  We consider the development of hyperbolic transport models for the
propagation in space of an epidemic phenomenon described by a classical
compartmental dynamics. The model is based on a kinetic description at discrete
velocities of the spatial movement and interactions of a population of
susceptible, infected and recovered individuals. Thanks to this, the unphysical
feature of instantaneous diffusive effects, which is typical of parabolic
models, is removed. In particular, we formally show how such reaction-diffusion
models are recovered in an appropriate diffusive limit. The kinetic transport
model is therefore considered within a spatial network, characterizing
different places such as villages, cities, countries, etc. The transmission
conditions in the nodes are analyzed and defined. Finally, the model is solved
numerically on the network through a finite-volume IMEX method able to maintain
the consistency with the diffusive limit without restrictions due to the
scaling parameters. Several numerical tests for simple epidemic network
structures are reported and confirm the ability of the model to correctly
describe the spread of an epidemic.
