Vaccination: A Numbers Game

Yesterday evening, NPR ran a story on the recent judgment that the original study linking vaccination with autism was fraudulent. The charge is that the researcher in that initial study, Andrew Wakefield, falsified data to support the conclusion he wanted.

I posted the story to my wall on Facebook, and received several strident comments from an acquaintance who is convinced that vaccination is in fact harmful and that the new conclusions about the Wakefield study are themselves suspect.

Regardless of the merits of charge and counter-charge regarding the Wakefield, my exchange with this particular interlocutor soon turned to the question of whether the idea of "herd immunity" has any validity. My interlocutor asserted simply that there is "no evidence" to support the idea of herd immunity.

I attempted to reply from a number of angles, including pointing out that the claim of "no evidence" sounds suspiciously like similar claims made by creationists about Darwinian evolution and by free-market ideologues about climate change.

Acknowledging that I am a little out of my depth, since I have only casual acquaintance with the concrete details of epidemiology, I suggested that "herd immunity" might serve as a theoretical concept, like "natural selection", for which direct evidence of the kind that simply confirms or falsifies it may not be available. Rather it is an organizing concept, one that has validity to the extent that it helps a community of inquirers makes sense of a wide range of observations, in both everyday and experimental contexts.

My interlocutor replied by citing a study:

A review of all (then published) studies of all human populations where over 95% of the population had received the vaccine showed that more people who got infected with the disease had received the vaccine than not.

Unpack the logic of this claim: A number of vaccinated people who get infected is larger than the number of non-vaccinated people who get infected! Therefore, vaccination is ineffective!

Well, not so fast.

On the face of it, the most you could conclude from the study is that vaccination is not 100% effective.  But then, I don't think anyone would claim that it is 100% effective.  Instead, the claim is that vaccination reduces risk, both to the individual and to the population at large.

In fact, I suspect the study cited may actually support the effectiveness of vaccination, once the numbers are worked out and given a more plausible interpretation.

I don't have access to the study in question - my interlocutor did not provide documentation - so I don't know the actual numbers. So, I plugged in some hypothetical numbers, which is sufficient to prove a logical point about my interlocutor's intended argument.

It was stipulated that "over" 95% of the populations in question had been vaccinated. For illustration, let's just say it's 95% on the dot.

Assume a population of 1000 of which 950 have been vaccinated against measles.

If 5% of those vaccinated get infected with measles, that comes to roughly 48 cases of measles among the vaccinated.

If 50% of those not vaccinated get infected with measles, that comes to 25 cases of measles among the non-vaccinated.

48 is a larger number than 25, nearly by a factor of 2! So, non-vaccination wins! 

Except . . .


. . . 95% of those who received the vaccine avoided infection, while only 50% of those who did not receive the vaccine avoided infection.

. . . . rates of infection among the non-vaccinated are likely to be lower than they would otherwise be, since they are less likely to be exposed to the disease. Consider that 926 of their fellow citizens have not been infected (902 of the non-infected having been vaccinated) and so cannot pass it on to them.


(As it happens, the study, as cited by my interlocutor, seems to have little bearing on the validity or non-validity of the idea of herd immunity. At least, the idea of herd immunity seems perfectly consistent with these results.)

The numbers are made up, but the logic is inescapable: The mere fact that the majority of those infected with a disease in the circumstances described were vaccinated against it does not imply that vaccination is ineffective, only that it is not 100% effective . . . which we already knew.

Frankly, given the choice between a 5% chance of infection and a 50% chance of infection, I would take the 5%. 

More than this, since there does seem to be some validity to the idea that vaccination reduces infection rates for the population as a whole, reducing the risks of epidemics or even pandemics, then it may well be a moral and civic obligation to get myself and my children vaccinated.


But then, I'm trying to be rational about risk. The fact is, people are likely to continue to see risks in all the wrong places, exaggerating risks posed by the medical establishment and the pharmaceutical industry while minimizing risks posed by the diseases themselves . . . a luxury we have only because of the widespread use of vaccines.

But then, as the NPR story has it: 

‎". . .more revelations about Wakefield aren't likely to make the fear of vaccines go away. But David Ropeik, is [sic] an instructor at Harvard, says something else eventually will. 'As more and more people get measles and kids die, which is happening around the world. Eventually the threat of the disease will come back and surmount our fear of the vaccine.'"