Post #982: Herd Immunity IV: The simplified version.

Posted on January 28, 2021

Judging from the feedback I’ve gotten, most people can’t make head or tail out of my last three posts on herd immunity.  (Post #978, concept; Post #979, empirical estimate; Post #981, U.K. variant estimate).

At my wife’s suggestion, let me just tell it as a story, and see if that’s clearer.

You’ll hear that 70% of the population must be immune to COVID-19 before we can end the pandemic in the U.S. 

That’s wrong.   Or, at the very least, that depends on what you mean by “end”. 

By “wrong”, I don’t mean that there’s some uncertainty around that number.  There is, to be sure.  But by “wrong”, I mean the 70% figure is conceptually wrong.

If we’re talking about the problem we are facing right now — ending this current wave of the pandemic — then that’s the wrong number to look at.  It’s far too high.  And that’s because most people who use that 70% figure don’t have their thinking straight about what, exactly, that 70% number represents.

And, ironically enough, clarifying that last point was what those confusing posts were about.  So let me try to fix that with this post.

What do you mean by “end”?  End versus remission.

If by “end”, you mean a complete return to normalcy — no masks, no distancing, no restrictions — then yes, 70% immune or something like that is right.  Right in concept, although there’s some uncertainty about the exact value.  (And even then, there’s an important nuance there that a reader has prompted me to explain, and I’ll do that in a separate section below).

I don’t know about you, but “complete return to normalcy” is not what’s on my mind, just right now.  I’m a little more focused on, say, not dying during this particular wave of the pandemic.  And, more generally, getting this third wave under control, so that we don’t have to (e.g.) relax air quality standards because we have to burn so many bodies.  And maybe getting roughly 75M kids back in grade school and college.

Little things like that.  That’s what I’m thinking about.  Full return to normalcy isn’t even within my planning horizon.  Being able to go grocery shopping without a mask again?  Eat at a restaurant indoors?  Go to the movies?  Those aren’t on my to-do list at the moment.

And so, by contrast, if by “end” you mean “get us off this current peak, and end this U.S. third wave of COVID”  — or “end” means “get the virus out of general circulation in the public” — then “70% need to be immune” is wrong.  Not a little wrong, but wrongedy-wrong-wrong.  Not even close.  Simply wrong in concept.

And those definitions of “end” are what I have mind, just right now.  End the immediate danger to the general welfare of the U.S. public. Given the bigger picture, I don’t really give a crap about getting to that final end point, where everything is completely back to normal.

I’ll settle for ending this third wave, getting the virus beaten back to a vastly lower level of incidence than it is now.  Later, I can fret about when I can take the mask off in public.

For want of a better term, let me call that remission.  All I want, right now, is to put the third wave of COVID in the U.S. into remission.  Just get the incidence levels vastly lower than they are now.  And I’ll worry about full return to normalcy later.

Immune percentage required for “remission of the third wave” is lower than the 70% required for complete return to normalcy.

That’s really all I was trying to say, in those posts.  And then to work out just how much lower.  And the further implications of that.  But if you now get the headline above, I’ve done my job.

And that’s how I got into all this.  I kept thinking that we needed to reach 70% immune to end the third wave.  Because all anybody talks about is that figure.  And that led to a kind of hopelessness, because I could do the math on vaccines, and realize that if 70% was required, we’d be stuck in this third-wave hell for months and months yet.

But when I sat down to do work through that, and do the math, I realized that 70% figure was wrong-in-concept.  That’s not the right figure if all you are trying to do is put the third wave into remission, and aren’t requiring a full return to normalcy just yet.

And the reason for that is simple enough.  That 70%-required-for-herd-immunity derives from models of literal herds, as in, like, cows and such.  Where the ONLY thing preventing spread of the virus is the members of the herd that are immune.  And that’s also why the 70% figure corresponds to “full return to normalcy”, because if we go back to living as we were before the pandemic, the only thing keeping the virus in check will be that high percentage of immune individuals.  Because by the definition of normal, we’ll have stopped doing all the masking/distancing/restrictions.

But for mere remission of the third wave — just getting us through this current crisis — we aren’t depending on immunity alone to break up transmission of the virus.  We’ve got immunity plus all the other tools — masks, distancing, and other restrictions — all of which ALSO serve to suppress transmission of the virus.

The upshot is that if you combine population immunity with the other methods of suppressing spread of the virus, you need less total population immunity in order to tip us into remission.  And so end the third wave.  You don’t need 70%.  You need something less than 70%, when combined with masking/distancing/restricting.

If that’s now obvious, then I’ve done my job here.  The entire rest of the analysis — all that math in Parts II and III – was just trying to get a handle on roughly how much less.  And what that implied for resistance to the soon-to-be-dominant U.K. variant of the virus.

If it’s so obvious, why isn’t anybody discussing it?

