The Path to Herd Immunization

Assumptions about the current situation:

  • Around 2.8 percent of Americans are currently infected with covid.
  • The dominant strain of covid-19 has a basic reproduction rate (R0) of 2.4; i.e., left unchecked, each person infected with the virus will in turn infect an average of 2.4 other people.
  • Covid infection rates are currently at a steady state; i.e., each newly infected person is in turn infecting one other person. Steady-state contagion is attributable in part to preventive measures (masks, social distancing, etc.), in part to the reduction in the number of people who can contract the disease because they’ve already been infected.
  • About 18 percent of the US population has already been infected with covid, leaving 82 percent vulnerable to contagion.
  • Therefore, the current effective contagion rate of the virus (Rt) is 1.0/0.82 = 1.25. I.e., people infected with the virus infect on average 1.25 other people who have not previously been infected.

Assumptions about the first three months of 2021:

  • Current levels of prevention  (masks, social distancing, etc.) will persist; i.e., effective Rt will remain at 1.25.
  • People remain infected for around 2 weeks before becoming immune (or dying). So, over the next 3 months = 13 weeks, an additional 2.8 x (13/2) = another 18 percent of Americans will have become infected, bringing the cumulative total to around 36 percent.
  • Multiply 1.25 (Rt) by .64 (Americans still vulnerable to contagion) = 0.8. I.e., by the beginning of April 2021 the rate of new covid infections would be decreasing by 20% every two weeks.
  • At that rate, the pandemic would run its course in about a year; i.e., by April 2022.

Now factor in assumptions about the covid vaccine over the next three months:

  • The FDA-approved vaccines are 95% effective against the dominant strain of the virus.
  • By the end of March 2021, 40 percent of the US population will have been vaccinated.
  • Now, by early April, the percentage of Americans still vulnerable to infection would be around .6 (unvaccinated) x .7 (not yet infected) = 40 percent.
  • Multiply 1.25 x .4 = 50% reduction in new covid infections every two weeks beginning in April.
  • At that rate, the pandemic would have run its course in 5 months; i.e., by about September 2021.

Other variables not factored into these assumptions that would increase infection rates and delay the extinction of the pandemic:

  • New, more contagious strains of the virus, for which the vaccines might not be as effective, might spread rapidly.
  • The general public, becoming totally reliant on the vaccine, might abandon other preventive behaviors that suppress contagion.
  • Logistical and behavioral and psychosocial fuckups slowing down the initial vaccine rollout might persist, lowering the percentage of people getting the shots.



Ratio of Case Counts to New Infections

My last post took shape less as a communique than as an exercise in thinking out loud. In the midst of that ordeal I did run a new analysis that enables a more direct and streamlined method of estimating the rate of covid contagion through the US population.

It’s well known that daily dx-positive case counts underestimate new infections. In June the CDC estimated that, in the US, there had been ten times as many infections as case-positives. But that ratio isn’t a constant: early in the pandemic, when very little testing was being done, the infections-to-cases ratio was far higher. As testing increased over the months the ratio decreased.

The other day as part of my convoluted last post I ran week-by-week analyses on data from the past four months looking for the best predictor of the subsequent week’s covid death counts. It turned out that the prior week’s death count and the case count from 3 weeks prior were the two best predictors. They’re not interchangeable predictors, so I used an average of the two in my analyses. However, as individual predictors each proved to be pretty accurate.

I’ve been using death counts as a 3-week lagging indicator of infections, and that algo was supported by the 4 months’ worth of data. But it turned out that the inverse relationship was also supported: changes in case counts are a good leading indicator of changes in death counts 3 weeks later.

Given the stable relationship over the past four months between case counts and subsequent deaths, along with the stable relationship between deaths and infections, it seemed likely that the relationship between case counts and infections had also stabilized. It’s still true that case counts underestimate infections, but based on the data it seemed possible to compensate for the case-count underestimate by means of an empirically supported conversion formula.

The method for deriving the conversion formula was based on three variables:

  1. Take the weekly total death and divide it by the estimated covid fatality rate of .0065.
  2. Take the weekly total case count from three weeks prior to variable 1.
  3. Divide variable 1 by variable 2: that’s the ratio of estimated infections to cases.

I ran the numbers week by week across the last four months of data, and the value for variable 3 proved quite stable, averaging around 2.7. So that’s the proposed conversion formula:

New infections =  2.7 x new cases

So, from 27 November to 11 December, there were 2.82 million new dx-positive cases reported in the US. Multiply 2.82 million by 2.7 = 7.61 million new cases, or about 2.3 percent of the US population. That estimate falls snugly within the 2.15% – 2.75% I estimated in the last post.

On the go-forward I’m going to use this conversion ratio for estimating infection rates in the US, which are likely to continue climbing even as the vaccine is being rolled out. I’ll also keep track of the week-to-week data to see if the ratio needs adjusting.