Traveling Waves with Critical Speed in a Delayed Diffusive Epidemic Model

In a recent paper [K. Zhou, M. Han, Q. Wang, Math. Method. Appl. Sci. 40 (2016) 2772-2783], the authors investigated the traveling wave solutions of a delayed diffusive SIR epidemic model. When the basic reproduction number R0 > 1 and the wave speed C = C* ( C* is the critical speed), they obtained the existence of a non-trivial and non-negative traveling wave solution. When R0 > 1 and 0 < C < C*, they established non-existence of the non-trivial and non-negative traveling wave solutions. When R0 > 1 and C = C*, the existence of traveling waves was left as an open problem. The aim of this paper is to solve this problem by applying upper-lower solution method and Schauder's fixed point theorem.