Dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate

This paper is concerned with the dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate. We first establish the well-posedness of this model. Then we clarify the relationship between the local basic reproduction number R̃ and the basic reproduction number R₀. It could be seen that R₀ plays an important role in determining the global dynamics of this model. In fact, we show that the disease-free equilibrium is globally asymptotically stable when R₀<1. If R₀=1, then the disease-free equilibrium is globally asymptotically stable under some assumptions. In addition, the phenomena of uniform persistence occurs when R₀>1. We also consider the local and global stability of endemic equilibrium when all the parameters of this model are constant. In the case R₀>1, we further establish the existence of traveling wave solutions of this model. Moreover, we provide an example and numerical simulations to support our theoretical results. Our model extended some known results.