Skip to main content

A Brief History of R 0 and a Recipe for its Calculation

Acta Biotheoretica volume 50, pages 189–204 (2002)Cite this article

Abstract

In this paper I present the genesis of R 0 in demography, ecology and epidemiology, from embryo to its current adult form. I argue on why it has taken so long for the concept to mature in epidemiology when there were ample opportunities for cross-fertilisation from demography and ecology from where it reached adulthood fifty years earlier. Today, R 0 is a more fully developed adult in epidemiology than in demography. In the final section I give an algorithm for its calculation in heterogeneous populations.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

REFERENCES

  • Aitchison, J. and G.S. Watson (1988). A not-so-plain tale from the Raj. In: The Influence of Scottish Medicine. D.A. Dow (ed.). Parthenon, Carnforth, pp 113-128.

    Google Scholar 

  • Anderson, R.M. (1981). Population ecology of infectious diseases. In: Theoretical Ecology. R.M. May (ed.). Blackwell, Oxford. pp. 318-355.

    Google Scholar 

  • Anderson, R.M. and R.M. May (1979a). Population biology of infectious diseases. Part I. Nature 280: 361-367.

    Google Scholar 

  • Anderson, R.M. and R.M. May (1979b). Population biology of infectious diseases. Part II. Nature 280: 455-461.

    Google Scholar 

  • Anderson, R.M. and R.M. May (1982a). Directly transmitted infectious diseases: control by vaccination. Science 215: 1053-1060.

    Google Scholar 

  • Anderson, R.M. and R.M. May (eds.) (1982b). Population Biology of Infectious Diseases. Springer-Verlag, Berlin.

    Google Scholar 

  • Anderson, R.M. and R.M. May (1991). Infectious Diseases of Humans: transmission and control. Oxford University Press, Oxford.

    Google Scholar 

  • Bailey, N.J.T. (1953). The total size of a general stochastic epidemic. Biometrika 40: 177-185.

    Google Scholar 

  • Bailey, N.J.T. (1957). Mathematical Theory of Epidemics. Griffin, London.

    Google Scholar 

  • Bartlett, M.S. (1955). An Introduction to Stochastic Processes. Cambridge University Press, Cambridge.

    Google Scholar 

  • Barucha-Reid, A.T. (1956). On the stochastic theory of epidemics. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Volume IV: Biology and Problems of Health, J. Neyman (ed.). University of California Press, Berkeley. pp. 111-119.

    Google Scholar 

  • Becker, N. (1975). The use of mathematical models in determining vaccination policies. Bulletin of the International Statistics Institute 46: 478-490.

    Google Scholar 

  • Begon, M., J. L. Harper and C. R. Townsend (1998). Ecology: Individuals, Populations and Communities. 3rd edition. Blackwell Science, Oxford.

    Google Scholar 

  • Bockh, R. (1886). Statistsiches Jahrbuch der Stadt Berlin, Volume 12, Statistik des Jahres 1884. P. Stankiewicz, Berlin.

    Google Scholar 

  • Diekmann, O. and J.A.P. Heesterbeek (2000). Mathematical Epidemiology of Infectious Diseases: model building, analysis and interpretation. John Wiley and Sons, Chichester.

    Google Scholar 

  • Diekmann, O., J.A.P. Heesterbeek and J.A.J. Metz (1990). On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology 28: 365-382.

    Google Scholar 

  • Dietz, K. (1975). Transmission and control of arboviruses. In: Epidemiology, D. Ludwig and K.L. Cooke (eds.). SIAM, Philadelphia. pp. 104-121.

    Google Scholar 

  • Dietz, K. (1988). The first epidemic model: a historical note on P.D. En'ko. Australian Journal of Statistics 30A: 56-65.

    Google Scholar 

  • Dietz, K. (1993). The estimation of the basic reproduction number for infectious diseases. Statistical Methods in Medical Research 2: 23-41.

    Google Scholar 

  • Dietz, K. (1995). Some problems in the theory of infectious disease transmission and control. In: Epidemic Models: their Structure and Relation to Data, D. Mollison (ed.). Cambridge University Press, Cambridge, pp. 3-16.

    Google Scholar 

  • Dietz, K., L. Molineaux and A. Thomas (1974). A malaria model tested in the African Savannah. Bulletin of the World Health Organization 50: 347-357.

    Google Scholar 

  • Dublin, L.I. and A.J. Lotka (1925). On the true rate of natural increase. Journal of the American Statistical Association 150: 305-339.

    Google Scholar 

  • En'ko, P.D. (1889). On the course of epidemics of some infectious diseases. Vrach. St. Petersburg, X, 1008-1010, 1039–1042, 1061–1063 (in Russian). English translation by K. Dietz, 1989. International Journal of Epidemiology 18: 749–755.

    Google Scholar 

  • Feller, W. (1941). On the integral equation of renewal theory. Annals of Mathematical Statistics 12: 243-267.

    Google Scholar 

  • Fine, P.E.M. (1975). Ross's a priori pathometry -a perspective. Proceeding of the Royal Society of Medicine 68: 547-551.

    Google Scholar 

  • Fine, P.E.M. and J.A. Clarkson (1982). Measles in England and Wales. International Journal of Epidemiology 11: 5-14 (part I), 15–25 (part II).

    Google Scholar 

  • Harvey, W.F. (1943). Anderson Gray McKendrick. Edinburgh Medical Journal 50: 500-506.

    Google Scholar 

  • Heesterbeek, J.A.P. (1992). R 0. PhD Thesis. University of Leiden.

