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# The Math on WuFlu

I’m finding myself in a shrinking minority with respect to my view of the WuFlu. To me, it still appears to be an irrational panic. Heather MacDonald still seems to be on my side, at least as of yesterday (article here). But even the Daily Wire guys have been convinced that there is something serious to fear, other than fear itself. By Monday, MacDonald and I may be the only skeptics left standing. (I would find her to be good company in such an event.)

I’ve done a bit of digging into the facts, and I still can’t understand the cause for alarm. I would particularly value the input of our Ricochet docs and other medical professionals, as I certainly realize that I could be wrong.

**I. The Severity of the Symptoms**

In round numbers, based on the Chinese experience thus far, it appears that about 80% of WuFlu cases are mild, about 15% are “serious,” and about 5% are “critical.” The main problem with the WuFlu seems to be pneumonia. My impression is that “serious” cases might require hospitalization and oxygen treatment, while “critical” cases might require ICU treatment such as intubation. My source is here, from the same Worldometer site that our friend Rodin is relying upon for his daily posts. These estimates are based on information from China through Feb. 11.

Even these figures seem too high to me. Rodin’s daily post today (here) generally shows lower rates of serious/critical cases than the 20% combined figure noted above. In Italy, it is less than 10% (1,518 serious/critical out of 17,750 active). In South Korea, it is less than 1% (59 serious/critical out of 7,300 active). In the US, it is less than 0.5% (10 serious/critical out of 2,395 active).

My suspicion is that the rates of serious or critical illness is much lower, and that the rates appear high because very few people have been tested. This makes sense, as I would expect that initial testing would be limited to people exhibiting WuFlu symptoms. South Korea seems to have done the most extensive testing to date, and its very low rate of serious/critical cases is consistent with the hypothesis that wider testing will show a higher prevalence of the WuFlu, with the vast majority of cases being so mild as to be almost unnoticeable.

**II. The Math on the Hospital Bed Crisis**

I’ve seen news reports that the WuFlu has overwhelmed the health care system, in Italy in particular. Here is an article from The Atlantic on Wednesday, March 11, stating:

Today, Italy has 10,149 cases of the coronavirus. There are now simply too many patients for each one of them to receive adequate care. Doctors and nurses are unable to tend to everybody. They lack machines to ventilate all those gasping for air.

This NYT article from Thursday, March 12 similarly claims, in its headline:

We don’t have enough ventilators and I.C.U. beds if there’s a significant surge of new cases. As with Italy, the health system could become overwhelmed.

I know that I’m just a country lawyer, though I did once study math through the graduate level, with a focus on probability, statistics, and mathematical modeling. But it doesn’t take grad-level math to question these figures. It takes middle-school algebra.

The NYT article linked above says that Italy has 3.2 hospital beds per 1,000 people (and the US has only 2.8 beds per 1,000 people). Italy has a population of about 60 million, so this implies about 192,000 hospital beds.

As noted above, the number of serious or critical WuFlu cases reported in Italy, according to Rodin’s post today, is 1,518. That is **0.79% of the number of hospital beds **in Italy.

Think about that. We’re supposed to believe that an influx of about 1,500 new patients has overwhelmed the medical system of a nation that has 192,000 hospital beds.

Put this in perspective. Let’s round up the Italian number to 1% — that is, assume that the number of serious or critical WuFlu cases in Italy is equal to 1% of the country’s hospital beds. Imagine that you run a hospital with 200 beds. This means that you can expect **two (2) extra patients** as a result of WuFlu. Are people seriously suggesting that a 200-bed hospital will be “overwhelmed” if it has to take in an additional two patients?

You all can believe anything you like. I’m staying in the skeptic camp with Heather MacDonald, at least for the moment.

Now let’s apply these figures to the US. Recall that, according to the NYT article linked above, the US has 2.8 hospital beds per 1,000 people. With a population of 327 million, that’s about 915,000 beds.

How many serious or critical cases are there in the US? Ten (10), according to Rodin’s post today. But let’s assume that the WuFlu spread rapidly in the US over the next month. How rapidly? Well, China has had 80,000 cases over several months, so let’s make the extreme assumption that the US has 100,000 new cases over the next month — a vastly faster spread than in China. And let’s use the Chinese figures for serious and critical cases, rather than the much lower figures from South Korea (more than 20 times lower).

