Re reports of a COVID-19 vaccine that was over 90% effective in experimental trials. An explanation: a virus lives or dies based on how effectively it's transmitted. If each infected person infects an average of more than 1 person, we get exponential growth. If fewer: extinction. https://twitter.com/moebius_strip/status/1325855698186428416

The average number of new infections per person in an uncontrolled setting is given by the basic reproductive number, R_0. R_0 is difficult to approximate, and this is a case where averages only tell a very smart part of the story.

MOST infected people infect NO ONE else; others infect many people. R_0 is also difficult to approximate; if you look it up, you'll see a lot of hedging & huge margins of error. "Somewhere between 2 and 3" seems to be the consensus: every infection leads to an avg of 2-3 new ones

Let's go with 3. This means that if, say, 100 people in a certain region have COVID-19 and everyone goes about business as usual, then after one incubation period we can expect 300 new infections. ("Expect" has a very precise meaning in probability, referring to averages.)

This is why we implement restrictions that limit the number of contacts we have. Suppose by closing universities, stadiums, etc, we're able to limit people's contacts to 25% of what they are now. Then those 100 infected people are exposed to 25% as many people as they were before

So we'd expect (again the precise meaning, referring to averages) that only 25% of the 300 people who were infected under the zero-restrictions scenario would be infected this time. That is, the 100 current infections are expected to lead to an average of 75 new ones.

Which is exactly what we're going for - fewer new infections than existing ones. The average number of new infections per old one at time t is R_t, the reproductive number at that time. We need to get it below 1 in order for the virus to die out.

We did get it below 1, during the summer, but now it's higher, which means the current restrictions aren't enough.

Anyway, the punchline here is that reducing contacts is our best bet for bringing R_t down and stopping the spread of COVID-19 *for now*, but once we have a vaccine, instead of having to reduce contacts, the vaccine reduces the chance of each contact resulting in a new infection.

So let's go back to our R_0=3 case, and a vaccine that's 90% effective and miraculously has 100% buy-in. Now, each contact of an infected person is 90% less likely than they were before to contract the virus themselves.

Previously, 100 infections led to an average of 300 new ones. Now that's 90% fewer - 30 new cases instead of 300. Repeat that for a few more incubation periods and the virus is gone. And that's going about business as usual ("usual" in the pre-mid-March sense).

Of course, we're not going to get 100% buy-in for the vaccine. Say we get 50% buy-in. Then, of the 300 people who would have become infected pre-vaccine, say only 150 got vaccinated. 10% of them, or 30 get infected now. And the other 150 non-vaccinated people all get sick.

So, 50% buy-in for a vaccine that's 90% effective would mean that 100 existing cases could be expected to lead to 150+30=180 new cases. So that means more more cases than existing ones, ie, the virus is still out of control.

Somewhere between 50% buy-in and 100% buy-in you get the virus remaining under control (=fewer new cases than existing ones) & dying out slowly. The exact number is left as an exercise to the reader, because I have a class to prepare and "just read my Twitter" isn't a lesson plan