Abstract: The current paper is devoted to investigate the existence of traveling waves in a spatial infectious disease model with vaccination. First, we propose a spatial infectious disease model with vaccination, and then study the well-posedness of it. Second, based on constructing a pair of the vector-value upper and lower solutions and the applications of Schauder's fixed point theorem, we show that the model admits nontrivial and positive traveling waves connecting the disease free equilibrium and the endemic equilibrium. Third, by Laplace transforms, we establish the non-existence of traveling waves for the model, in which the prior estimate of the exponential decay of the traveling wave solutions is obtained by the Stable Manifold Theorem. The approach in this paper provides an effective method to deal with more general high dimensional non-cooperative reaction-diffusion systems.

Key words: Traveling waves, Infectious disease model, Vector-value upper and lower solutions, Schauder's fixed point theorem, Laplace transform