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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

Information

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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References

Andersson, H. andBritton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis (Lecture Notes in Statistics 151). Springer, New York.Google Scholar
Ball, F. andDonnelly, P. (1995). Strong approximations for epidemic models. Stoch. Process. Appl. 55, 1–21.CrossRefGoogle Scholar
Ball, F. andNeal, P. (2010). Applications of branching processes to the final size of SIR epidemics. In Workshop on Branching Processes and their Applications, eds. M. González, I. M. del Puerto, R. Martínez, M. Molina, M. Mota and A. Ramos (Lecture Notes in Statistics 197). Springer, Berlin, pp. 207–223.Google Scholar
Ball, F., González, M., Martínez, R. andSlavtchova-Bojkova, M. (2014). Stochastic monotonicity and continuity properties of functions defined on Crump–Mode–Jagers branching processes, with application to vaccination in epidemic modelling. Bernoulli 20, 2076–2101.CrossRefGoogle Scholar
Bootsma, M. C. J, Wassenberg, M. W., Trapman, P. andBonten, M. J. M. (2011). The nosocomial transmission rate of animal-associated ST398 meticillin-resistant Staphylococcus aureus. J. R. Soc. Interface 8, 578–584.Google ScholarPubMed
Costa, C., Scotto, M. G. andPereira, I. (2010). Optimal alarm systems for FIAPARCH processes. Revstat - Stat. J. 8, 37–55.Google Scholar
Daley, D. andGani, J. (1999). Epidemic Modelling: An Introduction. Cambridge University Press.Google Scholar
Das, S. andKratz, M. (2012). Alarm system for insurance companies: A strategy for capital allocation. Insurance Math. Econom. 51, 53–65.CrossRefGoogle Scholar
Durrett, R. (2015). Branching Process Models of Cancer. Springer, Cham.10.1007/978-3-319-16065-8CrossRefGoogle Scholar
Greenwood, P. E. andGordillo, L. F. (2009). Stochastic epidemic modeling. In Mathematical and Statistical Estimation Approaches in Epidemiology, eds. G. Chowell, J. M. Hyman, L. M. A. Bettencourt and C. Castillo-Chavez. Springer, Dordrecht, pp. 31–52.CrossRefGoogle Scholar
Haccou, P., Jagers, P. andVatutin, V. A. (2005). Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.CrossRefGoogle Scholar
Kimmel, M. andAxelrod, D. E. (2015). Branching Processes in Biology, 2nd edn. Springer, Heidelberg.Google Scholar
Lambert, A. andTrapman, P. (2013). Splitting trees stopped when the first clock rings and Vervaat’s transformation. J. Appl. Prob. 50, 208–227.CrossRefGoogle Scholar
Lefèvre, C. andUtev, S. (1999). Branching approximation for the collective epidemic model. Methodology Comput. Appl. Prob. 1, 211–228.CrossRefGoogle Scholar
Lefèvre, C. andPicard, P. (2017). On the outcome of epidemics with detections. J. Appl. Prob. 54, 890–904.CrossRefGoogle Scholar
Pakes, A. G. (2003). Biological applications of branching processes. In Handbook of Statistics, Vol. 21, Stochastic Processes: Modeling and Simulation, eds. D. N. Shanbhag and C. R. Rao. Elsevier, Amsterdam, pp. 693–773.Google Scholar
Trapman, P. andBootsma, M. C. J. (2009). A useful relationship between epidemiology and queueing theory: The distribution of the number of infectives at the moment of the first detection. Math. Biosci. 219, 15–22.10.1016/j.mbs.2009.02.001CrossRefGoogle ScholarPubMed