Skip to main content

Research Repository

Advanced Search

An epidemic in a dynamic population with importation of infectives

Ball, Frank; Britton, Tom; Trapman, Pieter


Professor of Applied Probability

Tom Britton

Pieter Trapman


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.


Ball, F., Britton, T., & Trapman, P. (2017). An epidemic in a dynamic population with importation of infectives. Annals of Applied Probability, 27(1),

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
Keywords Branching process, Regenerative process, SIR epidemic, Skorohod metric, Weak convergence
Public URL
Publisher URL


euclid.aoap.1488790828.pdf (522 Kb)

Copyright Statement
Copyright information regarding this work can be found at the following address:

You might also like

Downloadable Citations