The SIS model with diffusion of virus in the environment

In this paper, we propose an SIS-type reaction-diffusion equations, which contains both direct transmission and indirect transmission via free-living and spatially diffusive bacteria/virus in the contaminated environment, motivated by the dynamics of hospital infections. We establish the basic reproduction number R-0 which can act as threshold level to determine whether the disease persists or not. In particular, if R-0 < 1, then the disease-free equilibrium is globally asymptotically stable, whereas the system is uniformly persistent for R-0 > 1. For the spatially homogeneous system, we investigate the traveling wave solutions and obtain that there exists a critical wave speed, below which there has no traveling waves, above which the traveling wave solutions may exist for small diffusion coefficient by the geometric singular perturbation method. The finding implies that great spatial transmission leads to an increase in new infection, while large diffusion of bacteria/virus results in the new infection decline for spatially heterogeneous environment.