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

Ball, Frank

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 deter-ministic, 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 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.

Citation

Ball, F. (2021). Central limit theorems for SIR epidemics and percolation on configuration model random graphs. Annals of Applied Probability, 31(5), 2091-2142. https://doi.org/10.1214/20-AAP1642

• Journal Article Type: Article
• Acceptance Date: Oct 29, 2020
• Online Publication Date: Oct 29, 2021
• Publication Date: Oct 1, 2021
• Deposit Date: Nov 5, 2020
• Publicly Available Date: Oct 1, 2021
• Journal: Annals of Applied Probability
• Print ISSN: 1050-5164
• Electronic ISSN: 1050-5164
• Publisher: Institute of Mathematical Statistics (IMS)
• Peer Reviewed: Peer Reviewed
• Volume: 31
• Issue: 5
• Pages: 2091-2142
• DOI: https://doi.org/10.1214/20-AAP1642
• Keywords: Bond and site percolation , central limit theorem , configuration model , density dependent population process , random graph , SIR epidemic , size of epidemic
• Public URL: https://nottingham-repository.worktribe.com/output/5019209
• Publisher URL: https://projecteuclid.org/journals/annals-of-applied-probability/volume-31/issue-5/Central-limit-theorems-for-SIR-epidemics-and-percolation-on-configuration/10.1214/20-AAP1642.short