# The Statistics Behind “Breakthrough” Infections

Consider a theoretical scenario in which 100% of a population is vaccinated. Since vaccines are not perfect, there will be some “breakthrough” infections, and in such a case, 100% of infections will be among vaccinated people! That does not mean vaccines do not work.

Recently we have seen a rash of headlines along the lines of “75% of infected people were fully vaccinated” atop stories describing a “super-spreader” event in Massachusetts. While that statistic is correct, it needs to be put into context. First, let’s note that the outbreak, in which 360 out of 469 cases (75%) were among vaccinated individuals, was linked with densely packed indoor and outdoor events on the July 4th weekend with few people wearing masks. Next, let’s examine that scary 75% number.

Consider a theoretical scenario in which 100% of a population is vaccinated. Since vaccines are not perfect, there will be some “breakthrough” infections, and in such a case, 100% of infections will be among vaccinated people! That does not mean vaccines do not work. To determine vaccine efficacy in this case, we would need to know the total number of vaccinated and unvaccinated people who gathered that weekend. If this were known, then the percent of infections in the two populations could be calculated and the efficacy of the vaccine determined. In other words, it is important to know the denominator in such a calculation!

If out of the 469 cases, 360 (75%) were vaccinated, to find the percent of infections among vaccinated people we would have to make the calculation 360/a X 100 where “a” is the total number of vaccinated people. Similarly, for the unvaccinated, it would be 109/b, X 100 where “b” is the total of unvaccinated individuals. But we do not know “a” or “b,” so the relative effectiveness cannot be calculated. However, given what we know about rates of vaccination in the state, which is roughly 65%, and in Provincetown, the epicentre, a reported 85%, it is a good bet that “a” is much larger than “b,” meaning that the percent of infection among vaccinated people is much less than among the unvaccinated. Such data is available from other studies and formed the basis for the approval of the various vaccines.

What all this means is that given the high vaccination rate, the chilling 75% number is not surprising and is basically meaningless. The really important number is 1.07, which is the percent of infected people (5 out of 469) who ended up in the hospital! One of these was unvaccinated, and two had prior health conditions. This means that the vaccines keep people out of the hospital! We can cope with flu-like symptoms, we just do not want to end up in the ICU!

To further buttress the point that vaccines work, CDC reports about 35,000 symptomatic cases a week out of 162 million vaccinated Americans, which is 0.02%. Also, more than 90% of patients hospitalized with COVID are unvaccinated. Furthermore, vaccination is associated with a 40-50% reduction in COVID cases among an infected person’s household contacts.

The delta variant is now responsible for most infections, and it is nasty, spreading much more easily than the original virus. Studies show that vaccinated individuals carry as much of this variant in their nose as the unvaccinated, suggesting that they can spread the virus. They can indeed, but are still less likely to do so than the unvaccinated since they are less likely to be infected in the first place. Another boost for vaccines comes from just-released Israeli data demonstrating that a third dose of COVID-19 vaccine among the over 60 offers greater protection than after two vaccines by a factor of four!

What should we then take away from the Massachusetts event? That this is not the time for unnecessary gatherings or travel. And neither is it time to toss away the masks, no matter how unpleasant they are to wear. And of course, for the yet unvaccinated, it is time to get the jab. Unvaccinated people are incubators for variants.

@JoeSchwarcz