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A model for an epidemic with contact tracing and cluster isolation, and a detection paradox

Part of: Markov processes

Published online by Cambridge University Press:  03 March 2023

Jean Bertoin Open the ORCID record for Jean Bertoin [Opens in a new window]
Jean Bertoin*
Affiliation:
University of Zurich
*
*Postal address: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zürich, Switzerland. Email address: jean.bertoin@math.uzh.ch
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Abstract

We determine the distributions of some random variables related to a simple model of an epidemic with contact tracing and cluster isolation. This enables us to apply general limit theorems for super-critical Crump–Mode–Jagers branching processes. Notably, we compute explicitly the asymptotic proportion of isolated clusters with a given size amongst all isolated clusters, conditionally on survival of the epidemic. Somewhat surprisingly, the latter differs from the distribution of the size of a typical cluster at the time of its detection, and we explain the reasons behind this seeming paradox.

Keywords

Crump–Mode–Jagers branching processstructured population model

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D25: Population dynamics (general)
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 1079 - 1095
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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