European and US lockdowns and second waves during the COVID-19 pandemic

Abstract

This paper investigates the lockdowns to contain the spread of the SARS-CoV-2 coronavirus in France, Germany, Italy, Spain, the UK and the US and also recent developments since these lockdowns have been relaxed. The analysis employs a two-stage SEIR model with different reproductive numbers pre- and post-lockdown. These parameters are estimated from data on the daily number of confirmed cases in a process that automatically detects the time at which the lockdown became effective. The model is evaluated by considering its predictive accuracy on current data and is then extended to a three-stage version to explore relaxations. The results show the extent to which each country was successful in reducing the reproductive number and demonstrate how the approach is able to model recent increases in the number of cases in all six countries, including the second peak in the US. The results also indicate that the current levels of relaxation in all five European countries could lead to significant second waves that last longer than the corresponding first waves. While there is uncertainty about the implications of these findings at this stage, they do suggest that a lot of vigilance is needed.

Fig. 1

Results obtained by applying the two-stage SEIR model to the daily confirmed cases for each of the six countries. Note that the results for the US are on a different scale.

Fig. 2

The effect on the daily number of confirmed cases of keeping the lockdown fully in place ( ), relaxing it by 25% ( ) and relaxing it by 50% ( ). Shaded regions represent 95% confidence intervals. Data points include those from day zero to 12/05/20 (1/06/20 for the UK) used to learn the model ($•$) and subsequent numbers of cases up to 31/07/20 ( ). In each case the relaxation is assumed to take effect by 01/06/20, which corresponds to the implementation of the relaxation around 20/05/20. Note that the results for the US are on a different scale.

Fig. 3

Results obtained by applying the three-stage model to learn the parameters for the time and degree of relaxation from data up to 08/08/20 ($•$) and compared with subsequent numbers of cases up to 22/08/20 ( ). Shaded regions represent 95% confidence intervals.

Fig. 4

As for Fig. 3, but now parameters are learned from data up to 22/08/20 and results are presented over the following six weeks up to 3/10/20.

Fig. 5

(a) Results obtained by applying the three-stage model to US data on the number of cases up to 22/06/20 ($•$) and then compared with data up to 08/08/20 ( ). (b) Results from applying the four-stage model to US data up to 08/08/20 ($•$) and compared with data up to 22/08/20 ( ), with results presented over the following six weeks up to 3/10/20.