  In this paper, we are concerned with an epidemic reaction-diffusion system
with nonlinear incidence mechanism of the form $S^qI^p\,(p,\,q>0)$. The
coefficients of the system are spatially heterogeneous and time dependent
(particularly time periodic). We first establish the $L^\infty$-bounds of the
solutions of a class of systems, which improve some previous results in [58].
Based on such estimates, we then study the long-time behavior of the solutions
of the system. Our results reveal the delicate effect of the infection
mechanism, transmission rate, recovery rate and disease-induced mortality rate
on the infection dynamics. Our analysis can be adapted to some other types of
infection incidence mechanisms.
