An epidemic in a dynamic population with importation of infectives

TL;DR

This paper analyzes the limiting behavior of a Markovian SIR epidemic model with importation in a large, dynamic population, revealing a simplified regenerative process that describes the susceptible fraction over time.

Contribution

It introduces a novel limiting process for the epidemic dynamics in large populations with importation, under specific asymptotic conditions.

Findings

The susceptible fraction process converges to a 1-dimensional regenerative process.

The process exhibits deterministic growth punctuated by random jumps.

Stationary distributions and jump properties are explicitly characterized.

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.