By the Numbers

The Curmudgeon does not like misinformation, no matter the source or the slant. He did not like it when covid cases and deaths were sensationalized through fuzzy reporting, or when the PCR cycle thresholds were set at high levels to increase positive test results. He especially hates it when fuzzy math is used to misinform, as it was with the positivity rate, or unemployment data last April and May.

So he is going to be equally deliberate in addressing misinformation fueled by bad math no matter what it relates to. In this post, it will relate to vaccine effectiveness. Specifically, claims that breakthrough infections show that vaccine effectiveness is low, or waning, or that the number of cases among the vaccinated is about the same as among the unvaccinated. But first, some disclaimers.

I am not a vaccine crusader. I have stated before my own fence-sitting position regarding the vaccines, and if and when that changes, I will keep it to myself. I will not proclaim my vaccination status on social media; if one derives one's virtue from having gotten a shot, one is rather devoid of virtue. The same is true if one has to signal one's virtues on social media in general.

I believe strongly that it is each individual's choice whether to get any vaccine: covid, influenza, shingles. (Although I do believe there is a special corner of the loony bin for those blanket anti-vaxxers who believe that childhood vaccines cause autism; as the grandfather of an exceptional child on the spectrum, I know better.) I do not believe in mandates, although as a free-market capitalist I defend the right of any business to make its own decisions regarding them. Just as I defend any consumer's or employee's right to make their own decisions regarding patronage or employment.

And I absolutely am opposed to vaccine shaming - the practice of using phrases like "what we have is a pandemic of the unvaccinated," or using the fact that the vast majority of infections are among the unvaccinated to attempt to marginalize those who have exercised their right to make their own health decisions, and not get a vaccine. Nobody tries to shame people who don't get the flu shot every season, and the flu kills a few hundred kids a year. (Spare me the false equivalencies over numbers.)

Now, on to the topic of this post. Amid the spikes in cases that we've seen from the spread of the Delta variant in the U.S., we've all seen the reports from doctors that "99% of infections are among the unvaccinated." Those, I believe, are indeed slanted toward the vaccine-shaming narrative; the more reasonable reports are probably those that say "97-99%," and that's likely even on the high side.

However, in recent days I've seen several articles floating around, reporting on breakthrough cases in various locations, and using bad math to imply that infections among the vaccinated are as common as among the unvaccinated. I'll look at each of these in turn, address the problems posed by the assumptions made and any conclusions offered, and explain the math that refutes the argument that cases among the vaccinated are equal to those among the unvaccinated. Finally, I'll present some verified statistics that support that math.

Provincetown, MA

Several sources reported a CDC finding that 74% of people infected in an outbreak stemming from several events and gatherings in Provincetown in July were fully vaccinated. (This, in part, was the basis for the CDC's ridiculous reversal of its mask guidance.) Provincetown sits at the tip of Cape Cod, and hosts a number of summer events.

However, here's what's important about the Provincetown data. First, the CDC study was based on a total of 469 covid cases that resulted from those events. That's 0.03% of the total covid cases in the U.S. in the month of July. So to extrapolate the Provincetown experience to the entire U.S, you'd have to multiply the Provincetown breakthrough infections by 3,333.33. In other words, given that there were 347 breakthrough infections of fully vaccinated individuals from the Provincetown events, that would mean that there were 1,156,866 breakthrough infections of fully vaccinated individuals in the U.S. in July. Don't you think that we'd have heard about that, were it true? Even if one is so predisposed to conspiracy theories as to believe that the mainstream media would bury such a story, consider that the mainstream media reported on the Provincetown study. At a minimum, Tucker Carlson of Fox News would have let us know if there were over a million breakthrough infections in July.

So the first thing the Provincetown case suffers from is the law of large (or in this case small) numbers. It just isn't statistically significant. It cannot be extrapolated.

The second thing that's noteworthy - and this is important, so pay attention, because this is the crux of the math behind all of these breakthrough reports - is that Massachusetts has the second-highest rate of vaccination among U.S. states, at more than 64% fully vaccinated. The highest vaccination rate in the state is on Cape Cod (other than the sparsely populated island communities), at 77% of the eligible population.

Why is this so important? Here's how the math works. Let's take it to the extreme and assume a hypothetical country (or state or county) where 100% of the population is vaccinated. At that point, every infection will be a breakthrough infection. But why will there be any infections at all, you ask? Because the vaccines' efficacy - their ability to prevent infection - is less than 100%. In trials and in actual experience since emergency use authorization was granted back in December 2020, the efficacy rate has been proven to be 90-95%. So 5-10% of those vaccinated could still get infected.

