We have conducted a poll to see how long it may take for case numbers to quadruple in New Zealand from 100,000 to 400,000. See:
To quadruple the numbers need to double twice.
To quadruple in March would mean doubling about twice in each 15-day period.
We see that an average of almost 4.73% daily increase in total case numbers is required over the month.
1.0473 ^ 30 = 4.000673468
2^(1/15) = 1.047294123 or 4.7294123%
March has 31 days so doubling every 15.5 days will work.
2^(1/15.5) = 1.045734148 or 4.5734148%
Will it be another month of quadrupling (1.6m people infected) before the daily increase in case numbers gets this low consistently before a peak is (almost)reached in this outbreak? Will we run out of population to infect?
We look at ways to estimate/ calculate the percentage increase required for this to happen. We will look at the rule of 72, often (falsely) attributed to Einstein. See: http://www.moneychimp.com/features/rule72_why.htm
Instead of yearly interest (Y) it also works for daily compounding (d).
Instead of 69, usually 72 is used. Obviously 72 has more factors than 69.
So we can use the formula d = 72/r or r = 72/d
Just divide 72 by r to get the number of days (d) to double, or divide 72 by d to get the average increase in case numbers required.
For the daily compounded interest we use
If we just need 1.08 instead of 8%, for example, then we can use 2^(1/d).
2^(1/9) = 1.080060 or 8.006%
72/9 = 8%
We see that the rule of 72 provides a reasonable result when d or r is between 8 and 10.
We are interested in d = 15 days. Below we compare the Rule of 72 estimation with the actual daily compound for 8 and 9 days.
Remember that 1.08 corresponds to 8%.
We see the Rule of 72 gives close to the exact answer for these values.