Journal of Mathematical Biology

, Volume 28, Issue 4, pp 451–462 | Cite as

Models for the spread of universally fatal diseases

  • F. Brauer


In the formulation of models of S-I-R type for the spread of communicable diseases it is necessary to distinguish between diseases with recovery with full immunity and diseases with permanent removal by death. We consider models which include nonlinear population dynamics with permanent removal. The principal result is that the stability of endemic equilibrium may depend on the population dynamics and on the distribution of infective periods; sustained oscillations are possible in some cases.

Key words

Epidemiology Stability of endemic equilibrium Distributed delays AIDS 


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  1. 1.
    Anderson, R. M., Jackson, H. C., May, R. M., Smith, A. M.: Population dynamics of fox rabies in Europe. Nature 289, 765–771 (1981)Google Scholar
  2. 2.
    Brauer, F.: Epidemic models in populations of varying size. In: Castillo-Chavez, C., Levin, S. A., Shoemaker, C. (eds.) Mathematical approaches to problems in resource management and epidemiology. (Lect. Notes Biomath., vol. 81, pp. 109–123) Berlin Heidelberg New York: Springer 1989Google Scholar
  3. 3.
    Castillo-Chavez, C., Cooke, K. L., Huang, W., Levin, S. A.: On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrom (AIDS). Part 1: Single population models. J. Math. Biol. 27, 373–398 (1989)Google Scholar
  4. 4.
    Feller, W.: On the integral equation of renewal theory. Ann. Math. Statist. 12, 243–267 (1940)Google Scholar
  5. 5.
    Hethcote, H. W.: Note on determining the limiting susceptible population in an epidemic model. Math. Biosc. 9, 161–163 (1970)Google Scholar
  6. 6.
    Hethcote, H. W.: Asymptotic behavior and stability in epidemic models. In: Driesse, P. van den (ed.) Mathematical problems in biology. (Lect. Notes Biomath., vol. 2, pp. 83–92) Berlin Heidelberg New York: Springer 1974Google Scholar
  7. 7.
    Hethcote, H. W.: Qualitative analyses of communicable disease models. Math. Biosci. 28, 335–356 (1975)Google Scholar
  8. 8.
    Hethcote, H. W., Stech, H. W., Driessche, P. van den: Stability analyses for models of diseases without immunity. J. Math. Biol. 13, 185–198 (1981)Google Scholar
  9. 9.
    Hethcote H. W., Tudor, D. W.: Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37–47 (1980)Google Scholar
  10. 10.
    Kermack, W. O., McKendrick A. G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond., Ser. A. 115, 700–721 (1987)Google Scholar
  11. 11.
    Thieme, H. R., Castillo-Chavez, C.: On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic. In: Castillo-Chavez, C. (ed.) Mathematical and statistical approaches to AIDS epidemiology. (Lect. Notes Biomath., vol. 83, pp. 157–177) Berlin Heidelberg New York: Springer 1989Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • F. Brauer
    • 1
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA

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