US county waves
471 county epidemic waves from 2020 to mid-2021: nearly all turned over far below the textbook threshold, with an immunity factor of 2 to 3.5 in winter 2020-21.
The data
Daily COVID-19 deaths by county from the New York Times, February 2020 to June 2021, with New York City as one unit, and Census 2019 populations. Deaths are used rather than cases because testing changed so much over the period. Only counties with more than 50,000 people are kept.
A wave is a peak in the three-week average of weekly deaths that rises at least 30% of the county maximum above its surroundings and reaches 10 deaths a week.
What is measured
For each wave:
- the early growth rate $r$, from the contiguous rise between about 10% and 60% of the peak, converted to a reproduction number with a gamma generation interval of mean 5.5 days and standard deviation 2.1 days, $$R = (1 + r\,\theta)^k, \qquad k = (5.5/2.1)^2, \quad \theta = 2.1^2/5.5;$$
- the infected share when the wave began and when it turned over, from cumulative deaths divided by an infection fatality rate times the population. Deaths follow infections by about three weeks, so the death peak marks the infection peak.
A wave that starts with a susceptible share $S_0$ and effective reproduction number $R$ should, in a homogeneous population, turn over at $S_0/R$. The immunity factor of the wave is
$$\lambda = \frac{\ln R}{-\ln\big(S_{\text{turn}}/S_0\big)} .$$The level
At a fatality rate of 0.7%, 95-100% of waves in every season turned over below the homogeneous threshold.
| season | waves | median $R$ | median homogeneous threshold | median infected at the turnover |
|---|---|---|---|---|
| spring 2020 | 61 | 1.51 | 34% | 4% |
| summer 2020 | 69 | 1.37 | 29% | 6% |
| winter 2020-21 | 341 | 1.29 | 30% | 16% (cumulative) |
Spring and summer 2020 carry the lockdowns and closures. In winter, the constant immunity factor that fits the waves is about 3.5 at a 0.7% fatality rate, 2.4 at 0.5% and 5.3 at 1%, or about 2.1 at 0.7% when growth is read from cases instead of deaths.
The slope cannot be measured
Grouped by their growth, the winter waves infected nearly the same share of their susceptibles whatever their $R$ (the turnover rule). A flat relation is what time-varying activity predicts, since faster waves outrun the refilling of busy phases.
But growth read from deaths and growth read from cases, three weeks earlier, barely agree: their correlation is 0.23 in winter and 0.57 in spring. Random error in a growth estimate flattens any fitted slope.
Using case growth as an instrument for death growth, the slope of log depletion on $\log\ln R$ is 0.91 in spring (90% interval 0.46 to 1.50). In winter the estimate is 2.9 with an interval from 0.7 to 12.5, because the instrument is weak. Grouping by case growth instead gives a flat relation again.
With these growth measures the slope is not identified, and the county data cannot separate fixed from time-varying heterogeneity. The level is robust: waves turn over far below the homogeneous threshold.
Many places, each switching
County transmission does vary strongly and non-persistently. The spread of weekly log growth across counties has standard deviation 0.25 to 0.47, and a county's excess growth does not carry over to the next week.
The cross-county variance of log incidence grows by 0.051 a week, a Green–Kubo number of 0.026 per week. That is too small to matter for the paradox at county scale: it moves $R$ by about 0.02.
Code: analysis scripts 02, 03 and 06-09. Data: nytimes/covid-19-data, Census population estimates.