Now, if we back that off a bit, and take a place like Cape Cod, where the vaccination rate isn't 100%, but rather 77%, you're going to get some infections among unvaccinated people, and some breakthrough infections among vaccinated people. I'll provide a graphic below that neatly illustrates how this works, but for now, let's stick to the math.

So let's put a bunch of people, vaccinated and unvaccinated, together at some large events in Provincetown, all in close proximity. (Provincetown is a pretty small place.) If we assume that about 77% of them are vaccinated, in line with the Cape Cod vaccination rate, and we wind up with 74% of the people who wind up getting infected being fully vaccinated ... are you starting to see how the numbers might line up?

It's important to note that among those 347 breakthrough cases, 274 (79%) were symptomatic, four (1.2%) were hospitalized, and none died.

But another really important thing to note is that there will be fewer infections, the higher the vaccination rate. This also belies the notion that there would be no difference in cases between the vaccinated and the unvaccinated. We'll address that toward the end of the post.

Israel.

There's a video making the rounds on social media of an interview on Israeli TV of a doctor from a Jerusalem hospital talking about breakthrough infections there. The interview is in Hebrew, with subtitles also in Hebrew, but the video has English subtitles. I question the veracity of the English subtitles, because having searched all over the internet for the video (I originally saw it on a pretty questionable site), I could only find it on some rather obviously dubious conspiracy theory sites, plus on the YouTube page of a guy going by "Mr. Ace."

I eventually landed on the hospital's English-language Facebook page, which posted the interview along with some quotes from it in English. However, that video did not have the English subtitles, hence my suspicion. (The Curmudgeon has a rule: if I can't verify it, I can't believe it. Of course, that doesn't mean it isn't true. But it earns bonus suspicion points if it appears on a lot of questionable sites.)

The subtitles say that the doctor is claiming that the vaccines' effectiveness is waning, hence the breakthrough infections in Israel. Now, even if he never said that, I will say that I have confirmed that Israeli officials have approved a third booster of the Pfizer vaccine, and are encouraging it for those over 60 or who are immunocompromised. However, they acknowledge that they don't know whether the vaccine's effectiveness is actually waning, or whether it's just not as effective against the Delta variant as against previous variants, so they're willing to give the booster a shot (pun intended). They also have stated that even absent the booster, the severity of illness among the vaccinated is much less than among the unvaccinated.

But in Israel, there aren't very many of those left. Like Massachusetts, Israel has a very high vaccination rate. They got an early start in their vaccination program, and they had mandatory vaccines for students as a requirement to sit for exams in the Spring (something I also don't agree with). Nearly 60% of the population is fully vaccinated. So again, the math is going to follow the pattern noted above.

A recent study published in the New England Journal of Medicine found a 2.6% breakthrough infection rate among fully vaccinated Israeli healthcare workers. Of those, two-thirds were mild and a third were asymptomatic, although, among the elderly who have been hospitalized in Israel, there have been serious breakthrough infections. Still, a 2.6% breakthrough rate is pretty low.

As early as June, the Israeli health minister noted that about half of infections were occurring in vaccinated individuals. At that time, 85% if Israeli adults were vaccinated. The graphic below illustrates how 50% of cases among vaccinated individuals can be distorted given that high a vaccination rate:

Each circle represents one person assuming a hypothetical population of 100, with a vaccination rate of 85%. Two of the vaccinated people got infected, and two of the unvaccinated people got infected, so yes, 50% of the infections were among the vaccinated population. What that statistic doesn't reveal is that the infection rate among the vaccinated was only 2/85, or 2.3% (not dissimilar to the NEJM study's finding), while the infection rate among the unvaccinated was 2/15, or 13.3%. In other words, the infection rate among the unvaccinated is about six times higher than among the vaccinated.

I'm going to pull a Jen Psaki and circle back to the Israeli government's decision to recommend a third booster dose. The CDC doesn't recommend that, not that we can trust their guidance much. Nor does Pfizer yet, even though they have a financial incentive to do so. So maybe Israel is onto something; maybe they know something we don't know. Or maybe they're just throwing a dart here (pun intended again). Too soon to tell. But even they admit that they can't conclusively say the vaccine's effectiveness is waning - they're hoping that if it's not that, if it's just that the efficacy against Delta isn't as high (and we've seen that to be true), another shot can't hurt.

The UK.

You may have heard of Alex Berenson. He's been making the rounds on some cable shows. He's written some booklets on covid, related to the ineffectiveness of the lockdowns, the inaccuracies of death count reporting, and the ineffectiveness of masks. He's also written about vaccines.

