Impact of population mixing between vaccinated and unvaccinated subpopulations on infectious disease dynamics: implications for SARS-CoV-2 transmission

We appreciate the opportunity to respond to the concerns voiced by Doidge and Lungu with respect to our paper. We have responded to Doidge’s comments on our model structure and parameters in an earlier letter. We reject Doidge’s comment that our paper has “potential to…foster social division and misplaced anger and blame”. We think that the stoking of anger around use of remarkably effective vaccines during a public health emergency is being done, quite intentionally, by others. Indeed, as federal agencies have noted, the use of vaccination as a wedge to foment social division has been identified as a key goal of threat actors, likely including hostile states (1).

Our paper, by contrast, is using the utilitarian (i.e., “greatest good for greatest number”) lens commonly applied in public health to identify policies that protect the health and wellbeing of Canadians. Balancing the rights of individuals with the rights of the wider community is a key tension in public health practice, as Doidge and co-authors would likely know. Canadian public health statutes do contain provisions to limit the freedoms of individuals when this is necessary for protection of the wider community from virulent communicable diseases. Identification of sources of risk does not imply stigmatization or moral condemnation; we are not morally condemning or stigmatizing an impaired driver when we point out that they are a danger to others on the road. Identification of sources of risk does, however, allow us to protect the populations we serve.

We disagree with the apparent suggestion by Lungu that, because we used a modeling approach to answer a policy question, we should have noted the study design in our paper's title. Mathematical models represent an important tool in the scientific toolbox; for example, mathematical models deserve credit for the non-collapse of our skyscrapers and bridges, as well as our ability to explore space. Readers can find the phrase “we constructed a simple susceptible–infectious–recovered compartmental model of a respiratory infectious disease with 2 connected subpopulations” in the sixth line of our abstract.

We appreciate the opportunity to respond to correspondence related to our paper. Several correspondents such as Schabas, Hsiang, and Aumond-Beaupre, suggest that we have used unduly favorable estimates of vaccine efficacy in the face of Omicron. In fact, the best and most recent Canadian data (from Carazo et al.) are remarkably consistent with our base case parameter values for vaccine efficacy in the face of Omicron infection (VE 70-80%). Furthermore, our findings don’t change in the face of wide-ranging sensitivity analyses on vaccine efficacy.

While it is true, as Doidge notes in his letter, that it would be possible to create scenarios where vaccinated individuals were less protected than unvaccinated individuals, this would necessitate the use of nonsense parameters without relation to real-world data, and would also (for example, in the scenario where there is far more widespread immunity among the unvaccinated than among the vaccinated) require that we treat the unvaccinated as though they have already come through an epidemic (while ignoring the costs and risks of their having done so), while treating the vaccinated as though their epidemic is yet to come. This too, we believe, would be nonsensical.

In other correspondence, Schanzer and Strobel both correctly point out that our model does not include waning immunity, which appears to be an important limitation of mRNA vaccine-derived immunity, as well as immunity conferred by prior infection. Importantly, boosted mRNA vaccination results in substantially higher antibody titers than are seen following natural infection. Based on the work of Townsend et al., it can be inferred that the durability of immune protection following vaccination will be greater than that seen with infection, and this observation can be used to shape post-pandemic vaccine policies. We hope to provide a follow-up analysis on this point in the weeks ahead.

We do agree that our model, combined with the work of Carazo, Chu, Townsend and others, strongly suggests that Canada needs to upwardly revise the number of doses that constitute full vaccination; for mRNA vaccines, “fully vaccinated” should be defined as receipt of at least a 3-dose vaccine series.

We are pleased to see that our paper stimulated so much thought and reflection. The criticisms on this page can be divided into three main categories: (i) our interpretation of our model results is incorrect; (ii) our choice of parameters is incorrect; and (iii) our model is stoking hatred against those who choose to remain unvaccinated against SARS-CoV-2. We respond to the first criticism below, and others in subsequent letters, due to a character limit on letter responses.

