Traveling Wave Solution with the Critical Speed for a Diffusive Epidemic System with Spatio-Temporal Delay

The existence of a critical traveling wave solution for a reaction-diffusion epidemic system with spatio-temporal delay is established, whose existence of super-critical traveling wave solutions is obtained in [Z. Zhen et al., Math. Method. Appl. Sci. 41 (2018) 7074-7098]. Meanwhile, the non-existence of traveling waves with non-positive speeds are derived. Two open problems left in the paper [Z. Zhen et al., Math. Method. Appl. Sci. 41 (2018) 7074-7098] are solved. Moreover, some properties of super-critical traveling waves are improved.