From exact stochastic to mean-field ODE models: a new approach to prove convergence results

Kiss, Istvan and Simon, Peter L (2012) From exact stochastic to mean-field ODE models: a new approach to prove convergence results. IMA Journal of Applied Mathematics. ISSN 0272-4960

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Abstract

In this paper, the rigorous linking of exact stochastic models to mean-field approximations is studied. Using a continuous-time Markov chain, we start from the exact formulation of a simple epidemic model on a certain class of networks, including completely connected and regular random graphs, and rigorously derive the well-known mean-field approximation that is usually justified based on biological hypotheses. We propose a unifying framework that incorporates and discusses the details of two existing proofs and we put forward a new ordinary differential equation (ODE)-based proof. The more well-known proof is based on a first-order partial differential equation approximation, while the other, more technical one, uses Martingale and Semigroup theory. We present the main steps of both proofs to investigate their applicability in different modelling contexts and to make these ideas more accessible to a broader group of applied researchers. The main result of the paper is a new ODE-based proof that may serve as a building block to prove similar convergence results for more complex networks. The new proof is based on deriving a countable system of ODEs for the moments of a distribution of interest and proving a perturbation theorem for this infinite system.

Item Type: Article
Additional Information: Online first edition
Keywords: epidemic model network mean-field approximation countable system of ODEs Markov chain
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Istvan Kiss
Date Deposited: 03 Dec 2012 10:02
Last Modified: 16 Jun 2015 11:06
URI: http://sro.sussex.ac.uk/id/eprint/29821
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