Advertisement

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

Abstract.

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations