My latest column for Quanta Magazine breaks down the mathematics of “herd immunity”. By vaccinating a critical percentage of a population against a disease, the potential spread of the disease through the population will proceed at a linear, not exponential, rate. This herd immunity can mean the difference between a handful of illnesses and a catastrophe.

We start by thinking about how rumors spread.

Let’s say you hear a juicy rumor that you just can’t keep to yourself. You hate rumormongers, so you compromise by telling only one person and then keeping your mouth shut. No big deal, right? After all, if the person you tell adopts the same policy and only tells one other person, the gossip won’t spread very far. If one new person hears the rumor each day, after 30 days it will have spread to only 31 people, including you.

So how bad could it be to tell two people? Shockingly bad, it turns out. If each day, each person who heard the rumor yesterday tells two new people, then after 30 days the rumor will have reached more than a quarter of the world’s population (2,147,483,647 people, or 231 − 1, to be exact). How can such a seemingly small change — telling two people instead of one — make such a big difference? The answer lies in rates of change.

A similar mathematical model can be used to understand the spread of disease. And by unpacking the mathematics behind the *basic reproduction number *of a disease, we can compute the critical cutoff for herd immunity.

Learn more by reading the full article, which comes with a classroom-ready worksheet and is freely available here.