Existence of Quasi-Stationary Distributions. A Renewal Dynamical Approach

Ferrari, Pablo A.; Kesten, Harry; Martı́nez, Servet; Picco, Pierre

doi:10.1214/aop/1176988277

Cited by 151 publications

(155 citation statements)

References 24 publications

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“…The notion of a quasi-stationary (QS) distribution has proved to be a powerful tool in this context. For such models, the QS distribution describes the asymptotic (long-time) properties of a finite system conditioned on survival [10,11,12]. The quasi-stationary properties converge to the stationary properties when the system size N → ∞.…”

Section: Introductionmentioning

confidence: 99%

Quasi-stationary distributions for models of heterogeneous catalysis

Oliveira

Dickman

2004

Physica A: Statistical Mechanics and its Applications

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We construct the quasi-stationary (QS) distribution for two models of heterogeneous catalysis having two absorbing states: the ZGB model for the oxidation of CO, and a version with noninstantaneous reactions. Using a mean-field-like approximation, we study the quasi-stationary surface coverages, moment ratios and the lifetime of the QS state. We also derive an improved, consistent one-site mean-field theory for the ZGB model.

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Section: Introductionmentioning

confidence: 99%

Quasi-stationary distributions for models of heterogeneous catalysis

Oliveira

Dickman

2004

Physica A: Statistical Mechanics and its Applications

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“…A deterministic version of our model can be defined by 6) where Nî(t) and Nŵ(t) respectively correspond to the numbers of infected hosts and freeliving infectious stages present at time t ≥ 0 in a population of N hosts. The deterministic system (1.6) has equilibrium points at (î,ŵ) = (0, 0) and…”

Section: We Further Suppose That S(t) = N − I (T)mentioning

confidence: 99%

“…For two-dimensional population processes, such as our SIS/W model, it is often necessary to simply assume the existence of the limiting conditional distribution before considering methods for its approximation (see, for example, Nåsell [12] and Clancy et al [4]). In the present case, our SIS/W process may be treated using the results of Ferrari et al [6]. The details are as follows.…”

Section: Existence Of a Quasi-stationary Distributionmentioning

confidence: 99%

“…In order to apply the results of Ferrari et al [6], we relabel the states by defining X(t) = I (t) + (N + 1)W (t), so that {X(t) : t ≥ 0} is a Markov chain with state space {0, 1, 2, . .…”

mentioning

confidence: 99%

“…Defining R = inf{t ≥ 0 : X(t) = 0}, we first need to check the 'asymptotic remoteness' condition (1.4) of [6]. That is, we require that lim x→∞ Pr(R < t | X(0) = x) = 0 for all t ≥ 0.…”

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confidence: 99%

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A stochastic SIS infection model incorporating indirect transmission

Clancy

2005

J. Appl. Probab.

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We describe a stochastic susceptible-infective-susceptible (SIS) model for transmission of infectious disease through a population, incorporating both direct host-host transmission and indirect transmission via free-living infectious stages (e.g. environmental bacteria). Existence of a quasi-stationary distribution conditional upon nonextinction of infection is established. A bivariate Ornstein-Uhlenbeck approximation is used to investigate the long-term behaviour of the process conditional upon nonextinction of infection. We show that indirect transmission leads to lower variability in the number of infected hosts present in quasi-stationarity and, consequently, to a greater tendency of infection to persist, compared with a model with direct transmission only and the same average individual infectivity. Some numerical work illustrating these results is presented.

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Length Biasing and Laws Equivalent to the Log-Normal

Pakes

1996

Journal of Mathematical Analysis and Applications

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Let X ) 0 denote a generic lifetime of a renewal process having unit mean Ž . lifetime, let X denote the stationary total lifetime and let q g 0, 1 be a fixed constant. We consider anew the scale invariance problem: For which laws does qX have the same distribution as X ? Our setting is more probabilistic than those presented hitherto, and we explore connections with the log-normal moment problem. In particular it is shown that all explicitly known laws which have log-normal moments solve our problem. The notion of remaining lifetime is generalized and its scaling invariance is investigated using the notion of total Ž . lifetime. Two moment equivalent laws of Askey are shown to have a simple representation in terms of laws equivalent to the log-normal. The representation involves a q-gamma law, which we explore in its own right. An affine extension of our basic scale invariance relation, arising in the theory of orthogonal polynomials, is shown to be equivalent to the latter.

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Existence of Quasi-Stationary Distributions. A Renewal Dynamical Approach

Cited by 151 publications

References 24 publications

Quasi-stationary distributions for models of heterogeneous catalysis

Quasi-stationary distributions for models of heterogeneous catalysis

A stochastic SIS infection model incorporating indirect transmission

Length Biasing and Laws Equivalent to the Log-Normal

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