Vaccine doses ... again

Welcome to Plugging the Gap (my email newsletter about Covid-19 and its economics). In case you don’t know me, I’m an economist and professor at the University of Toronto. I have written lots of books including, most recently, on Covid-19. You can follow me on Twitter (@joshgans) or subscribe to this email newsletter here.

UPDATE (5th December): In an earlier version of this post, I made a mathematics error that led me to conclude that “I believe that debate has neglected some basic logistics of supply constraint which tips the balance of the debate in favour of two doses at their initial levels of 3-4 weeks apart rather than extending that period to 12 weeks or beyond.” That conclusion no longer stands and so I have amended my post below. In particular, the changes that I suggested to model the supply issues, do not change the basic argument of the “first doses first” people. My apologies for the confusion.

Today’s post revisits the “first doses first” debate that I anticipated back in November, which has now come to the fore. Here I want to outline the current argument as I see it.

The new push has its origins a few weeks ago. Then, despite previously strong recommendations from experts that two doses were necessary, cracks started to appear. This culminated in calls for more trials to determine whether we could get away with one dose. Here is Zeynep Tufecki and Michael Mina:

Crucially, though, we should begin immediate single-dose trials, recruiting volunteers from low-risk populations who are first in line for the vaccinations. For example, among health care workers protective equipment works, rates of infection among this group have fallen sharply and severe disease is much more rare. Younger essential workers without risk factors are less likely to be severely affected if they are exposed since this disease’s impact rises steeply with age. Just as tens of thousands of people volunteered for the earlier vaccine trials, many may well volunteer to test a placebo against a second dose, allowing us to quickly ascertain questions of durability and effectiveness of the single dose.

As it turns out, the world isn’t going to wait. All three of the vaccines approved by the UK, have shown solid efficacy from just a single dose.

Previously, many authorities would only distribute the first dose to a person if the second dose was available and in reserve. The idea here is that they did not want to risk a delay in that second dose. As I wrote earlier, there was less concern about a delay in the second dose than there would be in, say, too short a time between doses.

News, this week, that Ontario had decided to relax the reserve requirement. That is, they would give someone a first dose without requiring the second dose to be held in reserve. That means that dose could be given to someone else. There would be a risk that when 3-4 weeks had passed there was not a second dose but they have decided to take that risk. Interestingly, I do wonder what occurs if there is a shortage of doses. Will they prioritise first or second doses? At the moment, I suspect it is the second dose.

The UK has now gone further. They will provide the second dose 12 rather than 3-4 weeks after the first. This allows them to greatly increase the initial number of first doses given. However, if supply constraints are hit, then it will not mean much in a steady-state although every person may well be vaccinated with their first dose almost two months before they might otherwise have been. This potential turns out to play an important role.

Pfizer’s Response

One thing that the proponents of “first doses first” have right is that advocates of “sticking with the plan” have not been doing a proper risk assessment.

The vaccine manufacturers are not happy about plan deviations.

Pfizer/BioNTech said that their vaccine was not designed to be used in two shots 12 weeks apart. In a statement, the firms said there was no evidence the first shot continued to work beyond three weeks.

“Data from the phase 3 study demonstrated that, although partial protection from the vaccine appears to begin as early as 12 days after the first dose, two doses of the vaccine are required to provide the maximum protection against the disease, a vaccine efficacy of 95%. There are no data to demonstrate that protection after the first dose is sustained after 21 days,” they said.

Let me lift the passage that most reflects are strange bias: “there was no evidence the first shot continued to work beyond three weeks.” This is true. No one has designed a trial to solicit this information and it will take time to do so especially since the trial has time baked in. But “no evidence” is a red herring. The issue is what expert beliefs are.

Let me point out two things. First, they have to say that. Liability laws and litigation risk being what they are, Pfizer cannot advocate or even look like recommending something other than the use for which they generated evidence for. Moreover, there is an economic conflict here. Pfizer makes money as a mark-up on each dose. If this results in fewer doses being sold, I can understand them not liking that. (Indeed, I think the UK government needs to mitigate this risk in their purchase guarantees).

Second, yes, we don’t know if, after three weeks plus one day, the vaccine suddenly stops working. But all of our priors suggest that it will last longer and that three to four months is far from an ‘out there’ hope. Moreover, we DO KNOW that if someone gets no shots, they are vulnerable to Covid-19. In other words, every dose you have as a second dose rather than a first dose is a trade-off between the risk one person doesn’t have long enough immunity versus the risk the other person contracts Covid-19 while waiting to be vaccinated. You would have to be pretty pessimistic about the ability of one dose to give immunity for a good period of time to require EVERYONE to get two doses.

The Maths

The advocates of “first doses first” argue that we need to do the maths. Here is their argument. Suppose that the efficacy (3 months out) if a single dose is x(1) and of two doses is x(2) with x(1) < x(2). If we have y doses available per month, then we can either give y/2 people full courses or y people one course each month. If we want to maximise the total number of people protected at the end of three months then we will insist on two doses if:

x(2)3y/2 > x(1)3y or x(2) > 2x(1).

