What does the efficacy of a vaccine mean, i.e. what does it model?

Question

I am aware that the efficacy of a vaccine is calculated with $${\displaystyle VE={\frac {ARU-ARV}{ARU}}\times 100\%,} \text{with}\\ \text{VE=Vaccine efficacy}\\ \text {ARU= Attack rate of unvaccinated people}\\ \text {ARV= Attack rate of vaccinated people}$$ but besides echoing the definition of efficacy, I cannot explain even to myself what does it model. Initially I thought that the efficacy is the probability of you not getting the disease in question. However a close look at the formula reveals that efficacy cannot be a mathematical probability, since it can be a negative value: to my understanding during the pandemic negative efficacies have sometimes been reported regarding proposed vaccines. So what does efficacy model at an individual level, and what, if any, connection efficacy has to probabilities and probability theory?

Answer 1

Suppose you have two near-identical populations of $N$ people. The only difference is that one of those populations gets vaccinated and the other population does not. Also suppose that $n_U$ people from the unvaccinated population get the disease while $n_V$ people from the vaccinated population get the disease. Then

$$ n_U=ARU \cdot N\\ n_V=ARV \cdot N $$

and so

$$ n_V=\frac{ARV}{ARU} n_U=(1-VE) \cdot n_U \, . $$

That is, you would expect that only $(1-VE) \cdot n_U$ people from the vaccinated population would get the disease, meaning that on average the vaccine reduces the number of people from the vaccinated population who got the disease by $VE \cdot n_U$.

Equivalently, suppose you either get vaccinated or don't. In either case, you then go about your life for a while in such a way that the probability of you getting the disease would have been $p$ had you not been vaccinated. Imagine a large collection of $N$ alternate universes where you got vaccinated, and $N$ where you didn't, and construct populations like the above out of these universes.

You expect to get the disease in $n_U=N \cdot p$ of the unvaccinated universes, by assumption. So, as above, you'd expect to get the disease in $(1-VE) \cdot N \cdot p$ vaccinated universes, meaning that the probability of vaccinated-you getting the disease is $(1-VE) \cdot p$.

In other words, getting vaccinated reduces the absolute probability that you get the disease by $VE \cdot p$: roughly speaking, the probability of you not getting the disease as the result of being vaccinated is $VE \cdot p$. More carefully, this is the probability that you don't get the disease as a result of being vaccinated, minus the probability that you do get the disease when you otherwise wouldn't have as a result of being vaccinated.

In particular, if the "vaccine" somehow makes you more susceptible to the disease, then $VE$ will be negative.