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COVID-19: The Cold Equations
This is my attempt to explain some of the dilemmas of ‘restart’. It pulls together a string of comments I made on this post. (My thanks to @rodin for not only that post but his whole series tracking the pandemic and its implications.) There will be math, but nothing worse than simple algebra. Estimates are sourced with inline links. Qualifications and elaborations are at the bottom in footnotes. Off we go:
1. There is no pre-existing pool of COVID immunity in the population. There’s a theory that COVID arrived here earlier than officially recognized, and was the cause of some of the nastier upper respiratory ‘flus’ of the mid-winter. This has been pretty thoroughly debunked by testing of patient samples that were preserved from that time. See the Twitter stream here (RTWT) with additional discussion here (post 7/ and onward). The CDC has a standing program to collect samples from ‘influenza like illnesses’, see here and here. This is done for flu vaccine planning and evaluation on an annual basis, and predated COVID. So there’s a large pool of existing samples that would have disclosed any earlier infections.* There’s no magic, we have to go on what’s shown by current testing and cases.
2. The case infection fatality rate in the US is somewhere around .6%
As of this writing, there are 646,300 positive tests in the US, 28,640 deaths.
Assuming a 7/1 ratio of total cases including asymptomatic & minimally symptomatic (nonpresenting) cases to positive tests (based on early Wuhan data) implies 4524100 actual infections.
28640/4524100 ~ .6%
This is probably a bit low, since some currently unresolved cases will end in death.
3. Trying to reach ‘herd immunity’ would be a catastrophe
In its pure form, herd immunity is the idea that an epidemic can be stopped by immunity in the population. The level of immunity required is implied by the R0 ‘R-nought’ characteristic of the infection, which is the average number of persons infected by each current carrier. If R0 is below one, the epidemic will eventually die out. If some of those who would otherwise be infected are already immune, R0 can be effectively reduced. To use a simple example, if the raw R0 is two, then having more than half the population already immune will take it below one. For more discussion, see this paper and this twitter stream. **
The paper linked above contains an estimated R0 for the onset of the US infections of 3.29. This implies a critical herd immunity threshold of about 70%.
My current estimated total cases is 4,524,100. The US Census estimates the current US population at about 329.5 million. So our current case percentage is 1.37%.
To get to pure herd immunity we need 70/1.37 more cases that result in immunity, which is about 51 x.
51 * 28,640 current deaths = 1.46 million deaths to reach herd immunity in the US.
We aren’t going to do this.
4. Antibody tests will not become a ‘pass’ to return to work or socializing
Current case percentage is about 1.4%, see above. Let’s assume we are about halfway through the first wave, which is consistent with the IHME modeling, so we have about 2.8% previously infected and hopefully immune by its end.
You can’t restart an economy with 2.8% of the workers and customers, and no one’s going to put up with rights infringements implied by a pass system to get that level of benefit. ***
Antibody testing will be very useful, but more in finding out what the actual case count has been as opposed to those formally tested positive, which can turn my back of the envelope estimates into hard numbers.
5. Distancing, masking, and isolation will be with us for the duration
R-nought can sound like it’s an immutable constant, but it’s not. It’s an effective rate that is roughly:
(A true constant that is characteristic of the virus) times
(The average number of exposures from each carrier) times
(An average probability of transmission per exposure)
Herd immunity goes after the last term, in a brute force manner, but using it alone is a disaster. Stay-at-home and shutdowns are a brute force way of affecting the second term, but using them alone is an economic disaster.
Masking is an attempt at mitigating the third term – cut down the chance of transmission per contact. Distancing affects both, fewer people encountered and less chance of infection per each. Isolation gets known cases out of circulation.
Unless and until a vaccine is found and distributed, we’re going to be trying to get R-nought below one by a variety of such measures, pushing the virus into small enough populations that it can be traced and victims isolated. Since we don’t have hard data on what measures have what results, the initial response has often been to take it to the limit. Now we have to back off and learn what’s real, what this article calls ‘the Dance’.
If there’s a policy lesson here, it’s that a uniform, nationally mandated response is the wrong approach. This a case where the ‘Laboratory of Democracy’ in the states will be all too real, and necessary to find combinations that work.
