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Thresholds and travelling waves for the geographical spread of infection

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Summary

A nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic is derived and analysed. Particular attention is paid to the hair-trigger effect and to the travelling wave problem.

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References

  1. Aronson, D. G., Weinberger, H. F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial Differential Equations and Related Topics (J. A. Goldstein, ed.), Lecture Notes in Math.446, 5–49. Berlin: Springer, 1975

    Google Scholar 

  2. Aronson, D. G., Weinberger, H. F.: Multidimensional nonlinear diffusion arising in population genetics. To appear in Advances in Math.

  3. Atkinson, C., Reuter, G. E. H.: Deterministic epidemic waves. Math. Proc. Camb. Phil. Soc.80, 315–330 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  4. Diekmann, O.: Limiting behaviour in an epidemic model. J. Nonl. Anal.-Theory Meth. Appl.1, 459–470 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  5. Diekmann, O., Kaper, H. G.: On the bounded solutions of a nonlinear convolution equation. To appear in J. Nonl. Anal.-Theory Meth. Appl.

  6. Essén, M.: Studies on a convolution inequality. Ark. Mat.5, 113–152 (1963)

    Google Scholar 

  7. Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II. New York: Wiley, 1966

    MATH  Google Scholar 

  8. Hadeler, K. P., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol.2, 251–263 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  9. Hale, J. K.: Ordinary Differential Equations. New York: Wiley, 1969

    MATH  Google Scholar 

  10. Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics and Epidemics. SIAM Regional Conference Series in Applied Mathematics, Vol. 20. Philadelphia: SIAM, 1975

    MATH  Google Scholar 

  11. Kendall, D. G.: Discussion of ‘Measles periodicity and community size’ by M. S. Bartlett. J. Roy. Statist. Soc. A120, 64–67 (1957)

    Article  Google Scholar 

  12. Kendall, D. G.: Mathematical models of the spread of infection. In: Mathematics and Computer Science in Biology and Medicine, pp. 213–224. London: Medical Research Council, 1965

    Google Scholar 

  13. Kermack, W. O., McKendrick, A. G.: A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. A115, 700–721 (1927)

    MATH  Google Scholar 

  14. Metz, J. A. J.: The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections. To appear in Acta Biotheoretica

  15. Mollison, D.: Possible velocities for a simple epidemic. Advances in Applied Prob.4, 233–257 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mollison, D.: The rate of spatial propagation of simple epidemics. In: Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability (L. M. le Cam, J. Neyman, E. L. Scott, eds.), Vol. III, pp. 579–614. Berkeley: Univ. of California Press, 1972

    Google Scholar 

  17. Titchmarsh, E. C.: Introduction to the Theory of Fourier Integrals. Oxford: Clarendon Press, 1937

    MATH  Google Scholar 

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  1. Mathematisch Centrum, 2e Boerhaavestraat 49, Amsterdam, The Netherlands

    O. Diekmann

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  1. O. Diekmann
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Diekmann, O. Thresholds and travelling waves for the geographical spread of infection. J. Math. Biology 6, 109–130 (1978). https://doi.org/10.1007/BF02450783

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