Monday, March 30, 2020

The Micro and the Macro of Covid-19

Or, The Case of the Invisible Denominator

I've been chatting with Jay Bhattacharya, Professor at the Stanford School of Medicine, and co-author of a very thought-provoking piece on mortality rates under covid-19. Briefly, these authors study incidence rates in three cases where either a full census of tests was conducted, as in the Italian town of VĂ² in Padua, or where a random sample was drawn, as in Iceland. Using incidence rates for covid from these cases, they suggest that the fatality rate for covid is far lower than we think, or fear. 
Going beyond these cases, the problem, of course, is a missing denominator: just how many are infected? Assuming we accurately know the deaths from covid-19 --- possibly not, but setting that aside --- the unknown denominator of actual cases moves us along an "isoquant" of fatality rates and contagion rates, which multiply out to the known deaths. The higher the contagion, the lower must be the fatality rate, for some given number of deaths. Where we are on that isoquant matters immensely for policy, especially in poor countries. (For instance, while I certainly hope that these musings do not form the basis of a dramatic Trumpian re-opening any time soon, I would also hope that they could be used to inform the wisdom of a lockdown in a poor country like India.)
In this post, I argue that a bit of common sense and some data can go some way towards unearthing the true fatality rate from covid-19. I will do that first, and then turn to the implications.
Here is the sequence of cumulative deaths for New York State, starting March 14 and ending March 29: 2, 6, 10, 17, 27, 30, 57, 80, 122, 159, 218, 325, 432, 535, 782, 965. (Thanks to the New York Times.) I think it is fair to say that as the numbers settle in, there is a doubling every 2 days.
(I am sure this doubling pattern is going to die out soon --- that last number 965 pertains to incidence on March 8-15, 14-21 days ago, and after that arrived the lockdown, upon which transmission rates will probably slow significantly, but we will have to any case, I am not basing any of what follows on these projections.)
If the death rate is assumed constant, then incidence in New York has probably been doubling every 48 hours as well, say over all of February to mid-March. Maybe earlier? Who knows? On the other hand, maybe the doubling conclusion every 48 hours is too much, because initial covid deaths were probably attributed to other causes, but still, 2-3 days does not seem like a crazy observation.
If that last number --- 965--- is attributed to the deaths from covid incidence 14-21 days ago, and if we assume a fatality rate of 1%, then the number of active cases on March 8-15 was around 100,000, which would mean that there were only 3 real cases in New York around February 8-15, assuming 48-hour doubling, and fewer than 100 cases with 72 hour doubling. Can that be? That looks unacceptably small, given that NY was almost surely seeded by January. I can therefore conclude that a fatality rate of 1% is way too high.
On the other hand, if the fatality rate is around 0.1%, which is that of the flu, then that same number 965 suggests that 1m people were infected in New York State around mid-March, and around 30-1000 in early- to mid-Feb, depending on whether you go with a two- or a three-day doubling and a 14- or 21-day lag. Even with a three-day doubling, the upper number of 1000 could look a tad on the low side, but here I am far less sure. That calls for a fatality rate around 0.1%.
There is no way we can go far below this estimate. If we get super-optimistic with a fatality rate of 0.01%, we have 10m infected on Match 8-15, close to complete saturation for New York State, which has 20m people. That is plausibly implausible.
We are playing with exponentially changing objects, which are incredibly delicate. But I don't think that fatality rates around the flu are ruled out at all.
I can imagine that a perfectly reasonable reaction to all this would be: let's wait for population level sero-prevalence tests instead of speculating. And indeed, Bhattacharya is involved in conducting them as we speak. But I am personally convinced by the above mixture of reasoning and common sense that what we are dealing with here is a virus with mortality rates that are roughly comparable to the flu. They could be somewhat higher, but not by much: we will only know for sure when the dust settles.
Which then brings us to the obvious question: if the above is correct, what explains the swamping of hospital capacity, the deaths we see, the horror stories of triage that we read? The answer is simple: Covid is seriously more contagious than the flu.
According to the Center for Disease Control, "influenza has resulted in between 9 million – 45 million illnesses, between 140,000 – 810,000 hospitalizations and between 12,000 – 61,000 deaths annually since 2010." That's, on average, less than 10% of the population every year. (To be honest, these estimates seem a bit low to me.) This is nothing compared to covid-19, which can easily infect half the population, and probably more. Imagine the comparison between 10% and 50%. Imagine we had a really nasty flu year, with the same benign household flu, but affecting half the population. There would be an unimaginable crisis, and people would be spilling out of hospital beds just as they are so tragically doing now. That is the macro picture.
At the same time, what would it look like to you, trapped in this nasty flu season? You and I would probably go, "Oh man, this year looks bad, it's really going around, everybody has it." That is the micro picture. 
There is, of course, still the same social case for a lockdown. But in our current panic-stricken way? Not really.
Ok, you might respond. But what about these objections?
1. Covid is nasty. (You know who said that.) People die horrible deaths from it.
That may or may not be true. People who die of the flu die of the pneumonia that the flu causes. In the end, one drowns to death, which is not a pleasant way to go. But barring data to the contrary, I don't see the difference.
2. Covid is particularly bad for those with pre-existing conditions. That is most definitely true. But if disease C and F have the same overall fatality rate, and one of them, say C, has greater variable impact depending on pre-existing conditions, then it could be easier, not harder, to deal with C, because you can tailor your behavior to your conditions, and you can't do that with F. (You could feel anxious about that, I agree, but it gives you more wiggle room.) 
3. Covid is more likely to send you to hospital. Let's go back to my CDC quote above: the flu results in 140,000 – 810,000 hospitalizations and between 12,000 – 61,000 deaths. That's about 12 hospitalizations per death. Covid appears to result in about 10 hospitalizations per death. Now, that does not tell us what the hospitalization rate is (that pesky missing denominator once again), but if you believe my arguments about a comparable death rate, you will have to agree that the hospitalization rate is not much different for the flu (the latter looks slightly higher, actually).
What implications does all this have for our daily lives? 
First, what I see around me is a mass freaking-out (yes, I'm included) that just makes no sense. This is assuredly not a freaking-out from the desire to be socially responsible and not overwhelm the hospital system, which is a thoroughly laudable objective. It is a primeval freaking-out, from the fear of being sick oneself.
Second,  and related, think of our current reticence to go to the doctor now. Again, if we have something that's significantly worse than having the flu (say, abdominal pain or a broken toe), we would go to the doctor in a bad flu season. Well then, we should also go now. Or at the least, our fear of catching covid-19 should not stop us.
Third, we should rethink lockdowns in poor societies where the implications are far worse than they are here. Here, we should compensate for those who lose their livelihoods. In a country like India, we cannot, even if we should. We need to rethink those policies.
One can reconcile macro-mayhem with micro-sanity.
This post was edited for an elementary arithmetical error, thanks to my dear friend S. Subramanian.


