Department of Epidemiology, University of Michigan, 1415 Washington Heights, Ann Arbor, MI, 48109, USA.
Bull Math Biol. 2022 Aug 5;84(9):96. doi: 10.1007/s11538-022-01057-9.
The basic reproduction number [Formula: see text] is a fundamental concept in mathematical epidemiology and infectious disease modeling. Loosely speaking, it describes the number of people that an infectious person is expected to infect. The basic reproduction number has profound implications for epidemic trajectories and disease control strategies. It is well known that the basic reproduction number can be calculated as the spectral radius of the next generation matrix, but why this is the case may not be intuitively obvious. Here, we walk through how the discrete, next generation process connects to the ordinary differential equation disease system of interest, linearized at the disease-free equilibrium. Then, we use linear algebra to develop a geometric explanation of why the spectral radius of the next generation matrix is an epidemic threshold. Finally, we work through a series of examples that help to build familiarity with the kinds of patterns that arise in parameter combinations produced by the next generation method. This article is intended to help new infectious disease modelers develop intuition for the form and interpretation of the basic reproduction number in their disease systems of interest.
基本繁殖数 [公式：见正文] 是数学流行病学和传染病建模中的一个基本概念。通俗地说，它描述了一个传染病患者预计会感染的人数。基本繁殖数对传染病的轨迹和疾病控制策略有着深远的影响。众所周知，基本繁殖数可以通过下一代矩阵的谱半径来计算，但为什么会这样可能并不直观。在这里，我们将探讨离散的下一代过程如何与感兴趣的常微分方程疾病系统联系起来，在无病平衡点处线性化。然后，我们使用线性代数来发展一个几何解释，说明为什么下一代矩阵的谱半径是传染病的阈值。最后，我们通过一系列的例子来帮助建立对下一代方法产生的参数组合中出现的模式的熟悉程度。本文旨在帮助新的传染病建模者在他们感兴趣的疾病系统中培养对基本繁殖数的形式和解释的直觉。
