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
Bailey, N. T. J.: The Mathematical Theory of Infectious Diseases, Second Edition. New York: Hafner Press, 1975
Birkhoff, G., Rota, G-C.: Ordinary Differential Equations, Second Edition. New York: John Wiley, 1969
Cooke, K. L., Yorke, J. A.: Some equations modelling growth processes and gonorrhea epidemics. Math. Biosci. 16, 75–101 (1973)
Hale, J. K.: Ordinary Differential Equations. New York: Wiley-Interscience, 1969
Grossman, Z.: Oscillatory phenomena in a model of infectious diseases, preprint
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
Hethcote, H. W.: Qualitative analyses of communicable disease models. Math. Biosci. 28, 335–356 (1976)
Hethcote, H. W.: An immunization model for a heterogeneous population. Theor. Pop. Biol. 14, 338–349 (1978)
Hethcote, H. W., Stech, H. W., van den Driessche, P.: Nonlinear oscillations in epidemic models, preprint.
Hethcote, H. W., Waltman, P.: Optimal vaccination schedules in a deterministic epidemic model. Math. Biosci. 18, 365–382 (1973)
Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics and Epidemics. Philadelphia: Society for Industrial and Applied Mathematics, 1975
Hoppensteadt, F., Waltman, P.: A problem in the theory of epidemics II. Math. Biosci 12, 133–145 (1971)
Kermack, W. O., McKendrick, A. G.: Contributions to the mathematical theory of epidemics, part I. Proc. Roy. Soc., Ser. A 115, 700–721 (1927)
Lajmanovich, A., Yorke, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci. 28, 221–236 (1976)
Ludwig, D.: Final size distributions for epidemics. Math. Biosci. 23, 33–46 (1975)
Miller, R. K.: On the linearization of Volterra integral equations. J. Math. Anal. Appl. 23, 198–208 (1968)
Miller, R. K.: Nonlinear Volterra Integral Equations. Menlo Park: Benjamin, 1971
Tudor, D. W.: Disease transmission and control in an age structured population, Ph.D. Thesis. University of Iowa, 1979
Waltman, P.: Deterministic Threshold Models in the Theory of Epidemics. Lecture Notes in Biomathematics 1, New York: Springer, 1974
Wang, F. J. S.: Asymptotic behavior of some deterministic epidemic models. SIAM J. Math. Anal. 9, 529–534 (1978)
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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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DOI: https://doi.org/10.1007/BF00276034