A hundred thousand ignorant people

Recently, the term “vaccination rate” has increasingly been mentioned in the public debate, to put it mildly. Apparently, a value of 85% is necessary for a return to normality.[1,2] What this would actually look like is not described in detail, and this percentage is not derived in a way that makes scientific sense. The only thing that seems clear is that not enough people have been vaccinated, and that anyone who “refuses” is now to blame for the never-ending public health emergency, catchphrase: “pandemic of the unvaccinated”.[3]

Rather than going into the pros and cons of vaccination, here we simply want to analyse what vaccination rate can realistically be attained.

The evolution over time of the cumulative number of doses administered can be seen as a growth process that is limited by an upper value, in this case the number of doses that will eventually be administered to willing vaccinees. Such processes take place in very different contexts, for example in biology, epidemiology and economics, and are described by the so-called logistic function [4]:

    \[ f(t) = \frac{A}{1+e^{-(t-t_{mid})/k}} \]

If you take the number of 1st and 2nd vaccine doses administered as reported in the weekly data from the Ministry of Health, and present them cumulatively, the following graph emerges.

By logistic regression using the R-package “SSlogis” [5] we arrive at the following coefficient values:

A = 822.000     k=4,63     und     t_{mid} = 22,1

What does that actually mean?

The first coefficient gives us the value for the asymptote, i.e., the value that the number of vaccine doses administered will approach in the future without exceeding it. t_{mid} is the time of maximum growth, which thus already took place in the 22nd calendar week (beginning of June).

In other words: A great number of doses will probably no longer be administered, and we can assume a final level of around 820,000. 

Even in the 12-17 age group, for which the vaccine was most recently released, growth has already slowed down (Source: ECDC)[6].

Additional information on the percentage of fully vaccinated in the different age groups was provided on 28 September in a reply to a parliamentary question [7]:

And then?

As at 30 September, according to the daily report from the Ministry of Health, 405,343 people had been fully vaccinated. Of these, 37,712 people had had one dose of Janssen, and 367,631 people had been given 2 doses of one of the other 3 vaccines. A total of 786,645 doses had been administered.

Since we are interested in the proportions of both minors and adults in the “vaccine-eligible” group, we will consider the following 3 age groups:

Age group
0-11 years12-17 years18+ yearstotal
Persons in age group81.24239.752513.736634.730
Percentage of total population12,8%6,26%80,94%100%

As at the beginning of October 2021, we thus have the following situation:

Age group
0-11 years12-17 years18+ yearstotal
Percentage of fully vaccinated in age group0%51,1%74,95%63,86%
Fully vaccinated persons020.313385.030405.343

Our regression showed that we arrive at about 820,000 vaccine doses, so that around 34,000 doses will still be administered between now and the end of the year. If we simply assume that 2,000 doses of Janssen and 32,000 doses of the other vaccines will be administered, this means that an additional

2.000 + 32.000 / 2 = 18.000 persons

will be vaccinated. However, we do not yet know how this number will be distributed among the individual age groups. Let us therefore assume that we will have a similar rate for the 12-17 age group as in the 18-19 group, i.e. 64%.

Thus, by the end of the year 2021, we would have:

Age group
0-11 years12-17 years18+ yearstotal
Percentage of fully vaccinated in age group0%64%77,45%66,7%
Fully vaccinated persons025.441397.902423.343
Unvaccinated persons

This leaves about 116,000 unvaccinated adults; for the population as a whole, we arrive at 211,000 unvaccinated people, which corresponds to about one third.

In the political debate, the vaccination rate is usually regarded as a proportion of the population eligible for vaccination. Currently, this is all persons over 12 years of age, which amounts to:

634.730 – 81.242 = 553.488

As at the beginning of October, the rate of vaccinated people is then:

405.343 / 553.488 = 73,23 %

Accordingly, by the end of the year, the rate will be:

423.343 / 553.488 = 76,49 %

It is thus clear that, in all likelihood, the targeted rate of 85% will not be achieved.

On the other hand, there will probably remain a substantial 6-digit number of people who are not vaccinated, a proportion of the total population that our politicians cannot easily marginalise.


[1] RTL (07.09.2021): Dr. Schockmel plädéiert, fir net just d’Infektiounszuelen ze kucken https://www.rtl.lu/news/national/a/1782926.html

[2] RTL (02.10.2021): Patronat fuerdert d’Aféiere vum CovidCheck-System an de Betriber https://www.rtl.lu/news/national/a/1782926.html

[3] RTL (01.09.2021): Vum 15. September u keng gratis PCR-Tester méi https://www.rtl.lu/news/national/a/1779765.html

[4] Wikipedia: Logistische Funktion

[5] SSlogis: Self-Starting Nls Logistic Model https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/SSlogis

[6] ECDC: Covid-19 Vaccine Tracker

[7] RTL (01.09.2021): Impfquot vun de geimpfte Persounen nei gerechent https://www.rtl.lu/news/national/a/1793926.html