Keeping the lid on infection spread From February to April 2020, many countries introduced variations on social distancing measures to slow the ravages of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Publicly available data show that Germany has been particularly successful in minimizing death rates. Dehning et al. quantified three governmental interventions introduced to control the outbreak. The authors predicted that the third governmental intervention—a strict contact ban since 22 March—switched incidence from growth to decay. They emphasize that relaxation of controls must be done carefully, not only because there is a 2-week lag between a measure being enacted and the effect on case reports but also because the three measures used in Germany only just kept virus spread below the growth threshold. Science, this issue p. eabb9789 Modeling and Bayesian inference reveal the time dependence of SARS-CoV-2 interventions on the number of new infections. INTRODUCTION When faced with the outbreak of a novel epidemic such as coronavirus disease 2019 (COVID-19), rapid response measures are required by individuals, as well as by society as a whole, to mitigate the spread of the virus. During this initial, time-critical period, neither the central epidemiological parameters nor the effectiveness of interventions such as cancellation of public events, school closings, or social distancing is known. RATIONALE As one of the key epidemiological parameters, we inferred the spreading rate λ from confirmed SARS-CoV-2 infections using the example of Germany. We apply Bayesian inference based on Markov chain Monte Carlo sampling to a class of compartmental models [susceptible-infected-recovered (SIR)]. Our analysis characterizes the temporal change of the spreading rate and allows us to identify potential change points. Furthermore, it enables short-term forecast scenarios that assume various degrees of social distancing. A detailed description is provided in the accompanying paper, and the models, inference, and forecasts are available on GitHub (https://github.com/Priesemann-Group/covid19_inference_forecast). Although we apply the model to Germany, our approach can be readily adapted to other countries or regions. RESULTS In Germany, interventions to contain the COVID-19 outbreak were implemented in three steps over 3 weeks: (i) Around 9 March 2020, large public events such as soccer matches were canceled; (ii) around 16 March 2020, schools, childcare facilities, and many stores were closed; and (iii) on 23 March 2020, a far-reaching contact ban (Kontaktsperre) was imposed by government authorities; this included the prohibition of even small public gatherings as well as the closing of restaurants and all nonessential stores. From the observed case numbers of COVID-19, we can quantify the impact of these measures on the disease spread using change point analysis. Essentially, we find that at each change point the spreading rate λ decreased by ~40%. At the first change point, assumed around 9 March 2020, λ decreased from 0.43 to 0.25, with 95% credible intervals (CIs) of [0.35, 0.51] and [0.20, 0.30], respectively. At the second change point, assumed around 16 March 2020, λ decreased to 0.15 (CI [0.12, 0.20]). Both changes in λ slowed the spread of the virus but still