Journal of Mathematical Biology

, Volume 41, Issue 6, pp 559–580 | Cite as

Stochastic epidemics in dynamic populations: quasi-stationarity and extinction

  • Håkan Andersson
  • Tom Britton


Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for τ, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how τ depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours.

Key words: Critical community size – Diffusion approximation – Persistence – Quasi-stationary distribution – SIR epidemics – Stochastic fade-out – Vaccination 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Håkan Andersson
    • 1
  • Tom Britton
    • 2
  1. 1.Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.SE
  2. 2.Department of Mathematics, Uppsala University, P.O. Box 480, 751 06 Uppsala, Sweden. e-mail: tom.britton@math.uu.seSE

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