Dynamics Behaviour Of SIR Diffusion Epidemic Model With Relapse

Relapse has a huge impact on the spread,diffusion and control of the epidemic.Relapse easily lead to the second diffusion of the disease.Relapse is an important factor and it can not be ignored in the study of the global dynamic behavior of epidemic.Hence,the dynamics behaviour of SIR(Susceptible-Infective-Removal)diffusion epidemic model with relapse is the topic of great concern and research interest.In this thesis,we shall study the global stability,the persistence,traveling waves and the optimal control problem of SIR diffusion epidemic model with relapse.In Chapter 1,we mainly introduce the background and development situation of epidemiology with relapse,and give some theoretical tools in the paper.In Chapter 2,several sufficient and necessary conditions to the existence of global exponential attract set for dissipative evolution equations are proved.These necessary and sufficient conditions are the reliable theoretical basis and convenient and feasible decision method for discussing the global asymptotic stability and the persistence of epidemic.Then,the model of ours consider the relapse in spatial heterogeneous environment.The existence of positive solutions and non-constant positive equilibria for the are proved by using comparison principle of parabolic equation and super-and sub-solution methods comprehensively.By selecting mul-tiple sets of data for numerical simulation,we can find that the effect of spatial heterogeneous environment on the non-constant endemic equilibrium of disease is very large.In Chapter 3,it is concerned with the traveling waves of a reaction-diffusion SIR epidemic model with relapse.The result shows that the existence and nonexistence of traveling waves are determined by the basic reproduction number of the system and the minimal wave speed.This threshold dynamics is proved by Schauder's fixed theorem combining with the theory of asymptotic spreading.Using matlab we simulate the existence of traveling wave solutions when c>c*and c = c*.In Chapter 4,we discuss the traveling wave solutions of a nonlocal dispersal SIR model with relapse.Under the condition that the kernel function is symmetric and compactly supported,it is proved that the similar results to Chapter 3.How-ever,when the kernel function is not compactly supported,numerical simulation shows that nonlocal dispersal has a large influence on the existence of traveling wave solutions.In Chapter 5,we conducted a analysis of the optimal control strategy for a reaction-diffusion SIR epidemic model with relapse.The bang-bang control con-ditions of the diffusion system are obtained by using the maximum principle,and several optimal control strategies for the diffusion epidemic are given based on the results we obtained.