# On some mortality rate processes and mortality deceleration with age

- 201 Downloads
- 7 Citations

## Abstract

A specific mortality rate process governed by the non-homogeneous Poisson process of point events is considered and its properties are studied. This process can describe the damage accumulation in organisms experiencing external shocks and define its survival characteristics. It is shown that, although the sample paths of the unconditional mortality rate process are monotonically increasing, the population mortality rate can decrease with age and, under certain assumptions, even tend to zero. The corresponding analysis is the main objective of this paper and it is performed using the derived conditional distributions of relevant random parameters. Several meaningful examples are presented and discussed.

## Keywords

Gompertz law of mortality Fixed heterogeneity Evolving heterogeneity Nonhomogeneous Poisson process Mortality rate Mortality process## Mathematics Subject Classification

62P10 (Applications to biology and medical sciences) 62N05 (Reliability and life testing)## Notes

### Acknowledgments

The authors would like to thank the Editor and the referees for helpful comments and suggestions, which have improved the presentation of this paper. The work of the first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0017338). The work of the first author was also supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0093827). The work of the second author was supported by the NRF (National Research Foundation of South Africa) grant IFR2011040500026.

## References

- Anderson JJ (2000) A vitality-based model relating stressors and environmental properties to organism survival. Ecol Monogr 70:445–470CrossRefGoogle Scholar
- Beard RE (1959) Note on some mathematical mortality models. In: Wolstenholme CEW, Connor MO (eds) The lifespan of animals. Little, Brown, Boston, pp 302–311Google Scholar
- Carey JR, Liedo P, Orozco D, Vaupel JW (1992) Slowing of mortality rates at older ages in large medfy cohorts. Science 258:457–461CrossRefGoogle Scholar
- Cha JH, Mi J (2007) Study of a stochastic failure model in a random environment. J Appl Probab 44:151–163zbMATHMathSciNetCrossRefGoogle Scholar
- Finkelstein M (2008) Failure rate modeling for reliability and risk. Springer, LondonzbMATHGoogle Scholar
- Finkelstein M (2012a) On ordered subpopulations and population mortality at advanced ages. Theor Popul Biol 81:292–299CrossRefGoogle Scholar
- Finkelstein M (2012b) Discussing the Strehler–Mildvan model of mortality. Demogr Res 26:191–206MathSciNetCrossRefGoogle Scholar
- Gampe J (2010) Supercentenarians. Demographic Research Monographs, Ch. III. In: Maier H, Gampe J, Jeune B, Robine JM, Vaupel J et al (eds) Human mortality beyond age 110. Springer, Heidelberg, pp 219–230Google Scholar
- Kannisto V, Lauritsen J, Thatcher AR, Vaupel JW (1994) Reduction in mortality at advanced ages: several decades of evidence from 27 countries. Popul Dev Rev 20:793–810CrossRefGoogle Scholar
- Lemoine AJ, Wenocur ML (1986) A note on shot-noise and reliability modeling. Oper Res 34:320–323zbMATHCrossRefGoogle Scholar
- Li T, Anderson JJ (2009) The vitality model: a way to understand population survival and demographic heterogeneity. Theor Popul Biol 76:118–131zbMATHCrossRefGoogle Scholar
- Missov TI, Finkelstein M (2011) Admissible mixing distributions for general class of mixture survival models with known asymptotics. Theor Popul Biol 80:64–70CrossRefGoogle Scholar
- Moolgavkar SH (2004) Commentary: fifty years of the multistage model: remarks in a landmark paper. Int J Epidemiol 33(6):1182–1183CrossRefGoogle Scholar
- Moolgavkar SH, Luebeck EG (2003) Multistage carcinogenesis and the incidence of human cancer. Genes Chromosom Cancer 38:302–306CrossRefGoogle Scholar
- Ross S (1996) Stoch Process. Wiley, New YorkGoogle Scholar
- Shaked M, Shanthikumar J (2007) Stoch Orders. Springer, New YorkCrossRefGoogle Scholar
- Strehler L, Mildvan AS (1960) General theory of mortality and aging. Science 132:14–21CrossRefGoogle Scholar
- Vaupel JW, Manton KG, Stallard E (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16:439–454CrossRefGoogle Scholar
- Vaupel JW, Yashin AI (1985) Heterogeneity’s ruses: some surprising effects of selection on population dynamics. Am Stat 39:176–185MathSciNetGoogle Scholar
- Yashin AI, Iachine IA, Begun AS (2000) Mortality modeling: a review. Math Popul Stud 8:305–332zbMATHMathSciNetCrossRefGoogle Scholar