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Stochastic Monotonicity and Continuity Properties of the Extinction Time of Bellman-Harris Branching Processes: An Application to Epidemic Modelling

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

M. González ,
R. Martínez  and
M. Slavtchova-Bojkova
M. González*
Affiliation:
University of Extremadura
R. Martínez*
Affiliation:
University of Extremadura
M. Slavtchova-Bojkova*
Affiliation:
Bulgarian Academy of Sciences
*
∗Postal address: Department of Mathematics, University of Extremadura, Badajoz 06006, Spain.
∗Postal address: Department of Mathematics, University of Extremadura, Badajoz 06006, Spain.
∗∗∗Postal address: Department of Probability and Statistics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria.
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Abstract

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The aim of this paper is to study the stochastic monotonicity and continuity properties of the extinction time of Bellman-Harris branching processes depending on their reproduction laws. Moreover, we show their applications in an epidemiological context, obtaining an optimal criterion to establish the proportion of susceptible individuals in a given population that must be vaccinated in order to eliminate an infectious disease. First the spread of infection is modelled by a Bellman-Harris branching process. Finally, we provide a simulation-based method to determine the optimal vaccination policies.

Keywords

Bellman-Harris branching processextinction timevaccination policiesSIRMonte Carlo method

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D30: Epidemiology
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 58 - 71
Copyright
Copyright © Applied Probability Trust 2010 

References

Agresti, A. (1974). Bounds on the extinction time distribution of a branching process. Adv. Appl. Prob. 6, 322–335.Google Scholar
Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis (Lecture Notes Statist. 151). Springer, New York.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.Google Scholar
Axelrod, D. E. and Gusev, Y. and Gamel, J. W. (1997). Ras oncogene-transformed and nontransformed cell population are each heterogeneous but respond differently to the chemotherpeutic drug cytosine arabinoside ({Ara-C}). Cancer Chem. Pharm. 39, 445–451.CrossRefGoogle Scholar
Axelrod, D. E. and Gusev, Y. and Kuczek, T. (1993). Persistence of cell cycle times over many generations as determined by heritability of colony sizes of ras oncogene-transformed and non-transformed cells. Cell Prolif. 26, 235–249.Google Scholar
Ball, F. and Donnelley, P. (1995). Strong approximations for epidemic models. Stoch. Process. Appl. 55, 1–21.CrossRefGoogle Scholar
Barbour, A. D. (1975). The duration of the closed stochastic epidemic. Biometrika 62, 477–482.Google Scholar
Becker, N. G. and Britton, T. (2004). Estimating vaccine efficacy from small outbreaks. Biometrika 91, 363–382.Google Scholar
Billingsley, P. (1986). Probility and Measure, 2nd edn. John Wiley, New York.Google Scholar
Farrington, C. P. and Grant, A. D. (1999). The distribution of time to extinction in subcritical branching processes: applications to outbreaks of infectious disease, J. Appl. Prob. 36, 771–779.CrossRefGoogle Scholar
Farrington, C. P. and Kanaan, M. N. and Gay, N. J. (2003). Branching process models for surveillance of infectious diseases controlled by mass vaccination. Biostatist. 4, 279–295.CrossRefGoogle ScholarPubMed
Gusev, Y. and Axelrod, D. E. (1995). Evaluation of models of inheritance of cell cycle times: computer simulation and recloning experiments. In Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 2, Wuerz Publications, Winnipeg, pp. 97–116.Google Scholar
Haccou, P. and Jagers, P. and Vatutin, V. A. (2007) Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.Google Scholar
Heinzmann, D. (2009). Extinction times in multitype Markov branching processes. J. Appl. Prob. 46, 296–307.Google Scholar
Isham, V. (2005). Stochastic models for epidemics. In Celebrating Statistics (Oxford Statist. Sci. Ser. 33), Oxford University Press, pp. 27–54.Google Scholar
Johnson, R. A. and Susarla, V. and Van Ryzin, J. (1979). Bayesian nonparametric estimation for age-dependent branching processes. Stoch. Process. Appl. 9, 307–318.Google Scholar
Kimmel, M. (1985). Nonparametric analysis of stathmokinesis. Math. Biosci. 74, 111–123.CrossRefGoogle Scholar
Kimmel, M. and Axelrod, D. E. (2002). Branching Processes in Biology. Springer, New York.Google Scholar
Kimmel, M. and Traganos, F. (1986). Estimation and prediction of cell cycle specific effects of anticancer drugs. Math. Biosci. 80, 187–208.Google Scholar
Martínez, R. and Slavtchova-Bojkova, M. (2005). Comparison between numerical and simulation methods for age-dependent branching models with immigration. Pliska Stud. Math. Bulgar. 17, 147–154.Google Scholar
Mode, C. J. and Sleemam, C. K. (2000). Stochastic Processes in Epidemiology. World Scientific, Singapore.CrossRefGoogle Scholar
Nåsell, I. (2002). Stochastic models of some endemic infections. Math. Biosci. 179, 1–19.CrossRefGoogle ScholarPubMed
Pakes, A. G. (1989). On the asymptotic behaviour of the extinction time of the simple branching process. Adv. Appl. Prob. 21, 470–472.Google Scholar
Pakes, A. G. (2003). Biological applications of branching processes. In Stochastic Processes: Modelling and Simulation (Handbook Statist. 21), eds Shanbhag, D. N. and Rao, C. R., North Holland, Amsterdam, pp. 693–773.Google Scholar
R Development Core Team (2009). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.Google Scholar
Yakovlev, A. and Yanev, N. (2006). Branching Stochastic Processes with immigration in analysis of renewing cell populations. Math. Biosci. 203, 37–63.CrossRefGoogle ScholarPubMed
Yakovlev, A. and Yanev, N. (2007). Age and residual lifetime distributions for branching processes. Statist. Prob. Lett. 77, 503–513.Google Scholar