The Wall Street Journal recently reported on the apparent effectiveness of the Operation Warp Speed COVID-19 vaccines:
The U.S. Centers for Disease Control and Prevention has identified a small cohort of approximately 5,800 cases of Covid-19 infection among more than 66 million Americans who have completed a full course of vaccination.
These so-called breakthrough cases, which are defined as positive Covid-19 test results received at least two weeks after patients receive their final vaccine dose, represent 0.008% of the fully vaccinated population.
Officials said such cases are in line with expectations because the approved vaccines in the U.S. are highly effective but not 100% foolproof.
For judging the effectiveness of a vaccine, that's the wrong measure. What we really want to do is compare how well the vaccines protect the portion of the public who have been fully vaccinated with the portion of the population that has either not been vaccinated or has only been partially vaccinated against COVID-19.
So we ran some rough numbers to find out. The following chart reveals a back-of-the-envelope estimate of how well the COVID-19 vaccines are so far working in the U.S.
At first glance, the Operation Warp Speed vaccines would appear highly effective.
Running Through the Rough Numbers
The first COVID-19 vaccincations of the U.S. public began on 14 December 2020.
On 15 December 2020, the U.S. Census Bureau estimated the size of the U.S. population on 1 April 2020 to be 332,601,000. The Census Bureau based this estimate on demographic analysis, independently of the actual 2020 U.S. Census results, where they estimated the actual population of the U.S. would fall in between a low estimate of 330,730,000 and a high estimate of 335,514,000.
As of 8 April 2021, 66,203,123 Americans (about 19.9% of the population) had been fully vaccinated for COVID-19. Using these figures, that would mean roughly 266,398,000 Americans would be considered either unvaccinated or partially vaccinated on that date .
From 15 December 2020 through 8 April 2020, the CDC reported a total of 14,232,649 new COVID-19 cases in the U.S. . If 5,800 is the number of "breakthrough" cases among the 66,203,123 fully vaccinated Americans, subtracting 5,800 from 14,238,649 let's us arrive at the estimated number of cases among the "less-than-fully-vaccinated" population of 14,232,849.
That's how you get the numbers we used to calculate the percentages in the chart. 5,800 breakthrough cases among 66,203,000 fully vaccinated Americans works out to be a percentage of 0.0088%.
Meanwhile, the 14,232,849 newly reported cases among the 266,398,000 unvaccinated or less-than-fully vaccinated portion of the U.S. population during the period of time vaccinations have been available for the U.S. public represents 5.3% of that portion of the population.
Based on this rough reckoning, it would appear the vaccines are working very well in limiting the spread of new infections from the strains of SARS-CoV-2 coronavirus active within the United States .
 None of this back-of-the-envelope analysis considers the number of Americans who already had and recovered from COVID-19, who have gained at least partial if not full immunity to the infection for at least some period of time as a result. As of 8 April 2021, that potential additional total is 30,181,371 (30,737,477 cases minus 556,106 deaths). We're also not considering the 46,843,488 Americans the CDC reports have received at least one dose of the COVID-19 vaccines as of 8 April 2021 either, who it would consider partially vaccinated.
We opted to not consider these additional categories because the CDC has not indicated how many of the fully or partially vaccinated portions of the population in the U.S. has previously tested positive for SARS-CoV-2 coronavirus infections. By grouping them in the unvaccinated or partially-vaccinated category however, our rough estimates understate the relative benefits of vaccination. Even so, the apparent relative benefit is massive.
 This figure almost certainly includes the results of positive COVID-19 test results whose samples were collected days and weeks before the Operation Warp Speed vaccines began to be introduced to the U.S. public. The CDC does not report test results by sample collection date, which would provide higher quality information.
 We're also not considering political factors that have affected how the U.S. vaccination program has been rolled out by various state governments.