Asymptotic Behavior Of Stochastic Epidemic Models

Dynamics of infectious diseases is the study of the spread and development of diseases with the purpose to trace factors that are contribute to their occurrence. In recent decades, epidemiology modeling by mathematical research has been recieved a great attention within the academia. This paper study some stochastic epidemic models, applying the theory of stochastic differerntial equation. In this paper, we extended some deterministic model for the epidemics by inducing random perturbations in the models, and study the property of dynamics for the stochastic epidemic models.This paper consists of four parts.In the first chapter, some definitions and preminary theorems which will be used in this paper are introduced exhaustively.In the second chapter, we explore stochastic SIR and SIRS models, respectively. First, we show the two models both exit the unique global positive solution. Furthermore, we investigate the asymptotic behavior of the positive solution. There is neither disease-free equilibrium nor endemic equilibrium for the two stochastics models after adding stochastics perturbation in the corresponding deterministic models. Hence in order to show the stability to some extent, we discuss the behavior around disease-free equilibrium and endemic equilibrium of the deterministic models, respectively. Finally, numerical simulations are present to illustrate our mathematical findings.In the third chapter, we present a S-DI-R epidemic model with two categories stochas-tic perturbations. For the one issue, taking into account the effect of randomly fluctuating environment in the S-DI-R epidemic model by the approach in. For this stochastic model, we show it has the nonnegative solution for any nonnegative initial value. Next, the long time behavior of this stochastic systems are studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of de-terministic system under different conditions. For the other issue, we set up the other stochastic model by adding white noise stochastic perturbations around the endemic equi-librium of the deterministic S-DI-R models. Subsequently, we show the solution of this stochastic model is stochastically asymptotically stable by Lyapunove function, and the stability condition is obtained.In the fourth chapter, we study stochastic multi-group SIR model, multi-group SEIR model and multi-group SIR model with time delay, respectively, allowing stochastic fluc-tuation around the endemic equilibrium. At first, We prove the endemic equilibriums of the stochastic multi-group SIR and SEIR models both are stochastic asymptotically stable and obtain the stability conditions by the construction of Lyapunov function and graph theory. In addition, we give the numerical simulations. Finally, We prove the en-demic equilibrium of the stochastic multigroup SIR model with time delay is stable in probability.In the last chapter, we made a summary of full paper.