What can we use key figures for?

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Every day, we might use different websites to see how many new confirmed Covid-19 infections are there since yesterday – in our country and other countries as well. Often, we realize that the number of cases has increased again by a few hundred or even a few thousand, depending on which country or region we look at.

Our question, which led to the creation of this website, was: where do we stand at all? After more restrictions on freedom of movement and assembly being applied in Germany and many other countries, can we already see that the increase in the number of cases is slowing down? How can we judge this?

We believe that simple key figures can help in assessing the situation, i.e. the current spread rate of the virus in our country. In what follows, we introduce some of these of figures.

A simple key figure that is often used is the so-called doubling number. It shows in how many days the number of confirmed Covid-19 infections doubles. This figure is very well suited to describe the growth of the number of cases and it is very easy to interpret: If I know, for example, that the number of cases doubles every two days, then I can expect the double number of cases in two days, four times the number in four days, eight times the number in six days, and so on, and in 20 days finally about 1000 times the current number of cases. Hence, a doubling number of 2 obviously indicates that the country in question is threatened with catastrophic development and the rate of the virus spread does not slow down. Even a doubling number of 3 is not much better. Although it will take around 30 days until the case numbers are 1000 times higher than today, there is a risk of catastrophic further development.

In addition to the doubling number, another indicator is of interest in our view (which, by the way, is very closely linked to the doubling number). This is the average daily increase in the number of confirmed cases. This is also a figure that is relatively easy to interpret: If I know that the number of confirmed cases increases by 50% every day (which is a factor of 1.5), then after two days I can expect 2.25 times today’s number of cases and after three days 1.5 * 1.5 * 1.5 = 3.375 times today’s number. After 20 days we would then be 1.5^20 times (1.5 to the power of 20), that means approximately 3325 times today’s numbers.

This average daily increase in confirmed Covid-19 infections is also mentioned by some media. Here, we consider (otherwise, we would not speak of a “mean” increase) the mean increase over a few days, for example over the last 3 days, 5 days, 7 days or whatever period of time. This seems to make sense in order to reduce random influences on the values of the individual days (such cases arise for days where tests were not completed or the test results were not reported).

We suggest that when looking at the average daily increase, we not only look at snapshots, e.g. the average daily increase based on the last 7 days, but always at time series, e.g. the progression of the indicator “average daily increase of the last 3 days” over the last days or weeks.

Take Italy, for example. The course of the average daily increase in the number of cases (based in each case on the observation of the last three days) looks like this:

Diagram 1

We can therefore see that the average daily increase in the number of cases has decreased significantly in the course of March 2020. This means that the social distancing measures taken are working. As of 24.03.2020, there is still a significant increase in case numbers every day, but the increase has weakened. A statement such as “The pandemic is accelerating”, as it was recently reported in some media, does not apply in this sense at least – and certainly not to Italy.

However, the daily increase in case numbers in Italy is still far too high. How would the course of the key figure “average daily increase in the number of infected persons” look like?

Let us take a look at the course of this indicator for South Korea, from the point of view of many – including ours – a model for combating the pandemic:

Diagram 2

However, we see here that the key figure “average increase in case numbers” has been rising again for a few days now. This looks comparatively undramatic on the chart above, although there have been more than 100 new infected persons per day in the past few days. This is due to the fact that the key figure “Average increase in case numbers” always refers to all those infected so far. With more than 8,000 infected persons, some of whom have already been recovering for some time, an increase of, say, 100 infected persons does not lead to any effect that is apparent at first glance in the time course of the key figure “average increase in case numbers”.

For this reason, we propose a further indicator n/m Growth Indicator to two suitably selected values n and m, which we define as follows

What’s this formula all about? And how can we interpret this number?

The idea here is first of all that we do not look at the total number of infected persons, but at the new infections detected (more precisely: the number of positively tested cases) in the last n days and the last m days and put these in relation to each other and then “normalize” the result appropriately.

For example, if we choose n = 3 and m = 14, the result is as follows:

For these parameters n and m we count the number of new infections (for the sake of simplicity we will refer to new infections, i.e. the number of detected infections) once in the last 3 days and once in the last 14 days.

Let us assume that there were 1,200 new infections in the last 3 days and a total of 1,400 new infections in the last 14 days (including the last 3 days). This means that a large part of the number of new infections of the last 14 days took place in the last 3 days. The 3/14 Growth Indicator gives the result 4.0, as the following simple calculation shows:

How is this number 4.0 to be interpreted?

If the 1,400 new infections of the last 14 days were more or less evenly distributed over the days, we would have about 100 new infections per day. We would have considered a situation in which the number of new infections does not increase or decrease significantly from day to day, but rather there would only be random fluctuations around the value of 100.

For the last 3 days we could therefore expect approximately 300 new infections. However, in our example we have assumed 1,200 new infections in the last 3 days, which is 4 times as many. This is exactly what the 4 that resulted in the above calculation indicates, i.e., how many times is the number of new infections in the last 3 days higher than it would be if we took the average number of new infections per day based on the last 14 days? In our case, by a factor of 4.

Values greater than 1 thus indicate that the number of new infections per day tends to increase over time. A value of 1 means that the number of new infections per day remains approximately the same and a value less than 1 means that the number of new infections per day tends to decrease from day to day.

Let us now have a look at the course of the 3/14 Growth Indicator for South Korea:

Diagram 3

In this diagram we now see – from our point of view better than in diagram 2 above – that the number of new infections per day has tended to rise again since March 18. If this value were to rise above 1 again, and to be well above 1 again, this would mean that a new wave of infections could begin. If this were the case, the 3/14 Growth Indicator would point to such a situation earlier than the average daily increase.

Conclusion: Indicators such as the doubling number as well as the two indicators mentioned in this article can help to assess where we stand in the fight against the pandemic.

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