Extinction and Quasi-stationarity in the Verhulst Logistic Model

Nåsell, Ingemar

doi:10.1006/jtbi.2001.2328

Cited by 166 publications

(136 citation statements)

References 34 publications

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“…where λ ext (t) := χ(1)P 1 (t) (Bailey, 1964;Nåsell, 2001). The variable λ ext (t) is an extinction rate in the sense that the conditional probability of extension in an "infinitesimal" interval (t,t + dt] is given by…”

Section: Transient Distributionsmentioning

confidence: 99%

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Singh

Hespanha

2007

Bull. Math. Biol.

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Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of degree higher than n. For analysis purpose, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function.In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at-least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximation of the actual moment dynamics.A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.

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Section: Transient Distributionsmentioning

confidence: 99%

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Singh

Hespanha

2007

Bull. Math. Biol.

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“…A particular example of birth-death processes is, for instance, the classical SIS model, with λ i = βi(N − i)/N for the birth rate and μ i = γ i for the death rate, where β is the contact rate and γ is the recovery rate per individual. A more general model is the Verhulst model (see [29] for details), with infection rates…”

Section: Birth-death Processmentioning

confidence: 99%

“…As is to be expected, the value of f L i (0) obtained numerically is in agreement with the theoretical Tauberian result given in Section 2.1. In order to explain why the density f L i (x) is not bimodal, we recall that the extinction time of the SIS model, when the epidemic persists for a very long time, can be approximated by an exponential distribution with a very small rate [29,32]. The flat decreasing shape of such an exponential density helps to explain why the overall density of the extinction time is not bimodal.…”

Section: Sis Model: Analysis Of the Modesmentioning

confidence: 99%

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Stochastic epidemic models revisited: analysis of some continuous performance measures

Artalejo

Economou

Lopez-Herrero

2012

Journal of Biological Dynamics

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We deal with stochastic epidemic models having a set of absorbing states. The aim of the paper is to study some continuous characteristics of the epidemic. In this sense, we first extend the classical study of the length of an outbreak by investigating the whole probability distribution of the extinction time via Laplace transforms. Moreover, we also study two almost new epidemic descriptors, namely, the time until a non-infected individual becomes infected and the time until the individual is removed from the infective group. The obtained results are illustrated by numerical examples including an application to a stochastic SIS model for head lice infections.

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“…Two widely used methods are moment closure techniques Nåsell 2001Nåsell , 2002Nåsell , 2003a and diffusion approximations (Kurtz 1970(Kurtz , 1971Pollett 1990; van Kampen 1992;Jacquez & Simon 1993;Andersson & Britton 2000;Ross 2006a,b). However, such methods are generally most accurate when the population size is sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

On methods for studying stochastic disease dynamics

Keeling

Ross

2007

J. R. Soc. Interface.

174

182

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Models that deal with the individual level of populations have shown the importance of stochasticity in ecology, epidemiology and evolution. An increasingly common approach to studying these models is through stochastic (event-driven) simulation. One striking disadvantage of this approach is the need for a large number of replicates to determine the range of expected behaviour. Here, for a class of stochastic models called Markov processes, we present results that overcome this difficulty and provide valuable insights, but which have been largely ignored by applied researchers. For these models, the so-called Kolmogorov forward equation (also called the ensemble or master equation) allows one to simultaneously consider the probability of each possible state occurring. Irrespective of the complexities and nonlinearities of population dynamics, this equation is linear and has a natural matrix formulation that provides many analytical insights into the behaviour of stochastic populations and allows rapid evaluation of process dynamics. Here, using epidemiological models as a template, these ensemble equations are explored and results are compared with traditional stochastic simulations. In addition, we describe further advantages of the matrix formulation of dynamics, providing simple exact methods for evaluating expected eradication (extinction) times of diseases, for comparing expected total costs of possible control programmes and for estimation of disease parameters.

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Extinction and Quasi-stationarity in the Verhulst Logistic Model

Cited by 166 publications

References 34 publications

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Stochastic epidemic models revisited: analysis of some continuous performance measures

On methods for studying stochastic disease dynamics

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