  In this paper we study a nonlinear reaction-diffusion system which models an
infectious disease caused by bacteria such as those for cholera. One of the
significant features in this model is that a certain portion of the recovered
human hosts may lose a lifetime immunity and could be infected again. Another
important feature in the model is that the mobility for each species is allowed
to be dependent upon both the location and time. With the whole population
assumed to be susceptible with the bacteria, the model is a strongly coupled
nonlinear reaction-diffusion system. We prove that the nonlinear system has a
unique solution globally in any space dimension under some natural conditions
on the model parameters and the given data. Moreover, the long-time behavior
and stability analysis for the solutions are carried out rigorously. In
particular, we characterize the precise conditions on variable parameters about
the stability or instability of all steady-state solutions. These new results
provide the answers to several open questions raised in the literature.
