Quarantine Day 20: Fundamental Law of Numerical Methods

It’s critical during a crisis for experts to stay in their swim lanes. Could you imagine letting your hairstylist repair a broken boiler spewing hot water all over the house? Would you rely on your lawyer to prepare cuisine on the all important night you plan to propose? Should you trust mathematical projections made by MDs?

FYI, MDs are people who chose to become experts in medicine (i.e., patient care) rather than science, in case you’re wondering about the distinction between MDs and PhDs.

Years ago I taught graduate classes in numerical methods at the University of New Mexico. In my opening lecture each semester, I’d present students with what I called, “Dolin’s fundamental law of numerical methods,” which stipulated that, “a well compiled algorithm always renders an answer.” This assertion was quickly followed by the corollary; “any answer rendered by a numerical algorithm is always wrong.” With that as our starting point my students and I would spend the rest of the semester discussing why this fundamental law, along with it’s supporting corollary, was always true. Along the way, we explored concepts such as accuracy and uncertainty.

What does any of this have to do with COVID, and why am I drawing a distinction between experts in medicine versus experts in science? Well first, Doctors Anthony Fauci and Deborah Brix are both MDs. Fauci is currently director of the National Institute of Allergy and Infectious Disease (NIAID), and Brix is part of the White House Coronavirus Task Force. Prior to that, she worked as a diplomat within the global HIV/AIDS medical community. Both Fauci and Brix are highly respected physicians with distinguished track records who have done an outstanding job raising public awareness and advising President Trump in their areas of expertise.

That being said, I am increasingly becoming skeptical of their ability to speak credibly about predictive modeling, which is something outside their swim lanes. While things like describing Coronavirus behavior, developing hygiene protocols, and recommending patient care strategies are all withing the expertise of an MD, mathematical modeling is not, that’s the domain of science. While I’m certain deep with the bowels of NIAID and the White House Coronavirus Task Force good statisticians are diligently developing the models Fauci and Brix present to both the President and public, I’m less certain they appreciate the cautions and caveats those statisticians are no doubt providing.

One aspect of my fundamental law of numerical methods is that any numerical algorithm (i.e., model), depends on input parameters to generate output results and those input parameters can be tweaked to provide a wide range of outputs. Based on the growing disparity between federal model projections and actual data, I suspect Fauci and Brix don’t understand the complex mathematics underlying their models or how sensitive those models are to slight variations to input parameters. In other words, I’m not certain they understand that when it come to mathematical models,

Garbage in = equals garbage out.

Consider the simple model we discussed a few days ago when I demonstrated that in order to get from 3,000 COVID deaths at that time to 250,000 deaths byApril14th as Fauci and Brix were inferring, we could use a mathematical model based on the exponential function f(x) = a^x, where a=2.43, and x=14. Recall it was April 1^st when we discussed this, so the number of days until April 14^th, is x=14.

Just as easily, we can exercise this model with an input parameter 11% lower (why 11% will be discussed below). In other words, with a = 2.165, the output projection for the number of COVID deaths by April 14^th decreases to 50,000,

f(x) = a^x = 2.165^14 = 49,706.

This means an 11% decrease in input parameter a, results in a 500% decrease in the output value for f(x). We could just as easily increase the input value of our base variable 11% to a = 2.7. This increases the output estimate for f(x) to 1,078,970, where

f(x) = a^x = 2.7^14 = 1,078,970.

This means an 11% increase in the input parameter a, results in a 432% increase in f(x). You should now understand why I began my opening lecture each semester in numerical methods by telling students a mathematical model always provides an answer and that the answer is always wrong. It’s always wrong here because variables like a, can never be known exactly, there is always uncertainty.

In the above example as it relates to COVID modeling, the true value of the input parameter a, cannot be known, only inferred. When we look over current data however, we can calculate what the value of a, was at any point during this pandemic and use that as a basis for inferring what variable a, is likely to be in the future. However, the caveat remains that there will always be an error, or uncertainty, associated with whatever value we infer.

If we’re feeling rascally, we could utilize a random number generating algorithm, like Monte Carlo, to run a million simulations using different values of a, bounded within an upper and lower limit to tell us the most likely value of a based on whatever criteria we give the simulation. But again, whatever value is numerically inferred, has uncertainty (i.e., is wrong).

This is a long winded way of saying that somewhere between a=2.165, and a=2.7, Fauci and Brix, along with their teams, decided the “best” value to infer for the input variable a, would be a=2.43.

I inferred the lower bound value of a=2.165, based on the US experiencing the same COVID death rate scenario as Italy (ref. 4/2 post), i.e., 50,000 deaths, which I maintain is too high. I inferred the upper bound value of a=2.7, based on a media report, i.e., one million deaths. How Fauci and Brix arrived at their decision to use the value a = 2.43, is unknown, but one has to believe it’s based on a combination of past virus’s performance and whatever evidence is providing current clairvoyance. The problem is that the scientific community can’t peer review their inferences because no one in the federal government has provided their basis.

It’s increasing clear, that Fauci and Brix do not understand the sensitivities of their models to slight variations in input parameters. It’s also clear that the value they inferred for variable a, could just as likely be higher or lower (fyi, smart money’s on lower).

As I mentioned, the future value of variable a, can never be known with certainty. However, existing data can be used to calculate exactly what the value of variable a, was at any prior point in this pandemic. It then becomes the domain of science to use that knowledge to infer a likely value for variable a, going forward. And it becomes the responsibility of the scientific community to vet assumptions and conclusions.

Yesterday we saw an 11% day-to-day increase in the number of infected American’s who died from COVID, far below the 243% increase required by the Fucci/Brix’s model. Today’s data indicates that the day-to-day rate of change is 18%, which while higher than yesterday is still well below federal model projections.

Based on current data, we can use our exponential growth model to calculate the most recent value for variable a. To do so we first need to assign a value for the input variable x. To do that, we can count the number of days from when the first COVID death occurred (i.e., February 29^th) to today, April 4^th, and arrive at x = 36. We know based on today’s cumulative death data that the output function f(x) = 7,146. What remains is determining a value for variable a that satisfies our equation.

Rather than delay the drama by solving for a, I’ll just tell you a = 1.2759. So, based on current data along with our mathematical model,

f(x) = a^x = 1.2759^36 = 7,136

What value we can infer for variable a, going forward is anyone’s guess; including Fauci, Brix, and all the intelligent statisticians working for them. We know there will be more COVID deaths in the future. At the same time, I continue to believe the future value for variable a, is lower than the Fauci/Brix inference of 2.43; let’s hope a lot lower.