I think we’ve made some progress here.  Time for a pop quiz:

Statement:  If we use herd immunity alone to control the virus, we need to get 70% of the population immune to achieve our goal.  But if we combine herd immunity with other tools to limit spread of the virus, we need less than 70% to achieve the same goal.

Was that statement:

  • A: Obvious and clear.
  • B: Confusing and unclear.
  • None of the above.

If you answered A, then you may proceed.  If you didn’t, you might as well stop here, because this gets more complicated going forward.

Here’s the next question that should occur to you, now that you see how obvious this is:  If it’s so obvious, then why is nobody talking about it? Let alone working out what this means for U.S. health care policy?

Part of the answer is that it’s not obvious until it’s pointed out.  And that’s in large part because all of us — me included — were simply parroting the phrase “herd immunity” and the figure of 70% without really knowing exactly what they meant.  But embedded in that 70% figure is the assumption that herd immunity is the ONLY factor suppressing the transmission of the virus.  And I’d guess that virtually nobody who wasn’t an epidemiologist fully realized that.

But a second part of the answer is that whole “herd immunity” concept was poisoned, early on, by those who suggested using that as the sole U.S. strategy.  AKA, the public-health equivalent of kill them all and let God sort it out.  So the whole concept is just toxic to the professional public health community, and they simply don’t discuss it.  Except to dismiss it.

A third factor that I think must play into this is that to have an intelligent discussion, you have to admit to how many people in the U.S have probably already had COVID.  And, for sure, if it was your job to help prevent that, then that’s not something you’re all that keen to discuss.  And so the estimate of the true fraction of the U.S. population that has recovered from a COVID infection is another subject that is fairly toxic for the Federal agencies tasked with keeping this under control.

And a final point is that even if it’s obvious, and you’re willing to discuss it, you don’t know whether it matters or not.  I mean, less than 70%, so what?  What if it’s 69% + other tools, or 70% in isolation?

Even if it is obvious, you have to hang some plausible numbers on it, and tell people what the implications are, before they’ll take it seriously.  See next section.

Numerical estimate and policy implications

My contention is that to end the third wave (put it into remission, so to speak), you only need 40% of the population immune to COVID-19.  That’s a rough cut, that’s based on a single set of numbers, and caveat, caveat, caveat, blah blah blah.*

* I’m a little jaundiced about this from my long exposure to staff of the Congressional Budget Office (CBO).  As I used to hear from CBO staff,  OK, that’s my number, where’s yours.  Meaning, you don’t get tell me all the things that are wrong with mine, if you don’t even have one.  When I worked for the Federal government, I used to call that The CBO Rules.  I’ll show you mine, you show me yours, and we’ll decided whose number is better.  In the policy world, you don’t have the luxury of not having a number, because not having a number always ends up being equivalent to assuming a really stupid number, you just don’t realize it.  As here, where if you don’t have a separate number for herd immunity plus other tools, separate from herd immunity in isolation, you are implicitly assuming that you need 70%,.  Which is a really stupid number.  But you just don’t realize it.

And, as discussed above, that’s less than 70% for the obvious reason that that’s being used in conjunction with other tools to limit viral spread.  While the 70% number is a stand-alone number for herd immunity.  It’s what you need when herd immunity is the only factor limiting viral spread.

If you want to see how I derived that, see Part II.

For the mathematically inclined, it may seem like I pulled a magic trick in Part II by solving for two unknowns, using just one equation.  I got the effects of immune persons, and the effects of all other factors (masking etc.) out of just one equation linking the currently observed viral replication rate to the original (R-nought) viral replication rate.  But that’s not true, because the parameter for the effectiveness of immune persons is 1.0 by definition.  There was really only one free parameter in that equation to begin with.

So now let me state the obvious implications for policy.

