Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

General compartmental epidemic models


The age of infection approach introduced by Kermack and Mckendrick in 1927 gives a unified way of describing and analyzing a variety of epidemic models, including models with multiple stages, treatment, and heterogeneous mixing. The author gives a description of the main results for such models, emphasizing the use of the final size relation to estimate the size of the epidemic.

This is a preview of subscription content, log in to check access.


  1. [1]

    Arino, J., Brauer, F., van den Driessche, et al, A final size relation for epidemic models, Math. Biosc. Eng., 4, 2007, 159–176.

  2. [2]

    Arino, J., Brauer, F., van den Driessche, et al, Simple models for containment of a pandemic, J. Roy. Soc. Interface, 3, 2006, 453–457.

  3. [3]

    Brauer, F., Age-of-infection and the final size relation, Math. Biosc. Eng., 5, 2008, 681–690.

  4. [4]

    Brauer, F. and Watmough, J., Age of infection epidemic models with heterogeneous mixing, J. Biol. Dyn., 3, 2009, 324–330.

  5. [5]

    Diekmann, O. and Heesterbeek, J. A. P., Mathematical Epidemiology of Infectious Diseases, Wiley, Chichester, 2000.

  6. [6]

    Diekmann, O., Heesterbeek, J. A. P. and Metz, J. A. J., On the definition and the computation of the basic reproductive ratio R 0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, 1990, 365–382.

  7. [7]

    Dietz, K., Overall patterns in the transmission cycle of infectious disease agents, Population Biology of Infectious Diseases, R. M. Anderson and R. M. May (eds.), Life Sciences Research Report, 25, Springer-Verlag, Berlin, Heidelberg, New York, 1982, 87–102.

  8. [8]

    Erdös, P. and Rényi, A., On random graphs, Publicationes Mathematicae, 6, 1959, 290–297.

  9. [9]

    Erdös, P. and Rényi, A., On the evolution of random graphs, Pub. Math. Inst. Hung. Acad. Science, 5, 1960, 17–61.

  10. [10]

    Erdös, P. and Rényi, A., On the strengths of connectedness of a random graph, Acta Math. Scientiae Hung., 12, 1961, 261–267.

  11. [11]

    Feng, Z., Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Biosc. Eng., 4, 2007, 675–686.

  12. [12]

    Feng, Z., Xu, D. and Zhao, W., Epidemiological models with non-exponentially distributed disease stages and applications to disease control, Bull. Math. Biol., 69, 2007, 1511–1536.

  13. [13]

    Gumel, A., Ruan, S., Day, T., et al, Modeling strategies for controlling SARS outbreaks based on Toronto, Hong Kong, Singapore and Beijing experience, Proc. Roy. Soc. London, 271, 2004, 2223–2232.

  14. [14]

    Heesterbeek, J. A. P. and Metz, J. A. J., The saturating contact rate in marriage and epidemic models, J. Math. Biol., 31, 1993, 529–539.

  15. [15]

    Heffernan, J. M. Smith, R. J. and Wahl, L. M., Perspectives on the basic reproductive ratio, J. Roy. Soc. Interface, 2, 2005, 281–293.

  16. [16]

    Hyman, J. M. and Li, J., Infection-age structured epidemic models with behavior change or treatment, J. Biol. Dyn., 1, 2007, 109–131.

  17. [17]

    Hyman, J. M., Li, J. and Stanley, E. A., The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155, 1999, 77–109.

  18. [18]

    Kermack, W. O. and GMcKendrick, A. G., A contribution to the mathematical theory of epidemics, Proc. Royal Soc. London, 115, 1927, 700–721.

  19. [19]

    Lloyd, A. L., Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics, Theor. Pop. Biol., 60, 2001, 59–71.

  20. [20]

    Mena-Lorca, J. and Hethcote, H. W., Dynamic models of infectious diseases as regulators of population size, J. Math. Biol., 30, 1992, 693–716.

  21. [21]

    Newman, M. E. J., The spread of epidemic disease on networks, Phys. Rev. E, 66, 2002, 016128.

  22. [22]

    Newman, M. E. J., The structure and function of complex networks, SIAM Review, 45, 2003, 167–256.

  23. [23]

    Newman, M. E. J., Strogatz, S. H. and Watts, D. J., Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E, 64, 2001, 026118.

  24. [24]

    Nold, A., Heterogeneity in disease transmission modeling, Math. Biosc., 52, 1980, 227–240.

  25. [25]

    Riley, S., Fraser, C., Donnelly, C. A., et al, Transmission dynamics of the etiological agent of SARS in Hong Kong: impact of public health interventions, Science, 300, 2003, 1961–1966.

  26. [26]

    Strogatz, S. H., Exploring complex networks, Nature, 410, 2001, 268–276.

  27. [27]

    Wearing, H. J., Rohani, P. and Keeling, M. J., Appropriate models for the management of infectious diseases, PLOS Medicine, 2, 2005, 621–627.

  28. [28]

    Yang, C. K. and Brauer, F., Calculation of R 0 for age-of-infection models, Math. Biosc. Eng., 5, 2008, 585–599.

Download references

Author information

Correspondence to Fred Brauer.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Brauer, F. General compartmental epidemic models. Chin. Ann. Math. Ser. B 31, 289–304 (2010). https://doi.org/10.1007/s11401-009-0454-1

Download citation


  • Epidemic models
  • Treatment models
  • Basic reproduction number
  • Final size relation

2000 MR Subject Classification

  • 92D30