The Arithmetic of an Anti-Vaccine Propagandist

In recent days, after  the Disneyland measles outbreak, anti-vaccine campaigners have received an amount of bashing that has been much harsher and much more extended than their usual share. As a consequence, there have been a number of voices that — while being clearly and thoroughly in favor of vaccines — tell us that the battering might be exaggerated, misplaced, ineffective and even dangerous; examples of this can be read here , here and here . I do think that all of them have very good points. And yet, here I am, joining the battering, placing my two cents in the service of the lynching mob. There is a reason for this, and a very particular one, that I will make clear soon. Also, I will be focusing exclusively to a particular article written by a particular anti-vaxxer.

I am certain that, in any debate, one should listen to all the parties involved; most of all, paying attention to the underdog is of particular importance, as not doing so would pretty surely give an unwanted bias to one’s views. That surely includes listening to people who would only listen to themselves and to closely akin ideologues. In the vaccine debate, this meant that I was really curious and eager to find out what on earth the people in the anti-vaccination camp could be saying in their defense, or whether many of them were already throwing the towel. So, when via facebook I came across this article I didn’t hesitate to click in and read it attentively. The author is a guy named Sayer Ji (after a quick search I found out that he is a self-proclaimed “widely recognized researcher”, and the founder of the site The basic premise of the article is that the MMR vaccine is largely ineffective in preventing measles, and that this fact can be derived from the observation of the recent epidemics. Everyone is entitled to personal views, and there are many subtleties in the world to make diverging opinions correct in their own way. But, bending the truth to conclude and proclaim something blatantly false, is a very different matter; that is something to be exposed whenever the opportunity is given, especially when the issue at hand is of relevance, and when all that is involved is a little school level mathematics gone wrong.

Both inside the body of the post, and in the links that it provides for support, we can find several inaccuracies (to put it very mildly), half truths, and a few downright self-contradictions. However, I think it’s sufficient to concentrate in one single paragraph, which seems to me is central to the whole idea of the article; this one tells us that:

‘This erroneous thinking has led the public, media and government alike to attribute the origin of measles outbreaks, such as the one recently reported at Disney, to the non-vaccinated, even though 18% of the measles cases occurred in those who had been vaccinated against it hardly the vaccine’s claimed “99% effective.”  The vaccine’s obvious fallibility is also indicated by the fact that that the CDC now requires two doses. [the boldface remark is from the original article; the underlining is mine]

So, hardly the vaccine’s claimed “99% effective.” Well, not so hardly. In fact, very plausible. If we need to be precise, it would not be possible to conclude anything about the effectiveness of the vaccine just from the 18% figure given; having data on the total number of people that were exposed to the virus, as well as on the proportions of vaccinated versus non-vaccinated is mandatory. Of course, we don’t have those numbers, but since the outbreak took place in a public place with thousands of daily visitors from all over the place, and from the fact that measles is an extremely contagious disease, we can be pretty sure of two facts:

1)There were a whole lot more of people exposed to the virus, compared to those infected.
2)The proportion of non-vaccinated among them is small.

Now, for the sake of clarity, let’s consider a specific example. Assume that from the set of people that were exposed to the virus 10% didn’t have the MMR shot. This might be much higher than the actual figure, as it is more than twice the proportion of school children in California without an MMR shot (detailed information on this has been made public by the California Department of Public Health), but we’ll stick to that number, just to be restrained. In this case, 20 exposed individuals for each one infected would be sufficient to have the 1% of infections among the vaccinated, making the 99% effective hold. The 20 to 1 proportion seems very moderate to me, given the circumstances.  But, then again, there is no way  be completely sure on this one. However, there is one fact that we can conclude from the 18% figure given:

It is far more likely to get the measles if you’re not vaccinated

Not a shocking finding, I know; but it is kind of amusing to get it as a bold remark from a post that intended to show the opposite. How much more likely? Well, again it is not possible to know with precision only from the knowledge of the 18% percentage (even if we knew that it correct and representative), but certainly it is much larger than the 82 to 18 not very insightful first look at the numbers could suggest (this would be the case if half the population were not vaccinated, which thankfully is far away from the truth). It is a moderate claim to assert that you’re at least 41 times more likely to get the measles without a vaccine; for instance, that would be the case with the 10% of non-vaccinated already considered. If we look at it in percentages, “41 times more likely” means a 4100% increase in relative risk. We would need to know the increase in absolute risk to get the whole picture, but  increases in relative risk of say 20 or 30% with marginal absolute risk change are abundant and often considered relevant in the medical literature. Four thousand percent is a massive increase that belongs to a very different league.

Hereby, I will do the computations to justify the numbers I gave (maybe you just trust myself blindly, but you shouldn’t; not me, not anyone). As I said before, the math involved is very simple, the kind of stuff that everyone should be able to follow; anyway, feel free to skip this paragraph if you want to do the homework yourself, or if you just don’t like this sort of stuff, or if for whatever reason that’s what you want to do. Well, let’s get to it: Call M the size of the exposed population, and N the infected ones. According to Sayer Li, 0.18N is the number of vaccinated people who got the disease; then 0.82N would be the number of non-vaccinated people that were infected. Call r the proportion of vaccinated people in the whole exposed population (so, rM is the number of vaccinated and (1-r)M the non-vaccinated). Then, if we consider these numbers to be representative, the chance of getting the infection with the vaccine would equal 0.18N/rM, which with the added assumption (made above as example) given by r=0.9, becomes 0.2N/M. So if the vaccine were to achieve the mentioned “99% effectiveness” what we need is 0.2N/M less or equal to 0.01, or equivalently M/N greater or equal to 20, as stated. Evidently, larger values for r would translate in smaller necessary proportions to get the required effectiveness. Now, for the comparison between the chances of infection for the non-vaccinated versus the vaccinated: The chance of infection for the non-vaccinated would be 0.82/(1-r)M, which divided by the chance of infection of the vaccinated cancels out the M, giving us that the chance of infection for the non-vaccinated only depends on the proportion r, being equal to 82r/18(1-r). The value of r=0.9 means that this quotient, which is the increase in relative risk, becomes 41. Again, an increase of r would imply an increase on the risk increase; just for the record, if we take r=0.05 (rounding the percentage of school children in California without an MMR shot) the increase in risk rises to a staggering 86.5.

That will do for the emphasized line. The next line (The vaccine’s obvious fallibility is also indicated by the fact that that the CDC now requires two doses) requires no number crunching to refute. Because we all now that everything that is useful in preventive medicine, and in life in general, has to be done only once.


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