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This study introduces a mathematical framework for analyzing "nonequilibrium transient centers" in continuous-time dynamical systems, which are points in state space where system observables remain nearly constant for extended periods before rapid transitions occur. The authors develop finite-time escape criteria to identify these centers and predict when trajectories will exit slow-evolving regimes. They demonstrate the framework using an epidemic model where disease levels can stay near zero for long periods before sudden outbreaks, illustrating delayed stability loss.
Why it matters
This framework is particularly relevant for ecological and epidemiological modeling where short-term transient behavior often matters more than long-term equilibria for management decisions and interventions. The ability to identify and predict sudden transitions from slow-evolving states could improve disease outbreak prediction and inform timing of public health interventions.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: Analysis of mathematical models in ecology and epidemiology often emphasizes asymptotic dynamics, such as stable equilibria, periodic orbits, and attractors. However, many biologically relevant trajectories exhibit long transient dynamics in which a relevant observable evolves slowly for an extended period before a comparatively rapid transition occurs. On finite time scales relevant to data collection, management, and intervention, such transient behavior can be more informative than the eventual asymptotic outcome. In this work, we study long-lasting and slowly varying transient dynamics in continuous-time systems through the framework of nonequilibrium transient centers. These centers are points in the state space near which the observable can remain nearly constant for a long time, even when the full state is not near an equilibrium. We develop finite-time escape criteria for identifying such centers and their reachable counterparts. These criteria provide practical conditions under which nearby trajectories can remain in a slow regime for a long period of time before eventually leaving it. We apply the theory to an epidemic model with delay of stability loss. In this model, trajectories may remain close to the disease-free manifold for a long time before the infectious population resurges. The framework developed here describes this behavior as slow evolution near a nonequilibrium transient center followed by finite-time escape from the slow regime.
Source: Nonequilibrium Transient Centers in Continuous-Time Systems