# COVID vaccine math – it seems to be confusing the anti-vaxxers

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I have been seeing the same old bad math from COVID-19 vaccine deniers that I’ve been seeing from anti-vaxxers for years. They don’t seem to understand the basic principle of “incidence,” that is the proportion of a population that contracts a disease.

I’ve heard and read several times the claim that more people contract COVID-19, despite having received the vaccine, than those who aren’t vaccinated. I sincerely doubt this math is true, it appears the number of hospitalizations for unvaccinated individuals far exceeds vaccinated individuals.

However, that doesn’t mean that zero individuals with the COVID-19 vaccine end up hospitalized. That would be bad math.

But what we’re getting to support these bad math claims from COVID-19 vaccine deniers is based on bias, avoiding the rate of incidence, and other issues. So, let’s try to clarify what’s going on with a couple of simple graphics.

## COVID-19 vaccine math

Take a look at the graphic above. Let’s look at the numbers on the right, and you will begin to understand that the incidence matters.

Let’s assume that in a population of 1,000,000 individuals 70% are vaccinated. Thus, 700,000 are vaccinated, and 300,000 are not.

Assuming a 2% infection rate, that would mean that there would be 6,000 positive COVID-19 in the unvaccinated group. Assuming that there is a 10% chance of being hospitalized after becoming infected, that would mean that 600 unvaccinated people would be admitted to a hospital.

Assuming the same infection rate and hospitalization rate PLUS an 80% vaccine effectiveness (which is quite low, the Moderna and Pfizer vaccines are close to 95% effective), that would mean that 84 vaccinated individuals would be admitted to a hospital.

So, that means over 87.7% of hospitalizations are in the unvaccinated group. And the hospitalization incidence is 2 out of 1000 in the unvaccinated group, but only 0.12 out of 1000 vaccinated individuals.

In other words, the incidence of hospitalization after exposure to COVID-19 is substantially larger in the unvaccinated population. And even here, we’re making assumptions that might not be true – for example, the infectivity rate may actually be much lower in vaccinated groups than in vaccinated ones. And vaccinated individuals might still wear masks and practice other pandemic-related public healthcare recommendations.

Of course, the numbers are even more powerful in the left column, where that same population of 1 million is 92% fully vaccinated (which is what we have for many vaccines like measles).

In this case, 160 unvaccinated individuals and 110 vaccinated ones would be predicted to require hospitalization. And I know someone from the back of the room is going to scream out “you see, the more you vaccinate, the more vaccinated people require hospitalization.”

That is true, but that’s bad COVID-19 vaccine math. In the 92% vaccinated case, there are nearly 12X more people who are vaccinated than are not. Once again, the risk of hospitalization in the unvaccinated group is still over 20X greater than in the vaccinated group. It’s just that there are so many more vaccinated individuals than unvaccinated ones in the population, even the rare risk of a breakthrough infection may mean numbers that appear similar between both groups.

But the risk remains unchanged –hospitalization incidence is 2 out of 1000 in the unvaccinated group, but only 0.12 out of 1000 vaccinated individuals.

But there’s something more amazing about these calculations. There would be 684 hospital admissions in a population that is 70% vaccinated compared to 270 admissions in a 92% vaccinated population. That would be a nearly 61% decrease in hospitalizations as a result of a high level of COVID-19 vaccination.

## Summary

The anti-vaccine crowd is trying to use bad math about the COVID-19 vaccine to sow fear, uncertainty, and doubt. Their use of improper statistics along with a misunderstanding about epidemiological incidence should be stamped out, but you know it will continue.

There will be numerous memes that push the narrative that breakthrough infections in the vaccinated group are large and it’s “safer” to not be vaccinated.

That’s false.

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