In this talk, I will report  some recent results  on a diffusive host-pathogen model with  spatially heterogeneous parameters and distinct  dispersal rates  for the susceptible and infected hosts. In addition to global existence of solution, existence of  a global attractor, we also discuss the threshold dynamics in terms of the basic reproduction number R0 which is identified as the spectral radius of a linear operator in the appropriate functions space.  We show that if R0<1, the pathogen free equilibrium is globally stable, and if R0>1, the solution of the model is uniformly persistent and there exists a positive steady state.  In the latter case, we also explore the asymptotic profiles of the endemic steady state  as the dispersal rate of the susceptible or infected hosts approaches zero. The results reveal some difference between the roles that the diffusions of susceptible and infectious hosts can  play. This is a joint work with Dr. Yixiang Wu (Vanderbilt University).