Impact of public health measures to control SARS-CoV-2Outbreak: a data-driven analysis

With the rapid spread of the SARS-CoV-2 virus since Fall 2019, governments took variousmeasures to contain the propagation of the pandemic, declared on March, 2020. This study introduces a novel method to estimate the reproductive number using Bayesian inference with time-dependent priors. By inferring the infection dates from incidence time series, the developed approach allows direct comparison between reproductive number and introduction of public health measures in a specific country. First a specific period between the onset of the symptoms and a case being declared as dead is derived on data available in Switzerland. Focussing on the measures taken by 31 European countries, this study shows that most countries required tough state interventions with a stringency index equal to 83.6 out of 100 to reduce the reproductive number below one and hence control the development of the epidemy. In addition, it is shown that there is a direct correlation between the time taken to introduce restrictive measures and the time required to contain the spread of the epidemy with a median time of 8 days between the introduction of initial restrictive measures and the reproductive rate reducing below one.

effective and basic reproduction rate. The basic reproduction R 0 refers to the evolution of the disease when the population is fully susceptible to the disease while the effective reproductive rate R t factors the immunity acquired within the population 3 . The reproduction rate is a key parameter to evaluate the evolution of an epidemic. Any value below 1 indicates that the spread is decreasing, any value above one indicates that the spread is increasing. In addition, the reproductive number allows a direct comparison of the epidemiologic profiles observed in different countries with largely different characteristics (population, testing methods, etc), thus considering temporality and populational characteristics.
Numerous methods have been developed to compute the reproductive rate and its evolution over time 4 . Initial methods derived the reproductive rate from transmission model similar to the SIR model [5][6][7][8][9] . However, these models require assumptions about the epidemiology of the disease and are dependent on the recovered cases which are very often difficult to evaluate. Later models, including the Wallinga and Teunis approach 10 , use a likelihood-based estimation procedure to reconstruct infection patterns. These methods were however found to exhibit large variations when using daily data 11 . Approaches which aimed to correct these fluctuations were very sensitive to the selected smoothing parameters 11, 12 . Cory et al. developed an additional method to mitigate these drawbacks and their method showed special robustness to underreporting 13 .
Since the start of the pandemic, various studies have looked at the impact of public health interventions on the evolution of the COVID-19 at a regional or national level. Studies first focussed on China, demonstrating the importance of control strategies to reduce the reproductive rate 14,15 . These studies used mechanistic transmission models to obtain the reproductive number with the drawbacks associated with these models described earlier. Further studies focussed on how state interventions prevented ICU capacity to be overwhelmed as well as the impact of these measures on fatalities in the UK 16 , Germany 17 and France 18 . While these researches focussed on individual country, a report published by Flaxman et al. 19 aimed to demonstrate the impact of non-pharmaceutical interventions in 11 European countries. This study assumed that the various interventions had the same effect across countries and that their impact was independent of the timing of the measure. In addition, this study assumed the reproductive number is fixed between the different measures. However, a recent research shows that community changes also play a role in slowing the evolution of the virus 20 .
The aim of this work is to extend previous researches by focussing on the effects of state 2 . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 11, 2020. . https://doi.org/10.1101/2020.06.10.20126870 doi: medRxiv preprint interventions in 31 European countries. As the evolution of the reproductive number is a function of both the introduction of restrictive measures as well as changes in behaviours with specific societal properties, we do not aim to quantify the effect of each measure. Instead, we aim to show how these combined interventions and their temporality have influenced the spread of the virus.
Evaluating the reproductive number from incidence data The number of infected individuals at a given time can be estimated as follows: where w s is the serial interval. The distribution of w s for the SARS-CoV-2 virus was found by Nishiura et al. 21 to have a mean of 4.8 days and a standard deviation of 2.3 days.
Given the time at which an infection I(t) occurred is not available, the number of confirmed cases and deaths on a given day are used as proxies, keeping in mind that the developed method is able to deal with underreporting. Linton et al. 22   From the latter periods it is possible to calculate a posterior distribution of R t based on the inferred infection dates extracted from the cases being reported as confirmed or dead. For the cases declared on each day, a shift following a gamma distribution between the defined cases (confirmed or dead) and the time of infection is randomly generated. For each case, the new date of infection 3 . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

(which was not certified by peer review)
The copyright holder for this preprint this version posted June 11, 2020. . https://doi.org/10.1101/2020.06.10.20126870 doi: medRxiv preprint then becomes the date at which the case was reported minus the generated shift. This procedure is then performed iteratively with the mean of the obtained infections on each day over the simulations being retained thus increasing the robustness of our proposed method by naturally smoothing the incidence data. This step is critical in order to deal with infrequently published data. In addition, the infections for the most recent days are corrected (Scire et al. 23 ) to factor the fact that some of the infections which occurred on this day will be reported in the future (see Methods).
Our method to estimate the reproductive rate is a variation of the one proposed by Cori et al. 13 .
While their method assumes a constant gamma distribution for the prior distribution, the presented model takes advantage of the information gained in time by updating the prior distribution for each window with the previous posterior (see Methods). R t was estimated using incidence data for con- 4 . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

(which was not certified by peer review)
The copyright holder for this preprint this version posted June 11, 2020. . https://doi.org/10.1101/2020.06.10.20126870 doi: medRxiv preprint A gamma distribution with a mean and a standard deviation equal respectively to 10.7 and 6.73 days was found to best fit the data from infection to death. The distribution along the extracted data are shown in figure 1. This distribution was then combined with the incubation period provided by Linton et al. 22 to obtain the period between onset and death shown in table 1. Austria provides a good analysis case as it was one of the first country in Europe to impose lockdown measure, as early as on the 13th of March, but also to ease restrictions in mid-April. It is interesting to note that the reproductive rate started to decline before the introduction of restrictive measures. The reductions was however steeper after a set of measure was introduced between the 13th and 17th of March. The reproductive rate then plateaued at around 0.65 during the lockdown.

