Post #979: The two distinct levels of herd immunity, Part II

Posted on January 27, 2021

Edit:  Read Post #982 first.

This post presents a calculation to match the herd immunity discussion of the just-prior post.  Read Post #978 first, then this one.

Here, I back-solve for the level of immunity in the population that should bring the effective COVID-19 viral reproduction factor below 1.0 (i.e., end the third wave of the pandemic), as long as we maintain masking, distancing, and other behaviors limiting viral spread.

This is a simple calculation, based on one point in the progress of the pandemic in North Dakota.  That point being the two weeks when North Dakota saw its sharpest increase in cases.

So there’s not a lot of accuracy here.  And it’s not an estimate, in the sense of being a statistic calculated from pooling a lot of data.  It’s really just a round-numbers (but data-based) illustration.  It shows that the two different herd immunity concepts defined in the prior post lead to two very different levels of required population immunity.  And that we may already be hitting the lower level in some states.

Bottom line:  40%.  Once something like 40% of the population has been infected, that ought to be enough to set the third wave of COVID on a downward trajectoryAs long as we maintain masking, distancing, limits on social gatherings, and other such controls.   But we’d still need the classic “70% herd immunity” to return to normalcy, meaning, life without those controls.

The upshot is that the uniformly downward trajectory seen in the U.S. Midwest probably isn’t a fluke, or luck.  It’s probably just a matter of arithmetic.

The clear policy implication is that there is a more efficient way to use the COVID vaccines, if the goal is to bring the U.S. third wave of COVID to a close.  You should concentrate vaccination in those states that have had the fewest infections so far.   You shouldn’t aim for an equal share of the population vaccinated in each state, as we are now.  You should aim for an equal share immune in each state, either via vaccination or via prior infection.  That means shifting vaccine from states that have already had widespread COVID infection, to states where a higher fraction of the population still lacks immunity to the virus.

Background and importance.

In my just-prior post, I clarified exactly what “herd immunity” means.  Classically, that’s the level of immunity in the population that, by itself, will suppress the spread of an infectious disease.  Restated, that’s the level you need if you are going to return to “normal” and thereby rely on immunity alone to suppress continued spread of the disease.

That level has typically been estimated to be about 70%, for the variants of COVID-19 that are currently prevalent in the U.S.

But if you combine herd immunity with other factors that limit the spread of the virus, you shouldn’t need 70% immune to drive the effective viral replication number below 1.0 (and so begin to reduce new infections).  You’ll need something less than 70%.  How much less is the subject of this post.  And, shown below, my crude estimate is that you only need around 40%.

Let me borrow a term from oncology: remission.  In a sense, the level of population immunity needed to drive the pandemic into remission is less than the level required to end it permanently.  You need the full 70% to end it, and allow return to full normalcy.  But you only need 40% or so (crude estimate) to put it into remission, as long as you keep up all the other infection-suppression strategies.

Up to now, herd immunity has gotten no serious attention from mainstream analysts.  And that’s reasonable.  Between that high value required (70%), and the low number of diagnosed infections (about 7.5 percent of the population), it looks like herd immunity just doesn’t matter.  Yet.  Today (1/27/2021), you’re looking at maybe 25M formally diagnosed with COVID-19 in the U.S., or something like 7.5 percent of the population.  To which we can now add some fraction of the nearly 20M who have been vaccinated so far, with varying levels of immunity for partial or full immunization.  No matter how you slice it, that total (infected plus fraction of vaccinated) will be small relative to the presumed 70% needed for classic herd immunity.

But, first, total persons actually infected is a large multiple of the formally diagnosed COVID-19 cases.  I presented the rationale for my working estimate — five total cases for every diagnosed cases — in Post #940.  In a nutshell, that splits the difference between two estimates based on work by CDC staff.  So, while only about 7.5% have been diagnosed, it’s likely that about ~38% have actually been infected.  Between infections and vaccinations, as of a couple of days ago, you could reasonably say that a little over 40% of the U.S. population is probably currently immune to COVID-19 (Post #976).

OK, well, that’s a different story then.  Now you’re getting into the ballpark of that 70% figure.

And now, second, if you combine herd immunity with other factors that limit the spread of the virus, you shouldn’t need 70% immune to drive the pandemic into remission.  You’ll need something less than 70%.  How much less is the subject of this post.  And, shown below, my crude estimate is around 40%.

