## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 1 (2017), 242-274.

### An epidemic in a dynamic population with importation of infectives

Frank Ball, Tom Britton, and 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 $S=\{S(t);t\ge0\}$ 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 1 (2017), 242-274.

**Dates**

Received: June 2015

Revised: February 2016

First available in Project Euclid: 6 March 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1488790828

**Digital Object Identifier**

doi:10.1214/16-AAP1203

**Mathematical Reviews number (MathSciNet)**

MR3619788

**Zentralblatt MATH identifier**

1380.92062

**Subjects**

Primary: 92D30: Epidemiology

Secondary: 60F05: Central limit and other weak theorems 60J28: Applications of continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K05: Renewal theory

**Keywords**

Branching process regenerative process SIR epidemic Skorohod metric weak convergence

#### Citation

Ball, Frank; Britton, Tom; Trapman, Pieter. An epidemic in a dynamic population with importation of infectives. Ann. Appl. Probab. 27 (2017), no. 1, 242--274. doi:10.1214/16-AAP1203. https://projecteuclid.org/euclid.aoap/1488790828