A SIR model on a refining spatial grid: Law of Large Numbers

TL;DR

This paper analyzes a spatial SIR disease spread model on a grid, proving convergence to deterministic and diffusion models as population size and grid resolution increase, respectively.

Contribution

It introduces a spatial SIR model on a refining grid and establishes laws of large numbers linking stochastic, deterministic, and diffusion limits.

Findings

Stochastic model converges to deterministic patch model as population grows.

Refined grid limit yields a diffusion SIR model.

Laws of large numbers validated for spatial disease modeling.

Abstract

We study in this paper a compartmental SIR model for a population distributed in a bounded domain D of $\mathbb{R}^d$, d= 1, 2, or 3. We describe a spatial model for the spread of a disease on a grid of D. We prove two laws of large numbers. On the one hand, we prove that the stochastic model converges to the corresponding deterministic patch model as the size of the population tends to infinity. On the other hand, by letting both the size of the population tend to infinity and the mesh of the grid go to zero, we obtain a law of large numbers in the supremum norm, where the limit is a diffusion SIR model in D.