The mathematics of fear

Numbers have probably never played such a dominant role in the public eye as they have since the beginning of the pandemic. Who does not almost instinctively consult the daily results of PCR tests and the occupancy of hospital beds? The statistical interpretation of all these data should make it possible to assess the current evolution of the infection incidence or, for example, answer the question of the extent to which herd immunity already exists. This determines and legitimises political decisions and thus has a far-reaching influence on the fate of many people.

A term that keeps popping up and has become synonymous with the worst case scenario in this crisis is that of exponential growth:

The challenge was also different every day. In the beginning it was maintaining health. You have to remember that we had days with a hundred new infections, which increased exponentially. We always managed to avoid situations like those in the overburdened health systems abroad. But the biggest challenge was that we had to anticipate and also react at the same time. And as long as vaccination is not available, this challenge will remain. We should not think that it is over.

Xavier Bettel[1]

The drama of this crisis thus takes place, at least apparently, in the field of tension between a virus that needs to be tamed and ever-changing countermeasures that, if they fail, will lead to disaster. The role of the sword of Damocles is played by this potentially unchecked increase in new infections at any time, which would cause the health system to collapse.

So what exactly is this exponential growth ?

Exponential growth is when a quantity multiplies by a constant factor (greater than one) at each step. A classic example of the factor two is the legend according to which a king wanted to show his gratitude to the inventor of the game of chess and asked him to make a wish. The latter now wanted a grain of rice on the first square on the chessboard, two on the second square, four on the third square and so on.

Checkerboard rice
Picture: shutterstock

At first, the king thought this was a modest wish, but very quickly realised that he was incapable of fulfilling it: in fact, on the 64th square, the number of grains of rice is 9,223,372,036,854,775,808.

Here is the table with the corresponding calculations for the first 6 fields:

Field number xNumber of rice grains on field x
11 = 20
22 = 21
34 = 22
48 = 23
516 = 24
632 = 25

However, exponential growth generally exists when, as mentioned, this factor is greater than one. In the following graph, the factors 3, 2, and 1.5 are shown for comparison.

In the graph on the right, the vertical axis is scaled logarithmically. The curves then become straight lines with different slopes. This is a good method to examine the “goodness” of a supposedly exponential growth.

So we see: exponential growth represents very strong growth, which also accelerates indefinitely as it progresses.

What is the significance of this exponential growth in an epidemiological context?

For a better understanding, we will look at two examples which initially have nothing to do with the context under consideration here.

Example 1:

Speed checks will be carried out on three consecutive days. We assume that the checks will be carried out over the same period of time and at locations with a comparable traffic situation.

We can now analyse this data in different ways, depending on whether we take the number of controls into account or not, and whether we cumulate the number of drivers or not. The results are summarised in the following table:

Motorists per control20 / 1 = 2020 / 1 = 2040 / 2 = 20
Motorists cumulated2020+20=4020+20+40=80
Drivers per control cumulated2020+20=4020+20+20=60

We can now graphically display the lines “Motorists”, “Motorists per check”, “Motorists cumulated” and “Motorists per check cumulated”:

Example 2:

This differs from the first only in one respect: on the third day, 40 motorists are detected in a single check.

Motorists per control20 / 1 = 2020 / 1 = 2040 / 1= 40
Motorists cumulated2020+20=4020+20+40=80
Drivers per control cumulated2020+20=4020+20+40=80

We conclude:

  1. The first graph gives an overview of the absolute numbers per day, and does not take into account the number of checks that led to this result. If we are interested in the evolution of the data, where a comparison of the individual days must be possible, this representation is not suitable.
  2. In the second graph, the number of drivers is set in relation to the number of checks. This relative number describes the reality much more accurately: in example 1, the majority of checks led to a higher number of drivers, but there were not generally more drivers who did not observe the speed limit.
  3. If one accumulates the absolute figures as shown in graph 3, there is obviously a danger of misjudging the situation: both a constant behaviour of motorists with an increase in checks and an increase in speeding violations with the same number of checks can lead to an exponential growth!
  4. Point 3 can be remedied by cumulating the relative numbers (Figure 4). If the behaviour of car drivers does not change, this becomes apparent in a linear growth (example 1), in the other case in an exponential growth (example 2). The disadvantage of this representation, however, is that the values on the vertical axis can no longer be assigned to any real variable. We will therefore not consider them further.

