I haven’t written about the pandemic in a while because, well, we have vaccines that work pretty damn well – even against the incredibly contagious delta variant. People just need to get vaccinated. I could do a post every day that just repeats that 500 times I suppose.
But I was talking to someone and they asked just how bad the delta variant could be for the United States. First off, what is absolutely clear is that:
Delta is so contagious that, until we get to herd immunity, if you don’t have some sort of immunity or don’t take strong precautions, i.e. N95 masks, social distancing, you will catch it.
So the most important question is when we will reach herd immunity? That’s actually not an easy question to answer and what answer you get depends on the model you use for herd immunity. And all the models depend on questions we don’t yet have complete answers to, for example: how rare is it that a vaccinated person gets reinfected and if they do get reinfected, how likely will it be that they can transmit it? Similarly, if a person already had a version of Covid and isn’t vaccinated, how likely are they to get reinfected and then transmit delta? And you can also ask: how likely is a child under 12 who catches delta to transmit the virus etc. The point is, the number of groups you can use and how they transmit delta in your model can grow, and then the model becomes very complicated. At that point, large-scale computer simulations are often the best way to get an answer for your model.
But there is some reason to believe that the naive model for calculating herd immunity I discussed here https://garycornell.com/2020/10/27/herd-immunity-1/ using an R0 of roughly 7 (https://www.thelancet.com/pdfs/journals/lanres/PIIS2213-2600(21)00328-3.pdf) will work pretty well. One can see for example how well it matches up with the Institute for Health Metrics projections which are based on very sophisticated computer simulations.
Using the model I described in my blog and an R0 of 7, we need that 85.71% (1-1/7) of the population to be immune – to not be transmitting the virus to other people – before herd immunity kicks in. So when will 85.71% of the population not be transmitting the virus?
I want to explain how one might get a handle on this number in the rest of this blog. I’m going to make the following simplifying assumption:
- I will assume that herd immunity happens when 85.71% of the population over 12 is vaccinated or has had a version of Covid.
This assumes vaccinated people and people who have had Covid are not contributing significantly to the transmission of delta and the transmission from children under 12 also isn’t significant to blocking herd immunity. If these assumptions are false and people in these groups do contribute to transmission significantly, it will make herd immunity happen much later, but based on what I have read so far, current thinking seems to be that this is unlikely.
There are about 329 million people in the United States and about 280 million of them are over the age of 12.
So since 85.71% of the 280 million people over 12 is about 240 million (.8571*240mil=239,988.000), we have to find out when 240 million people are not transmitting it to the remaining 40 million people over 12.
According to the CDC, as I write this, about 185 million people 12 and older have received at least one shot and about 161 million are fully vaccinated. I’m going to assume therefore that we can take 185 million people out of the equation. Let call these people category “A”. Category A lets us remove a lot of people from our 240 million goal – if only it were more.
Our goal shrinks to:
240mil – A = 240mill- 185mil = 55 million
So we are down to a goal of 55 million more people being or becoming immune before we get to herd immunity.
This 55 million people goal is made up of two groups in our model. Those who have already had Covid and those who are vulnerable and will get it in the months to come.
To analyze this number, we need to first figure out how many people have gotten Covid and aren’t vaccinated. Let’s call the number of people that are 12 and older, aren’t vaccinated, but have been infected by Covid, B. This means the number of people who will get sick going forward before we get to our goal of herd immunity is:
55mill – B
Let’s call this number “V” for vulnerable.
V = 55mill – B
Now we get to the joys of modeling. I have searched for good information on how many people are in group B (have gotten Covid but aren’t vaccinated), but have come up short. There just doesn’t seem to be any good numbers on the size of group B.
But all is not lost: there are good estimates on the total number of people who have been infected by Covid, we just don’t know how to distribute them between groups A (vaccinated) and B (unvaccinated).
The best estimates I have seen are that between two and three times the number of people who have tested positive (roughly 33mil) actually have had COVID. This means it is reasonable to assume between 67 and 100 million people in the United States have had a version of Covid. Let’s be as optimistic as possible and assume that 100 million people over the age of 12 have had some version of Covid.
