Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-07-05T00:20:04.903Z Has data issue: false hasContentIssue false
  • English
  • Français

Smallpox and the double decrement table: a piece of actuarial prehistory

Published online by Cambridge University Press:  20 April 2012

R. H. Daw
Get access Rights & Permissions [Opens in a new window]

Extract

More than 200 years ago, on 30 April 1760, Daniel Bernoulli (1766) read a memoir to the Royal Academy of Sciences in Paris entitled Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir (see Bradley, 1971, for a translation). In this remarkable memoir Bernoulli produced the first double decrement life table and one of the related single decrement tables, as well as deriving a mathematical model of the behaviour of smallpox in a community. This model was the forerunner of considerable developments in the mathematical theory of infectious diseases, a description of which is given in N. T. J. Bailey (1975). During the half century following Bernoulli's memoir there were a number of papers by other authors on the subject of that memoir; these, and the original memoir, seem to be little known to actuaries and are the subject of the present paper. They could have been the starting point of the actuarial development of exposed-to-risk formulae, but in fact were not.

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 106 , Issue 3 , December 1979 , pp. 299 - 318
Copyright
Copyright © Institute and Faculty of Actuaries 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bailey, N. T. J. (1975). The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn (London: Griffin).Google Scholar
Bailey, W.G. & Haycocks, H. W. (1946). Some Theoretical Aspects of Multiple Decrement Tables (Edinburgh: Institute of Actuaries and Faculty of Actuaries).Google Scholar
Benjamin, B. & Haycocks, H. W. (1970). The Analysis of Mortality and other Actuarial Statistics (London: Cambridge University Press).Google Scholar
Bernoulli, Daniel (1766) Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir. Mém. de l'Acad. Roy. des Sciences de l'Année 1760 pp. 1–45 (Paris) (U).Google Scholar
Bradley, L. (1971) Smallpox Inoculation: An Eighteenth-Century Mathematical Controversy (University of Nottingham, Adult Education Department) (A, S).Google Scholar
Burridge, A. R. (1903). Vaccination and the Act of 1898. J.I.A. 37, 245–312.Google Scholar
Cournot, A. (1843). Exposition de la théorie des chances et des probabilitiés (Paris: Hachette) (S).Google Scholar
D'Alembert, J. De R. (1761). Sur l'application du calcul des probabilités à l'inoculation de la petite vérole. Opuscules Mathématiques 2, 26–95 (U).Google Scholar
D'Alembert, J. De R. (1768a). Sur un mémoire de M. Bernoulli concernant l'inoculation. Opuscules Mathématiques 4, 98–105 (U).Google Scholar
D'Alembert, J. De R. (1768b). Sur les calculs relatifs à l'inoculation. Opuscules Mathématiques 4, 310–41 (U).Google Scholar
D'Alembert, J. De R. (1768c). Sur les tables de mortalité. Opuscules Mathématiques 5, 228–31. (relates to 1768b) (U).Google Scholar
D'Alembert, J. De R. (1768d). Sur les calculs relatifs à l'inoculation. Opuscules Mathématiques 5, 508–10 (an addition to 1768b) (U).Google Scholar
Duvillard, E. E. (1806). Analyse et Tableaux de l'influence de la petite vérole sur la mortalité à chaque age, et de celle qu'un préservatif tel que la vaccine peut avoir sur la population et la longévite (Paris) (A. S).Google Scholar
Greenwood, M. (1948). Medical Statistics from Graunt to Farr (London. Cambridge University Press; originally published in Biometrika 32, 101–27, 203 25; and 33, 1–24).Google Scholar
