Crudely Modeling Herd Immunity


So what is this “herd immunity” that people keep talking about?

It works something like this:

Adam becomes infected with some disease. We’ll call it Batpox for this example. It turns out that the statistics of Batpox’s transmissibility are such that it is about as contagious as measles. Measles has a basic reproduction number (R0) of about 12. When Adam goes to visit his friends Betty and Chuck, the odds are high that they will both become infected as well—unless they are already immune because of a previous encounter with the disease or a vaccine. OTOH, if enough of the people Adam contacts while he’s sick are immune to Batpox, the disease isn’t likely to spread any further. A population has reached herd immunity for a disease when enough of the population is immune to prevent the disease from easily spreading.

The percentage of population required for herd immunity is greater for larger values of R0. The formula for the approximate percentage of immune individuals necessary for herd immunity is

X = (1 – 1/Ro) X 100

For measles X is about 92 %. That’s why it’s important for kids to be vaccinated in order to get the number of immune individuals as high as possible.

The initial estimate of R0 for the Wuhan virus was around 2.7. That would imply that we’d need about 63 % of the population to be immune in order to achieve herd immunity. However, the Real World data for Covid-19 shows much lower values for R0. That’s values, plural, because different places have different factors that affect transmissibility.

Take a look at these charts of how R0 has varied over time in various states. (Source: rt.live) The solid lines represent the calculated values for R0 and the shaded areas around the lines show the confidence intervals for the calculations based on the amount and quality of the data. These plots are for entire states; the New York and Michigan numbers would be even lower with the effects of New York City and Detroit removed.

Note that these states have all achieved an R0 of about 1. Plugging that value into our formula for herd immunity gives a required immunity percentage of … pokes at calculator …  zero.

Now, I’ve been engaged in modeling here, and we know how problematic that can be, but I believe this gives us a hint about why people are ready to get back to their normal lives in large swaths of the country. Certainly, a value of R0 below 1 explains why the death toll hasn’t spiked in Georgia.

There are still places in the country struggling to contain the Wuhan virus outbreak, and they should be supported in their efforts. However, the data support letting the rest of the country get on with our lives.

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