Leaps-MPS: exploring properties of linear compartmental models

California State University, Stanislaus received a $249,996 Project Grant from the National Science Foundation's Mathematical and Physical Sciences program (CFDA 47.049), awarded on September 15, 2025, with completion targeted for August 31, 2027. This LEAPS-MPS (Low Energy Astrophysics and Plasma Sciences) project explores the mathematical and practical applications of Linear Compartmental Models (LCMs) to understand complex systems involving flow and movement, such as disease transmission, nutrient cycling, and pharmaceutical distribution. The investigator will develop visual, computationally efficient analytical tests to determine when meaningful system parameters—such as flow rates—can be reliably extracted from data and when different models may be indistinguishable in practice.

The project delivers both research and educational products. Primary research objectives include investigating irreducibility in LCMs to extend identifiability theory beyond current single-input, single-output frameworks, with particular focus on skeletal path models as candidates for broader theoretical generalization. The educational component provides undergraduate research participation opportunities in cutting-edge mathematics and establishes a new colloquium series to expose students to graduate-level research and support preparation for advanced study. These integrated research and educational activities align with the Mathematical and Physical Sciences program's emphasis on advancing scientific knowledge while developing the next generation of researchers.

Abstract

This project explores how mathematical diagrams called linear compartmental models (lcms) can help researchers understand the structure and behavior of systems involving movement or flow, including how diseases spread, how nutrients cycle through ecosystems, or how medications are processed in the body. These models represent systems as interconnected compartments, with arrows indicating how quantities move between them. The investigator aims to identify when meaningful information about these systems, such as flow rates, can be reliably extracted from data, and when different models might appear identical in practice. The project?s goal is to create simple, visual tests for analyzing these characteristics of lcms, making the process more accessible and less computationally intensive. This project provides undergraduates with opportunities to participate in cutting-edge mathematical research and equips them with background needed to pursue further education. A new colloquium series will further expand students? Exposure to graduate-level research and support their preparation for advanced study. This project addresses fundamental questions of identifiability and indistinguishability in linear compartmental models, which are widely used to represent dynamical systems with flow. Identifiability, the ability to recover model parameters from input-output data, has been well-studied for single-input, single-output, strongly connected lcms, and this project extends the scope of current theory. The first research objective is to investigate irreducibility in lcms, laying the groundwork for identifiability results for models with multiple inputs/outputs and weaker graph structures. A second objective focuses on a class of models known as skeletal path models, which are not strongly connected but have a well-understood structure and serve as promising candidates for generalizing identifiability theory. The final objective is to examine indistinguishability, a property in which structurally distinct models produce indistinguishable output behavior. The investigator and undergraduate collaborators will explore this question through the study of cycle models, which have been previously analyzed for identifiability but not for indistinguishability. The project emphasizes visual and combinatorial approaches to model analysis, enhancing accessibility for undergraduate researchers. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the foundation's intellectual merit and broader impacts review criteria.- subawards are not planned for this award.