Dynamical Analysis Of The Infectious Disease Models With Nonlinear Incidence Rate

Infectious diseases have brought great troubles to most people s lives,thus it is of great importance to study the pathogenesis,transmission routes,prevention and control measures of these diseases.Epidemic dynamics,by establishing mathematical models,is a way to study the spread of infectious diseases,it can effectively reflect the law of dis-ease transmission as well as the global behavior of the prevalence of disease.According to the infectious process and immune mechanism of infectious diseases,we establish two kinds of epidemic models with non-linear disease incidence.One is about latent delay SEIRS epidemic model,and the other is about hepatitis B dynamic model with vaccination.We discuss the dynamical behavior and biological significance of the two models in this thesis.In chapter 1,we first introduce the background information of infectious diseases,including the characteristics of patients,the transmission principles,transmission routes and prevention measures of the diseases as well as the impact they exert on people s lives.Secondly,we summarize the research progress of infectious disease dynamics model at home and abroad and also the preliminary knowledge required for this thesis.In chapter 2,according to the infectious mechanism and characteristics of infectious diseases,we establish the SEIRS epidemiological model with nonlinear incidence rate and latent delay.Firstly,we work out the basic reproduction number R0 and prove the nonnegative and boundedness of the solution,as well as the existence of endemic dis-eases.Secondly,by constructing Lyapunov functional,we obtain the global stability of disease-free equilibrium,the endemic equilibrium point s local stability and the system is uniformly persistent.Finally,some numerical simulations are presented to illustrate the conclusions.In chapter 3,in consideration of the effect of vaccination on late hepatitis B in-fection,we establish a hepatitis B model with the incidence of nonlinear diseases and vaccination on the basis of the original model.In the first place,we prove the nonnegativ-ity and boundedness of the system solutions,the basic reproduction number is obtained through analyzing the system.Then,we analyze the existence of positive equilibrium,and by constructing a suitable Lyapunov function,the global stability of disease-free equilibrium was obtained at R0?1 and the endemic equilibrium is globally stable at R0>1.In the end,the relevant conclusions of this chapter are explained vividly by numerical simulation.In chapter 4,we briefly summarize the work of this thesis and expounds the biolog-ical significance and practical value of this mathematical models.At the same time,we put forward some shortcomings of this thesis and the problems that need to be further explored.