Coronavirus Math

Early 2020 is a truly Interesting time for decision making. We have learned what an inexact science risk communication is and had Robin Hogarth’s statement reemphasized:

“Human achievement is lower when there are nonlinearities in the ecology.” (What has Brunswik’s Lens Model Taught?).


As we all now know, coronavirus (covid-19) spread needs to be expressed as a power function. Power functions are those where an exponent determines the slope of the line. On average a person who has a positive case of coronavirus will, if carrying on a normal life with a normal number of human contacts, give it to more than 1 person and each of those people will also give it to more than one person. Whether that exponent is 1.5 or more, the numbers jump more quickly than the untrained person believes possible. These nonlinear functions explain a diverse range of socially driven phenomena including rank and number of book sales, rank and frequency of citations for scientific papers, and the rank and number of deaths in terrorism and war.  Alas, cause and effect are not clear.  We see very few large values and lots of small values, so the average has no meaning for our linear driven brains. The bottom line seems to be that linear implies cause and effect with more or less input of one variable creating more or less output of another.

powerdownloadThere is the story of the savvy seven year old who asked her dad at the beginning of the month for a birthday present. She said, “Dad, I will provide the dollar and you just multiply it by one and a half times the new total each day until my birthday at the end of the month and only if my behavior is not too bad for that day.” On her birthday, the daughter agreed with her dad that she had had 3 bad days so told her dad, the check only had to be for $56,815.

Governor Mike Parson of Missouri seems likely to have fallen for that one. Even on April 3, 2020, he is saying he does not want to do a state wide stay at home order because many counties only have a few cases and a couple have none. Of course, it would be silly to have a stay at home order if you were absolutely certain there was no coronavirus and that none would be introduced during the stay at home period.  Mike Parson might try to tell you that it just does not make common sense to inconvenience many people when there are so few cases,

Common Sense

Kenneth Hammond  defined  it:  “Common sense (robust flexibility) means engaging in as much analytical work as required, and in as much intuition as will suffice, because intuition is by far the easiest.”  Finding the right mix of rules and discretion varies, and successful cognition must be adapted to the structure of information in the environment.  Common sense is easy to appeal to and to tell stories about, but not so easy to implement. (See post Coherence from the Default Mode Network.)

Mike Parson does not appear to have done the analytical work needed to know that, in this case, common sense would be to impose the order.  It seems likely that he is struggling with nonlinear relationships. This is kind of a big deal if John von Neumann’s analogy is correct:  that studying nonlinear relationships is similar to studying non elephants.

The common sense is to know that we are not good at nonlinear relationships and to see that as a red flag and a time to seek help and avoid your intuition.  An example of this for me is the base rate epidemiology brain teaser. Conditional probability may be good for statistics, but our linear brains need to think of natural frequencies. Here is the brain teaser and the solution is at the end of the post:

Breast Cancer. The probability of breast cancer
is 1% for a woman at age 40 who participates in
routine screening. If a woman has breast cancer,
the probability is 90% she will get a positive
mammography. If a woman does not have breast
cancer, the probability is 9% she will also get
a positive mammography. A woman in this age
group gets a positive mammography test result in
a routine screening. What is the probability she
actually has breast cancer?

Presented with this for the umpteenth time, I still want to blurt out my intuitive wrong answer. I note that the brain teaser is probably nonlinear only in its presentation.

Nonlinear–exponential growth–too scary

In the March 30, New Yorker, the article: “The Contrarian Coronavirus Theory That Informed the Trump Administration,” by Isaac Chotiner appeared. Richard Epstein, a much cited attorney and professor at New York University, was interviewed in the article after having written a piece, “Coronavirus Perspective”, that projected the likely path of the coronavirus. Epstein projected 500 deaths for the United States in that piece on March 16.  The interview included this response:

…There’s an underlying, standard model that you want to use, and the question is how you stuff it full of parameters. That is, numbers you add into it to make what’s going on. And, so, the situation that you get is you cannot use any exponential system because essentially then everybody is going to be dead, because things just keep doubling, doubling, and doubling.

So you have to develop a model which is going to explain why there’s a fairly rapid increase at the outset, and then why the thing starts to turn flat, ultimately down, and then disappears. That’s the strategy that you have to do…

I have bolded the most interesting part, where Epstein indicates he knows about exponential growth and that in explaining the world it cannot go on forever. He is correct in this that doubling cannot go on forever. California cannot have more 44 million cases.  Unfortunately, you have to have real reasons for deciding when that exponential growth will end. He missed on that.  Huge underestimates by public health officials kill people.


It is interesting to wonder if the brain or large portions of it can be nonlinear….logical extreme– quantum computer…while apparently functioning much better in a linear ecology.  Of course, a linear ecology is probably the simplest to perform well in, so we are best at the exception rather than the rule.