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

Ball, Frank; Britton, Tom; Neal, Peter

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) = λn 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.

Citation

Ball, F., Britton, T., & Neal, P. (2016). On expected durations of birth-death processes with applications to branching processes and SIS epidemics. Journal of Applied Probability, 53(1), https://doi.org/10.1017/jpr.2015.19

• Journal Article Type: Article
• Acceptance Date: Jan 14, 2015
• Publication Date: Mar 24, 2016
• Deposit Date: Jun 22, 2016
• Publicly Available Date: Jun 22, 2016
• Journal: Journal of Applied Probability
• Print ISSN: 0021-9002
• Electronic ISSN: 1475-6072
• Publisher: Applied Probability Trust
• Peer Reviewed: Peer Reviewed
• Volume: 53
• Issue: 1
• DOI: https://doi.org/10.1017/jpr.2015.19
• Keywords: Birth–death process; branching processes; SIS epidemics; insensitivity results
• Public URL: https://nottingham-repository.worktribe.com/output/779199
• Publisher URL: http://journals.cambridge.org/abstract_S0021900215000194
• Contract Date: Jun 22, 2016