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We study the qualitative properties of a spatial diffusive heterogeneous SIRmodel, that appears in mathematical epidemiology to describe the spread of aninfectious disease in a population. The model we consider consists in a systemof parabolic PDEs. In the first part of the paper, we give a criterion thatensures whether or not an epidemic propagates in a given population. We showhow the features of the disease and of the population (rates of infection andof recovery, localisation and diffusivity of individuals) influence thepropagation of the epidemic. In particular, we prove that there are situationswhere "slowing down" the individuals can trigger an epidemic that would notpropagate otherwise. In the second part of the paper, we show how the spatialdiffusive SIR model qualitatively differs from the usual, purely temporal, SIRmodel.

Qualitative properties of spatial epidemiological models