Wave propagation in a diffusive epidemic model with demography and time-periodic coefficients.

In this paper, the periodic traveling wave solution for a reaction–diffusion SIR epidemic model with demography and time-periodic coefficients is investigated. Because the traveling wave system of non-autonomous reaction–diffusion model is a partial differential equation system, some traditional methods using only the theory of ordinary differential equations are no longer applicable. To overcome these difficulties, the traditional methods are extended and improved, and some new techniques are introduced. The research results show that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number R 0 and the minimal wave speed c ∗ . Specifically, when R 0 > 1 and the wave speed c > c ∗ the existence of periodic traveling wave solutions is proved by means of auxiliary system, upper–lower solutions, fixed-point theorems and some limit arguments. Otherwise, when R 0 < 1 , for any wave speed c > 0 the nonexistence of periodic traveling wave solutions is proved. Lastly, the numerical examples are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]