Physical distancing is what we are engaged in right now. Spreading out the population and minimizing interactions is the strategy to minimize new infections so that the hospital system has a chance to care for the wave of people that are starting to hit the ER and then the ICU. It is a strategy where we will not see immediate success and the first success will “merely” be a slowdown in the rate of increase of infections and hospitalizations.

Anyone showing up to the hospital today was infected at the end of February or the first week of March.

Anyone showing up at the hospital next Monday will have been infected in the first half of March when there were instructions to wash your hands and cough in your elbow and little else.

Anyone showing up at the hospital at the end of March will have been infected as everyone was heading to the bars and restaurants and Spring Break for one last hurrah before everything started to lock down.

Incremental social distancing in the first half of March will have helped compared to the counterfactual scenario of business as usual. The big actions of governors like Cuomo, Murphy, Dewine, Baker, Inslee and others shutting down significant elements of their states in the past couple of days won’t show up in the hospitalization numbers until the last few days of March or the first couple of days of April. The next two weeks are already baked in.

And when we start to see results of physical distancing, the first results will merely be changes in the second derivative. Case counts per day may still increase, but at a slower rate. It is hard to get excited against a grim counterfactual. In other words, when things are bad, it’s hard to get excited about things not being as bad as they would have been had nothing been done. And that is what we have to wait for over the next several weeks; success as measured by a second derivative (rate of increase) shrinking at some point.

WereBear

I still remember the graph put up, here and elsewhere — 11 days away from Italy.

Tattooed on my brain now. And moving forward. The worst is yet to come.

Matt McIrvin

I’ve seen the models. If we do everything right, the system will still be completely overwhelmed and a million Americans will die. If we do nothing, two million. It will feel like total failure even if we save a million lives. It’s going to be hard to stay motivated.

New Deal democrat

At the rate that diagnoses have grown, there will be between 20,000 to 25,000 cases *reported* by a week from Friday. That probably means between 200,000 to 1,000,000 non-diagnosed carriers before the effects of social distancing start to sink in.

Meanwhile, I had two encounters with neighbors yesterday who are *still* in denial.

burnspbesq

@Matt McIrvin:

Given the unreliability of the inputs, no model tells us anything useful, yet. Get back to me when you models are based on data.

Baud

@Matt McIrvin:

We all are going to die. More so earlier than expected for the sake of enforcing email management best practices and deterring wall street speeches.

If we’re only motivated by success, we’re always going to fail. FIDO.

evodevo

THANK YOU FOR THIS…i have a nephew-in-law, a BIL and numerous MAGAt acquaintances who are still in denial, and this will help my case a lot…what’s even more frustrating, the NIL IS IN ITALY RIGHT NOW, and is still in denial about the epidemiology and IS AN ENGINEER WHO SHOULD BE ABLE TO UNDERSTAND MATH lol. So this, along with this graph, is how I presented my case…

https://twitter.com/melindacmills/status/1239142819903746049

EveryDayIHaveTheBlues

Isn’t rate of increase of counts the first derivative of the curve? Is there something I’m doing wrong?

1st derivative = velocity, 2nd is acceleration…

HuCat

We have been fortunate (though no longer the vacation we intended) to be in Japan, where schools closed the first of March and those who could were encouraged to work from home. Midweek, daycare also pushed kids who had a parent (or grandparents) who could care for them at home to stay home. So we’re pressed into monitoring (or herding cats) through 3rd grade distance learning and just being 3-years old.

This requires us to make a daily journey to their house, in the opposite direction of crowds commuting to/from work, and we walk to the grocery store where we practice not hoarding once a week. So far, so good, and we don’t have to get on a big ol’ jet aeroliner for another month.

David C

CDC reports about 3500 cases yesterday. If we continue in the exponential phase for the next two weeks we will look longingly back to mid-March.

Matt McIrvin

@Baud: It’s true, everything we ever do is striving for better outcomes on the margin. And it might be that the reason so many seniors aren’t taking this seriously is that they figure they haven’t got a lot of time left anyway.

I see the same kind of problem (but with different parameters) in talk about climate change. People want to hear a target threshold of warming or CO2 below which everything is OK, or at least not too bad. Well, you can estimate such a number and it’s hopeless; we’re never going to make it. So they then figure everyone is doomed and there’s no point in even trying. But it’s possible to distinguish between bad outcomes.

Ukko

@EveryDayIHaveTheBlues: the sad irony is that because this is an exponential it’s derivative is exponential, as is its second derivative. It is exponentials all the was down…

EveryDayIHaveTheBlues

@Ukko: Absolutely.

As David says, actions taken days ago are only now coming to bear fruit.

It is important to get the terminology right. I think the first differential is the rate of increase of infections, but would be glad to learn more if I’m wrong.

Matt McIrvin

@EveryDayIHaveTheBlues: Well, everything is the first derivative of *something*, that’s the fundamental theorem of calculus.

Matt McIrvin

@Ukko: But the point is to make it less than an exponential, and that becomes obvious in the higher derivatives first.

