The quasi-stationary distribution of the closed endemic sis model

The quasi-stationary distribution of the closed stochastic SIS model changes drastically as the basic reproduction ratio R 0 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 R 0 distinctly below 1 and by a normal distribution (continuous!) for R 0 distinctly above 1. Uniformity of the approximation with respect to R 0 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.

[1]  R. May,et al.  Infectious Diseases of Humans: Dynamics and Control , 1991, Annals of Internal Medicine.

[2]  O. A. Herwaarden,et al.  Stochastic epidemics: major outbreaks and the duration of the endemic period , 1995, Journal of mathematical biology.

[3]  I. Nåsell.  On the quasi-stationary distribution of the Ross malaria model. , 1991, Mathematical biosciences.

[4]  D. E. Stewart,et al.  An Efficient Procedure for Computing Quasi-Stationary Distributions of Markov Chains by Sparse Transition Structure , 1994, Advances in Applied Probability.

[5]  E. Seneta,et al.  On quasi-stationary distributions in absorbing continuous-time finite Markov chains , 1967, Journal of Applied Probability.

[6]  Richard J. Kryscio,et al.  On the Extinction of the S–I–S stochastic logistic epidemic , 1989 .

[7]  George H. Weiss,et al.  Stochastic theory of nonlinear rate processes with multiple stationary states , 1977 .

[8]  George H. Weiss,et al.  On the asymptotic behavior of the stochastic and deterministic models of an epidemic , 1971 .

[9]  D. J. Bartholomew,et al.  Continuous time diffusion models with random duration of interest , 1976 .

[10]  J. A. Cavender,et al.  Quasi-stationary distributions of birth-and-death processes , 1978, Advances in Applied Probability.

[11]  F. Olver.  Asymptotics and Special Functions , 1974 .

[12]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[13]  N. Temme.  Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function , 1975 .

[14]  J A Jacquez,et al.  The stochastic SI model with recruitment and deaths. I. Comparison with the closed SIS model. , 1993, Mathematical biosciences.

[15]  A. J. Hall.  Infectious diseases of humans: R. M. Anderson & R. M. May. Oxford etc.: Oxford University Press, 1991. viii + 757 pp. Price £50. ISBN 0-19-854599-1 , 1992 .

[16]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume I: Fundamental Algorithms, 2nd Edition , 1997 .

[17]  R. H. Norden.  On the distribution of the time to extinction in the stochastic logistic population model , 1982, Advances in Applied Probability.

[18]  P. Flajolet,et al.  On Ramanujan's Q-function , 1995, Journal of Computational and Applied Mathematics.

[19]  N. Temme.  The asymptotic expansion of the incomplete gamma functions : (preprint) , 1977 .