Time periodic travelling waves for an advection–reaction–diffusion SIR epidemic model with seasonality and bilinear incidence

Abstract

In order to investigate the influence of spatial convection effect and the periodic environment on the spreading behaviour of epidemic, in this paper, an SIR epidemic model with time varying coefficients and diffusion advection is proposed. The periodic travelling waves satisfying certain boundary conditions are discussed by constructing operators on bounded closed convex sets consisting of periodic upper and lower solutions, utilizing the twice fixed point theorem and some limit techniques. The results show that the existence of travelling waves depends on the reproduction number $R_0$ and the critical wave speed $c^{\ast }$. Specifically, when $R_0>1$ and $c>c^{\ast }$, the existence of periodic travelling waves satisfying some boundary condition is obtained, and the nonexistence of such travelling waves for two cases (i) $R_0>1$ and $0<c<c^{\ast }$, (ii) $R_0\le 1$ and $c\ge 0$ are also obtained. Finally, some brief simulations are shown to verify the theoretical results, and the effects of spread speed and advection phenomena on disease spread were further investigated.