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 = %/100To 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 = 90/100So, 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:
13 = (90 * Y)/100
13 * 100 = 90 * Y
(13 * 100)/90 = Y
1,300/90 = Y
14.4444... = Y
15/Y = 2/100We'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 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.
15 = (2 * Y)/100
15 * 100 = 2 * Y
(15 * 100)/2 = Y
1,500/2 = Y
750 = Y
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.