The spreading fronts of an infective environment in a man-environment-man epidemic model

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.