Daniel Bernoulli's epidemiological model revisited

Mathematical Biosciences

Volumn 180, Issue 1-2, 2002, Pages 1-21

  Dietz, Klaus ^a   Heesterbeek, J A P ^b  

^a UNIVERSITY OF TÜBINGEN   (Germany)

^b UTRECHT UNIVERSITY   (Netherlands)

Author keywords

Competing risks; Daniel Bernoulli; Epidemiological models; Inoculation; Life table; Smallpox

Indexed keywords

DISEASES; EPIDEMIOLOGY; HEALTH RISKS; IMMUNOLOGY; MATHEMATICAL MODELS;

INOCULATIONS;

BEHAVIORAL RESEARCH;

EPIDEMIOLOGY; MODEL; SMALLPOX; VACCINATION;

HISTORY OF MEDICINE; IMMUNITY; INFECTION RISK; LIFE EXPECTANCY; LIFESPAN; MORTALITY; REVIEW; RISK ASSESSMENT; SMALLPOX; VACCINATION;

VARIOLA VIRUS;

EID: 0036847088     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0025-5564(02)00122-0     Document Type: Article

References (49)

1. J.A. Jacquez, Compartmental Analysis in Biology and Medicine, 3rd Ed., BioMedware, Ann Arbor, MI, 1996.
2. J.A. Jacquez, J.S. Koopman, C.P. Simon, I.M. Longini Jr., Role of the primary infection in epidemics of HIV infection in gay cohorts, J. Acquir. Immune Defic. Syndr. 7 (1994) 1169.
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.)
4. 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.)
5. 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.)
6. 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.)
7. J. le R. d'Alembert, Sur l'application du calcul des probabilités à l'inoculation de la petite vérole, in: J. le R. d'Alembert (Eds.), Opuscules Mathématiques t. 2, David, Paris, 1761, p. 26.
8. A.G. Carmichael, A.M. Silverstein, Smallpox in Europe before the seventeenth century: Virulent killer or benign disease?, J. Hist. Med. Allied Sci. 42 (1987) 147.
9. A.K.C. Leung, 'Variolation' and vaccination in late imperial China ca 1570-1911, in: S. Plotkin, B. Fantini (Eds.), Vaccinia Vaccination and Vaccinology: Jenner, Pasteur and Their Successors, Elsevier, Amsterdam, 1996, p. 65.
10. 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.
11. C. Huygelen, The early years of vaccinology: Prophylactic immunization in the eighteenth and nineteenth centuries, Sartoniana 10 (1997) 79.
12. A.C. Klebs, Bibliography of Variolation, Lausanne 1913.
13. A.A. Rusnock, The Quantification of Things Human: Medicine and Political Arithmetic in Enlightenment England and France, dissertation, Princeton University, 1990.
14. 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.
15. A. Deparcieux, Essai sur les Probabilités de la Durée de la Vie Humaine, Paris, 1746.
16. 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.
17. C.M. de la Condamine, Histoire de l'Inoculation, Paris, 1776.
18. 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.
19. O. Spiess, Basel Anno 1760. Nach den Tagebüchern der Ungarischen Grafen Joseph und Samuel Teleki, Birkhäuser, Basel, 1936.
20. 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).
21. 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).
22. E.C. Zeeman, Controversy in science: On the ideas of Daniel Bernoulli and René Thom, Nieuw-Arch. -Wisk. 11 (1993) 257.
23. L.J. Daston, D'Alembert's critique of probability theory, Historia Math. 6 (1979) 259.
24. 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).
25. 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).
26. J. Trembley, Recherches sur la mortalité de la petite vérole, Mém. de l'Acad. Roy. des Sciences pour 1796 (Berlin) (1799) 17.
27. J. Trembley, 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) (1807) 80.
28. E.E. Duvillard, 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 longévité, Imprimerie Impériale, Paris, 1806.
29. J.O. Kephart, D.M. White, Computers and epidemiology, IEEE Spectrum 30 (1993) 20.
30. H.L. Seal, Studies in the history of probability and statistics XXXV multiple decrements or competing risks, Biometrika 64 (1977) 429.
31. R.H. Daw, Smallpox and the double decrement table: a piece of actuarial pre-history, J. Inst. Actuaries 106 (1979) 299.
32. F. Brambilla, Modelli deterministici e stocastici in epidemiologia, Bollettino del Centro per la Ricerca Operativa, Serie Metodologica 4 (1960) 3.
33. K. Dietz, Epidemics and rumours: a survey, J. R. Statist. Soc. A 130 (1967) 505.
34. H. Muench, Catalytic Models in Epidemiology, Harvard University, 1959.
35. N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, Griffin, London, 1975.
36. H. Muench, Derivation of rates from summation data by the catalytic curve, J.A.S.A. 29 (1934) 25.
37. 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.)
38. 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.)
39. 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.)
40. A. Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, New York, 1990.
41. M.N. Karn, An inquiry into various death-rates and the comparative influence of certain diseases on the duration of life, Ann. Eugen. 4 (1931) 279.
42. de Eeuw, een Sociaal-Historische en Historisch-Demografische Studie, A.A.G. Bijdragen Nr. 36, Landbouwuniversiteit Wageningen, 1997.
43. K. Dietz, J.A.P. Heesterbeek, Bernoulli was ahead of modern epidemiology, Nature 408 (2000) 513.
44. C.P. Farrington, M.N. Kanaan, N.J. Gay, Estimation of the basic reproduction number for infectious diseases from age-stratified serological survey data with discussion, Appl. Statist. 50 (2001) 251.
45. K. Dietz, Transmission and control of arbovirus diseases, in: D. Ludwig, K.L. Cooke (Eds.), Epidemiology, SIAM, Philadelphia, PA, 1975, p. 104.
46. R. Gani, S. Leach, Transmission potential of smallpox in contemporary populations, Nature 414 (2001) 748.
47. M.J. Crowder, Classical Competing Risks, Chapman and Hall, Boca Raton, FL, 2001.
48. F. Shann, Non-specific effects of vaccines in developing countries, British Med. J. 321 (2000) 1423.
49. 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.