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

Péter L. Simon, Istvan Z. Kiss

Research output: Contribution to journalArticle

14 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)945-964
Number of pages20
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Volume78
Issue number5
DOIs
Publication statusPublished - Oct 1 2013

Keywords

  • Countable system of ODEs
  • Epidemic model
  • Markov chain
  • Mean-field approximation
  • Network

ASJC Scopus subject areas

  • Applied Mathematics

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