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

TL;DR

This paper derives formulas for expected durations and other metrics of birth-death processes, including branching and SIS epidemic models, showing some expectations are insensitive to lifetime distributions, with applications to epidemic thresholds.

Contribution

It provides explicit expressions for key process metrics and demonstrates insensitivity of certain expectations to lifetime distribution in birth-death processes.

Findings

Expectations of extinction time and total lifetime are insensitive to lifetime distribution Q.

Expected time to extinction from endemic state depends on Q.

Threshold parameter R_* in household SIS epidemic is insensitive to Q.

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 \alpha(n). We focus on two important examples, namely \alpha(n)=\lambda n being a branching process, and \alpha(n)=\lambda n(N-n)/N which corresponds to an SIS 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, A_n, 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. The main results of the paper give expressions for the expectation of all these quantities, and shows 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 its threshold parameter R_* is insensitive to the distribution of Q, contrary to the household SIR epidemic, for which R_* does depend on Q.