Friday, 28 May 2021

Exponential Growth and the Virus: why it's hard understand (even if you think it's easy)

According to Dominic Cummings recent revelations, a lot of people in charge of the pandemic failed to grasp the idea of exponential growth. As Mike Bird says, we often ask kids what pocket money they would prefer: a pound every week or a penny that doubles every week. It should be a simple thing for people to understand but I would argue it isn’t.


If you are reading this you will probably have a good grasp of exponential growth. But I think for a lot of people it isn’t so obvious. This is what I would call “surely” knowledge. As in, you will say to me: “surely, people know this, it’s easy”.*


I think the reason for this is that there are not many things in life where we directly observe exponential growth, in real-time. The only thing I can think of is fire. You wouldn’t just leave a smouldering fire in the corner of the room, as you know it could burn down your home. But what is less intuitive is exactly how this occurs. I would highly recommend watching a video of it, just to see how it happens. 


At the start, the fire takes ages for it to do anything. It takes a minute for it to double in size. At this point, you think you can easily deal with it. And then, suddenly, it really starts to take off. Three minutes in and the room is completely on fire. This, to me, is what makes exponential growth so dangerous. It can lull you into a false sense of security that the situation is manageable before getting completely out of hand. 


I do think exponential growth is one of those things where you have to actually write down the numbers and work through an example to see what is going on. So let’s give it a go.

A quick recap, R represents the rate of how a virus (or anything) grows. An R number of 1.1 means that for every 10 people who have the virus, they will pass it on to 11 people. 


So if R is the growth rate per day, and there were 100 cases of infectious people, what would be the difference in case numbers when the R number was 1.1, 1.3 and 1.5, after 20 days? Before looking at the tables below or doing any calculation, write down your initial guess. When I did this, I was quite far off. I think it is a useful thing to do (especially if you think exponential growth is easy).

Let's look at the first 5 days of the virus.

Day

Cases R=1.1

Cases R=1.3

Cases R=1.5

Day 1

100

100

100

Day 2

110

130

150

Day 3

121

169

225

Day 4

133

220

338

Day 5

146

286

507


We can see that the case where R=1.5 is obviously bigger than 1.1, it is not quite double the number of cases but it is certainly not very large. However, at this stage in growth, things are still moving pretty slowly. This is the false sense of security phase where you think things are manageable. You may even think it's just a linear relationship and so anyone saying the sky is going to fall pretty soon are just Chicken Little. Now let’s look at the next 5 days.


Day

Cases R=1.1

Cases R=1.3

Cases R=1.5

Day 6

161

372

761

Day 7

177

484

1142

Day 8

195

629

1713

Day 9

215

818

2570

Day 10

237

1063

3855



On day 10, with an R number of 1.1, we have barely seen a doubling over the original value of cases. With an R of 1.3, it is only 5 times more than we originally started with. R=1.5, however, has really started to take off. This is the moment we would say a loud swear word. It is nearly 40 times its original size.


Day

Cases R=1.1

Cases R=1.3

Cases R=1.5

Day 11

261

1382

5783

Day 12

287

1797

8675

Day 13

316

2336

13013

Day 14

348

3037

19520

Day 15

383

3948

29280

Day 16

421

5132

43920

Day 17

463

6672

65880

Day 18

509

8674

98820

Day 19

560

11276

148230

Day 20

616

14659

222345


After 20 days you can really see the difference. At R=1.1, cases have only increased by a factor of 6. R=1.3 has just started its take-off phase and is looking pretty unmanageable.
But R=1.5 has increased by 222 times its original size. This is the point where you say many swear words in a row.

A difference of 0.4 seems so trivially small but has absolutely huge consequences.** I really do not think this is obvious to a lot of people until you have written it down. You may disagree, but my proof is in the pandemic pudding. If it really was so obvious, why did people fail to grasp it?



*When Tom and I were writing How to Read Numbers, I said we should put a brief explainer of the difference between percentages and percentage points. Tom said, “surely, people know the difference”. However, many people do not.

**I wonder if it may have been better to report a different number other than R. Imagine a number called C which stood for the number of cases we would expect if the virus goes unchecked for 20 days, based on 100 people having the virus. This way, an R of 1.1 would be C=616 and an R=1.5 would be C=222,345. This way, the difference doesn’t seem so small and in the end, what we really care about is cases, not the rate of growth per