Guy, W. A. (1882). Two hundred and fifty years of smallpox in London, together with a Supplement relating to England and Wales. J. Roy Statist. Soc. 45, 399–444.Google Scholar
Halley, E. (1693). An Estimate of the Degrees of Mortality of Mankind drawn from curious Tables of the Births and Funerals at the City of Breslaw. Phil. Trans. 17, 596 610 (reprinted J.I.A. 18, 251).CrossRefGoogle Scholar
Heysham, J. (1797). An abridgment of Observations on the Bills of Mortality in Carlisle, from the year 1779 to the year 1787 inclusive. And also a catalogue of Cumberland animals (Carlisle).Google Scholar
Karn, M. N. (1931). An inquiry into various death-rates and the comparative influence of certain diseases on the duration of life. Ann. Eugen. 4, 279–326.CrossRefGoogle Scholar
Karn, M. N. (1933). A further study of methods of constructing life tables when certain causes of death arc eliminated. Biometrika 25, 91–101.CrossRefGoogle Scholar
Kendall, Maurice Sir & Plackett, R. L. (eds) (1977). Studies in the History of Statistics and Probability, II (London and High Wycombe: Griffin).Google Scholar
Lambert, J. H. (1772). Die Tödlichkeit der Kinderblattern. Bey träge zum Gebrauche der Mathematik und deren Anwendung. Theil 3, pp. 568–99 (Berlin) (B, U).Google Scholar
Laplace, P. S., Marquis, De (1812). Théorie analytique des Probabilités, pp. 412–15 (Paris) (A, U).Google Scholar
Laplace, P.S., Marquis, De (1886–1912). Æuvers complètes de Laplace (14 vols) (Paris: Gautheier-Villars) (U).Google Scholar
Makeham, W. M. (1867) On the Law of Mortality. J.I.A. 13, 325–58.Google Scholar
Makeham, W. M. (1875) On an application of the theory of the composition of decrementai forces. J.I.A. 18, 317–22.Google Scholar
Milne, J. (1815). A Treatise on the Valuation of Annuities and Assurances on Lives and Survivorships (London) (A).Google Scholar
Neill, A. (1977). Life Contingencies (London: Heinemann).Google Scholar
QuiQuet, A. (1934) Duvillard (1755- 1832), notes et références pouvant servir à sa biographie. Bull Trim. Inst. Actu, Franç. 40, 49 105, 121–74, 210–97 (A).Google Scholar
Seal, H. L. (1977) Studies in the history of probability and statistics, XXXV. Multiple decrements or competing risks. Biometrika 64, 429–39.Google Scholar
Seventeen Offices' experience tables (1843). Tables exhibiting the Law of Mortality deduced from the combined experience of seventeen life assurance offices, embracing 83, 905 policies; of which 40, 616 are distinguished by denoting the sex of the lives assured, and by classing them into Town, Country and Irish assurances (London) (A).Google Scholar
Stigler, S. M. (1977). Laplace's early work: chronology and citations. Technical Report 493 (University of Wisconsin) (S).Google Scholar
Süssmilch, J. P. (1761) Die göttliche Ordnung in den Veränderungen des menschlichen Geschlechts, aus der Geburt, dem Tode und der Fortpflanzung desselben erwiesen (Berlin) (A, S).Google Scholar
Todhunter, I. (1865). A History of the Mathematical Theory of Probability (Cambridge: Macmillan) (A, S).Google Scholar
Trembley, J. (1799). Recherches sur la mortalité de la petite vérole. Mém. de l'Acad. Roy. des Sciences pour 1796 pp. 17–38 (Berlin) (U).Google Scholar
Trembley, J. (1807). Eclaircissement relatif au Mémoire sur la mortalité de la petite vérole qui se trouve dans le volume de 1796. Mém. de l'Acad. Roy. des Sciences pour 1804, pp, 80–2 (Berlin) (U).Google Scholar
Woolhouse, W. S. B. (1839) Investigation of Mortality in the Indian Army (London). (Bound as No. 13 in Volume 38 of ‘Tracts’ in the Library of the Institute of Actuaries.) (A).Google Scholar
Woolhouse, W. S. B. (1867). On the construction of tables of mortality. J.I.A. 13, 75–102.Google Scholar