* There’s some speculation that a recent heavy case of common cold might confer some resistance, since colds are also caused by coronavirus of a different sort. Any such effect would, however, be incorporated in the existing caseload.
* For those who want more depth on the analysis of the virus’ arrival and variations, see here for an amazing display of what can be done with current RNA sequencing technology.
** For simplicity, this assumes that immunity is an all-or-nothing thing, typically not the case. So it’s going to be an average, not a hard number.
*** To make matters worse, while there are effective antibody tests being produced, we don’t actually know what level (IgG titer) of antibody confers effective immunity. It’s known that asymptomatic or mild cases of COVID-19 produce lower levels of antibody, whether that creates immunity is also unknown. We’re going to be figuring this out on the fly.
On #2 are you describing the Case Fatality Rate or the Infection Fatality Rate? It seems at points that you are using the term “case” both in the test-confirmed diagnoses and in the numbers of infected whether or not diagnosed.
Great summary of what we know and don’t know at the moment.
I lack sufficient knowledge to weigh competing claims about the virus against each other. But I appreciate the input for consideration.
I am off to the range (it’s essential as far as I’m concerned) so won’t be able to respond to comments for a few hours.
A terrific summary of what the true state of play is, shorn of left wing pieties and right wing complacency.
51 * 28640 current deaths = 1.46 million deaths to reach herd immunity in the US
This is wrong
If you look at the diamond princess cruise ship the infection was less than 20% and the case fatality rate was slightly above 1%.
population = 330 million
20% infection rate = .2 * 330 million = 66 million test positive
1% case fatality rate = .01 * 66 million = 660,000 deaths
worst case scenario is 660,000 deaths
This is a discouraging report. I appreciate the information.
I was hoping for a larger number of undetected/asymptomatic cases. You estimate 7x, but that seems to be based on early information from Wuhan. How much variability might there be in this multiple?
What is the end game? The lockdown and other social distancing may slow the spread, but I don’t see any way to stop it. Is this correct, or do you see some way to stop it?
Because if we can’t stop it, the math is inexorable and it’s only a matter of time. The 1.46 million that you estimate — or whatever the number turns out to be — are going to die no matter what we do. If not this month, then in 6 months or 12 months. Am I missing something?
There’s not even any assurance that a vaccine is possible. Am I incorrect about this? I’ve heard that no one ever developed a vaccine for SARS or MERS, but I don’t know any details. Has there ever been a successful vaccine for a coronavirus?
My concern is that this is exactly what we need to do, because we can’t stop the virus. I have seen no feasible plan that gives us any prospect of stopping the virus. I see a plan that delays the inevitable, and throws us into a great depression that may make the 1930s look mild.
I think you calculated “infection fatality rate”.
Then key is accurately assessing the variables in the R0 value against preventive measures. We should be able to more precisely estimate the effects of various measures on the third factor in your equation based on different distancing protocols in various countries. Can we achieve and accept a slow herd immunity until a vaccine is approved? I think we must. Turn on the spigot, slowly, be relentless in washing hands, maintaining distance, disinfecting surfaces, avoiding face-touching, wearing masks (although I question the efficacy), and avoiding large group gatherings (no concerts, rallies, conventions for the foreseeable future), but get back to work. Be prepared to tighten down again (maintain an adequate TP inventory?).
Yes, you and @rodin are correct. I had conflated infection fatality rate and case fatality rate for the sake of simplicity, and then edited out the passage that made that necessary. Fixed.
There could be a lot. That’s the only figure I can find that appears to have some real analysis behind it. Yeah, it’s from China, but it appeared before the CCP started censoring publications.
I am no expert but I’ve seem comments from those working on that they are cautiously optimistic. A coronavirus apparently is simpler than influenza and is not likely to develop as many different strains as the flu, if we can get it stopped in the not too distant future. SARS and MERS efforts were dropped because they were contained with other methods. Apparently some of the current COVID efforts (at least one Israeli one) are being bootstrapped from previous work on those.
The way I see it, we are playing for a combination of data and time. Easing off restrictions in a variety of ways in different circumstances, see what happens = data on what effects R-nought with minimized economic impact. Get the antibody tests up and running and RNA tests faster and more widely distributed = data on what the actual infection rate is out there, and hence the true death rate. Play for time for the docs to try out new palliative therapies and run some statistically valid trials. Play for time for vaccines to be tested. Play for time to have some of the mitigating techniques become social habits and business expectations.