If you step back from it, I think that has some face validity.  It has just been under-recognized in discussion of the pandemic.  If having 70% of the population immune will, by itself, end the pandemic, then having 40% immune plus masking/distancing/other restrictions should (plausibly) also drive the pandemic into remission.  That’s all I’m trying to say here.

Practical upshot:  I think that most of the Midwest already has had enough COVID cases to exceed that threshold.  I think the slow and steady declines being seen in the Midwest are, in fact, the end of the third U.S. COVID wave there.  They’ve already reached what I’m calling He , the fraction of the population that you need to be immune, in order to drive the pandemic into remission, when combined with all your other infection-suppression strategies.  (That’s in contrast to what I’ll call H0 , the level you need to return to normalcy and drop all those other infection-suppression strategies.)

The calculation.

I’m just going to walk through it.  I don’t think you’ll need a lot of background to follow this, but I’m not sure.  The algebra is simpler than the jargon.

Overview:  I’m going to use an assumed initial viral reproduction rate R0  of 3.33.  I’m going to estimate an effective viral replication rate Re of about 1.1 during the height of the North Dakota outbreak.  And I’ll use those two values to parse out the effect of population immunity, separate from the effect of masking/distancing/limiting.  Finally, given the impact of masking/distancing/limiting, I’m going to calculate the level of population immunity required to bring Re down to 1.0

First, assume that the basic viral reproduction number — before anybody is immune, or anybody takes precautions — is 3.33.  That is, on average, with no precautions, each infected person would go on to infect an average of 3.33 others.  As discussed in the last post, that’s the number that corresponds to “70% must be immune to achieve herd immunity”.

Second, assume that each generation of infections takes 4.5 days.  I think that’s termed either the generation time or the serial interval, depending on how, exactly, you measure it.  That’s the average time between when a person gets infected, and when that person goes on to infect others.  You need to know that in order to work out the viral replication rate R from the observed weekly growth in new cases.  (I.e., one week is 7/4.5 = 1.56 replications of the virus.  The value of 4.5 is around the median of what has been estimated for COVID-19 (see Figure 2 in this reference).

Now turn to the North Dakota COVID outbreak, and estimate what the effective viral replication rate was at the steepest part of the up-slope.  By eye, I’m taking the two weeks starting 11/1/2020. (Two weeks, and round-number boundary to avoid over-optimizing this.)

I have a vague reason for picking this high-growth section for the calculation.  I don’t think you get a clean estimate of potential viral replication, from an area-level growth number, unless you have the entire population of that area heavily involved.  That is, you can’t get an estimate of the limits to growth unless you are actively pushing against those limits.  Otherwise, the observed average growth rate reflects those limits plus whatever slack is in the system, e.g., sub-populations or sub-areas that are not actively in contact with infected individuals, and so are not reflecting the infection transmission process.  In other words, you can’t expect to get a good estimate of the constraints on growth if there’s a lot of slack in the system.  You really want to find an instance where the system appears to completely up against those constraints.  Hence, I chose the highest-growth-rate scenario I could find.

Well, I said it was vague.

Now the numbers and the algebra.

For the two weeks starting 11/1/2021, I estimate the observed viral replication rate in North Dakota as follows.

  • Initial new infections:  138/100,000/day.
  • Final new infections:  181/100,000/day.
  • Increase:  1.32-fold (181/138).
  • Number of generations: 3.11 (14 days / 4.5 days per generation).
  • Observed replication rate 1.09 (= 1.32 ^ (1/3.11)), where ^ is the power function (“raised to the”).

For the sake of round numbers, I’m going to call that 1.1.  Under these assumptions, the effective (observed) viral replication rate was about 1.1 per generation.

One more fact you need to know:

  • Fraction of ND population diagnosed with COVID as of 11/1/2020: 6%.
  • Assumed true fraction that had been infected:  30% (= 5 x 6%).
  • See Post #940  for the justification for the factor of 5.

The basic equation:  Observed viral replication = initial viral replication adjusted by all the factors that reduce infection.  I’m going to use VX to represent all the factors that keep people from getting infected.  That follows tradition (V for vaccine), but I want to be clear that V here is a combination of the effects of the fraction of the population immune via infection, and the impact of masking/distancing/restrictions.

So:  The observed viral replication number (right) is the original replication number, reduced by all the factors that are blocking spread of disease.

  • R0 * (1 – Vx) = Re
  • 3.33 * (1 – Vx) = 1.1

And the factors blocking transmission are:

  • the 30% of the population that is already immune
  • plus the impact of masking etc. (denoted by X here)
  • less the overlap of the two.

