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 R0 for the non-spatial epidemic model is defined and the basic reproduction number R0F(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 R0 ≤ 1, the bacteria always vanish, and if R0F(t0)≥1 for some t0 ≥ 0, the bacteria must spread, while if R0F(0)<1