Tuesday, February 28, 2012

Of Maths and Measles

There are times that I really despair for our country. Specifically, the educational system troubles me from time to time. Take simple mathematics, as an example. By the time one graduates high school and becomes a, presumably, productive adult in our society, regardless of whether said individual goes on to college or jumps right into the workforce, there are certain simple skills that they should have. All the basics should be well in hand: addition, subtraction, multiplication, division. They should have a good understanding of decimals, fractions and percentages. Even if someone needs to use a calculator, rather than doing it in their head or scratching a problem out on paper, they should at least have an understanding of how these things work and how to use them.

This woe for the state of education came bubbling up again as I read a comment on a PBS article about the measles outbreak in Indiana. Allie Morris, the author of the article, wrote that at least 13 of the individuals who had contracted measles had a history of MMR vaccine refusal. The commenter, Mmavallet, couldn't believe this number, suggesting that it was statistically impossible for there to be 13 individuals who were all unvaccinated, coupled with a belief that the vaccine couldn't be 95% effective (it's not, BTW, being >99% effective):
I don't buy that all 13 people spread over various counties were all unvaccinated. And the MMR does not have a 95% efficacy rate or else you wouldn't be having these outbreaks in a population where 98% of people vaccinate. It's statistically impossible. If you really research the numbers on these outbreaks, about 50% of those infected are vaccinated! If they really worked, you wouldn't have to continually get revaccinated for the same illness. It's not immunity if it wears off...only natural immunity is permanent.
Let's take a closer look at the issue, shall we?

Basically, the problem involves figuring out a couple of fractions and percentages. We might all remember story problems asking us to figure out that X is what percentage of Y? Or, alternatively, X is Z percent of what? And yet another variation, what is Z percent of Y? This can be boiled down into a simple equation:
X/Y = %/100
To address this individual's comment, we would need to start with a few basic assumptions. First, that the vaccine is 95% effective (we'll do this with a 99% figure, as well). Second, we need to know about how virulent measles is; in other words, what percentage of vulnerable people would be infected if exposed? According to the CDC, that attack rate is about 90%. Finally, we need to know what the approximate uptake rate is for the vaccine. The commenter claims that 98% of individuals in the area vaccinate. We'll use that number for our calculations.

Let's get to work, then. Assuming that the 13 individuals were unvaccinated, and that they are 90% of the total unvaccinated population who were exposed (remember the attack rate for measles), then we find that there would have to be at least 14.4 people total who were not vaccinated (let's round that up to 15, since we can't have partial people). How'd I get that? Using the formula above:
13/Y = 90/100
13 = (90 * Y)/100
13 * 100 = 90 * Y
(13 * 100)/90 = Y
1,300/90 = Y
14.4444... = Y
So, if there are a total of at least 15 individuals who are not immunized, how big would the population need to be? Let's use Mamavallant's 98% MMR vaccine uptake rate and the formula above:
15/Y = 2/100
15 = (2 * Y)/100
15 * 100 = 2 * Y
(15 * 100)/2 = Y
1,500/2 = Y
750 = Y
We'd need a population of at least 750 people in order for there to be 15 unvaccinated individuals in an area with a 98% vaccine uptake rate. That's not a whole lot of people, and my gut is telling me that there are probably more than 750 people living in the affected areas. In fact, if we take a look at the U.S. Census Bureau's QuickFacts for Hamilton County (one of the two counties affected by the outbreak), we see an estimated total state population of 6,516,922 for 2011. Even if we take only individuals under the age of 18, we still have 1,607,982.896 (round up to 1,607,983) people using the 2010 census numbers. If there is a 98% uptake rate, then there are 32,160 people under 18 who are not immunized.

It's been a while since I was in math class, but 32,160 appears to be larger than 750. But, says the vaccine critic, that's for the whole state! What happens if you limit it to the city (or cities) where cases have been identified.

