Pfizer’s antiviral Covid19 drug (aka Paxlovid or Ritonavir) was shown to be 89% effective in reducing hospitalization and death when administered within three days of the onset of symptoms. As a reasonable person, you may now wonder whether it is safer to get a vaccine now, or that antiviral after you showed Covid19 symptoms. This is hard to tell with the information currently available, but here I show how to think at this problem and get closer to an answer.

In this post, I will generally talk about Adverse Effects referring to either hospitalization, death, long covid, or anything that you may be worried about, as well as the health risks caused by the treatment (vaccine or antiviral) itself. I will mostly be talking in qualitative terms, therefore a precise definition of adverse effects is not necessary.

TL;DR: Not taking the vaccine results in a higher chance of having a symptomatic infection. Therefore, in order to have an overall lower chance of adverse effects, the antiviral has to be more effective at preventing adverse effects in symptomatic individuals than the vaccine. Current data suggests to me that neither treatment is obviously better than the other and more detailed studies are needed.

The antiviral case

The main advantage of antiviral drugs is that you only take them when you show symptoms. We can show this graphically as follows:

Where events of interest are circled, and arrows indicate what directly influences what. For example, the arrow between Symptoms and Antiviral indicates that your choice of taking the antiviral depends on the occurrence of symptoms, because you would take the antiviral only in presence of symptoms. Arrows indicate direct dependencies, that is why there is no arrow from Infection to Antiviral: you need to know that you are infected (through symptoms) before you decide to take the antiviral. You would not realize that you are infected if you did not show symptoms, hence you would not take the antiviral. Adverse Effect is connected to everything to indicate that they may arise as a result of the symptoms, because of an asymptomatic infection (e.g., months or years later), or because of the antiviral treatment itself.

If we now use $p(A|B,C)$ to indicate the probability of an event $A$ occurring given that events $B$ and $C$ have also occurred, we can compute the probability of having Adverse Effects as follows:

\[\begin{align} p(\text{Adverse Effects}) &=p(\text{Adverse Effects}\vert \text{Antiviral},\text{Symptoms},\text{Infection}) \\ &\quad\times p(\text{Antiviral}\vert \text{Symptoms}) \times p(\text{Symptoms}\vert \text{Infection})\times p(\text{Infection}) \\ &= p(\text{Adverse Effects}\vert \text{Antiviral},\text{Symptoms},\text{Infection}) \\ &\quad\times p(\text{Symptoms}\vert \text{Infection})\times p(\text{Infection}) \end{align}\]

Where I used the fact that $p(\text{Antiviral}\vert \text{Symptoms})=1$ (i.e., you are certain to take the antiviral as soon as you observe symptoms).

The vaccination case

The case for vaccination is different because you would take the vaccine immediately and no matter what. The graphical model then looks like this:

Which means that the vaccine affects whether you get infected, whether you get symptoms, and whether you get adverse effects (because of Covid19, the vaccine itself, or both).

The probability of adverse effects under this model is:

\[\begin{align} p(\text{Adverse Effects}) &=p(\text{Adverse Effects}\vert \text{Vaccine},\text{Symptoms},\text{Infection}) \\ &\quad\times p(\text{Symptoms}\vert \text{Vaccine},\text{Infection}) \\ &\quad\times p(\text{Infection}\vert \text{Vaccine}) \end{align}\]

Where I omitted $p(\text{Vaccine})$ at the end because it equals one (i.e., you take the vaccine no matter what).

Comparing the risk from antivirals and vaccines

We can compare the probability of adverse effects under the two models by taking their ratio (I explicitly added No Vaccine and No Antiviral where appropriate):

\[\begin{align} &r= r(\text{Adverse Effects}\vert\text{Symptoms})\times r(\text{Symptoms}) \\ &\quad= \frac{ p(\text{Adverse Effects}\vert \text{Antiviral},\text{No Vaccine},\text{Symptoms},\text{Infection}) }{ p(\text{Adverse Effects}\vert \text{Vaccine},\text{No Antiviral},\text{Symptoms},\text{Infection}) } \\ &\qquad\times \frac{ p(\text{Symptoms}\vert \text{Infection},\text{No Vaccine})\times p(\text{Infection}\vert\text{No Vaccine}) }{ p(\text{Symptoms}\vert \text{Infection},\text{Vaccine}) \times p(\text{Infection}\vert \text{Vaccine}) } \end{align}\]

So that a ratio $r$ larger than one indicates a higher probability of adverse effects following the antiviral, while a ratio smaller than one indicates a higher risk because of the vaccine. I split the ratio in two parts: $r(\text{Adverse Effects}\vert\text{Symptoms})$ is the relative risk of having adverse effects given that symptoms occurred, while $r(\text{Symptoms})$ is the relative risk of developing symptoms in the first place.

