Skip to main content
NCBI home page
Springer Nature - PMC COVID-19 Collection logoLink to Springer Nature - PMC COVID-19 Collection
. 2006 Apr 8;68(3):679–702. doi: 10.1007/s11538-005-9047-7

Generality of the Final Size Formula for an Epidemic of a Newly Invading Infectious Disease

Junling Ma 1,✉, David J D Earn 1
PMCID: PMC7088645  PMID: 16794950

Abstract

The well-known formula for the final size of an epidemic was published by Kermack and McKendrick in 1927. Their analysis was based on a simple susceptible-infected-recovered (SIR) model that assumes exponentially distributed infectious periods. More recent analyses have established that the standard final size formula is valid regardless of the distribution of infectious periods, but that it fails to be correct in the presence of certain kinds of heterogeneous mixing (e.g., if there is a core group, as for sexually transmitted diseases). We review previous work and establish more general conditions under which Kermack and McKendrick's formula is valid. We show that the final size formula is unchanged if there is a latent stage, any number of distinct infectious stages and/or a stage during which infectives are isolated (the durations of each stage can be drawn from any integrable distribution). We also consider the possibility that the transmission rates of infectious individuals are arbitrarily distributed—allowing, in particular, for the existence of super-spreaders—and prove that this potential complexity has no impact on the final size formula. Finally, we show that the final size formula is unchanged even for a general class of spatial contact structures. We conclude that whenever a new respiratory pathogen emerges, an estimate of the expected magnitude of the epidemic can be made as soon the basic reproduction number ℝ0 can be approximated, and this estimate is likely to be improved only by more accurate estimates of ℝ0, not by knowledge of any other epidemiological details.

Keywords: Epidemic models, Final size, Arbitrary stage durations, Integro-differential equations 1991 MSC:92D30

Contributor Information

Junling Ma, Email: junlingm@math.mcmaster.ca.