I have no idea if it is feasible to estimate the effectiveness of various distancing protocols or other measures on R0. From what I’ve seen so far, the epidemiologists don’t seem to be able to get a very solid estimate of R0 itself, generally giving a range (sometimes like “2.0-3.9,” which is pretty wide).
Lock On, I appreciate your response #10.
I am at a loss to understand why widespread antibody testing has not been carried out. I do not know how hard it is to do. It seems to me that this is one of the most important things that CDC (or whoever) should have been planning to do from the start, so that they (and we) would have decent information. I would think that a random testing of, say, 10,000 New Yorkers could answer a lot of questions right now.
I did see a report (from a National Review article here) that 15% of pregnant women giving birth at a NYC hospital tested positive (on the antigen test, meaning current infection). The paper is in the New England Journal of Medicine (here), and it reports on testing of 214 women between March 22 and April 4. As of April 4, reported cases in NYC were 63,306, which is 0.76% of the city’s population.
The pregnant women are not a random sample, but this is an extraordinarily high figure. If we assume that 15% of all New Yorkers were infected on April 4, this implies a multiplier of around 20x (higher than your 7x, Lock On).
This is similar to the 30x multiplier from the Gangelt study in Germany, except that to get this figure, I had to assume that the “confirmed” cases in Gangelt was the same as for its entire district (Heinsberg), and the information was limited.
Back to your comment #10: I agree that we are playing for time, but to what end? I don’t think that we can play for time for 18 months, or even 3-4 months. I worry that the economic damage is going to be unbelievably devastating. People seem to cavalierly assume that we can just re-start the economy, and I suspect it’s going to be much harder.
I’ve seen the terrible downward cycle before, back in 2008 — economic decline, people can’t pay the rent or the mortgage, property values decline, more properties become marginal, the foreclosures start, property values decline more, and on and on. It’s going to be a lot worse with 15% unemployment, or worse.
I do hope that someone has a good answer. I don’t see one. I see a bullet that we have to bite, and everyone trying to find some way to avoid having to do it. I take no joy in this, as I understand that the consequences could be 500,000 deaths or more.
The 2nd Quarter GDP numbers will dwarf any single quarter seen in the 1930s. Current estimates are from a 25% decline to over 40%. Those are annualized averages, so the if we are at no growth for the 1st, 3rd and 4th quarters, the annual drop would be between 8 and 10 percent. That works out to between $1.7 and $2.2 trillion in economic output, and assumes we go back to business as usual on June 1, as if all this never happened.
It is correct to point out that the 1.46 million figure is not the relevant number. That number is how many more die because the health system is overwhelmed. That is the number you need to calculate the cost of a life saved by the shutdown. If that number is 100,000 the best case scenario is that the cost of each life saved is between $17 and $21 million.
One last thing that will be considered in the decision that our nation will ultimately take, is that the case fatality rate is not uniform across all groups. This suggests that much more effective and less costly measures can be taken to protect vulnerable populations without shuttering the entire economy. If so, the herd immunity percentage can be reached without getting anywhere near 1.46 million deaths.
I appreciate the relatively sober tone of this post, and I absolutely endorse the take-home message that what we need to do is start opening up society in incremental but also different manners in different states/jurisdictions. There is almost certainly a treasure trove of information about this virus and its transmission and treatment waiting to be unlocked, but the only way to get there is to deliberately set a goal of non-zero growth in cases while keeping the exponential factor below the explosive rates that rapidly overwhelm not only health care services but also our data collection capabilities.
All that being said, I find the figures in this post to be on the pessimistic end of the “realistic range” of possibilities. Figures like 3 million deaths in the US or, conversely, infection fatality rates of 0.01% (a la Jay Bhattacharya) are truly absurd yet occupy far too much public attention. However, even within the range of realistic figures there is a huge amount of variation.
