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Home >Science >News >Opinion | When it’s exponential, we worry. As we should

Three small countries born of my imagination each has a population of just about 1 million people. Each detected their first cases of people infected with novel coronavirus on the same day: last Friday. As the days go by, the count of cases in all three rises—as you no doubt expect—but in different ways.

One country, Borduria, finds they have 500 new cases every day. Thus on day 2, they have a total of 501 cases. Day 3: 1,001. Day 4: 1,501, and so on.

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The second, San Theodoros, finds that their count of cases is the square of the number of days since Day Zero. This may seem strange, but this is the way with Theodoros, which on a map is even a perfect square. Thus on day 2, they have 4 cases. Day 3: 9. Day 4: 16, and so on.

The third, Middle Earth, has twice as many cases every day as the day before. Thus on day 2, they have 2 cases. Day 3: 4. Day 4: 8, and so on.

Now these countries are known to be notoriously lax about any health precautions. (Lucky that they are also fictional). Relevant here is that they have no way at all to check the spread of Covid-19 through their populations. Their only hope is a vaccine to fight the disease, but they know that’s unlikely to be available for at least several years. So the virus spreads without hindrance in each country.

Question: which country will first see its entire population infected with the novel coronavirus? If you don’t have an idea already, take a quick guess before working out the mathematics.

If you went only by the numbers above, you’d probably choose Borduria as your answer—why, on Day 4 they have over 1,500 cases compared to San Theodoros’s 16 and Middle Earth’s 8! Surely there’ll be blanket Covid-19 coverage there before the other two?

Well, actually no. To start with, San Theodoros’s count will overtake Borduria’s count on Day 500—250,000 to 249,500. Going on from there, all of San Theodoros’s million residents will be infected on Day 1,000, but it will take till Day 2,001 for the virus to reach every one of Borduria’s million. That is, despite the impression those few numbers above give us, the growth of the infection is actually faster in San Theodoros than in Borduria.

Ah, and what about Middle Earth, whose numbers above are far lower than the other two countries’? Well, appearances are deceptive indeed. Because on Day 7 — today! — Middle Earth will have more cases than San Theodoros—64 to 49. On Day 14— next Friday! — Middle Earth will have overtaken Borduria as well—8,192 to 6,501 cases. And on Day 20—the following Thursday!—all of Middle Earth will be infected. Yes, every one of its million people.

Think of that: 20 days, compared to 1,000 and 2,001.

That’s the power, the danger, of the kind of growth Middle Earth will experience in its brush with coronavirus. It’s called exponential growth and it is faster than any other kind of growth we know about. In this case, it is certainly faster than the arithmetic and geometric growth in Borduria’s and San Theodoros’s case counts, respectively.

And while the three countries are fictional, and the numbers I’ve used are fictional as well, what isn’t fiction by any means is this: coronavirus’s spread so far, in country after country, is exponential. Fundamentally, this is why it is such a threat to the world. Yes, we don’t yet have a vaccine to fight the virus and that’s part of the reason too. But the speed it has shown in spreading so far deprives us of the time we need to find that vaccine, and that’s the spectre we’re staring at.

So what is exponential growth? Essentially, it means that the number of cases multiplies daily by a fixed factor that’s greater than 1. In Middle Earth’s case above, that factor is 2, so each day’s count is twice the previous day’s. In comparison, Borduria’s numbers grow by the daily addition of a fixed number (not a factor). San Theodoros’s, by an amount that depends on the particular day. This would be an excellent place for a diagram: for if you plot a graph with all three of these growth curves on it, it’s easy to see that Middle Earth swiftly outstrips the other two.

And in fact, the daily increments themselves tell the story. They are an indication of what, in calculus, is called the (first) derivative of the growth function, and this derivative tells us how fast the growth is. (Think of acceleration: it tells you how fast your speed is changing). For Borduria, this derivative is the fixed number 500 — each day, 500 more cases are detected. For San Theodoros, it is twice the serial number of the day we are considering, plus 1; thus on Day 4, they detect 9 more cases; on Day 10, 21 more cases; and so on.

But here’s the kicker. The derivative of Middle Earth’s exponential growth function is—are you ready?—itself exponential. Check: on Day 1, they detect 1 more case. On Day 2, 2 more cases. On Day 3, 4 more. Day 4, 8 more. You can see that the increments form the same series of numbers as the counts themselves. Put another way, it’s as if each case detected immediately produces another; the growth depends on the number of existing cases. And this is why exponential growth eventually outstrips anything else.

Let’s be clear: while the growth of coronavirus has been exponential, the growth factor is not Middle Earth’s 2. In China, until they managed to slow the spread, the daily factor was about 1.25. But that’s just a detail. Even that kind of growth—for that matter, growth with a factor greater than 1 by any amount however tiny—will outstrip anything else.

Indeed and again, that’s the power and danger of exponential growth.

Now India’s infection count is still relatively small, and we don’t have enough data yet to nail down how it is growing. But the early indications suggest that it is exponential. There’s no reason to think it wouldn’t be so — after all, new infections happen from existing infections, the very prescription for exponential growth. Nor is there reason to think our experience with its growth will be different from China’s, or Italy’s, or anywhere else.

This is hardly the time or place for more than just this much: Covid-19 is too much of a Sword of Damocles hanging over us right now. But let me leave you with at least a hint of the fascination this kind of analysis has been for mathematicians for many centuries.

At one point in their The Mathematical Experience, the mathematicians Philip Davis and Reuben Hersh write about just this kind of growth, observing that studying it has “ancient origins" just like studying trigonometric ratios does. There’s a reason they mention those two, because they are linked in a truly deep way. In a short section titled “Unity in Diversity", Davis and Hersh write:

“Unification, the establishment of a relationship between seemingly diverse objects, is at once one of the great motivating forces and one of the great sources of aesthetic satisfaction in mathematics. It is beautifully illustrated by the formula of Euler which unifies the trigonometric functions with the ‘power’ or ‘exponential’ functions."

When we’ve emerged from under the coronavirus menace, I will attempt a column explaining the “aesthetic satisfaction"—yes! in mathematics!—of Euler’s almost mystical formula. That’s a promise.

Until then, wash your hands regularly and don’t hoard stuff.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun