Qualitative Analysis Of A Class Of Infectious Disease Models

The epidemic model is an important mathematical model in biomathematics.The importance of the model comes from the impact of infectious diseases on human health.Infectious diseases were called plagues in ancient times,which is a frightening word.Although in today's technologically advanced society,there are various infectious diseases like influenza,tuberculosis,hepatitis,black death,AIDS and so on threatens human life and health.Infectious diseases have become a major problem that people need to solve.Then the research of infectious disease model is of great significance.This paper studies a class of epidemic models with diffusion in a space heterogeneous environment.The existence of equilibrium solutions is studied based on the reaction diffusion equation theory,the maximum value principle,the comparison principle,the degree theory,and the comparison principle of parabolic equations.we obtain the existence and stability of the equilibrium solution.The first chapter introduces the history of infectious disease models and the work done at home and abroad.In chapter two,the dynamic behavior of an epidemic model with homogeneous Neumann boundary and Dirichlet boundary was studied.Firstly,the stability of the model's only semi-trivial equilibrium state solution under homogeneous Neumann boundary conditions by using linear stability theory.Then according to the properties of the function,we obtain the existence and uniqueness of the equilibrium solution of normal number.Besides,we proved that the equilibrium solution of normal number is global asymptotic stability by constructing Lyapunov function.Then the maximum value principle and comparison method are used to analyze the prior estimate of the upper bound of the classical solution of the model under the boundary conditions of Dirichlet.The existence of positive solutions is analyzed with the knowledge of degree theory.In Chapter 3,when considering the parameter-dependent space,firstly according to Co-semigroup theory,we obtained the existence of a positive solution.Then we found the threshold R0,if R0<1,then the semi-trivial equilibrium solution is globally asymptotically nearly stable.Finally,this article summarizes and analyzes the work that can be continued in the future.