Research Of Dynamical Behaviors Of A Nonlocal SIS Model In Heterogeneous Environment

Among many kinds of biomathematics problems,epidemic ones have always drawn extensive attention and have been widely studied.The study of mathematical models of epidemic dynamics is of great significance,for it can not only provide mathematical support and basis for epidemiology,but also help work out measures to prevent the epidemic disease from spreading.The linear diffusion term in reaction-diffusion equations can only describe movements of objects between adjacent spatial locations,while the nonlocal dispersal term can describe movements and interactions of objects between non-adjacent spatial locations.For that reason,nonlocal differential-integral equations can be more accurate in describing real world biological problems.This paper is devoted to the study of dynamical behaviors of a kind of nonlocal dispersal susceptible-infected-susceptible?SIS?epidemic model in heterogeneous environment with Neumann boundary condition,where the total population is assumed to be constant and the disease transmission rate and recovery rate are spatially heterogeneous.We first introduce the definition,properties and expression of the basic reproduction number₀R,then discuss the existence,uniqueness and stability of steady states of the model.We prove that if the basic reproduction number is less than 1,then the disease-free equilibrium is globally asymptotically stable,hence the disease will die out eventually.Then discuss the existence conditions and nonexistence conditions of the endemic equilibrium.Under the assumption that the basic reproduction number is larger than 1 and the diffusion coefficient of susceptible individuals and the diffusion coefficient of infected individuals are equal,we prove that if the endemic equilibrium exists,then it is globally asymptotically stable.The results we obtained provide a clearer understanding of the transmitting mechanism of this particular kind of epidemic problems.