With COVID running rampant throughout the US, I’ve seen a bunch of discussions about herd immunity, and questions about what it means. There’s a simple mathematical concept behind it, so I decided to spend a bit of time explaining.

The basic concept is pretty simple. Let’s put together a simple model of an infectious disease. This will be an extremely simple model – we won’t consider things like variable infectivity, population age distributions, population density – we’re just building a simple model to illustrate the point.

To start, we need to model the infectivity of the disease. This is typically done using the name . is the average number of susceptible people that will be infected by each person with the disease.

is the purest measure of infectivity – it’s the infectivity of the disease in ideal circumstances. In practice, we look for a value , which is the actual infectivity. includes the effects of social behaviors, population density, etc.

The state of an infectious disease is based on the expected number of new infections that will be produced by each infected individual. We compute that by using a number S, which is the proportion of the population that is susceptible to the disease.

• If R S < 1, then the disease dies out without spreading throughout the population. More people can get sick, but each wave of infection will be smaller than the last.
• If R S = 1, then the disease is said to be endemic. It continues as a steady state in the population. It never spreads dramatically, but it never dies out, either.
• If R S > 1, then the disease is pandemic. Each wave of infection spreads the disease to a larger subsequent wave. The higher the value of in a pandemic, the faster the disease will spread, and the more people will end up sick.

There are two keys to managing the spread of an infectious disease

1. Reduce the effective value of . The value of can be affected by various attributes of the population, including behavioral ones. In the case of COVID-19, an infected person wearing a mask will spread the disease to fewer others; and if other people are also wearing masks, then it will spread even less.
2. Reduce the value of . If there are fewer susceptible people in the population, then even with a high value of , the disease can’t spread as quickly.

The latter is the key concept behind herd immunity. If you can get the value of to be small enough, then you can get to the sub-endemic level – you can prevent the disease from spreading. You’re effectively denying the disease access to enough susceptible people to be able to spread.

Let’s look at a somewhat concrete example. The for measles is somewhere around 15, which is insanely infectious. If 50% of the population is susceptible, and no one is doing anything to avoid the infection, then each person infected with measles will infect 7 or 8 other people – and they’ll each infect 7 or 8 others – and so on, which means you’ll have epidemic spread.

Now, let’s say that we get 95% of the population vaccinated, and they’re immune to measles. Now . The disease isn’t able to spread. If you had an initial outbreak of 5 infected, then they’ll infect around 3 people, who’ll infect around 2 people, who’ll infect one person, and soon, there’s no more infections.

In this case, we say that the population has herd immunity to the measles. There aren’t enough susceptible people in the population to sustain the spread of the disease – so if the disease is introduced to the population, it will rapidly die out. Even if there are individuals who are still susceptible, they probably won’t get infected, because there aren’t enough other susceptible people to carry it to them.

There are very few diseases that are as infectious as measles. But even with a disease that is that infectious, you can get to herd immunity relatively easily with vaccination.

Without vaccination, it’s still possible to develop herd immunity. It’s just extremely painful. If you’re dealing with a disease that can kill, getting to herd immunity means letting the disease spread until enough people have gotten sick and recovered that the disease can’t spread any more. What that means is letting a huge number of people get sick and suffer – and let some portion of those people die.

Getting back to COVID-19: it’s got an that’s much lower. It’s somewhere between 1.4 and 2.5. Of those who get sick, even with good medical care, somewhere between 1 and 2% of the infected end up dying. Based on that , herd immunity for COVID-19 (the value of S required to make R*S<1) is somewhere around 50% of the population. Without a vaccine, that means that we’d need to have 150 million people in the US get sick, and of those, around 2 million would die.

(UPDATE: Ok, so I blew it here. The papers that I found in a quick search appear to have a really bad estimate. The current CDC estimate of is around 5.7 – so the S needed for herd immunity is significantly higher – upward of 80%, and so the would the number of deaths.)

A strategy for dealing with an infection disease that accepts the needless death of 2 million people is not exactly a good strategy.