Qualitative analysis on a reaction-diffusion host-pathogen model with incubation period and nonlinear incidence rate

In this paper, a degenerate reaction-diffusion host-pathogen model with an incubation period and a nonlinear incidence rate in a spatially heterogeneous environment is proposed. We analyze the dynamics of this model on a bounded domain. Firstly, we establish the well-posedness, including the global existence of solutions and the existence of a global attractor by defining a noncompact measure. Then, the basic reproduction number is given and a threshold dynamics is established. Finally, when there is a positive steady state, we investigate the asymptotic profiles of the positive steady state when host individuals disperse at small or large rate. Our results show that the incubation period can significantly enhance the persistence of the disease if the dispersal rate of susceptible hosts or exposed hosts is small or large.