Dynamics Analysis On An SIRS Reaction-diffusion Epidemic Model With Logistic Growth And Nonlinear Incidence Rate

In this thesis,a reaction-diffusion SIRS epidemic model with logistic growth and nonlinear incidence rate is considered.We analyze the global dy-namic behavior of the model,The main works of this thesis are as follows.Firstly,the research background of infectious disease model are described,and the research status of Logistic growth infectious disease model and SIRS infectious disease model with diffusion are introduced.Secondly,based on previous studies,we consider a class of SIRS reaction-diffusion models with logistic growth and nonlinear incidence rate,and discuss the well posedness of the solutions.By using classical theory of the parabolic equations,the existence and uniqueness of solutions are obtained.We prove the uniform boundedness of the system by mathematical induction and Young’s inequality.The basic reproduction numberR₀of the system is defined by the next generation operator method,the eigenvalues of the system are studied by spectral theory,and some properties of the eigenvalues are given.Then,the persistence and extinction of the system are studied by using the upper and lower solution method and persistence theory,that is,whenR₀<1,the disease is extinct;whenR₀>1,the disease is persistent and the system has at least one positive stendy state solution.We also study the asymptotic properties of the steady-state solution as the diffusion coefficient approaches 0 or infinity,respectively,The asymptotic prop-erties of steady-state solutions are obtained by Harnack-type inequality,maxi-mum theorem,comparison principle and upper and lower solution theory.Finally,we summarize the work of this paper and look forward to the future research works.