A threshold limit theorem for the stochastic logistic epidemic

Abstract

The time until extinction for the closed SIS stochastic logistic epidemic model is investigated. We derive the asymptotic distribution for the extinction time as the population grows to infinity, under different initial conditions and for different values of the infection rate.

Exact epidemic models on graphs using graph-automorphism driven lumping

“…The S I S type dynamics (Anderson and May 1991) is considered on a network with N nodes and with adjacency matrix G = (g i j ) i, j=1,2,...,N ∈ {0, 1} N 2 where g i j = 1 if node i and j are connected, and g i j = 0 otherwise. Here, we only consider networks with bi-directional edges and without self-loops.…”

“…county etc), disease characteristics and various control programmes that are aimed at halting disease transmission or bringing infection prevalence to as low a level as possible (Anderson and May 1991). The main aim of many models is to gain insight into how diseases transmit and to identify the most effective strategies for their prevention and control.…”

“…The early work by Kermack and McKendrick (1927) forms the basis of differential-equation-based models which lie at the heart of modern quantitative epidemiology. Traditionally, mathematical epidemiology is based on differential equation models (Anderson and May 1991;Diekmann and Heesterbeek 2000) and these operate on the basis of some strong simplifying assumptions about the behaviour of the individuals and the biology of the disease. While these models do not explicitly model contact-network structure, they allow us to improve our understanding of the disease transmission process and to derive threshold quantities such as the basic reproduction number R 0 and critical vaccination threshold (Diekmann and Heesterbeek 2000) in terms of key biological parameters such as rate of infection and infectious period.…”

The intermediate dispersal principle in spatially explicit metapopulations

“…As for the stochastic logistic model, according to Na˚sell (2001) its study goes back to Feller (1939); however, approximations for the mean time to extinction have been determined only recently (Na˚sell, 1996;Andersson and Djehiche, 1998;Na˚sell, 2001;Ovaskainen, 2001). In particular, Na˚sell (2001) showed that the mean time to extinction starting from quasi-stationarity (that is, the distribution of local abundance conditional on non-extinction) increases exponentially with the carrying capacity K and scales as 1=r if the corresponding deterministic model corresponds to a viable population, namely if r ¼ nð0Þ À mð0Þ40.…”

The limiting behaviour of a mainland-island metapopulation

“…The stochastic logistic model provides a typical example. Under these simplifying assumptions, results have been obtained concerning the quasi-stationary distribution (Ovaskainen 2001), time to extinction of the metapopulation (Andersson and Djehiche 1998) and the limiting behaviour (Ethier and Kurtz 1986;Pollett 2001) which connects the stochastic logistic model to Levins's model (1969). More recently, researchers have tried to bring the modelling assumptions closer to the ecological reality.…”