How (not) to break a wave

It has been said many times that the measures had no impact. This is not true, the measures have had an impact and we see it when we look at the timeline. First, the exponential increase has been broken and this is very important. I cannot say exactly which measures were taken that triggered this. I am not capable of telling you this and no scientist is able to do so either.

Paulette Lenert [1]

At the same time, we are in the process of generally stepping back from this “emergency” article that we made for the government and concentrating more on academic publications.

Taskforce-Member Dr. Alexander Skupin [2]

The “Updates for second wave” by Research Luxembourg

Considering the effort with which the “deconfinement” and the subsequent Large Scale Testing were accompanied by the Taskforce, this step can certainly be seen as a turning point. In the midst of a pandemic, it seems at least strange when science classifies its own policy advisory “contribution” as no longer necessary.

Let’s take a look at the weekly reports from Research Luxembourg [3] from September to December 2020.

We find in each case a graph of the effective reproduction number, the absolute as well as percentage number of positive PCR tests, the active infections as well as the weekly case numbers per 100 000 population. With the exception of the latter graph, this information is not exclusive and is also available, for example, on the official Covid-19 website [4].

A current trend (linear, exponential growth) should be identified on a graph with the cumulative number of positive PCR tests. Without going into depth, it should be pointed out that it is problematic to cumulate absolute numbers, that were collected under different circumstances (total number of tests) [5]. In this case, the term “model” is used, but it is in fact a regression that attempts to put the current test numbers into a numerical context. An underlying principle is not abstracted in this way and a prediction is therefore not possible.

This task is fulfilled by a midterm projection. An attempt is made to determine the future course of the number of positive PCR tests using a SIR model [6]. We have compared the prognosis and the actual numbers in the table. In the months from September to November, Research Luxembourg was apparently not able to reliably predict the development of case numbers.

Report fromPrognosisFor time intervalactual valueDifference (%)
11/09/202055“mid october”200-145 (-73%)
18/09/2020220“5 november”700-480 (-69%)
25/09/202085“beginning november”700-615 (-88%)
02/10/2020120“beginning november”700-580 (-83%)
09/10/2020170“mid november”700-530 (-76%)
16/10/2020370“beginning december”700-330 (-47%)
23/10/20201400“mid november”650750 (115%)
30/10/20201300“beginning november”700600 (86%)
05/11/2020950“mid november”650300 (46%)
12/11/2020780“next days”650130 (20%)
19/11/2020720“next days”600120 (20%)
26/11/2020300“christmas”3000 (0%)
Predictions in the weekly reports of Research Luxembourg

The second wave and the measures

In order to eliminate the influence of the fluctuations in the total number of tests, we consider the value of the percentage of positive tests averaged over 7 days in the following graph.

The start dates of the various measures are marked. We briefly introduce them again here:

  • from October 30, 2020: curfew from 11 p.m. to 6 a.m., visits by a maximum of 4 people, 4 people at the table in restaurants, mask requirement inside and outside if there are more than 4 people, etc.
  • from 26.11.2020: closure of the Horesca sector, cinemas and fitness centers
  • From 12/26/2020: curfew brought forward to 9:00 p.m. to 6:00 a.m., all non-“essential” shops closed, homeschooling during the week of January 4, 2021
  • from 11.01.2021: curfew again from 11 p.m. to 6 a.m., reopening of the non-“essential” shops, the Horesca sector remains closed

In order to be able to decide whether a measure has an influence on the development of the number of Covid-19 cases, we apply the following criterion: a variation in the tendency must be observed after 8 days at the earliest. We take into account the fact that a change in the infection rate can only be noticeable in the statistics after this period at the earliest [8]. The number of cases rose by October 26, 2020, and reached its maximum. This period corresponds to the “exponential development” cited at the beginning. It is therefore clear that none of the measures that began later “broke” this growth.

According to our criterion, the measure of October 30, 2020, should have made itself noticeable in the statistics from November 6, 2020. At this point, however, there was a slight increase again. An intermediate maximum was reached on December 1, 2020, after which the numbers fell again. The measure of November 26, 2020 cannot, therefore, have been the initial reason to initiate this trend. The same is true for the third measure on December 26, 2020: an intermediate maximum on December 28, 2020 with a subsequent decrease, for which this measure cannot be assigned as a cause.

