Time periodic traveling waves in a three-component non-autonomous and reaction-diffusion epidemic model

In this paper, we propose a non-autonomous and diffusive SIR epidemic model based on the fact that the infection rate, the removal rate and the death rate often vary in time. The explicit formulas of the basic reproduction number R0 and the minimum wave speed c are derived. Applying upper-lower solution method and Schauder's fixed point theorem, we show that when R0 > 1, c > c and the diffusion rates satisfy a certain condition, a time periodic traveling wave solution exists in the model. By the method of contradiction analysis and the comparison arguments together with the properties of the spreading speed of an associated subsystem, we prove that when R0 <= 1 and c > 0 or R0 > 1 and 0 < c < c, the model possesses no time periodic traveling wave solutions.