Family trees of continuous-time birth-and-death processes. (English) Zbl 1054.60088

Müller, Johannes; Möhle, Martin

J. Appl. Probab. 40, No. 4, 980-994 (2003).

Consider a simple continuous-time birth-and-death process in which the members of the population have an exponentially distributed life span during which they give birth to a certain Poisson process. (Thus, the model is a particular case of the Crump-Mode-Jagers process with immigration.) Let \(Z(t)\) be the number of (living) individuals in the process at time \(t\) and \(Y(t)\) be the number of such (living) individuals at this moment whose parent individuals died. Let us call these individuals as grandmothers. Viewing a grandmother as the root of the tree generated by the (living) descendants of the grandmother and having directed edges connecting mothers of this subpopulation with daughters, the authors study various properties of the obtained stochastic graph (a forest in general). Among them are: the distribution of the size of a random connected component, probability of appearing a given tree and the age distribution of roots

Reviewer: Vladimir Vatutin (Moskva)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
05C80	Random graphs (graph-theoretic aspects)

Keywords:

birth-and-death processes; stochastic graph; family tree

References:

[1]	Abramowitz, M. and Stegun, I. A. (eds) (1965). Handbook of Mathematical Functions . Dover, New York. · Zbl 0171.38503
[2]	Athreya, K. B. and Ney, P. E. (1972). Branching Processes . Springer, New York. · Zbl 0259.60002
[3]	Feller, W. (1939). Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung. Acta Biometrika 5, 11–40. · JFM 65.1365.01
[4]	Goel, N. S. and Richter-Dyn, N. (1974). Stochastic Models in Biology . Academic Press, New York.
[5]	Harris, T. E. (1963). The Theory of Branching Processes . Springer, Berlin. · Zbl 0117.13002
[6]	Jagers, P. (1975). Branching Processes with Biological Applications . John Wiley, London. · Zbl 0356.60039
[7]	Kendall, D. G. (1949). Stochastic processes and population growth. J. R. Statist. Soc. B 11, 230–264. · Zbl 0038.08803
[8]	Müller, J., Kretzschmar, M. and Dietz, K. (2000). Contact tracing in deterministic and stochastic models. Math. Biosci. 164, 39–64. · Zbl 0965.92027 · doi:10.1016/S0025-5564(99)00061-9
[9]	Small, P. M. \et (1994). The epidemiology of tuberculosis in San Francisco. New England J. Med. 330, 1703–1709. · doi:10.1056/NEJM199406163302402