Asymptotic Behavior Of Stochastic Epidemic Models

In natural ecological system, various stochastic perturbations appear. Epidemic model can much closer to the actual system by adding the random disturbance to the deterministic model of infectious diseases. This paper mainly discusses the asymptotic behavior of the system of infectious diseases by adding random white noise interference and interference with Lévy jumps. Detailed discussion are arranged as follows:Chapter 1 mainly introduces the research of epidemic models in recent years,and the main discussion of this paper is given.Chapter 2 considers the qualitative analysis of a class of the stochastic SIR epidemic model and the stochastic SEIR epidemic model with Lévy jumps. we firstly prove that the existence of the unique global positive solution of the system.Then we acquire the qualitative analysis of the corresponding stochastic models with Lévy jumps around the equilibrium points of the deterministic models.Lastly the extinction of the stochastic systems is offered.Chapter 3 discusses a nonlinear stochastic SIS epidemic model with double epidemic hypothesis. We obtain the thresholds of extinction and permanence in mean of the stochastic epidemic model respectively by the Doob’s martingale inequality,Burkholder-Davis-Gundy inequality and Borel-Cantelli lemma. And we present a series of numerical simulations to illustrate the theoretical results by using mathematical software MATLAB.Chapter 4 summarizes the contents and main conclusions of this paper, and offers the prospects of further research.