# Central limit theorems for SIR epidemics and percolation on configuration model random graphs

## TL;DR

This paper establishes central limit theorems for the final size of SIR epidemics on configuration model graphs, providing explicit variance formulas, and connects these results to percolation theory on such graphs.

## Contribution

It introduces new CLTs for epidemic final sizes on configuration models, including explicit variance expressions, covering various initial infective scenarios and linking to percolation properties.

## Key findings

- CLTs for epidemic final size with explicit variance formulas
- Results apply to both Molloy-Reed and Newman-Strogatz-Watts graphs
- Connections established between epidemic sizes and percolation cluster sizes

## Abstract

We consider a stochastic SIR (susceptible $\to$ infective $\to$ recovered) epidemic defined on a configuration model random graph, in which infective individuals can infect only their neighbours in the graph during an infectious period which has an arbitrary but specified distribution. Central limit theorems for the final size (number of initial susceptibles that become infected) of such an epidemic as the population size $n$ tends to infinity, with explicit, easy to compute expressions for the asymptotic variance, are proved assuming that the degrees are bounded. The results are obtained for both the Molloy-Reed random graph, in which the degrees of individuals are deterministic, and the Newman-Strogatz-Watts random graph, in which the degrees are independent and identically distributed. The central limit theorems cover the cases when the number of initial infectives either (a) tends to infinity or (b) is held fixed as $n \to \infty$. In (a) it is assumed that the fraction of the population that is initially infected converges to a limit (which may be $0$) as $n \to \infty$, while in (b) the central limit theorems are conditional upon the occurrence of a large outbreak (more precisely one of size at least $\log n$). Central limit theorems for the size of the largest cluster in bond percolation on Molloy-Reed and Newman-Strogatz-Watts random graphs follow immediately from our results, as do central limit theorems for the size of the giant component of those graphs. Corresponding central limit theorems for site percolation on those graphs are also proved.