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Understanding exponential growth

March 08, 2020

The past few days it dawned on the entire world that the Coronavirus is here to stay. It’s going to be a long battle. A few people have been claiming that the spread of the virus is going to follow an exponential growth. Exponential growth as a subject is a bit confusing to understand so I will try to understand it a little bit better. Hopefully it helps someone else too.

Man sitting in front of window

Exponential growth

Let’s start with a simple definition of what is exponential growth? According to Investopedia we can define exponential growth as followed:

Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function.

Exponential functions

Interesting, but this is introducing an important subject to us. Exponential functions. So lets try to find out what is an exponential function? Mathisfun to the rescue, according to it a basic form of an exponential function looks like this:

f(x)=axf(x) = a^x

In that function x is called the exponent. It’s what school taught us as power of. So an exponential function can be any function where the variable of it is in the power of part of it. Cool. I remember now.

Combining the two

OK, now that we understand exponential functions lets get back to exponential growth. Looking back to Investopedia again, and in various other resources too, we can find the following function being mentioned a lot:

V=S(1+R)TV = S * (1 + R) ^ T

V (for value) is the current value, and it’s equal to S (from starting) which is your starting value, times 1 plus R (for Rate), which is the growth rate, raised in the power of T which is the time. So if we wanted this to make it as an exponential function it would look like this:

f(x)=S(1+R)x,xεTf(x) = S*(1+R)^x, x \varepsilon T

We got a bit fancy, but all it does is representing the growth as a function with time being the variable. In normal people terms, all it’s saying is that your current growth rate (our value V) is directly proportional to our current size (S).

What is a rate? 😭

Wait, where is the time in the above description, and what is a rate? According to Merriam-Webster dictionary the rate is defined as follows:

A quantity, amount, or degree of something measured per unit of something else.

So in our case the rate is the time that passes. So that’s where the time is, in the word rate.

How do we calculate the rate though? To make things more confusing, when it comes to growth rates we need to calculate first the rate and then the growth rate. The calculation is fairly simple and it’s as follows (that gives us a percentage %):

R=100(FP)/PR = 100*(F - P)/P

Where R is the rate (in percentages), F is the future value and P is the present value. So that’s our rate. What about our growth rate? Here’s how to calculate that:

G=R/TG = R / T

Where G is our growth rate percent, R is the rate percent we calculated above and T is a number of time units (for example 12 months, 10 years, 24 hours etc).

If all that is too complicated there is this useful online rate calculator. Also further explanation can be found here too.

An example

OK, lets get some practical examples regarding exponential growth. Let’s say that you live in a flat in London and you have two mice living under your cupboard. Mice have an average of 5 babies per litter and they can reproduce every month. So according to our above equation we have:

  • S = 2, the original number of mice
  • R = 20%, the growth rate for 12 months is (R = (100*(7-2)/2)/12 = 20.83% = 20%)
  • T = 1..12, which is the time in months.

Now that we have all the numbers let’s see how our exponential growth equation works on our mouse infestation problem:

  • On month 1 we have V = 2*1.2^1 = 2.4 mice under our cupboard
  • On month 2 we have V = 2*1.2^2 = 2.88 mice under our cupboard
  • On month 12 we have V = 2*1.2^12 = 17.8 mice under our cupboard.

That might sound to you as not a big problem, but if you calculate the number for the 2 year mark you’ll find out that you have about 159 mice, and in the 3 year mark you have more than 1400 mice. This example is fictional and nature doesn’t work exactly like that but it shows the power of the exponential growth.

If you want to do your own calculation here’s an interesting calculator. Works nicely with the rate calculator I linked in the previous section.

On coronavirus

We can very clearly see how powerful exponential growth can be. Will Coronavirus be the same? I have no idea, I can barely understand exponential growth but I like my mice example. If you just calculate the numbers, base on some growth rate (which I personally think it’s the biggest assumption everyone makes) you will start hoarding toilet paper like everyone else. You can see that very quickly we can get to millions of people infected.

But the rate is very hard to calculate. A lot of the people infected with the virus have mild symptoms and they never present themselves. Therefore the numbers are slightly skewed towards the more serious cases. To give an understanding on how the rate can impact the result lets look my mice example above:

  • With a growth rate of 20% month over month, on year 3 we will have 1400 mice.
  • With a growth rate of 10% month over month, on year 3 we will have 61 mice.
  • With a growth rate of 5% month over month, on year 3 we will have 11 mice.
  • With a growth rate of 25% month over month, on year 3 we will have 6162 mice.

You can clearly see how big of a difference the rate makes. So yeah, take everything with a pinch of salt, be calm and wash your freaking hands. ALWAYS.


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