David Anderson

@Matt McIrvin: 1st derivative is the slope of the line

2nd derivative is the change in the slope of the line

Drdavechemist

@EveryDayIHaveTheBlues:

First derivative is the slope of the curve, which is the rate of increase: how many new infections per day. Second derivative is the rate of change of the slope, which is related to the curvature. David is focusing on the latter, since the slope can still be positive (more infections today than yesterday) but if that number is getting smaller (only a few new infections rather than dozens or hundreds), then the strategies are working and that will be visible in the data.

And David got there first…

EveryDayIHaveTheBlues

@David Anderson: Exactly! So rate of infections per day is 1st derivative, correct?

MattF

@David Anderson: More to the point, the sign of the second derivative tells you whether a curve is concave-up (positive) or concave-down (negative).

EveryDayIHaveTheBlues

@Drdavechemist: the text says, “And when we start to see results of physical distancing, the first results will merely be changes in the second derivative. Case counts per day may still increase, but at a slower rate.”

This rate is 1st derivative, right?

SW

The numbers are meaningless until everyone who is suspected of having the illness or who came in contact with someone who has the illness is routinely tested. Until then, the numbers are just a function of the number of tests performed.

Ohio Mom

Is this where I add: I was told there would be no math on this blog?

Don’t worry, I get the general gist. My mother once won a radio contest by knowing the answer to the question, If you had a penny on the first day of the month, and two cents the next day, four the next, and it kept on doubling like that, how much would you have at the end of the month?

The answer, as I recall is something like, over five million dollars.

(She was taking a continuing ed course, as then required for NYC teachers to earn raises, and that had been a class exercise the day before.)

GC

In more intuitive terms: we’re moving faster and faster toward the cliff, and still accelerating.

If we are lucky we should have now started to lift our foot off the accelerator and may eventually begin braking (“hopefully turning negative”). At some later point, as a result of braking, we may even stop moving toward the cliff.

Cam-WA

@EveryDayIHaveTheBlues:

@EveryDayIHaveTheBlues:

Yup, the rate of increase is the 1st derivative. The 2nd derivative is the “rate of increase of the rate of increase.”

Anderson has his last sentence wrong, too. When the “rate of increase” is decreasing, the 2nd derivative already is negative.

I wish more people really understood the implications of “exponential growth” (and no, you don’t have know Calculus or have any idea about derivatives to understand the concept). We use the phrase informally to mean “really really fast growth,” which is true enough. But, no, it’s even faster than that (assuming a reasonable base for the growth rate).

Cam-WA

@MattF:

While correct, this doesn’t make things very clear for those who haven’t studied calculus.

In (more or less) plain English: The 2nd derivative being negative means the “rate of increase of the rate of increase is negative.”

Here’s a concrete example from the world we are currently in. Let’s say that today the # of cases is increasing by 1000 cases per day. So, if there were 10,000 cases at the beginning of the day, there would be 11,000 cases by the end of the day. By tomorrow, the rate of increase in the # of cases might be 3,000/day, so by the end of tomorrow there will be 14,000 cases. 1,000 is the first derivative today, and 3,000 is the first derivative tomorrow. The 2nd derivative is the “rate of increase of the rate of increase.” The rate of increase today is 1,000 day/day, and the rate of increase tomorrow is 3,000 cases/day, so the “rate of increase of the rate of increase is 2,000 cases/day/day (that’s not a misprint), or 2,000 cases/day^2 (“2000 cases per day squared,” and yeah, I know I’m losing folks here).

So, right now now in my example we’re in a bad way…Not only is the # of cases going up, but tomorrow the rate of increase in the # of cases is higher, so things are getting worse. The day after tomorrow things are going to be even worse. It won’t be 2,000 cases/day anymore. It will be higher (barring more social isolation, etc.).

The first piece of good news is when we hear that while yesterday the rate of increase was 100,000 cases/day, today the rate of increase is only 90,000 cases/day. The number of cases is still going up a lot, but the rate of increase is now decreasing, i.e., the 2nd derivative is negative. The “rate of increase of the rate of increase” is -1000 cases/day^2.

Clear as mud? Hope this helps at least somebody.

Cam-WA

@Matt McIrvin:

Well, no. The fundamental theorem of calculus doesn’t say this. Explaining the fundamental theorem of calculus is beyond the scope of this post (and isn’t relevant to the discussion).

Let’s say that by “everything is the first derivative of something” we mean “every function is the first derivative of some other function.” That’s isn’t the case. Some functions aren’t integrable. “Well behaved” functions are. Pretty much all functions one is likely to encounter in a first-year calculus course are integrable, so, let’s say that “Within the context of a first year calculus course, every function is the first derivative of some other function” is a true statement.

Cam-WA

@GC:

Good metaphor! Alas, I think it is probably a more accurate metaphor is to say that the rate at which we are falling to our deaths after going over the cliff we hope will be slowing down.

Cam-WA

@EveryDayIHaveTheBlues:

You nailed it!

John Revolta

@Cam-WA: This did help me. Thanks.

low-tech cyclist

Upon reflection, it seems we’re looking for the point of inflection.