Traveling waves for a diffusive SIR model with delay

Abstract

In this paper, we study traveling waves connecting the infection-free equilibrium state and the endemic equilibrium state for a diffusive SIR model with delay and saturated incidence rate. Since this system does not enjoy the comparison principle, we will use an iteration process to construct a pair of upper and lower solutions. With the aid of the pair of upper and lower solutions, we can apply the Schauder fixed point theorem to construct a family of solutions of the truncated problems, which, via the limiting argument, can generate the traveling wave. Indeed, we show that there exists c ⁎ > 0 such that this system admits a traveling wave solution with speed c iff c ≥ c ⁎ .