“”Wisconsin is now experiencing unprecedented, near-exponential growth of the number of COVID-19 cases in our state,” said Governor Tony Evers in a video message.“
Dear Governor Evers,
“Near exponential growth” is not a thing. You either have exponential growth or you don’t – and I’m sorry to tell you, you do. I promise I’ll write you a longer blog letter to explain why this is true and go into much greater depth about the difference between exponential and non-exponential growth, but in a nutshell, here’s what is the thing: exponential growth (and if we could only have it, exponential decay) is characterized by one thing and one thing only: the rate of new cases is proportional to the number of present cases. (Another common term for this is compound growth). The only question is how fast is your doubling rate. If every day you have even 1% more cases then the previous day, you have exponential growth. Your doubling time with only a 1% daily increase will be longer (it is 69.6607168936 days approximately, please trust me on this) but eventually, you will get to the situation described in my “Parable of the Lily Pond” (https://garycornell.com/2020/04/20/the-parable-of-the-lily-pond-or-why-you-should-shelter-in-place/).
Best regards
Gary Cornell PhD
Dear Gary Cornell PhD,
your definition of exponential growth (” the rate of new cases is proportional to the number of present cases.”) is wrong. The correct definition would be “the rate of change of cases is at all times in fixed proportion to the number of cases”.
The only question is therefore whether or not the constant of proportionality stays the same over time or not. When the constant of proportionality is almost constant, it makes sense to speak of “almost exponential growth”.
Best regards,
Matty Wacksen
df/dt = kf no?
No. Take f(x) = x^2, with “present” defined as “t=1”. Then f'(1) = 2*f(1), i.e. “the rate of new cases is proportional to the number of present cases”. This equality does not hold in general though, and so we do not have exponential growth. The crucial point is that k should not depend on t or f(t).