Skip to main content
SpringerLink
Log in

Stochastic epidemics: major outbreaks and the duration of the endemic period

  • Letter to the Editor
  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

A study is made of a two-dimensional stochastic system that models the spread of an infectious disease in a population. An asymptotic expression is derived for the probability that a major outbreak of the disease will occur in case the number of infectives is small. For the case that a major outbreak has occurred, an asymptotic approximation is derived for the expected time that the disease is in the population. The analytical expressions are obtained by asymptotically solving Dirichlet problems based on the Fokker-Planck equation for the stochastic system. Results of numerical calculations for the analytical expressions are compared with simulation results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • Courant, R. and D. Hilbert (1962) Methods of Mathematical Physics, Vol. 2, Wiley, New York

    Google Scholar 

  • Edelstein-Keshet, L. (1988) Mathematical Models in Biology, Random House, New York

    Google Scholar 

  • Erdélyi, A. (1956) Asymptotic Expansions, Dover, New York

    Google Scholar 

  • Feller, W. (1952) The parabolic differential equations and the associated semigroups of transformations, Ann. Math., 55, 468–519

    Google Scholar 

  • Gardiner, C. W. (1983) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer-Verlag, Berlin

    Google Scholar 

  • Goel, N. S. and N. Richter-Dyn (1974) Stochastic Models in Biology, Academic Press, New York

    Google Scholar 

  • Kamin, S. (1979) On elliptic singular perturbation problems with turning points, SIAM J. Math. Anal., 10, 447–455

    Google Scholar 

  • Karlin, S. (1966) A First Course in Stochastic Processes, Academic Press, New York

    Google Scholar 

  • Kendall, D. G. (1956) Deterministic and stochastic epidemics in closed populations, Proc. Symp. Math. Stat. Prob., 3rd, Berkeley, 4

  • Kevorkian, J. and J. D. Cole (1981) Perturbation Methods in Applied Mathematics, Springer-Verlag, New York

    Google Scholar 

  • Ludwig, D. (1975) Persistence of dynamical systems under random perturbations, SIAM Rev., 17, 605–640

    Google Scholar 

  • Roozen, H. (1989) An asymptotic solution to a two-dimensional exit problem arising in population dynamics, SIAM J. Appl. Math., 49, 1793–1810

    Google Scholar 

  • Roughgarden, J. (1979) Theory of Population Genetics and Evolutionary Ecology: an Introduction, Macmillan, New York

    Google Scholar 

  • Schuss, Z. (1980) Theory and Applications of Stochastic Differential Equations, Wiley, New York

    Google Scholar 

  • Wazwaz, A. M. and F. B. Hanson (1986) Matched uniform approximations for a singular boundary point and an interior turning point, SIAM J. Appl. Math., 46, 943–961

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Agricultural University, Dreijenlaan 4, 6703 HA, Wageningen, The Netherlands

    Onno A. van Herwaarden & Johan Grasman

Authors
  1. Onno A. van Herwaarden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Johan Grasman
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

van Herwaarden, O.A., Grasman, J. Stochastic epidemics: major outbreaks and the duration of the endemic period. J. Math. Biology 33, 581–601 (1995). https://doi.org/10.1007/BF00298644

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00298644

Keywords

Search

Navigation