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

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.

Dynamics of epidemic spreading on connected graphs

“…The first biological limitation of our model comes from the purely diffusive behavior imposed on each edge of the graph. It would be biologically relevant to incorporate some kind of directed or ballistic motion (drift term in ( 2.2 )) given that many individuals set off on transportation lines travel from one specific city to another preferentially, see Bertaglia and Pareschi ( 2021 ) for a purely hyperbolic model. This would result in an equation on each edge of the form for some speed whose sign will determine the preferred direction of transportation.…”

Multi-fidelity methods for uncertainty propagation in kinetic equations

“…In Section 4.4 we will discuss strategies for quantifying uncertainty for the above model while also introducing an alternative low-fidelity model based on a simpler two-velocity dynamic [6,7].…”

“…The low-fidelity model, is based on considering individuals moving in two opposite directions (indicated by signs "+" and "-"), with velocities ±λ S for susceptible, ±λ I for infected and ±λ R for removed. This dynamics of the population through the two-velocity epidemic model [7] is given by…”

“…Concerning spatially heterogeneous environments, we assume the contact rate [7,100] depends on space and contains uncertainties β(x, z) = β 0 (z) 1 + 0.05 sin 13πx 20 , where β 0 (z) = 11(1 + 0.6z 1 ). The uncertain recovery rate is given by γ(z) = 10(1 + 0.4z 2 ).…”

Identification of Time Delays in COVID-19 Data

“…A full review of the relevant literature is beyond the current work. However, some general classes of models include data-driven and machine learning approaches [1,2,3,4,5,6], models based on partial differential equation (PDE) systems [7,8,9,10,11,12,13,14], agent-based models [15], and models based on ordinary differential equation (ODE) systems. This last category is by far the most common such model, with such articles numbering in the thousands.…”