Understanding exponential functions in the times of the corona virus

Neil deGrasse Tyson spoke for the entire teaching community recently, when he said the following:

So if you were nodding off or were otherwise engaged when exponential functions were taught  in your class – or you just feel like a refresher – here’s some links to help you understand what exponential functions are, and why they matter so much where the corona virus is concerned:

First, from yours truly, in plain simple English: exponential functions essentially imply that y is going to change pretty darn quickly, even for very small changes in x.

“x” thoda si bhi change hone pe “y” legendray change kar jaayega

It means more than that, and there are exceptions, but if you are a non-math person, that line above is what you need to take away.

Here’s Wikipedia on the same topic, and here’s a short videoby Khan Academy:

(The link before the YouTube video will also have other, related videos and a practice set. Recommended)

If you want to play around with exponential graphs yourself, try Desmos:

Do you see the little “play” buttons next to a, b and c? Try clicking them and see what happens. “a” and “b” are crucial for social distancing. The lower those values, the slower the spread. Try it for yourself! (Note, this will work best on a desktop/laptop, rather than a phone)

So why does this matter in times of the corona virus?

On Monday, March 15, the US had about 4,000 confirmed cases. You might have said “Hey, that’s a tiny fraction of the country’s population. What’s all the fuss?” By Wednesday it had grown to around 8,000. So then you might think the total will grow by 4,000 every two days. That would be wrong; that’s linear thinking. It’s much worse than that.

That is from a Wired article, from which I will continue to quote below as well. So if four thousand becomes eight thousand, eight thousand becomes twelve thousand, and twelve thousand becomes sixteen thousand… not so bad, right?


With exponential growth, the number of new cases each day constantly increases—graph the total over time, and you’ll see that the line curves upward—and that can get you into big numbers real fast. What you need to look at is the percentage increase. In this case, it doubled (an increase of 100 percent) in two days. At that rate, it will grow from 8,000 on Wednesday to 16,000 on Friday, and 32,000 by Sunday.

Please read the rest of the article, and take your time doing so. This is important!

By the way, I’ve said this before on these pages, and I’ll say it again – don’t be confused when you look at a graph that shows a linear growth, but the chart says growth is exponential. First look at the axes!

For example, take a look at the picture below, taken from this post:

Please read the entire post, here is the link again.

Now, every article we’ve read so far has given us cause to worry, right?

Go back to the start of this article:

…exponential functions essentially imply that y is going to change pretty darn quickly, even for very small changes in x.

I used the word “changes” quite deliberately. You see, exponentials go up in a hurry, it is true – but they also come down in a hurry!

And since the number of new cases also depends on the number of infectious people (which declines as folks recover), that will also be exponential, but exponentially decreasing.

Bottom line; While the bad news grows rapidly, the good news will also evolve rapidly. So let’s just hope that by invoking sheltering-in-place and other strategies, we can reduce the infection rate and cause the inevitable bell-shape curve of the number that are sick to turn over sooner.

Think of the corona virus as us chugging up at the start of a roller-coaster ride. That’s kind of where we are right now. The good news is, when we start to “come down”, that’ll be pretty quick too.

But until then, one thing, and one thing only: social distancing!

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