We study the traveling wave solutions for a discrete diffusive
epidemic model. The traveling wave is a mixed of front and pulse types.
We derive the existence and non-existence of traveling wave solutions of
this model. The proof of existence is based on constructing a suitable
pair of upper and lower solutions and the application of Schauder’s fixed
point theorem. By passing to the limit for a sequence of truncated
problems, we are able to derive the existence of traveling waves by a
delicate analysis of wave tails. Some open problems are also addressed.