Edit: Read Post #982 first.

I think this is an important post. So, no kidding around. Minimal entertainment value. And you’re going to have to follow a little bit of arithmetic in the followup post.

And at the end, you’re going to say, well, that’s obvious. And yet, nobody seems to have grasped this. Yet. Or, at least, I have yet to find a single discussion that makes this point.

**The point being that there are two distinct herd immunity levels that matter.** One is the fraction of the population that needs to be immune, to stop a pandemic,* if immunity is the only tool used to limit spread of the virus.* That’s our classic “70% required for herd immunity”. The other is the level of immunity needed to stop a pandemic *while all the other infection limiting tools (masking, distancing, limits on gatherings) are still in place.* That’s a much lower limit that I have not seen discussed anywhere. But that’s the level of immunity that’s relevant to ending this third U.S. COVID wave. And I crudely estimate that second version of herd immunity to require something like 40% of the population to be immune.

That’s a little controversial, I think. Or, at least, under-discussed.

So here goes. Words today, crude numerical estimates tomorrow.

### The basic math

**I decided to work through the basic arithmetic of herd immunity. That took all of about two minutes, because the math is really easy.** If a virus is such that each infected person infects *two others* on average, then *half* the people have to be immune to bring it under control. If the average infected person infects *three others*, then *two thirds* have to be immune to bring it under control. Ten others, then 90% have to be immune. And so on.

The percentage required for herd immunity is simply the number that generates a net viral reproduction of 1.0 or less. Because once you get the reproduction below 1.0, the number of infected people shrinks. So if the raw viral reproduction number is 2, you have to cut that in half. If it’s three, you have to cut it by two-thirds. And so on.

Just as a matter of arithmetic, then, you now know how we got the often-discussed “70% required for herd immunity”. That’s based on somebody’s estimate that the initial reproduction number for COVID (“R_{0}“) is about 3.33. Because 30% (1 – 70%) of 3.33 is 1.

### The full definition matching that basic math

That’s easy enough to grasp. But the arithmetic isn’t the point.

**Here’s the point. Everybody parrots the standard definition of herd immunity. But nobody actually says the full, entire definition.** It’s easy enough to calculate the 70% figure. But what does it actually mean?

So here it is, in full. This is the full, technically correct definition of the fraction of the population required to achieve herd immunity, per the math above.

**The often-discussed “70% required for herd immunity” is the fraction of the population that must be immune to COVID in order to stop the pandemic, *** if immunity is the only tool used to limit the spread of the virus. *Another way of looking at it is that it’s the level of population immunity required to keep the pandemic at bay

*and allow us to return to normal, where “normal” is defined as life without any of the tools we are currently using to limit spread of the virus.*Just to drive that home, that 70% figure is what you get *when immunity to the virus is used as a stand-alone tool. * When it’s *the sole factor limiting spread of the virus.* Because — at the end of the day, if you want to return to “normal”, that will be the only thing keeping the virus in check. Get it?

### The empirically important implication

Obvious, right?

But that implies the following. It means that **the 70% figure is what we ultimately need to be able to return to normal, but it overstates the level of immunity required merely to bring the pandemic under control.**

**More specifically, that 70% figure is NOT the fraction of the population needed to be immune to bring the pandemic under control **** while we continue to use all the other tools that limit spread of the virus. **Putting that another way, it’s NOT the fraction that we need to bring the pandemic under control in the period before we return to “normal”, that is, in the period where we continue masking, distancing, and limiting social activities.

Just to drive that home, that 70% figure is NOT what you need to bring the pandemic under control, *when immunity to the virus is used in conjunction with other tools that limit viral spread. *When it’s j*ust one of many factors being used to limit infections from the virus.* Because — right now, as we try to get the third wave under control, we’re going to keep using masks, distancing, and controls on social gatherings as we accumulate immunity via infections and vaccines.

And, as you would expect, if herd immunity is just one of several tools limiting infections, **then you need far less than 70% to achieve control of the pandemic.**

**How much less?** I’m going to work out the arithmetic in the next post (tomorrow AM), but my quick back of the envelope using a one-point-in-time estimate from the worst of the North Dakota episode **yields 40%.** Once you get over 40% truly infected, and you have all the other tools in place, that ought to be adequate to produce a net viral reproduction number below 1.

And so, my contention is that you need 70% to be immune, in order to bring the net (effective) viral reproductive number below 1.0, if you start with an initial reproductive factor of 3.33, *and if immunity is the only factor limiting viral reproduction.* But if you take the observed reproductive number at the height of the North Dakota episode, and back-solve, you only need about 40% immunity to bring the net (effective) viral reproductive number below 1.0, *if you keep all the other factors limiting viral spread in place.*

That’s the arithmetic I’ll present in the next post, tomorrow morning.

### Further implications

So, how does any state manage to go over 40%, and why is the epidemic still going on in the Midwest? That 40% estimate is an equilibrium estimate. It doesn’t really account for the dynamics, and it doesn’t account for disequilibrium (And separately, it doesn’t account for heterogeneity with in the state.)

It’s a rough cut, in many ways. But if you work it through, I think it explains this picture below. That sure looks like those states are on the glide-path toward the end of the third wave of COVID. After having had a high fraction of the population infected with COVID.

Source: Calcluated from NY Times COVID Github repository. Data reported throu 1/25/2021.

But they sure seem to be taking their time getting there. And that’s the other thing I’ll try to explain tomorrow. But that’s not rocket science either: **You won’t necessarily get to that 70% “pure” herd immunity level if other factors are being used to limit viral spread, in addition to immunity.** Why? Because the rate of new infections slows when the combined effect of herd immunity, plus those other factors, reduces the net (effective) viral reproductive factor to 1.0. **With immunity plus masking-distancing-limiting, you should stop the spread of infection well before you reach that stand-alone herd immunity level of 70%.**

And so, absent changes in the infectiousness of the virus, for the Midwest, at least, the end of the pandemic should proceed in two discrete steps. First, they’ll work their way down to a trivial number of daily new cases. But maintaining that will depend critically on maintaining the masking-distancing-limiting that, along with immunity, keeps the virus in check. And then, they’ll get enough immunizations to raise the overall immunity level up enough that they can drop the masking-distancing-limiting, and rely solely on herd immunity to keep the pandemic in check.

**The upshot of this is that there are two distinct herd immunity levels of interest.** One is the lower level required to bring the pandemic to a halt when used in combination with other infection-limiting tools. The other, higher level is the classic stand-alone herd immunity, the 70% figure, where immunity is the only tool keeping the pandemic in check. The level required to return to full normalcy.

For the time being, call those H_{e} and H_{0}, corresponding to R_{e} and R_{0}. A crude estimate of H_{e}, for North Dakota, is the subject of tomorrow morning’s post.