Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T21:26:05.799Z Has data issue: false hasContentIssue false
  • English
  • Français

Extinction Times in Multitype Markov Branching Processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Dominik Heinzmann
Dominik Heinzmann*
Affiliation:
University of Zurich
*
∗Postal address: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland. Email address: dominik.heinzmann@math.uzh.ch
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

Keywords

Multitype branching processextinction timeconvergence rate

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D30: Epidemiology
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 296 - 307
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
[2] Ball, F. (1983). The threshold behaviour of epidemic models. J. Appl. Prob. 20, 227–241.CrossRefGoogle Scholar
[3] Ball, F. and Donnelly, P. (1995). Strong approximations of epidemic models. Stoch. Process. Appl. 55, 1–21.Google Scholar
[4] Barbour, A. D. (2007). Coupling a branching process to an infinite dimensional epidemic process. Preprint. Available at http://arxiv.org/abs/0710.3697v1.Google Scholar
[5] Barbour, A. D. and Utev, S. (2004). Approximating the Reed–Frost epidemic process. Stoch. Process. Appl. 113, 173–197.CrossRefGoogle Scholar
[6] Eckert, J., Friedhoff, K. T., Zahner, H. and Deplazer, P. (eds) (2005). Lehrbuch der Parasitologie für die Tiermedizin. Enke Verlag, Stuttgart.Google Scholar
[7] Galambos, J. and Simonelli, I. (1996). Bonferroni-type Inequalities with Applications. Springer, New York.Google Scholar
[8] Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Chem. Phys. 81, 2340–2361.CrossRefGoogle Scholar
[9] Gillespie, D. T. and Petzold, L. R. (2003). Improved leap-size selection for accelerated stochastic simulation. J. Chem. Phys. 119, 8229–8234.Google Scholar
[10] Gronwall, T. H. (1919). Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20, 292–296.Google Scholar
[11] Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
[12] Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, New York.Google Scholar
[13] Jagers, P., Klebaner, F. C. and Sagitov, S. (2007). On the path to extinction. Proc. Nat. Acad. Sci. USA 104, 6107–6111.Google Scholar
[14] Levinson, N. (1948). The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126.Google Scholar
[15] Montoya, J. G. and Liesenfeld, O. (2004). Toxoplasmosis. The Lancet 363, 1965–1965.CrossRefGoogle ScholarPubMed
[16] Seneta, E. (1973). Non-negative Matrices. Halsted Press, New York.Google Scholar
[17] Sewastjanow, B. A. (1974). Verzweigungsprozesse. Akademie, Berlin.Google Scholar
[18] Whittle, P. (1955). The outcome of a stochastic epidemic—a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar