Abstract
In 1825 British actuary Benjamin Gompertz made a simple but important observation that a law of geometrical progression pervades large portions of different tables of mortality for humans. The simple formula he derived describing the exponential rise in death rates between sexual maturity and old age is commonly referred to as the Gompertz equation—a formula that remains a valuable tool in demography and in other scientific disciplines. Gompertz’s observation of a mathematical regularity in the life table led him to believe in the presence of a law of mortality that explained why common age patterns of death exist. This law of mortality has captured the attention of scientists for the past 170 years because it was the first among what are now several reliable empirical tools for describing the dying-out process of many living organisms during a significant portion of their life spans. In this paper we review the literature on Gompertz’slaw of mortality and discuss the importance of his observations and insights in light of research on aging that has taken place since then.
The text of this article is only available as a PDF.
References
Abernathy, J.D. (
1979
). The Exponential Increase in Mortality Rate with Age Attributed to Wearing-Out of Biological Components
. Journal of Theoretical Biology
, 80
, 333
–54
. 10.1016/0022-5193(79)90098-5Barinaga, M. (
1992
). Mortality: Overturning Received Wisdom
. Science
, 258
, 398
–99
. 10.1126/science.1411534Beard, R.E. (
1959
). Note on Some Mathematical Mortality Models
. In Wolstenholme, G.E.W., & O’Connor, M. (Eds.), CIBA Foundation Colloquia on Ageing
(pp. 121
–127
). Boston
: Little, Brown, and Company
.Berlin, N.I. (
1960
). An Analysis of Some Radiation Effects on Mortality
. In Strehler, B.L., Ebert, J.D., Glass, H.B., & Shock, N.W. (Eds.), The Biology of Aging
(pp. 121
–27
). Washington, DC
: American Institute of Biological Sciences
.Brody, S. (
1924
). The Kinetics of Senescence
. Journal of General Physiology
, 6
, 245
–57
. 10.1085/jgp.6.3.245Brooks, A., Lithgow, G.J., & Johnson, T.E. (
1994
). Mortality Rates in a Genetically Heterogeneous Population of Caenorhabditis elegans
. Science
, 263
, 668
–71
. 10.1126/science.8303273Brownlee, J. (
1919
). Notes on the Biology of a Life-Table
. Journal of the Royal Statistical Society
, 82
, 34
–77
. 10.2307/2340732Brues, A.M., & Sacher, G.A. (
1952
). Analysis of Mammalian Radiation Injury and Lethality
. In Nickson, J.J. (Ed.), Symposium on Ra-diobiology
(pp. 441
–65
). New York
: John Wiley and Sons
.Carey, J.R., Liedo, P., Orozco, D., & Vaupel, J.W. (
1992
). Slowing of Mortality Rates at Older Ages in Large Medfly Cohorts
. Science
, 258
, 457
–61
. 10.1126/science.1411540Carnes, B.A., & Olshansky, S.J. (
1993
). Evolutionary Perspectives on Human Senescence
. Population and Development Review
, 19
, 793
–806
. 10.2307/2938414Carnes, B.A., Olshansky, S.J., & Grahn, D.A. (
1996
). Continuing the Search for a Law of Mortality
. Population and Develop ment Review
, 22
, 231
–64
. 10.2307/2137434Charlesworth, B. (
1994
). Evolution in Age-Structured Populations
. Cambridge
: Cambridge University Press
.Clark, R.D. (
1950
). A Bio-Actuarial Approach to Forecasting Rates of Mortality
. Proceedings of the Centerary Assembly of the Institute of Actuaries
, 2
, 12
–27
.Connor, A., Weiss, K.M., & Weeks, S.C. (
1993
). Evolutionary Models of Quantitative Disease Risk Factors
. Human Biology
, 65
, 917
–40
.Deevey, E.S. (
1947
). Life Tables for Natural Populations of Animals
. Quarterly Review of Bioiogy
, 22
, 283
–314
. 10.1086/395888Doubal, S. (
1982
). Theory of Reliability, Biological Systems and Aging
. Mechanisms of Ageing and Development
, 18
, 339
–53
. 10.1016/0047-6374(82)90037-9Economos, A. (
1980
). Kinetics of Metazoan Mortality
. Journal of Social Biology
, 3
, 317
–29
.Economos, A. (
1982
). Rate of Aging, Rate of Dying and the Mechanism of Mortality
. Archives of Gerontological Geriatrics
, 1
, 3
–27
. 10.1016/0167-4943(82)90003-6Failla, G. (
1958
). The Aging Process and Cancerogenesis
. Annals of the New York Academy of Sciences
, 71
, 1124
–40
. 10.1111/j.1749-6632.1958.tb46828.xFailla, G. (
1960
). The Aging Process and Somatic Mutations
. In Strehler, B.L., Ebert, J.D., Glass, H.B., & Shock, N.H. (Eds.), The Biology of Aging
(pp. 170
–75
). Washington, DC
: American Institute of Biological Sciences
.Finch, C.E. (
1990
). Longevity, Senescence, and the Genome
. Chicago
: University of Chicago Press
.Finch, C.E., Pike, M.C., & Witten, M. (
1990
). Slow Mortality Rate Accelerations During Aging in Some Animals Approximate that of Humans
. Science
, 249
, 902
–905
. 10.1126/science.2392680Fukui, H.H., Xiu, L., & Curtsinger, J.W. (
1993
). Slowing of Age-Specific Mortality Rates in Drosophila melanogaster
. Experi-mental Gerontology
, 28
, 585
–99
. 10.1016/0531-5565(93)90048-IGavrilov, L.A., & Gavrilova, N.S. (
1991
). The Biology of Life Span: A Quantitative Approach
. Switzerland
: Harwood Academic Publishers
.Gompertz, B. (
1820
). A Sketch on an Analysis and the Notation Applicable to the Estimation of the Value of Life Contingencies
. Philosophical Transactions of the Royal Society of London
, 110
, 214
–94
. 10.1098/rstl.1820.0018Gompertz, B. (
1825
). On the Nature of the Function Expressive of the Law of Human Mortality and on a New Mode of Determining Life Contingencies
. Philosophical Transactions of the Royal Society of London
, 115
, 513
–85
. 10.1098/rstl.1825.0026Gompertz, B. (
1862
). A Supplement to Two Papers Published in the Transactions of the Royal Society on the Science Connected with Human Mortality, This is a Supplement Published in 1862 to Papers Published in 1820 and 1825
. Philosophical Transactions of the Royal Society of London
, 152
, 511
–59
. 10.1098/rstl.1862.0026Gompertz, B. (
1872
). On One Uniform Law of Mortality from Birth to Extreme Old Age, and on the Law of Sickness
. Journal of the Institute of Actuaries
, 16
, 329
–44
.Grahn, D. (
1970
). Biological Effects of Protracted Low Dose Radiation Exposure of Man and Animals
. In Fry, R.J.M., Grahn, D., Griem, M.L., & Rust, J.H. (Eds.), Late Effects of Radiation
(pp. 101
–36
). London
: Taylor and Francis
.Greenwood, M. (
1928
). Laws of Mortality from the Biological Point of View
. Journal of Hygiene
, 28
, 267
–94
. 10.1017/S002217240000961XGumbel, E.J. (
1954
). Statistical Theory of Extreme Values and Some Practical Applications: A Series of Lectures
. Washington, DC
: National Bureau of Standards
.Hamilton, W.D. (
1966
). The Moulding of Senescence by Natural Selection
. Journal of Theoretical Biology
, 12
, 12
–45
. 10.1016/0022-5193(66)90184-6Harmon, D. (
1992
). Free Radical Theory of Aging
. Mutation Research
, 275
, 257
–66
. 10.1016/0921-8734(92)90030-SHeligman, L., & Pollard, J.H. (
1980
). The Age Pattern of Mortality
. Journal of the Institute of Actuaries
, 107
, 49
–80
.Horiuchi, S., & Coale, A.J. (
1990
). Age Patterns of Mortality for Older Women: An Analysis Using Age-Specific Rate of Mortality Change with Age
. Mathematical Population Studies
, 2
(4
), 245
–67
. 10.1080/08898489009525312Horiuchi, S. and J. Wilmoth. 1996. “Deceleration in the Age Pattern of Mortality at Older Ages.” Unpublished manuscript. Laboratory of Populations, Rockefeller University.
Lawless, J.F. (
1982
). Statistical Models and Methods for Lifetime Data
. New York
: John Wiley and Sons
.Lederberg, J., Shope, R.E., & Oaks, S.C. (
1992
). Emerging Infections: Microbial Threats to Health in the United States
. Washington, DC
: National Academy Press
.Loeb, J., & Northrop, J.H. (
1916
). Is There a Temperature Coefficient for the Duration of Life?
. Proceedings of the National Academy of Sciences
, 2
, 456
–57
. 10.1073/pnas.2.8.456Loeb, J., & Northrop, J.H. (
1917
). What Determines the Duration of Life in Metazoa?
. Proceedings of the National Academy of Sciences
, 3
, 382
–86
. 10.1073/pnas.3.5.382Loeb, J., & Northrop, J.H. (
1917
). On the Influence of Food and Temperature upon the Duration of Life
. Journal of Biological Chemistry
, 32
, 102
–21
.Lorenz, E. (
1950
). Some Biologic Effects of Long-Continued Irradiation
. American Journal of Roentgenol Radium Therapy
, 63
, 176
–85
.Makeham, W.M. (
1860
). On the Law of Mortality and the Construction of Annuity Tables
. Journal of the Institute of Actuaries
, 6
, 301
–10
.Makeham, W.M. (
1867
). On the Law of Mortality
. Journal of the Institute of Actuaries
, 13
, 325
–58
.Makeham, W.M. (
1872
). Explanation and Example of a Method Constructing Mortality Tables with Imperfect Data; and of the Extension of Gompertz’s Theory to the Entire Period of Life
. Journal of the Institute of Actuaries
, 16
, 344
–54
.Makeham, W.M. (
1889
). On the Further Development of Gompertz’s Law
. Journal of the Institute of Actuaries
, 28
, 152
–60
.Makeham, W.M. (
1890
). On the Further Development of Gompertz’s Law
. Journal of the Institute of Actuaries
, 29
, 316
–32
.Manton, K.G., Patrick, C.H., & Stallard, E. (
1980
). Population Impact of Mortality Reduction: The Effects of Elimination of Major Causes of Death on the “Saved’ Population
. International Journal of Epidemiology
, 9
, 111
–20
. 10.1093/ije/9.2.111Manton, K.G., & Soldo, B. (
1985
). Dynamics of Health Changes in the Oldest Old: New Perspectives and Evidence
. Milbank Memorial Fund Quarterly/Health and Society
, 63
, 206
–85
. 10.2307/3349881Manton, K.G., Stallard, E., Creason, J.P., Riggan, W.B., & Woodbury, M.A. (
1985
). Compartment Model Approaches for Estimating the Parameters of a Chronic Disease Process Under Changing Risk Factor Exposures
. Environmental Health Perspectives
, 19
, 151
–69
.Manton, K.G., Stallard, E., & Vaupel, J.W. (
1981
). Methods for Comparing the Mortality Experience of Heterogeneous Populations
. Demography
, 18
, 389
–410
. 10.2307/2061005Mildvan, A., & Strehler, B.L. (
1960
). A Critique of Theories of Mortality
. In Strehler, B.L., Ebert, J.D., Glass, H.B., & Shock, N.W. (Eds.), The Biology of Aging
(pp. 216
–35
). Washington, DC
: American Institute of Biological Sciences
.Olshansky, S.J., & Carnes, B.A. (
1994
). Demographic Perspectives on Human Senescence
. Population and Development Review
, 20
, 57
–80
. 10.2307/2137630Olshansky, S.J., B.A. Carnes, R. Rogers, and L. Smith. Forthcoming. ”Epidemiologic Transitions: The Re-Emergence of Infectious and Parasitic Diseases.” Population Bulletin.
Pakin, Y.V., & Hrisanov, S.M. (
1984
). Critical Analysis of the Applicability of the Gompertz-Makeham Law in Human Populations
. Gerontology
, 30
, 8
–12
. 10.1159/000212600Patz, J.A., Epstein, P.R., Burke, T.A., & Balbus, l.M. (
1996
). Global Climate Change and Emerging Infectious Diseases
. Journal of the American Medical Association
, 275
, 217
–23
. 10.1001/jama.275.3.217Pearl, R. (
1921
). Experimental Studies on the Duration of Life
. The American Naturalist
, 55
, 481
–509
. 10.1086/279836Pearl, R. (
1922
). A Comparison of the Laws of Mortality in Drosophila and in Man
. The American Naturalist
, 56
, 398
–405
.Pearl, R., & Miner, J.R. (
1935
). Experimental Studies on the Duration of Life. XIV. The Comparative Mortality of Certain Lower Organisms
. Quarterly Review of Biology
, 10
, 60
–79
. 10.1086/394476Perks, W. (
1932
). On Some Experiments in the Graduation of Mortality Statistics
. Journal of the Institute of Actuaries
, 63
, 12
–57
.Perls, T. (
1995
). The Oldest Old
. Scientific American
, 272
, 70
–75
. 10.1038/scientificamerican0195-70Pinner, R.W., Teutsch, S.M., Simonsen, L., Klug, L.A., Graber, J.M., Clarke, M.J., & Berkelman, R.L. (
1996
). Trends in Infectious Diseases Mortality in the United States
. Journal of the American Medical Association
, 275
, 189
–93
. 10.1001/jama.275.3.189Pollard, J.R. and K. Streatfield. 1979. “Factors Affecting Mortality and the Length of Life.” Presented at the Joint Convention of The Institute of Actuaries of Australia, Christchurch, New Zealand.
Pollard, J.H., & Valkovics, E. (
1992
). The Gompertz Distribution and its Applications
. Genus
, 48
, 15
–27
.Riggs, J.E. (
1993
). The Gompertz Function: Distinguishing Mathematical from Biological Limitations
. Mechanisms of Ageing and Development
, 69
, 33
–36
. 10.1016/0047-6374(93)90069-4Sacher, G.A. 1950a. “The Survival of Mice Under Duration-of-Life Exposure to X-Rays at Various Dose Rates.” Working paper CH-3900, Metallurgical Laboratory, University of Chicago.
Sacher, G.A. 1950b. “Preliminary Report: A Comparative Study of Radiation Lethality in Several Species of Experimental Animals Irradiated for the Duration-of-Life.” Pp. 105–22 in quarterly report number ANL-4488, Argonne National Laboratory.
Sacher, G.A. (
1955
). A Comparative Analysis of Radiation Lethality in Mammals Exposed at Constant Average Intensity for the Duration of Life
. Journal of the National Cancer Institute
, 15
, 112
–543
.Sacher, G.A. 1956a. “Survival of Mice Under Duration-of-Life Exposure to X-Rays at Various Rates.” Pp. 435–63 in Biological Effects of External X and Gamma Radiation, Part 2, edited by R.E. Zirkle. Washington, DC: AEC TID-5220.
