Abstract
We describe and analyze compartmental models for disease transmission. We begin with models for epidemics, showing how to calculate the basic reproduction number and the final size of the epidemic. We also study models with multiple compartments, including treatment or isolation of infectives. We then consider models including births and deaths in which there may be an endemic equilibrium and study the asymptotic stability of equilibria. We conclude by studying age of infection models which give a unifying framework for more complicated compartmental models.
Keywords: Compartmental Model, Reproduction Number, Epidemic Model, Contact Rate, Endemic Equilibrium
Contributor Information
Fred Brauer, Email: brauer@math.ubc.ca.
Pauline van den Driessche, Email: pvdd@math.uvic.ca.
Jianhong Wu, Email: wujh@mathstat.yorku.ca.
Fred Brauer, Email: brauer@math.ubc.ca.
References
- 1.Anderson R.M., Jackson H.C., May R.M., Smith A.M. Population dynamics of fox rabies in Europe. Nature. 1981;289:765–771. doi: 10.1038/289765a0. [DOI] [PubMed] [Google Scholar]
- 2.Anderson R.M., May R.M. Infectious Diseases of Humans. Oxford: Oxford Science Publications; 1991. [Google Scholar]
- 3.Brauer F., Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology. New York: Springer; 2001. [Google Scholar]
- 4.Busenberg S., Cooke K.L. Vertically Transmitted Diseases: Models and Dynamics. Berlin Heidelberg New York: Springer; 1993. [Google Scholar]
- 5.Castillo-Chavez C., Blower S., van den Driessche P., Kirschner D., Yakubu A.A., editors. Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction. Berlin Heidelberg New York: Springer; 2001. [Google Scholar]
- 6.Castillo-Chavez C., Blower S., van den Driessche P., Kirschner D., Yakubu A.A., editors. Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory. Berlin Heidelberg New York: Springer; 2001. [Google Scholar]
- 7.Castillo-Chavez C., Cooke K.L., Huang W., Levin S.A. The role of long incubation periods in the dynamics of HIV/AIDS. Part 1: Single populations models. J. Math. Biol. 1989;27:373–98. doi: 10.1007/BF00290636. [DOI] [PubMed] [Google Scholar]
- 8.C. Castillo-Chavez, H.R. Thieme: Asymptotically autonomous epidemic models. In: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (eds.) Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 1: Theory of Epidemics. Wuerz, Winnipeg, pp. 33–50 (1993)
- 9.Daley D.J., Gani J. Epidemic Modelling: An Introduction. Cambridge Studies in Mathematical Biology. Cambridge: Cambridge University Press; 1999. [Google Scholar]
- 10.K. Dietz: Overall patterns in the transmission cycle of infectious disease agents. In: R.M. Anderson, R.M. May (eds.) Population Biology of Infectious Diseases. Life Sciences Research Report, Vol. 25. Springer, Berlin Heidelberg New York, pp. 87–102 (1982)
- 11.Dietz K. The first epidemic model: a historical note on P.D. En’ko. Aust. J. Stat. 1988;30:56–65. doi: 10.1111/j.1467-842X.1988.tb00464.x. [DOI] [Google Scholar]
- 12.Ellner S., Gallant R., Theiler J. Detecting nonlinearity and chaos in epidemic data. In: Mollison D., editor. Epidemic Models: Their Structure and Relation to Data. Cambridge: Cambridge University Press; 1995. pp. 229–247. [Google Scholar]
- 13.Gumel A., Ruan S., Day T., Watmough J., Driessche P, Brauer F., Gabrielson D., Bowman C., Alexander M.E., Ardal S., Wu J., Sahai B.M. Modeling strategies for controlling SARS outbreaks based on Toronto, Hong Kong, Singapore and Beijing experience. Proc. R. Soc. Lond. B Biol. Sci. 2004;271:2223–2232. doi: 10.1098/rspb.2004.2800. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 14.Heesterbeek J.A.P., Metz J.A.J. The saturating contact rate in marriage and epidemic models. J. Math. Biol. 1993;31:529–539. doi: 10.1007/BF00173891. [DOI] [PubMed] [Google Scholar]
- 15.Hethcote H.W. Qualitative analysis for communicable disease models. Math. Biosci. 1976;28:335–356. doi: 10.1016/0025-5564(76)90132-2. [DOI] [Google Scholar]
