  The importance of spatial networks in the spread of an epidemic is an
essential aspect in modeling the dynamics of an infectious disease.
Additionally, any realistic data-driven model must take into account the large
uncertainty in the values reported by official sources, such as the amount of
infectious individuals. In this paper we address the above aspects through a
hyperbolic compartmental model on networks, in which nodes identify locations
of interest, such as cities or regions, and arcs represent the ensemble of main
mobility paths. The model describes the spatial movement and interactions of a
population partitioned, from an epidemiological point of view, on the basis of
an extended compartmental structure and divided into commuters, moving on a
suburban scale, and non-commuters, acting on an urban scale. Through a
diffusive rescaling, the model allows us to recover classical diffusion
equations related to commuting dynamics. The numerical solution of the
resulting multiscale hyperbolic system with uncertainty is then tackled using a
stochastic collocation approach in combination with a finite-volume IMEX
method. The ability of the model to correctly describe the spatial
heterogeneity underlying the spread of an epidemic in a realistic city network
is confirmed with a study of the outbreak of COVID-19 in Italy and its spread
in the Lombardy Region.
