The exponential growth of cases is rather like a fractal: zoom in or out, you’ll see essentially the same kind of curve, a gradual increase that turns to steep
The numbers keep flying about, in this surreal Corona-tion time. How many infected in which country, what kind of spread the virus is showing, how many tested and how many should be tested, on and on. A pandemic comes with its numbers—well it’s partly because of the numbers that we decide to call it a pandemic.
But what’s a citizen to do with all these numbers? Which should she pay attention to, which should she ignore? And what do they all mean?
I can’t answer every question that’s out there. But let’s examine some of these numbers.
First: In my last column, I explored the idea of exponential growth, and how it makes Corona the dangerous spectre it is. The growth is most visible when you plot it on a graph of the number of cases versus time. Almost invariably, the curve you get will follow a familiar path: gradual growth for a fairly long time, then the curve starts to steepen, almost imperceptibly but soon almost vertiginously. By the time it reaches the right edge of the graph, the curve is close to vertical.
The worry, of course, stems from that very near-vertical nature of the curve. As it should, of course. After all, what does that steepening mean? That it takes ever shorter slices of time for the same increment in the count.
For example, I’ve been keeping track of the coronavirus infection numbers in India over the last several days, and this is how they went up. (These are from three different trackers, one of which seems to have folded and another went through long spells without updates, so your numbers might vary).
(The Prime Minister made the same point when he spoke to us on Tuesday night, speaking of the first 100,000 cases in the world and then the next 100,000 cases).
Call the 300-400 increase (40 hours) a blip. The rest show that the increment of 100 cases is taking less and less time: a classic characteristic of exponential growth. And this is one reason this kind of growth is frightening. Maybe there’s a time coming when in the five minutes it takes you to read this column, 100 more Indians will have been infected. Maybe soon after that it will happen in the time you take to read this sentence. For all our sakes, it must not get there.
What this also means is that in some ways, the exponential growth of coronavirus cases is rather like a fractal. Zoom in or zoom out and you’ll see essentially the same kind of curve: gradual increase that turns to steep. Suppose we have the same growth over the next few weeks, say, resulting in thousands of cases. Then the growth that looks steep to us today—400 to 500 in 11 hours, for example—will seem gradual then, only because it is the precursor to the much steeper growth we’ll be seeing then. In the same way, the 30 hours it took to go from 200 to 300 cases seems slow now, but check how much faster it was than the rise from 0 to 200.
Which leads us to…
Second: Thus with exponential growth, the usual way we draw graphs—to a scale of one centimetre (cm) for 100 cases, for example—quickly becomes hard to comprehend. That’s because the growth gets so vertiginous that you have to re-scale—one cm for 1000 cases, perhaps—and re-scale again. This is the problem with using linear scales for a growth curve that isn’t linear at all.
Luckily, there’s a more reasonable way to draw exponential curves: use a different kind of scale. Instead of every centimetre representing a fixed increment of cases, we let it stand for a fixed ratio between case counts. That is, each additional centimetre represents not 100 more cases, but 10 times more cases (or some factor other than 10, if more convenient). So the labels are not 0, 100, 200, 300 and so on, but 1, 10, 100, 1000 and so on.
It’s called a logarithmic scale. Plotted this way, the exponential curve no longer rises steeply, but may instead look like a straight line and therefore, be easier to understand. Again, take a look at the numbers above. The case count went from 100 to 200—it doubled—in 4 days. The next doubling, from 200 to 400, took about 3 days (70 hours). Call those about the same, really. What about from 400 to 800, how long will that take?
At the time I write this, that remains to be seen. If it stays at 3 or 4 days, the coronavirus is likely spreading in India at about the same rate as it has till now. If it decreases, that spells trouble: the growth itself will be accelerating and may zoom out of control. But if—as I suspect and I hope I’m right—it takes longer, that may just be a glimmer of some light at the end of this Corona tunnel. For it will mean that while absolute numbers are still growing, the growth itself is slowing.
Third: consider Covid-19’s mortality rate. That’s the fraction of those who contract the disease who die. How do we calculate this? Simple: by dividing the number of deaths by the number of detected cases. Across the world as I write this, nearly 425,000 people are known to have been infected. So far, nearly 19,000 have died. Do the division: that’s about a 4.5% mortality rate.
Now admittedly, that is a high number compared to other flus—their mortality rates tend to hover at a maximum of 1%, usually less. This comparison may worry you. But remember that these are early days for Covid-19: we don’t have a vaccine yet, and we understand those other flus much better. And in fact, even with the lower mortality rates, other flus and other diseases kill many more every year than Covid-19 has so far—only because they infect that many more people.
And even so, there’s more to understand behind that Covid-19 mortality rate. Take Italy’s tragedy: nearly 70,000 cases, nearly 7,000 deaths: a 10% mortality rate. Or Spain: 42,000 cases and 3,000 deaths for a 7% mortality rate. France is at about 5%, Iran at nearly 8%, China at 4%. Even within China, Wuhan province has seen nearly 6% of its infected people die, whereas in the rest of the country it’s less than 1%. The US has the third-highest number of detected cases, nearly 55,000, but so far less than 800 deaths, or less than 1.5%. India? Ten deaths in just under 600 cases: 1.7%.
The point is, mortality rates vary widely by region and country right now. (They also vary widely by age). That’s the nature of a new disease with its new numbers. Besides, it’s possible that there are people with milder cases of Corona infection that they choose not to report, and then they recover. If there was a way to count those, we will find mortality rates reducing significantly. Some public health experts believe it will, like with other flus, eventually stabilize at around 1% or less.
Does this mean we are over-reacting to this virus, that it is no different from other flus that we don’t pay much attention to? Not at all, precisely because we don’t yet fully understand the disease, and certainly don’t yet have a cure for it. But it does argue for some perspective: With reasonable care, the great majority of those who are infected will recover.
And fourth: do these numbers make a case for a lockdown? I’ll stay locked down over the next three weeks, certainly, but I think the numbers available to us so far might at least offer some food for thought. For me, it might go like this: if I’m in good health, clearly I have a better chance of battling this or any disease. But will my good health be compromised by spending day after day indoors? If so, does that make me more vulnerable to the disease? How does that stack up against the risk of contracting the disease if I step out, or the risk of passing it on if I am already a carrier, and should Covid-19’s mortality rate figure in this exercise of weighing risks?
No easy answers, but then this is not an easy time. Yet poring over the numbers as I have been doing these last several days, and not meaning to downplay the tragedies Covid-19 has already wrought, I feel a faint but distinct optimism. About India, about our world.
I hope you will too.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun
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