Orac highlighted here a post over at Vox Populi which doubted the effectiveness of the mumps vaccine, in light of the recent epidemic in Iowa. I was prepared to write a whole post on the math of it, but Mark at Good Math, Bad Math saved me some work. Nevertheless, I have a few things to add after the jump.

As has been mentioned, the given efficacy rate for the mumps vaccine is 95%. This is actually likely a bit high; previous outbreaks have suggested it’s more like 85-90% effective, so that as many as 15% of the vaccinated population won’t actually be immune. The key to telling whether the vaccine is helpful, then, is to look at the attack rate–the percent of the population that develops disease–in the vaccinated versus unvaccinated population. So, some more math to follow.

For the sake of simplicity, say you have a population of 100,000 people. Vaccination coverage in the population is 95%, meaning that 5,000 will be unvaccinated and lacking in immunity. Additionally, let’s say the vaccine is only 90% effective. So, of your vaccinated population of 95,000 people, you’ll have 9,500 people who remain susceptible–“vaccinated but not effectively so,” let’s call them.

Now let’s assume, again for the sake of simplicity, that susceptible people are equally likely to become infected with mumps, whether they’re in the “vaccinated but not effectively so” or “unvaccinated” group. (Real life is actually messier, leading to a skew in one group or the other, but we’ll ignore that for now). Therefore, if you have an outbreak of 500 cases–roughly the size of Iowa’s right now–in an ideal world they’d be divided randomly between the two groups. The “vaccinated but not effectively so” group is roughly twice as large as the “never vaccinated” group, so figure they get 333 of the 500 cases, and the remaining 167 cases are in the unvaccinated population.

Still following? Now it’s time to calculate the attack rate. In the vaccinated population, we ended up with 333 cases of disease in a total population of 95,000. **So, the attack rate = 333/95,000 = .35% **

In the unvaccinated population, we ended up with 167 cases of disease in a total population of 5000. **So, the attack rate = 167/5,000 = 3.34%**: TEN TIMES the rate of the vaccinated population.

This is the key to the whole thing. Yes, there’s disease in the vaccinated population. Of the cases in this little hypothetical, by the numbers alone, 66% were vaccinated–lower, but similar to our numbers here in Iowa. Yet as you can see, this doesn’t mean that “the vaccine isn’t working:” in our scenario, it means it’s working at a 90% effectiveness rate, which is pretty good. The unvaccinated population acquired disease at 10 times the rate of the vaccinated population overall, so while being vaccinated was no guarantee of protection, it’s a damn good gamble.

Now, as I mentioned, real life is messier. It’s unlikely that the spread of disease is completely random. For example, most of the cases thus far have occurred on college campuses, which are very highly vaccinated populations. As such, it’s more likely that there will be more of the disease in the “vaccinated but not effectively so” population than the “unvaccinated” population, simply because there is less assortment with the unvaccinated folks. And this is what we’ve seen–instead of the 66% of cases that were vaccinated, up to 80% of the mumps cases here in Iowa have received at least one dose of vaccine. I’ve not crunched the actual numbers, but this would mean that in our real-life outbreak, the difference in attack rate in vaccinated vs. unvaccinated would be less than a factor of 10–perhaps 7, or 5–but still significantly better than not having the vaccine at all.

Soooo, next time you read stuff like this:

And isn’t it at least somewhat doubt-inspiring that the health authorities continue to insist that the vaccine is working in the face of direct evidence that, at least in some cases, it is not?

…ask them if they know what the difference in the attack rate is between the two populations. If they don’t know, educate them. (But if it’s Vox, don’t bother to point them here…apparently, being XX disqualifies me from having a say in it).