The Free Boundary Problem Of A Class Of SIR Epidemic Model With Two Diffusion Modes

In the study of infectious disease model,the reaction-diffusion equation plays a crucial role.As research has progressed,scholars have found that in practical problems,infectious diseases do not immediately spread to the whole space,but concentrate in one area and spread to other areas along with the trajectory of the infected person.Moreover,some infectious diseases are contained in the initial outbreak area when they are initially discovered,and will not spread.In response,scholars use free boundary to overcome these two deficiencies.Inspired by the free boundary problem,this paper studies the free boundary problem of the SIR infectious disease model under local diffusion.However,when the population density of the species involved in the model is very large or the areas where the population lives are not connected,the Laplace operator,which is often used to describe the population diffusion,is not very accurate.For local diffusion,scholars often use non-local operators to study the dynamic behavior of such populations.Based on the free boundary problem of the SIR model under local diffusion,the situation of non-local diffusion is studied.In this thesis,the free boundary problem of a class of SIR epidemic model with nonlinear incidence of Sf(I)is studied in homogeneous space,and the dynamic behaviors of the SIR epidemic model under local diffusion and nonlocal diffusion are considered respectively.The first chapter introduces the research background and present situation of this SIR model,and the conception of the model and the main contents of this paper are briefly described.The second chapter is concerned with the free boundary problem of SIR epidemic model under local diffusion.We prove the existence and uniqueness of local solution to the model by using the Lp theory,Sobolev embedding theorem and contraction mapping principle.Then,by estimating the boundedness of the solution,we obtain the existence and uniqueness of the global solution.Finally,Based on the monotonicity of the free boundary,a comparison principle and upper and lower solutions are constructed to obtain that the infectious disease vanishes when basic regeneration number R0<1.The third chapter focus on the free boundary problem of the SIR model with nonlocal diffusion,which is based on the local diffusion model.By using the contraction mapping principle,fundamentals of the solution of differential equations and the property that the non-local operator corresponds to the free boundary condition,the global existence and uniqueness of positive solution of the model is proved.Then,based on the comparison principle and the up-and-down solution method,sufficient conditions for the vanishing of the disease are investigated.The results show that the disease vanishes when the fundamental regeneration number R0<1.