Epidemiology
Many COVID-19 waves peaked when far fewer people had been infected than their early growth implied under a homogeneous model. Contact surveys predict how much of that gap comes from people with many contacts being infected first.
In 341 US county epidemic waves of winter 2020-21, the median wave reached its peak of incidence when about 16% of the population had been infected. The growth at the start of those waves implies a median threshold of 30% under a homogeneous model.
Infections are estimated from deaths at an infection fatality rate of 0.7%, and growth is converted to a reproduction number with a generation interval of mean 5.5 days. The peak of incidence is called the turnover.
In the spring of 2020 case counts doubled in under a week, which a homogeneous model reads as a reproduction number near 2.5 and a threshold of 60%. Cotton (May 2020) called the gap between such thresholds and the observed peaks the herd-immunity paradox and traced it to variation in transmission across people, places and time.
The rule
Let $R$ be the reproduction number at the start of a wave. The infected share at the turnover is approximately
$$\text{turnover} \approx 1 - R^{-1/\lambda},$$where $\lambda$ is the immunity factor. People with many contacts are infected sooner, and while infectious they generate more exposure, so the effective reproduction number falls faster than the number of susceptible people. $\lambda$ measures how much faster.
A homogeneous population has $\lambda = 1$, and the rule reduces to the textbook $1 - 1/R$. For fixed gamma-distributed activity the rule is exact; elsewhere it approximates the peak from the initial slope. The turnover rule gives the derivation, the table and its accuracy.
Where the immunity factor comes from
- Age. The contact matrices of Prem et al. give $\lambda = 1.1$ to $1.7$ without fitting to epidemic outcomes. Age alone moves the threshold at $R = 2.5$ from 60% only to about 51-56%.
- Activity within age groups. One-day contact diaries show that the number of people someone meets varies widely within an age group. Combined with the age matrices, and holding each person's activity fixed, this gives $\lambda = 2.3$ to $3.0$.
- How long busy phases last. Surveys that ask the same people about their contacts every two weeks for several months separate lasting differences from passing ones. Under a model of persistent and changing activity, 26-30% of the variation in daily contacts persists over the months each person is observed and 13-14% changes over three to five weeks. The predicted immunity factor is 4.6-5.2 with daily contacts capped at 50, the survey's own convention, 3.4-3.7 at 20 and 5.9-6.8 at 100.
- Activity that changes over time. When busy and quiet phases alternate, the immunity factor in the growth phase exceeds its long-run value by about the growth rate times the Green–Kubo number of activity.
- Mixtures of places. In a population made of many independent places, early growth is dominated by the fastest of them, and each place turns over on its own. Two convexity adjustments, one on growth and one on the turnover, separate the two.
What the data show
If contacts stayed at their early-wave level, a wave's growth and its infected share at the peak would imply a value of $\lambda$ through the rule. This implied peak index $\hat\lambda$ equals the immunity factor only when contacts did not change during the wave. When contacts fell, it absorbs the fall.
| place | contacts | infected at the turnover | implied peak index $\hat\lambda$ |
|---|---|---|---|
| England, 9 regions | lockdown | 1-5% | 17-94 |
| Geneva | partial lockdown | 5% | 23 |
| Sweden, Stockholm and 2 other regions | no lockdown; large voluntary drop | 1-3% | 20-45 |
| Spain, spring 2020, 52 provinces | national lockdown | 0.3-4.6% | about 60 |
| Spain, autumn 2020 | curfew | 2.6% of those still susceptible | about 7 |
| US counties, winter 2020-21 | mixed | about 7% of those still susceptible | 2.4-5.3 (3.5 at IFR 0.7%) |
| Manaus | little effective control | about 17% | 2.5 |
Where contacts were cut, by law or voluntarily, the first wave peaked with 1-5% infected. Those indices are consistent with the reduction in contacts and cannot be read as immunity.
Where control was weaker, the index is 2.4 to 5.3 over fatality rates from 0.5% to 1%. The contact surveys predict 3.4 to 6.8 depending on the cap, and Gomes et al. fit about 3.5 to England and Scotland. The ranges overlap.
Both sides depend on their assumptions: the cap on the contact side, the fatality rate and the growth proxy on the epidemic side.
Manaus reportedly went on to 66-76% infected, which is difficult to reconcile with a fixed-activity model of this size. Changing activity is one possible explanation; the contact distribution in Manaus and the blood-donor estimates are additional uncertainties. The details are on US county waves and Serology by country.
Cite
P. Cotton (2020). Addressing the Herd Immunity Paradox Using Symmetry, Convexity Adjustments and Bond Prices. arXiv:2006.07341.
Working paper: The Immunity Factor from Contact Surveys. Analysis code and a dated log of results: research/epidemic-hit. Comments welcome.