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Daniel Bernoulli’s epidemiological model revisited. (English) Zbl 1019.92028

Summary: A seminal paper by Daniel Bernoulli published in 1766 is put into a new perspective. After a short account of smallpox inoculation and of Bernoulli’s life, the motivation for that paper and its impact are described. It determines the age-specific equilibrium prevalence of immune individuals in an endemic potentially lethal infectious disease. The gain in life expectancy after elimination of this cause of death can be explicitly expressed in terms of the case fatality and the endemic prevalence of susceptibles. D’Alembert developed in 1761 an alternative method for dealing with competing risks of death, which is also applicable to non-infectious diseases.
Bernoulli’s formula for the endemic prevalence of susceptibles has so far escaped attention. It involves the lifetime risk of the infection, the force of infection and the life expectancy at birth. A new formula for the basic reproduction number is derived which involves the average force of infection, the average case fatality and the life expectancy at the time of infection. One can use this estimate to assess the gain in life expectancy if only a fraction of the population is immunized.

MSC:

92D30 Epidemiology
92-03 History of biology
01A50 History of mathematics in the 18th century
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[1] Jacquez, J. A., Compartmental Analysis in Biology and Medicine (1996), BioMedware: BioMedware Ann Arbor, MI · Zbl 0703.92001
[2] Jacquez, J. A.; Koopman, J. S.; Simon, C. P.; Longini, I. M., Role of the primary infection in epidemics of HIV infection in gay cohorts, J. Acquir. Immune Defic. Syndr., 7, 1169 (1994)
[3] D. Bernoulli, Essai d’une nouvelle analyse de la mortalité causée par la petite vérole. Mém. Math. Phys. Acad. Roy. Sci., Paris, (1766) 1. (Reprinted in: L.P. Bouckaert, B.L. van der Waerden (Eds.), Die Werke von Daniel Bernoulli, Bd. 2 Analysis und Wahrscheinlichkeitsrechnung, Birkhäuser, Basel, 1982, p. 235. English translation entitled ‘An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it’ in: L. Bradley, Smallpox Inoculation: An Eighteenth Century Mathematical Controversy, Adult Education Department, Nottingham, 1971, p. 21. Reprinted in: S. Haberman, T.A. Sibbett (Eds.) History of Actuarial Science, vol. VIII, Multiple Decrement and Multiple State Models, William Pickering, London, 1995, p. 1.); D. Bernoulli, Essai d’une nouvelle analyse de la mortalité causée par la petite vérole. Mém. Math. Phys. Acad. Roy. Sci., Paris, (1766) 1. (Reprinted in: L.P. Bouckaert, B.L. van der Waerden (Eds.), Die Werke von Daniel Bernoulli, Bd. 2 Analysis und Wahrscheinlichkeitsrechnung, Birkhäuser, Basel, 1982, p. 235. English translation entitled ‘An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it’ in: L. Bradley, Smallpox Inoculation: An Eighteenth Century Mathematical Controversy, Adult Education Department, Nottingham, 1971, p. 21. Reprinted in: S. Haberman, T.A. Sibbett (Eds.) History of Actuarial Science, vol. VIII, Multiple Decrement and Multiple State Models, William Pickering, London, 1995, p. 1.)
