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Home » Vaccine effectiveness – correcting misunderstandings about COVID-19 vaccines

Vaccine effectiveness – correcting misunderstandings about COVID-19 vaccines

I keep reading how people think that when a COVID-19 vaccine is reported to have 95% effectiveness it must mean that they have a 5% chance of catching COVID-19 after being vaccinated. But effectiveness doesn’t quite mean that, and I wanted to clarify.

Part of the reason this needs clarification is that until we have herd immunity against COVID-19 (along with all of the variants), it’s important to practice all of the public health measures like wearing a mask and social distancing even if you have been vaccinated.

So, this will be a little bit of mathematical analysis of vaccine effectiveness, so that you know what it means.

Vaccine effectiveness math

Vaccine effectiveness (or efficacy, a term used for non-biological drugs) is a pharmacological term that describes the difference in an observed outcome (in this case COVID-19) between the vaccinated and unvaccinated (placebo) groups.

The basic formula is written as:

{\displaystyle VE={\frac {ARU-ARV}{ARU}}\times 100\%,}


  • {\textstyle VE} = Vaccine efficacy,
  • {\displaystyle ARU} = Attack rate of unvaccinated people,
  • {\displaystyle ARV} = Attack rate of vaccinated people.

So, when a vaccine is claimed to have 95% effectiveness does that mean you have a 5% chance of contracting COVID-19 after being vaccinated?

The simple answer is no. When you see claims that the Moderna or Pfizer vaccines have around 95% effectiveness, the results are calculated from clinical trial results. It’s a basic calculation:

  • In a clinical trial of 20,000 people, 10,000 will randomly receive the vaccine and the other 10,000 will receive the placebo.
  • The two groups will be monitored for lab-confirmed COVID-19, and the numbers counted. For the sake of this exercise let’s say that there were 50 cases in the vaccinated group, and 1000 in the unvaccinated group.
  • The ARU was 1000/10000, or 0.1.
  • The ARV was 50/10000, or 0.005
  • We then calculate vaccine effectiveness as (0.1-0.005)/0.1 or 95%. In other words, there was a 95% reduction in cases from the unvaccinated to the vaccinated groups.

Using these numbers vaccine effectiveness measures the reduction in risk of COVID-19 after vaccination. It is not a poker game where you have a 5% chance of getting a full house if you’re holding two jacks and two 10s (sorry, I don’t play poker). It means you have a lower risk of contracting COVID-19.

And that number is fungible:

  • We do not know what the level of exposure to COVID-19 is between the groups. Now if they are randomized, there should be equal exposure, but it’s possible that the non-vaccinated group had more exposure to wild SARS-CoV-2 viruses.
  • Exposure to variants against which the vaccine may not be as effective.
  • Each individual will experience a different level of immune response to the vaccine. I may have a healthy immune system that gives a powerful response to the vaccine, while someone else might have type 2 diabetes and has a less powerful response. The latter group may have lower vaccine effectiveness.
  • Slight variations in dosage or delivery might have an impact on the overall vaccine effectiveness for an individual.
  • Length of time after all doses are given – shortly after being given the last dose, your immune response may not be robust enough to stop an infection. Usually, maximum immunity is reached after a few days.
  • Randomness – all biological systems have inherent randomness in them, so it is plausible to hypothesize that the vaccine just didn’t induce an immune response.
vaccine effectiveness
Photo by Markus Spiske on Unsplash.

Moderna phase 3 clinical trial results

The Moderna phase 3 clinical trial results are a good way to look at this. The Moderna trial included over 30,000 volunteers roughly divided into equal sized groups of 15,000 who received two doses of the vaccine and 15,000 who received two doses of the placebo. The patients were randomized to each group, and it was double-blinded, meaning neither the patient nor the clinician knew who was getting what vaccine (although, given the frequency of minor effects post-vaccination, I don’t think it was a big secret).

The volunteers were then instructed to live their lives as they did prior to the vaccine – wear masks, socially distance from others, and other public health measures. Participants were monitored closely, and if they didn’t feel well, they were tested for COVID-19. In fact, if they were symptomatic, the researchers called the participant every day.

Because Moderna was going to seek an Emergency Use Authorization (EUA), the effectiveness numbers were calculated two months after the average participant had received their second dose. Let’s calculate the vaccine effectiveness for the Moderna vaccine

  • ARU (attack rate for unvaccinated) = 185/15000 = 0.01233
  • ARV (attack rate for vaccinated) = 11/15000 = 0.00073
  • Vaccine effectiveness = (0.01233-0.00073)/0.01233 = 94.1%

But once again, these are results from a clinical trial with all of the caveats I mentioned above that may affect the real-world results for you. And it doesn’t mean that if you have had the Moderna vaccine that each time you encounter COVID-19 in the wild that you have 5.9% chance of catching it.

If the vaccine gave you immunity, and you have a 94.1% chance of that, you are probably immune as long as other factors don’t come into play like a new variant. Once you’re immune, you’re immune (though immunity can wane over time).

Thus ends my Ted Talk on vaccine effectiveness. If you don’t mind contributing to this discussion through the Patreon link below, I will appreciate it.


  • Orenstein WA, Bernier RH, Dondero TJ, Hinman AR, Marks JS, Bart KJ, Sirotkin B. Field evaluation of vaccine efficacy. Bull World Health Organ. 1985;63(6):1055-68. PMID: 3879673; PMCID: PMC2536484.


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