Resistance is futile: COVID-19 - flattening the curve
My last post was in September 2015. Somehow, after that, I lost interest in
writing my sermons and stepped off the soap box. But the way in which
politicians and journalists speak abut "flattening the curve" when talking
about COVID-19 as if it were obvious to the vast majority of the public
prompted me to attempt an explanation of what "flattening the curve" actually
means. It still involves logarithmic plots, which many will find confusing,
but it might help.
I want to post this article as quickly as possible. My apologies for the
typos I will inevitably make.
Here are the curves the politicians are talking about, drawn for China,
Korea, Iran, and Japan:
The numbers at the bottom indicate the days since the World Health Organisation
has started reporting data on COVID-19 on 2020-01-21. You can see them on the
WHO web site. For this plot, I have used all the reports till #63,
published today (2020-03-23).
The numbers on the left tell you how many new cases were reported for each day.
In fact, this is not entirely true, because the plot shows weekly averages
to avoid wild fluctuations. That is, every point of each curve is averaged
with the preceding and following three points. Therefore, these plots are
useful to see the trends, rather than individual values.
The numbers of daily new cases of all countries represented in this plot
reached a maximum before starting to decline. It means that the drastic
measures taken in those countries managed to bring the contagoin under
control. Notice that Korea and Japan have an initial "bump" followed by
a systematic increase. This could be due to the transition from imported
cases to community-transmitted cases, but it is only my speculation and I
could be completely wrong.
More importantly, note that China and Korea are experiencing a resurgence
of new cases in the past week or so. This could be due to the relaxing
of the containment measures or, as China has stated in several occasions,
to infected residents returning home from abroad, thereby carrying the
virus back home. In any case, unless great attention is paid, the contagion
could flare up again, like a non-completely extinguished bush fire.
While China, Korea, and Iran experienced a rapid increase in new cases,
Japan quickly managed to bring the increases under control, as shown by
the fact that the curve is "flatter" (first hint at what "flattening the
curve" means, although it will become clear at the end).
Let's have a look at Germany, Italy, and Spain:
The curves are bent but haven't reached a maximum. This means that the
measures adopted by these countries has started to bite, but the situation
will become worse before beginning to improve. In other words, the bending
of the curves indicate that the number of daily new cases is still increasing,
although less rapidly. The days in which the number of new infections will
begin to decrease is still to come.
Finally, let's have a look at Australia and the United States:
Do you see how the lines are straight up? These countries are
still in the "exploding" phase of the contagion. In semi-logarithmic plots,
straight lines mean exponential growth. It means that the number of new cases
is growing exponentially. The situation in the USA is worse than
in Australia because in Australia the daily increase is around a couple of
hundred, while in the USA they get several few thousand new cases per day.
The last plot I want to show you is of the total number of cases, rather than
of the number of daily new cases:
First of all, notice that the numbers on the left now reach 100,000. For
those with knowledge of Mathematics, I will say that these curves are the
integral of those shown in the first three plots. That is, these curves
show the areas under the previous curves. Perhaps not surprisingly, the bottom
curve of this fourth plot is that of Japan, which is the country with the
lowest number of daily new infections.
As I already said, a straight line represents an exponential growth. The thin
grey lines are there for reference, and tell you in practical terms how to
read the country-specific curves. The slope of the lowest (dashed) thin line
represents a doubling of the total number of cases every 10 days. As you
can see, Japan managed to contain their total number of cases around that
figure, as the curve for Japan is almost parallel to the 10-day-doubling line.
The other thin lines, closer and closer to the vertical, represent doublings
of total number of cases every 5, 4, 3, and 2 days. As you can see, the curves
of most of the countries shown are clustered around the 2-day-doubling line,
the only exception being Australia, which is close to the 3-day-doubling line.
These are the curves that the governments try to flatten with their measures
(some might refer to the curves shown in the first three plots, but if you
flatten one, you also flatten the other).
Here, like in the first three plots, you can clearly see that China
and Korea have managed to flatten their curves, while the USA and Australia
are still shooting straight up.
To give you a better idea of what a 2-day-doubling means, consider that
each 100 infected people become 1131 after one week, 12,800
after two weeks, and 144,815 cases after three weeks. Staggering numbers.
With 10-day-doublings, the initial 100 cases become 162 after one week,
264 after two weeks, and 429 after three weeks. This is the difference
betweem Italy, overwhelmed by the sick, and Japan.