So if the US has 100,000 new cases over the next month, 15% will be serious (15,000) and 5% will be critical (5,000), for a total of 20,000. This would be about 13 times the number of serious or critical cases currently existing in Italy.

20,000 new cases in the US would represent about **2.2%** of the hospital beds in the country. A hypothetical hospital with 200 beds would have to take in **about 4 new patients** over the next month.

Is the medical profession seriously maintaining that their capabilities are so marginal, their ability to adapt so limited, as to be unable to cope with an increase in their patient load of about 2%?

I fully understand the graph about the capacity of the health system. Here is one example:

I do not dispute this graph in theory. I dispute the dashed red line about the “healthcare system capacity.” Based on my calculations above, the dashed red line is nowhere near as low as indicated. It is far, far higher — literally off the chart, in this graph.

As noted above, I don’t just understand mathematical modeling. I am a lawyer. I know how to mislead — in my case, I endeavor not to mislead myself, but I am ever vigilant about how my opposition can mislead. This is precisely the way that one can generate a panic — with a graph that is correct in theory, with just one small misleading element.

I see no evidence whatsoever of any serious danger that the WuFlu will overwhelm our healthcare system capacity, even with no protective measures.

Another way to mislead, incidentally, is to assume that the number of cases will continue to grow exponentially. The very early stages seem exponential, but the number of cases eventually follows an S-curve. Continuing to project an exponential growth rate — say for an entire month — is contrary to the facts, and will lead to a vast overestimate of the number of cases that we can expect.

**III. Expanding capacity**

The calculations above assume that we have no ability to increase our capacity to handle patients needing hospitalization. Obviously, we have such capacity. I haven’t looked into the precise figures, but my recollection from the hospitalization of family and friends over the years is that most hospital rooms are either single or double occupancy. In a crisis, it does not seem, to me, that it would be difficult to add an additional bed in each room. This would probably increase the availability of hospital beds by 30-40%.

This would be enough to hospitalize every American needing it, even if the number of cases increased to about 900,000, and even assuming the very high, 20% rate of serious or critical cases based on reporting from China, and not the rate of about 1% in South Korea and 10% in Italy.

This suggests that we could handle, without too much trouble, the health care needs of Americans even if the WuFlu spreads **10 or 20 times faster** in the US than it has spread in China.

And we haven’t even talked about setting up emergency medical facilities. You know, schools are closing. Why not set up temporary hospitals in school gyms or auditoriums? How hard could it be? Bring in about 100 beds and some oxygen masks. Have 4 nurses or orderlies monitor the patients, administering oxygen when necessary. They could check each patient every 30 minutes or so. If there aren’t enough pulse oximeters for each patient, have the nurse carry it around. Patients who need critical care could be sent to a hospital.

As I understand it, even the serious WuFlu cases are essentially moderate-grade pneumonia. Patients may need an oxygen mask, but they won’t immediately die without it. They can take the mask off to eat, or to go to the bathroom. They can basically lie there, in relative comfort with an oxygen mask, and watch TV. Except that they can’t watch March Madness.

**IV. About That March Madness**

Actually, perhaps these hypothetical WuFlu patients will be able to watch March Madness. Because, it seems to me, the term is being redefined.

I was expecting to watch March Madness on CBS Sports and ESPN. It was going to involve a bunch of college basketball players. Now, I seem to be watching a different kind of March Madness on CNN, MSNBC, and Fox News. It involves a bunch of talking heads, politicians, and medical experts telling me that we’re all going to die unless we shut down the world. I find this extremely unlikely.

I would appreciate any corrections to my analysis.

If I turn out to be correct, I am going to prepare a huge plate of crow for everyone who disagreed. :)

Published in Healthcare

As far as I know, most S-curves used in epidemiology or any other study of population growth are based on a function that is exponential, but which is modified by other factors that are fairly insignificant when the population size (e.g. population of infected persons) is low, but become more significant when the population size is greater. I’m thinking of Lotka-Volterra type equations. (A criticism of Lotka-Volterra equations, which I used a lot in my Master’s thesis, is that it’s hard to ascribe a biological significance to the parameters, even though they may do a useful job of describing the shape of the observed population growth curves.)

Clearly not correct, since we’ve had 62 deaths while that was stuck at 10, and I assume that those 62 had some kind of hospitalization going on. Although 40 of those are all in Washington State (and I am assuming most of those were from that one nursing home).

