PDE approximation of large systems of differential equations

Bátkai, András; Havasi, Ágnes; Horváth, Róbert; Kunszenti-Kovács, Dávid; Šimon, Peter

doi:10.7153/oam-09-08

Cited by 4 publications

(6 citation statements)

References 11 publications

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“…For the large N limit to exist, it is known [21,23,11,29,4] that the rates of the master equation need to satisfy the following density condition:…”

Section: Fokker Planck Equation As a Limit Of The Birth-death Processmentioning

confidence: 99%

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PDE-limits of stochastic SIS epidemics on networks

Lauro¹,

Croix²,

Berthouze³

et al. 2020

Preprint

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Stochastic epidemic models on networks are inherently high-dimensional and the resulting exact models are intractable numerically even for modest network sizes. Mean-field models provide an alternative but can only capture average quantities, thus offering little or no information about variability in the outcome of the exact process. In this paper we conjecture and numerically prove that it is possible to construct PDE-limits of the exact stochastic SIS epidemics on regular and Erdős-Rényi networks. To do this we first approximate the exact stochastic process at population level by a Birth-and-Death process (BD) (with a state space of O(N ) rather than O(2 N )) whose coefficients are determined numerically from Gillespie simulations of the exact epidemic on explicit networks. We numerically demonstrate that the coefficients of the resulting BD process are density-dependent, a crucial condition for the existence of a PDE limit. Extensive numerical tests for Regular and Erdős-Rényi networks show excellent agreement between the outcome of simulations and the numerical solution of the Fokker-Planck equations. Apart from a significant reduction in dimensionality, the PDE also provides the means to derive the epidemic outbreak threshold linking network and disease dynamics parameters, albeit in an implicit way. Perhaps more importantly, it enables the formulation and numerical evaluation of likelihoods for epidemic and network inference as illustrated in a fully worked out example.

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“…For the large N limit to exist, it is known [21,23,11,29,4] that the rates of the master equation need to satisfy the following density condition:…”

Section: Fokker Planck Equation As a Limit Of The Birth-death Processmentioning

confidence: 99%

“…In the density-dependent case, it can be shown [21,29,4] that f (t, x) satisfies the following forward Fokker-Planck equation:…”

Section: Fokker Planck Equation As a Limit Of The Birth-death Processmentioning

confidence: 99%

PDE-limits of stochastic SIS epidemics on networks

Lauro¹,

Croix²,

Berthouze³

et al. 2020

Preprint

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“…Following the papers [3,7], we will use systems with only two particle states Q and T . Thus the state space of the whole system will be {0, 1, .…”

Section: Model Formulationmentioning

confidence: 99%

“…The aim of this paper is to expand the ideas presented in [3] and [7], where operator semigroup methods were used to provide a second-order PDE that can be used as a good approximation to a large Markovian stochastic system of two-state particles. Both papers provided a rough sketch of how to obtain bounds on the error of approximation.…”

Section: Introductionmentioning

confidence: 99%

“…Then the master equations can be considered to be the discretisation of the Fokker-Planck equation in an appropriate sense. This was presented in [3], but since the main focus was a different type of second order PDE, very little detail was provided for the case we wish to study here, and some statements were inaccurately formulated. In [7], an overview of the main building blocks of the proofs were given, but the details were left for this present paper.…”

Section: Introductionmentioning

confidence: 99%

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On the error of Fokker-Planck approximations of some one-step density dependent processes

Kunszenti-Kovács¹

2016

Preprint

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Using operator semigroup methods, we show that Fokker-Planck type secondorder PDE-s can be used to approximate the evolution of the distribution of a one-step process on N particles governed by a large system of ODEs. The error bound is shown to be of order O(1/N ), surpassing earlier results that yielded this order for the error only for the expected value of the process, through mean-field approximations. We also present some conjectures showing that the methods used have the potential to yield even stronger bounds, up to O(1/N 3 ).

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