具有参数异质性的流行病模型：封闭群体的确定性理论
Epidemiological models with parametric heterogeneity: Deterministic theory for closed populations
摘要：
我们提出了一个统一的数学方法，流行病模型的参数异质性，即，这些模型将群体中的个体描述为具有特定的参数（性状）值，这些参数值在个体之间存在差异。这是一个自然的建模框架，例如，个体易感性或传染性的异质性。我们审查，随着必要的理论，使用所讨论的方法获得的结果。特别是，我们制定和分析SIR模型分布的易感性和传染性，表明封闭人群的流行病学模型非常适合建议的框架。一些已知的结果，从文献中导出，包括最终的流行病规模方程的SIR模型分布的易感性。它被证明，自下而上的方法的理论与参数异质性的异质种群允许推断的人口水平的描述，这是以前使用的没有一个坚实的机械基础，特别是，幂律传递函数被证明是一个后果的初始伽马分布的易感性和传染性。我们讨论了如何将一般理论应用于建模目标，包括异质接触人口结构，并提供分析的SI模型与异质接触。最后，我们与一些开放的问题和有前途的方向，异质种群的理论可以导致重要的简化和概括。
英文摘要：
We present a unified mathematical approach to epidemiological models with parametric heterogeneity, i.e., to the models that describe individuals in the population as having specific parameter (trait) values that vary from one individuals to another. This is a natural framework to model, e.g., heterogeneity in susceptibility or infectivity of individuals. We review, along with the necessary theory, the results obtained using the discussed approach. In particular, we formulate and analyze an SIR model with distributed susceptibility and infectivity, showing that the epidemiological models for closed populations are well suited to the suggested framework. A number of known results from the literature is derived, including the final epidemic size equation for an SIR model with distributed susceptibility. It is proved that the bottom up approach of the theory of heterogeneous populations with parametric heterogeneity allows to infer the population level description, which was previously used without a firm mechanistic basis; in particular, the power law transmission function is shown to be a consequence of the initial gamma distributed susceptibility and infectivity. We discuss how the general theory can be applied to the modeling goals to include the heterogeneous contact population structure and provide analysis of an SI model with heterogeneous contacts. We conclude with a number of open questions and promising directions, where the theory of heterogeneous populations can lead to important simplifications and generalizations.
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