Incidence for everybody

In the search for a “reliable” indicator for assessing the current epidemiological situation, pretty much every possible and impossible variable has been used over the years: daily new infections and/or death figures, utilisation of hospital capacities or, for example, the reproduction figure.

In recent weeks, the good old “7-day incidence per 100,000 inhabitants” has been recalled. In our neighbouring country Germany, it is now to determine “the severity of the lockdown and the measures to contain the pandemic” [1] as part of the third wave. For the upper limit, it was finally agreed on 100 new infections; if this value is not reached, “cautious openings are possible”.

A vigilant observer drew our attention to the fact that in Luxembourg, too, the word “incidence” has been used conspicuously more often in the media headlines for some time [2,3,4]. The Ministry of Education has been thinking in terms of weekly incidence for a long time and also sees the latter as a decision-making criterion: the upper limit up to which face-to-face teaching should be possible has been set at 300 new infections per week in the school community [5]. 

The fact that incidence is not an undisputed variable became clear last summer, among other things: Luxembourg, as the “test world champion”, was classified as a risk area several times due to its excessively high incidence, although the incidence of infection would hardly have justified this in itself: After all, the positivity rate, for example, was not higher than in countries that were considered more harmless. Luxembourg then unceremoniously removed the 200,000 commuters from the statistics in order to at least gain some breathing space. Since this group, with a population that is only about three times larger, certainly contributes a non-negligible share to the incidence of infection, such an act can really only be described as sleight of hand. The fact that this nevertheless went through already showed the whole absurdity of the situation at this point.

Be that as it may, there is a real probability that incidence could determine our everyday life in the future, in whatever way. We should therefore take a closer look at this epidemiological variable.

Incidence – past and present

In medical statistics, incidence refers to the number of new cases of disease occurring in a certain period of time. In keeping with the meaning of the Latin verb incidere: to occur, to happen, one assumes that these manifest themselves “of their own accord”, for example that a person with symptoms seeks medical treatment, or that a certain disease is determined as the cause of death post mortem.

In Corona times, everything is different: one no longer waits for a case of illness to occur in the health system, but also wants to identify asymptomatic carriers of the virus in order to break chains of infection. As is well known, this is done on the one hand by the Large Scale Test (LST) and on the other hand by tracing the contacts of those tested positive with symptoms. The latter are listed by Santé in the statistics under “sur ordonnance”.

The PCR tests used for this purpose will be supplemented in the near future by the widespread use of rapid antigen tests. However, both methods have their limitations and it is now considered proven that a positive test alone cannot be equated with a disease, i.e. not with a case of disease, which could then contribute to the incidence in statistics. In the case of asymptomatic persons, this also applies to infectivity. In this context, we refer to the article in The Lancet [6]:

It is a net loss to the health, social, and economic wellbeing of communities if post-infectious individuals test positive and isolate for 10 days. In our view, current PCR testing is therefore not the appropriate gold standard for evaluating a SARS-CoV-2 public health test.

or the WHO guideline [7], which questions the significance of positive PCR tests in asymptomatic persons. The Santé also advises not to carry out a second test after the isolation phase [8]:

The risk of transmission of a person who is symptom-free is extremely low after the 10th day. However, he or she may remain positive for a longer period of time. 

Nevertheless, all those who test positive are treated equally in the statistics: As a case of illness.

The situation in Luxembourg

Let’s look at the figures for Luxembourg. We look at the period from the 36th calendar week (CW) 2020 (31/08 – 06/09). You can easily see the “second wave” in the number of positive cases (in red, left scale), which builds up to reach its maximum in week 44 and has almost completely subsided again in week 52. From week 3 onwards, the cases then rise steadily again. It should be noted that the latter also applies to the total number of tests carried out (in purple, right-hand scale).

The incidence differs only by a fixed factor from the number of positive tests per week:

    \[ \text{Incidence} =  \frac{100000}{ 626000} \cdot \text{Number of positive Tests per week} \]

It is therefore irrelevant for the trend development whether we look at the positive tests or the incidence.

Let us now take a closer look at this epidemiological parameter, analyse how it comes about and find out how it behaves when other parameters involved in the infection process change. To do this, we use three different perspectives.

1. composition of those tested positive

Let us look at the composition of the caseload according to the categories: LST, tracing and “sur ordonnance” (diagnostic):

For calendar weeks 46/2020 to 4/2021, no data are available separately in the “Tracing” category; these were offset together with “Diagnostic”. Let us therefore look at the percentage shares of the number of cases for the available data:

It is noticeable 
that the share of tracing has increased in 
recent weeks and, together with LST, accounts for about 2/3 of the cases. Since it can be assumed that in cases where someone is symptomatic, a positive test does not first appear in the LST or during tracing, 2/3 of all those who test positive are asymptomatic and, to put it bluntly, only find out that they have come into contact with the virus through a test. These people therefore do not represent a burden for the hospitals and the legitimate question remains why they are counted as cases and included in the incidence.