  1. Hello Debraj. Thought provoking post. I see some potential problems in your logic, however. Why are you ignoring the growth in deaths (or lack thereof) in New York before March 14th? If it's true that there were no deaths before then, then why wouldn't it follow that cases before then were scarce? Of course, likely there were few deaths before March 14th because there was little testing -- if you die of flu symptoms with no test for Covid available, it's likely you are not marked as dying of coronavirus. To the extent this is true, the growth rate of deaths is slower that what you have merely assumed. And, if it turns out that the rate of growth doubles every four days, then that would imply that either New York had it's first case before Wuhan did, or the death rate is even higher than 1%.

    Lastly, Bhattarchaya's stuff has already been debunked. The death rate in Vo was 1%. The Chinese data should not be taken at face value. The NBA is not a random sample of the US. The Stanford couple mention .01% as a plausible death rate. But, some Italian provinces have death rates already as high as .065%, and that is assuming a 100% infection rate, which can't be right, as these provinces are still recording masses of new cases despite a draconian quarantine. In other Italian towns, already .1% of the population has died, and many people have also tested negative.

    After all this, I should mention that I'm actually in agreement that the death rate could well be much less than 1%. .4-.6% seems plausible to me, but spreads easier since there is no vaccine. On the other hand, .1% seems ruled out by what we know already. This could explain why Italian hospitals were overwhelmed so early, why they are not by the flu.

    Also, you write that people have freaked out too much. Have you paid any attention, at all, to what has happened in Italy? The freak out there has not prevented ICU units to become completely overwhelmed. They are essentially letting people over the age of 65 die for lack of ventilators. You really think this mass "freaking out" in Italy makes no sense?

    Very Respectfully Submitted,
    Doug Campbell

    1. Hi Doug, thanks. I wish we knew the truth behind fatality rates. You are right that it could be as high as 0.4%. My post is not written with a sense of dramatic certainty. That said (1) could you give me references to the Vo fatality rate? (2) I don't know what you mean by trusting the "Chinese data" --- the data has to do with airlifts out of Wuhan where everyone was tested, (3) the flu rate is for a particular distribution of ages (say, the US distribution), and we need to correct for that age distribution when studying regions with a high proportion of elderly, and (4) please read carefully before commenting on my interpretation. My post is perfectly consistent with OVERALL deaths being higher by a factor of 3 or more. I don't know what you want to include under the term "freaking out," but I am not including the terrible sense of sorrow that people feel for losing their loved ones. I simply refer to one's *own* fear of being fatally sick. Thanks.

  2. Hello Raj. Thanks for the reply. I posted a response over at my blog:

    FYI -- the death rate in NYC, assuming the entire population is infected, and no one left the city, is already .1%, roughly. Given a 60% negative test rate of those with severe symptoms, that gives us a lower bound for the death rate much higher than flu. Wouldn't you agree?

    1. Hi I didn’t see this. I am so sorry. Yes I agree it would be significantly higher. Just how much I don’t know. Once again my apologies for not replying earlier.

    2. That said, I don't think --- even with the benefit of hindsight --- that those early calculations were way off the wall, and I am still "voting" for a rate way below 1%. Could be 0.5%. But I don't think it is as low as 0.1% (which was my lower bound).


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