In my opinion, the realistic range for infection fatality rates is somewhere between about 0.25%-0.6% give or take. The OP provides a range for R0 from the literature of between 2 and 3.9. Within those two ranges is a huge variety of outcomes that would have different implications for public policy, and at the moment we simply don’t have enough data to refine either of these figures into a useful narrow band.
With that I mind, I again endorse the notion of finding the cojones and logistics to slowly turn on the tap, not only to start recovering our economy but also to allow us to start defining these key figures with more reliable data.
An added bonus that gets rarely talked about is the possibility that additional experience with the disease will allow doctors to get better at treating severe cases of Covid-19 (or preventing mild cases from becoming severe) even in the absence of effective medications. However, that also requires a rate of hospitalization somewhere between the two extremes of “trickle” and “deluge” that have been the norm to date.
But not in line with Diamond Princess data. I would trust the Japanese over the Chinese in this regard.
As of April 11, 2020 – 712 confirmed, 331 asymptomatic.
The main reasons for the general optimism about a vaccine are because
a) there are currently almost a dozen different proven vaccine technologies (and another half-dozen or so in advance development/testing) that can be used. The consensus is that with so many to choose from, at least one can be expected to pan out.
b) Traditionally, we’ve had much more success at vaccinating against viruses than treating them with medications. The few viruses for which multiple attempts at creating a vaccine have proven difficult tend to be rather unusual viruses like HIV or dengue fever.
The reason flu viruses are less effective is partially because of the number of strains that circulate, but even more because the protection conferred by the flu vaccine attenuates much more rapidly than most other vaccines. That’s a theoretical possibility for any coronavirus vaccine as well – but even if it occurred, we would still likely have enough protection to drop social distancing (having to get a booster every year would be a small price compared with annual social distancing).
Yes, a vaccine can be developed in that time.
This frustrates me to no end as well.
I understand why it’s taking a while to roll tests out – I am intimately familiar with the development process for antibody tests and I even know a few people personally who are working at companies/institutions developing them. There’s a basic problem with deciding which “target” to use that’s similar to drug discovery: a major decision on which target to choose has to be made early in the development process using limited tools (like banked patient sera), and it’s only after a major investment in late-stage production and development that large-scale testing can take place. At that point, it’s not uncommon to get a rude surprise that the chosen target wasn’t as specific as hoped. The production process can also be finicky.
That’s why many of the first “quick tests” (akin to pregnancy tests) that have appeared on the market have been so unreliable – they rushed the selection and validation process. A series of lab tests are slowly coming online and supposedly will be shipping at the end of April/beginning of May. These tests (by much larger and more reputable pharmaceutical companies than the rapid tests) have presumably been tested much more thoroughly. What’s more, once established, they can both be manufactured much more rapidly and performed much more rapidly by well-equipped diagnostic labs: under ideal circumstances, many of the systems can test up to 300 samples/hour.
The reason I’m frustrated, though, is because there are several old-school methods to test serology which, while not necessarily as accurate or convenient as commercial tests, can be spun up DIY-style by many reasonably competent molecular biology labs without any special reagents or equipment. Tests of this nature would likely be more than sufficient for basic surveillance purposes and gathering data on undetected cases. But they’re very labor-intensive.
And this is why I’m mad: research labs around the world have been shutting down due to Covid-19. Many of those labs are more than competent enough to perform such DIY serology testing, but for some reason they’ve all been sent home to sit on their thumbs instead of working around the clock to grind out serology results the hard way. There’s simply no excuse for this in my opinion.
It certainly can, but there’s no guarantee. Vaccine development is often very hit-or-miss, so a 6-month to 18-month window is the most realistic time frame at the moment.
On the other hand, there’s this, which is rather astounding and suggests a much higher ratio of asymptomatic to symptomatic.
My guess is – and it’s just that – is that the Diamond Princess data will turn out to be a combination of an early uncontained spread on board followed by a mostly effective lock down, also on board. Remember this happened before it was realized that asymptomatic can spread the virus, and those who become symptomatic are effective spreaders before the symptoms show up. There was a lot of talk at the time about COVID spreading through the ventilation system onboard, but doesn’t match up with what we’ve since learned. So my guess is that DP was not truly a ‘saturation’ event where everyone was exposed.
