Painting by numbers

What has happened so far

After it seemed clear from an astonishingly early stage that only a vaccine could end the Covid-19 pandemic, as soon as the saviour was available, they naturally wanted to prove “evidence-based” that its effectiveness was beyond all doubt.

The first attempts in Luxembourg to do this were still a bit clumsy: without considering the different size of the vaccinated/unvaccinated groups, they initially (from calendar week 29/2021) looked at the dismal percentages of those testing positive in each of these two groups in the total number of positive people[1]:

Weekly report from 28 July 2021

The weekly report of 20 October 2021 represented a quantum leap: The incidence rate (taux d’incidence) had been discovered and was now being used without restraint[2]:

Weekly report from 20 October 2021

All seemed well with the world, at least if one did not question too much how these figures were arrived at. The group of “personnes non-vaccinées” included the “partially vaccinated”, i.e., persons who were known to have often had non-negligible vaccination reactions immediately after the first dose, thus unjustifiably driving up the incidence rate among the “unvaccinated”. The fact that the unvaccinated were tested relatively more frequently also did not necessarily contribute to an objective presentation[3].

The end of the quarantine regulation by the law of 11 February 2022 finally also ended the euphoria: in calendar weeks 11 to 13, the incidence rate of the vaccinated turned out to be higher than that of the unvaccinated for 3 weeks in a row (which has also become the rule since then).

The Ministry of Health then threw in the towel and announced in the weekly report of 13 April 2022 that the incidence rates of vaccinated/unvaccinated would no longer be published[4].

Weekly report from 13 April 2022

A similar reaction was observed at the RKI in Germany a little later. Here, the vaccination effectiveness had been calculated from the incidence rates of the vaccinated/unvaccinated. For example, for the age group 5-11 years with basic immunization (green curve, left graph), the vaccination effectiveness in the raw data had already been negative for weeks, but this was not shown in the graph: from week 12 onwards, a horizontal line is drawn instead[5].

In the weekly report of 5 May 2022, one could then read on page 22 under the item “2.2. Effectiveness of COVID-19 vaccination”[6]:

As of Thursday, 26th May, the RKI’s COVID-19 weekly report will no longer include regular information on the effectiveness of COVID-19 vaccination.[…] Instead, it is planned that the RKI will publish separate evaluations regarding COVID-19 vaccination/vaccination effectiveness at regular intervals, which allows a more detailed examination of individual aspects than is possible within the framework of the weekly report.

As we believe that these data are important for forming an opinion on Covid-19 vaccination in general and compulsory vaccination in particular, we requested these data in writing from the Ministry of Health on 22 April 2022[7]. We referred to the law of 14 September 2018 “relative à une administration transparente et ouverte”.

We have not received a reply by this date (27.05.2022), which is why we have now written a reminder. We will keep you informed in this regard.

DIY (do it yourself)

Even though the incidence rates are no longer published, the case numbers of the vaccinated and unvaccinated are still listed separately in the weekly reports.

We would like to calculate the no longer published incidence rates ourselves and, of course, want to orient ourselves as far as possible on the previous calculation method. For this we need the:

  • Total population N and
  • the weekly vaccination rate q.

Since it is not initially clear which figures are used here, we carry out “reverse engineering” in this regard. For the first weeks of 2022, we know both the absolute numbers of people testing positive in the vaccinated and unvaccinated groups, as well as the corresponding incidence rates.

Let i_V and i_U be the incidence rates of the vaccinated and unvaccinated, respectively, and p_V and p_U be the number of positive test persons, respectively.

The following then applies to the incidence rates per 100,000 inhabitants:

    \[ i_V = 100.000 \cdot \frac{p_V}{q \cdot N} \hspace{1cm} \text{and} \hspace{1cm} i_U = 100.000 \cdot \frac{p_U}{(1-q) \cdot N} \]

If you solve the equations according to the denominator, you get:

    \[ q \cdot N = 100.000 \cdot \frac{p_V}{i_V} \hspace{1cm} \text{and} \hspace{1cm} (1-q) \cdot N = 100.000 \cdot \frac{p_U}{i_U} \]

Thus, one obtains a system of equations with the unknowns N and q:

    \[ \begin{cases} \begin{array}{rrrcll} & & q \cdot N & = & 100.000 \cdot \dfrac{p_V}{i_V} & (1) \\ \\ N & - & q \cdot N & = & 100.000 \cdot \dfrac{p_U}{i_U} & (2) \end{array} \end{cases} \]

Adding (1) and (2), it follows:

    \[ N = 100.000 \cdot  \left( \frac{p_V}{i_V} + \frac{p_U}{i_U} \right) \]

According to (1), q then holds:

    \[ q = \frac{1}{N} \cdot  100.000 \cdot \frac{p_V}{i_V} = \frac{100.000}{100.000 \cdot \left( \frac{p_V}{i_V} + \frac{p_U}{i_U} \right) } \cdot \frac{p_V}{i_V} = \frac{p_V}{i_V \cdot \left( \frac{p_V}{i_V} + \frac{p_U}{i_U} \right)} \]

