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

Published online by Cambridge University Press:  24 March 2016

Frank Ball ,
Tom Britton  and
Peter Neal
Frank Ball
Affiliation:
School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, UK.
Tom Britton
Affiliation:
Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden.
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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.

Keywords

Birth–death processbranching processesSIS epidemicsinsensitivity results
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 203 - 215
Copyright
Copyright © Applied Probability Trust 2016 

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