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

Frank Ball

doi:10.1214/20-aap1642

Abstract

We consider a stochastic SIR (susceptible → infective → 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→∞. 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→∞, while in (b) the central limit theorems are conditional upon the occurrence of a large outbreak (more precisely one of size at least logn). 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.

Highlights

There has been considerable work in the past two decades on models for the spread of epidemics on random networks; see, for example, the recent book Kiss et al (2017)
The arguments are not fully rigorous and the result for a constant number of initial infectives is based purely on the existence of equivalent results for other SIR epidemic models. Another limitation of Ball et al (2019) is the assumption that I is exponentially distributed, which is unrealistic for most real-life diseases. We address these shortcomings and derive fully rigorous central limit theorems for the final size of SIR epidemics on MR and NSW random graphs having bounded degrees, when the infectious period I follows an arbitrary but specified distribution
To extend the present proof to models with unbounded degree would require a functional central limit theorem for density dependent population processes with countable state spaces

Summary

Introduction

There has been considerable work in the past two decades on models for the spread of epidemics on random networks; see, for example, the recent book Kiss et al (2017). Keywords and phrases: Bond and site percolation, central limit theorem, configuration model, density dependent population process, random graph, SIR epidemic, size of epidemic. The most-studied type of epidemic model is the SIR (susceptible → infective → recovered) model. If a susceptible individual is contacted by an infective it too becomes an infective and remains so for a time, called its infectious period, that is distributed according to a non-negative random variable I having an arbitrary but specified distribution. An infective contacts its susceptible neighbours in the graph independently at the points of Poisson processes having rate λ. The main aim of this paper is to develop central limit theorems for the final size of an SIR epidemic on configuration model graphs as the population size n → ∞

Objectives

Conclusion