Global threshold analysis on a diffusive host–pathogen model with hyperinfectivity and nonlinear incidence functions

In this paper, we are concerned with the mathematical analysis of a host–pathogen model with diffusion, hyperinfectivity and nonlinear incidence. We define the basic reproduction number ℜ0 by the spectral radius of the next generation operator, and study the relation between ℜ0 and the principal eigenvalue of the problem linearized at the disease-free steady state (DFSS). Under some assumptions, we show the threshold property of ℜ0: if ℜ0<1, then the DFSS is globally asymptotically stable (GAS), whereas if ℜ0>1, then the system is uniformly persistent and a positive steady state (PSS) exists. Moreover, for the special case where all parameters are constants, we show that the PSS is GAS for ℜ0>1. Numerical simulation suggests that the spatial heterogeneity could enhance the intensity of epidemic, whereas the diffusion effect could reduce it.