Containment measures implemented by some countries to suppress the spread of COVID-19 have resulted in a slowdown of the epidemic characterized by time series of daily infections plateauing over extended periods of time. We prove that such a dynamical pattern is compatible with critical susceptible-infected-removed (SIR) dynamics. In traditional analyses of the critical SIR model, the critical dynamical regime is started from a single infected node. The application of containment measures to an ongoing epidemic, however, has the effect to make the system enter in its critical regime with a number of infected individuals potentially large. We describe how such nontrivial starting conditions affect the critical behavior of the SIR model. We perform a theoretical and large-scale numerical investigation of the model. We show that the expected outbreak size is an increasing function of the initial number of infected individuals, while the expected duration of the outbreak is a nonmonotonic function of the initial number of infected individuals. Also, we precisely characterize the magnitude of the fluctuations associated with the size and duration of the outbreak in critical SIR dynamics with nontrivial initial conditions. Far from herd immunity, fluctuations are much larger than average values, thus indicating that predictions of plateauing time series may be particularly challenging.

中文翻译：

---

具有非平凡初始条件的临界易感感染去除动力学中的流行高原

一些国家实施的遏制措施以抑制COVID-19的传播，导致该流行病的速度下降，其特征是每日感染的时间序列在较长时期内处于平稳状态。我们证明了这种动态模式与临界易感感染去除（SIR）动力学兼容。在对关键SIR模型的传统分析中，关键动态机制是从单个受感染节点开始的。但是，对持续进行的流行病采取遏制措施的效果是，使系统进入临界状态，大量受感染的个体可能会因此而庞大。我们描述了这种非平凡的开始条件如何影响SIR模型的关键行为。我们对该模型进行了理论上和大规模的数值研究。我们表明，预期的爆发规模是受感染个体初始数量的增加函数，而预期的爆发持续时间是受感染个体初始数量的非单调函数。同样，我们精确地描述了在具有非平凡初始条件的关键SIR动态中与爆发的大小和持续时间相关的波动幅度。波动远非群体免疫，其波动远大于平均值，因此表明平稳时间序列的预测可能特别具有挑战性。我们精确地描述了在初始条件不重要的情况下，临界SIR动态中与爆发规模和持续时间相关的波动幅度。波动远非群体免疫，其波动远大于平均值，因此表明平稳时间序列的预测可能特别具有挑战性。我们精确地描述了在初始条件不重要的情况下，临界SIR动态中与爆发规模和持续时间相关的波动幅度。波动远非群体免疫，其波动远大于平均值，因此表明平稳时间序列的预测可能特别具有挑战性。