Stability Analysis Of Two Classes Of Reaction-diffusion SIR Epidemic Systems With Nonlinear Incidence

Epidemic systems have been widely studied due to their significant importance and applications in humans and biology.When considering the random movement of disease individuals in space,such as diffusion factor,the biological system evolves into a reactiondiffusion epidemic system;By increasing the advection factor,the biological system evolves into a reaction-diffusion advection epidemic system.Practice shows that the control of infectious diseases depends on the stability of the system.Therefore,the research on the stability of reaction-diffusion epidemic system has important theoretical significance and practical application value.This paper investigates the stability of a reaction-diffusion SIR epidemic system with nonlinear incidence and logistic source,as well as a reactiondiffusion advection SIR infectious disease system with nonlinear incidence and saturation treatment rate.The main research content and results of the paper are given as follows:1.A reaction-diffusion SIR epidemic system with nonlinear incidence and logistic sources is established by considering the influence of nonlinear incidence and logistic sources on disease transmission.Firstly,using regularization estimation and limit thought,we discuss the well-posedness of reaction-diffusion epidemic system.Secondly,when all coefficients of the system are constant,the existence conditions of two kinds of equilibrium state(endemic disease equilibrium state and disease-free equilibrium state)of the system are given,and then we study the global asymptotic stability of these two kinds of equilibrium state by constructing four Lyapunov functions and some reasonable assumptions.In addition,based on the defined basic regeneration number R0 and the local basic regeneration number,we obtain the threshold dynamics of reaction diffusion epidemic system.Specifically,when R0=1 and under some reasonable assumptions,we give the global asymptotic stability of the disease-free equilibrium state of the system.Finally,numerical simulation are used to verify the correctness of the theoretical results in chapter 2.2.The stability of a reaction-diffusion advection SIR epidemic system with nonlinear incidence and saturation treatment rate is studied.Firstly,we investigate the wellposedness of the system.Secondly,based on the defined basic regeneration number R’0,combining with the principal eigenvalue problem,comparison principle,and R’0>1,we give the existence and consistent persistence of the equilibrium state of the system.In addition,when considering the diffusion rate of susceptible individuals approaching zero and the saturation incidence rate approaching infinity,we discuss the asymptotic distribution for the solution of the system.Finally,numerical simulation are used to verify the correctness of the theoretical results in chapter 3.The stability of the reaction-diffusion epidemic system studied in this paper provides theoretical support for epidemic control.Epidemic system with nonlinear incidence and logistic sources can better describe the real situation,and the complex dynamics include global stability,the persistence and extinction of disease,which provide theoretical basis for effective disease control.In addition,the results of the stability study of the reactiondiffusion advection SIR epidemic system with nonlinear incidence and saturation treatment rate can also provide a theoretical basis for the spread control of diseases in society under the two factors of diffusion and advection.