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A multitype decomposable age-dependent branching process and its applications

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Chinsan Lee  and
Grace L. Yang
Chinsan Lee*
Affiliation:
National Sun Yat-Sen University
Grace L. Yang*
Affiliation:
University of Maryland, College Park
*
∗Postal address: Applied Mathematics Department, National Sun Yat-Sen University, Taiwan, ROC.
∗∗Postal address: Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
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Abstract

Asymptotic formulas for means and variances of a multitype decomposable age-dependent supercritical branching process are derived. This process is a generalization of the Kendall–Neyman–Scott two-stage model for tumor growth. Both means and variances have exponential growth rates as in the case of the Markov branching process. But unlike Markov branching, these asymptotic moments depend on the age of the original individual at the start of the process and the life span distribution of the progenies.

Keywords

COUNTING PROCESSTWO-STAGE TUMOR GROWTH MODELRENEWAL EQUATIONLIFESPAN DISTRIBUTIONASYMPTOTIC MOMENTS

MSC classification

Secondary: 60J85: Applications of branching processes

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 591 - 608
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Research partially supported by the US EPA.

References

Asmussen, S. and Keiding, N. (1978) Martingale central limit theorems and asymptotic estimation theory for multitype branching processes. Adv. Appl. Prob. 10, 109–129.CrossRefGoogle Scholar
Farber, E. and Sarma, O. S. R. (1987) Hepatocarcinogenesis: a dynamic cellular perspective. Lab. Invest. 56, 4–22.Google ScholarPubMed
Goldie, J. H. and Goldman, A. J. (1979) A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Ca. Treat. Rep. 63, 1727–1733.Google ScholarPubMed
Goldie, J. H. and Goldman, A. J. (1983) Quantitative model for multiple levels of drug resistance in clinical tumors. Ca. Treat. Rep. 67, 923–931.Google ScholarPubMed
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Jagers, P. (1974) Almost sure convergence of general branching processes and functionals thereof. J. Appl. Prob, 11, 471–478.CrossRefGoogle Scholar
Kendall, D. (1960) Birth and death processes and the theory of carcinogenesis. Biometrika 47, 13–21.CrossRefGoogle Scholar
Mode, C. (1971) Multitype Branching Processes. American Elsevier, New York.Google Scholar
Moolgavkar, S. and Venzon, D. (1979) Two-event models for carcinogenesis: incidence curves for childhood and adult tumors. Math. Biosci. 47, 55–77.CrossRefGoogle Scholar
Moolgavkar, S. H., Dewanji, A. and Venzon, D. J. (1988) A stochastic two stage model for cancer risk assesment. 1. The hazard function and the probability of tumor. Risk Analy. 8, 383–392.CrossRefGoogle Scholar
Neyman, J. and Scott, E. (1967) Statistical aspect of the problem of carcinogenesis. 5th Berkeley Symp. Math. Statist. Prob. 4, 745–776.Google Scholar
Pitot, H. C. and Sirica, A. E. (1980) The stages of initiation and promotion in hepatocarcinogenesis. Biochim. Biophys. Acta 605, 191–197.Google ScholarPubMed
Smith, J. A. and Martin, L. (1973) Do cells cycle? Proc. Nat. Acad. Sci. USA 70, 1263–1267.CrossRefGoogle ScholarPubMed
Srinivasan, S. and Ranganathan, C. (1981) An age-dependent stochastic model for carcinogenesis. Math. Biosci. 57, 155–174.CrossRefGoogle Scholar
Tucker, H. G. (1961) A stochastic model for a two-stage theory of carcinogenesis. 4th Berkeley Symp. Math. Statist. Prob. 4, 387–403.Google Scholar