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The authors propose a fresh concept called nonequilibrium transient centers—points in a system where an observable stays nearly constant for a long time even though the full state is far from equilibrium. The idea comes from studying models in ecology and epidemiology that show long “slow” phases before a rapid change, a behavior that can be more relevant for real data than final steady states.
To find these centers, the authors develop finite‑time escape criteria. By examining how the speed of the observable grows, they write explicit conditions (see Theorems 3.1, 3.3, 3.4) that guarantee a point in a candidate set is a transient center.
When the simple escape criteria cannot be applied directly, the authors introduce a transport lemma (Lemma 3.5). This lemma shows that a trajectory that remains slow must reach a specific auxiliary level set in finite time, after which the immediate escape tests apply.
The theory is illustrated by an SIR epidemic model with a slowly varying host population. The model includes densities for susceptible, infectious, and total population, with a small parameter ε describing the slow population growth. Near the disease‑free equilibrium at the carrying capacity, the authors prove that transient centers exist.
| Symbol | Description |
|---|---|
| ε | small positive parameter determining the slow time scale of host population growth |
| q(N) | density-dependent host population growth function, assumed to satisfy q′′(N)<0, q(0)=q(N∗)=0, and q(N)>0 for 0<N<N∗ |
| N∗ | carrying capacity of the host population |
| h(S,N) | disease transmission function, assumed to be smooth, increasing in S, and to satisfy h(0,N)=0 |
| ζ | natural death rate of the host population |
| p | vaccination rate of susceptible individuals |
| γ | recovery rate of infectious individuals |
| μ | disease-induced death rate |
| a | total removal rate from the infectious class, given by a=ζ+μ+γ |
A Lyapunov‑type construction (Lemma 4.1) is used to establish necessary invariance and growth conditions in a neighbourhood of that equilibrium, enabling the application of Theorem 3.4. This mathematical framework explains how slow dynamics can persist in epidemiological systems before a rapid outbreak or recovery.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| ζ | 0.2 | p | 0.01 |
| μ | 0.048 | β | 1 |
| γ | 0.75 | K | 0.1 |
| N∗ | 400 | ε | 10−4 |
The papers also present detailed tables of model parameters and numerical example values, sharpening the practical understanding of these transient phenomena.
Together, these results give a new lens for interpreting prolonged slow periods in biological systems, with potential implications for data analysis, management, and intervention strategies.
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[Table 1: Parameters and functions appearing in the SIR model] [Table 2: Parameter values used in the numerical illustration].