The construction of next-generation matrices for compartmental epidemic models
The construction of next-generation matrices for compartmental epidemic models
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DOI：
10.1098/rsif.2009.0386
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发表时间：
2010-06-06
影响因子：
3.9
通讯作者：
Roberts, M. G.
中科院分区：
文献类型：
--
作者：
Diekmann, O.;Heesterbeek, J. A. P.;Roberts, M. G.
基本繁殖数R(0)可以说是传染病流行病学中最重要的数量。下一代矩阵（NGM）是定义和计算R(0)的自然基础，其中可以识别有限多个不同类别的个体。我们澄清了文献中一直存在的关于这个矩阵的构造的困惑，特别是对于最常用的所谓的区室模型。我们提出了一个详细的简单配方，从直接从模型规格中导出的基本成分构建NGM。我们证明了存在两个相关的矩阵，我们将它们定义为大定义域的NGM和小定义域的NGM。这三个矩阵一起反映了文献中遇到的表征R(0)的可能性范围。我们展示了它们是如何连接的，以及它们的构造如何遵循基本模型成分，并建立了它们具有相同的非零特征值，其中最大的是基本复制数R(0)。虽然我们提出了基于线性代数的正式配方，但我们鼓励通过直接流行病学推理的方式构建NGM，使用对NGM元素和模型成分的清晰解释。我们列举了一些例子作为我们方法的实用指南。在附录中，我们给出了一个初等但完整的证明，即定义为区室系统NGM的显性特征值的R(0)与爆发早期的马尔萨斯参数R(0)通过以下性质相联系：R(0) >当且仅当R > 0， R(0) = 1当且仅当R = 0。
The basic reproduction number R(0) is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of R(0) where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain. The three matrices together reflect the range of possibilities encountered in the literature for the characterization of R(0). We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number R(0). Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that R(0) defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r, the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that R(0) > 1 if and only if r > 0, and R(0) = 1 if and only if r = 0.