Dynamics Of Several Stochastic SEIR Epidemic Models With Infectivity In Incubation Period And Homestead Isolation

Infectious diseases have always been the public enemy that endangers human health,and each outbreak has brought serious disasters to human beings.Using mathematical means and methods to establish corresponding epidemic mathematical models have become an important and powerful tool to understand the law of disease transmission.In the real world,environmental noises will inevitably influence the spread of epidemic.It is very meaningful to use stochastic differential equations to study the dynamic behavior of epidemic model.This paper will consider three types of environmental noises(white noise,color noise and Lévy noise),and the dynamic properties of several stochastic epidemic models are studied to reveal the influence of environmental noises on disease transmission.The research content of this paper is arranged as follows:Chapter 1 summarizes the biological background,research significance and research status of stochastic epidemic models,and briefly introduces the main research content of this topic.In Chapter 2,several definitions,lemmas and inequalities which will be used in proof of conclusions are introduced,respectively.In Chapter 3,two kinds of stochastic SEIR epidemic models with white noise are studied.For the autonomous stochastic SEIR model,sufficient conditions for disease extinction and persistence are given respectively,and then the existence and uniqueness of the ergodic stationary distribution of model is obtained by using Khasminskii ergodic theorem and Lyapunov function.For periodic stochastic SEIR model,with the help of Khasminskii periodic theorem,sufficient conditions for the existence of nontrivial periodic solutions are obtained.Finally,numerical simulation is given to verify the feasibility of the theoretical results obtained.In Chapter 4,a stochastic SEIR epidemic model disturbed by white noise and color noise is analyzed.By using Khasminskii theory and Markovian switch theory,sufficient conditions for the existence and uniqueness of the ergodic stationary distribution of model are proved.Numerical simulation is also given to verify the feasibility of the theoretical results.In Chapter 5,a stochastic SEIR epidemic model with Lévy noise is obtained.Firstly,the persistence and extinction of the disease are studied.Then the asymptotic behaviors of solutions of the stochastic model near disease-free equilibria and endemic equilibria of the deterministic model are discussed by constructing Lyapunov functions,respectively.Finally,several numerical examples are conducted to verify the feasibility of the theoretical results.