Many biological and medical questions can be modeled using time-to-event data in finite-state Markov chains, with the phase-type distribution describing intervals between events. We solve the inverse problem: given a phase-type distribution, can we identify the transition rate parameters of the underlying Markov chain? For a specific class of solvable Markov models, we show this problem has a unique solution up to finite symmetry transformations, and we outline a recursive method for computing symbolic solutions for these models across any number of states. Using the Thomas decomposition technique from computer algebra, we further provide symbolic solutions for any model. Interestingly, different models with the same state count but distinct transition graphs can yield identical phase-type distributions. To distinguish among these, we propose additional properties beyond just the time to the next event. We demonstrate the method's applicability by inferring transcriptional regulation models from single-cell transcription imaging data.

Ovidiu Radulescu

Dima Grigoriev

Matthias Seiss

Maria Douaihy

Mounia Lagha

Edouard Bertrand

Bulletin of Mathematical Biology

BULLETIN OF MATHEMATICAL BIOLOGY 是数学生物学学会的官方期刊，传播原始研究成果和其他与生物学和数学科学接口相关的信息。贡献应该与这两个领域相关。为了适应广泛的新发展，该期刊接受各种投稿，包括：原创研究文章侧重于借助数学科学工具或新的数学工具和方法获得的新生物学见解，这些工具和方法已证明适用于生物学调查研究数学生物学教育评论评论观点，以及讨论对专业重要的问题的贡献所有贡献都经过同行评审。

The Bulletin of Mathematical Biology is the official journal of the Society for Mathematical Biology, dedicated to disseminating original research findings and information related to the interface between biology and mathematical sciences. Contributions should be relevant to both fields. The journal welcomes a variety of submissions, including original research articles that focus on new biological insights gained through mathematical tools or novel mathematical methods that have proven applicable to biological investigations; reviews on mathematical biology education; and discussions and viewpoints on issues of professional significance. All contributions undergo peer review.

《Bulletin of Mathematical Biology》是数学生物学学会的官方期刊，旨在传播原始研究成果及与生物学和数学科学交叉领域相关的信息。我们欢迎与这两个领域相关的各种投稿，包括：利用数学科学工具或新方法获得的新生物学见解的原创研究文章，这些工具和方法已被证明适用于生物学研究；数学生物学教育的评论；以及对专业重要问题的讨论和观点。所有投稿均经过同行评审。

Category	Quartile
数学	4
数学, 生物学	4
数学, 数学与计算生物学	4

Identifying Markov Chain Models from Time-to-Event Data: An Algebraic Approach

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