Kinetic SIR equations and particle limits

TL;DR

This paper introduces two particle-based models for infection spread, proves their convergence to kinetic equations as particle number grows, and analyzes their long-term behavior with numerical comparisons.

Contribution

It presents new particle systems with binary and multi-body interactions and rigorously connects them to kinetic equations, justifying numerical methods.

Findings

Convergence of particle systems to kinetic equations as N approaches infinity

Rigorous analysis of the long-term behavior of the kinetic models

Numerical comparison of the models' dynamics

Abstract

We present and analyze two simple $N$-particle particle systems for the spread of an infection, respectively with binary and with multi-body interactions. We establish a convergence result, as $N \to \infty$, to a set of kinetic equations, providing a mathematical justification of related numerical schemes. We analyze rigorously the time asymptotics of these equations, and compare the models numerically.