On The Double II
How do we calculate doubling times from growth rates?
A doubling time is the amount of time, assuming constant growth, it takes for a quantity to double. Researchers often calculate this figure for population growth in cells, animals, and humans.
How should we think about exponential growth? Exponential growth means something increases in relation to its current value. Imagine you covered the squares of a 64-square chess board with game pieces. Each step doubles the number of filled squares. After four rounds, eight squares have a piece. Three generations later, pieces fill the whole board.
It does not mean ‘fast’ — savings accounts with low interest rates have this kind of growth. One way is to describe the changing volume with a simple mathematical model. Here, there is a constant, representing volumes at the start. We multiply that constant factor by a base number, raised to the power of a fixed growth rate multiplied by time.
For positive whole numbers, exponentiation is the act of repeated multiplication. ‘Two to the power of three’ is the same as two times two times two. The natural base for this exponentiation process is Euler’s number, which is about 2.718. That number has many special properties. Its exponentiation function is equal to its own derivative.
The UK Health Security Agency report on 10th December 2021 estimated Omicron advantage. The growth rate in odds was 0.35 per day.
That estimate came from a proxy of spike protein target failures in viral tests. Their method adjusted for Omicron proportions after sequencing.
Working through the formula, that growth rate implies daily rises of around 42%.
How do we work out doubling times? Thinking about the formula: find the natural logarithm of two, then divide by the growth rate.
Taking the logarithm is the inverse of exponentiation, like division inverts multiplication. Plugging in 0.35 gets about 2.0. That means the odds of cases being Omicron were doubling every two days.
This is also where the ‘rule of 72’ in finance comes from. For an annual interest rate of 2%, 72 divided by two means the doubling time is about 36 years. The number 72 is about 100 times the natural logarithm of two.
That simple model of exponential growth can give poor performance in future extrapolations. Exponential growth cannot continue forever.