implied exponential growth (see red and orange traces in the figure). To contain the disease spread, i.e., to turn exponential growth into a decline of new cases, the spreading rate has to be smaller than the recovery rate μ = 0.13 (CI [0.09, 0.18]). This critical transition was reached with the third change point, which resulted in λ = 0.09 (CI [0.06, 0.13]; see blue trace in the figure), assumed around 23 March 2020. From the peak position of daily new cases, one could conclude that the transition from growth to decline was already reached at the end of March. However, the observed transient decline can be explained by a short-term effect that originates from a sudden change in the spreading rate (see Fig. 2C in the main text). As long as interventions and the concurrent individual behavior frequently change the spreading rate, reliable short- and long-term forecasts are very difficult. As the figure shows, the three example scenarios (representing the effects up to the first, second, and third change point) quickly diverge from each other and, consequently, span a considerable range of future case numbers. Inference and subsequent forecasts are further complicated by the delay of ~2 weeks between an intervention and the first useful estimates of the new λ (which are derived from the reported case numbers). Because of this delay, any uncertainty in the magnitude of social distancing in the previous 2 weeks can have a major impact on the case numbers in the subsequent 2 weeks. Beyond 2 weeks, the case numbers depend on our future behavior, for which we must make explicit assumptions. In sum, future interventions (such as lifting restrictions) should be implemented cautiously to respect the delayed visibility of their effects. CONCLUSION We developed a Bayesian framework for the spread of COVID-19 to infer central epidemiological parameters and the timing and magnitude of intervention effects. With such an approach, the effects of interventions can be assessed in a timely manner. Future interventions and lifting of restrictions can be modeled as additional change points, enabling short-term forecasts for case numbers. In general, our approach may help to infer the efficiency of measures taken in other countries and inform policy-makers about tightening, loosening, and selecting appropriate measures for containment of COVID-19. Bayesian inference of SIR model parameters from daily new cases of COVID-19 enables us to assess the impact of interventions. In Germany, three interventions (mild social distancing, strong social distancing, and contact ban) were enacted consecutively (circles). Colored lines depict the inferred models that include the impact of one, two, or three interventions (red, orange, or green, respectively, with individual data cutoff) or all available data until 21 April 2020 (blue). Forecasts (dashed lines) show how case numbers would have developed without the effects of the subsequent change points. Note the delay between intervention and first possible inference of parameters caused by the reporting delay and the necessary accumulation of evidence (gray arrows). Shaded areas indicate 50% and 95% Bayesian credible intervals. As coronavirus disease 2019 (COVID-19) is rapidly spreading across the globe, short-term modeling forecasts provide