Traveling Wave Solutions For Two Types Of Reaction-Diffusion Infectious Disease Models With Time Delay

Throughout history,infectious diseases have been threatening the physical and mental health of human beings,destroying the stability of social order and curbing the rapid economic growth.With the continuous growth of the global economy and the increasing mobility of the population,it is also more difficult,for relevant departments to effectively prevent and control infectious diseases.In recent years,many researchers have applied mathematical methods to explore the spread of infectious diseases and analyze the control strategies of infectious diseases.Among them,the reaction-diffusion equation is the most widely used,because it can be better to explain and predict some practical problems related to space and time.Therefore,we study a class of time-delayed reaction-diffusion SIR model with general nonlinear incidence and a class of time-delayed reaction-diffusion SIR model with demographic structure and non-local incidence.In the epidemic model,the traveling wave solutions could better describe the spread of the disease in space with a constant speed.The boundedness and asymptotic behavior of traveling wave solutions can judge the speed of the disease outbreak and whether the disease will be eliminated eventually,which provides theoretical value and practical guidance for the prevention and control of infectious diseases.The main contents of this thesis are as follows:In chapter 1,the background and significance of the research in this thesis are described,and the research history and current situation of two types of reaction-diffusion infectious disease models with time delay are introduced.In chapter 2,we discuss the traveling waves for a class of time-delayed reaction-diffusion SIR model with general nonlinear incidence.Through transformation,we transform the traveling waves problem of this model into a fixed point prob lem.When the basic reproduction number R0>1,the exponential solution of the model is substituted into its linearization equation,the minimum wave speed c*is obtained by analyzing the properties of the equation.Under the conditions of R0>1 and c>c*,the existence of non-trivial non-negative traveling waves in the model is proved by using the Schauder’s fixed point theorem.And then the asymptotic behavior of the traveling waves is proved on the basis of its existence.On the other hand,the non-existence of traveling waves is proved when R0>1,0<c<c*or R0≤1 by the analyticity of the bilateral Laplace transform.In chapter 3,we study the traveling waves of a class of time-delayed reaction-diffusion SIR model with demographic structure and non-local incidence.Firstly,the existence of traveling waves is proved by defining bounded upper and lower solutions combined with the Schauder’s fixed point theorem.Secondly.an appropriate Lyapunov functional is constructed by the boundedness of the solutions,and the asymptotic behavior of the traveling wave solutions at infinity is proved.In addition,the non-existence of traveling waves is obtained by using the asymptotic propagation theory.Since the demographic structure will cause the model to generate an endemic equilibrium point,then the traveling waves of this model are a continuous function connecting the disease-free equilibrium point to the endemic equilibrium point.Finally,we conclude that the existence of traveling wave solutions is only det,ermined by the basic reproduction number R0 and the minimum wave speed c*of the corresponding spatially homogeneous system.