  1.  A lot of states have probably already exceed that 40% threshold.  (See Post #958 for a list that’s a couple of weeks out-of-date.)
  2. And so, what looks like the slow end of the third wave, in much of the Midwest, probably is.  I’m going to talk about that “slow” part below, at the prompting of one reader.
  3. With vaccination ramping up, a lot more will exceed that 40% threshold in the near future.  At (say) 1M/day, we ought to see almost another 10% of the entire population vaccinated (31M vaccines / 330M people) by March 1.  If we’re taking about the adult US population — and I think we are — then we’ll see another (31M/250M =) another 12.5 percent of the U.S. adult population vaccinated by March 1.  (But bear in mind that it takes six weeks, from the first shot, until full development of antibodies to COVID-19. Now do you begin to see how much this new threshold calculation matters for framing policy?  That additional 12.5% is nowhere near enough to “end the pandemic”, meaning, full return to normalcy with 70% immune.  But it’s more than enough to “put the third wave into remission”, assuming I have correctly estimated the 40% threshold.  It flips the current vaccination effort from “yeah, sure, that’ll be useful after another 200K have died” to “we can put the third wave firmly on a downward path by the end of this month”.  (Again, abstracting from the six week lag between first shot and full development of antibodies.)
  4. If the goal is to end the third wave, then supplying vaccine to states that have already exceeded 40% is a waste.  (Vaccinating individuals who have already had a COVID infection is a waste, but that’s a separate issue.) We should be sending the vaccine to states that have a low fraction of the population that is already immune.
  5. The softer version of that would be that we should be aiming for equal levels of total immunity (infections plus vaccinations) across the states, not equal levels of vaccination as we are now.  And that would imply shifting more of the vaccine supply to states that have had relatively few infections up to this point.
  6. We can also solve for the likely impact of the new U.K. variant, and find that we need something like 54% of the population to be immune to suppress spread of that more infectious variant.  And that gives us some kind of guess as to where the (inevitable) spread of the U.K. variant is going to give us a 4th wave, and where it isn’t.   Again, allowing the Federal government to target scarce resources more accurately.

And I’m going to stop with that.  I think this has been clear enough, and short enough, that I don’t need a formal summary section.  Let me end with a brief recap.

We don’t need to have 70% of the population immune to put this third wave into remission.  People who think that way are confused about how that 70% number was derived.  They don’t understand that it embodies the assumption that the only thing preventing viral spread is herd immunity.  If we combine herd immunity with our other existing COVID hygiene, we need far less than 70%.  And that has serious implications for health care policy going forward.  Most notable of which is that, to put the third wave firmly into remission, we ought to be aiming for equal immunity across the states, not equal rates of vaccination.

The slow decline to the end of the pandemic.

A reader asked that I explain what that threshold number means in the context of the actual end of the pandemic.  And what the likely path of the pandemic would be, once we reach that.  The main point being that we have to exceed that threshold (not meet it) to end the pandemic.

If you look at the math, the threshold number is the point at which the viral replication rate (“R”) reaches 1.0.  That’s the point where the pandemic is neither growing nor shrinking.  Each infected person infects, on average, one other, and we reach a steady fraction of the population showing up newly infected each day.

You have to go over that threshold to get the pandemic to shrink.  So even if 70% was estimated with perfect accuracy, you’re going to end up with more than 70% immune before the complete return to normalcy.

And then, when you hit 70% (for immunity with no other tools), or more realistically 40% (for immunity combined with all our other COVID hygiene practices), how fast can you expect the virus to disappear from general circulation? 

I’m pretty sure the answer is, not very fast.  But that will depend almost entirely on the pace of vaccinations at that point.  If people are still gung-ho to get vaccinated, we’ll see a swift end.  If not, then not.

It will depend on vaccinations because once you get to the point where the pandemic is winding down, you are increasing the infected-but-recovered population quite slowly.

Let me just hang some crude numbers on that.  Each cycle of viral transmission takes an average of 4.5 days (references given back in Part II of this series).  So you get about 7 generations of viral transmission per month.  So far, the steepest rate of decline I think I have observed is North Dakota, where I believe the decline managed to hit an R value of about 0.91.  That is, with each generation, there were only 91% as many new infections as in the last generation.  One month’s decline, at that rate, is (.91 ^ 7 = ) 0.52, or 52% of the new-case rate that existed at the start of the month.

In other words, at best, you can hope to cut the number of new cases in half, in a month.  Once you are firmly into the remission phase of the pandemic.  (Because this is a crude cut, I haven’t bothered to check that against the real data, but it’s certainly ballpark.)

And then, as these things go, after two months, you’d still have about one-quarter of the peak cases left.  And so on.

Meanwhile, the rate of decline would speed up a little bit, because each generation of infections adds to the immune population.  But only a little bit.

But of you add a lot of vaccinations to that — even at our current rate of just 12.5% of the adult population per month — you speed up the end game enormously.  You aren’t relying on just the residual rate of infections to generate newly immune persons.  You’re artificially goosing that number up by vaccinating people.

And in the end game, you’d need to compare that 12.5% not against the total, but against the 30% that would be remaining (1 – 70%).  So, in the end, a continued high rate of vaccination can bring a swift end of the pandemic.  And if vaccination rates peter out — well, then so will the pandemic.  Refusal to vaccinate simply prolongs the agony of this for us all.

So let me end with that.  It is possible to believe that herd immunity matters, and that vaccination matters.  While there is some truth in the popular perception that “herd immunity” and “vaccination” are two opposing camps, that’s sociology, not biology.  Biologically, nature cares about immunity.  And to a close approximation, nature doesn’t give a damn how you acquired that immunity.

And if Mother Nature doesn’t care, our Federal policy makers shouldn’t, either.  And right now, that doesn’t appear to be true.