We would now like to transfer these considerations to the epidemiological situation in Luxembourg. In our imaginary examples, we have assumed that the number of drivers found to be speeding increases with the number of checks. For this, it is necessary to assume that there is a sufficiently high number of unreported cases, which is then also reflected in the number of drivers during the (additional) checks.

If we apply this to the PCR tests, it must be shown that here too there is a sufficiently high number of people who could potentially be tested positive, so that a higher number of tests also leads to more people testing positive. (It is irrelevant whether a disease is actually present in this case).

The statistics [2] available since 13 July 2020 on the number of tests carried out in the framework of the LST and “tracing” confirm this: in the week from 13 to 19 July, of 685 people tested positive, only 244 were tested for “ordonnance” (i.e. suspicion due to clinical symptoms), the rest in the categories “LST”, “tracing” and “voucher aeroport”. Persons in the latter 3 categories would thus not have been statistically recorded and would have been part of the unreported cases.

It is interesting in this context that even Research Luxembourg is not sure whether an increase in the number of positive tests cannot also be caused by a simultaneous increase in the number of tests carried out. In its report of 2 July 2020 [3] it writes:

To evaluate whether this behaviour could be due to the increased number of tests, Figure 6 shows the relative number of positive cases per day and per number of tests […] .

Thus, even the Task Force considers the consideration of relative numbers, as described under point 2, to be the more reliable method to assess the development of the infection incidence.

This is all the more surprising since the alleged exponential growth of the infection figures was always derived from the presentation of the cumulative absolute figures. Since the beginning of the pandemic, such a development has been postulated 3 times so far.

First wave in March 2020

In the Research Luxembourg press conference of 7 May 2020 [4], a graph is shown in which the cumulative absolute numbers of positive PCR tests are represented as red dots. An exponential regression up to about 22 March with a doubling time of 2.28 days is drawn in grey and labelled “no lockdown scenario”. It is thus suggested that without measures, the development of infections would have taken this course.

We now look at the absolute numbers of positive PCR tests (blue, as bars, moving average over 7 days as a line, left-hand scale) as in the examples above, and additionally at the total number of tests (grey, as bars, moving average over 7 days as a line, right-hand scale) as well as the numbers of positive tests relative to the total number of tests (positive rate) in another graph.

We see immediately that both the number of positive tests and the total number are increasing, so we are in the situation of example 1: increase in tests without any exponential growth in the positive rate.

The next graph shows the reconstructed points of the cumulative absolute numbers, as well as the postulated exponential progression.

When the logarithmic scaling of the vertical axis is used, it becomes clear that the regression, which appears satisfactory at first glance, does have significant shortcomings: it actually only gives a satisfactory representation of the course for the points from 18 to 22 March.

Conclusion: the first wave does not show an exponential growth of the positive rate and even for the cumulative absolute numbers this is only true to a very limited extent.


As we had already shown in an article [5], the positive rate already reaches its maximum on 21 March, so the lockdown cannot be the cause of the decline in positive tests.

2. the wave in summer 2020

The second exponential growth is detected during the second wave in summer 2020. The publication of 28 August [6] shows the following graph.

The exponential growth is expected to extend until 16 July. We again look at the absolute and relative numbers of positive PCR tests:

We again note an increase in the total number of tests from 26 June to 19 July. The positive rate varies only minimally during this period and barely exceeds the 1 percent mark. Moreover, with such low prevalence values, one must also assume a not insignificant proportion of false positives. So there can be no talk of exponential growth. Only the cumulative absolute figures again show such a behaviour.

