Journal of Mathematical Biology

, Volume 34, Issue 1, pp 1–16 | Cite as

The effects of health histories on stochastic process models of aging and mortality

  • Anatoli I. Yashin
  • Kenneth G. Manton
  • Max A. Woodbury
  • Eric Stallard


A model of human health history and aging, based on a multivariate stochastic process with both continuous diffusion and discrete jump components, is presented. Discrete changes generate non-Gaussian diffusion with time varying continuous state distributions. An approach to calculating transition rates in dynamically heterogeneous populations, which generalizes the conditional averaging of hazard rates done in “fixed frailty” population models, is presented to describe health processes with multiple jumps. Conditional semi-invariants are used to approximate the conditional p.d.f. of the unobserved health history components. This is useful in analyzing the age dependence of mortality and health changes at advanced age (e.g., 95 +) where homeostatic controls weaken, and physiological dynamics and survival manifest nonlinear behavior.

Key words

Gaussian processes Semi-invariants Aging and mortality Physiological dynamics, mortality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Brémaud, P.: Point processes and queues: martingale dynamics. New York: Springer 1981Google Scholar
  2. Brooks, A., Lithgow, G., Johnson, T.: Mortality rates in a genetically heterogeneous population of Caenorhabditis clegans. Science 263, 668–671 (1994)Google Scholar
  3. Carey, J. R., Liedo, P., Orozco, D., Vaupel, J. W.: Slowing of mortality rates at older ages in large Medfly cohorts. Science 258, 457–460 (1992)Google Scholar
  4. Curtsinger, J. W., Fukui, H. H., Townsend, D. R., Vaupel, J. W.: Demography of genotypes: failure of the limited lifespan paradigm in Drosophila melanogaster. Science 258,461–463 (1992)Google Scholar
  5. Dellacherie, C.: Capacit éset processus stochastiques. Berlin New York: Springer 1972Google Scholar
  6. Hadjiev, D.: On the filtration of the semi-Margingale in the case of point processes observations. Theory Probab. Appl. 23, 175–194 (1978)Google Scholar
  7. Lew, E., Garfinkel, L.: Mortality at ages 75 and older in the cancer prevention study. (CPSI). Cancer J Clinicians 40, 210–224 (1990)Google Scholar
  8. Liptser, R. S., Shiryayev, A. N.: Theory of martingales London New York: Springer 1988Google Scholar
  9. Manton, K. G., Stallard E.: Chronic disease modeling: measurement and evaluation of the risks of chronic disease processes. London: Charles Griffin 1988Google Scholar
  10. Manton, K. G., Stallard, E., Singer, B. H.: Projecting the future size and health status of the U.S. elderly population. In: Wise D. (ed.) The economics of aging. Chicago: University of Chicago Press 1994aGoogle Scholar
  11. Manton, K. G., Stallard, E., Woodbury, M. A., Dowd, J. E.: Time varying covariates of human mortality and aging: multidimensional generalization of the Gompertz. J. Gerontology: Biological Sci. 49, B169-B190 (1994b)Google Scholar
  12. Orchard, G., Woodbury, M. A.: A missing information principle: Theory and application. In: L. LeCam eds et al.: Sixth Berkeley Symposium on Mathematical Statistics and Probability. Berkeley, CA: University of California Press 1971Google Scholar
  13. Woodbury, M. A., Manton, K. G.: A random walk model of human mortality and aging. Theoret. Population Biology 11, 37–48 (1977)Google Scholar
  14. Woodbury, M. A., Manton, K. G., Yashin, A. I.: Estimating hidden morbidity via its effects on mortality and disability. Statist. Med. 7, 325–326 (1988)Google Scholar
  15. Yashin, A. I.: Conditional Gaussian estimation of dynamic systems under jumping observations. Automation Remote Control 5, 618–626 (1980)Google Scholar
  16. Yashin, A. I.: Dynamics of survival analysis: Conditional Gaussian property versus Cameron Martin formula. In: Krylov N. V. et al. (eds). Statistics and control of stochastic processes: Steklov seminar, 1984 New York: Optimization Software 1985Google Scholar
  17. Yashin, A. L: On filtering of jumping processes. Automation Remote Control 725–730 (1970)Google Scholar
  18. Yashin, A. L: On some constructive algorithms of optimal and nonlinear filtration I, II. Automation Remote Control 11–12, 1775–1781 (1975)Google Scholar
  19. Yashin, A. I., Manton, K. G.: Modification of the EM-algorithm for survival influenced by an unobserved stochastic process.. Stochastic Processes Appl, in press (1994)Google Scholar
  20. Yashin, A. I., Manton, K. G., Stallard, E.: Dependent competing risks: a stochastic process model. J. Math. Biology 24, 119–140 (1986a)Google Scholar
  21. Yashin, A. I., Manton, K. G., Stallard, E.: Evaluating the effects of observed and unobserved diffusion processes in survival analysis of longitudinal data. Math. Modeling 7, 1353–1363 (1986b)Google Scholar
  22. Yashin, A. I., Manton, K. G., Vaupel, J. W.: Mortality and aging in heterogeneous populations: a stochastic process model with observed and unobserved variables. Theoret. Population Biology 27, 159–175 (1985)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Anatoli I. Yashin
    • 1
  • Kenneth G. Manton
    • 2
  • Max A. Woodbury
    • 2
  • Eric Stallard
    • 2
  1. 1.Institute of Community HealthOdense UniversityDenmark
  2. 2.Center for Demographic StudiesDuke UniversityDurhamUSA

Personalised recommendations