Virus models and mad prophets

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“There is no evidence to suggest that anyone ever asked a prophet home for supper more than once…Most prophets went a little mad before they were through, if they weren’t a little mad to begin with.” — Frederick Buechner, Wishful Thinking.

Novelist/theologian Frederick Buechner was talking about the Old Testament guys, but mathematical models and their purveyors have some of the characteristics of those odd ducks. On March 15, I wrote this very inaccurate prediction regarding the coronavirus confirmed case count, at a time when 3500 cases had been reported in the U.S. and there was an exponential growth rate of confirmed cases in the range of 36% per day:

“Even if we can reduce this growth by, say, 1% per day (from 36% to 35% to 34%, etc.) we will still have twenty times as many cases as currently reported by April 1 [two weeks later] as we know about today.”

Well, twenty times that March 15 case count would have been about 70,000 total cases. The total case count on April 1 was instead over 210,000! Thus, the problem with trying to be prophetic when exponential daily growth rates have been so crazy-high.

Granted, my 70,000 forecast was an “even if” best case scenario. If we had continued instead on that mid-March 36% daily growth rate path, we could have seen well over 500,000 cases by April 1. We landed somewhere in between those two forecasts. It was not until March 27th that the seven-day average growth rate in COVID-19 confirmed cases dropped below 30% per day. As of this writing, that daily growth rate has dropped below 5%, but as we will see later, this is not necessarily as good a number as you might think.

COVID-19 nationwide deaths had a similar early daily growth rate that climbed up to nearly 32% on a seven-day average in late March with no relief in sight. That is the number that had New York governor Andrew Cuomo rightfully scared. This growth number nationwide is now closer to 4% daily. However, “brush fires” keep breaking out in nursing homes, detention facilities and workplaces, each with their own wildfire exponential growth path, that continue to strain local medical facilities and hamper plans to open the economy.

Did the models overestimate COVID-19 deaths?

As the worst of the crisis appears to have passed in the U.S. (but not by any means eliminated), there is now a lot of criticism of the early models which predicted up to one million deaths. Comparisons have been made to the dire “Y2K” predictions from twenty years ago (although I have my own personal take on that – see Note 1 below).

I created the chart below to demonstrate that the models were mostly correct at the time given available information, and that the interventions were mostly effective. The thick blue line indicates COVID-19 deaths to date. [2] The orange line indicates the 30% daily exponential growth trend that was very accurate in predicting deaths right through to the first of April.

Covid death trends

How can you tell that those early weeks of March had exponential rates of growth in deaths? The compression of the vertical scale hides where we were in March. However, we can “zoom in” to look just at what I call “the bad ten days” following my mid-March post. From the perspective of where we were at that time (put yourself in a “prophet” position at that March 17 point on the graph below), there was a 32% daily exponential growth rate in national deaths, and the rate was likely even higher than that in New York City:

The bad ten days

By the third week of March, the growth in confirmed cases had started to drop off, but there was still over a 20% exponential daily growth (the gray line in the first chart above) in cases, on top of an ever-increasing base. The growth in deaths (which obviously lags behind confirmed cases) began to drop slightly about a week later. Deaths were still climbing at 10% per day as of the 15th of April and are still running at above 4% per day growth over the past week. That rate (or higher) has been “stuck” in some states just beginning to feel the effects of the virus, such as Iowa. In short, we have “bent the curve” nationwide but the relentless local growth continues.


So, “What if?”

The models have all been very sensitive to the slightest shifts in these growth rates, which is why they have varied all over the map. Try a little “What if?” here: Follow that orange 30% line and ask, “What if we had delayed for another week before major nationwide efforts took effect to stop the transmissibility of this virus?” Just follow the orange line in the first chart above for another week and see where it goes.

At the rate we were headed during those “bad ten days,” we could have easily passed the 30,000-death mark during the first week in April. As you can see, the orange line quickly “goes off the chart,” and this is where the “worst case” projections were coming from. Even if we had then slowed the growth just a week later, we would have been working from a much higher base, and we could easily be now looking at twice as many deaths to date.

Or, go the other way: “What if we had rolled out mitigation efforts one week sooner, by early March, instead of experiencing the well-documented ‘wasted February’?” If you re-run those same “bad ten days” at the lower 20% growth rate reached just a week later, almost two-thirds of those deaths disappear by the end of March and many thousands more from that point until now! That is the ugly power of exponential growth and why early intervention saves lives.

Some critics have suggested that it was natural immunity or other factors that slowed the growth rates. However, successes at mitigation in places like South Korea and New Zealand, as well as early mitigation failures in Italy and recent “brush fires” in some prisons and nursing homes, demonstrate both the power of effective contravention and the incredible virulent (literally) nature of this disease when unmitigated.

The experience with those “successful” countries illustrates the very number that drives exponential-curve prophets mad. Each day of new cases at a high exponential rate launches the rocket on an ever-steeper path into the air. But hitting the brakes “hard and early” keeps the rocket close to the ground. Many in the U.S. still have not figured that out.

About that base

I noted above that the confirmed case daily growth rate has dropped below 5%. One not-good reason for this is that we still are not testing at the levels that would tell us where the “hot spots” are and how many asymptomatic cases are out there driving new infections.

Another confusing feature about exponential growth rates like these is that they hide absolute growth in cases and deaths amidst that declining growth rate. At the end of those “bad ten days” in March, that 32% growth rate was producing almost 500 new deaths per day. However, by April 20, a much lower 5% daily growth in deaths was producing about 2000 new deaths per day. Five percent on a base of 40,000 deaths is far higher than thirty percent on the much-smaller late-March base of 1500 deaths.

We see the implications of this in the states that interpret these declining growth rates as a reason to open the economy up. In almost all of those states, the absolute number of new confirmed cases continues to rise as more testing is being done. And a still unknown but tragically high percentage of those new cases will result in death.

As I have noted in an earlier post, many of those new deaths will show up in the natural “Petri dishes” of nursing homes, correctional facilities and workplaces found even in rural areas. Politicians seem to be “blaming the Petri dishes” for their own infections, not understanding that these cases are not spontaneously generated. They are always brought in from the outside, where many thousands of asymptomatic, but infectious, people have not been tested.

Until a comprehensive program for testing directly for the virus and for its antibodies is in place, even in rural areas, the risk of exponential “break out” infections remain. And meanwhile, millions of elderly Americans, as well as those with immune deficiencies and lung problems, will be effectively incarcerated in their homes under substantial risk of death if they leave.

And that’s enough to make this old modeler go a bit mad.


  1. I moved to the U.K. in April of 2000 to manage a sizable IT operation. The programmers there were still putting out “brush fires” of Y2K date problems that kept popping up in old programs, 18 months after rooting out known bugs at considerable cost. Their post mortem analysis demonstrated that, had they not spent that time identifying, correcting and testing hundreds of millennial date change issues, the company would have been completely non-operational that January. This company was not alone. Y2K was a real crisis that was averted so well that only the programmers who booked lots of overtime saw any major effects.
  2. The data I am using comes from a compilation of official state sources. Some estimates are significantly higher.


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