Sacher, G.A. (
1956
). On the Statistical Nature of Mortality, with Especial Reference of Chronic Radiation Mortality
. Radiology
, 67
, 250
–57
.Sacher, G.A. (
1960
). Problems in the Extrapolation of Long-Term Effects from Animals to Man
. In Watson, B.B. (Ed.), Symposium on the Delayed Effects of Whole-Body Radiation
(pp. 3
–10
). Bethesda
: Operations Research Office, Johns Hopkins University
.Sacher, G.A. (
1966
). The Gompertz Transformation in the Study of the Injury-Mortality Relationship: Application to Late Radiation Effects and Ageing
. In Lindop, P.J., & Sacher, G.A. (Eds.), Radiation & Ageing
(pp. 411
–41
). London
: Taylor and Francis
.Sacher, G.A., & Grahn, D. (
1964
). Survival of Mice under Duration-of-Life Exposure to Gamma Rays. I. The Dosage-Survival Relation and the Lethality Function
. Journal of the National Cancer Institute
, 32
, 277
–320
.Sacher, G.A., & Trucco, E. (
1962
). The Stochastic Theory of Mortality
. Annals of the New York Academy of Sciences
, 96
, 985
–1007
. 10.1111/j.1749-6632.1962.tb54116.xShock, N.W. (
1961
). Physiological Aspects of Aging in Man
. Annual Review of Physiology
, 23
, 97
–122
. 10.1146/annurev.ph.23.030161.000525Simms, H.S. (
1948
). Logarithmic Increase in Mortality as a Manifestation of Aging
. Journal of Gerontology
, 1
, 13
–26
.Stearns, S.C. (
1992
). Reproductive Life Span and Ageing
The Evolution of life Histories
(pp. 180
–205
). London
: Oxford University Press
.Strehler, B.L. (
1959
). Origin and Comparison of the Effects of Time and High-Energy Radiations on Living Systems
. Quarterly Review of Biology
, 34
, 117
–42
. 10.1086/402632Strehler, B.L. (
1960
). Fluctuating Energy Demands as Determinants of the Death Process (A Parsimonious Theory of the Gompertz Function)
The Biology of Aging
(pp. 309
–14
). Washington, DC
: American Institute of Biological Sciences
.Strehler, B.L., & Mildvan, A.S. (
1960
). General Theory of Mortality and Aging (A Stochastic Model Relates Observations on Aging, Physiologic Decline, Mortality, and Radiation)
. Science
, 132
, 14
–19
. 10.1126/science.132.3418.14Szilard, L. (
1959
). On the Nature of the Aging Process
. Proceedings of the National Academy of Sciences
, 45
, 30
–45
. 10.1073/pnas.45.1.30Vaupel, J.W., Manton, K.G., & Stallard, E. (
1979
). The Impact of Heterogeneity on Individual Frailty on the Dynamics of Mortality
. Demography
, 16
, 439
–54
. 10.2307/2061224Weibull, W. (
1951
). A Statistical Distribution Function of Wide Applicability
. Journal of Applied Mechanics
, 18
, 293
–97
.Weismann, A. (
1891
). Essays Upon Heredity and Kindred Biological Problems
. Oxford
: Clarendon Press
.Weiss, K. (
1989
). Are the Known Chronic Diseases Related to the Human Lifespan and its Evolution?
. American Journal of Human Biology
, 1
, 307
–19
. 10.1002/ajhb.1310010311Weiss, K. (
1990
). The Biodemography of Variation in Human Frailty
. Demography
, 27
, 185
–206
. 10.2307/2061448Weiss, K., Ferrell, R.E., & Ranis, C.T. (
1984
). A New World Syndrome of Metabolic Diseases with a Genetic and Evolutionary Basis
. Yearbook of Physical Anthropology
, 27
, 153
–78
. 10.1002/ajpa.1330270508Williams, G.C. (
1957
). Pleiotropy, Natural Selection, and the Evolution of Senescence
. Evolution
, 11
, 298
–311
. 10.2307/2405793Wilson, M.E., Levins, R., & Spielman, A. (
1994
). Disease in Evolution: Global Changes and Emergence of Infectious Diseases
. New York
: New York Academy of Sciences
.Witten, M. (
1988
). A Return to Time, Cells, Systems, and Aging: V. Further Thoughts on Gompertzian Survival Dynamics-The Geriatric Years
. Mechanisms of Aging and Development
, 46
, 175
–200
. 10.1016/0047-6374(88)90124-8© Population Association of America 1997
1997
