Traveling waves of a discrete diffusive waterborne pathogen model with general incidence

We study traveling waves of a discrete diffusive waterborne pathogen model with general incidence. The existence and non-existence of traveling waves depend on the basic reproduction number R₀ and minimum wave speed c^∗. When R₀>1 and c≥c^∗, applying the Schauder fixed point theorem, technique of Lyapunov function and the limiting argument, we establish the traveling waves connecting the disease-free equilibrium and endemic equilibrium. If R₀≤1 or R₀>1 and c<c^∗, the non-existence of traveling waves can be verified by using the comparison principle and the method of Laplace transform. From our result one can see the diffusion rates of infectious individuals and bacteria in water can increase the minimum wave speed.