The is for professionals only. It has significant math. This is not the continuation of COVID Resurgent: Of Hares and Foxes; Primer for Policy Makers, Part 1, which will follow shortly. If you’re not a pro, skip this article.
“Something’s not working,” Fauci said of the nation’s current approach in an interview with The Post. “I mean, you can do all the diagramming you want, but something is not working.”
All testing, pooled or no, is good, and the more testing the better. This article looks beyond the testing strategy, for problems in the model. All models, even of simple systems, are wrong. A model seldom deals with fundamental processes. It works at the higher level of convenient simplification, or even fiction. The only criteria is how well it works. Albert Einstein: “Everything should be made as simple as possible, but no simpler.”
If a model does not produce useful predictions, we can try complicating it with features of the situation which might have been thought unimportant. The mainstream of thought centers around these parameters:
- A(t) is the number of active cases
- R_o, the basic reproduction number, is the number of cases infected by each active case
- C(t) is the increment of cases that recover.
- S(t), which I will divulge after you have a look at the next equation.
With each time step, the model predicts,
A(t+1) = R_o*A(t) + A(t) – C(t) + S(t)
where C(t) is proportional to A.
This is the standard equation for a mostly naive population. Later on, when herd immunity has a significant influence, it is replaced by a model of higher dimension. It’s all quite familiar, except for a new term, S(t), which is the source of our problems, why the predictions of testing on the model fail so miserably.
S is a guess that even at this early stage, COVID-19 is not adequately simulated by a piecewise linear model. “S” could stand for “secret” or “seed”:
- Secret, because it characterizes the 90% of COVID-19 cases that are invisible.
- Secret also because we know as much about how it works as dark matter.
- Seed, because, with asymptomatic transmission, the secret process seeds the one we know about.
Cynically, adding S to the model could make it work better simply because there are more parameters to play with. But it allows us to incorporate a reference to the fundamental processes of infection which the standard model cannot accommodate. It is motivated by the prominent relationship of drinking establishments with COVID spikes.
The number of COVID virions required to produce infection may have no threshold greater than zero, but it is still thought to influence disease severity. The simple classification scheme of infected/clear handicaps modeling. Defining a series of disease states makes it possible to incorporate a threshold function based on the number of virions in the exposure:
- Not infected.
- Latent, viral load undetectable.
- Asymptomatic, non communicable, sheds non viable viral detritus.
- Asymptomatic, communicable, sheds live virus.
- Symptomatic, communicable. The 10% which have been mistaken for the main event.
We could graph these states on the Y-axis, from 0 to 4, against an X-axis of virions per exposure. Do we expect a straight line? Linear systems are very special and rare. Without knowing anything more than the way the universe works in general, we can say the curve is not linear. Occam’s Razor has two more suggestions:
- The graph is monotonically increasing.
- There is a threshold, a level of exposure that sharply increases severity.
Would a threshold pop out with the above scheme of 5 disease states? Maybe, maybe not. But as serology becomes more attuned, it’s likely that someone looking at some graph will see it. This refines the model.
Louis Pasteur said, “In the fields of observation chance favors only the prepared mind.” So let’s prepare, by writing a model that can approximate more of reality. In the secret process S, the simplicity of a constant R_o will not exist. R_o is replaced by F(S). The model becomes nonlinear. This is an inevitable consequence of a threshold. So where:
- S is the number of people in the secret state of infection.
- F(S), a random variable, is analogous to R_o for the linear model.
- C_s(t) is the increment of S who recover completely.
- I is the increment who transition to obvious infection, adding to A.
We might write,
S(t+1) = F(S)*S(t) + S(t) – C_s(t) – I(t)
Now Pasteur gives his attention to the bar, the innumerable social interactions within, and the COVID explosion that comes out. He goes in with a sampling gadget that looks beyond the transient life of suspended droplets. It could be something like flypaper that captures droplets as they fall out of suspension. Our modern Pasteur goes to that bar for a week, takes a sequence of measurements, and discovers something amazing. Over the course of a week, the sequence COVID concentration is (M-T-W-TH-F-SA-SU):
0, 0, M, M^2, M^5, M^7, M^(off-scale)
where M is the measurement on the first non-zero day. This is enough for Pasteur to speculate that there is a significant threshold on the 3rd day. Before then, bar patrons were pretty safe. By Thursday, they are doomed. This situation cannot be modeled by R_o. It requires F(S), dependent on the above sequence.
What goes into F(S)? We may as well make it a random variable, which requires a probability distribution. Pasteur’s observations require a threshold. Since we are still far away from fundamental processes, aesthetics count. If we can forgive Vilfredo Pareto for advising Mussolini to march on Rome, the Pareto distribution is beguiling. It served Pareto well, even though he didn’t know why either.
The Pareto is actually a family of distributions:
- It has a threshold, adjustable from a vertical cliff to a rolling rise.
- The higher order moments are large or infinite, reflecting the association between bars and COVID black swan events.
- It has only two independent parameters, the median, and alpha, which allows us to substitute faith for the inscrutable.
- Pareto invented it for problems in the social sciences, where it has been repeatedly validated, even where the fundamental processes are unknowable.
The Pareto distribution predicts a constant seeding of black swan events.The fundamentals will remain unknowable, relying in detail on every aspect of society and life that make the U.S. different from China. An enterprising mathematician might use the Pareto for a few plausibility arguments:
In a contest between testing and masks, masks win hands down.
Close the bars until COVID-19 is history.