Hurtado Paul J, Richards Cameron
Department of Mathematics and Statistics, University of Nevada-Reno, 1664 North Virginia Street; University of Nevada, Reno/0084, Reno, 89557-0001, NV, USA.
Bull Math Biol. 2025 Jun 3;87(7):89. doi: 10.1007/s11538-025-01467-5.
Reproduction numbers, like the basic reproduction number , play an important role in the analysis and application of dynamic models, including contagion models and ecological population models. One difficulty in deriving these quantities is that they must be computed on a model-by-model basis, since it is typically impractical to obtain general reproduction number expressions applicable to a family of related models, especially if these are of different dimensions (i.e., differing numbers of state variables). For example, this is typically the case for SIR-type infectious disease models derived using the linear chain trick. Here we show how to find general reproduction number expressions for such model families (which vary in their number of state variables) using the next generation operator approach in conjunction with the generalized linear chain trick (GLCT). We further show how the GLCT enables modelers to draw insights from these results by leveraging theory and intuition from continuous time Markov chains (CTMCs) and their absorption time distributions (i.e., phase-type probability distributions). To do this, we first review the GLCT and other connections between mean-field ODE model assumptions, CTMCs, and phase-type distributions. We then apply this technique to find reproduction numbers for two sets of models: a family of generalized SEIRS models of arbitrary finite dimension, and a generalized family of finite dimensional predator-prey (Rosenzweig-MacArthur type) models. These results highlight the utility of the GLCT for the derivation and analysis of mean field ODE models, especially when used in conjunction with theory from CTMCs and their associated phase-type distributions.
繁殖数，如基本繁殖数(R_0)，在动态模型的分析和应用中起着重要作用，这些动态模型包括传染病模型和生态种群模型。推导这些量的一个困难在于，它们必须逐个模型地计算，因为通常不可能获得适用于一族相关模型的一般繁殖数表达式，特别是如果这些模型具有不同的维度（即状态变量的数量不同）。例如，使用线性链技巧推导的SIR型传染病模型通常就是这种情况。在这里，我们展示了如何使用下一代算子方法结合广义线性链技巧（GLCT）来找到此类模型族（其状态变量数量不同）的一般繁殖数表达式。我们进一步展示了GLCT如何使建模者能够通过利用连续时间马尔可夫链（CTMC）及其吸收时间分布（即相位型概率分布）的理论和直觉，从这些结果中获得见解。为此，我们首先回顾GLCT以及平均场常微分方程模型假设、CTMC和相位型分布之间的其他联系。然后，我们应用这种技术来找到两组模型的繁殖数：一族任意有限维度的广义SEIRS模型，以及一族有限维捕食者 - 猎物（Rosenzweig - MacArthur型）模型。这些结果突出了GLCT在推导和分析平均场常微分方程模型方面的效用，特别是当它与CTMC及其相关的相位型分布的理论结合使用时。
