Exponential Growth

November 22, 2020

We’ll have to wait till tomorrow to see some B.C. numbers, so until then, let’s shift our attention eastward by one province and look at Alberta, who unfortunately is giving us a textbook example of what exponential growth looks like.

For today, I’ve added a third row of graphs. The top row of the three is each province’s journey through this pandemic, from day one. The bottom two rows represent only the 2nd wave; first, logarithmically… and, the bottom one, normally.

They say a picture is worth a thousand words, but sometimes, it’s not because the picture is so indescribably beautiful… but because the picture is difficult to describe. If math isn’t your thing, hearing a sentence like “the plot an exponential curve on a logarithmic scale will be a perfectly straight line.”… might sound confusing. But when you look at the pictures, it makes perfect sense.

Have a look at Alberta, and have a look at the bottom two graphs. They are displaying the same dataset, but on the bottom one, the Y-axis is linear, ie normal, ie… perfectly spaced out. The one above it is logarithmic, which “squashes” the Y-axis the bigger it gets.

The way a logarithmic scale works is that it perfectly compensates for that exponential growth… which is why those smooth, increasing curves of the bottom graph (TTDs of 20 and 25) show up as perfectly straight lines on the graph above it.

Accordingly, the seven-day moving average of daily case-counts of Alberta, the thicker black line, follows the curve on the bottom graph and follows the straight line on the upper graph. While the logarithmic graphs tend to minimize the growth as things get worse (the steepness gets squashed), the real-data graphs tell the truth. It’s clear from looking at these graphs exactly where Alberta is heading, if things don’t change. It’s clear as well that Saskatchewan’s recent increases are far worse than TTDs of 20 or 25; closer to 15 over the last few days.

Exactly 8 months ago, we were heading into the last week of March wondering the same thing we are today… I wonder what the week ahead will look like. It feels like a lifetime ago, March 22nd… that was the 6th day of me writing about all of this (here it is if you’re interested: https://kemeny.ca/2020/03/22/march-22-2020/). The U.S. had 32,000 cases. Canada had less than 1,500. B.C. had less than 500.

Here we are today (Day 251) – and the sentiment hasn’t changed. Just the numbers, which are all a lot bigger.

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Day 250 – November 21, 2020

No B.C. numbers today, but here’s a brief look elsewhere…

Parts of Ontario (Toronto and Peel) are in a lockdown of the sort we saw around here at the start of this pandemic. Very tight constraints with respect to with whom you can get together, and strict rules around what that needs to look like. Everything else is pretty-much closed, except the essentials.

Saskatchewan saw a huge increase in numbers today, something they saw coming; measures were put in last week, but they’re now dealing with the effects of what came before. As we know, it can take a couple of weeks to realize the effects of these measures.

Alberta also set its record for new cases in the last 24 hours.

Today’s lesson in exponential growth comes from Nunavut… where, for the longest time (like till November), they’d seen zero cases. They got their first one Nov. 6th… their second one Nov. 7th… and then two more Nov. 8th. Then 8, 18, 26, 60… and they’re now over 100. Their graph is not a gentle slope or a hockey stick… it’s a literal cliff wall which they slammed into, after 7 months of flat road. That’s how this thing can take off.

In the spring, it was all about flattening the curve. For those late to the game, like Nunavut and Saskatchewan, where they never got a first wave, that’s where they’re at.

For places like Ontario and Quebec, it’s not just about flattening the potential frightening growth… it’s that the numbers, as flat as they may be (which they’re not) are already really big.

What’s worse… if you have 100 hospital beds available… to see cases go from 2 to 8 to 20 to 50 in a few days? Or to see them go 98, 99, 103, 98?

The answer is… it depends… on what measures are in place. Drastic measures are needed in example A, but example B is just as frantic, because it’s evidence of a problem that’s stressing the limits and that’s not going away unless something is done about it. Roughly speaking, example A is Nunavut and example B is Ontario.

The rate of growth is interesting to look at, on an apples-to-apples basis. I’ve added Time To Double (TTD) lines to the provincial graphs, and I’ve set them all (for now) to 20 and 25, so you can compare the data against those straight lines… and across provinces. Don’t worry too much about where those lines cross, just look at the slope of the data compared to the TTD lines. B.C. and Alberta are examples of consistent growth… you can see the recent growth is virtually parallel to the TTDs of 25. And at their steepest recent points, both Saskatchewan and Manitoba had recent TTDs approaching 10.

You’ll notice that Quebec and Ontario are a lot flatter. Indeed, their TTDs are 50 and 77 respectively. Their issue isn’t so much exponential growth… it’s just that any growth is already putting pressure on a system that at some point won’t be able to handle it.

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November 12, 2020

Today’s update is about numbers, because I’m looking at them… and they’re not great. At all. Dr. Bonnie is not pleased. B.C. has just gone over 20,000 cases. By tomorrow, Manitoba will have gone over 10,000… and Ontario over 90,000… and Quebec over 120,000.

And the U.S… wow. They blew through 10 million cases recently, but every day their growth is increasing sharply. Today’s new-case number of +164,878 is by far their biggest ever.

The pictures reflect all of this better than the words. Those are steep ramps everywhere, and even the logarithmic graphs are slanted upward… the U.S., Canada, everywhere. Around the word, daily, 10,000 people are dying.

Here are two little examples of exponential growth:

Imagine a chessboard… put a grain of salt on the first square. Put 2 on the 2nd square. Put 4 on the 3rd square… and so on. By the end, you’ll probably have a pretty big pile of salt, right? Enough to fill the room? Enough to salt the road from here to Whistler?

Well… after a couple of rows of the chessboard, it’d be about 3lbs of salt. Not a big deal.

At the end of the next row, you’d have enough to coat the floor of a big room. Hmm… perhaps more than you thought. I’ll cut to the chase… by the end, you’d have 18 trillion dollars worth of salt, and you’d need a box a mile long, wide and high to store it all.

Here’s a better one, and a chance to make some money! Imagine a piece of paper… you fold it in half. Fold it in half again… no big deal. How thick would it be if you could fold it 20 times? The answer is… 1km. Crazy, eh? You can’t come even close. Not even halfway close. So here’s a challenge… send me a video of you folding a piece of paper successfully in half 8 times… that’s it, just 8 simple little folds… any piece of paper you want. But it has to be in half every time, because that’s true exponential growth. Do it successfully and I will send you $1,000. Go for it.

That’s the thing with exponential growth… it’s simple and dismissable to begin with, and suddenly, it hits a tipping point, and it’s drastic. The latter half of the chess board is hugely problematic. The last 4 folds you’re about to attempt are a lot more difficult than the first 4. Like, incomparably more difficult.

And that’s exactly where we are now. I’m not sure where we are on the chessboard, nor on which paper-fold we’re at. But it feels like we’re pretty close to jumping from “this isn’t so bad” to… “Oh oh.”

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