Over at my friend Pal’s blog, in a discussion about vaccination, a commenter came up with the following in an argument against the value of vaccination:

Mathematical formula:

100% – % of population who are not/cannot be vaccinated – % of population who have been vaccinated but are not immune (1-effective rate)-% of population who have been vaccinated but immunity has waned – % of population who have become immune compromised-(any other variables an immunologist would know that I may not)

What vaccine preventable illnesses have the result of that formula above the necessary threshold to maintain herd immunity?

I don’t know if the population is still immune to Smallpox, but I would hope that that is just a science fiction question. Smallpox was eradicated, but that vaccine did have the highest number of adverse reaction (I’m sure PAL will correct me if that statement is wrong)

It’s a classic example of what I call *obfuscatory mathematics*: that is, it’s an attempt to use fake math in an attempt to intimidate people into believing that there’s a real argument, when in fact they’re just hiding behind the appearance of mathematics in order to avoid having to really make their argument. It’s a classic technique, frequently used by crackpots of all stripes.

It’s largely illegible, due to notation, punctuation, and general babble. That’s typical of obfuscatory math: the point isn’t to use math to be comprehensible, or to use formal reasoning; it’s to create an appearance of credibility. So let’s take that, and try to make it sort of readable.

What he wants to do is to take each group of people who, supposedly, aren’t protected by vaccines, and try to put together an argument about how it’s unlikely that vaccines can possibly create a large enough group of protected people to really provide herd immunity.

So, let’s consider the population of people. Per Chuck’s argument, we can consider the following subgroups:

- is the percentage of the population that does not get vaccinated, for whatever reason.
- is the percentage of people who got vaccinated; obviously equal to .
- is the percentage of people who
*were*vaccinated, but who didn’t gain any immunity from their vaccination. - is the percentage of people who were vaccinated, but whose immunity from the vaccine has worn off.
- is the percentage of people who were vaccinated, but who have for some reason become immune-compromised, and thus gain no immunity from the vaccine.

He’s arguing then, that the percentage of effectively vaccinated people is . And he implies that there are other groups. Since herd immunity requires a very large part of the population to be immune to a disease, and there are so many groups of people who can’t be part of the immune population, then with so many people excluded, what’s the chance that we really have effective herd immunity to *any* disease?

There’s a whole lot wrong with this, ranging from the trivial to the moderately interesting. We’ll start with the trivial, and move on to the more interesting.