### 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 language | English |
---|---|

Pages (from-to) | 945-964 |

Number of pages | 20 |

Journal | IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) |

Volume | 78 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**From exact stochastic to mean-field ODE models : A new approach to prove convergence results.** / Simon, L. P.; Kiss, Istvan Z.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - From exact stochastic to mean-field ODE models

T2 - A new approach to prove convergence results

AU - Simon, L. P.

AU - Kiss, Istvan Z.

PY - 2013/10

Y1 - 2013/10

N2 - 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.

AB - 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.

KW - Countable system of ODEs

KW - Epidemic model

KW - Markov chain

KW - Mean-field approximation

KW - Network

UR - http://www.scopus.com/inward/record.url?scp=84885010813&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84885010813&partnerID=8YFLogxK

U2 - 10.1093/imamat/hxs001

DO - 10.1093/imamat/hxs001

M3 - Article

AN - SCOPUS:84885010813

VL - 78

SP - 945

EP - 964

JO - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

SN - 0272-4960

IS - 5

ER -