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The point-process approach to age- and time-dependent branching processes

Published online by Cambridge University Press:  01 July 2016

Marek Kimmel
Marek Kimmel*
Affiliation:
Silesian Technical University
*
∗Postal address: Institute of Automation, Silesian Technical University, 44–100 Gliwice, Pstrowskiego 16, Poland.
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Abstract

The multitype age-dependent branching process in varying environment is treated as a random stream (point process) of the birth and death events. We derive the point-process version of the Bellman–Harris equation, which provides us with a symbolic method of writing the equations for the arbitrary finite-dimensional distribution of the process. We also derive a new recurrence relation, the ‘principle of last generation'. This result is applied to obtain new moment equations for the process with immigration also. General considerations are illustrated by a simple cell kinetics model.

Keywords

VARYING ENVIRONMENTIMMIGRATIONPROBABILITY GENERATING FUNCTIONALFUNCTIONAL EQUATIONSYMBOLIC METHODPRINCIPLE OF LAST GENERATIONMOMENT EQUATIONSCANCER CELL CYCLE
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 1 , March 1983 , pp. 1 - 20
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

This work was supported in part by the Polish Academy of Sciences grant No. 10.4.3.01.4.

References

Athreya, K. and Ney, P. (1972). Branching Processes. Elsevier, New York.CrossRefGoogle Scholar
Cameron, R. H. and Martin, W. T. (1941) An unsymmetric Fubini theorem. Bull. Amer. Math. Soc. 47, 121–125.CrossRefGoogle Scholar
Cox, D. R. and Lewis, P. A. W. (1972) Multivariate point processes. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 401–448.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes, ed. Lewis, P. A. W., Wiley, New York, 299–383.Google Scholar
Fisher, L. (1972) A survey of the mathematical theory of multidimensional point processes. In Stochastic Point Processes, ed. Lewis, P. A. W., Wiley, New York, 468–513.Google Scholar
Harris, T. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Jagers, P. (1969) A general stochastic model for population development. Skand Aktuar. Tidskr. 52, 84–103.Google Scholar
Jagers, P. (1974) Branching processes in varying environments. J. Appl. Prob. 11, 174–178.CrossRefGoogle Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, New York.Google Scholar
Keiding, N. and Nielsen, J. E. (1975) Branching processes with varying and random geometric offspring distribution. J. Appl. Prob. 12, 135–141.Google Scholar
Kimmel, M. (1980) Cellular population dynamics: I, II. Math. Biosci. 48, 211–224, 225-239.Google Scholar
Kimmel, M. (1981) General theory of cell cycle dynamics basing on branching processes in varying environment. In Biomathematics and Cell Kinetics, ed. Rotenberg, M., Elsevier-North-Holland, Amsterdam, 357–375.Google Scholar
Kimmel, M. (1982) An equivalence result for integral equations with applications to branching processes. Bull. Math. Biol. 44, 1–15.Google Scholar
Klein, B. and Macdonald, P. D. M. (1980) The multitype continuous-time Markov branching process in a periodic environment. Adv. Appl. Prob. 12, 81–93.Google Scholar
Mode, C. (1971) Multitype Branching Processes. Elsevier, New York.Google Scholar
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.CrossRefGoogle Scholar
Westcott, M. (1970) The probability generating functional. J. Austral. Math. Soc. 10, 448–466.Google Scholar
Westcott, M. (1971) On existence and mixing results for cluster point processes. J. R. Statist. Soc. B 33, 290–300.Google Scholar
Whittaker, C. and Feldman, R. M. (1980) Moments for a general branching process in a semi-Markovian environment. J. Appl. Prob. 17, 341–349.Google Scholar