The Stability Analysis Of Several Epidemic Models With Spontaneous Mutation Of Virus

In recent years,the spread of COVID-19 seriously threatens the safety of human life and the development of world civilization,the spread and control of infectious diseases have been the focus of human attention.Since the research of infectious diseases cannot be carried out experiments,the mathematical model is established for the dynamic analysis of infectious diseases.It is of great significance to understand the laws of disease transmission and predict their development trends.However,the virus may mutate during the transmission process,resulting in the disease getting out of control.Therefore,research on infectious diseases model with virus mutation can better prevent and control the epidemic of diseases.This paper studies the problem of infectious diseases with virus mutation.The main contents are as follows:In chapter 1,we briefly introduces the background of topic selection,research significance and status at home and abroad in this paper.The definitions,lemmas and theorems that will be used in the article are given.In chapter 2,an SIR epidemic model with different incidences is studied.Firstly,the basic reproduction number is found by using the next generation matrix method.Secondly,the local asymptotic stability of equilibria are carried out through the use of Hurwitz criterion and Jacobi matrix.Finally,the sufficient conditions for global asymptotic stability of equilibria are determined through suitable Lyapunov function and Lasalle invariant set principle.In chapter 3,a virus mutation epidemic model with age structure is established.The existence and uniqueness of the non-negative solution are proved by using the fixed point theorem.The expressions for the basic reproductive number and the reproductive number with vaccination are calculated through the characteristic equation.By using the theory and methods in integral equation,the asymptotic stability of the disease-free equilibrium point is proved,the existence and local asymptotic stability of the endemic equilibrium point are demonstrated.The optimal age vaccination strategy is given with the help of linear optimization problem.In chapter 4,taking into account the situation in which the loss of immunity of vaccinated and recovered people are reinfected,we propose a class of virus mutation epidemic model with immune failure and age structure.The threshold of epidemic prevalence is studied,the existence of disease-free equilibrium point is proved by normalization,and the expression of basic regeneration number is obtained.we prove sufficient conditions for the stability of the disease-free equilibrium point and the existence of the endemic equilibrium point.