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  • Open Access

Effective One-Dimensional Reduction of Multicompartment Complex Systems Dynamics

Giorgio Vittorio Visco1,*,‡, Johannes Nauta1,‡, Tomas Scagliarini1, Oriol Artime2,3,4,§, and Manlio De Domenico1,5,6,†,§

  • *Contact author: giorgiovittorio.visco@phd.unipd.it
  • †Contact author: manlio.dedomenico@unipd.it
  • ‡These authors contributed equally to this work.
  • §These authors jointly supervised this work.
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Phys. Rev. X 15, 031017 – Published 15 July, 2025

DOI: https://doi.org/10.1103/whdb-2lgl

Abstract

A broad class of systems, including ecological, epidemiological, and sociological ones, are characterized by populations of individuals assigned to specific categories, e.g., a chemical species, an opinion, or an epidemic state, that are modeled as compartments. Because of interactions and intrinsic dynamics, the system units are allowed to change category, leading to concentrations varying over time with complex behavior, typical of reaction-diffusion systems. While compartmental modeling provides a powerful framework for studying the dynamics of such populations and describe the spatiotemporal evolution of a system, it mostly relies on deterministic mean-field descriptions to deal with systems with many degrees of freedom. Here, we propose a method to alleviate some of the limitations of compartmental models by capitalizing on tools originating from quantum physics to systematically reduce multidimensional systems to an effective one-dimensional representation. Using this reduced system, we are able not only to investigate the mean-field dynamics and their critical behavior, but we can additionally study stochastic representations that capture fundamental features of the system. We demonstrate the validity of our formalism by studying the critical behavior of models widely adopted to study epidemic, ecological, and economic systems.

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Popular Summary

Many real-world problems—from disease outbreaks to infrastructure failure and ecosystem collapse—can be studied using compartmental models. In these models, the system is broken down into categories, or compartments, and the dynamics of how individuals or units move between these compartments are analyzed. This approach is particularly useful for studying phase transitions, where a small change can lead to a large, often irreversible shift in the system. However, existing mathematical tools struggle to fully capture the complexity of systems with many compartments, especially when randomness plays a big role. To overcome this, we develop a new, unified framework using the Doi-Peliti path integral method, a technique borrowed from quantum field theory.

Our approach maps complex multicompartment systems onto a simpler Hamiltonian system, a formalism often used in physics to describe energy dynamics. In the stationary limit, our method simplifies the entire system to a 1D model that still captures the critical behavior. This allows us to analyze complex models more easily, and we confirm that our method recovers known results. We test it on several examples, especially in ecology and epidemiology, showing that it works well even in challenging cases.

This framework provides a general and physically grounded way to study random behavior in complex systems. Looking ahead, our work opens the door to more accurate and comprehensive studies of phase transitions in a wide variety of systems.

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Outline

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Phys. Rev. X 15, 031017– Published 15 July, 2025
  • Received 23 April 2024
  • Accepted 3 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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