Dynamics of epidemic spreading on connected graphs

TL;DR

This paper introduces a PDE-ODE model for epidemic spread on connected graphs, combining SIR dynamics at vertices with heat equations on edges, and provides numerical schemes and simulations.

Contribution

It develops a novel coupled PDE-ODE framework for epidemic modeling on networks, including analysis and numerical methods.

Findings

The model accurately captures epidemic dynamics on complex networks.

The numerical scheme preserves key properties like positivity and conservation.

Simulations demonstrate the model's applicability to various graph structures.

Abstract

We propose a new model that describes the dynamics of epidemic spreading on connected graphs. Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connexions between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex. We describe the main properties of the system, and also derive the final total population of infected individuals. We present a semi-implicit in time numerical scheme based on finite differences in space which preserves the main properties of the continuous model such as the uniqueness and positivity of solutions and the conservation of the total population. We also illustrate our results with a selection of numerical simulations for a selection of connected graphs.