Glad you asked, hypothetical naysayer! Noblesville is one of the cities in which at least one case has been identified. In 2010, again, according to the U.S. Census, the city had a population of 51,969. Of those, 15,694.638 (rounding up to 15,695) are under 18. Again, that's quite a bit larger than the total population needed to have at least 15 unimmunized people. If the uptake rate for Noblesville is 98%, that means there would still be 314 people under 18 years old who had not received the vaccine.

Alright. So we have 314 unvaccinated (all of whom are vulnerable) and 15,381 immunized (770 of whom are vulnerable because the vaccine did not take, assuming 95% efficacy). Just in Noblesville. It is actually a testament to the work of the state health officials that we have only seen 16 cases to date, since we very easily could have seen nearly a thousand in just the under-18 population in this one city.

The argument could be made, I suppose, that the at least 314 unvaccinated individuals were spread out and unlikely to come into contact with one another. However, people tend to group with those who share similar views. This is true for most people, and vaccine refusers are no exception. While average numbers for states and cities tend to reflect higher rates of vaccine uptake, when you focus in on smaller regions, you begin to see that there are pockets, where like-minded parents reinforce each others convictions that vaccines are bad and to be avoided. So you get groupings of unvaccinated. We saw it in last year's outbreak in Minnesota, as one stark example.

All of this is basically a lengthy way of saying that, yes, it actually is statistically possible for there to be 13 individuals who are all unvaccinated in an outbreak. Setting population clustering aside, looking at only the sheer numbers involved illustrates that there are likely to be plenty of unimmunized people ripe for infection. Mamavallant fell victim to the difficulty that humans have intuitively dealing with large numbers. A 98% uptake rate seems like you shouldn't have very many people at all who are vulnerable, but when you actually plug the numbers in, you see that there are actually a lot. There are a couple other errors she made (e.g., you only need two MMR shots, not repeated boosters), but this mathematical error was the biggest and primary focus of her comment.

Perhaps if we improve the educational standards in this country, people would be able to understand how these outbreaks occur, and we would see uptake rates improve.


  1. I'm not surprised. The new anti-vax thing, wherever I seem to go, is that most of the people in an outbreak are vaccinated.

  2. Yep. Just about every outbreak reported has an unvaccinated individual as the index case. Likewise, the majority of people in the outbreaks are either unvaccinated or have an unknown immunization status.

    BTW, aboutpediatrics, great work on your posts following all the various outbreaks of the past few years!

  3. The "most of the people in an outbreak are vaccinated" thing comes from a couple of outbreaks where the number of vaccinated has been larger than unvaccinated, but the proportions are way off. More fun with math...

    If there is an outbreak of 200 people, where 105 are vaccinated and 95 are unvaccinated, but the population is, say, 10,000 with a vaccine uptake of 98% and effectiveness of 95%, then there are 9,800 vaccinated and 200 unvaccinated. Of those 9,800 vaccinated, 490 had a vaccine "failure". (So it's perfectly possible that 105 of those 490 would be involved in the outbreak, given the probabilities.)

    All in all, of the people in the outbreak, 105/9800 = 1.07% (vaccinated) while 95/200 = 47.5% (unvaccinated). In epidemiological terms, you are about 44 times more likely to be part of the outbreak if you are not vaccinated versus being vaccinated, even if the absolute numbers show a larger number of vaccinated people in an outbreak.

    Heck, it even works if 150 out of 200 (very lopsided and not likely to be seen in real life) in the outbreak are vaccinated. 150/9800 = 1.53%, while 50/200 = 25%, making you 16 times more likely to be part of the outbreak if you are not vaccinated versus being vaccinated.

    This lesson brought to you by the number 3.

  4. Thanks, Ren. When I started writing the post, I had meant to touch on that, but forgot. Kinda puts things into perspective, no?

  5. Anti-vaccine folks drive me batshit sometimes.

    They claim all sorts of 'big pharma' influence on medicine, but run right over to Mercola, Tenpenny and Adams for all their 'factual' information - along with the supplements, vitamins and 'probiotics' they need to stay healthy - sold by Mercola, Tenpenny and others.

    Yeah, there's a willful ignorance there that's alive and well.

  6. I share L. Tock's views on this...


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