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(which was not certified by peer review)
The copyright holder for this preprint this version posted June 11, 2020. . Recently the reproductive rate has been oscillating around one. This last phase shows the difficulty to reduce the number of cases below a certain threshold with the emergence of localised clusters whose identification will be critical to contain the evolution of the virus.
The stringency index developed by Petherick et al. 24 was used to assess the role of state interventions in controlling the epidemy. This index was compared with the evolution of the reproductive rate rather than the incidence of confirmed or dead cases. Indeed, differences in testing or reporting policies between countries make it very difficult to compare these variables directly.
While the reproductive rate is also subject to variations in these policies, it depends on the change in confirmed and death cases therefore allowing comparison between countries with different policies. The measures as well as the stringency index each country had in place when their reproductive rate based on the confirmed cases dropped below one is analysed. The intention is to determine which measures allowed countries to contain the evolution of the virus. The stringency index of the countries included in our analysis are presented in figure 3. When countries managed to reduce their reproductive rate below one they had a mean stringency index of 83.6 out of 100 with a standard deviation of 13.5. When R t dropped below one, the median severity of the measures for each category defined in the OxCGRT dataset was the following. Countries had completely closed their school as well as non-essential workplace. All public events were cancelled and gathering 6 . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

(which was not certified by peer review)
The copyright holder for this preprint this version posted June 11, 2020. . https://doi.org/10.1101/2020.06.10.20126870 doi: medRxiv preprint with more than 10 people were banned. Finally, the median country required not to leave the house apart than for specific activities and internal movements were severely restricted. In order to assess the impact of taking restrictive measures early in the crisis, the time taken to introduce initial restrictive measures was compared to the period taken to control the epidemy.
The time until the introduction of restrictive measure was defined as the period between the 5th death in a given country and the stringency index reaching a score of 35. The stringency index corresponds to the lowest score observed when Andorra reached a R t smaller than one. The time required to control the epidemy was then defined as the period between the 5th death and the reproductive rate based on the confirmed cases dropping below one. The results along a linear regression are presented in figure 4. A Pearson correlation coefficient of 0.762 was found between the two variables indicating that there is indeed a positive benefit of taking early measures to control the epidemy. The United Kingdom can serve as an interesting example. The UK had initially planned to build "targeted herd immunity" delaying the introduction of restrictive measure.
As a results of this delay, the UK were only able to contain the epidemy 30 days after the fifth death occurred in the country when the median time for the countries included in our analysis was of eight days. There are three outliers in our analysis being Andorra, Sweden and Iceland.
Sweden has decided not to introduce a complete lockdown. While true that Sweden has achieved a reproductive rate oscillating around one, it currently stands with one of the highest daily death incidence in Europe having 5.34 deaths per million people per day on the 23th of May when the mean for the countries included in our analysis is 0.82 on the same day (as retrieved from Our World in Data).

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The copyright holder for this preprint this version posted June 11, 2020. . https://doi.org/10.1101/2020.06.10.20126870 doi: medRxiv preprint Figure 4: Period required to contain the epidemy as a function of the period to introduce initial restrictive measures.

Conclusions
First, an improved estimation of the time spent between a positive test and the death of a patient was derived using 1430 cases reported in Switzerland. This is an essential parameter to predict the occupancy of ICU units and models are very sensitive to it. Second, we propose a new method, that allows to estimate a sequence of reproduction numbers, where information acquired by previous reproduction numbers is transferred to more recent reproduction number estimates via Bayesian inference. By retrieving the infection date, this method allows direct comparison between the introduction of restrictive measures and their effect on the reproduction rate. Our analysis on 30 European countries shows that all countries apart from three required state interventions to control the epidemy. It was finally shown that countries which took restrictive measures earlier managed to control the epidemy in a shorter time frame indicating these measures were indeed effective in limiting the number of casualties in Europe as of the 23rd of May 2020.

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(which was not certified by peer review)
The copyright holder for this preprint this version posted June 11, 2020. . https://doi.org/10.1101/2020.06.10.20126870 doi: medRxiv preprint . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

Methods
Correcting the number of infections The infections for the most recent days are corrected (Scire et al. 23 ) to factor the fact that some of the infections which occurred on this day will be reported in the future: whereF is the cumulative distributive function of the period between an infection and a case being reported as positive or dead, l is the time between t and the last reported case,Î(t)) and I(t) are respectively the corrected and initial infections which took place on a given day.
Estimation of the reproductive rate The method presented in this report is a variation of the one proposed by Cori et al 13 . Assuming the incidence at time t, I t , is Poisson distributed so that the likelihood of the incidence I t given R t and conditional on previous incidence I 0 , . . . , I t−1 : The posterior of the reproductive number R t conditional on previous incidences is: While the method developed by Cori et al 13 assumes a constant gamma distribution for the prior distribution, the presented model takes advantage of the information gained in time by updating the prior distribution for each window with the previous posterior: The 95% CI is then derived by computing the 2.5% and 97.5% quantiles.
The reproductive number R t based on the confirmed cases is reported up to 9 days before the last date at which results are available. This corresponds to the median time for confirmed cases to be reported. Using the same method, R t based on the cases reported as dead is reported up to 19 days before the last day on which deaths were reported for a given country.
Code availability The codes are available upon request to the corresponding author.

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