This makes things simple. We know that for the Pfizer and Moderna vaccines, x(2) = 0.95 (maybe a little more). Thus, we will choose to give two doses if x(1) < 0.95/2 = 0.475.

So what is x(1)? We actually do not know. Pfizer’s own study suggests a single dose may be as high as 92% effective for three weeks. Because no one received only one dose after that, we don’t know what happens beyond that point. But if it was the case that the vaccine fully wore off — and was 0% effective by 3 months, then x(1) would be too low. In other words, x(1) is an expectation and different experts have different options about the underlying probability distribution.

This is the calculation most people are doing but it has a problem. The calculation makes sense if you are choosing to give ‘first doses first’ until you have inoculated everyone in a population. But that is not what is currently being proposed. Instead, what is being proposed is that instead of three weeks between doses, it is three months.

To see why this matters, suppose that, in the first three months, with two doses, you can innoculate, 3y/2 people or y/2 people per month on-going. But, with just one dose, you can innoculate 3y people in the first three months and then 0 in the three months thereafter. The reason is that in month 4, you have to give those who had a first dose in month 1, their second dose. That sucks up available capacity. In other words, all of the savings are right at the start. There is no relief on supply capacity thereafter if that capacity is fixed. At the end of six months, with either plan, you have innoculated the same number of people. In the first doses first plan, you have shifted half the people earlier while if you stick to the plan, you have ‘lost’ y/2 people in the first month in the sense that those doses are wasted.

Maybe that is still a good idea. Well, suppose that, if a vaccine is ineffective or you are not vaccinated, your probability of contracting Covid-19 in a given month is p. Then, a person who receives one dose has a probability of contracting Covid-19 while they only have one dose of p(1 - x(1)) which falls to p(1 - x(2)) when they receive their second dose. (Yes, I am aggregating up to a month which is a shortcut.) To make things easier for myself I am going to set x(2) = 1 and just call x(1), x. However, we also need another variable which takes into account the argument that efficacy of a single dose after 3 months may be less than that after one month. Let me make an extreme assumption to make my life easier. Suppose that efficacy after one month remains at 1, after two months it is X and after 3 months it is x < X.

Under the “stick with the plan” scenario, here are the number of protected people from a population of 3y which is the number of people who can receive the vaccine at the end of 6 months.

6y/2 + 5y/2 + 4y/2 + 3y/2 + 2y/2 + y/2 = y10.5

The first six terms are the number of people/months vaccinated (21(y/2)). If efficacy during one month was not 1, there would be an additional risk. Moving to a three month gap doesn’t change that though. If you conduct “first doses first” over those six months, you get:

6y + 5y + 4y - (1-X)3y - (1-x)3y = y(15 - 3(1 - X + 1 - x))

Notice that the total number of person-months protected is 15y (rather than 10.5y in the initial plan) but the exposure risk is now higher (3(2 - X - x))) for those receiving the vaccine.

Comparing these two, you want to “stick with the plan” if

y10.5 > y(15 - 3(1 - X + 1 - x)) or x + X < 1/2

In other words, a single dose has to average 50% effectiveness for three months to justify moving away from “sticking to the plan.” Importantly, if you believe there is a quick drop off in efficacy, you want to stick with the plan. If you believe there is a slow drop off in efficacy, you want first doses first.

Nonetheless, unless something changes, all of the impact of this is in shifting around innoculations over a three month period. That said, you could consider a longer gap but that would impact both on your ability to shift things around but also cause a potential increase in the risk of drop-off.

Some caveats

There are three reasons why you might want to shift doses earlier that are not taken into account by the calculations thusfar.

First, capacity may be growing. In that case, by giving people doses quickly you may be able to move back to the plan after temporarily dealing with the relatively low number of doses early on. I believe (although I have not checked) that this may tilt you back towards first doses first. (That is, if there is a risk of shortages down the track, that tips you back in the other direction).

Second, there may be higher prevalence (p) during the next three months than afterwards. In that case, front-loading protection becomes relatively more important.

Finally, the above calculations look only at protection from Covid-19 and not the vaccine’s role in spread. If a single dose more widely distributed could reduce the reproduction rate that would favour first doses first. That said, we still don’t know how well two doses achieves this, let alone a single dose. In addition, the current prioritisation favours people who are at risk but, for the most part, stationary. If you wanted to reduce infection spread, you would target non-stationary populations.


In the end, I think more work is needed to really sort this out. I think there are some complexities here although the basic trade-offs highlighted by those advocating first doses first appear robust.

[UPDATE: The original emailed version had a mistake (noticed by Joshua Blake on Twitter) and originally overstated the threshold for x as less 93%. It has been corrected for less than 50% for stick with the plan to be optimal.]

What did I miss?