Post #1117: William and Mary, another good day, the math of herd immunity.

Posted on April 22, 2021

Today’s test results were 1-for-1772.  Three days of census testing are highlighted in yellow below.  In short, they’ve tested three-quarters of campus enrollment and found one COVID-19 case.

Again, obviously good news.

This is a follow-up to my last post, and in this I try to explain why you might not want to, or be able to, relax the William and Mary COVID-19 hygiene rules just yet.  Despite this good news.

You’ll have to trust me on this, but I’m not one of those people who pushes for all possible protections.  I just want to make a simple point.  The success, right now, shows that the combination of vaccination and COVID-19 hygiene appears successful in preventing spread of COVID-19.  It doesn’t show that vaccination alone will do that.

In this post, I back that last statement up with some numbers, below.

For sure, the risk from relaxing some COVID-19 restrictions is much lower now, than it was just a few weeks ago.  But there is still a risk.  It’s not a slam-dunk.  Somebody will have to make a judgement call that the potential benefits exceed the potential risks, in order to relax those rules for the remaining month of the semester.


Let me start with the law.

Whether or not it’s safe to relax some of the COVID-19 hygiene rules at William and Mary, I’m pretty sure that William and Mary can’t legally drop its mask requirements.  The reason being that there’s still a mask mandate in force in Virginia.

You can read the literal text of it as Executive Order #63, as amended,  on this Commonwealth of Virginia web page.  It’s pretty broad, and is clearly intended to require mask use in  ” … indoor settings to which the public has access, ...”  In addition to a long list of specific settings (including food and beverage establishments), it includes “… Any other indoor place shared by groups of people who are in close proximity to each other.”

Executive Order 72 (same internet page as above) gives much more detailed sets of limitations for specific types of businesses.  Examples include gyms, sports venues, and so on.  It reiterated the requirement to wear a mask as stated in Executive Order #63.  And it added a further requirement, that masks must be worn in outdoor settings when individuals cannot maintain a 6′ social distance.

There are exceptions.  This does not apply within a private residence, or while eating, drinking, or exercising, or to (e.g.) religious rituals.  In addition, you can exempt yourself simply by claiming to have a medical condition that prevents mask use.  Nobody can require you to prove it, and in fact, nobody even has the right to require you to state what the condition is.

I guess William and Mary could claim that they are a private institution, and therefore campus buildings are not “indoor settings to which the public has access”, but I think that’s a stretch.  Particularly for a state school, where many buildings beyond dorms don’t seem to have much in the way of restricted access.  But I don’t think it’s reasonable to expect that they would do that.

I’m not a lawyer, so I’m not sure sure exactly how far this goes.  E.g., could you claim that dorm common areas, accessible only with a key card, constitute part of a private residence for purposes of the law?  I can’t even guess, and I don’t think that’s fruitful thing to ask.

Let me put it to you this way:  Even Liberty University still requires face masks, as of today, as you can see on this web page.  Given their constituency, it’s hard for me to imagine that William and Mary would be the one to push the envelope on removal of COVID-19 restrictions, while Liberty University continues to walk the straight and narrow.

My takeaway is that any notion that William and Mary will relax mask use requirements in response to the recent drop in cases is probably wishful thinking.  I don’t think they have the option to do much, if anything, in that area, if they aim to continue to comply with the law.

In the context of the law, William and Mary doesn’t matter.  The law is set for the entire state of Virginia, of which W&M is a small part.  Best guess, masks rules will remain in place at William and Mary until such time as the Governor removes the mandate for all of Virginia.

Why, aside from legal issues?

The point of this section is to show that there are sound epidemiological reasons for keeping up COVID-19 hygiene measures.  That’s true even if a large fraction of the student body has been vaccinated.

The sole point here is that vaccination is not a guarantee.  The decision to relax the COVID-19 hygiene rules is NOT cut-and-dried.  You might think it is, given the way the test results are coming in.  But it’s going to be a judgment call, based on somebody’s perception of risks and benefits.

It’s obvious that the case for rigid hygiene measures is, in fact, far less under the current situation (high fraction of students vaccinated, few to no active cases known on campus) than it was in the recent past.   But it’s not the slam-dunk that some might think.  And that’s the only point I’m trying to make.

Here’s the first key fact:  If we continue to see zero active new cases on campus, the only thing we (probably) know with confidence is that the combination of vaccination and COVID-19 hygiene was enough to bring us past the point of herd immunity.

Worse, depending on the actual numbers (percent of students vaccinated, with which vaccines) if you run the numbers, it’s plausible that vaccination alone actually isn’t enough to suppress COVID-19 infections on campus yet.  So this isn’t some theoretical musing.  This is a plausible result.  And, if that’s the case, if you rely on vaccination alone, and drop all COVID-19 hygiene, you’ll see COVID-19 begin to spread in the student body again.  Particularly as the more-infectious U.K. variant B.1.1.7 becomes the prevalent strain of virus in Virginia.

The simple arithmetic of herd immunity.

The whole herd-immunity concept is pretty simple, as long as you don’t get into the details.  In order to have an epidemic go away, each infected person has to go on to infect less (fewer) than one additional person.  As long as that’s true, each additional group of newly-infected individuals will be smaller than the last, and the epidemic shrinks away.

