Thresholds and travelling waves for the geographical spread of infection

  • O. Diekmann


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.


  1. 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, 1975Google Scholar
  2. 2.
    Aronson, D. G., Weinberger, H. F.: Multidimensional nonlinear diffusion arising in population genetics. To appear in Advances in Math.Google Scholar
  3. 3.
    Atkinson, C., Reuter, G. E. H.: Deterministic epidemic waves. Math. Proc. Camb. Phil. Soc.80, 315–330 (1976)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Diekmann, O.: Limiting behaviour in an epidemic model. J. Nonl. Anal.-Theory Meth. Appl.1, 459–470 (1977).MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Diekmann, O., Kaper, H. G.: On the bounded solutions of a nonlinear convolution equation. To appear in J. Nonl. Anal.-Theory Meth. Appl.Google Scholar
  6. 6.
    Essén, M.: Studies on a convolution inequality. Ark. Mat.5, 113–152 (1963)Google Scholar
  7. 7.
    Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II. New York: Wiley, 1966MATHGoogle Scholar
  8. 8.
    Hadeler, K. P., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol.2, 251–263 (1975).MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Hale, J. K.: Ordinary Differential Equations. New York: Wiley, 1969MATHGoogle Scholar
  10. 10.
    Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics and Epidemics. SIAM Regional Conference Series in Applied Mathematics, Vol. 20. Philadelphia: SIAM, 1975MATHGoogle Scholar
  11. 11.
    Kendall, D. G.: Discussion of ‘Measles periodicity and community size’ by M. S. Bartlett. J. Roy. Statist. Soc. A120, 64–67 (1957)CrossRefGoogle Scholar
  12. 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, 1965Google Scholar
  13. 13.
    Kermack, W. O., McKendrick, A. G.: A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. A115, 700–721 (1927)MATHGoogle Scholar
  14. 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 BiotheoreticaGoogle Scholar
  15. 15.
    Mollison, D.: Possible velocities for a simple epidemic. Advances in Applied Prob.4, 233–257 (1972)MATHMathSciNetCrossRefGoogle Scholar
  16. 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, 1972Google Scholar
  17. 17.
    Titchmarsh, E. C.: Introduction to the Theory of Fourier Integrals. Oxford: Clarendon Press, 1937MATHGoogle Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • O. Diekmann
    • 1
  1. 1.Mathematisch CentrumAmsterdamThe Netherlands

Personalised recommendations