Building mean field ODE models using the generalized linear chain trick & Markov chain theory.

IF 1.8 4区 数学 Q3 ECOLOGY Journal of Biological Dynamics Pub Date : 2021-05-01 Epub Date: 2021-04-13 DOI:10.1080/17513758.2021.1912418
Paul J Hurtado, Cameron Richards
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引用次数: 5

Abstract

The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on [0,) that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.

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利用广义线性链技巧和马尔可夫链理论建立平均场ODE模型。
众所周知的线性链技巧(LCT)允许建模者通过在子状态链中依次转换个体,得出假设伽马(Erlang)分布通过时间的平均场ode。在这些子状态中花费的时间是k个指数分布的随机变量的和,因此是伽马分布的。广义线性链技巧(GLCT)将该技术扩展到更广泛的相类型分布族,其中包括指数分布、Erlang分布、次指数分布和Coxian分布。相型分布是在[0,∞)上的矩阵指数分布族,表示有限状态连续时间马尔可夫链(ctmc)的吸收时间分布。在这里,我们回顾了ctmc和相位类型分布,然后说明了如何使用GLCT从底层随机模型假设有效地构建ODE模型。利用GLCT对Rosenzweig-MacArthur捕食者-猎物模型和SEIR模型进行了推广,引入了两个新的模型族。我们通过多个示例说明了这种模型可以捕获的各种复杂性。我们还展示了使用基于glct的模型公式来加速此类模型数值解的计算的好处。这些结果突出了使用GLCT从第一原理派生ODE模型的直观性质和实用性。
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来源期刊
Journal of Biological Dynamics
Journal of Biological Dynamics ECOLOGY-MATHEMATICAL & COMPUTATIONAL BIOLOGY
CiteScore
4.90
自引率
3.60%
发文量
28
审稿时长
33 weeks
期刊介绍: Journal of Biological Dynamics, an open access journal, publishes state of the art papers dealing with the analysis of dynamic models that arise from biological processes. The Journal focuses on dynamic phenomena at scales ranging from the level of individual organisms to that of populations, communities, and ecosystems in the fields of ecology and evolutionary biology, population dynamics, epidemiology, immunology, neuroscience, environmental science, and animal behavior. Papers in other areas are acceptable at the editors’ discretion. In addition to papers that analyze original mathematical models and develop new theories and analytic methods, the Journal welcomes papers that connect mathematical modeling and analysis to experimental and observational data. The Journal also publishes short notes, expository and review articles, book reviews and a section on open problems.
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