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The number of individuals alive in a branching process given only times of deaths

Part of: Markov processes

Published online by Cambridge University Press:  31 January 2025

Frank Ball Open the ORCID record for Frank Ball [Opens in a new window]  and
Peter Neal Open the ORCID record for Peter Neal [Opens in a new window]
Frank Ball*
Affiliation:
University of Nottingham
Peter Neal*
Affiliation:
University of Nottingham
*
* Postal address: School of Mathematical Sciences, University of Nottingham, NG7 2RD, United Kingdom.
* Postal address: School of Mathematical Sciences, University of Nottingham, NG7 2RD, United Kingdom.
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Abstract

The study of many population growth models is complicated by only partial observation of the underlying stochastic process driving the model. For example, in an epidemic outbreak we might know when individuals show symptoms to a disease and are removed, but not when individuals are infected. Motivated by the above example and the long-established approximation of epidemic processes by branching processes, we explore the number of individuals alive in a time-inhomogeneous branching process with a general phase-type lifetime distribution given only (partial) information on the times of deaths of individuals. Deaths are detected independently with a detection probability that can vary with time and type. We show that the number of individuals alive immediately after the kth detected death can be expressed as the mixture of random variables each of which consists of the sum of k independent zero-modified geometric distributions. Furthermore, in the case of an Erlang lifetime distribution, we derive an easy-to-compute mixture of negative binomial distributions as an approximation of the number of individuals alive immediately after the kth detected death.

Keywords

Phase-type distributionnegative binomial distributionSIR epidemic models

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D30: Epidemiology
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 36
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

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