Hostname: page-component-6bf8c574d5-9nwgx Total loading time: 0 Render date: 2025-02-17T12:26:55.776Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ball, F. G., Knock, E. S. and O’Neill, P. D. (2011). Threshold behaviour of emerging epidemics featuring contact tracing. Adv. Appl. Prob. 43, 1048–1065.CrossRefGoogle Scholar
Bansaye, V., Gu, C. and Yuan, L. (2022). A growth–fragmentation–isolation process on random recursive trees. Available at arXiv:2109.05760. To appear in Ann. Appl. Prob.Google Scholar
Barlow, M. (2020). A branching process with contact tracings. Available at arXiv:2007.16182.Google Scholar
Britton, T. and Pardoux, E. (2019). Stochastic Epidemic Models with Inference (Lecture Notes Math. 2255). Springer, Cham.Google Scholar
Deijfen, M. (2010). Random networks with preferential growth and vertex death. J. Appl. Prob. 47, 1150–1163.CrossRefGoogle Scholar
Holmgren, C. and Janson, S. (2017). Fringe trees, Crump–Mode–Jagers branching processes and m-ary search trees. Prob. Surv. 14, 53–154.Google Scholar
Huo, X. (2015). Modeling of contact tracing in epidemic populations structured by disease age. Discrete Contin. Dyn. Syst. Ser. B 20, 1685–1713.Google Scholar
Iksanov, A., Kolesko, K. and Meiners, M. (2021). Asymptotic fluctuations in supercritical Crump–Mode–Jagers processes. Available at arXiv:2109.00867.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications (Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics). John Wiley, London, New York, Sydney.Google Scholar
Jagers, P. (1989). General branching processes as Markov fields. Stoch. Process. Appl. 32, 183–212.CrossRefGoogle Scholar
Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. Appl. Prob. 16, 221–259.CrossRefGoogle Scholar
Kot, M. (2003). Elements of Mathematical Ecology. Cambridge University Press.Google Scholar
Lambert, A. (2021). A mathematical assessment of the efficiency of quarantining and contact tracing in curbing the COVID-19 epidemic. Math. Model. Nat. Phenom. 16, 53.CrossRefGoogle Scholar
Müller, J. and Kretzschmar, M. (2021). Contact tracing: old models and new challenges. Infect. Disease Model. 6, 222–231.CrossRefGoogle ScholarPubMed
Müller, J., Kretzschmar, M. and Dietz, K. (2000). Contact tracing in stochastic and deterministic epidemic models. Math. Biosci. 164, 39–64.CrossRefGoogle ScholarPubMed
Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.CrossRefGoogle Scholar
Okolie, A. and Müller, J. (2020). Exact and approximate formulas for contact tracing on random trees. Math. Biosci. 321, 108320.10.1016/j.mbs.2020.108320CrossRefGoogle ScholarPubMed
Simon, H. A. (1955). On a class of skew distribution functions. Biometrika 42, 425–440.CrossRefGoogle Scholar
Yule, G. U. (1925). A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F.R.S. Phil. Trans. R. Soc. London B 213, 21–87.Google Scholar