Individual-Based Foundation of SIR-Type Epidemic Models: mean-field limit and large time behaviour

TL;DR

This paper develops a kinetic framework for SIR epidemic models using Boltzmann-type equations, deriving macroscopic parameters from microscopic interactions, and analyzes the large-time behavior and equilibrium convergence.

Contribution

It introduces a kinetic approach to epidemic modeling, linking microscopic interactions to macroscopic SIR dynamics, and rigorously studies the large-time behavior and equilibrium stability.

Findings

Derivation of SIR-type models from kinetic equations

Convergence to equilibrium proven in Sobolev space

Large-time behavior characterized by dissipative dynamics

Abstract

We introduce a kinetic framework for modeling the time evolution of the statistical distributions of the population densities in the three compartments of susceptible, infectious, and recovered individuals, under epidemic spreading driven by susceptible-infectious interactions. The model is based on a system of Boltzmann-type equations describing binary interactions between susceptible and infectious individuals, supplemented with linear redistribution operators that account for recovery and reinfection dynamics. The mean values of the kinetic system recover a SIR-type model with reinfection, where the macroscopic parameters are explicitly derived from the underlying microscopic interaction rules. In the grazing collision regime, the Boltzmann system can be approximated by a system of coupled Fokker-Planck equations. This limit allows for a more tractable analysis of the dynamics, including the large-time behavior of the population densities. In this context, we rigorously prove the convergence to equilibrium of the resulting mean-field system in a suitable Sobolev space by means of the so-called energy distance. The analysis reveals the dissipative structure of the dynamics and the role of the interaction terms in driving the system toward a stable equilibrium configuration. These results provide a multi-scale perspective connecting kinetic theory with classical epidemic models.