Summary
A delay-integral equation, proposed by Cooke and Kaplan in [1] as a model of epidemics, is studied. The focus of this work is on the qualitative behavior of solutions as a certain parameter is allowed to vary. It is shown that if a certain threshold is not exceeded then solutions tend to zero exponentially while if this threshold is exceeded, periodic solutions exist. Many features of the numerical studies in [1] are explained.
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Cooke, K. L., Kaplan, J. L.: A periodicity threshold theorem for epidemics and population growth. J. Math. Biosciences (to appear).
Cooke, K. L., Yorke, J. A.: Some equations modelling growth processes and gonorrhea epidemics. Math. Biosci. 16, 75–101 (1973).
Gatica, J. A., Smith, H. L.: Fixed point techniques in a cone with applications (in preparation).
Hethcote, H. W.: Asymptotic behavior in a deterministic epidemic model. Bull. Math. Biology 35, 607–614 (1973).
Hoppensteadt, F., Waltman, P.: A problem in the theory of epidemics. Math. Biosci. 9, 71–91 (1970).
Hoppensteadt, F., Waltman, P.: A problem in the theory of epidemics II. Math. Biosci. 12, 133–145 (1971).
Krasnosel'skii, M. A.: Positive solutions of operator equations. Groningen: Noordhoff 1964.
Krein, M. G., Rutman, M. A.: Linear operators leaving invariant a cone in a Banach space. Uspehi Mat. Nauk 3, no. 1 (23), 3–95 (1948). (A.M.S. Translation Number 26, 1950.)
London, W. P., Yorke, J. A.: Recurrent outbreaks of measles, chickenpox, and mumps I: seasonal variation in contact rates. Amer. J. Epidemics 98, 453–468 (1973).
Nussbaum, R.: A periodicity threshold theorem for some nonlinear integral equations (to appear).
Rogers, F. B.: Epidemiology and Communicable Disease Control. New York: Grune & Stratton1963.
Rudin, W.: Functional Analysis. New York: McGraw-Hill 1973.
Yorke, J. A., London, W. P.: Recurrent outbreaks of measles, chickenpox, and mumps II: systematic differences in contact rates and stochastic effects. Amer. J. Epidemics 98, 469–482 (1973).
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Smith, H.L. On periodic solutions of a delay integral equation modelling epidemics. J. Math. Biology 4, 69–80 (1977). https://doi.org/10.1007/BF00276353
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DOI: https://doi.org/10.1007/BF00276353