But we still don’t know how to split these 100 million people between groups A and B is. We have to do this because if they are in group A we have already removed them from the equation, we don’t want to count them twice! How do we proceed? Here’s what we are assuming:
- The “odds” of having been infected with Covid if you are over 12 is
100mil/280mill = .357
So, of the 185 million people in our vaccinated group A, we will assume 35.7% of them have already had Covid:
.357*A = .357*185mil = about 66 million people in group A have had Covid
(Yes, I know that people in group A probably took better precautions, or got vaccinated before they could catch Covid, so their infection rates are lower than group B’s, but you can change this number to take this into account if you want.)
The rest of these 100 million people are exactly the people who have had Covid but aren’t vaccinated i.e. group B. So
B = 100mil – 66 million
This means that, with our assumptions, group B has about 34 million people!
So now let’s calculate V – the people who will get sick from delta before we get to herd immunity with our assumptions. In our model, since V is equal to:
V = 238mil – A – B
or
V = 238mil – 185mil – 34mill
so
V = 55mil – 34mil = 21mil
Our model predicts 21 million more people over 12 will get delta before we get to herd immunity!
Let’s check our simple model against the very sophisticated Institute for Health Metrics (IHME) model which goes until November 2021 (https://covid19.healthdata.org/global). Their model predicts there will be another 50,000 deaths by November 1st and since the death rate is roughly 1/10 of the hospitalization rate, their projections imply 500,000 or so hospitalizations (https://jamanetwork.com/journals/jamanetworkopen/fullarticle/2778237) by November 1.
Now compare this to the result of our analysis. What we got was 21 million more people who have no immunity will get sick from delta before we get to herd immunity. This implies that there will be roughly 610,000 hospitalizations from delta (best knowledge is that 3% of those infected are hospitalized) and 61,000 more deaths before herd immunity kicks in (using the current knowledge that mortality is 1% of hospitalized cases). This number is consistent with the IHME numbers and probably means our model, simple though it may be, is realistic. Also, if you believe the IHME model and this analysis, we probably won’t get to herd immunity until the beginning of 2022. And, alas if our analysis is right:
hospitals in areas with low vaccination rates i.e. where most of the people in group V live, will not just be overwhelmed by sick patients, they will break completely under the burden.
Feel free to make your own assumptions in this model and change the values for the variables accordingly. But I believe this isn’t a bad model and it gives a good picture, how &^%$ things are going to get in the United States because of the number of people over 12 who are unvaccinated.
As you & I have discussed under separate cover, it looks to me like you’re re-inventing the family of SIR models of epidemiology. I think your model shares with SIR models the assumption that the populations are homogenous and well-mixed. That is, there is little compartmentalization where, say, lots of unvaccinated people live in the American South while lots of vaccinated people live in New England or in the coastal cities. The model assumes there is equal contact within and between those communities, and that the vaccination fractions are comparable.
This is, of course false.
I did some SIR modeling last year that predicted the pandemic would burn out in about 6 months, assuming we kept containment in Wuhan City or Hubei province. Since we obviously messed that up, my model was wildly wrong.
Your model, while perhaps wildly wrong to describe the US in general (or the world in general), will likely describe regions accurately (with separate parameters for each region): the coastal cities of the US are largely ok now, while the American South and the inter-mountain West not so much. Africa is, of course, a human-made disaster.
So it’s important to remember that the overall vaccination rate is irrelevant, but the vaccination rate in each particular community is highly relevant.
As to your assumption of R0 ~ 7: this post contains a terrifying plot of US vaccination rates vs Trump/Biden popular vote, on a county-by-county basis. What’s interesting is that it puts herd immunity to the Delta variant at 85% vaccination (or immunity from previous COVID-19). That translates to R0 ~ 6.7. I would have thought that a bit high, but it is based on some data coming out of the Yale School of Medicine, so maybe that’s the case.
So a while ago I would have thought your estimate of R0 too pessimistic. But now I fear that you are correct: this is probably the case. (“The optimist proclaims that we live in the best of all possible worlds; and the pessimist fears this is true.” — James Branch Cabell, The Silver Stallion: A Comedy of Redemption)