Qualitatively speaking, there is now plenty of evidence showing that vaccines reduce the risk of symptomatic infection, making $r(\text{Symptoms})$ a positive number, say $x$. By choosing the antiviral and foregoing the vaccine, one is also accepting a $x$ higher chance of symptomatic infection. Therefore, to have a lower overall risk, the antiviral must be at least $x$ more effective than the vaccine at reducing adverse effects in symptomatic individuals, i.e. $r(\text{Adverse Effects}\vert\text{Symptoms})$ must be lower than $1/x$.

So what is safer?

The conceptual part ends here. From now on I will try to populate these formulas with numbers found online. A warning is thus obligatory:

These are unreliable results, because I used different sources that collected data with different methodologies, at different locations, at different time points, with different inclusion criteria and definitions of “symptomatic” and “adverse effects”. It is totally fine if you discredit these results as completely rubbish. Actually, you probably should. Nonetheless, I believe this is an useful example to get an idea of the quantities involved. Interpret at your own risk.

With this clear and out of the way, let’s go. The relative risk of developing symptoms is (and less controversial) to estimate, so let’s start from there. According to the CDC,1 vaccination reduced the risk of infection by 91 percent, i.e. there where 9 vaccinated infected for every 100 non-vaccinated infected. This means that:

\[\frac{ p(\text{Infection}\vert\text{No Vaccine}) }{ p(\text{Infection}\vert \text{Vaccine}) }=\frac{100}{100-91}\approx 11.1\]

The same report1 also claims that risk of symptoms after infection was 60% lower, meaning that:

\[\frac{ p(\text{Symptoms}\vert \text{Infection},\text{No Vaccine}) }{ p(\text{Symptoms}\vert \text{Infection},\text{Vaccine}) }=\frac{100}{100-60}=2.5\]

In other words, you are 11 times more likely to be infected without a vaccine, and a further 2.5 times more likely to show symptoms once infected. Overall, this means that as a non-vaccinated you are almost 28 times more likely to show Covid19 symptoms:

\[\begin{align} \hat{r}(\text{Symptoms})&= \frac{ p(\text{Symptoms}\vert \text{Infection},\text{No Vaccine}) }{ p(\text{Symptoms}\vert \text{Infection},\text{Vaccine}) }\times\frac{ p(\text{Infection}\vert\text{No Vaccine}) }{ p(\text{Infection}\vert \text{Vaccine}) } \\ &\approx 2.5\times 11.1=27.75 \end{align}\]

This is not entirely accurate due to the waning protection of vaccines over time, but I will use this number anyways. Feel free to use estimates that are more appropriate for the scenario you are considering.

Whether vaccines are safer in the end still depends on the relationship between the probabilities of adverse effects occurring to symptomatic people who took the vaccine and the antiviral The difficulty in analyzing this quantity stems for the precise definition of Adverse Effects, but, regardless of the exact definition, the antiviral is safer when it reduces the probability of adverse effects by 28 times or more compared to the vaccine! This is intuitive: if you are 28 times more likely to have symptoms without a vaccine, you must have at least 28 times less chance of developing adverse effects when treated with the antiviral compared to the vaccine, otherwise the vaccine is still safer in the end

It is unfortunately impossible to estimate this ratio by putting together data from the internet without committing gross mistakes due to all the biases hidden in the data, but we can still get an idea of what it would take for the antiviral to be overall as safe as the vaccine. If we take the same meaning of Adverse Effects as in the Pfizer’s press release,2 i.e., hospitalization or death, we get that the antiviral reduces the risk by 89%. This was estimated based on the outcomes for symptomatic patients which got either a placebo or the antiviral, meaning that:

\[\frac{ p(\text{Adverse Effects}\vert \text{No Antiviral},\text{No Vaccine},\text{Symptoms},\text{Infection}) }{ p(\text{Adverse Effects}\vert \text{Antiviral},\text{No Vaccine},\text{Symptoms},\text{Infection}) }=\frac{100}{100-89}\approx 9.1\]

Let us now assume that vaccine and antiviral have the same overall risk of causing adverse effects; given what we computed above, we must have

\[\frac{ p(\text{Adverse Effects}\vert \text{Antiviral},\text{No Vaccine},\text{Symptoms},\text{Infection}) }{ p(\text{Adverse Effects}\vert \text{Vaccine},\text{No Antiviral},\text{Symptoms},\text{Infection}) }\approx \frac{1}{28}\]

