Quasi-stationary distributions for discrete-state models

Doorn, Erik A. van; Pollett, P. K.

doi:10.1016/j.ejor.2013.01.032

Cited by 115 publications

(120 citation statements)

References 128 publications

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“…[23][24][25][26] and references therein). Having two processes in the game, we can distinguish three possible outcomes: any of the processes wins or we have a dynamical equilibrium between them.…”

Section: Discussionmentioning

confidence: 99%

Competing contact processes in the Watts-Strogatz network

2016

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Abstract. We investigate two competing contact processes on a set of Watts-Strogatz networks with the clustering coefficient tuned by rewiring. The base for network construction is one-dimensional chain of N sites, where each site i is directly linked to nodes labelled as i ± 1 and i ± 2. So initially, each node has the same degree ki = 4. The periodic boundary conditions are assumed as well. For each node i the links to sites i + 1 and i + 2 are rewired to two randomly selected nodes so far not-connected to node i. An increase of the rewiring probability q influences the nodes degree distribution and the network clusterization coefficient C. For given values of rewiring probability q the set N (q) = {N1, N2, . . . , NM } of M networks is generated. The network's nodes are decorated with spin-like variables si ∈ {S, D}. During simulation each S node having a D-site in its neighbourhood converts this neighbour from D to S state. Conversely, a node in D state having at least one neighbour also in state D-state converts all nearest-neighbours of this pair into D-state. The latter is realized with probability p. We plot the dependence of the nodes S final density n .

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Section: Discussionmentioning

confidence: 99%

Competing contact processes in the Watts-Strogatz network

2016

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“…For a review of results related to quasi-stationary distributions and two alternative choices (i.e., the ratio of means distribution and a doubly-limiting conditional distribution) see van Doorn and Pollett [33], and Darroch and Seneta [10, Sections 2 and 4], respectively. In the case of an absorbing CTMC with a single communicating class of transient states, which is our case here, the sub-matrix Q * recording transition rates among transient states has all its eigenvalues with negative real parts.…”

Section: The Special Case λ I = 0 Extinction Times and Quasi-stationmentioning

confidence: 99%

A stochastic SIS epidemic model with heterogeneous contacts

Economou

Gómez‐Corral

López‐García

2015

Physica A: Statistical Mechanics and its Applications

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A stochastic model for the spread of an SIS epidemic among a population consisting of N individuals, each having heterogeneous infectiousness and/or susceptibility, is considered and its behavior is analyzed under the practically relevant situation when N is small. The model is formulated as a finite timehomogeneous continuous-time Markov chain X . Based on an appropriate labeling of states, we first construct its infinitesimal rate matrix by using an iterative argument, and we then present an algorithmic procedure for computing steadystate measures, such as the number of infected individuals, the length of an outbreak, the maximum number of infectives, and the number of infections suffered by a marked individual during an outbreak. The time till the epidemic extinction is characterized as a phase-type random variable when there is no external source of infection, and its Laplace-Stieljtes transform and moments are derived in terms of a forward elimination backward substitution solution. The inverse iteration method is applied to the quasi-stationary distribution of X , which provides a good approximation of the process X at a certain time, conditional on non-extinction, after a suitable waiting time. The basic reproduction number R 0 is defined here as a random variable, rather than an expected value.

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“…This is particularly relevant in the setting of a birth-death process for which absorption at -1 is certain (that is, in view of (4), the setting of Theorem 1), since positivity of the decay parameter is necessary and sufficient for the existence of a quasistationary distribution (see [10,Section 5.1] for detailed information).…”

Section: Introductionmentioning

confidence: 99%

Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

Doorn

2015

Journal of Applied Probability

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Abstract. We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, . . . }, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer Theorem for eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

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Quasi-stationary distributions for discrete-state models

Cited by 115 publications

References 128 publications

Competing contact processes in the Watts-Strogatz network

Competing contact processes in the Watts-Strogatz network

A stochastic SIS epidemic model with heterogeneous contacts

Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

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