The shutdown, while painful, appears to have saved a LOT of lives so far

If you recall, the growth of the new corona virus disease in the US (and many other countries) at first looked to be exponential, meaning that the number of cases (and deaths) were rising at an alarming, fixed percent each and every single day.

Even if you slept through your high school or middle school math lessons on exponential growth, the story of the Shah and the chessboard filled with rice may have told you that the equation 2^x gets very, very hairy after a while. Pyramid schemes eventually run out of suckers people. Or perhaps you have seen a relatively modest credit-card bill get way out of hand as the bank applies 8 percent interest PER MONTH, which ends up multiplying your debt by a factor of 6 after just 2 years!

(If the total number of deaths were still increasing by 25 percent per day, as they were during the middle of March, and if that trend somehow continued without slowing down, then every single person residing inside America’s borders would be dead before the end of May. Not kidding! But it’s also not happening.)

However, judging by numbers released by the CDC and reported by my former colleague Ron Jenkins, I am quite confident that THE NUMBER OF CASES AND DEATHS FROM COVID-19 ARE NO LONGER following a fixed exponential curve. Or at least, the daily rate of increase has been going down. Which is good. But it’s still not zero.

Let me show you the data and fitted curves in a number of graphs, which often make complex things easier to visualize and understand.

My first graph is the total reported number of deaths so far in the US, compared to a best-fit exponential graph:

Deaths in US are not growing exponentially

During the first part of this pandemic, during the first 40 or so days, the data actually fit an exponential graph pretty well – that is, the red dotted line (the exponential curve of best fit) fit the actual cumulative number of deaths (in blue). And that’s not good. However, since about day 50 (last week) the data is WAY UNDER the red dots. To give you an idea of how much of a victory that is: find day 70, which is May 9, and follow the vertical line up until it meets the red dotted line. I’ll wait.

Did you find it? If this pandemic were still following exponential growth, now and into the future, at the same rate, we would have roughly a MILLION PEOPLE DEAD BY JUNE 9 in just the US, just from this disease, and 2 million the week after that, and 4 million the next week, then 8 million, then 16 million, and so on.

THAT AIN’T HAPPENIN’! YAY! HUZZAH!

As you can see — the blue and red graphs have diverged. Ignore the relatively high correlation value of 0.935 – it just ain’t so.

But what IS the curve of best fit? I don’t know, so I’ll let you look for yourself.

Is it linear?

Deaths in US are not growing in a linear fashion

This particular line of best doesn’t fit the data very well; however, if we start at day 36 or thereabouts, we could get a line that fits the data from there on pretty well, like so:

maybe this purple line

 

The purple line fits the blue dots quite well after about day 37 (about April 6), and the statistics algorithms quite agree. However, it still calls for over 80,000 Americans dead by May 8. I do not want the slope of that line to be positive! I want it to turn to the right and remain horizontal – meaning NOBODY ELSE DIES ANY MORE FROM THIS DISEASE.

Perhaps it’s not linear? Perhaps it’s one of those other types of equations you might remember from some algebra class, like a parabola, a cubic, or a quartic? Let’s take a look:

Deaths might be growing at a 2nd degree polynomial rate - still not good

This is a parabolic function, or a quadratic. The red dots do fit the data pretty well. Unfortunately, we want the blue dots NOT to fit that graph, because that would, once again, mean about a hundred thousand people dead by May 8. That’s better than a million, but I want the deaths to stop increasing at all. Like this piecewise function (which some of you studied). Note that the purple line cannot go back downwards, because generally speaking, dead people cannot be brought back to life.

maybe this purple line - nah, prefer horizontal

Well, does the data fit a cubic?

deaths fit a cubic very well

Unfortunately, this also fits pretty well. If it continues, we would still have about a hundred thousand dead by May 8, and the number would increase without limit (which, fortunately, is impossible).

How about a quartic (fourth-degree polynomial)? Let’s see:

4th degree polynomial is impossible - people do NOT come back to life

I admit that the actual data, in blue, fit the red calculated quartic red curve quite well, in fact, the best so far, and the number of deaths by Day 70 is the lowest so far. But it’s impossible: for the curve to go downwards like that would mean that you had ten thousand people who died, and who later came back to life. Nah, not happening.

