  A reaction-diffusion model is investigated to understand infective
environments in a man-environment-man epidemic model.
  The free boundary is introduced to describe the expanding front of an
infective environment induced by fecally-orally transmitted disease. The basic
reproduction number $R^F_0(t)$ for the free boundary problem is introduced, and
the behavior of positive solutions to the reaction-diffusion system is
discussed. Sufficient conditions for the bacteria to vanish or spread are
given. We show that, if $R_0\leq 1$, the bacteria always vanish, and if
$R^F_0(t_0)\geq 1$ for some $t_0\geq 0$, the bacteria must spread, while if
$R^F_0(0)<1<R_0$, the spreading or vanishing of the bacteria depends on the
initial number of bacteria, the length of the initial habitat, the diffusion
rate, and other factors. Moreover, some sharp criteria are given.
