On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations


The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.

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


  1. 1.

    Andreasen, V., Christiansen, F. B.: Persistence of an infectious disease in a subdivided population. Math. Biosci. 96, 239–253 (1989)

    Google Scholar 

  2. 2.

    Bailey, N. T. J.: The biomathematics of malaria. London: Griffin 1982

    Google Scholar 

  3. 3.

    Barbour, A. D.: MacDonald's model and the transmission of bilharzia. Trans. Roy. Soc. Trop. Med. Hyg. 72, 6–15 (1978)

    Google Scholar 

  4. 4.

    Blythe, S. P., Castillo-Chavez, C.: Like-with-like preference and sexual mixing models. Math. Biosci. 96, 221–238 (1989)

    Google Scholar 

  5. 5.

    Dietz, K.: Models for vector-borne parasitic diseases. In: Barigozzi, C. (ed.) Vito Volterra Symposium on Mathematical Models in Biology. (Lect. Notes Biomath., vol. 39, pp. 264–277) Berlin Heidelberg New York: Springer 1980

    Google Scholar 

  6. 6.

    Dietz, K., Schenzle, D.: Proportionate mixing models for age-dependent infection transmission. J. Math. Biol. 22, 117–120 (1985)

    Google Scholar 

  7. 7.

    Heijmans, H. J. A. M.: The dynamical behaviour of the age-size distribution of a cell population. In: Metz, J. A. J., Diekmann, O. (eds.) The dynamics of physiologically structured populations. (Lect. Notes Biomath., vol. 68, pp. 185–202) Berlin Heidelberg New York: Springer 1986

    Google Scholar 

  8. 8.

    Hethcote, H. W., Yorke, J. A.: Gonorrhea, transmission dynamics and control. (Lect. Notes Biomath., vol. 56) Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  9. 9.

    Hethcote, H. W., van Ark, J. W.: Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs. Math. Biosci. 84, 85–118 (1987)

    Google Scholar 

  10. 10.

    Hyman, J. M., Stanley, E. A.: Using mathematical models to understand the AIDS epidemic. Math. Biosci. 90, 415–473 (1988)

    Google Scholar 

  11. 11.

    Jacquez, J. A., Simon, C. P., Koopman, J., Sattenspiel, L., Perry, T.: Modeling and analyzing HIV transmission: the effect of contact patterns. Math. Biosci. 92, 119–199 (1988)

    Google Scholar 

  12. 12.

    Jagers, P., Nerman O.: The growth and composition of branching populations. Adv. Appl. Probab. 16, 221–259 (1984)

    Google Scholar 

  13. 13.

    May, R. M., Anderson, R. M.: The transmission dynamics of human immunodeficiency virus (HIV). Philo. Trans. R. Soc. Lond., B 321, 565–607 (1988)

    Google Scholar 

  14. 14.

    Mode, C. J.: Multitype branching processes, theory and applications. New York: Elsevier 1971

    Google Scholar 

  15. 15.

    Nold, A.: Heterogeneity in disease-transmission modeling. Math. Biosci. 52, 227–240 (1980)

    Google Scholar 

  16. 16.

    Schaefer, H. H.: Some spectral properties of positive linear operators. Pacific J. Math. 10, 1009–1019 (1960)

    Google Scholar 

  17. 17.

    Schaefer, H. H.: Banach lattices and positive operators. Berlin Heidelberg New York: Springer 1974

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Diekmann, O., Heesterbeek, J.A.P. & Metz, J.A.J. On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990). https://doi.org/10.1007/BF00178324

Download citation

Key words

  • Epidemic models
  • Heterogeneous populations
  • Basic reproductive number
  • Invasion