Qualitative Spatial Heterogeneity Of A Class Of SIS Epidemic Model

Mathematics, as a basic subject, has been widely used in various fields of natural science, such as physics, chemistry, biology. Biomathematics, which is a marginal discipline between biology and mathematics, has been one of the most well recognized subjects in modern applied mathematics. There are plenty of interesting models in biology, and most of these models can be described by nonlinear differential equations. Epidemic dynamics, which is closed to our daily life, is a branch of biomathematics. It establishes mathematical model which can reflect the disease’s changing regulation, according to the development of the disease and the change of environment, etc. Epidemic dynamics can also analyze the reason why the disease has become popular and provide theoretical basis for people to make decisions to prevent and cure disease.Considering the impact of spatial heterogeneity of environment and random dispersal of individuals on the persistence and extinction of a disease, in this paper we study an SIS epidemic reaction-diffusion model with the null Neumann boundary condition. To be more specific, we are mainly interested in the existence, uniqueness and stability of the disease-free equilibrium and the endemic equilibrium. For this SIS PDE model, we first introduce the definition of the basic reproduction number Ro. It is shown that if R0<1, then the unique disease-free equilibrium is globally asymptotic stable, and there is no endemic equilibrium; while if R0>1, then the unique disease-free equilibrium becomes unstable, and there appears a unique endemic equilibrium. Moreover, by assuming that the susceptible individuals move as fast as the infected individuals, we then find that the disease will be eliminated in the low-risk domain, and in the high-risk domain the susceptible individuals and the infected individuals will inevitably coexist. Also, we find that the global stability of the disease-free equilibrium can be proved under the condition that the disease transmission rate is less than the disease recovery rate, and a reversed condition implies the global stability of the endemic equilibrium. In particular, by assumting the disease transmission rate and the disease recovery rate to be two positive constants, we obtain the global stability of the endemic equilibrium.In Chapter1, we give a brief introduction about the background of some traditional and classical epidemic models, and the derivation for our SIS reaction-diffusion model. Also, here we present an overview of our main work.In Chapter2, motivated by several pioneering works, we establish a more general SIS epidemic model in a bit more general sense, and the existence and uniqueness of the disease-free equilibrium are discussed here.In Chapter3, we define the basic reproduction number R0, and try to explore the connection between R0and the principal eigenvalue of the linearized problem. Based on these results, we then give sufficient conditions for the local stability or instability of the disease-free equilibrium. Moreover, the existence and uniqueness of the endemic equilibrium are discussed.In Chapter4, by the super-lower solution method, we further study the global stability of the disease-free equilibrium. Also, with some additional conditions, we investigate the global stability of the endemic equilibrium.In Chapter5, we use Matlab to simulate the local stability of the disease-free equilibrium and the endemic equilibrium. Our numerical results help one to better understand the theoretical conclusions obtained in former chapters.In Chapter6, we first summarize our results, and then propose some work for future consideration.