Model And Study Of SIS Reaction-diffusion Equation With Saturated Treatment

Biological dynamics is one of major branches of biomathematical.The conventional method of biological dynamics studying infectious diseases is to establish appropriate mathematical model,analysis dynamic behaviors of model,explain the popular principle of infectious diseases,and put forward feasible strategies finally.At present,the epidemic models are very extensive,and mainly ordinary differential equation model based.Reaction-diffusion model is a new model for epidemic researching.Generally,treatment is the major factor affecting the spread of disease,and is used to control the infectious disease.Based on the above developments,and considering the influence of spatial heterogeneity,migration,external treatment,in this paper,we study the SIS reaction-diffusion epidemic models with two types of treatment,one class for uniform saturated treatment,and the other heterogeneous saturated treatment.Two class models above are the essence of nonlinear reaction diffusion equations with Neumann boundary conditions.Basic reproduction number of two models will be defined in our paper,and to discusses the dynamics properties of solutions.We analyze mainly the existence,uniqueness and stability of the disease free equilibrium and the endemic equilibrium when R₀<1 and R₀>1.We verify the theoretical results and discuss the influence of treatment by doing numerical simulation.In paper,when R₀<1,the disease free equilibrium of the two models are unique existence and local stability;and when R₀>1,disease-free equilibrium is unstable,at this time endemic equilibrium exists.For a uniform saturation treatment,there is a R*and satisfies R₀<R*<1,the disease free equilibrium is global stability,and when o>1 and the parameters satisfy certain conditions,the endemic equilibrium is unique.By the numerical simulation,we obtain that to treatment timely and to avoid delays can help to control the spread of the disease effectively.In chapter 1,we introduces historical background and research dynamic of the epidemic model,and the origin of the problem.We also present the main work of this paper briefly.In chapter 2,we introduces the preliminary knowledge required later,which are the definitions and properties of the minimum eigenvalue and the basic reproduction number.In chapter 3,we mainly study the SIS reaction-diffusion epidemic model with uniformly saturated treatment.We defined the basic reproductive number,analyzes the nature of the solutions of the model,and do numerical simulation to verify the theoretical results.In chapter 4,we mainly study the SIS reaction-diffusion epidemic model with heterogeneous saturated treatment.We defined the basic reproductive number,analyzes the nature of the solutions of the model,and do numerical simulation to verify the theoretical results.In chapter 5,we summarize our results,and propose some words for future consideration.