Converging towards the optimal path to extinction

Schwartz, Ira B.; Forgoston, Eric; Bianco, Simone; Shaw, Leah B.

doi:10.1098/rsif.2011.0159

Cited by 42 publications

(49 citation statements)

References 53 publications

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“…More information on the above procedure can be found in [35,38]. The method has also been applied to classic epidemiological models [369] and to a variant of the Verhulst logistic model [370] (exact results on this model with added immigration have been obtained in [371] using the generating function technique).…”

Section: Wkb Approximations and Related Approachesmentioning

confidence: 99%

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Master equations and the theory of stochastic path integrals

2017

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Abstract. This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of masters equations most often rely on lownoise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a "generating functional", which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a "forward" and a "backward" path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit. arXiv:1609.02849v2 [cond-mat...

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Section: Wkb Approximations and Related Approachesmentioning

confidence: 99%

“…A HamiltonJacobi equation can be solved by the method of characteristics [368], with the characteristic curves q(s) and x(s) obeying Hamilton's equations 2 ) to the "passive state" (0, 0) constitutes the "optimal path to extinction" (see below) [35,38,369].…”

Section: Wkb Approximations and Related Approachesmentioning

confidence: 99%

Master equations and the theory of stochastic path integrals

2017

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“…However, if the typical population size is not large, internal fluctuations can lead to the extinction of the population [14]. The effects of internal fluctuations have been studied in predator-prey models [15,16], epidemic models [17][18][19][20][21][22][23], cell biology [24], and ecological systems [13]. In particular, extinction of a stochastic population [11,25,26], which is a crucial concern for population biology [27] and epidemiology [28,29], has also attracted scrutiny in cell biochemistry [30] and in physics [31,32].…”

Section: Introductionmentioning

confidence: 99%

Stochastic dynamics and logistic population growth

et al. 2015

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The Verhulst model is probably the best known macroscopic rate equation in population ecology. It depends on two parameters, the intrinsic growth rate and the carrying capacity. These parameters can be estimated for different populations and are related to the reproductive fitness and the competition for limited resources, respectively. We investigate analytically and numerically the simplest possible microscopic scenarios that give rise to the logistic equation in the deterministic mean-field limit. We provide a definition of the two parameters of the Verhulst equation in terms of microscopic parameters. In addition, we derive the conditions for extinction or persistence of the population by employing either the "momentum-space" spectral theory or the "real-space" Wentzel-KramersBrillouin (WKB) approximation to determine the probability distribution function and the mean time to extinction of the population. Our analytical results agree well with numerical simulations.

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“…In Figure 7b, the analytically determined optimal path is overlaid and is observed to correspond to the ridge of maximal FTLE values. We have published two papers on this method for locating the optimal path to extinction [2,13].…”

Section: Epidemic Extinctionmentioning

confidence: 99%

Adaptive Networks Foundations: Modeling, Dynamics, and Applications

Shaw¹

2013

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The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. We are studying adaptive social networks, focusing on spread of infectious disease as our primary example and including terrorist recruitment as an additional example. In an adaptive network, individuals change their social connections in response to their neighbors' characteristics, and these changes in network topology affect subsequent properties of the individuals. The network adaptation can be disease avoidance or connecting to potential recruits. Major goals of the project included extending previous models to incorporate more realistic network structure, ABSTRACTWe are studying adaptive social networks, focusing on spread of infectious disease as our primary example and including terrorist recruitment as an additional example. In an adaptive network, individuals change their social connections in response to their neighbors' characteristics, and these changes in network topology affect subsequent properties of the individuals. The network adaptation can be disease avoidance or connecting to potential recruits. Major goals of the project included extending previous models to incorporate more realistic network structure, adding spread of information that affects human behavior, studying the extinction of diseases, developing control strategies for epidemics on adaptive networks, and developing tools to analyze and monitor adaptive network properties. We have extended models to include network community structure, information spread, and more realistic social adaptation. We developed the first adaptive network model for terrorist recruitment. Our analytic work includes new techniques for predicting extinction rates of epidemics and the trajectory to extinction, methods to apply this to extinction on a network, and new moment closure approximation techniques that lead to more accurate predictions. For monitoring and control, we developed a method to quantify network adaptation and studied vaccine control for epidemics in adaptive networks.(a) Papers published in peer-reviewed journals (N/A for none)Enter List of papers submitted or published that acknowledge ARO support from the start of the project to the date of this printing. List the papers, including journal references, in the following categories:

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Converging towards the optimal path to extinction

Cited by 42 publications

References 53 publications

Master equations and the theory of stochastic path integrals

Master equations and the theory of stochastic path integrals

Stochastic dynamics and logistic population growth

Adaptive Networks Foundations: Modeling, Dynamics, and Applications

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