Understanding Exponential Growth
The mathematics of pandemics
Exponential growth is a simple idea, with far-reaching consequences, and it’s something that everyone really ought to understand.
When something grows in proportion to how big it is already, it grows exponentially. So if it’s twice as big, that means it’ll grow twice as fast. It follows from this that it will regularly double in size. That doesn’t necessarily mean it will grow quickly — it might take 100 years to double in size! What it does mean is that however fast or slow it might be going now, as long as exponential growth continues, you can expect it to go much faster in future.
Exponential growth has a way of catching people out, because it tends to seem slow… until it really really doesn’t.
To take one example, outbreaks of a disease often undergo a period of exponential growth. The total number of known cases of COVID-19 in the UK increased by a factor of just over 1.25 each day during March, on average: every day, there were about a quarter again as many cases as the day before. That gives a doubling time of around three days. The doubling time has slowed since the start of April, probably because measures taken to implement physical distancing and shut down mass gatherings are finally starting to show up in the statistics — the virus has an incubation period of about 1–2 weeks, so it takes a while before we can assess the effects of actions taken.
While the rate of doubling has slowed down, the rate of new infections is still just about as high as it ever was. After all, there are more people with the disease than ever, it’s just that each one is not passing it on to so many people. It’s easier to feel optimistic about what’s happening if we look at a chart with a logarithmic scale.
You’ll see a lot of logarithmic graphs if you follow the news, because they really help to make sense of exponential growth, especially when it comes to comparing different countries. That means the numbers on the vertical axis double at regular intervals, so that periodic doubling becomes a straight line, rather than the hockey-stick-like shape you get if you plot the numbers directly. John Burn-Murdoch, the Financial Times’ senior data-visualisation journalist, explains why he uses these in this video, and regularly digs into the data on Twitter. Rather than the height of the line being relative to the number of cases or deaths, it relates to its logarithm.
Logarithms are the inverse of exponentials, and like exponentials, they are a fantastically powerful tool in maths and science. To give you some idea how they work, the logarithm of 10 is 1; log(1000) is 3; log(1,000,000) is 6. The logarithm of a number is the power you need to raise ten to, to get that number: 10³=10×10×10=1000, so log(1000)=3. Human perception is also based on logarithms, incidentally — this is how we manage to be so sensitive to small changes in small amounts, like hearing a pin drop in a silent room. The more intense the input is, the less sensitive we are to changes in it. Similar effects are at play across the senses, allowing us to make sense of a far greater range of variation than we could otherwise.
The root cause of exponential growth in infections is that each individual catching a disease tends to infect a number of other people. The average of this number is the effective reproduction number, sometimes called R.
We can say that ‘on average, one person infects two or three others’, but it’s vital to realise this is just the average — it is absolute nonsense to go on to say ‘You therefore have a very low probability of infecting a large number of people in a stadium.’ The number of people that any one individual passes it on to depends on how many they’re meeting, in what context, and with what hygiene precautions. ‘Super-spreader events’ can occur where one person infects dozens of others, and especially in the early stages of an outbreak, these events can make a huge difference to the rate of spreading. People spreading a disease often show no signs of it themselves.
Measures like physical distancing, handwashing and mask-wearing are so helpful because they tend reduce how many people each case passes the virus on to. By bringing the effective reproduction number down, they slow the time it takes for the number of new cases to double. If we can bring the effective reproduction number down below one, most people with the virus never pass it on, and we move from exponential growth to exponential decay: rather than going up in proportion to how many are infected, cases go down in proportion to how many are still infected. We can start talking about half life — the time it takes for something to reduce by half — rather than doubling time.
Immunity is another way to bring the effective reproduction number down — obviously, the higher the proportion of the population that is immune to an illness, the fewer people can get infected. This is the idea behind ‘herd immunity’; if enough of the population is immune, the effective reproduction number naturally drops down to one, or less than one with the help of a vaccine, and spread is limited. For extremely contagious diseases like measles, where each person is likely to infect a dozen or so others, this requires upwards of 90% of the population to be immune. Vaccination drives have made herd immunity for measles possible in many countries. Unfortunately, pseudoscientific panic about vaccines has led to many children going unvaccinated in recent years.
Herd immunity kicks in sooner for diseases with a lower reproduction number: if each person is infecting an average of 2.5 others in the absence of any immunity, the threshold for herd immunity to bring that down to 1 is 60% of the population. Vaccinations are the way to achieve real herd immunity for dangerous diseases. Allowing them to run their course through the population does too much damage, and takes too long — there is a danger that the disease will mutate into a new form, allowing it to get around existing immunity, which is more likely the more people are infected. There is also a risk that the human immune system may only develop short-lived defences.
Germs can grow exponentially on a more local level, too: bacteria reproduce by dividing in two, and under the right conditions they can do this on a regular basis. Depending on the bacterium, they could double in number every twenty minutes or so. After one hour, you’d have eight times as many as you started with; after two hours, sixty-four times as many. After twelve hours, about 69 billion times as many. After a full day you’d need a 21-digit number to write it down.
If you think something’s starting to look fishy there, you’re not wrong. Bacteria are very, very small, but the result we’ve just come up with would still suggest 5,000 tonnes of bacteria could descend from a single cell in just a day. In practice, exponential growth never goes on forever. Bacteria run out of food; economic growth hits up against planetary resource limits; disease outbreaks are either contained, or they infect so many people that they have nowhere else to go.
Exponential growth is all around us, and as we have seen in the COVID-19 pandemic, a failure to recognise it and to understand what it means in practice can have very serious real-world implications. I’ve skimmed over the mathematical details here, but if you’re at all interested in making sense of how the world works, I encourage you to dig in some time.