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2 results

  • The quasi-stationary distribution of the closed endemic sis model

  • Part of
  • Ingemar Nåsell
  • Journal:
    Advances in Applied Probability / Volume 28 / Issue 3 / September 1996
    Published online by Cambridge University Press:
    01 July 2016, pp. 895-932
    Print publication:
    September 1996

  • The quasi-stationary distribution of the closed stochastic SIS model changes drastically as the basic reproduction ratio R0 passes the deterministic threshold value 1. Approximations are derived that describe these changes. The quasi-stationary distribution is approximated by a geometric distribution (discrete!) for R0 distinctly below 1 and by a normal distribution (continuous!) for R0 distinctly above 1. Uniformity of the approximation with respect to R0 allows one to study the transition between these two extreme distributions. We also study the time to extinction and the invasion and persistence thresholds of the model.

  • Asymptotic Monotonicity of the Relative Extrema of Jacobi Polynomials

  • R. Wong, J.-M. Zhang
  • Journal:
    Canadian Journal of Mathematics / Volume 46 / Issue 6 / 01 December 1994
    Published online by Cambridge University Press:
    20 November 2018, pp. 1318-1337
    Print publication:
    01 December 1994
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  • If μk,n(α,β) denotes the relative extrema of the Jacobi polynomial P(α,β)n(x), ordered so that μk+1,n(α,β) lies to the left of μk,n(α,β), then R. A. Askey has conjectured twenty years ago that for for k = 1,…, n — 1 and n = 1,2,=. In this paper, we give an asymptotic expansion for μk,n(α,β) when k is fixed and n → ∞, which corrects an earlier result of R. Cooper (1950). Furthermore, we show that Askey's conjecture is true at least in the asymptotic sense.