Behind the Curve

The problem we have with the impending Covid-19 epidemic in the US is that the Chief Executive is neither a mathematician nor an aviator. Chances are that he’s never given a lot of thought to the math behind exponential growth, nor has he ever heard the expression “behind the curve.”

A pandemic spreads because of a factor called R0 (pronounced R-naught). R0 is the average number of people who will catch a disease from an infected person. So if R0 is 1, then 10 infected people will infect 10 more people. The original 10 people will get well or die, removing themselves from the pool of infectious people, to be replaced by 10 new infected people. So the number of infected people is stable, neither growing nor shrinking.

If R0 is 2.5, then 10 infected people will infect 25 more people. The original 10 people will get well or die, to be replaced by 25 new infected people. Therefore the number of infected people jumps from 10 to 15 (the 25 new people minus the original 10), an increase of 50%. That 50% increase might occur in a few days, or a week, or a month. Mathematicians call this “exponential growth.” It drives financial growth for people who invest their savings carefully and regularly. Over a long enough time period, their small monthly investments can become a very large amount of money because of compound interest. Now imagine that the “investment” is not money but sick people, and the interest rate is R0, and the compounding rate is how long it takes for an infectious group to infect a new group. The result is exponential growth of infection—an epidemic driven by the rules of compound interest. Except instead of money in bank accounts, we have bodies in cemeteries.

Another example I prefer from the world of aviation is the expression “behind the curve” or “on the back side of the curve.” The curve in this expression is the so-called power curve (or power required curve) of an airplane. The power curve is a curve on a chart that displays aircraft engine power on the vertical axis and airspeed on the horizontal axis. The curve is approximately U-shaped or J-shaped. Pilots try to keep their aircraft on the front side of the curve, ahead of the curve’s minimum point on the chart. This is the region where increased engine power produces increased airspeed and, conversely, decreased engine power produces decreased airspeed. But if the plane’s airspeed continues to drop, engine power will eventually reach a minimum value. A further decrease in speed requires more engine power to remain airborne. Further decreases in speed require further increases in engine power. This counterintuitive region of flight is called the region of reversed command. If the plane has plenty of altitude, it may experience an aerodynamic stall and begin to fall, which leads to increased airspeed, which leads to stall recovery and all is well. However, if the plane is close to the ground (as it is on takeoff) and the pilot does not have enough altitude to bring the nose down and get to the front of the power curve, then a crash is inevitable.

What does this have to do with the coronavirus? The reason doctors and medical experts have been aggressive about stopping the spread of the virus is because they know if they are not sufficiently aggressive now, then by the time it becomes obvious to everyone that we should have done more, sooner, it will already be too late. In the fight against the virus, the country will be “behind the curve” and a disaster becomes unavoidable.