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Rule of 72: One More Time

By Shlomo Maital

   Sorry, but – one more time for the Rule of 72.   Many print and cable journalists are rather innumerate – they took Philosophy 1 instead of Calculus 1 in college. So it is no wonder they struggle to interpret the COVID-19 numbers for us, and simply throw totals at us, without really explaining what they mean. So, it’s left to us citizens to “do the math”. And, alas, that math could involve logarithms.

     (Someone I know well, recently asked me, is a logarithm the same as an algorithm? Because, high school teachers themselves don’t seem to know what logarithms) are).

     So, first, a quibble. CNN, when you show your COVID-19 graphs, daily numbers, can you please do it on a logarithmic scale, not absolute numbers? (in Excel, you can use the log( ) function or the ln ( ) function, where ln is the log to base ‘e’ 2.71828 and log is the log to the base 10… ok, never mind about all that!   But the reason for logs is, the steepness of the logarithmic curve shows the rate of change, and you can easily tell if the curve is getting steeper (rate of change is rising) or less steep (rate of change is slowing), and this is of course CRUCIAL!).  

         So, some of you, maybe very few, maybe VERY VERY few, want to know, where does this Rule of 72 come from? (Divide 72 by the daily rate of change of COVID-19 patients, and you get the number of days it takes for the number of those infected to double). If it’s 10%, it doubles every week – disaster. If it’s 2%, it doubles every 36 days, about monthly – phew…we made it.  

       So here is the basic equation:

                 (1+R/100)T = 2    

Where R is the rate of (daily) change, in %, T is the number of days it takes to double, and 2, well,   that’s the doubling, e.g. a 10% daily spread rate will double in 7 days:   (1+0.1)7 = 2

       If you take the logarithm of each side of the equation, you get this:

             T ln(1+R/100) =   ln 2                     (trust me!)

So T, the number of days it takes for the virus to double the number infected, if the rate of spread growth is R,   is equal to

              T =   ln2 /   ln(1+R/100)

Now, mathematicians pull a neat trick out of their bag of tricks, and find a way to simplify this equation, so we don’t need calculators or log tables:

             T = 72/R

Hence Rule of 72: divide 72 by the rate of spread, you get ‘days to double’. (One last word, skip this is you wish —   you can do this approximation using a neat way to find approximations called a Taylor Function, neatly tailored by mathematicians to simplify our lives…).

           Normally, we use Rule of 72 to see how many years it will take for our money to double. Well-heeled people get 8% interest on their money, or more, usually through the stock market, so their millions double on their own every 72/8 = 9 years.

             Initially the virus was spreading at 25% growth rates daily, in many places, meaning the numbers infected were doubling every 72/25 = 3 days!   Roughly. Yikes. Say 100 people were infected initially.   At this rate, in 30 days, there will be 10 doublings. 10 doublings is 2 times 2 times 2,   ten times, or 1,024. So those 100 infected become 100 x 1,000 or 100,000!   THIS is why it was so absolutely crucial to jump on things early and lock everyone down. Alas, the US failed to do this. So did other nations.

       So if TV and media fail to use Rule of 72 – do it yourself. Figure out the daily % rate of change of those infected with virus (come on, you can do it…. Today’s number / Yesterday’s number, minus 1 and then times 100. Then divide 72 by this number. Presto: Days to double. Big number? Worry. Small number? Yeah!

Blog entries written by Prof. Shlomo Maital

Shlomo Maital

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