And when that happens, you are said to have reached herd immunity.

Suppose that the virus is such that, if nobody did anything to stop it, each person would infect three additional persons, on average.  In order to stop that viral epidemic, you’ve got to find a way to block at least two-thirds of those infections.  If you can do that — if the native infectiousness (R-nought) is 3, and you can manage to block (more than) two-thirds of those potential infections, then you’ll get less than one new infection, for every existing infection.  The epidemic will shrink.

(If the R-nought had been 10 — so that each infected person would infect 10 others, with no precautions taken — then you’d have to block 90 percent of those potential infections to bring the pandemic to an end.  The math really is that simple.)

Textbooks present the story of herd immunity in terms of vaccine.  In the first example, if 100% of the population is going to be vaccinated, then the vaccine has to be more than 66.7% effective to reach herd immunity.  Under those assumptions, you’d bock more than two-thirds of new infections, and the pandemic would shrink.

But that’s not the only way to block infection.  Classically, you’d also bring in the fraction of the population that has already been infected.  Those people are (largely) immune.  They serve the same function as vaccinated individuals.  And if that population gets large enough, that will bring you to herd immunity and the epidemic will end.

But vaccination and prior infection are NOT the only ways to block infection.  Mask use, social distancing, and so on — all those COVID-19 hygiene measures — also serve to prevent infections that would otherwise occur.  Those factors also contribute to shrinking the number of new infections, and so shrinking the pandemic.

When discussing herd immunity, the only distinction between vaccine, prior infection, and  COVID hygiene is one of semantics, not math.  We talk about herd immunity to mean a state where “things can go back to normal”.  We talk about herd immunity as allowing us to get rid of the masks.  And so we don’t think to include masks and other measures as part of the herd immunity calculation.  And under that definition — normal means no COVID-19 hygiene measures — that’s OK.

But if we’re just talking about getting the pandemic under control, or maybe wiping out all spread with in the W&M student body, then all of those factors count equally.  Anything that reduces the number of new infections, from each existing infected individual, counts on an equal footing.

And if we see success, as we are seeing now, we can’t say that the success is due to one factor or another.  All we know is that the combined effect of everything that blocks transmission of disease is adequate, at the moment, to prevent spread of COVID-19 within the student body.

Now let me hang a few numbers on this, to show just how close to the margin we would be with vaccine alone.  I’m not giving citation as to source, but these numbers are approximately correct.

The strains of COVID-19 prior to the U.K. strain were thought to have R-nought somewhere around 2.5.  That is, with no intervention, in a typical situation, each infected person would infect another 2.5 persons.  That’s obviously going to vary from place to place.  E.g., group living quarters such as you find on a college campus probably lead to a higher R-nought value than the U.S. as a whole.

The new U.K. variant –– which is now accounts for about two-thirds of new cases nationwide — is thought to be about 40% more infectious.  Doing the math, the U.K. variant, then, would tend to generate around 3.5 new infections for every existing infection, if no action were taken.

Key question:  If 100% of students were vaccinated with the J and J vaccine, would that, by itself, be enough to stop the spread of the U.K. COVID-19 variant?  The J and J vaccine is thought to be just over 70% effective.  So you reduce infections to just 30% of what they would have been.  And if you do the math — 30% x 3.5 = 1.05 > 1.  So the answer is NO, vaccinating 100% of students with the J and J vaccine would NOT be enough to eliminate the U.K. COVID-19 variant from circulation in the student population.  You’d need to do more.

Now, the other vaccines are more effective.  But on the other hand, we don’t have 100% of the student body vaccinated.  So, would vaccinating, say, 70% of the student body with the Pfizer vaccine, at a presumed effectiveness of 90%, be enough to stop the spread of the U.K. variant?   Again, doing the math, 70%*90% – = 63%, so 37% of potential infections would remain.  Which is more than in the example above, and so 37% x 3.5 = 1.3 > 1.  Again, the answer is NO, vaccinating 70% of students with the superior Pfizer vaccine would NOT be enough to he U.K. COVID-19 variant from circulation in the student population.  You’d need to do more.

Anyway, the “more” in the paragraphs above is all the COVID-19 hygiene.  It’s hard to get a good estimate of how effective those measures are, but a good lower bound guess is 50%.  They stop maybe half, maybe more, of the infections that would otherwise occur.   It’s a fair bet that, absent those “more” measures, we wouldn’t be seeing the good results above.

These are simple but realistic examples.  My point is, I’m not just blowing smoke when I say that we take a risk if we drop all COVID-19 hygiene now.  I didn’t make that up out of thin air, I’m not one of those people who presses for doing everything possible.   I’m saying that because that’s how the numbers shake out.  Put in some round-number estimates of how infectious the new strains are, and how effective the vaccines are, and, at best, if we rely solely on vaccines, we’re right on the margin of being able to suppress spread of COVID-19.

So, for now, with one month left, and COVID-19 still widely in circulation in Virginia, just don’t be surprised if W&M chooses to stay the course, and not drop any COVID-19 hygiene measures.  It’s not clear that they will or won’t.  It’s not clear that keeping them all in place is the best thing to do.  And I certainly don’t know what’s the right risk-return tradeoff here.  I just want you to know that if they relax those rules, that’s not going to be a simple, no-brainer decision.