Some of the letters, despite their vehemence, contain errors which may suggest lack of familiarity with infectious disease modeling. Rancourt, for example, states that "the model as presented is blind as to whether the “contacts” in the normalizing denominator of Ψ are infectious or benign, irrespective of vaccination status". In fact, the disproportionate infectivity of unvaccinated contacts can be inferred from our results (as well as obtained directly from the model, which we have made publicly accessible in Microsoft Excel format). It is the higher average infection prevalence among the unvaccinated that is the driver of the disproportionate burden of infection derived from contact with unvaccinated individuals. Note that we treat contacts with vaccinated and unvaccinated infectives as equally infectious in our model, which has the effect of biasing our results against vaccination. Recent data from Puhach et al. demonstrate reduced infectivity among fully vaccinated individuals, relative to unvaccinated individuals, when infected. Rancourt also suggests that we should not normalize the contribution to infection for contact numbers; in doing so, he reinforces the key take-home point of our model: decreasing mixing between vaccinated and unvaccinated individuals does indeed reduce risk to the vaccinated, as he states. This is both due to less direct infection by the unvaccinated themselves, and also because reduced mixing with unvaccinated individuals potentiates indirect (“herd”) effects among the vaccinated. We thank Rancourt for re-emphasizing the basic rationale and justification for the use of vaccine passports and mandates.

I thank Dr. Lynch for her feedback. Dr. Lynch is correct that a substantial fraction of hospitalizations are expected to occur among the vaccinated when the overwhelming majority of the population is vaccinated, as is the case in Canada. Conflation of risk (far lower among the vaccinated) with absolute burden of hospitalizations is referred to as the "base rate fallacy". I would encourage Dr. Lynch to learn more about this concept by accessing the second suggested reference below; as I would assume she has read our paper, I'd encourage her to revisit Figure 1, where this concept is also illustrated.

Indeed, her point, that risk persists in vaccinated individuals despite their attempts to protect themselves, and that that risk derives disproportionately from interaction with unvaccinated individuals, is the basic finding of our paper.

Dr. Lynch suggests that my declared service on one AstraZeneca and one Pfizer vaccine advisory board since the start of the pandemic somehow invalidates our model findings. I would point out that all journals require authors to disclose any potential or perceived conflict, which we have done. As a recognized expert in, and enthusiastic supporter of, vaccination, I am invited to serve on such boards, as are many other Canadian vaccine experts. Attempting to discredit a paper by attacking an author is termed "argumentum ad hominem", and I provide information on that logical fallacy in the resources below.

While David Fisman et al point out that, historically, "behaviours that create health risks for the community as well as individuals have been the subject of public health regulation,"(1) they appear to ignore the reality that, historically, minority groups have been scapegoated as carriers of new diseases and often harmed as a result.(2)

It is well established that particular demographic groups, in Canada and elsewhere, are disproportionately vaccine-hesitant. According to a study by the Ontario Covid-19 Science Table, "Vaccine uptake in Ontario has been lower among racialized communities, especially in areas with the highest proportions of refugees, recent immigrants, and recent OHIP registrants — communities that have also been the most impacted by SARS-CoV-2 infections." (3) There are good historical reasons for such hesitancy, which is not simply a product of demands for the "rights of the unvaccinated," as referenced in the article, or of "misinformation."

Certain demographic groups - including racialized and low-income Canadians, as well as recent immigrants - have disproportionately borne the burden of Covid-19.(4) As communities which tend to be more vaccine-hesitant, they have suffered further from coercive measures such as vaccine mandates and passports. These interventions have disproportionately benefited higher-income white individuals while disproportionately excluding lower-income, racialized Canadians, especially Black and indigenous people.

Other responses here have more than adequately identified the flaws in the model. Inflated estimates of vaccine effectiveness, complete disregard of waning immunity, and underestimation of natural immunity among the unvaccinated are all used to support the claim that the minority is dangerous to the majority and that "mixing" with such disease-carriers is risky.

While there is considerable historical precedent for such scapegoating, the CMAJ should reconsider whether it is prudent and ethical to publish models which support this type of discrimination. While retraction of the paper would be welcome, it would be even more helpful if the CMAJ would publish a paper which re-runs the model using realistic inputs and assumptions.

