Open Access
February 2017 An epidemic in a dynamic population with importation of infectives
Frank Ball, Tom Britton, Pieter Trapman
Ann. Appl. Probab. 27(1): 242-274 (February 2017). DOI: 10.1214/16-AAP1203

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 infective recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where n, keeping the basic reproduction number R0 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/logn. It is shown that, as n, 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);t0} 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 start of the regenerative cycle. Properties of the process S, including the jump size and stationary distributions, are determined.

Citation

Download Citation

Frank Ball. Tom Britton. Pieter Trapman. "An epidemic in a dynamic population with importation of infectives." Ann. Appl. Probab. 27 (1) 242 - 274, February 2017. https://doi.org/10.1214/16-AAP1203

Information

Received: 1 June 2015; Revised: 1 February 2016; Published: February 2017
First available in Project Euclid: 6 March 2017

zbMATH: 1380.92062
MathSciNet: MR3619788
Digital Object Identifier: 10.1214/16-AAP1203

Subjects:
Primary: 92D30
Secondary: 60F05 , 60J28 , 60J80 , 60K05

Keywords: branching process , Regenerative process , SIR epidemic , Skorohod metric , weak convergence

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 1 • February 2017
Back to Top