Rare-event extinction on stochastic networks

Lindley, Brandon; Shaw, Leah B.; Schwartz, Ira B.

doi:10.1209/0295-5075/108/58008

Cited by 20 publications

(21 citation statements)

References 38 publications

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“…Secondly, to evaluate the constant A in the dominant exponential term will, in general, require the solution of a boundary value problem for a 2k dimensional ordinary differential equation system, where k is the dimensionality of the original model. Recent work on numerical approaches to this problem includes [28,22], where systems in k = 2, 3 dimensions are analysed. An alternative is to seek approximations valid within certain regions of parameter space, such as the 'adiabatic approximation' of [13], where (for their model of interest) a simple explicit expression for A valid for R 0 close to 1 is derived.…”

Section: Discussionmentioning

confidence: 99%

Approximating Time to Extinction for Endemic Infection Models

Clancy

Tjia

2018

Methodol Comput Appl Probab

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Approximating the time to extinction of infection is an important problem in infection modelling. A variety of different approaches have been proposed in the literature. We study the performance of a number of such methods, and characterize their performance in terms of simplicity, accuracy, and generality. To this end, we consider first the classic stochastic susceptible-infected-susceptible (SIS) model, and then a multidimensional generalization of this which allows for Erlang distributed infectious periods. We find that (i) for a below-threshold infection initiated by a small number of infected individuals, approximation via a linear branching process works well; (ii) for an above-threshold infection initiated at endemic equilibrium, methods from Hamiltonian statistical mechanics yield correct asymptotic behaviour as population size becomes large; (iii) the widely-used Ornstein-Uhlenbeck diffusion approximation gives a very poor approximation, but may retain some value for qualitative comparisons in certain cases; (iv) a more detailed diffusion approximation can give good numerical approximation in certain circumstances, but does not provide correct large population asymptotic behaviour, and cannot be relied upon without some form of external validation (eg simulation studies).

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Section: Discussionmentioning

confidence: 99%

Approximating Time to Extinction for Endemic Infection Models

Clancy

Tjia

2018

Methodol Comput Appl Probab

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“…Ref. [138], the recent years have witnessed a surge of works on the long-time population dynamics on degree-heterogenous networks with strong heterogeneity, such as scale-free networks [139]. Refs.…”

Section: Population Extinction On Heterogeneous Networkmentioning

confidence: 99%

WKB theory of large deviations in stochastic populations

Assaf¹,

Meerson²

2017

J. Phys. A: Math. Theor.

169

229

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Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and people. Large deviations in stochastic population dynamics -such as those determining population extinction, fixation or switching between different statesare presently in a focus of attention of statistical physicists. We review recent progress in applying different variants of dissipative WKB approximation (after Wentzel, Kramers and Brillouin) to this class of problems. The WKB approximation allows one to evaluate the mean time and/or probability of population extinction, fixation and switches resulting from either intrinsic (demographic) noise, or a combination of the demographic noise and environmental variations, deterministic or random. We mostly cover well-mixed populations, single and multiple, but also briefly consider populations on heterogeneous networks and spatial populations. The spatial setting also allows one to study large fluctuations of the speed of biological invasions. Finally, we briefly discuss possible directions of future work.

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“…In the future, it would be desirable to consider the effect of interacting patterns among agents on the extinction of the generalized SIS model [24,25].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Epidemic extinction in a generalized susceptible-infected-susceptible model

Chen¹,

Huang

Zhang

et al. 2017

J. Stat. Mech.

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We study the extinction of epidemics in a generalized susceptible-infected-susceptible model, where a susceptible individual becomes infected with the rate λ when contacting m infective individual(s) simultaneously, and an infected individual spontaneously recovers with the rate µ. By employing the Wentzel-Kramers-Brillouin approximation for the master equation, the problem is reduced to finding the zero-energy trajectories in an effective Hamiltonian system, and the mean extinction time T depends exponentially on the associated action S and the size of the population N , T ∼ exp(N S). Because of qualitatively different bifurcation features for m = 1 and m ≥ 2, we derive independently the expressions of S as a function of the rescaled infection rate λ/µ. For the weak infection, S scales to the distance to the bifurcation with an exponent 2 for m = 1 and 3/2 for m ≥ 2. Finally, a rare-event simulation method is used to validate the theory.

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Rare-event extinction on stochastic networks

Cited by 20 publications

References 38 publications

Approximating Time to Extinction for Endemic Infection Models

Approximating Time to Extinction for Endemic Infection Models

WKB theory of large deviations in stochastic populations

Epidemic extinction in a generalized susceptible-infected-susceptible model

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