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A stochastic population projection system based on general age-dependent branching processes

Published online by Cambridge University Press:  14 July 2016

Charles J. Mode ,
Marc E. Jacobson  and
Gary T. Pickens
Charles J. Mode*
Affiliation:
Drexel University
Marc E. Jacobson*
Affiliation:
University of Pennsylvania
Gary T. Pickens*
Affiliation:
Drexel University
*
∗Postal address: Department of Mathematics and Computer Science, Drexel University, Philadelphia, PA 19104, USA.
∗∗Postal address: Department of Statistics, University of Pennsylvania, Philadelphia, PA 19104, USA.
∗Postal address: Department of Mathematics and Computer Science, Drexel University, Philadelphia, PA 19104, USA.
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Abstract

Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections to an arbitrary initial population size are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations.

Keywords

STOCHASTIC POPULATION PROCESSESALGORITHMSDISCRETE TIMEBIRTH COHORTSLESLIE MATRIXVARIANCE AND COVARIANCE FUNCTIONSUNIT POPULATION PROJECTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987 , pp. 1 - 13
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

†

Supported in part by NICHD Grant R01 HD 09571.

References

Cohen, J. E. (1976) Ergodicity of age-structures in populations with Markovian vital rates I: Countable states. J. Amer. Statist. Assoc. 71, 335–339.10.1080/01621459.1976.10480343Google Scholar
Cohen, J. E. (1977) Ergodicity of age-structure in populations with Markovian vital rates II: General states. Adv. Appl. Prob. 9, 18–37.Google Scholar
Crump, K. S. and Mode, C. J. (1968) A general age-dependent branching process I. J. Math. Anal. Appl. 24, 494–508.Google Scholar
Crump, K. S. and Mode, C. J. (1969) A general age-dependent branching process II. J. Math. Anal. Appl. 25, 8–17.Google Scholar
Fisher, R. A. (1958) The Genetical Theory of Natural Selection. Dover, New York.Google Scholar
Heyde, C. C. (1986) On the use of time series representations of population models. J. Appl. Prob. 23A, 345–353.Google Scholar
Heyde, C. C. and Cohen, J. E. (1985) Confidence intervals for demographic projections based on products of random matrices. Theoret. Popn. Biol. 27, 120–153.Google Scholar
Jagers, P. (1969) A general stochastic model for population development. Skand. Aktuarietidsk. 84–103.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
Jagers, P. (1982) How probable is it to be first born? and other branching processes applications to kinship problems. Math. Biosci. 59, 1–15.Google Scholar
Jagers, P. and Nerman, O. (1984) The growth and composition of branching population. Adv. Appl. Prob. 16, 221–259.Google Scholar
Leslie, P. H. (1945) On the use of matrices in certain population mathematics. Biometrika 33, 183–212.Google Scholar
Lotka, A. J. (1956) Elements of Mathematical Biology. Dover, New York.Google Scholar
Mode, C. J. (1971) Multitype Branching Processes — Theory and Applications. American Elsevier, New York.Google Scholar
Mode, C. J. (1985) Stochastic Processes in Demography and Their Computer Implementations. Springer-Verlag, Berlin.Google Scholar
Mode, C. J. and Jacobson, M. E. (1984) A parametric algorithm for computing model period and cohort human survival functions. Internat. J. Bio-Med. Comput. 15, 341–356.Google Scholar
Mode, C. J., Busby, R. C., Ewbank, D. C. and Pickens, G. T. (1984) A mathematical overview of a computer simulation model of maternity histories with illustrative examples. IMA J. Math. Appl. Med. Biol. 1, 107–121.Google Scholar
Nerman, O. (1981) On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.Google Scholar
Nerman, O. (1984) The stable pedigrees of critical populations. J. Appl. Prob. 21, 447–463.Google Scholar
Nerman, O. and Jagers, P. (1984) The stable doubly infinite pedigree process of supercritical branching populations. Z. Wahrscheinlichkeitsth. 65, 445–460.Google Scholar
Parzen, E. (1962) Stochastic Processes. Holden-Day, San Francisco, Ca.Google Scholar
Pickens, G. T., Busby, R. C. and Mode, C. J. (1983) Computerization of a population projection model based on generalized branching processes — a connection with the Leslie matrix. Math. Biosci. 64, 91–97.Google Scholar
Pollard, J. H. (1966) On the use of the direct matrix product in analysing certain stochastic models. Biometrika 53, 397–415.Google Scholar