Exponential growth

Exponential growth occurs like branches on a tree. Each branch grows more branches, which grow more branches, so the rate of growing branches gets faster every year.

People often assume that exponential growth is rapid growth. This isn't necessarily the case. Something can grow fast, but not exponentially, or something can grow slow and exponentially. Exponential growth means that size of something is multiplied by a number in any given time. This compares to steady growth where the size has a number added to it in a given time.

For example, if you have a garden and hate mowing your lawn, you may think your grass grows fast. Every week it grows by 2 centimetres. Although this is fast it is not exponential - it is always growing at the same speed. Every week, 2 centimetres. Imagine instead, if you had exponential grass, which doubled in length every week. If it starts of at 5 millimetres tall, then in the first week it only grows 5 millimetres - slower than our normal grass. In the second week it doubles again adding on 10 millimetres, then 20 millimetres. At this point it's only just as fast as the normal grass. But then it grows by 4 centimetres, then 8 centimetres. By week nine, the grass is growing more than a metre per week.

This is what defines exponential growth. It means that instead of growing at a steady rate, something grows by a multiple in any given time. This could be a small multiple or a big multiple, for example if you have a bank account earning 1% interest, then your money is growing by a multiple of 1.01 every year. This is exponential, but you may consider it quite slow. Importantly the number must always be bigger than one. If the number is exactly one then no growth occurs. If the number is less than one then exponential decrease is occurring - multiplying by a number less than one gives a smaller number.

Why does exponential growth happen?

Exponential growth happens because after something grows, it has more opportunity to grow even further. Grass grows at a roughly steady rate, because no matter how far a shoot grows upwards, it is still just one shoot growing upwards. By contrast, if you are worried about how many weeds grow in your lawn - these spread exponentially. Each weed releases seeds and these grow allowing even more seeds to be released, which grow and allow even more seeds to be releases, and so on.
This exponential growth means that something that starts of small can suddenly become very important

Virus outbreaks

A new virus outbreak will spread exponentially. One infected person infects a few other people nearby. Now the virus has multiple hosts so it can infect even more people, which continues over and over again. This is related to the reproduction number, which indicates how many people each carrier will infect.

The difference between the exponential growth rate and the reproduction number is that time is taken into account with exponential growth rate - it takes into account both how many people a carrier infects, and how quickly they pass the disease on.

Faster all the time

Importantly exponential growth gets faster all the time. This means that something can start out relatively low level and not be causing any sort of problem. Then as the speed cranks up, suddenly it becomes a problem. It can seem like the problem came from absolutely nowhere. However, it was possibly always there, but it just wasn’t noticed until the growth rate grew and grew and kept on growing.

Exponential decrease

In the same way that exponential growth is the growth by a multiple each time, exponential decrease is the decrease by a factor each time. This has the opposite effect to exponential growth, the decrease becomes slower and slower and slower as time goes on. This means that if something is decreasing exponentially it will never disappear completely.