Now Berenson is a novelist, of the spy genre, which I'm keen on. I've been reading his stuff since long before the Wuhan lab engineered covid. He has degrees in history and economics, and he used to write for The Street.com, which was founded by CNBC's Jim Cramer. I like Cramer, and I follow The Street. I also agree with much of what Berenson has had to say about lockdowns, death count reporting, and masks. But I have to part ways with him on his vaccine "analysis." Not because I'm a vaccine advocate. But because his math is flawed. And let's not forget that, by trade, he's a writer of fiction, not a researcher.

I'm not going to go back and find the Twitter post where he talked about the UK data, but it was very similar to the Israel data: he "found" that breakthrough infections accounted for about the same number of recent cases as infections among the unvaccinated. And he used that to conclude that the vaccines were ineffective.

But the UK also has a very high vaccination rate: 58.8% of the population fully vaccinated. So when I applied the math to his data, I found that the infection rate among the vaccinated was quite low compared to the infection rate among the unvaccinated, as illustrated above.

Now, let's look at some data from a place where the vaccination rate is low: Southwest Missouri. I'm picking this area because a) it's been a recent hotspot for infections and hospitalizations, and b) I personally know people who work in the two main hospitals in Joplin, Missouri.

Joplin straddles two counties: Newton and Jasper. In the month of July, those two counties had 2,379 covid cases. Compare that to the 469 cases from the Provincetown study. The two counties have a combined population of 179,654. So the infection rate was 1.3% of the population. 

As of July 31, 17.9% of the population of Newton County was fully vaccinated, and 33.6% of the population of Jasper County was fully vaccinated. A local news article interviewed officials from both major hospitals in Joplin, and both reported that 92-94% of hospitalizations were among the unvaccinated. They also noted that those hospitalized who were vaccinated had less severe cases and were discharged earlier.

Now, this is important: if the hospitalizations among the vaccinated are less severe, logic would tell us that the cases overall among the vaccinated are less severe. Thus you'd have fewer hospitalizations overall among the vaccinated. So if 93%, on average, of hospitalizations, are unvaccinated, then it may not be a stretch to say that 97% of all cases - hospitalized, not hospitalized but symptomatic, and asymptomatic - are among the unvaccinated.

But who really cares? The number, averaged across the U.S., with its average vaccination rate of just over 50%, is likely north of 90% among the unvaccinated, especially since many of the 21 states whose rates are above the national average aren't very populous. In the states with the highest vaccination rates, the number will be below 90% due to the math explained above; as we approach 100% in a given location, we'll approach 100% breakthrough infections. In states with very low vaccination rates, like Alabama, you're probably close to 99%.

Again, who cares? As I said before, I'm adamantly opposed to vaccine shaming. But I'm equally opposed to spreading misinformation just because it aligns with one's preconceived notions - because it purports to prove what one desperately wants it to prove. Don't want to get vaccinated? I wholeheartedly support your choice. Just don't use flawed "analysis" propped up by bad math to try to convince the world that your decision is rooted in "evidence" that the vaccines are ineffective. The math proves that they are - but you're still free to choose not to get one. You think the guys who choose not to wear motorcycle helmets post articles on Facebook claiming scientific evidence that they're not effective in preventing brain injury in a crash? No. They know that they are. They just make a personal health choice that is their right to make (where the law allows).

Let me also say that I do not believe that this spreading of misinformation is always willful or wanton. Sometimes, it's just that those who share it want so much to believe the narrative, to believe that there is scientific justification for their decision to not get vaccinated, that they latch onto anything that aligns with their pre-conceived notions, without questioning it. And maybe they don't understand the math well enough to question it in the first place. They think they've found a smoking gun.

But others see themselves as right-fighters, and do share the misinformation willfully and wantonly (not that they understand that it's misinformation, mind you - they just don't get the math). They decry the vaccine crusaders, while they crusade against the vaccines. They criticize those who would infringe upon the rights of those who make their own private health choice to not get vaccinated, while they attempt to influence the rights of those who might make the private health choice to go ahead and get the vaccine, by plying them with flawed information fueled by bad math. Just leave everybody alone, already, and let them make their own choice, just as you want to be free to make your own choice. You want to be a right-fighter? Fight for that.

One final bit of evidence that I alluded to earlier, related to the difference between cases in the vaccinated vs. the unvaccinated: in July, Massachusetts, with its 64% vaccination rate, had about 10,000 covid cases among its 6.9 million citizens. Missouri, with its 42% vaccination rate, had more than 50,000 cases - more than five times as many - among its 6.1 million residents.