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The effect of integral conditions in certain equations modelling epidemics and population growth

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Summary

Models of epidemics that lead to delay differential equations often have subsidiary integral conditions that are imposed by the interpretation of these models. The neglect of these conditions may lead to solutions that behave in a radically different manner from solutions restricted to obey them. Examples are given of such behavior, including cases where periodic solutions may occur off the natural set defined by these conditions but not on it. A complete stability analysis is also given of a new model of a disease propagated by a vector where these integral conditions play an important role.

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References

  1. Bellman, R., Cooke, K.: Differential-difference equations. New York: Academic Press 1963

    Google Scholar 

  2. Busenberg, S., Cooke, K.: Periodic solutions of a periodic non-linear delay differential equation. SIAM J. Applied Math. 35, 704–721 (1978)

    Google Scholar 

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

    Google Scholar 

  4. Cooke, K.: Stability analysis for a vector disease model. Rocky Mountain Math. J., 9, 31–42 (1979)

    Google Scholar 

  5. Cooke, K.: Functional-differential equations: Some models and perturbation problems. Differential equations and dynamical systems, pp. 167–183. New York: Academic Press 1967

    Google Scholar 

  6. Green, D.: Self-oscillations for epidemic models. Math. Biosciences 38, 91–111 (1978)

    Google Scholar 

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

  8. Hale, J.: Behavior near constant solutions of functional differential equations. J. Diff. Eq. 15, 278–294 (1974)

    Google Scholar 

  9. Hale, J.: Theory of functional differential equations. New York: Springer 1977

    Google Scholar 

  10. Hoppensteadt, F., Waltman, P.: A problem in the theory of epidemics. Math. Biosciences 9, 71–91 (1970)

    Google Scholar 

  11. Hoppensteadt, F.: Mathematical theories of populations: Demographics, genetics and epidemics, Philadelphia: SIAM 1975

    Google Scholar 

  12. Seifert, G.: Positively invariant closed sets for systems of delay differential equations. J. Diff. Eq. 22, 292–304 (1976)

    Google Scholar 

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Authors and Affiliations

  1. Harvey Mudd College, 91711, Claremont, CA, USA

    Stavros Busenberg

  2. Pomona College, 91711, Claremont, CA, USA

    Kenneth L. Cooke

Authors
  1. Stavros Busenberg
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  2. Kenneth L. Cooke
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Additional information

This work was partially supported by N.S.F. Grant MCS 7903497

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Busenberg, S., Cooke, K.L. The effect of integral conditions in certain equations modelling epidemics and population growth. J. Math. Biology 10, 13–32 (1980). https://doi.org/10.1007/BF00276393

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  • DOI: https://doi.org/10.1007/BF00276393

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