The majority of known early warning indicators of critical transitions rely on asymptotic resilience and critical slowing down. In continuous systems, critical slowing down is mathematically described by a decrease in magnitude of the dominant eigenvalue of the Jacobian matrix on the approach to a critical transition. Here, we show that measures of transient dynamics, specifically, reactivity and the maximum of the amplification envelope, also change systematically as a bifurcation is approached in an important class of models for epidemics of infectious diseases. Furthermore, we introduce indicators designed to detect trends in these measures and find that they reliably classify time series of case notifications simulated from stochastic models according to levels of vaccine uptake. Greater attention should be focused on the potential for systems to exhibit transient amplification of perturbations as a critical threshold is approached, and should be considered when searching for generic leading indicators of tipping points. Awareness of this phenomenon will enrich understanding of the dynamics of complex systems on the verge of a critical transition.

中文翻译：

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传染病临界点的瞬态指标

大多数已知的关键过渡早期预警指标依赖于渐近弹性和临界减速。在连续系统中，临界减速在数学上由雅可比矩阵的主要特征值在接近临界转变时的幅度减小来描述。在这里，我们展示了瞬态动力学的测量，特别是反应性和放大包络的最大值，也随着在一类重要的传染病流行模型中接近分叉而系统地改变。此外，我们引入了旨在检测​​这些措施趋势的指标，并发现它们根据疫苗接种水平可靠地对从随机模型模拟的病例通知时间序列进行分类。应更多地关注系统在接近临界阈值时表现出扰动瞬态放大的可能性，并且在搜索临界点的通用领先指标时应予以考虑。对这种现象的认识将丰富对处于临界转变边缘的复杂系统动力学的理解。