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The size of a Markovian SIR epidemic given only removal data

Part of: Markov processes

Published online by Cambridge University Press:  21 March 2023

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

During an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time $t \geq 0$, given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.

Keywords

Branching processestime-inhomogeneous birth–death processnegative binomial distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D30: Epidemiology
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 3 , September 2023 , pp. 895 - 926
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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