## Journal of Applied Probability

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

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

Frank Ball, Tom Britton, and Peter Neal

#### 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*) = λ*n*(*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*, *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. 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