Finally, the fourth measure of January 11, 2021, which is a relaxation of the previous one, led to a slight decrease after 8 days. We can conclude: None of the measures can be assigned a verifiable effect on the incidence of infection. The fact that “nonpharmaceutical interventions (NPI)” generally have little or no influence on the epidemiological course is stated in an article by the American Institute for Economic Research [9], which lists 29 scientific studies that come to this conclusion.

Maybe everything is different?

What could have determined the course of the second wave? We try to find a possible answer in this part

On the official Covid-19 website [4] we find the graph “Tests COVID-19”. The cumulative numbers of tests, people tested and positive tests are shown here. This data can be used to determine the percentage of those tested for the first time each day. Graph 2 shows the percentage of those tested for the first time (red) and positive PCR tests (blue) of the total number of daily tests. The proportion of those tested in the total population (626 108, as of 01/01/2021) is shown in green. This is currently 90% and will asymptotically approach 100% in the near future. As expected, the red curve starts at 100% (all people are tested for the first time). An ever-increasing proportion is tested for at least the second time over the course of time, the proportion of those tested for the first time decreases accordingly.

From mid-October 2020, there is a pronounced covariance of the first two variables. We consider a section of the period from October 2020 to January 2021. (The values averaged over 7 days are shown in dashed lines.)

Although the average of those tested for the first time (group A) is never greater than 20%, every change in this value leads to a comparatively large change in the proportion of positive tests. It follows that test positives are found almost exclusively in group A, while in complementary group B only a very small proportion of those tested at least once before receive a positive test result.

For this purpose a small sample calculation (January 11, 2021), assuming that group A contributes 90% to the positive tests:

Total number of tests: 10421 of which: tested for the first time (group A): 687 (6.59%) tested more than once (group B): 10421 – 687 = 9734 (93.41%)

tested positive: 154 (1.48%) of which: tested for the first time (group A): 90% of 154: 139 tested more than once (group B): 10% of 154: 15

Proportion of positive tests (prevalence) among those tested for the first time (group A): 100 – 139 / 687 = 20.23%. among those tested more than once (group B): 100 – 15 / 9734 = 0.15%. (If we were to assume a proportion of 80%, the proportion of positive tests would still be 17.9% in Group A and 0.32% in Group B).

The infection process would therefore take place separately in 2 groups, with one in group A that is many times higher in reproduction. The prevalence in group B, to which 90% of the population now belongs, would, on the other hand, be in the error range of the test and thus virtually negligible. Even if the reasons for this remain to be clarified, this circumstance could answer a few questions: 1. The contact restriction through the measures is useless for the vast majority of people since there is no infection anyway. Accordingly, they also have no effect.

2. The second wave reached its maximum with only a 6% positive rate on October 26, 2020. For comparison: the peak of the wave in March 2020 came to 20%.

3. The proportion of positive tests must necessarily decrease over time, as the proportion of those that have never been tested becomes smaller and smaller with ongoing tests. A decline in the number of cases can thus be explained even without measures.

We have thus presented an approach that could at least partially explain the course of the second wave.


The infection process cannot be explained with the methods used by Research Luxembourg, and the causal effect of the measures cannot be proven. In addition, politicians do not see themselves as obliged to answer these questions. The suspicion is strengthened that the strategy of fighting Corona is no longer based on scientific criteria. Sometimes no secret is made of the fact that the decisions are “politically” motivated, whatever is meant by that.


[1] RTL (15.12.2020 ): Gesondheetsministesch seet, d’Mesuren hu gegraff

[2] Tageblatt (18.09.2020): Einschätzung aus der Covid-19-Taskforce: „Das werden spannende Tage und Wochen“

[3] Research Luxembourg: Publications

[4] Offizielle Covid19-Webseite – Grafiken

[5] Expressis Verbis: Zahlen und Fakten

[6] Wikipedia: SIR-Model [7] WHO Information Notice for IVD Users 2020/05

[8] Research Luxembourg: Real time Rt estimation

[9] American Institute for Economic Research: Lockdowns Do Not Control the Coronavirus: The Evidence

The original language of this article is German. The English and French versions are translations.