time-critical information for decisions on containment and mitigation strategies. A major challenge for short-term forecasts is the assessment of key epidemiological parameters and how they change when first interventions show an effect. By combining an established epidemiological model with Bayesian inference, we analyzed the time dependence of the effective growth rate of new infections. Focusing on COVID-19 spread in Germany, we detected change points in the effective growth rate that correlate well with the times of publicly announced interventions. Thereby, we could quantify the effect of interventions and incorporate the corresponding change points into forecasts of future scenarios and case numbers. Our code is freely available and can be readily adapted to any country or region.

中文翻译：

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推断 COVID-19 传播的变化点揭示了干预措施的有效性

遏制感染传播 2020 年 2 月至 4 月，许多国家采取了不同的社会疏远措施，以减缓严重急性呼吸综合征冠状病毒 2 (SARS-CoV-2) 的破坏。公开数据显示，德国在降低死亡率方面尤其成功。德宁等人。量化了为控制疫情而采取的三项政府干预措施。作者预测，第三次政府干预——3 月 22 日以来的严格接触禁令——使发病率从增长转向衰退。他们强调，放松管制必须谨慎行事，不仅因为措施的实施与对病例报告的影响之间存在两周的滞后，还因为德国采用的三项措施只是将病毒传播控制在增长阈值以下。科学，本期第 14 页。eabb9789 建模和贝叶斯推理揭示了 SARS-CoV-2 干预措施对新感染数量的时间依赖性。简介 当面对 2019 年冠状病毒病 (COVID-19) 等新型流行病的爆发时，个人以及整个社会都需要采取快速应对措施，以减轻病毒的传播。在这个最初的、时间紧迫的时期，我们既不知道中心流行病学参数，也不知道取消公共活动、关闭学校或保持社交距离等干预措施的有效性。基本原理 作为关键流行病学参数之一，我们以德国为例，从确诊的 SARS-CoV-2 感染中推断出传播率 λ。我们将基于马尔可夫链蒙特卡罗采样的贝叶斯推理应用于一类分区模型[易感-感染-恢复（SIR）]。我们的分析表征了传播率的时间变化，并使我们能够识别潜在的变化点。此外，它还可以实现假设不同程度的社交距离的短期预测场景。随附的论文提供了详细的描述，模型、推理和预测可在 GitHub (https://github.com/Priesemann-Group/covid19_inference_forecast) 上找到。虽然我们将该模型应用于德国，但我们的方法可以很容易地适用于其他国家或地区。结果 在德国，遏制 COVID-19 疫情的干预措施在 3 周内分三步实施： (i) 2020 年 3 月 9 日左右，取消了足球比赛等大型公共活动；(ii) 2020 年 3 月 16 日左右，学校、托儿设施和许多商店关闭；(iii) 2020 年 3 月 23 日，政府当局实施了影响深远的接触禁令 (Kontaktsperre)；这包括禁止小型公共集会以及关闭餐馆和所有非必需品商店。根据观察到的 COVID-19 病例数，我们可以使用变化点分析来量化这些措施对疾病传播的影响。本质上，我们发现，在每个变化点，传播率 λ 下降约 40%。在第一个变化点（假设在 2020 年 3 月 9 日左右），λ 从 0.43 降至 0.25，95% 可信区间 (CI) 分别为 [0.35, 0.51] 和 [0.20, 0.30]。在第二个变化点（假设在 2020 年 3 月 16 日左右），λ 降至 0.15（CI [0.12，0.20]）。λ 的这两种变化都减缓了病毒的传播，但仍然意味着指数增长（参见图中的红色和橙色痕迹）。为了遏制疾病传播，即将指数增长转变为新病例下降，传播率必须小于恢复率 μ = 0.13 (CI [0.09, 0.18])。假设在 2020 年 3 月 23 日左右，第三个变化点达到了这一关键转变，导致 λ = 0.09（CI [0.06, 0.13]；参见图中的蓝色轨迹）。从每日新病例的峰值位置可以看出得出的结论是，3月底已经实现了从增长到下降的转变。然而，观察到的短暂下降可以用传播率突然变化引起的短期效应来解释（参见正文中的图2C）。只要干预措施和同时发生的个人行为经常改变传播率，可靠的短期和长期预测就非常困难。如图所示，三个示例场景（代表第一个、第二个和第三个变化点的影响）彼此迅速出现分歧，因此涵盖了相当大范围的未来案例数量。由于干预和新 λ 的第一个有用估计（从报告的病例数得出）之间大约有 2 周的延迟，推断和随后的预测变得更加复杂。由于这种延迟，前两周社交距离程度的任何不确定性都可能对接下来两周的病例数产生重大影响。超过两周后，病例数量取决于我们未来的行为，为此我们必须做出明确的假设。总之，未来的干预措施（例如取消限制）应谨慎实施，以尊重其影响的延迟可见性。结论 我们开发了一个关于 COVID-19 传播的贝叶斯框架，以推断中心流行病学参数以及干预效果的时间和程度。通过这种方法，可以及时评估干预措施的效果。未来的干预措施和取消限制可以建模为额外的变化点，从而可以对病例数进行短期预测。总的来说，我们的方法可能有助于推断其他国家采取的措施的效率，并告知政策制定者有关收紧、放松和选择适当措施来遏制 COVID-19 的信息。根据每日新发的 COVID-19 病例对 SIR 模型参数进行贝叶斯推断，使我们能够评估干预措施的影响。在德国，三种干预措施（轻度社交距离、强烈社交距离、和接触禁令）相继颁布（圆圈）。彩色线描绘了推断模型，其中包括一项、两项或三项干预措施（分别为红色、橙色或绿色，个别数据截止）或截至 2020 年 4 月 21 日的所有可用数据（蓝色）的影响。预测（虚线）显示如果没有后续变化点的影响，病例数将如何发展。注意由于报告延迟和必要的证据积累（灰色箭头）而导致的干预和首次可能的参数推断之间的延迟。阴影区域表示 50% 和 95% 贝叶斯可信区间。随着 2019 年冠状病毒病 (COVID-19) 在全球范围内迅速传播，短期建模预测为遏制和缓解策略的决策提供了时间关键的信息。短期预测的一个主要挑战是评估关键流行病学参数以及当首次干预措施显示效果时它们如何变化。通过将已建立的流行病学模型与贝叶斯推理相结合，我们分析了新感染有效增长率的时间依赖性。着眼于德国的 COVID-19 传播，我们发现了有效增长率的变化点，这些变化点与公开宣布的干预措施的时间密切相关。因此，我们可以量化干预措施的效果，并将相应的变化点纳入对未来情景和病例数的预测中。我们的代码是免费提供的，并且可以轻松适应任何国家或地区。