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On branching models with alarm triggerings

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  04 September 2020

Claude Lefèvre ,
Philippe Picard  and
Sergey Utev
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Philippe Picard*
Affiliation:
Université de Lyon
Sergey Utev*
Affiliation:
University of Leicester
*
*Postal address: Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine C.P. 210, B-1050 Bruxelles, Belgique. Email address: clefevre@ulb.ac.be
**Postal address: ISFA, Univ Lyon, Université Lyon 1, LSAF EA2429, 50 Avenue Tony Garnier, F-69007 Lyon, France. Email address: philippe.picard69@free.fr
***Postal address: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom. Email address: su35@le.ac.uk
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Abstract

We discuss a continuous-time Markov branching model in which each individual can trigger an alarm according to a Poisson process. The model is stopped when a given number of alarms is triggered or when there are no more individuals present. Our goal is to determine the distribution of the state of the population at this stopping time. In addition, the state distribution at any fixed time is also obtained. The model is then modified to take into account the possible influence of death cases. All distributions are derived using probability-generating functions, and the approach followed is based on the construction of families of martingales.

Keywords

Birth–death processalarm systemepidemic modelingmartingalesstopping times

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 60G44: Martingales with continuous parameter 92D30: Epidemiology
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 734 - 759
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Copyright
© Applied Probability Trust 2020

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