An Sei Epidemic Model With Diffusion And Corresponding Free Boundary Problem

Advances in biology provide opportunities for the development of mathematical ecology. Mathematics and Ecology are now no longer completely independent subjects. They are closely combined and current trends are becoming evident. In particular, epidemiology has attracted much attention in applied mathematics, mainly because that with much progress in modern science and technology, Mathematics in various fields of science is playing an increasingly important contribution, especially in ecology. It is worth mentioning that the ecological dynamics of infectious diseases has been developed significantly. Based on the incidence of disease and its spreading within the population, we can use mathematics to build models of the dynamics of infectious diseases and provide a reasonable explanation, in turn, the prevalence of infectious disease is used to test the rationality and correctness of the mathematical conclusion. Now epidemiology has been received a great attention, many scholars have done a lot of research work, and built mathematical models of different forms, for example, SIR model, SI model, SIS model, SEI model and SEER model.This paper is concerned about an SEI epidemic model and the corresponding free boundary problem. In particular, the paper considers the SEI model in which the disease is infectious in the latent period and the infected period. We first introduce the concepts related to the dynamics of infectious diseases, and then present the diffusive model, the partial differential system with homogeneous Neumann boundary condition is given and long time behaviors of the solution are discussed. The paper consists of six parts.Background and some related work are introduced, and related conclusions are given. Then ordinary differential model of the SEI is presented. Considering the diffusion of space and introducing the homogenious Neumann boundary conditions and initial conditions, the partial differential model of the SEI is proposed in this paper, that is, a nonlinear reaction diffusion problem.Following the first chapter, we first consider the SEI model in the fixed domain. The positivity and uniformly boundedness of the solution are then given.Chapter2deals with the asymptotic behavior of the steady states to the SEI ordinary differential systems. The results show that if contact rate is big or the average incubation period is long, the endemic equilibrium is locally asymptotically stable, while if the contact rate is very small or the average incubation period is short, the disease-free equilibrium is globally asymptotically stable.Based on the previous chapters, Chapter3focuses on the local stability and global stability of equilibrium solution to the corresponding partial differential equations. Our results are similar to those of ordinary differential systems.Chapter4is the main part of the paper. We consider the corresponding free boundary problem. Global existence and uniqueness of the solution are first given and then the properties of the free boundary are discussed. We prove that either the disease spreads or distinguishes. Sufficient conditions for spreading or extinction are given. Our results show that when the contact rate is very small or average incubation period is short, and the initial infected domain is small enough, then the disease distinguishes; and when the contact rate or the average incubation period is long, and the initial infected domain is sufficiently large, then the disease spreads.Finally, by using two sets of data and Matlab software, some simulation are presented. From the given graphs and data, numerical simulations illustrates the obtained theoretical results.