If you don’t grasp the need for taking out the overlap, read Post #976.  Heuristically, masks are wasted on people who’ve already recovered from infection.  That’s the reason for netting out the cross product, and the exact form of that embodies an assumption of independence between the two factors.

  • Vx = .3 + X – .3X

So, write out the full equation and solve for X:

  • 3.33 * (1 – (.3 + X – .3X)) = 1.1

I come up with X ~= 0.53.  Check that by plugging that in, if you wish.

(Which, as an aside, is interesting in its own right.  That says that the net effect of all those other restrictions is equivalent to having fully vaccinated half the population.)

OK, take that 0.53 as a given, and solve for the fraction of the population that needs to be immune in order to bring the observed viral replication down to 1.0.  Let that fraction be H.

  • 3.33 * (1 – (.3 + .53 – .3*.53) = 1.1
  • 3.33 * (1 – (H + .53 – .H*.53) = 1.0
  • Solve for H
  • H =~ 0.37

I actually did the calculation then rounded, but if you plug in the numbers as written here, you’ll come close.

The upshot is the following:  Given the estimated effectiveness of masking/distancing/restriction at preventing spread of the virus, you only need about 37% of the population to be immune in order to drive the observed viral replication factor down to 1.0.

Given how approximate this whole calculation is, and bowing in the direction of round numbers, let me just call that 40%.

And so, my one-point-in-time estimate of He , the fraction required for herd immunity when used in conjunction with all the other infection-prevention tools, is 40%.

So, best estimate, in North Dakota, in order to turn this around, you had, in round numbers:

  • A 40% percent reduction in infections, due to herd immunity (those already immune)
  • A 50% reduction in infections, due to masking/distancing/restricting.
  • Less a 20% overlap between the two.

Do that math, and … hey, that adds to 70%.  Which is, if you think about it, not really a coincidence.  It’s just math.

And I think that’s the real lesson of the algebra.  If the classical herd immunity number is 70%, then you will drive the effective viral replication factor below 1.0 however you go about achieving the equivalent of that 70%.  And to hang some numbers on that, all you need to do is translate your current infection-reduction efforts (masking/distancing/restricting) into an equivalent number of immune individuals.

Which is, in effect, what this calculation did.


First, I apologize if this has been already been worked out, in the public domain somewhere.  I’m not familiar with (e.g.) what is taught in epidemiology textbooks.

Second, I think this has some fairly positive implications for the pandemic, and for vaccination strategy. If you keep all the rest of the protections in place, you can put the third U.S. COVID wave into remission with 40% of the population immune, via a combination of infection and vaccination.

It also has some strong implications for how vaccine strategy could be made much more efficient.  And, fully acknowledging that this will never happen, I’ll say it anyway:

Don’t waste vaccine on states were a large fraction of the population has been infected.

Ain’t gonna happen, I get that.  But, if this analysis is right, once states have had a high enough prevalence of infections, they are on the glide-path toward remission, regardless.  Vaccinating on top of that is a waste.

Once you factor in the impact of herd immunity (the He variety, defined above), you should, instead, concentrate vaccinations in states that have had a low total prevalence of infection so far.  Crudely, you’d do all the vaccinating in states where less than 40% of the population has been infected.  That would produce the most rapid end of the COVID third wave, for the U.S. as a whole.

Again, not going to happen.  But if you step back from it, it makes common sense, doesn’t it?  The whole point of vaccination, from the standpoint of ending the pandemic, is to increase the fraction of the population that is immune.  Places that already have a large fraction immune via infection need less vaccine, to achieve some overall targeted fraction of the population immune.  All I’ve added to that, with this calculation, is a crude estimate of a 40% cutoff.  If the goal is to shut down the pandemic, once a state reaches 40% immunity by any means, anything more is a waste.

Addendum:  Appeal to authority information/about the author.

I suspect that a lot of people are going to recoil from this finding, and will judge it based on the fact that it’s being said by some random blogger.  So let me append the standard appeal-to-authority information, a.k.a, “about the author”.

The author, Christopher Hogan, is a retired Ph.D. health economist.  He spent ten years as a principal policy analyst for the Medicare Payment Advisory Commission (and predecessor organization), a Federal legislative-branch agency tasked with advising the Congress on Medicare policy.  There, he worked on issues including Medicare physician payment rates, and the cost of end-of-life care in the Medicare program.  Following that, he spent 20 years as a self-employed health economics consultant, working primarily for Fortune 500 health care manufacturers.  He retired in 2018.