Analysis of a Reaction-Diffusion System Modeling Man--Environment--Man Epidemics

In this paper an old model for the temporal and spatial evolution of orofecal transmitted disease is reexamined. It consists of a system of two coupled reaction-diffusion equations for the concentration of bacteria and infective humans, under the assumptions that the total population of humans is unaffected by the disease and only a small proportion of the population is affected at any one time. The force of infection on healthy humans is assumed to be a sigmoidal function of bacterial concentration tending to some finite limit, and with zero gradient at zero. (This last feature models an immune response to low concentrations of the infectious agent.) In practice the diffusion coefficient for infective humans is much smaller than that of bacteria and is therefore set to zero; a detailed analysis of the steady-state bifurcation pattern is then performed for the case of homogeneous Dirichlet boundary conditions applied at the endpoints of a one-dimensional interval. Particular attention is paid to the limiting case of small bacterial diffusivity. A partial analysis of the dynamical behavior of the system, based on monotone techniques, is carried out. It is speculated that the system, subject to homogeneous Dirichlet boundary conditions, has saddle point structure in the natural function space of the problem, similar to the ODE case in which the diffusivity of bacteria is also set to zero. This conjecture is supported by some numerical simulation on both one- and two-dimensional space domains.