We can now use these two equalities to find how the risk of adverse effects for the vaccinated would compare with the risk of adverse effects for the non-vaccinated, assuming the antiviral and vaccine

\[%\frac{ %p(\text{Adverse Effects}\vert \text{No Vaccine},\text{No Antiviral},\text{Symptoms},\text{Infection}) %}{ %p(\text{Adverse Effects}\vert \text{Vaccine},\text{No Antiviral},\text{Symptoms},\text{Infection}) %} \tilde{r}(\text{Adverse Effects}\vert\text{Symptoms}) \approx \frac{9.1}{28} \approx 0.325 \approx \frac{1}{3.08}\]

In other words, for the antiviral to be overall safer than the vaccine, adverse effects in vaccinated symptomatic individuals must be at least three times more likely than adverse effects in non-vaccinated symptomatic individuals. This is not as big of a stretch as it sounds like: while vaccination prevents most cases of Covid19, showing symptoms when being vaccinated implies having a relatively more severe disease, while most symptomatic cases among the non-vaccinated are mild cases that would be eliminated by the vaccine.

A rough (and unreliable) estimation

Things get really sketchy here, as I will estimate the relative risk of adverse effects conditioned on symptomatic infection using data from two sources of information that were collected in wildly different ways. While the previous section was somewhat reliable, you have to be very careful here.

We may get an idea of this relative risk by looking at a report3 by Public Health England, specifically at Table 5. We can estimate the probability of adverse effects in symptomatic individuals by dividing the number of deaths by the number of cases with an emergency care visit. The latter quantity is reported twice, by including and excluding the cases where the positive test was taken in the same date as the visit to emergency care. The rationale for excluding these cases is that they probably occurred during routine testing for other conditions, i.e. when Covid19 was not the primary cause of the visit.

We shall include these cases in our calculations, and assume that the non-vaccinated in this group would take the antiviral. We shall also focus on patients younger than 50 years old to be consistent with the data we used in our estimation of the risk of symptomatic infection based on the report by the CDC.1

The probability of adverse effects following symptoms in the vaccinated is thus:

\[p(\text{Adverse Effects}\vert \text{Vaccine},\text{No Antiviral},\text{Symptoms},\text{Infection}) =\frac{48}{3162}\approx 1.52 \%\]

Where I only considered fully vaccinated patients, i.e. at least 14 days from the second dose. For the un-vaccinated we have:

\[p(\text{Adverse Effects}\vert \text{No Vaccine},\text{No Antiviral},\text{Symptoms},\text{Infection}) =\frac{132}{14860}\approx 0.89 \%\]

Showing a lower chance of adverse effects in the non-vaccinated due to the prevalence of milder cases, as explained above. The probabilities of adverse effects in vaccinated and non-vaccinated are thus related as follows:

\[\frac{ p(\text{Adverse Effects}\vert \text{No Vaccine},\text{No Antiviral},\text{Symptoms},\text{Infection}) }{ p(\text{Adverse Effects}\vert \text{Vaccine},\text{No Antiviral},\text{Symptoms},\text{Infection}) }\approx\frac{\frac{132}{14860}}{\frac{48}{3162}}\approx 0.59 \approx\frac{1}{1.71}\]

And, finally, we can find the original ratio we set out to estimate:

\[%\frac{ %p(\text{Adverse Effects}\vert \text{Antiviral},\text{No Vaccine},\text{Symptoms},\text{Infection}) %}{ %p(\text{Adverse Effects}\vert \text{Vaccine},\text{No Antiviral},\text{Symptoms},\text{Infection}) %} \hat{r}(\text{Adverse Effects}\vert\text{Symptoms}) \approx \frac{0.59}{9.1}\approx 0.065 \approx\frac{1}{15.4}\]

Thus, the antiviral reduces the probability of adverse effects in symptomatic individuals by 15 times compared to vaccination, a rather large gap from the 28 times required to offset the increased risk of Covid19 infection of the non-vaccinated. The end result is that refusing to be vaccinated now and taking an antiviral in case of symptomatic infection gives a 80% higher probability of adverse effects:

\[\hat{r}=\hat{r}(\text{Adverse Effects}\vert\text{Symptoms})\times \hat{r}(\text{Symptoms}) \approx\frac{1}{15.4}\times 27.75\approx 180.2\%\]

Again, this is a dangerous and potentially reckless use of the published data, and the result is far from properly answering the original question. Only the gold standard of clinical research, a randomized, double-blind trial with enough participants could provide a reliable answer. You have been warned.