When I arrived in Princeton last Friday, I was greeted with this headline:

Emergency meningitis vaccine will be imported to halt Ivy League outbreak

Emergency doses of a meningitis vaccine not approved for use in the U.S. may soon be on the way to Princeton University to halt an outbreak of the potentially deadly infection that has sickened seven students since March.

Well then. Perfect timing. But seriously, it is actually a perfect time to reflect on why vaccines are necessary in the first place. And it’s not (just) for the reason you might think.

If you’re vaccinated against a disease, not only does it mean that you won’t get sick, it also means that you won’t pass that disease on to other people. Vaccinations protect the people around you too. And conversely, even if you’re not vaccinated yourself, the more people around you who are, the lower your chances of catching the disease from someone else.

Let me illustrate this with a simple model of how a disease spreads. Imagine a world where people live in apartments on a perfect grid and only ever talk to their four neighbors, once a day.

Suppose …