There are two false dichotomies that are easy to steer clear of in the debate about third doses of the Pfizer and Moderna vaccines.

1. The US government can keep pushing those who don’t yet have two doses to get them even as it offers third doses to those who already have two.

2. The US government can pay manufacturers to produce enough doses to cover all domestic demand and to provide protection for people throughout the world.

But even if we can avoid needless arguments about “which” when we can have “both”, a technical confusion about the information provided by the different endpoints researchers use to assess the protection provided by a vaccine could still prevent a consensus to move forward.

The natural first reaction is to doubt that the endpoint matters. “Who cares whether you look at how many people get infected, how many get symptoms, or how many develop severe disease? Sick is sick, right?” In fact, measurements against different endpoints reveal different types of information. We care the most about the performance of a vaccine against the worst endpoint, severe disease, but it is easier to detect a change in the protection that it provides against infection. For example, an easy to spot reduction by 20 percentage points in the effectiveness of a vaccine against infection might translate into a 4 percentage point reduction in the vaccine’s effectiveness against severe disease that gets lost in the noise.

It makes sense track breakthrough infections because of the harm they can cause even in cases that do not end in severe disease. But the real reason to watch for a fall in effectiveness against infection is because it can be the “canary in the coal mine” that provides the first warning of a serious erosion in the effectiveness of a vaccine against all outcomes.

A reaction along the lines of “Who cares whether the probability of breakthrough infections is increasing; we only care about preventing severe disease” amounts to saying “Who cares that the canary died; we only care about saving the lives of miners.”

The clarity of mathematics

The distinction between different endpoints is hard even to express using natural language, let alone to explain. The already verbose statements “the effectiveness of the vaccine against the endpoint of infection” and “the effectiveness against the endpoint of having symptoms” explain nothing about why the difference between the two measures might matter.

To understand endpoints, it helps to have in mind a model with different stages, and then to use the domain specific language of mathematics to describe the transitions between those stages.

The symbols of math can seem intimidating at first, but to understand them, all you have to do is read them out loud. For example, the symbol $$p_{I}$$ should be read aloud as “the probability that someone becomes infected.” The letter ||p|| suggests “probability”, the letter ||I||, “infected.”

This probability depends on a person’s vaccine status, which can be included as part of the symbol like so: $$p_{I}(vaxed = No) \ \ \text{or} \ \ p_{I}(vaxed = Yes).$$

These probabilities depend on common factors. When more people in the community are infected, the probability of infection increases proportionally for both those who are vaccinated and those who are not. The “relative risk of infection”, ||r_{I}||, takes the ratio of these two probabilities so that any common factors cancel out:

$$r_{I} = \frac{p_{I}(vaxed = Yes)}{p_{I}({vaxed = No)}}$$

What’s left is a pure indication of the effect of the vaccine. If the relative risk is less than one (in symbols, if ||r_{I} < 1||), vaccinations reduce the probability of infection.

For people who become infected, it makes sense to consider the probability that they move on to the stage of having symptoms. (The next section offers a conjecture about what might be going on in the body during these two stages.) The way to show this in symbols is to let ||S|| stand for “symptoms”, just as ||I|| stands for “infection”, and to use a vertical line “|| | ||” to mean “given that” or “conditional on”. Then one can read the symbol $$p_{S|I}$$ out loud as “the probability of symptoms given that someone is infected”; or equivalently, “the probability of symptoms conditional on infection.”

In parallel with the pairing of ||p_{I}|| and ||r_{I}||, the conditional relative risk of developing symptoms for someone who is infected can be defined as the ratio of the two conditional probabilities:

$$r_{S|I} = \frac{p_{S|I}(\text{vaxed = Yes})}{p_{S|I}(\text{vaxed = No)}}$$

If ||r_{S|I} < 1||, the vaccine reduces the probability that someone who is infected goes on to develop symptoms.

If the letter ||D|| represent severe disease (because the letter ||S|| is already taken), there is an analogous definition of the relative risk of severe disease, conditional on having symptoms:

$$r_{D|S} = \frac{p_{D|S}(\text{vaxed = Yes})}{p_{D|S}(\text{vaxed = No)}}.$$

If ||r_{D|S} < 1||, the vaccine reduces the probability that someone who has symptoms goes on to develop severe disease.

The underlying model of stages of disease

As noted in the beginning, the experience with the virus can vary across people. Before vaccines were available, the data showed that the probability that someone who is infected develops symptoms is less than ||70\%||. (For details, see here and here.) In symbols,

$$p_{S|I} \le 0.7.$$

This tells us something about the interaction between the human body and the virus. The body’s immune system can fail to prevent a colony of the virus from forming in the nasal passages, which means that this person will have a positive PCR test and can start shedding the virus. Yet the immune system may still prevent the virus in the nasal colony from spreading to other organ systems and causing symptoms.

