Strong convergence of an epidemic model with mixing groups

Ball, Frank; Neal, Peter

Abstract

We consider an SIR (susceptible → infective → recovered) epidemic in a closed population of size n, in which infection spreads via mixing events, comprising individuals chosen uniformly at random from the population, which occur at the points of a Poisson process. This contrasts sharply with most epidemic models, in which infection is spread purely by pairwise interaction. A sequence of epidemic processes, indexed by n, and an approximating branching process are constructed on a common probability space via embedded random walks. We show that under suitable conditions the process of infectives in the epidemic process converges almost surely to the branching process. This leads to a threshold theorem for the epidemic process, where a major outbreak is defined as one that infects at least log n individuals. We show further that there exists δ > 0, depending on the model parameters, such that the probability that a major outbreak has size at least δn tends to one as n → ∞.

Citation

Ball, F., & Neal, P. (2024). Strong convergence of an epidemic model with mixing groups. Advances in Applied Probability, 56(2), 430-463. https://doi.org/10.1017/apr.2023.29

• Journal Article Type: Article
• Acceptance Date: Jun 3, 2023
• Online Publication Date: Sep 1, 2023
• Publication Date: 2024-06
• Deposit Date: Jul 14, 2023
• Publicly Available Date: Sep 1, 2023
• Journal: Advances in Applied Probability
• Print ISSN: 0001-8678
• Electronic ISSN: 1475-6064
• Publisher: Applied Probability Trust
• Peer Reviewed: Peer Reviewed
• Volume: 56
• Issue: 2
• Pages: 430-463
• DOI: https://doi.org/10.1017/apr.2023.29
• Keywords: Branching process; coupling; random walk; SIR epidemic; size of epidemic; threshold theorem
• Public URL: https://nottingham-repository.worktribe.com/output/23003279
• Publisher URL: https://www.cambridge.org/core/journals/advances-in-applied-probability/article/strong-convergence-of-an-epidemic-model-with-mixing-groups/25897F275776DB109485DCC825FE739C
• Additional Information: Copyright: © The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust