Spatial propagation for a two component reaction-diffusion system arising in population dynamics (Q266382)

This work studies the propagation of positive solutions of a non-monotone reaction-diffusion system involving two densities (susceptible and infected individuals) in an epidemic, the domain of the spatial variable being the whole \(\mathbb{R}^N\).NEWLINENEWLINEWhen the so-called basic reproduction number \(R_0\) (which is characterized in terms of the parameters of the problem) is not larger than 1, the disease-free equilibrium is a global attractor. In the opposite case, the infected population persists and the system satisfies an inner spreading property, the propagation speed being linearly determined and explicitly given in terms of the parameters of the problem. An outer spreading property also holds true when the initial datum for the infected population is compactly supported.NEWLINENEWLINEAgain in the case \(R_0>1\), the author proves that, when the two populations have the same diffusion coefficient, the spreading occurs towards the so-called endemic equilibrium, i.e. the unique strictly positive spatially homogeneous steady state. In the general case the inner spreading is shown to occur towards a bounded entire solution which is uniformly far from the disease-free equilibrium, and every such solution is shown to satisfy a spatial and a temporal averaging properties around the endemic equilibrium. The author conjectures that even in this case the unique uniformly persistent entire solution is the endemic equilibrium.