So, exactly what is exponential growth and how can one tell if it is so?
By MUNGAI KIHANYA
The Sunday Nation
Nairobi,
05 April 2020
I saw a nice comment
on the internet recently: the amount of times people are spending
looking at exponential graphs is growing exponentially! Indeed, the
number of times the phrase “exponential growth” is being mentioned in
the mass media is also increasing exponentially. The reason is the
Covid-19 pandemic.
But exactly what is
exponential growth?
The basic feature of
exponential growth is that the rate of increment is proportional to the
time elapsed. In other words, the higher the value, the faster it
increases. This can be a dangerous trend in the case of an undesirable
event; for example, a disease outbreak.
Now, after the
confirmation of the first case of Covid-19 in Kenya, we waited three
days before that number changed. And then, on day four, it jumped
suddenly to three. Two days later it jumped to seven; then 15 and so on.
From then on, the
daily increment has been rising. At the time of writing this (Wednesday
the 1st of April), the government announced 22 new
confirmations in a single day! This trend – where the daily increment
rises day by day – points to an exponential trend.
However, from a
strict mathematical standpoint, it might not be a purely exponential
trend. To find out if it is, a mathematician would plot the data on a
logarithmic scale. Do you remember logarithms? Those things that you
thought would never be useful in real life? Well, it turns out that they
are!
Let me remind you: In
base ten, the logarithm of a number is the power to which we must raise
ten in order to get that number. For example, log1,000 = 3 because 10
raised to the power of 3 is 1,000 (10 x 10 x 10); similarly,
log1,000,000 is 6…and so on.
Thus the
mathematician works out the logarithms of the data (well, actually,
simply checks them out from a standard mathematical table!) and plots
the results on a graph against the time. If the it comes out as a
straight line, then the trend is a pure exponential.
If the logarithmic
plot curves to the upper side of a straight line, then it is increasing
faster than exponential. In the case of disease outbreak, such a trend
should get everyone very worried. It means the fight is being lost.
If the logarithmic
plot curves to the lower side of a straight line, then the rate of
growth is slowing down: which good news – it means we are winning! This
is what the experts are referring to when they talk about “flattening
the curve”. It is the logarithmic graph; not the direct (linear) one.
There is a website
that has been tracking the Covid-19 cases worldwide:
www.worldometers.info.
You can view both linear and logarithmic plots there. Unfortunately, it
hasn’t started showing the plots for Kenya just yet; but it is still a
good source of information.
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