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The size of a Markovian SIR epidemic given only removal data

Ball, Frank; Neal, Peter


Professor of Applied Probability

Professor of Statistics


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 that 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 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 ≥ 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. The extension to the case where some deaths are unobserved is considered. Application of the results to control measures and statistical inference are discussed.


Ball, F., & Neal, P. (2023). The size of a Markovian SIR epidemic given only removal data. Advances in Applied Probability,

Journal Article Type Article
Acceptance Date Aug 26, 2022
Online Publication Date Mar 21, 2023
Publication Date Mar 21, 2023
Deposit Date Aug 26, 2022
Publicly Available Date Sep 22, 2023
Journal Advances in Applied Probability
Print ISSN 0001-8678
Electronic ISSN 1475-6064
Publisher Applied Probability Trust
Peer Reviewed Peer Reviewed
Keywords Branching processes; time-inhomogeneous birth-death process; negative binomial distribution.
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