David J. D. Earn, Email: earn@math.mcmaster.ca

References

  1. Anderson D., Watson R. On the spread of a disease with gamma distributed latent and infectious periods. Biometrika. 1980;67(1):191–198. doi: 10.1093/biomet/67.1.191. [DOI] [Google Scholar]
  2. Anderson R.M., May R.M. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991. [Google Scholar]
  3. Anderson R.M., Medley G.F., May R.M., Johnson A.M. A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS. IMA J. Math. Appl. Med. Biol. 1986;3(4):229–263. doi: 10.1093/imammb/3.4.229. [DOI] [PubMed] [Google Scholar]
  4. Andersson, H., Britton, T., 2000. Stochastic epidemic models and their statistical analysis. Lecture Notes in Statistics, vol. 151. Springer-Verlag, New York.
  5. Andersson M. The asymptotic final size distribution of multitype chain-binomial epidemic processes. Adv. Appl. Prob. 1999;31:220–234. doi: 10.1239/aap/1029954274. [DOI] [Google Scholar]
  6. Andreasen V. Dynamics of annual influenza a epidemics with immuno-selection. J. Math. Biol. 2003;46(6):504–536. doi: 10.1007/s00285-002-0186-2. [DOI] [PubMed] [Google Scholar]
  7. Bailey N.T.J. The Mathematical Theory of Infectious Diseases and its Application. 2. London: Griffin; 1975. [Google Scholar]
  8. Brauer, F., Castillo-Chavez, C., 2001. Mathematical models in population biology and epidemiology. Texts in Applied Mathematics, vol. 40. Springer-Verlag, New York.
  9. Diekmann, O., Heesterbeek, J.A.P., 2000. Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, LTD, New York.
  10. Donnelly C.A., Ghani A.C., Leung G.M., Hedley A.J., Fraser C., Riley S., Abu-Raddad L.J., Ho L.M., Thach T.Q., Chau P., Chan K.P., Lam T.H., Tse L.Y., Tsang T., Liu S.H., Kong J.H.B., Lau E.M.C., Ferguson N.M., Anderson R.M. Epidemiological determinants of spread of causal agent of severe acute respiratory syndrome in Hong Kong. Lancet. 2003;361(9371):1761–1766. doi: 10.1016/S0140-6736(03)13410-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
  11. Dwyer G., Dushoff J., Elkinton J.S., Levin S.A. Pathogen-driven outbreaks in forest defoliators revisited: Building models from exprimental data. Am. Nat. 2000;156:105–120. doi: 10.1086/303379. [DOI] [PubMed] [Google Scholar]
  12. Earn D.J.D., Dushoff J., Levin S.A. Ecology and evolution of the flu. Trends Ecol. Evol. 2002;17(7):334–340. doi: 10.1016/S0169-5347(02)02502-8. [DOI] [Google Scholar]
  13. Earn D.J.D., Levin S.A., Rohani P. Coherence and conservation. Science. 2000b;290(5495):1360–1364. doi: 10.1126/science.290.5495.1360. [DOI] [PubMed] [Google Scholar]
  14. Earn D.J.D., Rohani P., Bolker B.M., Grenfell B.T. A simple model for complex dynamical transitions in epidemics. Science. 2000a;287(5453):667–670. doi: 10.1126/science.287.5453.667. [DOI] [PubMed] [Google Scholar]
  15. Feng Z.L., Thieme H.R. Endemic models with arbitrarily distributed periods of infection I: Fundamental properties of the model. SIAM J. Appl. Math. 2000;61(3):803–833. doi: 10.1137/S0036139998347834. [DOI] [Google Scholar]
  16. Gart, J.J., 1968. The mathematical analysis of an epidemic with two kinds of susceptibles. Biometrics 557–566. [PubMed]
  17. Hanski I.A., Gilpin M.E., editors. Metapopulation Biology: Ecology, Genetics, and Evolution. San Diego: Academic Press; 1997. [Google Scholar]
  18. Horn R.A., Johnson C.R. Matrix Analysis. Cambridge: Cambridge University Press; 1985. [Google Scholar]
  19. Hyman, J.M., Li, J.A.S.E., 1999. The differential infectivity and staged progression models for the transmission of HIV. Math. Biosci. 77–109. [DOI] [PubMed]
  20. Kermack, W.O., McKendrick, A.G., 1927. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A 115, 700–721.
  21. Kurtz T.G. Relationships between stochastic and deterministic population models. Lect. Notes Biomath. 1980;38:449–467. [Google Scholar]
  22. Lipsitch M., Cohen T., Cooper B., Robins J.M., Ma S., James L., Gopalakrishna G., Chew S.K., Tan C.C., Samore M.H., Fisman D., Murray M. Transmission dynamics and control of severe acute respiratory syndrome. Science. 2003;300(5627):1966–1970. doi: 10.1126/science.1086616. [DOI] [PMC free article] [PubMed] [Google Scholar]
  23. Lloyd A.L. Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R. Soc. Lond. Ser. B. 2001a;268(1470):985–993. doi: 10.1098/rspb.2001.1599. [DOI] [PMC free article] [PubMed] [Google Scholar]
  24. Lloyd A.L. Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics. Theor. Popul. Biol. 2001b;60(1):59–71. doi: 10.1006/tpbi.2001.1525. [DOI] [PubMed] [Google Scholar]
  25. London W., Yorke J.A. Recurrent outbreaks of measles, chickenpox and mumps. i. seasonal variation in contact rates. Am. J. Epidemiol. 1973;98(6):453–468. doi: 10.1093/oxfordjournals.aje.a121575. [DOI] [PubMed] [Google Scholar]
  26. Low D.E., McGeer A. SARS—one year later. N. Engl. J. Med. 2003;349(25):2381–2382. doi: 10.1056/NEJMp038203. [DOI] [PubMed] [Google Scholar]
  27. Poutanen S.M., Low D.E., Henry B., Finkelstein S., Rose D., Green K., Tellier R., Draker R., Adachi D., Ayers M., Chan A.K., Skowronski D.M., Salit I., Simor A.E., Slutsky A.S., Doyle P.W., Krajden M., Petric M., Brunham R.C., McGeer A.J. Identification of severe acute respiratory syndrome in Canada. N. Engl. J. Med. 2003;348(20):1995–2005. doi: 10.1056/NEJMoa030634. [DOI] [PubMed] [Google Scholar]
  28. Redfield R.R., Wright D.C., Tramont E.C. The Walter Reed staging classification for HTLV-III/LAV infection. N. Eng. J. Med. 1986;314:131–132. doi: 10.1056/NEJM198601093140232. [DOI] [PubMed] [Google Scholar]
  29. Scalia-Tomba G. Asymptotic final size distribution for some chain-binomial processes. Adv. Appl. Prob. 1985;17:477–495. doi: 10.2307/1427116. [DOI] [Google Scholar]
  30. Scalia-Tomba G. Asymptotic final size distribution of the multitype reed-frost process. Adv. Appl. Prob. 1986;23:563–584. doi: 10.1007/BF00275255. [DOI] [PubMed] [Google Scholar]
  31. Schenzle D. An age-structured model of pre- and post-vaccination measles transmission. IMA J. Math. Appl. Med. Biol. 1984;1:169–191. doi: 10.1093/imammb/1.2.169. [DOI] [PubMed] [Google Scholar]
  32. Schwartz I., Smith H. Infinite subharmonic bifurcation in an SEIR model. J. Math. Biol. 1983;18:233–253. doi: 10.1007/BF00276090. [DOI] [PubMed] [Google Scholar]
  33. Seligmann M., Pinching A.J., Rosen F.S.E.A. Immunology of human immunodeficiency virus and the acquired immune dificiency syndrome. Ann. Int. Med. 1987;107(2):234–242. doi: 10.7326/0003-4819-107-2-234. [DOI] [PubMed] [Google Scholar]
  34. van den Driessche P., Watmough J. Reproduction numbers and subthreadold endemic equilibria for compartmental models fo disease transmission. Math. Biosci. 2002;180:29–48. doi: 10.1016/S0025-5564(02)00108-6. [DOI] [PubMed] [Google Scholar]
  35. Von Bahr B., Martin-Lof A. Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 1980;12:319–349. doi: 10.2307/1426600. [DOI] [Google Scholar]
  36. Wallinga J., Teunis P. Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Am. J. Eoidemiol. 2004;160:509–516. doi: 10.1093/aje/kwh255. [DOI] [PMC free article] [PubMed] [Google Scholar]
  37. Weisstein, E.W., Lambert W-Function. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/LambertW-Function.html.
  38. WHO SARS Update 27, 2003. Severe acute respiratory syndrome (SARS), multi-country outbreak. Technical report, W.H.O. http://www.who.int/csr/sars/archive/2003_04_11/en/print.html.
  39. WHO SARS Update 33, 2003. Severe acute respiratory syndrome (SARS), multi-country outbreak. Technical report, W.H.O. http://www.who.int/csr/sars/archive/2003_04_18/en/print.html.
  40. Yorke, J.A., Hethcote, H.W., 1984. Gonorrhea: Transmission dynamics and control. Lecture notes in Biomathematics, 56, 1–105. Springer-Verlag, Berlin.

Articles from Bulletin of Mathematical Biology are provided here courtesy of Nature Publishing Group

RESOURCES