| Human beings have been fighting against infectious diseases since their birth.Although some achievements have been made in the prevention and control of certain diseases,with the increase of the range of human activities,we have gradually come into contact with many new pathogens,In particular,the coronavirus outbreak in 2019 is even more frightening.Therefore,we need to study the dynamic behavior of infectious diseases,find out the pathogen,epidemic law,and pathogenesis of infectious diseases,and so on,is the direction that we have been working hard for.There are various types of infectious disease models.This thesis mainly studies the stochastic SIRS infectious disease model with nonlinear incidence,whose exposure rate is interfered with by white noise.Meanwhile,it also considers the case with Markov switch and Logistic birth,and mainly discusses the extinction and persistence of the disease and so on.The first and second chapters introduce the research background and some preliminary knowledge used in the thesis and so on.In Chapter 3,a stochastic SIRS epidemic model disturbed by environment is established to analyze its dynamic behavior The existence and uniqueness of the global positive solution of the model are proved.The sufficient conditions for the extinction of the disease and the persistence of the disease in the mean sense are discussed.Finally,the numerical simulation is carried out.In chapter 4,we study a random SIRS epidemic model which is influenced by both Markov switching and white noise.According to Ito’s formula,local martingale’s law of strong numbers,and Birkhoff’s ergodicity theorem,the sufficient conditions for disease extinction are discussed.Then,the existence of an ergodic stationary distribution for the solution of this model is analyzed and the theoretical results are verified by numerical simulation.In chapter 5,a random SIRS infectious disease model with Logistic birth and Markov switching is studied.By constructing appropriate V function and using Ito’s formula,the existence and uniqueness of the global positive solution of the stochastic infectious disease model are analyzed.Then,the existence of an ergodically stationary distribution of the solution of the model and the sufficient conditions for the extinction of the disease are discussed.Finally,numerical examples are given to illustrate the conclusions obtained in this thesis. |
