Not too long ago I wrote about the silliness of all the hysteria surrounding the smidgen of ebola cases entering the United States. Of course, ebola’s gone from the news cycle (though Africans are still dying by the thousands, concerned Americans), but now hysteria has raised its ugly head again with the measles outbreak in California. A few dozens have been infected, all a lovely gift from Disneyland and the anti-vaccination fad.

Of course, everybody is looking for somebody to blame. The majority (all?) of the people who got sick were unvaccinated, as is the case in every measles outbreak for obvious reasons. Therefore, it’s those goddamn antivaxxers making everybody sick, right? It’s irrational child abuse not to vaccinate your kids, so let’s force them all to get their measles shots. Or maybe it’s the goddamn Mexicans, bringing their filthy third world diseases illegally across the Rio Grande. Let’s close the borders.

Whoa, pardner. First of all, those dirty foreigners apparently have higher vaccination rates than Americans. Second of all, before we go railing about how irrational the antivaxxers are, maybe we should step back and take a second look at that question.

Vaccination is a classic case of a Public Goods game. Vaccination obviously greatly reduces the risk of getting sick, and it also provides “herd immunity” — the more people around you have been vaccinated, the more protected you are even if you haven’t been vaccinated. There’s obviously a cost associated with being sick (as in you could possibly die), so it’s good to not get sick. But, as the antivaxxers remind us, there is also a potential cost to getting a vaccine, in that you might suffer a side-effect, also possibly killing you. Because of herd immunity, the *benefits* of vaccination are shared by the community, including the unvaccinated people who don’t accept the *costs* of vaccination.

Why it interests me is that there’s a clear partitioning of the benefit of vaccination between a private component — the near-complete immunity of vaccinated people to the disease — and the public component, the lower risk of being infected in a population containing many vaccinated people. Public goods that have that partitioning are what we call Black Queen functions, and they have the interesting property that they allow co-existence between “helpers” (in this case, vaccinated people) and “cheaters” (antivaxxers). In other words, it makes rational sense for a certain number of people to remain unvaccinated in society.

We can use some math to figure out what that number is. Now, let me preface this by saying I’m neither a mathematician nor an epidemiologist, so if this is all over-simplified or algebraically wrong, I apologize. I’ve definitely made a lot of simplifying assumptions for mathematical tractability. But here goes. Let’s define a system as containing three groups of people: iMmune people (*M*) who have been vaccinated and can’t get sick, Anti-vaxxers (*A*) who haven’t been vaccinated and *can* get sick, and iNfected people (*N*) who are currently sick. The whole population of people is thus *P = M+A+N*.

The population as a whole grows and shrinks by normal birth and death, with death being a constant function *d* of the current population size for each group, and births occurring at a certain maximum rate *b* that drops to 0 at a rate controlled by a constant, *k*. *M*‘s and *A*‘s grow and die at the same rate and always reproduce their own type. Absent disease, we can describe the normal disease-free growth of the population as a differential equation:

Now let’s add in the expectations of the Black Queen evolutionary system. First of all, there’s a cost to vaccination. Some number of *M*‘s will suffer severe consequences from vaccination, resulting either in death or an inability to reproduce. Let’s say this takes place at rate *s*; then we can describe the dynamics of *M* with:

Importantly, *M*‘s never get sick. *A*‘s, on the other hand, get sick at a rate determined by an infection constant *i* and the number of infected people — *N*‘s — in the community. *N*‘s on the other hand, either get better at rate *r* (thus reverting to A’s) or they die at rate *m*. Now we can describe the dynamics of *A*‘s and *N*‘s:

Lots of things can happen inside a model like this, but we’re only going to concern ourselves with its equilibrium point — or where everything stabilizes and quits moving around. We do that by setting the “dX/dt” side of the equations — the differential, which describes movement — to 0 and solving for the three variables representing our population members. If we assume that all the *N*‘s get over their illness reasonably quickly (relative to the long-term birth/death process of the population), we can say that *r+m=1*. Thus, the total of antivaxxers and infected people in the population at equilibrium becomes:

Okay, let’s break this down. This equation describes the number of antivaxxers we expect if everybody is acting rationally — it balances the costs of infection with the costs of immunization exactly. This number is largely controlled by the danger associated with the vaccine — *s*, which makes the number go up — and *i*, or how infectious the disease is, which makes the number go down. To put it in human terms, it makes sense not to get the vaccine if you see more people having side effects to the vaccine than you see people getting sick with measles.

*(Note that this is a generic problem with preventive measures: if they work, it starts to seem like they’re useless, because the thing they prevent becomes so rare as to seem inconsequential. A very similar argument could be made about the value of individually-owned firearms to society: nobody in Somalia would argue about the value of owning a rifle, but in whitebread suburbia — suffused with armed cops lurking in the wings — guns seem superfluous.)*

Let’s plug in some real numbers. As best I can tell, the probability of having a severe side effect from the measles vaccine is about 1 in a million, but let’s go crazy nuts and say it’s 100 times that high, and 1 in 10,000 people who get immunized are either killed or sterilized by the measles shot. Since you only need one shot in a whole lifetime (let’s call that 50 years), the rate constant *s* works out to 0.000002 per year. The death rate for people infected with measles is also quite low — roughly 2 in 1000 in the developed world — so *m* = 0.002. Finally, the infection rate is very high, so let’s set *i* = 0.95, representing a 95% chance of getting sick if you’re surrounded by sick people. With these values, the total number of antivaxxers and infected people supported by the system is: 1 per square mile. In fact, assuming *s* is very small, the abundance of antivaxxers reduces to 1/*i*, entirely dictated by the likelihood of being infected when you encounter a sick person.

Interestingly, this value doesn’t change if we raise or lower the total population size, although we can calculate how population size affects the *proportion* of the infection pool to the total population. The algebra is a little hairier but eventually you get:

So, the proportion of cheaters goes down as *k* (i.e. population size) goes up — meaning it’s more likely you’ll see the use of vaccinations when you’re in a big city with lots of people than out in the country by yourself. Makes sense, no? If we plug some numbers into this equation — *b* = 0.003 per month, *d* = 0.0007 per month, and *k*= 1000 people per square mile — we arrive at an equilibrium proportion of cheaters of about 0.03%. In the United States, that would work out to about 90,000 people.

We’re clearly not at this equilibrium point now — there are way more antivaxxers than that. It’s possible that right now, the number of *N*‘s is so low and health care is so effective at isolating illness (lowering *i*) that it’s really hard to get sick. Presumably, if the antivaxxer fad continues, the prevalence of measles will increase until more and more *A*‘s get sick, eventually causing their numbers to start to drop towards a stable equilibrium.

I know people like to attribute “insanity” to movements like antivaccination. However, I suspect that antivaxxers (and other conspiracy theorists) are normally rational people with mistaken beliefs. When confronted with obvious evidence in the form of sick antivaxxers, they’ll change their tune. But regardless, the equations suggest that the risk even to them is low, and to those of us who are vaccinated, the risk is extremely, vanishingly low.

We COULD force the antivaxxers to get shots. This would save a very very small number of lives. It would also cost a fortune and would involve using force to shoot drugs into people, which seems exceedingly creepy to me. Before I’m willing to sign on to a policy like that, you’ll have to show me a disease with a much higher *m* than measles. Till then, there are more important things to spend our money fighting.

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