An epidemic in a dynamic population with importation of infectives

Ball, Frank; Britton, Tom; Trapman, Pieter

Abstract

Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size $n$. A Markovian SIR (susceptible $\to$ infective $\to$ recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where $n\to\infty$, keeping the basic reproduction number $R_0$ as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than $1/\log n$. It is shown that, as $ n \to \infty$, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process $S=\{ S(t);t\ge 0\}$ describing the limiting fraction of the population that are susceptible. The process $S$ grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the previous jump. Properties of the process $S$, including the jump size and stationary distributions, are determined.

Citation

Ball, F., Britton, T., & Trapman, P. (2017). An epidemic in a dynamic population with importation of infectives. Annals of Applied Probability, 27(1), https://doi.org/10.1214/16-AAP1203

• Journal Article Type: Article
• Acceptance Date: Apr 17, 2016
• Publication Date: Mar 6, 2017
• Deposit Date: Jun 22, 2016
• Publicly Available Date: Mar 6, 2017
• Journal: Annals of Applied Probability
• Print ISSN: 1050-5164
• Electronic ISSN: 1050-5164
• Publisher: Institute of Mathematical Statistics (IMS)
• Peer Reviewed: Peer Reviewed
• Volume: 27
• Issue: 1
• DOI: https://doi.org/10.1214/16-AAP1203
• Keywords: Branching process, Regenerative process, SIR epidemic, Skorohod metric, Weak convergence
• Public URL: https://nottingham-repository.worktribe.com/output/848947
• Publisher URL: http://projecteuclid.org/euclid.aoap/1488790828