Asymptotic Behavior Of Stochastic Epidemic Models

Dynamics of infectious diseases is main to study of the spread and development of diseases with the purpose to trace factors that are contribute to their occurrence, so as to more effectively control epidemic of the disease.In recent decades, epidemiol-ogy modeling by mathematical research has been received a great attention within the academia,especially the stochastic model of infectious diseases. This paper study some stochastic epidemic models,applying the theory of stochastic differential equation to study the property of dynamics for the stochastic epidemic models.This paper consists of five parts.In the first chapter, we introduce the biological background and significance of the stochastic epidemic models,Then introduce some achievement has been achieved in stochastic epidemic models by mathematical workers, Finally,we introduced the important results has been achieved in this paper.In the second chapter, some definitions and lemmas which will be used in this paper are introduced exhaustively.In the third chapter, mainly discussed the asymptotic dynamics of stochastic SIS epidemic model,taking a stochastic disturbance into account in the deterministic model, in the form of Gaussian white noises, we obtained its invariant probability distribution by applying Fokker-Planck equation,then taking two stochastic disturbances into account in the deterministic model, in the form of Gaussian white noises, We mainly focuses on the influence of the size of the correlation coefficient of two stochastic perturbations on stochastic epidemic model, under two stochastic disturbance, the correlative of two stochastic disturbances can affect the dynamics of the epidemic model, and the effects are different if we introduce the stochastic disturbance in different ways. The results are presented using data from bovine tuberculosis in possum populations in New Zealand as an example.In the fourth chapter,we explore stochastic SIRS models with linear incidence and nonlinear incidence repectively.First, we show the two models both has the unique global positive solution.Furthermore, we investigate the asymptotic behavior of the positive so-lution. There is neither disease-free equilibrium nor endemic equilibrium for the two stochastic models after adding stochastic perturbation in the corresponding deterministic models. Hence in order to show the stability to some extent, we discuss the behav-ior around disease-free equilibrium and endemic equilibrium of the deterministic mod-els,respeetively.Finally,numerical simulations are present to illustrate our mathematical findings.In the fifth chapter,we did some discuss and conclusions about what we have studied in this paper.