Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods

TL;DR

This paper develops hyperbolic transport models for epidemic spread on networks, removing the unphysical instantaneous diffusion of classical models, and introduces numerical methods to simulate the process accurately.

Contribution

It introduces a kinetic hyperbolic model for epidemics on networks, connecting it to reaction-diffusion models and providing a numerical scheme that preserves the diffusive limit.

Findings

Model accurately describes epidemic spread on networks.

Numerical method maintains consistency with diffusive limit.

Tests confirm the model's effectiveness in simple network structures.

Abstract

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.