@article { ,
title = {An epidemic in a dynamic population with importation of infectives},
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.},
doi = {10.1214/16-AAP1203},
eissn = {1050-5164},
issn = {1050-5164},
issue = {1},
journal = {Annals of Applied Probability},
publicationstatus = {Published},
publisher = {Institute of Mathematical Statistics (IMS)},
url = {https://nottingham-repository.worktribe.com/output/848947},
volume = {27},
keyword = {Branching process, Regenerative process, SIR epidemic, Skorohod metric, Weak convergence},
year = {2017},
author = {Ball, Frank and Britton, Tom and Trapman, Pieter}
}