2. incidence versus positivity rate

next graph shows the weekly incidence per 100,000 inhabitants (red, left scale), as well as the positivity rate in percent (green, right scale). We obtain the latter with the formula:

    \[ \text{Positivity rate in percent} = 100 \cdot \frac{\text{Number of positive tests}}{\text{Total number of tests}} \]

What looks similar at first glance, however, differs in detail. Let’s look at the section from week 2 onwards, for example: while the positivity rate fluctuates around the 2% value, the incidence increases by an average of around 7 cases per week during this period.

With the positivity rate, we cut out the total number of tests, so to speak, and we get an indicator that is independent of this. The incidence, on the other hand, shows a covariance between these variables. If we test more, we will find more cases, because the number of unreported cases will be smaller. (Of course, this does not mean that no one is sick if no testing is done).

The positivity rate is therefore a much more reliable indicator than the incidence. The latter can take on high values, and even increase, without this corresponding to the incidence of infection.

Calculation example

What is the effect of a cluster of 60 positive test persons in a group of 300 persons (e.g. in a nursing home) on the incidence and positivity rate (per week and per 100,000)?

we take the average of the four weeks in March 2021, we get:

  • Total number of tests: 63.658
  • Tested positive: 1.420
  • Positivity rate: 2,22%
  • Incidence: 227

So we test an additional 300 people and the number of people testing positive increases by 60:

    \[ \text{new positivity rate} = 100 \cdot \frac{1420+60}{63658+300}  \approx 2,31\% \]

    \[ \text{neue Inzidenz} =  \frac{100.000}{626.000} \cdot (1.420+60) \approx 236 \]

Thus, the absolute changes for these two variables are:

Positivity rate: 2,31 − 2,22 = 0,09 % and incidence: 236 − 227 = 9

Even if the relative changes are only insignificantly different, the incidence increases by 9 cases, which could lead to more stringent measures 
depending on the upper limit.

3. Different countries, different incidences

A virus knows no national borders – so they say. In the following graph, we have visualised the incidences of our 3 neighbouring countries. The weekly number of tests is shown as a dashed line (right scale); it has been normalised to 100000 inhabitants, just like the incidence. Sources: Santé, RKI and

Obviously the incidences of the individual countries differ greatly. At the peak of the second wave, for example, Belgium reaches a value of over 1000, followed at the same time by Luxembourg with 752 and France with 553. Germany reaches a maximum here only 6 weeks later with only 196 cases, which is also very flat in comparison with the other countries.

contrast, a correlation between the incidence and the total number of tests per country can be observed: The peaks in the incidence of the second wave are more or less reflected in the development of the number of tests, both in terms of timing and breadth. Finally, in the last few weeks, the increase in the number of tests is also reflected in the incidences.

The question of meaning

Germany will go the way of making the tightening or easing of pandemic measures dependent on the incidence value. The value of one hundred is probably more “politically” motivated; in any case, a sound scientific justification for this has not yet been provided.

And Luxembourg? Do we follow suit, and if so, with what threshold value? Is there even a meaningful way to define this value?

If incidence happens “by itself”, i.e. through cases of illness or death with clear symptoms, then it undoubtedly has its justification and a fixed place in medical statistics. But that is not what we are talking about here. By testing for suspicion, asymptomatic “virus carriers” are found who might never have appeared in statistics in the past.

Moreover, a scientific principle is broken: the result of a measurement must not depend on the measurement itself. Obviously, this is at least not always the case here: the more one searches, the more one will find. We have repeatedly been able to establish a correlation between the development of incidence and the total number of tests.

Finally, if one starts to compare individual countries in terms of incidence in order to abstract some logic in the whole, the meaningfulness fails completely. The number of parameters that contribute to the value of incidence must be endless.

Maybe we should just try something new.


[1] (31.03.2021): So ist die Corona-Infektionslage in Ihrem Landkreis

[2] RTL (24.03.2021): De Corona-Inzidenz-Taux klëmmt op 250 Fäll op 100.000 Awunner

[3] RTL (31.03.2021): 7-Deeg-Inzidenz klëmmt zu Lëtzebuerg op 269 pro 100.000 Awunner

[4] RTL (07.04.2021): Inzidenztaux ass a praktesch allen Alterskategorien erofgaangen

[5] RTL (15.03.2021): No der Ouschtervakanz kommen d’Schnelltester an de Schoulen

[6] Clarifying the evidence on SARS-CoV-2 antigen rapid tests in public health responses to COVID-19

[7] WHO: Criteria for releasing COVID-19 patients from isolation

[8] Isolation, Quarantäne und Behandlung