Two Infectious Disease Models With Latency Time Delays

Infectious diseases have always been harmful to human life and health.Using mathematical models to study infectious diseases is helpful for decision-makers to formulate prevention and control measures to combat the spread of diseases.According to the transmission mechanism and characteristics of infectious diseases,researchers have established and analyzed many different types of infectious disease dynamic models.On the basis of existing studies,this thesis studies two types of infectious disease models with general disease incidence and latency time delays,considering asymptomatic infection,latency time delays,general disease incidence and vaccination and other factors.The main research contents are as follows:In Chapter 1,the research background of infectious diseases and the research status of infectious disease dynamics model at home and abroad are introduced,the main research content and the main theories used in this thesis are summarized.In Chapter 2,a class of infectious disease models with asymptomatic infected persons and latency time delay is studied.The nonnegativity and boundedness of the model solutions are proved.The basic reproduction number R₀is defined and the existence and stability of the equilibria of the model are discussed.The global stability of the disease-free equilibrium point and the endemic equilibrium point are verified and the impact of asymptomatic infected persons on the disease is analyzed by numerical simulations.The simulations results show that with the increase of the proportion of asymptomatic infected persons,the number of susceptible persons will gradually decrease,indicating that the presence of asymptomatic infected persons will make the disease spread more widely.In Chapter 3,on the basis of the model in Chapter 2,considering the effect of vaccination and according to the transmission characteristics of varicella,a class of varicella transmission dynamics model with vaccination and latency time delay is established.Firstly,the nonnegativity and boundedness of the model solutions are proved.Secondly,the expression of basic reproduction number R₀and the uniqueness of positive equilibrium are proved.Thirdly,the global stability of equilibria are obtained by constructing Lyapunov functionals and applying La Salle invariance principle.Finally,the theoretical results of this chapter are verified by numerical simulations and the effect of vaccination on disease is analyzed by changing the value of vaccination rateε.It is found that with the increase of vaccination rate,the number of varicella patients will gradually decrease.In Chapter 4,the main work of this thesis is summarized and the shortcomings and future research directions are discussed.