The JohnsHopkins site doesn’t seem to be counting any recoveries either.

So . . . the question remains: of the 3,000 cases, how many are just recovering at home? Minnesota is up to 21 today I see, but yesterday when it was at 19, the news reports said that

allthose people were just recovering at home.I think it’s worth asking whether the economic impact on Italy after “not doing enough” could possibly be worse than the economic impact of sufficiently aggressive preventative action would have been? (To say nothing of the social, spiritual, political or demographic effects…?)

I’m not sure the impact will “only” be felt by the elderly who would’ve died of something anyway. First of all, it depends on what you mean by “elderly.” A seventy two year old woman of my acquaintance contracted pneumonia, and was intubated in the ICU. Her life was saved, and she went on to work another eight years and live another fifteen, all but the last six months very happily and productively. Second of all, the “something” that elderly person would’ve died of is still out there, making the rounds. Wuhan is an add-on, not a substitute.

And a hospital filled with corona virus victims is a hospital that can’t accommodate nearly as many patients suffering from the Regular Old Flu, stroke and heart attack victims not to mention mothers giving birth, kids with broken arms, strep throat or concussions, people who have been injured on the job, in car accidents, fires, gun violence, maybe a terrorist attack…And then, if the doctors and nurses get sick, who is going to provide the care?

It’s not difficult to imagine that, should Wuhan explode here in my little neck of the woods, our local hospitals could easily be overwhelmed, and transporting people to other hospitals would get tricky, since the usual places where the acutely ill are sent (Portland, Boston) will have their own problems.

We’re lucky (?) enough to have unlucky Italians and Brits offering us real-time witnesses to the various approaches.

I’m glad there seems to be financial help being at least considered for folks whose incomes have just been abruptly interrupted.

I was just thinking how it would be nice if the John Hopkins data would tell us how many of the cases are hospitalized. A lot of the recent discussion here is about the possible overwhelming of our medical systems, so that would seem to be a very relevant number. But I’m sure there are problems in defining “hospitalized,” much less getting those data reported to a central database in a timely manner.

No, it depends on two things: The initial number of cases and the exponent. The bigger the exponent, the more quickly it grows. There is exponential growth in all kinds of systems (at least for a time) that don’t overwhelm us, because the exponents are low enough.

One way to write the exponential growth law is

where

Nis the number of cases on a given day, “^” means whatever follows is in the exponent,tis the number of days since the initial number, andTis the number of days the cases take to double. This paper from May estimates the doubling time ast= 7.31 days.Now, if the

truenumber of cases were really 71 at the end of February (dayt=0), then we would expect there to be just 294 cases by March 15:If true number of cases Feb 29 were 71

So the problem here is knowing the true number of cases. Until we get more widespread testing, we won’t really know.

Daily update from Italy came in recently. Cases up 17% over 24 hours. South Korea cases increased 11% over the past

week. Key for Italy is path over next 10 days. Lombardy quarantine was March 8; countrywide travel ban March 9, and country shutdown on the 11th. Given incubation time let’s see if this was effective.You’ve got it backwards. It is the exponent that is rising. Try 100 to the 3rd power => 1,000,000, then the 4th power => 100,000,000, then the fifth power => 10,000,000,000.

But thinking in terms of powers of 100 is hard to reason with. The same curve can be scaled to be powers of two. The growth is 2 to the n’th power, where n is steadily growing.

Um, I don’t think so. Not in the usual equations to model these things.

I don’t think so:

Since I’m still hazy*, I’ll get back to the original numbers about which I was puzzling. They went from 71 to 2,976 over two weeks, an increase of 207 per day, which is supposedly “exponential.” Merely “doubling” would get there in fewer than 5 days. I still believe language is being used to amp up the hysteria.

*Speaking of “hazy” — for you IPA fans, Solace Brewing of Dulles, VA makes an excellent hazy IPA called “Partly Cloudy.”

A little more detail would help to provide some understanding of what is going on with that class of infected people who require hospitalization. How many of those hospitalized require ICU and Ventilators or other breathing related assistance, for how long, and what are the outcomes, death or recovery and discharge.

No, this does not contradict my point. It is difficult to identify the inflection point from daily case information. As the number of total cases grows, the number of new cases daily also grows, even if the growth

rateis declining.I’ve done the calculations for Italy since it passed 200 cases on Feb. 24, 19 days ago. There is some noise in the daily figures (notable in your graph on March 10). Looking at 5-day increments, the daily rate of growth has been:

The latest daily increase, on March 14, was 19.8%, so it’s not showing a new acceleration.

