  In this paper we introduce a space-dependent multiscale model to describe the
spatial spread of an infectious disease under uncertain data with particular
interest in simulating the onset of the COVID-19 epidemic in Italy. While virus
transmission is ruled by a SEIAR type compartmental model, within our approach
the population is given by a sum of commuters moving on a extra-urban scale and
non commuters interacting only on the smaller urban scale. A transport dynamic
of the commuter population at large spatial scales, based on kinetic equations,
is coupled with a diffusion model for non commuters at the urban scale. Thanks
to a suitable scaling limit, the kinetic transport model used to describe the
dynamics of commuters, within a given urban area coincides with the diffusion
equations that characterize the movement of non-commuting individuals. Because
of the high uncertainty in the data reported in the early phase of the
epidemic, the presence of random inputs in both the initial data and the
epidemic parameters is included in the model. A robust numerical method is
designed to deal with the presence of multiple scales and the uncertainty
quantification process. In our simulations, we considered a realistic
geographical domain, describing the Lombardy region, in which the size of the
cities, the number of infected individuals, the average number of daily
commuters moving from one city to another, and the epidemic aspects are taken
into account through a calibration of the model parameters based on the actual
available data. The results show that the model is able to describe correctly
the main features of the spatial expansion of the first wave of COVID-19 in
northern Italy.

Populations and Evolution;Numerical Analysis;Numerical Analysis;Physics and Society