For those who do go on to the stage of having symptoms, they do so after a lag. For example, recent estimates for the delta variant show that for someone who is already infecting others, it takes about ||2|| days for symptoms to develop (if they develop at all.)

Given this multi-stage evolution of disease, it follows that a vaccine could have different effects on the body’s lines of defense against a transition to the next stage.

• If a vaccine helps the body stop the virus from colonizing the nasal passages, then $$r_{I} < 1.$$

• To take an unrealistic case that illustrates the meaning of the symbols, if the vaccine does nothing to prevent the virus from colonizing the nasal passages but does help the body keep virus particles that leave the nasal colony from starting new infections in other organ systems, then $$ r_{S|I} < 1, \text{ but }r_{I} = 1.$$

• In another unrealistic but revealing case, if the vaccine does nothing to help stop the initial infection and does nothing to prevent this infection from spreading throughout the body and causing symptoms, but does help the immune system keep the total number of virus particles of the systemic infection from growing so large that the person develops severe disease, $$r_{D|S} < 1, \text{ but } r_{I} = r_{S|I} = 1.$$

Why an increase in the risk of infection will be largest

The symbols of mathematics are uniquely well suited to the task of conveying interactions and quantitative magnitudes. They can show how a change in one relative risk induces a change in another, and why some relative risks are bigger than others.

The unconditional probability of having symptoms can be written as the product: $$p_{S} = p_{S|I} \cdot p_{I}$$ This means that the unconditional relative risk of symptoms is the product of the two relative risks: $$r_{S} = r_{S|I} \cdot r_{I}$$ The unconditional relative risk of severe disease is the triple product: $$r_{D} = r_{D|S} \cdot r_{S|I} \cdot r_{I}$$

We would not expect a vaccine that has gone through clinical trials to be harmful in the sense that it increases the probability of a transition to a more serious stage in the disease. Formally, this means that we expect the conditional relative risks ||r_{S|I}|| and ||r_{D|S}|| to both be less than or equal to ||1||.

It then follows that the relative risk of the three endpoints, ||r_{I}||, ||r_{S} = r_{S|I} \cdot r_{I}||, and ||r_{D} = r_{D|S} \cdot r_{S|I} \cdot r_{I}||, will have magnitudes that can be ranked:

$$ r_{D} \le r_{S} \le r_{I}$$

To see what this means in a specific example, suppose that ||r_{D|S} = 0.5,|| ||r_{S|I} = 0.4,|| and ||r_{I} = 0.2.|| Then the unconditional relative risk of severe disease is $$r_{D} = r_{D|S} \cdot r_{S|I} \cdot r_{I} = 0.5 \cdot 0.4 \cdot 0.2 = 0.04,$$ which corresponds to an effectiveness against severe disease of ||96\%.||

Suppose next that a new variant takes over and the relative risk of infection doubles from ||r_{I} = 0.2|| to ||r^\prime_{I} = 0.4.|| As in the previous post, the effectiveness of the vaccnie against severe disease seems to suffer only a modest reduction:

$$ e_{D} = 1 - r_{D} = 96\% \longrightarrow e^\prime_{D} = 1 - r^\prime_{D} = 1 - 0.5 \cdot 0.4 \cdot 0.4 = 92\%$$

Its effectiveness against infection falls more sharply:

$$ e_{I} = 1 - r_{I} = 80\% \longrightarrow e^\prime_{I} = 1 - r^\prime_{I} = 60\%$$

Against the noisy background of chance events, it will be easier to detect a statistically significant large change in ||e_{I}|| (or equivalently in ||r_{I}||) than the smaller change in ||e_{D}|| (or ||r_{D}||).

After the new variant emerges, the new value for ||e^\prime_{I}|| could be so low that people who want to avoid being infected will want a third dose of the vaccine. They might want it because a reduction in ||r_{I}|| will reduce ||r_{D}||. For something as bad as severe disease, low is good, but even lower is even better.

They may also want to reduce ||r_{I}|| to reduce the risk of symptoms that can cause a large reduction in the quality of life such as permanent damage to the senses of smell and taste.

Finally, they may want additional protection from the uncertain but worrisome possibility that when the virus infects the heart and lungs it could cause subtle but lasting damage without producing the urgent respiratory distress of severe disease. For someone concerned about this risk, uncertainty is a spur to action, not a reason to hesitate.