COVID-19

COVID-19 is officially a pandemic. It is up to all of us to do what we can to limit the spread of this disease. An earlier post shows how herd immunity can stop an outbreak of disease by making the effective reproductive rate, $$R$$, below 1. In that post, we looked at measles. In measles, the basic reproductive rate $$R_0$$ is around 15, but when over about 93% of the population is immune (either by vaccination or by having survived the disease) we get an $$R$$ below 1, meaning the likelihood of an outbreak is low. Please read that post to see how all that works.

Most estimates I’ve seen for COVID-19 estimate its $$R_0$$ at about 2 or 3 (I have been cautioned that there are wildly divergent estimates of this number) but since there is as yet no vaccine and few people have had the disease, without other interventions, $$R = R_0$$, and the disease continues to spread. Until there’s a vaccine, is there some other way to reduce $$R$$ below 1?

Recall from that earlier post that $$R_0$$ represents the average number of people that one infected person will infect. This number depends on many factors, including biological ones (the probability that an infected person infects someone in close contact, that I’ll call $$p_0$$) and also sociological ones (how many people the average person comes into close contact with, that I’ll call $$n_0$$). Assuming there are no other factors, $$R_0 = p_0 n_0$$.

Until there’s a vaccine or until there’s a significant fraction of the population that’s immune, there’s not much we can do about $$p_0$$. That means $$R = p_0 n$$, where $$n$$ is the average number of people we come into close contact with after we change our behavior. How much do we have to change our behavior? We want $$R < 1$$: $\begin{eqnarray*} R & < & 1 \\ R & < & \frac{R_0}{R_0} \\ p_0 n & < & \frac{p_0 n_0}{R_0} \\ n & < & \frac{n_0}{R_0} \end{eqnarray*}$

So here’s the good news: In theory, if each of us reduces the probability of coming into close contact with another person by over a factor of $$\frac{1}{R_0}$$, we can stop this epidemic. If we estimate $$R_0 = 3$$, that means we just need to come into close contact $$\frac{1}{3}$$ as much as we usually do. We don’t have to go into total isolation or do perfect hygiene. We just need to limit our interactions with other people and to use better hygiene.

In practice, of course, it’s much more complicated. A friend pointed me to this excellent and extremely detailed article, which recommends a “hammer” phase, in which $$R$$ must go much lower than 1, to get the exponential decay to happen much more quickly, and a later “dance” phase in which we can bring $$R$$ back closer to, but still below, one. The “hammer” phase, according to the author, will require much more stringent restrictions.