That’s certainly an eye-opener. Although I do wonder how many of those homeless who tested positive were truly asymptomatic vs. presymptomatic; i.e., most of them got infected around the same time and hadn’t been infected long enough to start showing symptoms.
Either way, I do expect that when all is said and done we’ll see that the symptomatic cases were merely the tip of the iceberg. It doesn’t change the fact that the number of speed of deaths in NYC and elsewhere is unprecedented, but it might give us a sense of whether that level of death was the upper limit of what the virus can do.
Mendel, I think that you hit the problem on the head. There’s this assumption that a vaccine can be developed quickly. We don’t even know if this is possible at all.
Instigutor, forgive me, but I don’t recall if you have special expertise in this area. I have none. If you have some special expertise, I’d appreciate a brief explanation of the reasons to believe that vaccine development is feasible.
There needs to be testing too, right? My presumption is that a vaccine can turn out to be dangerous, but again, I have no special expertise in this area. I don’t know how to quantify that risk. Perhaps no one does, but if any of you have better information on the issue, I’d appreciate it.
Yeah, and it’s in line with the pregnant-women report from NYC that I mentioned in #12, although the pregnant women were “only” 15% positive while the homeless were 37% positive.
This is just bizarre. It’s almost as if someone doesn’t want to know.
Mendel, you have inside information about the Gangelt testing. I got the impression that there was a great deal of hostility to those findings, now essentially repeated in the Boston homeless and NYC pregnant women testing (admittedly, not random samples). Is my impression about the level of hostility wrong?
If my impression is not wrong, what is the explanation for widespread hostility toward results that would be, frankly, extremely good news?
@mendel already answered this very well, but I wanted to point to a description of the development process that you view if you’d like: https://youtu.be/Xm76adKULY4?t=2680
This is from the Stanford Grand Rounds videos. This is a research doc talking to clinicians so it’s pitched at an understandable level for laymen. (You can also scroll back for a description of the RNA testing available at a top level research hospital, and a Q&A about how it’s being employed.)
Summarizing what I think I understood, to get to a useful antibody test you need to:
Except Dr. Bhattacharya is conducting an actual study on the matter, and will undoubtedly revise (or affirm) his estimation when that is completed. As of 7 days ago, he was expecting results early this week, so I am hoping that we hear about it soon (and, of course, that it is good news).
Actions to Take Now:
1. Antibody Testing – Aggressively pursue widespread testing to understand the magnitude of the problem.
2. Test for Infection – Develop more rapid tests so those who are at risk or live with those at risk (diabetes, lung disease, etc.) and those who work with at risk patients (senior care personnel, hospital workers) can be tested frequently. Lower R0.
3. Continue social distancing and sanitation protocols, but not full lockdown. Lower R0.
4. Share and refine treatment methods that work. Lower the mortality rate.
Bottom line: More deaths will occur, and we simply have to accept that reality. There are no solutions, only trade-offs.
“Simple algebra” is an oxymoron.
This info was posted in a comment elsewhere but is germane here, too:
This is not (yet?) clinically useful.
Sensitivity and specificity are not known yet, which means we don’t know how likely it is that the results for any individual’s test give a false positive or false negative. So right now, we cannot responsibly use this test to say “Yep, you’re immune (for now, anyway, because we don’t have that info either;) go, contage no more.”
Another downside to the shutdown (backup link here):
RTWT.
I don’t have medical knowledge about who might be immune to what. But, as a matter of logic, the text of section 1 does not necessarily support the headline of section 1. The text of section 1 asserts that the particular virus was not present in the United States before the virus was officially recognized. Does that really support the conclusion that there was no preexisting immunity in the population. Is it possible a large portion of the population has immunities that protect them against viruses that they have not previously personally had? If so, the herd immunity argument of section 3 becomes suspect.
I mention that possibility in the footnotes – some have speculated that a previous bad case of common cold, also caused by a coronavirus, might confer some resistance. I haven’t seen any results to back that up with data, and the behavior when it hits an unprepared population would seen to weight against it. (I sure wish it were true, as this was my worst winter on record for colds. Achoo!)
I’m more staking my hopes for good news on getting adequate testing into places that are not hammered by the virus, so the clinicians can do their thing and collect data on tests vs. outcomes while not just trying to stay alive and conduct triage.