October 2021 Central limit theorems for SIR epidemics and percolation on configuration model random graphs
Frank Ball
Author Affiliations +
Ann. Appl. Probab. 31(5): 2091-2142 (October 2021). DOI: 10.1214/20-AAP1642


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.


I thank Svante Janson and Etienne Pardoux for helpful correspondence, concerning simple graphs and density dependent population processes, respectively, and the referees for their careful reading of the paper and constructive comments, which have improved its presentation.


Download Citation

Frank Ball. "Central limit theorems for SIR epidemics and percolation on configuration model random graphs." Ann. Appl. Probab. 31 (5) 2091 - 2142, October 2021. https://doi.org/10.1214/20-AAP1642


Received: 1 October 2018; Revised: 1 May 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

MathSciNet: MR4332692
zbMATH: 1475.92147
Digital Object Identifier: 10.1214/20-AAP1642

Primary: 05C80 , 60K35 , 92D30
Secondary: 60F05 , 60J27 , 91D30

Keywords: Bond and site percolation , central limit theorem , configuration model , density dependent population process , random graph , SIR epidemic , size of epidemic

Rights: Copyright © 2021 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.31 • No. 5 • October 2021
Back to Top