The second derivative won’t save us!

  

David Leonhardt of the New York Times wrote a column yesterday that pointed out using a chart that the US, compared to other countries like China and Italy, has experienced a very small decline in total cases eleven days after that number had peaked. He believes this is because the experts are being ignored by the White House on issues like social distancing. The difference is very dramatic.

But there’s one problem with his analysis: Total cases haven’t peaked. They’ve been going up continually since I started tracking them on March 1, and yesterday rose by four percent (up from 3 percent the day before). However, the rate of increase has declined since peaking at 49% on March 19 – this might be what he had in mind, except that was much longer than 11 days ago.

If you’ve studied calculus, you’ll know the rate of change of the rate of change is the second derivative. The first derivative is the rate of change of the actual number (or more correctly the function that is generating that number), and the second derivative is the rate of change in the first derivative. I know that the other countries have had actual declines in total cases, meaning their first derivatives have turned negative. So I think including the US in this graph was simply the result of sloppy work by some assistant.

But that leads to the question: Why hasn’t the US total cases number peaked? After all, we’ve had massive social distancing in most states for close to a month now. In theory, cases should peak about two weeks after that happens (and that’s definitely why they peaked in China and Italy).

That answer is also obvious (and I can’t believe David doesn’t know this): We’re still doing far too little testing to have any good idea what the total number of cases is. There were estimates by epidemiologists a month ago that the total cases were 5-10 times the reported ones. Given what I’ve seen from comparing the total deaths numbers to cases lagged 2-4 weeks, even the range of 5-10 times may be too low.

However, there is one fact – which was reported in the Times but seems to have escaped Mr. Leonhardt’s attention – that provides a good clue to what total cases may be: Even though tests are in short supply and are supposed to be restricted to suspected actual cases, there have been some studies that tested a sampling of the population in various areas. These have consistently found that, even though the people being tested don’t show any symptoms of Covid-19, 20% of them actually have the disease, meaning they’re asymptomatic (some might develop symptoms later, of course). The equivalent number in South Korea – which has the virus very much under control – is 2% (in some states like New York, the number is even higher).

So it seems likely that around 20% of the US population, or about 60 million people, is infected, although the big majority of them are asymptomatic. In other words, total cases are no longer 5-10 times reported cases, but something like 60 times reported cases.

Now, someone who isn’t thinking too clearly (as was the case in the Wall Street Journal last week) might think the fact that total cases is so much larger than reported ones is a good sign, since it indicates the mortality rate is a lot lower than most people have been thinking. For example, I was using an estimate of 7% until two weeks ago, at which time I decided that estimating total deaths by multiplying reported cases by an estimated mortality rate was a waste of time. This is because reported cases are clearly a small percentage of actual cases. I could see this by looking at total deaths numbers, which are growing at a much higher rate than reported cases, even than reported cases 3 or 4 weeks ago.

However, it’s only good news if the mortality rate drops and the case number stays the same. As it is, the real case number is 60 times the reported one. The mortality rate would have to be 60 times lower, for there to be any net benefit to this.

And is that the case? Well, there is one measure of actual mortality rate that we have – that’s the ratio of total deaths to total closed cases (= deaths plus recoveries). As I’ve said before, that ratio has been oscillating between 35% and 41% since about a month ago (when the recoveries number first became available). And other countries, including Italy and Spain, have similar numbers. When the pandemic’s over, that number will of course be equal to the mortality rate.

Steve Parker of EnergySec estimated that total pandemic deaths if the entire population (now over 330 million) became infected would be over 6 million. 20% of that is 1.2 million, so let’s say that ends up being the final number of deaths. Given that the case number would be at least 60 million (since 20% of the population will be infected), this means the final case mortality rate would be 2%. This seems to be in the range for other countries like South Korea. So the reported number would have to come down from 36% (today’s value) to about 2% at the end of the pandemic (assuming it has an end – see this post). Could this happen? Maybe so, but it will be a stretch, and it’s disturbing it hasn’t come down at all in a month. And remember, even that number still leaves us with 1.2 million deaths.

What does this mean for the reported case number? As I’ve said before, this is now simply a measure of how available tests are (and of course not just the kits, but the supplies that are needed). If there were 60 times as many tests available as there are today, the reported number of cases would come close to equaling the actual one.

Of course, we’re a long way from having enough tests available, to allow us to come anywhere near measuring total cases, and we probably won’t ever have that many. Had tests been available at all in early February (as they were in other countries), we could have actually have identified close to the total number of cases. And had we followed up aggressively with contract tracing, we would have a far smaller number of cases (and deaths) today; then we could truly say we know within a reasonable margin of error what the real total is.

It is only in that situation that we would know if total cases were anywhere near peaking. As of now, the only thing we can say is they’re far, far from doing so. The only question is whether they’ll “peak” when the entire population is infected, or even when 20% of the population is infected (since further social distancing just doesn’t seem to be in the cards. Far from it). In either of these cases, we’re talking about Covid-19 deaths in the millions.

The numbers

These numbers are updated every day, based on reported US Covid-19 deaths the day before (taken from the Worldometers.info site, where I’ve been getting my numbers all along). No other variables go into these numbers – they are all projections based on yesterday’s 3-day rate of increase in total Covid-19 deaths, which was 14%.

Week ending	Deaths reported during week/month	Avg. deaths per day during week/month	Pct. Change from previous week/month
March 7	18	3
March 14	38	5	111%
March 21	244	35	542%
March 28	1,928	275	690%
Month of March	4,058	131
April 4	6,225	889	223%
April 11	12,126	1,732	95%
April 18	18,434	2,633	52%
April 25	17,031	2,433	-8%
Month of April	69,577	2,319	1715%
May 2	25,116	3,588	47%
May 9	38,423	5,489	53%
May 16	54,154	7,736	41%
May 23	77,858	11,123	44%
May 30	119,110	17,016	53%
Month of May	316,185	10,200	454%
June 6	167,877	23,982	41%
June 13	241,358	34,480	44%
June 20	369,237	52,748	53%
June 27	520,414	74,345	41%
Month of June	1,572,655	52,422	497%
Total March - June	1,962,476

Red = projected numbers

I. Total deaths

Total US deaths as of yesterday: 47,681

Increase in deaths since previous day: 2,338 (vs. 2,825 yesterday)

Percent increase in deaths since previous day: 5% (vs. 7% yesterday)

Yesterday’s 3-day rate of increase in total deaths: 18% (used to project deaths in table below – was 16% yesterday)

II. Total reported cases

Total US confirmed cases: 849,092

Increase in cases since previous day: 29,917

Percent increase in cases since yesterday: 4%

Percent increase in cases since 3 days previous: 11%

III. Reported case mortality rate so far in the pandemic in the US:

Total Recoveries in US as of yesterday: 84,050

Total Deaths as of yesterday: 47,681

Deaths so far as percentage of closed cases (=deaths + recoveries): 36% (vs. 35% yesterday) Let’s be clear. This means that, of all the coronavirus cases that have been closed so far in the US, 35% of them have resulted in death. Of course, this number will come down as time goes on and more cases are closed in which the victim recovered. But this number has gone down and up since Worldometers started publishing the recovery rate on March 26 (when it was 41%), and on about half the days, it’s gone up; there is still no sign of a downward trend, and other countries like Italy and France show comparable percentages.