Yes, yes they are. Tim did a better job than me in #94. One must remember that exponents don’t have to be integers.

If following OldPhil’s pattern, the curve is parabolic, not exponential. Parabolic curves are much less dramatic than exponential.

You are thinking that doubling must happen per day. It can happen per multiple days, or a fraction of a day, or a non-integer number of days. Current curves suggest the coronavirus is doubling per three-ish days.

For now, one’s best bet for finding hospitalization rates is to look at local or county sources. WA and CA have the most cases and make up a third of all confirmed US cases, but the CA Dept of Public Health just gives bare-bones stats. The WA Dept of Health provides case figures by county and links to each one’s public health department. In King and Snohomish counties, which have the highest numbers, the latter reports hospitalizations (32 of 176 as of 3/15), but not the former. Santa Clara county in CA also reports the number of cases hospitalized (48 of 114 as of 3/14).

Don’t understand a word y’all are saying, but you sure do sound sexy.

FYI – this is chart for South Korea

Extremely good news.

We make these kinds of judgments all the time. Every time I drive, I take your life in my hands. You are correct that the answer to the question, “What’s Prof. Rahe’s life worth?” is not zero. However, neither is it infinity. It seems to me like we already have plenty of irrational fear-mongering on one side of this question and we could do with a little unjustified complacency to sort of balance it out.

You made a point like that in a later comment, so maybe I’m just agreeing with you. Anything to combat the boredom of Seattle Coronapocalypse Day 2.

Yep.

Left uncheckedan epidemic would follow a logistic curve, the S-shaped curve you mentioned. A logistic curve is sort of the equivalent of exponential growth when the thing in question is a rate (i.e., the probability of being infected) rather than a magnitude (the number of dollars I wish were in my IRA).The analogy is like this: in exponential growth, the thing growing goes to infinity. In the case of a virus, what goes to infinity is the odds of being infected — p(infected)/p(not infected).

In reality, of course, lots of things are working to check the spread of a virus. It has to be that way, or we’d all already have everything. The question at hand is probably something like, “How many people are going to get this before exponential growth turns to exponential decay?”

Not quite. For every 100,000 cases we would require

an additional 2,000 ventilators (and 500 or so more qualified therapists) over and above the day in, day out utilization before the epidemic.Also, in many cases, ventilators are necessary but not sufficient. If all COVID-19 cases have been triaged to locations where the necessary level of total care is not possible (which is likely if an epidemic reaches mass casualty levels.) If appropriate care levels are possible but the number of staffable ventilators is the limiting factor, likely prognosis will be a triage factor and those unlikely to survive may not be put on ventilators.

In order to permit the use of unapproved equipment (several comments have suggested this in various ways) and/or unqualified staff, there will also need to be emergency regulations promulgated to allow this without incurring legal liability.

If they haven’t done it, states will need to pass enabling legislation to permit hospitals to provide patients to be admitted for care in wards set up in tents, gyms, hotels, etc.

Assume for the sake of argument that China’s numbers are correct.

Not even China ended up with 100,000 cases. And they have more than four times the population of the U.S.

China also instituted mandatory movement and public behavior restrictions for about a third of its population; this is credited with containing the spread.

I read this to my wife, and she’s says I

sound sexy. (:doWhen people speak of exponential growth, the exponent is usually the growth rate, and is a constant (although in real life, growth rate is not a constant forever). Tim’s example doesn’t express it that way, though the growth rate (doubling time) is in the exponent, as the denominator of the exponent. And in Tim’s example, the exponent is not rising. It’s falling as t gets larger, which it usually does except where time flows backwards. But I’m not quite sure what you were getting at when you were saying the exponent is rising, so maybe I’m missing something.

Perhaps the same is happening in AZ. As I poked around the Arizona website for coronavirus status in our state, I made the mistake of jumping out a level and found myself reading this influenza information, about which I had heard not a peep from our state officials [emphasis added]:

Quick.

Check Worldometers.

NOW.

This won’t last. It’s gotta be typos!

Screenshots.

Vatican City. Check it out. It’s got 568,000 cases, and 892,045 deaths, with -324,045 active cases.

Yeah, definitely something wrong there.

Got to be someone testing a revision to the database/spreadsheet.