From exact stochastic to mean-field ODE models: a new approach to prove convergence results
From exact stochastic to mean-field ODE models: a new approach to prove convergence results
复制标题
从精确随机模型到平均场 ODE 模型：证明收敛结果的新方法
DOI：
10.1093/imamat/hxs001
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发表时间：
2012
影响因子：
1.2
通讯作者：
Simon P
中科院分区：
文献类型：
--
作者：
Simon P
本文研究了精确随机模型与平均场近似的严格联系。利用连续时间马尔可夫链，我们从一类网络（包括完全连通和正则随机图）上的简单流行病模型的精确公式出发，严格推导出通常基于生物学假设的众所周知的平均场近似。我们提出了一个统一的框架，结合并讨论了两个现有证明的细节，并提出了一个新的基于常微分方程（ODE）的证明。更著名的证明是基于一阶偏微分方程近似，而另一个更专业的证明，使用了鞅和半群理论。我们提出了两种证明的主要步骤，以研究它们在不同建模环境中的适用性，并使这些想法更容易被更广泛的应用研究人员所接受。本文的主要成果是一个新的基于ode的证明，它可以作为一个构建块来证明更复杂网络的类似收敛结果。新的证明是基于对一个兴趣分布的矩导出一个可数的ode系统，并证明了这个无穷系统的一个摄动定理。
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.
生物生长和传播
影响因子：
1.7
作者：
W. Jäger;H. Rost;P. Tăutu
通讯作者：
P. Tăutu
影响因子：
4.3
作者：
Ball, Frank;Neal, Peter
通讯作者：
Neal, Peter