Why are Vaccinations Mandatory?

From an AP story:

"When you choose not to get a vaccine, you're not just making a choice for yourself, you're making a choice for the person sitting next to you," said Dr. Lance Rodewald, director of the CDC's Immunization Services Division.

This. Makes. No. Sense.

I thought the whole point of getting a vaccine was that it made you immune to the illness. If this is true, then it shouldn't matter that the person next to you has measles. Because you can't catch it.

So what collective purpose is served by mandating vaccinations? I don't have a lot of sympathy for religious objections to vaccinations on the merits, but according to the article, the present wave of vaccination opt-outs are driven by a rational calculation of the relative risks. Sure, those calculations may be wrong, but I do think that if the government is going to substitute its judgment for theirs, the government ought to show a better reason than this.

Update: Colin comments:

The issue is that a vaccination doesn't always work. The person sitting next to you may have been vaccinated, yet is still susceptible.

Childhood vaccinations are an example of the prisoner's dilemma. If one parent defects, his child gains and the other's don't lose much. However, if all parents defect, then we have outbreaks.

Makes sense up to a point. Mandatory vaccinations reduce both the risk of exposure and the risk of infection given exposure. If these probabilities are independent, then they multiply, and so become smaller.

However, it seems that this would put a "natural" ceiling on the number of defections, since as the defection rate goes up, so does the risk of exposure, which makes the value of the vaccination all that much higher.

It should be straightforward matter to construct a probabilistic model for the "optimum" participation rate, assuming voluntary participation and given the probabilities involved. (By optimum, I don't mean socially optimum necessarily, only individually optimum, which might not be the same.) I'll try to have a go at it in a future post.