The parable of the lily pond-or why you should shelter in place

Behavioral economists have shown our brains are hard wired to certain ways of thinking that are “wrong” – well at least to people who still seem to believe in “homo economicus”. We seem to have have hard wired ways of thinking, like the “prospect theory” that won Kahneman the Nobel Prize in Economics that shows the pain of a loss far outweighs the pleasure of a gain. It’s not symmetric at all.

Anyway, back when I used to teach exponential growth, it quickly became obvious to me that human beings are also not wired to “grok” it. It is yet another hard wired thing: we jump to conclusions about exponential growth that are just wrong. Yea, if you are have enough mathematical training. you can do the calculations and so hold your instinctive jumps to the wrong conclusion in check, but to get it instantly, to “grok it” as the saying go, forget that. I really believe we are just not wired to understand exponential growth – perhaps because nothing that our distant ancestors on the savannas of Africa encountered was likely to involve it.

So that brings me to the parable of the Lily pond which I use to use to try to get my students to understand exponential growth. I told them “Imagine a giant lake, so large it dwarfs any of the  great lakes. Then a child throws a lily seed into the lake. The next day they come back and see a lily plant. The next day they come back and see 2 lily plants. The day after that 4 lily plants. Cool they think.” So the doubling continues every day. On the 30^th and last day of the month this enormous lake is absolutely totally filled with lilies and what’s more, all the lilies are dying.

I then ask my students: “When was the lily pond ¼ full, when was it 10% full and so the lilies and so the lake seem to be thriving?

It was always surprising to me that only a few of my students get that the answer was the lily pond was a quarter full two days before. The 10% question is a little trickier though estimating it isn’t that difficult. You need to use logarithms to get the exact answer for when it was 10% full. It turns out it was 10% full about 80 hours before it was full.

Some people think of getting to this answer by asking what power of 2 is the same as multiplying by 10. That turns out to be (by definition) log₂(10) which is about 3.322 as your calculator will tell you. So you go from .1 to 1 (10% full to completely full); in 3.32 days or about 80 hours. Other people like to think of it as moving backwards in time, so you are solving log₂(.1) , which is -3.32.

In any case things happen fast with exponential growth. You go from ¼ full to all dead in just two more days and from 10% full to all dead takes around 2 days more than going from ¼ full to full.

In epidemiology, the doubling rate of a disease is determined by a number called R₀ (pronounced “R nought or R zero”) which is the number of people a single person can infect together with how long it takes for that person to infect that many people – and when R₀ is bigger than one you have exponential growth and eventually everybody who is susceptible will get the disease. Oh and pretty damn quickly when you have a doubling every three days – which seems to be how fast COVID 19 grows absent any social distancing into a population where almost everyone is susceptible. (A doubling rate of 3 in our lily pond means it doesn’t double every day, it does every three days. So instead of 30 days, it takes 90 days to death, because 30 doublings takes 30*3=90 days when a single doubling takes 3 days instead of one day…)

But if social distancing brings R₀ to <1 , the disease dies out eventually.

So stay at home damn it.

Added: Fixed the mistake of conflating R₀ with the doubling rate, it is a bit more complicated then that because R₀ has to be combined with the time to infect to get the actual doubling rate. More on that later.