Study On Some Stochastic Epidemic Models

In this paper,several types of stochastic epidemic models with general incidence and random perturbation are studied.This paper analyzes the influence of various stochastic factors to the process of the spreading of disease.Chapter 1 reviews the development of the infectious disease models,especially the stochastic epidemic models with several kinds of environmental factors.The preliminaries are given out.Chapter 2 studies the stochastic SIRS model and SIS model with general incidence.For stochastic SIRS model,this part shows the existence and uniqueness of the global positive solution with the stopping time approach.Then,the upper bound of p-th moment can be estimated.And,several methods are given out to study the almost surely globally stochastically exponential stability of the disease-free equilibrium and obtain some conditions for extinction of the disease.Then,this chapter derives the disease will prevail,which is measured through the difference between the solution and the endemic equilibrium of the deterministic model in time average and gives out the conditions for disease persistence and the existence of the stationary distribution.Finally,by adding unlinear control,this chapter proves the system has finite-time stability.After that,numerical simulations are given out to confirm the conclusions involved.Then this chapter introduces a new method to study the extinction and ergodic property of the stochastic SIS model with general incidence,and makes the corresponding numerical simulation supporting these conclusion.Chapter 3 discusses the stochastic epidemic model with Markov switching.Using a similar method,this chapter studies the global existence and uniqueness of positive solutions.Then it gives out the conditions to ensure the extinction and the persistence.Due to the special nature of Markov switching,this chapter finally proves that the system is stable in distribution.Chapter 4 studies the stochastic epidemic model with diffusion on network.Firstly this chapter uses stopping time to prove the global existence and uniqueness of positive solutions.Then it discusses the stability of disease-free equilibrium,and the asymptotic behavior around the endemic equilibrium model of the deterministic model.Finally,it deduces some conditions for the persistence of disease,and gives out the corresponding numerical simulation.Chapter 5 studies the stochastic epidemic model with Levy jump,which breaks the continuity of the solution.Stopping time is used to prove the existence and uniqueness of the positive global solution.Then the extinction and persistence are established under some conditions.Finally,the asymptotic behavior around the disease free equilibrium and the endemic equilibrium are studied.Chapter 6 summarizes and prospects the relevant research.