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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 8 March 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Birth-death process branching processes SIS epidemics insensitivity results


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.

Export citation


  • Aldous, D. J. (1985). Exchangeability and related topics. In École d'Été de Probabilités de Saint-Flour, XIII–-1983 (Lecture Notes Math. 1117), Springer, Berlin, pp. 1–198
  • Andersson, H. and Djehiche, B. (1998). A threshold limit theorem for the stochastic logistic epidemic. J. Appl. Prob. 35, 662–670.
  • Ball, F. (1999). Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci. 156, 41–67.
  • Ball, F. G. and Donnelly, P. (1995). Strong approximations for epidemic models. Stoch. Process Appl. 55, 1–21.
  • Ball, F. G. and Milne, R. K. (2004). Applications of simple point process methods to superpositions of aggregated stationary processes. Austral. N. Z. J. Statist., 46, 181–196.
  • Ball, F., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob., 7, 46–89.
  • Britton, T. and Neal, P. (2010). The time to extinction for a stochastic SIS-household-epidemic model. J. Math. Biol. 61, 763–779.
  • Ghoshal, G., Sander, L. M. and Sokolov, I. M. (2004). SIS epidemics with household structure: the self-consistent field method. Math. Biosci. 190, 71–85.
  • Hernández-Suárez, C. M. and Castillo-Chavez, C. (1999). A basic result on the integral for birth–death Markov processes. Math Biosci. 161, 95–104.
  • Kryscio, R. J. and Lefèvre, C. (1989). On the extinction of the S-I-S stochastic logistic epidemic. J. Appl. Prob. 26, 685–694.
  • Lambert, A. (2011). Species abundance distributions in neutral models with immigration or mutation and general lifetimes. J. Math. Biol. 63, 57–72.
  • Nåsell, I. (1999). On the time to extinction in recurrent epidemics. J. R. Statist. Soc. B 61, 309–330.
  • Neal, P. (2006). Stochastic and deterministic analysis of SIS household epidemics. Adv. Appl. Prob. 38, 943–968. (Correction : 44 (2012), 309–310.)
  • Neal, P. (2014). Endemic behaviour of SIS epidemics with general infectious period distributions. Adv. Appl. Prob. 46, 241–255.
  • Sevast'yanov, B. A. (1957). An ergodic theorem for Markov processes and its application to telephone systems with refusals. Theory Prob. Appl. 2, 104–112.
  • Whittle, P. (1955). The outcome of a stochastic epidemic–-a note on Bailey's paper. Biometrika 42, 116–122.
  • Whittle, P. (1985). Partial balance and insensitivity. J. Appl. Prob. 22, 168–176.
  • Zachary, S. (2007). A note on insensitivity in stochastic networks. J. Appl. Prob. 44, 238–248. \endharvreferences