[4] d’Alembert, J.le R., Sur l’application du calcul des probabilités à l’inoculation de la petite vérole, (d’Alembert, J.le R., Opuscules Mathématiques t. 2 (1761), David: David Paris), 26
[5] Carmichael, A. G.; Silverstein, A. M., Smallpox in Europe before the seventeenth century: virulent killer or benign disease?, J. Hist. Med. Allied Sci., 42, 147 (1987)
[6] Leung, A. K.C., ‘Variolation’ and vaccination in late imperial China ca 1570-1911, (Plotkin, S.; Fantini, B., Vaccinia Vaccination and Vaccinology: Jenner, Pasteur and their successors (1996), Elsevier: Elsevier Amsterdam), 65
[7] J.Z. Holwell, An account of the manner of inoculating for the small pox in the East Indies, with some observations on the practice and mode of treating that disease in those parts, London, 1767, p. 1; J.Z. Holwell, An account of the manner of inoculating for the small pox in the East Indies, with some observations on the practice and mode of treating that disease in those parts, London, 1767, p. 1
[8] Huygelen, C., The early years of vaccinology: prophylactic immunization in the eighteenth and nineteenth centuries, Sartoniana, 10, 79 (1997)
[9] A.C. Klebs, Bibliography of Variolation, Lausanne 1913; A.C. Klebs, Bibliography of Variolation, Lausanne 1913
[10] A.A. Rusnock, The quantification of things human: Medicine and political arithmetic in Enlightenment England and France, dissertation, Princeton University, 1990; A.A. Rusnock, The quantification of things human: Medicine and political arithmetic in Enlightenment England and France, dissertation, Princeton University, 1990
[11] J. Arbuthnot, Mr. Maitland’s account of inoculating the smallpox vindicated, From Dr. Wagstaffe’s misinterpretations of that practice, with some remarks on Mr. Massey’s Sermon, London, 1722; J. Arbuthnot, Mr. Maitland’s account of inoculating the smallpox vindicated, From Dr. Wagstaffe’s misinterpretations of that practice, with some remarks on Mr. Massey’s Sermon, London, 1722
[12] A. Deparcieux, Essai sur les probabilités de la durée de la vie humaine, Paris, 1746; A. Deparcieux, Essai sur les probabilités de la durée de la vie humaine, Paris, 1746
[13] J. Jurin, An Account of the Success of Inoculating the Smallpox in Great Britain with a Comparison Between the Miscarriages in that Practice, and the Mortality on the Natural Smallpox, 2nd Ed., London, 1724; J. Jurin, An Account of the Success of Inoculating the Smallpox in Great Britain with a Comparison Between the Miscarriages in that Practice, and the Mortality on the Natural Smallpox, 2nd Ed., London, 1724
[14] C.M. de la Condamine, Histoire de l’inoculation, Paris, 1776; C.M. de la Condamine, Histoire de l’inoculation, Paris, 1776
[15] O.B. Sheynin, D. Bernoulli’s work on probability, in: M. Kendall, R.L. Plackett (Eds.), Studies in the History of Statistics and Probability, vol. II, London, 1977, p. 105; O.B. Sheynin, D. Bernoulli’s work on probability, in: M. Kendall, R.L. Plackett (Eds.), Studies in the History of Statistics and Probability, vol. II, London, 1977, p. 105 · Zbl 0276.01012
[16] O. Spiess, Basel anno 1760. Nach den Tagebüchern der ungarischen Grafen Joseph und Samuel Teleki, Birkhäuser, Basel, 1936; O. Spiess, Basel anno 1760. Nach den Tagebüchern der ungarischen Grafen Joseph und Samuel Teleki, Birkhäuser, Basel, 1936
[17] D. Bernoulli, Réflexions sur les avantages de l’inoculation, Mercure de France (1760) 173. (Reprinted in L.P. Bouckaert, B.L. van der Waerden (Eds), Die Werke von Daniel Bernoulli, Bd. 2 Analysis und Wahrscheinlichkeitsrechnung, Birkhäuser, Basel, 1982, p. 268); D. Bernoulli, Réflexions sur les avantages de l’inoculation, Mercure de France (1760) 173. (Reprinted in L.P. Bouckaert, B.L. van der Waerden (Eds), Die Werke von Daniel Bernoulli, Bd. 2 Analysis und Wahrscheinlichkeitsrechnung, Birkhäuser, Basel, 1982, p. 268)
[18] Zeeman, E. C., Controversy in science: on the ideas of Daniel Bernoulli and René Thom, Nieuw-Arch. -Wisk., 11, 257 (1993) · Zbl 0802.01008
[19] Daston, L. J., D’Alembert’s critique of probability theory, Historia Math., 6, 259 (1979) · Zbl 0422.01010
[20] J.H. Lambert, Die Tödlichkeit der Kinderblattern. Beyträge zum Gebrauche der Mathematik und deren Anwendung, vol. 3, Berlin, 1772, p. 568. (English translation entitled ‘The mortality of smallpox in children’ in: S. Haberman, T.A. Sibbett (Eds.) History of Actuarial Science, vol. VIII, Multiple Decrement and Multiple State Models, William Pickering, London, 1995, p. 39); J.H. Lambert, Die Tödlichkeit der Kinderblattern. Beyträge zum Gebrauche der Mathematik und deren Anwendung, vol. 3, Berlin, 1772, p. 568. (English translation entitled ‘The mortality of smallpox in children’ in: S. Haberman, T.A. Sibbett (Eds.) History of Actuarial Science, vol. VIII, Multiple Decrement and Multiple State Models, William Pickering, London, 1995, p. 39)