  • Heesterbeek, J.A.P. and K. Dietz (1996). The concept of R 0 in epidemic theory. Statistica Neerlandica 50: 89-110.

    Google Scholar 

  • Hethcote, H.W. (1975). Mathematical models for the spread of infectious diseases. In: Epidemiology, D. Ludwig and K.L. Cooke (eds.). SIAM, Philadelphia. pp. 122-131.

    Google Scholar 

  • Kermack, W.O. and A.G. McKendrick (1927). Contributions to the mathematical theory of epidemics-I. Proceedings of the Royal Society of Medicine 115A: 700-721 (reprinted in Bulletin of Mathematical Biology 53: 33–55, 1991).

    Google Scholar 

  • Kingsland, S.E. (1985). Modeling Nature: episodes in the history of population ecology. University of Chicago Press, Chicago.

    Google Scholar 

  • Knibbs, G.H. (1917). Mathematical theory of population, of its character and fluctuations, and of the factors which influence them. Appendix to Census of the Commonwealth of Australia. McCarron, Bird & Co, Melbourne.

    Google Scholar 

  • Kuczynski, R.R. (1932). Fertility and Reproduction. Falcon Press, New York.

    Google Scholar 

  • Kuczynski, R.R. (1935). The Measurement of Population Growth. Sidgwick and Jackson, London.

    Google Scholar 

  • Lotka, A.J. (1907). Relation between birth rates and death rates. Science 26: 21-22.

    Google Scholar 

  • Lotka, A.J. (1913). A natural population norm. J. Washington. Acad. Science 3: 241-293.

    Google Scholar 

  • Lotka, A.J. (1919). A contribution to quantitative epidemiology. Journal of the Washington Academy of Science 9: 73-77.

    Google Scholar 

  • Lotka, A.J. (1923). Contribution to the analysis of malaria epidemiology. I: General part. Supplement to American Journal of Hygiene 3: 1-37 (parts II to V in same volume).

    Google Scholar 

  • Lotka, A.J. (1925a). The measure of net fertility. Journal of the Washington Academy of Science 15: 469-472.

    Google Scholar 

  • Lotka, A.J. (1925b). Elements of Physical Biology. Williams and Wilkins Company, Baltimore.

    Google Scholar 

  • Macdonald, G. (1952). The analysis of equilibrium in malaria. Tropical Diseases Bulletin 49: 813-829.

    Google Scholar 

  • Macnamara, F.N. (1955). Man as the host of yellow fever virus. Dissertation for M.D. degree, University of Cambridge.

  • May, R.M. (1976). Estimating r: a pedagogical note. American Naturalist 110: 469-499.

    Google Scholar 

  • May, R.M. (1981). Models for single populations. In: Theoretical Ecology. R.M. May (ed.). Blackwell, Oxford. pp. 30-52.

    Google Scholar 

  • Nye, E.R. and M.E. Gibson (1997). Ronald Ross, Malariologist and Polymath, a Biography. Macmillan Press, London.

    Google Scholar 

  • Ross, R. (1911). The Prevention of Malaria. 2nd edition. John Murray, London.

    Google Scholar 

  • Ross, R. (1916). An application of the theory of probabilities to the study of a priori pathometry -Part I. Proceedings of the Royal Society London A 42: 204-230.

    Google Scholar 

  • Ross, R. and H.P. Hudson (1917a). An application of the theory of probabilities to the study of a priori pathometry -Part II. Proceedings of the Royal Society London A 43: 212-225.

    Google Scholar 

  • Ross, R. and H.P. Hudson (1917b). An application of the theory of probabilities to the study of a priori pathometry -Part III. Proceedings of the Royal Society London A 43: 225-240.

    Google Scholar 

  • Samuelson, P.A. (1977). Resolving a historical confusion in population analysis. In: Mathematical Demography, selected papers, D. Smith and N. Keyfitz (eds.). Springer-Verlag, Berlin. pp. 109-130.

    Google Scholar 

  • Sharpe, F.R. and A.J. Lotka (1911). A problem in age distribution. Philosophical Magazine 21: 435-438.

    Google Scholar 

  • Smith, C.E.G. (1964). Factors in the transmission of virus infections from animal to man. Scientific Basis of Medicine Annual Review 1964. pp. 125-150.

    Google Scholar 

  • Smith, D. and N. Keyfitz (eds.) (1977). Mathematical Demography, selected papers. Springer-Verlag, Berlin.

    Google Scholar 

  • Thompson, W.R. (1923). La théorie mahématique de l'action des parasites entomophages. Revue Générale des Sciences Pures et Appliqueés 34: 202-210.

    Google Scholar 

  • Whittle, P. (1955). The outcome of a stochastic epidemic -a note on Bailey's paper. Biometrica 42: 116-122.

    Google Scholar 

  • Yorke, J.A., N. Nathanson, G. Pianigiani and J. Martin (1979). Seasonality and the requirements for perpetuation and eradication of viruses in populations. American Journal of Epidemiology 109: 103-123.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Veterinary Medicine, University of Utrecht, Quantitative Veterinary Epidemiology Group, Yalelaan 7, 3584 CL, Utrecht, The Netherlands

    J.A.P. Heesterbeek

Authors
  1. J.A.P. Heesterbeek
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Heesterbeek, J. A Brief History of R 0 and a Recipe for its Calculation. Acta Biotheor 50, 189–204 (2002). https://doi.org/10.1023/A:1016599411804

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1016599411804

Keywords

  • Final Section
  • Heterogeneous Population
  • Ample Opportunity
  • Adult Form