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Integral equation models for endemic infectious diseases. (English) Zbl 0433.92026

MSC:
92D25Population dynamics (general)
45E10Integral equations of the convolution type
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Full Text: DOI
References:
[1] Bailey, N. T. J.: The Mathematical Theory of Infectious Diseases, Second Edition. New York: Hafner Press, 1975 · Zbl 0334.92024
[2] Birkhoff, G., Rota, G-C.: Ordinary Differential Equations, Second Edition. New York: John Wiley, 1969 · Zbl 0183.35601
[3] Cooke, K. L., Yorke, J. A.: Some equations modelling growth processes and gonorrhea epidemics. Math. Biosci. 16, 75-101 (1973) · Zbl 0251.92011 · doi:10.1016/0025-5564(73)90046-1
[4] Hale, J. K.: Ordinary Differential Equations. New York: Wiley-Interscience, 1969 · Zbl 0186.40901
[5] Grossman, Z.: Oscillatory phenomena in a model of infectious diseases, preprint · Zbl 0457.92020
[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) · Zbl 0326.92017 · doi:10.1016/0025-5564(76)90132-2
[8] Hethcote, H. W.: An immunization model for a heterogeneous population. Theor. Pop. Biol. 14, 338-349 (1978) · Zbl 0392.92009 · doi:10.1016/0040-5809(78)90011-4
[9] Hethcote, H. W., Stech, H. W., van den Driessche, P.: Nonlinear oscillations in epidemic models, preprint. · Zbl 0469.92012
[10] Hethcote, H. W., Waltman, P.: Optimal vaccination schedules in a deterministic epidemic model. Math. Biosci. 18, 365-382 (1973) · Zbl 0266.92011 · doi:10.1016/0025-5564(73)90011-4
[11] Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics and Epidemics. Philadelphia: Society for Industrial and Applied Mathematics, 1975 · Zbl 0304.92012
[12] Hoppensteadt, F., Waltman, P.: A problem in the theory of epidemics II. Math. Biosci 12, 133-145 (1971) · Zbl 0226.92011 · doi:10.1016/0025-5564(71)90078-2
[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) · Zbl 53.0517.01 · doi:10.1098/rspa.1927.0118
[14] Lajmanovich, A., Yorke, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci. 28, 221-236 (1976) · Zbl 0344.92016 · doi:10.1016/0025-5564(76)90125-5
[15] Ludwig, D.: Final size distributions for epidemics. Math. Biosci. 23, 33-46 (1975) · Zbl 0318.92025 · doi:10.1016/0025-5564(75)90119-4
[16] Miller, R. K.: On the linearization of Volterra integral equations. J. Math. Anal. Appl. 23, 198-208 (1968) · Zbl 0167.40902 · doi:10.1016/0022-247X(68)90127-3
[17] Miller, R. K.: Nonlinear Volterra Integral Equations. Menlo Park: Benjamin, 1971 · Zbl 0448.45004
[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 · Zbl 0293.92015
[20] Wang, F. J. S.: Asymptotic behavior of some deterministic epidemic models. SIAM J. Math. Anal. 9, 529-534 (1978) · Zbl 0417.92020 · doi:10.1137/0509034