An Infectious Disease Model With Free Diffusion And Cross-diffusion Analysis

Epidemiology and ecology are traditionally treated as independent research areas, but there are a lot of commonalities between these two fields. It is usually observed in nature that epidemiology has an encroachment into the later. Wherefore, it changes the system dynamics significantly. The predator-prey interaction in presence of parasites can produce more complex dynamics, such as switching of stability, extinction and oscillations and so on. On the other hand, another variable—the harvesting practices may play an important role in the host-parasite system. Reasonable harvesting can remove a parasite, in principle, from their host.In this paper, we are concerned with the disease transmission in the predator-prey system, and consider the asymptotic behavior of the corresponding reaction -diffusion systems with homogeneous Neumann boundary conditions.In Chapter 1, the background and history about the related work are first introduced and the major work of this paper is presented.Chapter 2 is devoted to the prior estimates for the solutions of parabolic system and the sufficient conditions for the existence of each steady solution of the ODE system.In Chapter 3, the local asymptotic stabilities around each of the equilibrium are discussed by using the characteristic decomposition and linearization. Furthermore, some sufficient conditions for the global asymptotic stabilities around some equilibriums are given.The cross-diffusion is introduced in Chapter 4 to discuss the stability of the system at the equilibrium point, and the conditions for the instability caused by cross-diffusion coefficient are given. Our results show that cross-diffusion can induce the instability of an equilibrium which is stable for the kinetic system and for the self-diffusion reaction system.Chapter 5 deals with numerical simulations and some discussions are presented to illustrate the main result.