Generalities on a delayed spatiotemporal host-pathogen infection model with distinct dispersal rates

We propose a general model to investigate the effect of the distinct dispersal coefficient {for the} infected and susceptible hosts on the pathogen dynamics. The mathematical challenge lies in the fact that the investigated model is partially degenerate and the solution map is not compact. The spatial heterogeneity of the model parameters and the distinct diffusion coefficients induce infection in the low-risk regions. In fact, as infection dispersal increases, the reproduction of the pathogen particles decreases. The dynamics of the investigated model is governed by the value of the basic reproduction number $R_0$. {If $R_0\leq1$, then the} pathogen particles extinct, and for $R_0>1$ the pathogen particles persist, and we guarantees of the existence of at least one positive steady state. The asymptotic profile of the positive steady state is shown in the case when one or both diffusion coefficients for the host tends to zero or infinity.