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首页> 外文期刊>Mathematical Biosciences: An International Journal >Complete hierarchies of SIR models on arbitrary networks with exact and approximate moment closure
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Complete hierarchies of SIR models on arbitrary networks with exact and approximate moment closure

机译：任意网络上具有精确和近似矩闭合的SIR模型的完整层次

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摘要

We first generalise ideas discussed by Kiss et al. (2015) to prove a theorem for generating exact closures (here expressing joint probabilities in terms of their constituent marginal probabilities) for susceptible-infectious-removed (SIR) dynamics on arbitrary graphs (networks). For Poisson transmission and removal processes, this enables us to obtain a systematic reduction in the number of differential equations needed for an exact 'moment closure' representation of the underlying stochastic model. We define 'transmission blocks' as a possible extension of the block concept in graph theory and show that the order at which the exact moment closure representation is curtailed is the size of the largest transmission block. More generally, approximate closures of the hierarchy of moment equations for these dynamics are typically defined for the first and second order yielding mean-field and pairwise models respectively. It is frequently implied that, in principle, closed models can be written down at arbitrary order if only we had the time and patience to do this. However, for epidemic dynamics on networks, these higher-order models have not been defined explicitly. Here we unambiguously define hierarchies of approximate closed models that can utilise subsystem states of any order, and show how well-known models are special cases of these hierarchies. (C) 2015 The Authors. Published by Elsevier Inc.
机译：我们首先概括一下Kiss等人讨论的想法。 （2015年）证明了一个定理，用于为任意图（网络）上的易感性感染去除（SIR）动力学生成精确的闭包（此处以其构成的边际概率表示联合概率）。对于泊松传递和去除过程，这使我们能够获得系统地减少潜在随机模型的精确“矩闭合”表示所需的微分方程的数量。我们将“传输块”定义为图论中块概念的可能扩展，并表明减少精确矩闭合表示的顺序是最大传输块的大小。更一般而言，通常针对一阶和二阶屈服均值场模型和成对模型分别定义这些动力学矩方程的近似闭合。经常暗示，原则上，只要我们有时间和耐心这样做，就可以以任意顺序写下封闭模型。但是，对于网络上的流行病动态，尚未明确定义这些高阶模型。在这里，我们明确定义了可以使用任何顺序的子系统状态的近似封闭模型的层次结构，并说明了众所周知的模型是这些层次结构的特殊情况的方式。 （C）2015作者。由Elsevier Inc.发布

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