Kermack-McKendrick type models for epidemics with nonlocal aggregation terms

We propose an approach to model spatial heterogeneity in SIR-type models for the spread of epidemics via nonlocal aggregation terms. We first consider an SIR model with spatial movements driven by nonlocal aggregation terms, in which the inter-compartment and intra-compartment interaction terms are distinct, and modelled through smooth interaction kernels. For the Cauchy problem of said model we provide a well-posedness theory on R2 for L1 ∩ L∞ ∩ H1 initial conditions. Existence is achieved by considering an approximated model with artificial linear diffusion, for which existence and uniqueness are proven via Banach fixed-point, and by providing uniform estimates on the approximated solution in order to pass to the limit via classical compactness. To prove uniqueness, we use classical L2-stability, which relies on the H1-regularity of the solution. We then provide a brief, general discussion on the steady states for these type of models, and display a specific example of non-trivial steady states for an SIS model with aggregations (driven by a single repulsive-attractive potential), the existence of which is determined by a threshold condition for a suitable “space-dependent” basic reproduction rate. We complement the analysis with numerical simulations.