Effect of Vaccinations on R_0 – With graphs

This is an expansion of my last post, with graphs. For sharper graphs, see https://www.math.wustl.edu/wp/mccarthy/public_papers/EffectOfRequiringVaccinationGraphs.pdf

The CDC estimates that the three vaccines currently used in the US have an efficiency of between 74% (Johnson and Johnson) and 89% (Pfizer-BioNTech and Moderna) at preventing infection. They also say that preliminary data from Israel suggest that vaccinated people who develop COVID-19 have ΒΌ the viral load of unvaccinated people who get the disease.

So I shall make two assumptions going forward:

  1. Vaccination reduces the chance of becoming infected from a given exposure by 80%.
  2. Vaccinated people who are infected pose 25% the risk of infecting another person during an interaction than if they had not been vaccinated.

Neither of these assumptions is completely accurate, but they are close enough that one can draw qualitative conclusions. Let us estimate the expected number of infections that would arise from an event, like attending a class, sharing a table for a meal, or even going to a party, in which the only change from a baseline of 100% is how many people are vaccinated.

  1. You are vaccinated, nobody else is. Your chance of getting infected is reduced to 20%.
  1. You are not vaccinated, everybody else is. Your probability of getting infected goes down for two reasons: the expected number of interactions with an infected person is only 20% of what it would otherwise have been, and during any such interaction with someone who is infected, you have 25% the chance of being infected (from (ii) above). So your probability of becoming infected will decrease to 25% of 20%, which is 5%.
  1. Everybody is vaccinated. Then each person is doubly protected: their exposure is 5% what it would have been, and if exposed, they have only 20% the chance of becoming infected. So the expected number of infections is reduced to 1%.
  1. Most but not all people are vaccinated. For illustration, suppose 75% of people are vaccinated. Then virion emission is reduced to (.05 x .75 + .25) = 28.75% of what it was, and the probability that a random person becomes infected is reduced by this number times (.2 x .75 + .25) = 40%. So the overall reduction is to 11.5%. (If the fraction of the population vaccinated is p, then the risk is reduced to (1 – .95 p)(1 -.8 p).)

There are several conclusions to be drawn.

  1. Like mask-wearing, being vaccinated benefits both the recipient, and the people they interact with. But the latter benefit is much greater.
  • There is a big difference between most people being vaccinated and everybody being vaccinated, because of the quadratic nature of the benefit in (IV). Universal vaccination is more than 10 times better than 75% vaccination.
  • This analysis understates the benefit of universal vaccination, because with reduced transmission,  the fraction of the population that becomes infected will fall, further reducing risk. This will be particularly important for students, who largely interact with other students; as the semester goes on their risk of being exposed to another student who is infectious will decrease.

Section 2. Charts

First, let us consider a closed community. Then vaccination will reduce R0, the expected number of new cases caused by each infected person. How much it reduces it depends on the efficacy of the vaccine, E, which we set to 80% above; and how much the vaccine reduces the infectiousness of somebody who is vaccinated but still catches the disease. We shall denote this number by T; in Section 1 we assumed T = 25%. For higher values of levels of  p, and lower values of T, we get that R0   is reduced from its base value, which the CDC currently estimates to be 2.5.

Another possible reduction in R0 comes from considering the proportion of the population q that has recovered from infection, but is not vaccinated. We shall assume that they are as protected from catching COVID again as the vaccinated population (so E = 80% for them) but if they do get re-infected, they are just as infectious as a non-vaccinated person (so T = 1 for them).

Finally, let us consider a small community, like a University, within a larger group, the surrounding city. Let us assume that the infection level in the city is constant, so the number of community members being exposed to infections is constant. What is the effect of a vaccination level of p in the small community? The number of primary infections will decrease by p(1-E).

The number of secondary infections (people in the small community who are infected by this primary group) will decrease further.

For clarity, we will repeat the last graph just for the value T=0.25

Section 3: What if Variants are worse?

One scenario the CDC considers is if the virus became more infectious, with an unvaccinated R0 of 4.0.

What percentage of the population would need to be vaccinated to make the pandemic die out (which would occur once R0  is significantly smaller than 1). For one illustration, we assume that T =0.5.

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©2024 John McCarthy, Ph.D.