1,445 research outputs found
Quasi-stationary distributions
This paper contains a survey of results related to quasi-stationary distributions, which arise in the setting of stochastic dynamical systems that eventually evanesce, and which may be useful in describing the long-term behaviour of such systems before evanescence. We are concerned mainly with continuous-time Markov chains over a finite or countably infinite state space, since these processes most often arise in applications, but will make reference to results for other processes where appropriate. Next to giving an historical account of the subject, we review the most important results on the existence and identification of quasi-stationary distributions for general Markov chains, and give special attention to birth-death processes and related models. Results on the question of whether a quasi-stationary distribution, given its existence, is indeed a good descriptor of the long-term behaviour of a system before evanescence, are reviewed as well. The paper is concluded with a summary of recent developments in numerical and approximation methods
A scalable computational framework for establishing long-term behavior of stochastic reaction networks
Reaction networks are systems in which the populations of a finite number of
species evolve through predefined interactions. Such networks are found as
modeling tools in many biological disciplines such as biochemistry, ecology,
epidemiology, immunology, systems biology and synthetic biology. It is now
well-established that, for small population sizes, stochastic models for
biochemical reaction networks are necessary to capture randomness in the
interactions. The tools for analyzing such models, however, still lag far
behind their deterministic counterparts. In this paper, we bridge this gap by
developing a constructive framework for examining the long-term behavior and
stability properties of the reaction dynamics in a stochastic setting. In
particular, we address the problems of determining ergodicity of the reaction
dynamics, which is analogous to having a globally attracting fixed point for
deterministic dynamics. We also examine when the statistical moments of the
underlying process remain bounded with time and when they converge to their
steady state values. The framework we develop relies on a blend of ideas from
probability theory, linear algebra and optimization theory. We demonstrate that
the stability properties of a wide class of biological networks can be assessed
from our sufficient theoretical conditions that can be recast as efficient and
scalable linear programs, well-known for their tractability. It is notably
shown that the computational complexity is often linear in the number of
species. We illustrate the validity, the efficiency and the wide applicability
of our results on several reaction networks arising in biochemistry, systems
biology, epidemiology and ecology. The biological implications of the results
as well as an example of a non-ergodic biological network are also discussed.Comment: 31 pages, 9 figure
Ergodicity, Output-Controllability, and Antithetic Integral Control of Uncertain Stochastic Reaction Networks
The ergodicity and the output-controllability of stochastic reaction networks
have been shown to be essential properties to fulfill to enable their control
using, for instance, antithetic integral control. We propose here to extend
those properties to the case of uncertain networks. To this aim, the notions of
interval, robust, sign, and structural ergodicity/output-controllability are
introduced. The obtained results lie in the same spirit as those obtained in
[Briat, Gupta & Khammash, Cell Systems, 2016] where those properties are
characterized in terms of control theoretic concepts, linear algebraic
conditions, linear programs, and graph-theoretic/algebraic conditions. An
important conclusion is that all those properties can be characterized by
linear programs. Two examples are given for illustration.Comment: 29 pages. arXiv admin note: text overlap with arXiv:1703.0031
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