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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


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 $S=\{S(t);t\ge0\}$ 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.


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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.


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

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
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