It is worth mentioning in this context that this “storm in a teacup” prompted Research Luxembourg to warn of an impending second wave for almost 6 weeks:

  • Already in the report of 18 June [7], the possibility of a second wave was addressed for the first time: “Potential actors of a second wave”, “Countries with second waves of infection”.
  • Then in the 2 July report [3], where an “exponential” progression of the positive rate was noted, but this was due to a single outlier on 23 June:
  • In the press conference of 9 July [8], the conclusions (Slide 32) state: “Current figures are alarming and may indicate a 2nd wave.”
  • Furthermore, in the report of 15 July [9] we read: “The course of the curve corresponds to an exponential increase, which can be expected at the beginning of a second wave” and: “Thus, one would have to assume a general second wave on the basis of the currently available case numbers.”
  • The report of 19 July [10] reaches the climax of the doom and gloom, where, according to projections, a maximum of 2000 (!) deaths should be expected by the end of August in the worst case scenario: The report of 19 July [10] reaches the climax of the doom and gloom, where, according to projections, a maximum of 2000 (!) deaths should be expected by the end of August in the worst case scenario: ”These most recent analyses underline the notion that, in the absence of any additional measures, the numbers of cases will continue to rise exponentially and that Luxembourg is witnessing a second wave in SARS-CoV-2 infections within its population. If the current dynamics persist, a shortfall in available intensive care unit (ICU) beds may be expected already by the end of August.
  • The all-clear finally comes on 24 July [11]: “Taken together, the updated analysis on the second wave provides evidence for a potential weakening of the second wave during the past days but that the number of active infections and ICU demands will still increase during the next days and potentially weeks.”

3rd wave in October 2020

Finally, the third exponential growth is said to have taken place in the period from 1 to 31 October, shown in green in the following graph from the 20 November publication [12].

As in the cases before, however, this again only concerns the cumulative absolute numbers. There was again an increase in the total number of tests in October, at the same time as the number of positive tests.

The positive rate increased towards the end of the month to reach its maximum on 26 October. In our article “How (not) to break a wave” [13] we had shown that this increase was linear and weakened before the measures came into force on 30 October.


As we proved at the beginning, an exponential course of the cumulated absolute numbers cannot be concluded to be the same for the infection dynamics. For mathematically incomprehensible reasons, this parameter was nevertheless used instead of the positive rate to evaluate the epidemiological situation. We examined the individual time periods for which exponential growth was postulated with this argumentation and could not confirm this assessment in any case using the criterion of the positive rate.

Exponential growth has therefore never taken place.

However, if one wants to lend dramatic emphasis to the narrative of a virus that can only be kept under control with constant measures (“an der Gitt halen”), such a construct comes in very handy.


[1] Luxemburger Wort (17.09.2020): Bettel: „Haben das Schlimmste verhindert“ [2] COVID-19 Rapport hebdomadaire du 30 juillet 2020 [3] Research Luxembourg (02.07.2020): Report: Analyse der Covid-19 Fälle im Hinblick auf eine mögliche 2. Welle [4] Research Luxembourg (07.05.2020): COVID-19 press conference [5] Expressis Verbis: Weitergehen, es gibt hier nichts zu sehen ! [6] Research Luxembourg (28.08.2020): Update of the current situation in Luxembourg [7] Research Luxembourg (18.06.2020): COVID-19 press conference [8] Research Luxembourg (09.07.2020): COVID-19 press conference [9] Research Luxembourg (15.07.2020): Analyse der COVID-19 Fallzahlen in Luxemburg im Hinblick auf die derzeitige Lage [10] Research Luxembourg (19.07.2020): Report: Controlling the second wave [11] Research Luxembourg (24.07.2020): Update of the current situation in Luxembourg [12] Research Luxembourg (20.11.2020): Update of the current situation in Luxembourg [13] Expressis Verbis: Wie man (k)eine Welle bricht