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Integral equation models for endemic infectious diseases - Journal of Mathematical Biology

  • ️Tudor, David W.
  • ️Sat Mar 01 1980

Summary

Endemic infectious diseases for which infection confers permanent immunity are described by a system of nonlinear Volterra integral equations of convolution type. These constant-parameter models include vital dynamics (births and deaths), immunization and distributed infectious period. The models are shown to be well posed, the threshold criteria are determined and the asymptotic behavior is analysed. It is concluded that distributed delays do not change the thresholds and the asymptotic behaviors of the models.

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References

  1. Bailey, N. T. J.: The Mathematical Theory of Infectious Diseases, Second Edition. New York: Hafner Press, 1975

    Google Scholar 

  2. Birkhoff, G., Rota, G-C.: Ordinary Differential Equations, Second Edition. New York: John Wiley, 1969

    Google Scholar 

  3. Cooke, K. L., Yorke, J. A.: Some equations modelling growth processes and gonorrhea epidemics. Math. Biosci. 16, 75–101 (1973)

    Google Scholar 

  4. Hale, J. K.: Ordinary Differential Equations. New York: Wiley-Interscience, 1969

    Google Scholar 

  5. Grossman, Z.: Oscillatory phenomena in a model of infectious diseases, preprint

  6. Hethcote, H. W.: Asymptotic behavior and stability in epidemic models. In: Mathematical Problems in Biology, pp. 83–92. Lecture Notes in Biomathematics 2, New York: Springer, 1974

  7. Hethcote, H. W.: Qualitative analyses of communicable disease models. Math. Biosci. 28, 335–356 (1976)

    Google Scholar 

  8. Hethcote, H. W.: An immunization model for a heterogeneous population. Theor. Pop. Biol. 14, 338–349 (1978)

    Google Scholar 

  9. Hethcote, H. W., Stech, H. W., van den Driessche, P.: Nonlinear oscillations in epidemic models, preprint.

  10. Hethcote, H. W., Waltman, P.: Optimal vaccination schedules in a deterministic epidemic model. Math. Biosci. 18, 365–382 (1973)

    Google Scholar 

  11. Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics and Epidemics. Philadelphia: Society for Industrial and Applied Mathematics, 1975

    Google Scholar 

  12. Hoppensteadt, F., Waltman, P.: A problem in the theory of epidemics II. Math. Biosci 12, 133–145 (1971)

    Google Scholar 

  13. Kermack, W. O., McKendrick, A. G.: Contributions to the mathematical theory of epidemics, part I. Proc. Roy. Soc., Ser. A 115, 700–721 (1927)

    Google Scholar 

  14. Lajmanovich, A., Yorke, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci. 28, 221–236 (1976)

    Google Scholar 

  15. Ludwig, D.: Final size distributions for epidemics. Math. Biosci. 23, 33–46 (1975)

    Google Scholar 

  16. Miller, R. K.: On the linearization of Volterra integral equations. J. Math. Anal. Appl. 23, 198–208 (1968)

    Google Scholar 

  17. Miller, R. K.: Nonlinear Volterra Integral Equations. Menlo Park: Benjamin, 1971

    Google Scholar 

  18. Tudor, D. W.: Disease transmission and control in an age structured population, Ph.D. Thesis. University of Iowa, 1979

  19. Waltman, P.: Deterministic Threshold Models in the Theory of Epidemics. Lecture Notes in Biomathematics 1, New York: Springer, 1974

    Google Scholar 

  20. Wang, F. J. S.: Asymptotic behavior of some deterministic epidemic models. SIAM J. Math. Anal. 9, 529–534 (1978)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Iowa, 52242, Iowa City, IA, USA

    Herbert W. Hethcote

  2. Department of Mathematics, The College of Charleston, 29401, Charleston, South Carolina, USA

    David W. Tudor

Authors

  1. Herbert W. Hethcote

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  2. David W. Tudor

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Additional information

This work was partially supported by NIH Grant AI 13233.

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Hethcote, H.W., Tudor, D.W. Integral equation models for endemic infectious diseases. J. Math. Biology 9, 37–47 (1980). https://doi.org/10.1007/BF00276034

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  • Received: 24 April 1979

  • Revised: 09 July 1979

  • Issue Date: March 1980

  • DOI: https://doi.org/10.1007/BF00276034

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