On the stabilization of reaction-diffusion systems modeling a class of man-environment epidemics: a review

A two-component reaction–diffusion system modeling a class of spatially structured epidemic systems is considered. The system describes the spatial spread of infectious diseases mediated by environmental pollution. A relevant problem, related to the possible eradication of the epidemic, is the so-called zero-stabilization. In a series of papers, necessary conditions and sufficient conditions of stabilizability have been obtained. It has been proved that it is possible to diminish exponentially the epidemic process, in the whole habitat, just by reducing the concentration of the pollutant in a nonempty and sufficiently large subset of the spatial domain. In order to model the possible seasonal variability of the environmental conditions, the relevant parameters need to be assumed to be periodic, all with the same period. Corresponding results for the time homogeneous case are presented too.