The SIR epidemic model from a PDE point of view

TL;DR

This paper derives the classical SIR epidemic model from a PDE perspective, connecting discrete models to continuous equations and analyzing long-term behavior related to the basic reproduction number R0.

Contribution

It provides a PDE-based derivation of the SIR model and analyzes its long-term equilibrium using a hyperbolic Kolmogorov equation.

Findings

The PDE approach recovers the standard SIR model.

Long-term solutions concentrate on stable SIR equilibria.

The model's behavior depends on the basic reproduction number R0.

Abstract

We present a derivation of the classical SIR model through a mean-field approximation from a discrete version of SIR. We then obtain a hyperbolic forward Kolmogorov equation, and show that its projected characteristics recover the standard SIR model. Moreover, we show that the long time limit of the evolution will be a Dirac measure. The exact position will depend on the well-know $R_0$ parameter, and it will be supported on the corresponding stable SIR equilibrium.