[21] Trembley, J., Recherches sur la mortalité de la petite vérole, Mém. de l’Acad. Roy. des Sciences pour, 1796 (Berlin), 17 (1799)
[22] Trembley, J., 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 (Berlin), 80 (1807)
[23] E.E. Duvillard, Analyse et tableaux de l’influence de la petite vérole sur la mortalité á chaque age, et de celle qu’un préservatif tel que la vaccine peut avoir sur la population et la longévité, Imprimerie Impériale, Paris, 1806; E.E. Duvillard, Analyse et tableaux de l’influence de la petite vérole sur la mortalité á chaque age, et de celle qu’un préservatif tel que la vaccine peut avoir sur la population et la longévité, Imprimerie Impériale, Paris, 1806
[24] Kephart, J. O.; White, D. M., Computers and epidemiology, IEEE Spectrum, 30, 20 (1993)
[25] Seal, H. L., Studies in the history of probability and statistics XXXV multiple decrements or competing risks, Biometrika, 64, 429 (1977) · Zbl 0375.62096
[26] Daw, R. H., Smallpox and the double decrement table: a piece of actuarial pre-history, J. Inst. Actuaries, 106, 299 (1979)
[27] Brambilla, F., Modelli deterministici e stocastici in epidemiologia, Bollettino del centro per la ricerca operativa, Serie Metodologica, 4, 3 (1960)
[28] Dietz, K., Epidemics and rumours: a survey, J. R. Statist. Soc. A, 130, 505 (1967)
[29] Muench, H., Catalytic Models in Epidemiology (1959), Harvard University
[30] Bailey, N. T.J., The mathematical theory of infectious diseases and its applications (1975), Griffin: Griffin London · Zbl 0115.37202
[31] Muench, H., Derivation of rates from summation data by the catalytic curve, J.A.S.A., 29, 25 (1934)
[32] E. Halley, An estimate of the degrees of the mortality of mankind, drawn from curious tables of the births and funerals at the city of Breslaw; with an attempt to ascertain the price of annuities upon lives, Philos. Trans. 17 (1693) 596. Some further considerations on the Breslaw bills of mortality, Philos. Trans. 17 (1694) 654. (Reprinted in J. Inst. Actuaries 18 (1874) 251 and in facsimile 112 (1985) 278.); E. Halley, An estimate of the degrees of the mortality of mankind, drawn from curious tables of the births and funerals at the city of Breslaw; with an attempt to ascertain the price of annuities upon lives, Philos. Trans. 17 (1693) 596. Some further considerations on the Breslaw bills of mortality, Philos. Trans. 17 (1694) 654. (Reprinted in J. Inst. Actuaries 18 (1874) 251 and in facsimile 112 (1985) 278.)
[33] Hald, A., A History of Probability and Statistics and Their Applications Before 1750 (1990), Wiley: Wiley New York · Zbl 0731.01001
[34] Karn, M. N., An inquiry into various death-rates and the comparative influence of certain diseases on the duration of life, Ann. Eugen., 4, 279 (1931)
[35] W. Rutten, ‘De Vreselijkste aller Harpijen’, Pokkenepidemiën en pokkenbestrijding in Nederland in de \(XVIII^{de}^{de}\); W. Rutten, ‘De Vreselijkste aller Harpijen’, Pokkenepidemiën en pokkenbestrijding in Nederland in de \(XVIII^{de}^{de}\)
[36] Dietz, K.; Heesterbeek, J. A.P., Bernoulli was ahead of modern epidemiology, Nature, 408, 513 (2000)
[37] Farrington, C. P.; Kanaan, M. N.; Gay, N. J., Estimation of the basic reproduction number for infectious diseases from age-stratified serological survey data with discussion, Appl. Statist., 50, 251 (2001) · Zbl 1112.62304
[38] Dietz, K., Transmission and control of arbovirus diseases, (Ludwig, D.; Cooke, K. L., Epidemiology (1975), SIAM: SIAM Philadelphia, PA), 104 · Zbl 0322.92023
[39] Gani, R.; Leach, S., Transmission potential of smallpox in contemporary populations, Nature, 414, 748 (2001)
[40] Crowder, M. J., Classical Competing Risks (2001), Chapman and Hall: Chapman and Hall Boca Raton, FL · Zbl 0979.62078
[41] Shann, F., Non-specific effects of vaccines in developing countries, British Med. J., 321, 1423 (2000)
[42] H. Straub, Bernoulli, Daniel, in: C.C. Gillespie (Ed.), Dictionary of Scientific Biography, American Council of Learned Societies, Scribner, New York, Vol. 2, p. 36; H. Straub, Bernoulli, Daniel, in: C.C. Gillespie (Ed.), Dictionary of Scientific Biography, American Council of Learned Societies, Scribner, New York, Vol. 2, p. 36
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