Turing patterns in a reaction–diffusion epidemic model

Abstract

In this paper, we investigate the spatiotemporal dynamics of a reaction–diffusion epidemic model with zero-flux boundary conditions. The value of our study lies in two aspects: mathematically, by using maximum principle and the linearized stability theory, a priori estimates of the steady state system and the local asymptotic stability of positive constant solution are given. By using the implicit function theorem, the existence and nonexistence of nonconstant positive steady states are shown. Applying the bifurcation theory, the global bifurcation structure of nonconstant positive steady states is established. Epidemiologically, through numerical simulations, under the conditions of the existence of nonconstant positive steady states, we find that the smaller the space, the easier the pattern formation; the bigger the diffusion, the easier the pattern formation. These results are beneficial to disease control, that is, we must do our best to control the diffusion of the infectious to avoid disease outbreak.