation which can more ac-curately and truthfully describe the real world. Therefore, stochastic model has become one of hot research subjects in the mathematical theory. This paper, we we discuss the dynamic behavior of stochastic epidemic model by he theory of stochastic differential equations. 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 studied a class of stochastic SIRS epidemic model with non-linear incidence ?f(S)g(?). In this paper, we first obtain stochastic model by interfering the deterministic model with white noise. Second we study the existence of global pos-itive solution through Liapunov function and Ito formula. Then the threshold value widetildeRo is identified, we use auxiliary function and the theory of stochastic differ-ential equations to get the sufficient conditions for the extinction of the disease and the permanence in the mean of the model with probability one. Finally, considering model without disease-related death, study the system exists unique stationary distribution.In the fourth part, we mainly studies the stochastic dynamics of SIRS epidemic model with nonlinear incidence rate ?Sg(?) and immune. In this paper, we not only use Liapunov function, Ito formula to prove the existence of global positive solution, extinction of disease and persistence in the mean, but also use B-D-G inequalities, great inequality of martingale, the strong theorem to prove the extinction exponentially and persistence of the disease. Finally, through the previous proof of the theorem, we discuss the existence of the stationary distribution and ergodicity of the model.In the fifth part, we do some discussions and conclusions about the research results on this paper.