Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T04:36:34.401Z Has data issue: false hasContentIssue false
  • English
  • Français

The quasistationary distribution of the stochastic logistic model

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Otso Ovaskainen
Otso Ovaskainen*
Affiliation:
University of Helsinki
Get access Rights & Permissions [Opens in a new window]

Abstract

The stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology. Among the model predictions, the quasistationary distribution and the mean time to extinction are of major interest for most applications, and a number of approximation formulae for these quantities have been derived. In this paper, previous approximation formulae are improved for two mathematically tractable cases: at the limit of the number of individuals N → ∞ (with relative error of the approximations of the order 𝒪(1/N)), and at the limit of the basic reproduction ratio R0 → ∞ (with relative error of the approximations of the order 𝒪(1/R0)). The mathematically rigorous formulae are then extended heuristically for other values of N and R0 > 1.

Keywords

Stochastic logistic modelSIS modeltime to extinctionquasistationary distribution

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 898 - 907
Copyright
Copyright © by the Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andersson, H., and Djehiche, B. (1998). A threshold limit theorem for the stochastic logistic epidemic. J. Appl. Prob. 35, 662–670.Google Scholar
Bartholomew, D. J. (1976). Continuous time diffusion models with random duration of interest. J. Math. Sociol. 4, 187–199.CrossRefGoogle Scholar
Darroch, J. N., and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196.Google Scholar
Jacquez, J. A., and Simon, C. P. (1993). The stochastic SI model with recruitment and deaths I. Comparison with the closed SIS model. Math. Biosci. 117, 77–125.Google Scholar
Kryscio, R. J. and Lefèvre, C. (1989). On the extinction of the S-I-S stochastic logistic epidemic. J. Appl. Prob. 26, 685–694.CrossRefGoogle Scholar
Levins, R. (1969). Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Amer. 15, 237–240.Google Scholar
Nåsell, I. (1995). The threshold concept in deterministic and stochastic models. In Epidemic Models: Their Structure and Relation to Data, ed. Mollison, D., Cambridge University Press.Google Scholar
Nåsell, I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Prob. 28, 895–932.Google Scholar
Nåsell, I. (1999). On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156, 21–40.CrossRefGoogle ScholarPubMed
Norden, R. H. (1982). On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Prob. 14, 687–708.CrossRefGoogle Scholar
Oppenheim, I., Shuler, K. E., and Weiss, G. H. (1977). Stochastic theory of nonlinear rate processes with multiple stationary states. Physica A 88, 191–214.Google Scholar
Weiss, G. H., and Dishon, M. (1971). On the asymptotic behavior of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261–265.Google Scholar