Uniform sampling in a structured branching population
Uniform sampling in a structured branching population
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结构化分支总体中的均匀抽样
作者：
A. Marguet
我们感兴趣的是一个结构化的分支人口的动态，其中每个人的特点移动根据马尔可夫过程。每个个体的分裂率是其特征的函数，当分支事件发生时，后代出生时的特征取决于母亲的特征和后代的数量。在这篇文章中，我们明确地描述了惩罚马尔可夫过程，称为辅助过程，对应于动态的特点沿着脊柱，通过给出其相关的无穷小生成元。证明了分叉的一个多对一公式和一个多对一公式。此外，我们还证明了在大总体近似下，这个辅助过程精确地刻画了均匀抽样个体的性状过程。我们详细介绍了三个增长-破碎模型的例子：线性增长模型，指数增长模型和寄生虫感染模型。
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of the descendants at birth depends on the trait of the mother and on the number of descendants. In this article, we explicitly describe the penalized Markov process, named auxiliary process, corresponding to the dynamic of the trait along the spine by giving its associated infinitesimal generator. We prove a Many-to-One formula and a Many-to-One formula for forks. Furthermore, we prove that this auxiliary process characterizes exactly the process of the trait of a uniformly sampled individual in the large population approximation. We detail three examples of growth-fragmentation models: the linear growth model, the exponential growth model and the parasite infection model.