Time for a little bit of math. Suppose S represents the number of sick people, and H represents the number of people who are healthy, out of a total population of P. Then at any given moment, the number of healthy people is H = P — S. Now, when a sick person meets a healthy person, there is a certain probability that the healthy person catches the disease. A sick person meeting a sick person doesn’t change things. Neither does a healthy person meeting a healthy person. So the rate of change of the number of sick people is proportional to the product of the number of sick people times the number of healthy people. As an equation
dS/dt = a * S * H = a * S *( P — S )
(Yes, I slipped in a bit of calculus there. That’s a differential equation.) Here a is the interaction probability. When the number of sick people is small, this becomes approximately
dS/dt = a * S * P
since each sick person is likely to only encounter healthy people. The rate of change of the number of sick people grows proportional to the number of sick people. This is the exponential growth phase. The growth doubles every fixed time called the doubling time. So the number of sick people grows like 1, 2, 4, 8, 16, 32, 64, 128, 356, 512, 1024….. over a million at iteration 20…
The only thing we can control here is the interaction coefficient a. By not meeting people, we make a smaller, and can extend the doubling time to something our hospitals can handle.
At the other end of the epidemic, almost everyone will have gotten the disease. The number of people with it (or who have had it) S becomes large, and nearly equals the total population P. So the first equation then becomes
dS/dt = a * S * (P — S ) ~ a * S * 0 = 0
since P-S → 0. This is the ‘herd immunity’ stage, and I suspect its the way we all beat the common cold each year. We all get some version of it, about.
Now we’d like to control this. Which would get us into the discipline of optimal control. And that’s where you have to choose a ‘cost function’. What is it you want to minimize, or maximize? And there are two obvious choices being touted here. If you are a liberal, you probably want to minimize the total death count. (That would require making the model above more complex, including the portion of sick people who die. ) A slower doubling time would let the hospital network give every sick person the best chance of recovery, and that means lower ‘a’. On the other hand, if you want to maximize economic activity and get everyone ‘back to work’, you’d like to reach the final herd immunity state quickly, and you would want to increase the interaction coefficient. Get it over with, and reopen the country.
Did you hear NYC is going to dig mass graves in city parks to handle the large number of dead?
