Epidemics generally spread through a succession of waves that reflect factors on multiple timescales. On short timescales, superspreading events lead to burstiness and overdispersion, whereas long-term persistent heterogeneity in susceptibility is expected to lead to a reduction in both the infection peak and the herd immunity threshold (HIT). Here, we develop a general approach to encompass both timescales, including time variations in individual social activity, and demonstrate how to incorporate them phenomenologically into a wide class of epidemiological models through reparameterization. We derive a nonlinear dependence of the effective reproduction number Re on the susceptible population fraction S. We show that a state of transient collective immunity (TCI) emerges well below the HIT during early, high-paced stages of the epidemic. However, this is a fragile state that wanes over time due to changing levels of social activity, and so the infection peak is not an indication of long-lasting herd immunity: Subsequent waves may emerge due to behavioral changes in the population, driven by, for example, seasonal factors. Transient and long-term levels of heterogeneity are estimated using empirical data from the COVID-19 epidemic and from real-life face-to-face contact networks. These results suggest that the hardest hit areas, such as New York City, have achieved TCI following the first wave of the epidemic, but likely remain below the long-term HIT. Thus, in contrast to some previous claims, these regions can still experience subsequent waves.

流行病通常通过反映多个时间尺度因素的一系列波浪传播。在短时间内，超级传播事件会导致爆发和过度分散，而长期持续的易感性异质性预计会导致感染峰值和群体免疫阈值（HIT）降低。在这里，我们开发了一种涵盖两个时间尺度的通用方法，包括个体社会活动的时间变化，并演示如何通过重新参数化将它们在现象学上纳入广泛的流行病学模型中。我们推导出有效繁殖数的非线性相关性右e我们发现，在流行病的早期、高节奏阶段，出现了一种远低于 HIT 的短暂集体免疫 (TCI) 状态。然而，这是一种脆弱的状态，由于社会活动水平的变化，这种状态会随着时间的推移而减弱，因此感染高峰并不表明长期存在群体免疫：由于人口的行为变化，可能会出现后续的浪潮，而这些变化是由以下因素驱动的：例如，季节性因素。使用来自 COVID-19 流行病和现实生活中面对面接触网络的经验数据来估计暂时和长期的异质性水平。这些结果表明，纽约市等受灾最严重的地区在第一波疫情后已达到 TCI，但可能仍低于长期 HIT。因此，与之前的一些说法相反，这些地区仍然可能经历后续的波浪。