- 16.Hethcote H.W. An immunization model for a hetereogeneous population. Theor. Popul. Biol. 1978;14:338–349. doi: 10.1016/0040-5809(78)90011-4. [DOI] [PubMed] [Google Scholar]
- 17.Hethcote H.W. The mathematics of infectious diseases. SIAM Rev. 2000;42:599–653. doi: 10.1137/S0036144500371907. [DOI] [Google Scholar]
- 18.H.W. Hethcote, S.A. Levin: Periodicity in epidemic models. In: S.A. Levin, T.G. Hallam, L.J. Gross (eds.) Applied Mathematical Ecology. Biomathematics, Vol. 18. Springer, Berlin Heidelberg New York, pp. 193–211 (1989)
- 19.H.W. Hethcote, H.W. Stech, P. van den Driessche: Periodicity and stability in epidemic models: a survey. In: S. Busenberg, K.L. Cooke (eds.) Differential Equations and Applications in Ecology, Epidemics and Population Problems. Academic, New York, pp. 65–82 (1981)
- 20.E. Hopf: Abzweigung einer periodischen Lösungen von einer stationaren Lösung eines Differentialsystems. Berlin Math-Phys. Sachsiche Akademie der Wissenschaften, Leipzig, 94, 1–22 (1942)
- 21.Kermack W.O., McKendrick A.G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. B Biol. Sci. 1927;115:700–721. [Google Scholar]
- 22.Kermack W.O., McKendrick A.G. Contributions to the mathematical theory of epidemics, part. II. Proc. R. Soc. Lond. B Biol. Sci. 1932;138:55–83. [Google Scholar]
- 23.Kermack W.O., McKendrick A.G. Contributions to the mathematical theory of epidemics, part. III . Proc. R. Soc. Lond. B Biol. Sci. 1932;141:94–112. [Google Scholar]
- 24.L. Markus: Asymptotically autonomous differential systems. In: S. Lefschetz (ed.) Contributions to the Theory of Nonlinear Oscillations III. Annals of Mathematics Studies, Vol. 36. Princeton University Press, Princeton, NJ, pp. 17–29 (1956)
- 25.Mena-Lorca J., Hethcote H.W. Dynamic models of infectious diseases as regulators of population size. J. Math. Biol. 1992;30:693–716. doi: 10.1007/BF00173264. [DOI] [PubMed] [Google Scholar]
- 26.McNeill W.H. Plagues and Peoples. New York: Doubleday; 1976. [Google Scholar]
- 27.McNeill W.H. The Global Condition. Princeton, NJ: Princeton University Press; 1992. [Google Scholar]
- 28.Meyers L.A., Pourbohloul B., Newman M.E.J., Skowronski D.M., Brunham R.C. Network theory and SARS: predicting outbreak diversity. J. Theor. Biol. 2005;232:71–81. doi: 10.1016/j.jtbi.2004.07.026. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 29.Mollison D., editor. Epidemic Models: Their Structure and Relation to Data. Cambridge: Cambridge University Press; 1995. [Google Scholar]
- 30.Newman M.E.J. The structure and function of complex networks. SIAM Rev. 2003;45:167–256. doi: 10.1137/S003614450342480. [DOI] [Google Scholar]
- 31.Raggett G.F. Modeling the Eyam plague. IMA J. 1982;18:221–226. [Google Scholar]
- 32.Soper H.E. Interpretation of periodicity in disease prevalence. J. R. Stat. Soc. B. 1929;92:34–73. doi: 10.2307/2341437. [DOI] [Google Scholar]
- 33.Strogatz S.H. Exploring complex networks. Nature. 2001;410:268–276. doi: 10.1038/35065725. [DOI] [PubMed] [Google Scholar]
- 34.Thieme H.R. Asymptotically autonomous differential equations in the plane. Rocky Mt. J. Math. 1994;24:351–380. doi: 10.1216/rmjm/1181072470. [DOI] [Google Scholar]
- 35.Thieme H.R. Mathematics in Population Biology. Princeton, NJ: Princeton University Press; 2003. [Google Scholar]
- 36.H.R. Thieme, C. Castillo-Chavez: On the role of variable infectivity in the dynamics of the human immunodeficiency virus. In: C. Castillo-Chavez (ed.) Mathematical and Statistical Approaches to AIDS Epidemiology. Lecture Notes in Biomathematics, Vol. 83. Springer, Berlin Heidelberg New York, pp. 200–217 (1989)
- 37.Thieme H.R., Castillo-Chavez C. How may infection-age dependent infectivity affect the dynamics of HIV/AIDS? SIAM J. Appl. Math. 1989;53:1447–1479. doi: 10.1137/0153068. [DOI] [Google Scholar]
- 38.van den Driessche P., Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 2002;180:29–48. doi: 10.1016/S0025-5564(02)00108-6. [DOI] [PubMed] [Google Scholar]
- 39.Webb G.F. Theory of Nonlinear Age-Dependent Population Dynamics. New York: Marcel Dekker; 1985. [Google Scholar]