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An epidemic in a dynamic population with importation of infectives

Ball, Frank; Britton, Tom; Trapman, Pieter

Authors

Frank Ball

Tom Britton

Pieter Trapman



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.

Journal Article Type Article
Publication 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
APA6 Citation Ball, F., Britton, T., & Trapman, P. (2017). An epidemic in a dynamic population with importation of infectives. Annals of Applied Probability, 27(1), doi:10.1214/16-AAP1203
DOI https://doi.org/10.1214/16-AAP1203
Keywords Branching process, Regenerative process, SIR epidemic, Skorohod metric, Weak convergence
Publisher URL http://projecteuclid.org/euclid.aoap/1488790828
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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