Journal of Applied Probability

On expected durations of birth-death processes, with applications to branching processes and SIS epidemics

Frank Ball, Tom Britton, and Peter Neal

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Abstract

We study continuous-time birth-death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λ n being a branching process, and α(n) = λn(N - n) / N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T, An, C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth-death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible → infective → recovered) epidemic, its threshold parameter R* is insensitive to the distribution of Q.

Article information

Source
J. Appl. Probab., Volume 53, Number 1 (2016), 203-215.

Dates
First available in Project Euclid: 8 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.jap/1457470569

Mathematical Reviews number (MathSciNet)
MR3471957

Zentralblatt MATH identifier
1337.60207

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G10: Stationary processes 92D30: Epidemiology

Keywords
Birth-death process branching processes SIS epidemics insensitivity results

Citation

Ball, Frank; Britton, Tom; Neal, Peter. On expected durations of birth-death processes, with applications to branching processes and SIS epidemics. J. Appl. Probab. 53 (2016), no. 1, 203--215. https://projecteuclid.org/euclid.jap/1457470569


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