The quasistationary distribution of the stochastic logistic model

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.