An epidemic in a dynamic population with importation of infectives

Ball, Frank and Britton, Tom and Trapman, Pieter (2016) An epidemic in a dynamic population with importation of infectives. Annals of Applied Probability . ISSN 1050-5164 (In Press)

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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.

Item Type: Article
Keywords: Branching process, Regenerative process, SIR epidemic, Skorohod metric, Weak convergence
Schools/Departments: University of Nottingham UK Campus > Faculty of Science > School of Mathematical Sciences
Depositing User: Ball, Prof Frank Granville
Date Deposited: 22 Jun 2016 09:52
Last Modified: 19 Sep 2016 14:48

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