herd immunity‘ for the Quanta Magazine using something that cannot be denied: math.

Here’s where vaccination comes in. When vaccinated, an individual develops resistance to the disease: Success rates vary, but for simplicity we will assume that vaccination provides complete immunity to the disease. Vaccination directly benefits the vaccinated individual, but it also indirectly benefits the broader population. If many people in a community are vaccinated against a disease, the disease won’t spread as rapidly.

In effect, widespread vaccination can help reduce the effective reproduction number of the disease. And if enough individuals are vaccinated, the effective reproduction number can actually be reduced to one, ensuring that the disease will only spread at a linear rate. So how many individuals need to be vaccinated to bring a disease’s effective reproduction number down to one?

Let’s think about what the basic reproduction number really tells us. Consider an influenza epidemic with R0 = 2. This means that an infected person will, on average, infect two new people. This single number, R0 = 2, offers us a lot of information: how easily the disease is transmitted, the length of the infectious period, and the average number of people an infected person will interact with in a given period of time. By unpacking this number, we can easily see how vaccination can reduce it.

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