What about logarithmic growth? That would actually be sweet – it’s a situation where a number rises quickly at first, but over time rises more and more slowly. Like this, in red:

logarithmic growth

I wish this described the real situation, but clearly, it does not.

One last option – a ‘power law’ where there is some fixed power of the date (in this case, the computer calculated it to be the date raised to the 5.377 power) which explains all of the deaths, like so:

no sign of a power law

I don’t think this fits the data very well, either. Fortunately. It’s too low from about day 38 to day 29, and is much too high from day 50 onwards. Otherwise we would be looking at about 230,000 dead by day 70 (May 8).

But saying that the entire number of deaths in the US is no longer following a single exponential curve doesn’t quite do the subject justice. Exponential growth (or decay) simply means that in any given time period, the quantity you are measuring is increasing (or decreasing) by a fixed percentage (or fraction). That’s all. And, as you can see, for the past week, the daily percentage of increase in the total number of deaths has been in the range of three to seven percent. However, during the first part of March, the rate of increase in deaths was enormous: 20 to 40 percent PER DAY. And the daily percent of increase in the number of cases was at times over A HUNDRED PERCENT!!! – which is off the chart below.

daily percentages of increases in covid 19 cases and deaths, USA, thru April 25

The situation is still not good! If we are stuck at a daily increase in the number of deaths as low as a 3%/day increase, then we are all dead within a year. Obviously, and fortunately, that’s probably not going to happen, but it’s a bit difficult to believe that the math works out that way.

But it does. Let me show you, using logs.

For simple round numbers, let’s say we have 50,000 poor souls who have died so far from this coronavirus in the USA right now, and that number of deaths is increasing at a rate of 3 percent per day. Let’s also say that the US has a population of about 330 million. The question is, when will we all be dead if that exponential growth keeps going on somehow? (Fortunately, it won’t.*) Here is the first equation, and then the steps I went through. Keep in mind that a growth of 3% per day means that you can multiply any day’s value by 1.03, or 103%, to get the next day’s value. Here goes:

in 10 months we are all dead

Sound unbelievable? To check that, let us take almost any calculator and try raising the expression 1.03 to the 300th power. I think you’ll get about 7098. Now take that and multiply it by the approximate number of people dead so far in the US, namely 50,000. You’ll get about 355,000,000 – well more than the total number of Americans.

So we still need to get that rate of increase in fatalities down, to basically zero. We are not there yet. With our current highly-incompetent national leadership, we might not.

===================================================================

* what happens in cases like this is you get sort of an s-shaped curve, called the Logistic or logit curve, in which the total number levels off after a while. That’s shown below. Still not pleasant.

I have no idea how to model this sort of problem with a logistic curve; for one thing, one would need to know what the total ‘carrying capacity’ – or total number of dead — would be if current trends continue and we are unsuccessful at stopping this virus. The epidemiologists and statisticians who make models for this sort of thing know a lot more math, stats, biology, and so on than I do, but even they are working with a whole lot of unknowns, including the rate of infectiousness, what fraction of the people feel really sick, what fraction die, whether you get immunity if you are exposed, what is the effect of different viral loads, and much more. This virus has only been out for a few months…

logistic curve again

 

What’s the best approach – should we lock down harder, or let people start to go back to work? Some countries have had lockdowns, others have not. How will the future play out? I don’t know. I do know that before we can decide, we need to have fast, plentiful, and accurate tests, so we can quarantine just the people who are infected or are carriers, and let everybody else get back on with their lives. We are doing this lockdown simply because we have no other choice.

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One CommentLeave a comment

  1. If nothing else, you have refreshed my memory as to why I “learned” logarithms 50+ years ago! Granted, the most I know is that I can use them to simplify exponential equations, not an exercise I ever had much cause to use. I wish I hadn’t been so clueless and/or teachers had been better at helping me internalize their utility. Whatever the curve looks like, I am in an age group that is more likely to be part of that curve sooner rather than later. My survival is literally dependent on other people’s behavior unless I hibernate.

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