For the first 13 weeks of 2022, you thus get:

CW 2022p_Vp_Ui_Vi_UNqvaccine uptake (ECDC)
1391567651544,111990,2659344942,72 %69,00 %
2632354051429,802807,7863473169,67 %69,70 %
3816471291831,263769,7063492670,21 %70,20 %
4798969291782,123717,0063470070,63 %70,60 %
5627347301389,742580,2363469671,12 %71,10 %
634062465749,001368,8063482471,63 %71,60 %
724651183540,90661,1063466671,80 %71,80 %
833341336728,60754,3063470872,09 %72,10 %
932681526712,80866,0063468672,24 %72,20 %
1042401728922,80986,0063472572,39 %72,40 %
11583419451268,301119,6063370972,59 %72,50 %
12710024501541,201407,7063472372,58 %72,60 %
13618119291340,501111,0063472472,65 %72,60 %

The incidence rates of the first week were therefore obviously calculated incorrectly, for the other calendar weeks a value of around 634,700 results for the total population. We therefore assume that the population on 1 January 2021 was included in the calculation here. According to STATEC, the population at that time was 634,730.

As the table shows, the vaccination rate q corresponds to the best approximation of the values of the proportion of the total population fully vaccinated (Cumulative vaccine uptake in total population in Luxembourg—Uptake of the primary course) submitted by the Ministry of Health to the ECDC[8]. It can therefore be assumed that these values were also used.

Unfortunately, we have to leave the question of how these values for the vaccination rate come about unanswered. All attempts to derive them from the numbers of fully vaccinated persons reported in the weekly reviews[9] did not lead to satisfactory results.

Thus, we have all the data we need to calculate incidence rates conforming to the previous approach. We additionally consider the vaccine effectiveness E, which we have already described in a previous article[10]:

    \[ E = 1 - \frac{i_V}{i_U} \]

It represents a rough measure of the extent to which vaccination relatively reduces the risk of infection.

Calculation example

According to the weekly report of 18/05/2022, the following applies to the number of people tested positive among vaccinated and unvaccinated people[11]:

With the ECDC[7] vaccination rate of 72.8% (0.728) for the corresponding calendar week 19, the following therefore applies to the incidence rates:

    \[ i_V = 100.000 \cdot \frac{p_V}{q \cdot N} = 100.000 \cdot \frac{2.185}{0,728 \cdot 634.730} = 472,86 \]


    \[ i_U = 100.000 \cdot \frac{p_U}{(1-q) \cdot N} = 100.000 \cdot \frac{519}{(1-0,728) \cdot 634.730} = 300,61 \]

For the vaccine effectiveness we get:

    \[ E = 1 - \frac{i_V}{i_U} = 1 -  \frac{472,86}{300,61}  = -0,573  = -57,3\% \]

The incidence rates and vaccine effectiveness calculated in the way described have now been presented in an interactive online graphic:

It is also possible to integrate the graphic into your own website:

<iframe src="" style="width: 1000px; height: 350px; border:1px solid black; overflow: hidden; padding: 0;" scrolling="no"></iframe>


The situation is clear: vaccinated people also test positive for the SARS-CoV-2 virus proportionately more often than unvaccinated people, and this is increasingly true.

Meanwhile, these are almost exclusively symptomatic cases, as hardly any testing is still carried out within the framework of contact tracing. Occasional vaccination breakthroughs can therefore no longer be spoken of, and the same development in other countries further confirms the situation.

Since it is known that the Covid-19 vaccines do not produce sterile immunity, vaccination does not provide any kind of foreign protection. Nor does it contribute to the establishment of herd immunity (the main argument for vaccinating children!), it is even counterproductive in this context.

In the context of the corona crisis, those responsible always like to refer to science, which is supposed to determine political decisions. However, the incidence rates discussed here show (once again) that the political discussion has meanwhile decoupled itself from any evidence.


[1] Ministère de la Santé: Coronavirus – Rétrospective de la semaine 28.07.2021

[2] Ministère de la Santé: Coronavirus – Rétrospective de la semaine 22.10.2021

[3] Expressis Verbis: Pandemie der Geimpften

[4] Ministère de la Santé: Coronavirus – Rétrospective de la semaine 13.04.2022

[5] RKI: Wöchentlicher COVID-19-Lagebericht vom 28.04.2022

[6] RKI: Wöchentlicher COVID-19-Lagebericht vom 05.05.2022

[7] Brief von Expressis Verbis an das Gesundheitsministerium vom 22.04.2022é.pdf

[8] ECDC Vaccine Tracker

[9] Coronavirus – Neuigkeiten

[10] Expressis Verbis: Parallelwelt der Zahlen

[11] Ministère de la Santé: Coronavirus – Rétrospective de la semaine 18.05.2022