The Power of Exponential Growth
With the Coronavirus outbreak full under way in most European countries, it can be very helpful to understand the dynamics of such an outbreak. Let’s have a look at the cumulative number of confirmed cases from Germany over the last 15 days represented in the figure below.
As can be seen from the plot, the cumulative number of cases doubles roughly every 3 days: from 1908 cases on March 11th, over 4585 on March 14th and 9257 on March 17th, to 19848 on March 20th. This multiplication by a fixed factor (in this case 2) over equal periods of time (in this case 3 days) is typical for an exponential function. This is not a coincidence: epidemiologists have studied those types of outbreaks for a long time and it is well known that the initial phase of an epidemic follows exponential growth.
Exponential growth is characterized by the following formula:
This means that the number of cases on any given day i is equal to the number of cases on the previous day i – 1, times a constant g, also known as the growth factor. So the growth factor represents the relative increase of cases from one day to the next. In our example above, we have seen that the number of cases doubles roughly every 3 days. Knowing this, how can we determine g? The answer is simple: we just need to cascade the above formula 3 times:
From this we can conclude that for Germany
so
This means that the daily growth of cases in Germany is roughly 26%. This is a powerful piece of information, because it allows to extrapolate the number of cases into the future. Given the number of cases at given date x0 we can estimate the number of cases for any future date by using the formula:
We know that for example on March 14th, the number of cases was 4585. If we want to estimate the number of cases on March 20th (6 days later), we do it as follows:
which is not too far from the real value of 19848 cases (an error of 8%).
There are however better ways for estimating the growth factor. One widely used approach to find the growth factor from empirical observations is to use a statistical model called linear regression. Linear regression allows to estimate the best values for a and b in a linear relationship between the empirical observations y and x:
In the present case, y is the number of cases and x is the time. The formula looks similar to the exponential function but does not correspond exactly. There is however a trick to make both formulas match: the tool we need for this is logarithms. By applying the logarithm to both sides of the exponential function’s equation, the exponential functional can be linearised.
Now the formula matches the linear relationship, with the following correspondences between the two lines:
This means that we use the log of the number of infections instead of the number of infections and the log of the growth factor instead of the growth factor. Now instead of having to cope with exponential data, we are in the presence of linear data, for which it is extremely easy to estimate the unknown parameters a and b. To illustrate the idea, the same data as in the first figure is now plotted on a logarithmic scale (note the modified y-axis). The exponential curve has become a line!
Fitting a line to this data leads to the following result for the slope of that line:
Compared to the initial method, the linear regression provides a better estimate of the German growth rate: 28%. The estimation of the number of cases 6 days after March, 14th now leads to:
(an error of only 1.6% compared to the reported value). Projecting the fit back onto the original plot gives the following result. The fitted curve (dashed red curve) matches the input data quite well. We have now all we need to project our number of cases into the future.
Given that the effects of the counter-measures that are being put in place by the German government around the 20th of March will only be reflected in the number of confirmed cases with a delay of one to two weeks due to the incubation period (the time between the infection and the first symptoms), the number of cases is expected to continue growing with the computed growth rate for at least another 7 days. Taking March 20th as the reference date, this leads to an estimated number of
i.e. more than 10000 cases in Germany by March 27th.
The same approach can be applied to Luxembourg over the past 15 days, leading to the results shown in the following figures.
Obviously, Luxembourg has much less cases than Germany, which is of course due to the fact that the population is much smaller. The growth rate for Luxembourg however is estimated to 52%, i.e. a doubling in less than 1.5 days! This means that the cumulative number of confirmed cases is growing much faster than in Germany. The reason for this exceptionally high growth rate is currently not publicly known. It might be due to the ramping up of testing activities and it can only be hoped that the growth rate will decrease over the coming days. In any case, taking March 20th as the reference date and using the computed growth rate of 52%, the estimated number of cases by March 27th will be:
The estimated number of cases in Luxembourg will be almost 10000 until March 27th.
The following figure shows Germany and Luxembourg on the same plot to illustrate the respective projections into the future.
