The United States was unprepared for the scale of the pandemic, which has hit many counties and filled some hospitals to the max. A new paper in PNAS suggests that there may have been a mathematical method, somehow, to the madness of the early days of COVID.
It is testing a style that largely matches patterns of reported cases and deaths, county by county, in the United States between April 2020 and June 2021. The style suggests that unprecedented COVID spikes could, even now, overwhelm local jurisdictions.
Our most productive estimate, based on data, is that the number of cases and deaths for the county has infinite variation, meaning that a county can be affected by a large number of cases or deaths. We can’t expect a county to have the resources to deal with incredibly vital and infrequent events, so it’s very important that counties, as well as states and even countries, expand plans, ahead of time, to be consistent with percentage resources. “
Environmentalists might have guessed that the spread of COVID cases and deaths would be at least more or less consistent with Taylor’s Law, a formula that relates a population’s mean to its variance (a measure of dispersion around the mean). From fluctuating crop yields to the frequency of tornado outbreaks to the multiplication of cancer cells, the bureaucracy of Taylor’s Law, the backbone of many statistical models that experts use to describe thousands of species, adding to humans.
But when Cohen began investigating whether Taylor’s Law could also describe the grim COVID statistics provided through the New York Times, he was shocked.
99% of the county’s case-death counts between April 2020 and June 2021 conformed to a “lognormal” distribution of Taylor’s Law, which expects the variation of cases or deaths at the location to be proportional to the square mean of cases or deaths. For example, if the average number of instances consistent with the county is 50 in Arizona and one hundred in California, this edition of Taylor’s law would expect the instance discrepancy in California to be 4 times greater than the instance discrepancy in Arizona. Similarly, if the number of county-consistent instances in those two states were 50 and 150, respectively, the discrepancy would be nine times greater in California.
However, the first % of the number of instances and deaths did not fit the normal logarithmic distribution. Instead, the height numbers corresponded to the Pareto distribution; A style seen more occasionally in economics than biology, in which incredibly high values are rarely observed regularly (think: distribution of the source of income or wealth). What made this Pareto-specific distribution unique was that it also had infinite variation, implying that dispersion would accumulate beyond any finite limit, the greater the number of observed cases or deaths. The challenge was to perceive why even the most sensible 1% of the counts conformed to Taylor’s law with the same exponent as the bottom 99%.
“It’s a puzzle,” Cohen recalls. And I sat on this puzzle, taking it out from time to time, torturing it a little bit, and putting it away. Until the day I called the heavy artillery. “
Cohen sent his PC simulations and guesses to Richard A. Davis of Columbia University and Gennady Samorodnitsky of Cornell University, asking for their contribution. A few months later, the two sent him some theorems: the lack of evidence that Taylor’s law would be valid even for the maximum of 1% of counties distributed across Pareto, with the same exponent as 99% of counties distributed logarithmically normally. The pandemic has produced an orderly trend of cases per county and deaths per county. The unforeseen thing about this order was that, in most cases, there was no limit to the gravity of things. “
It’s unclear why the pandemic is following this hybrid (lognormal-Pareto) edition of Taylor’s law so strongly. One option is that Taylor’s Law, which describes the variation of many ecological systems, adding infectious diseases such as measles and Chagas disease, simply captures the nature of the infection. If one patient infects two other people (with some probability) and of those two patients infects two other people (with some probability), we would expect the cases to accumulate exponentially (with some probability), and occasionally the occasions can result in infinite variation.
Cohen hopes the study will sound alarm bells for policymakers. An infinite variety of cases and deaths depending on the county means there is a very unlikely but imaginable situation where a COVID spike causes each and every person in that county to get sick, or worse. the advent of vaccines makes such a situation increasingly unlikely, regions in the U. S. U. S. and overseas with low vaccination rates still face the option of spikes they can’t handle.
Calculations, Cohen says, suggest that COVID cases and deaths may exceed the ability of local jurisdictions to cope. “Governments had more to be prepared to call their friends,” he said.
Rockefeller University
Cohen, JE, et al. (2022) COVID-19 cases and deaths in the United States Taylor law for heavy tail distributions with infinita. PNAS. doi. org/10. 1073/pnas. 2209234119 variance.
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