Fisman et al conclude that "considerations around equity and justice for people who do choose to be vaccinated, as well as those who choose not to be, need to be considered in the formulation of vaccination policy." Surely an equity-and-justice approach requires taking special care to not use flawed assumptions to justify coercive and exclusive measures against a disproportionately-marginalized minority.

There are significant problems with this study.

Starting with the title: The title implies that there was actual mixing of populations in the study, this was not the case, this was a math modeling study not a clinical study.

There were inaccurate assumptions with this model rendering the conclusions inaccurate and not consistent with real world experience. The most glaring assumption is that the baseline immunity in unvaccinated people is only 0.2. There is evidence that even before the delta and omicron waves baseline immunity in a BC population was much higher than this (1). Given the current situation and that many have recovered from Covid infections and are now naturally immune, the immunity in the unvaccinated population is likely to be greater than 80% as a conservative estimate. Another assumption that is clearly inaccurate is that vaccine effectiveness is 80% (0.8 in table 1) this refers to relative risk rather than absolute risk and without adequate consideration of known and considerable waning immunity. Re-calculation with correction of these assumptions should be done.

Real world experience using the most recent Ontario statistics demonstrates that it is the vaccinated who are at most risk of having or being hospitalized with Covid (2).

There was also significant conflict of interest with the lead author having served on numerous advisory boards for the pharmaceutical industry, including those involved in the Covid response. This conflict was evident from the start including the first paragraph which implies the Covid vaccines disrupted transmission. This was not examined in the clinical trials and real-world experience has demonstrated that this is not the case. There was also a pejorative, inaccurate description of the population of people who have not been vaccinated, many of whom have serious medical reasons for this.

This article has received widespread media attention and is spreading inaccurate information with potential to further polarize Canadians and others. Consideration should be given to retraction with an apology.

We are writing to address the article written by Fisman et al. regarding the mixing of vaccinated and unvaccinated populations. We feel that the study has significant limitations that merit discussion, as they substantially hamper the validity of the study’s conclusions to suggest policy directions, most of which are justified neither by the study’s methodology or results.

The first limitation is the choice of an endpoint of minimal clinical interest. Ultimately, COVID-19 infection is inevitable for almost all people, independent of vaccine status. Waning immunity means that most infections currently occur in the vaccinated, with equivalent infection rates in vaccinated and unvaccinated people.1 If the expected infection rate of the population is ultimately close to 100% vaccinated or not, the vaccine status of the source of infection becomes irrelevant.

Secondly, the model erroneously assumes a high level of permanent immunity following vaccination. While COVID-19 vaccines retain excellent efficacy for preventing severe disease, they quickly wane for preventing infection.2 This is true of the primary series as well as booster doses. The model assumes an vaccine efficacy of 40-80%, which may have been true with previous variants, and immediately following vaccination. However, in the Omicron era, vaccine efficacy is substantially lower and wanes quickly. It would better be modeled between 0 and 30% for infection.2 Similarly, the team assumes low immunity in the unvaccinated population (20%), and low efficacy of infection for providing ongoing immunity. The estimates for infection in unvaccinated populations varies world-wide, but approaches 75% for North Americans.3 With protection against subsequent infection similar to vaccination, and relatively long lasting.4 Using numbers that are more realistic in the study model reverses the paper’s conclusions and better match the observed data.2 Readers can do so at https://figshare.com/search?q=10.6084%2Fm9.figshare.15189576.

A more nuanced but important point is the role of children who are disproportionately represented in the unvaccinated population, since many are ineligible for vaccination. Children are generally minor contributors to transmission when compared to adults,5 and are also more likely to have been previously infected.3 However, no correction is made to the model to control for this variable which overestimates the likelihood of transmission from unvaccinated populations.

Since models are inherently dependant on the accuracy of assumptions, the model with its current inputs and assumptions is unable to accurately predict or explain COVID-19 transmission in the current context and should therefore not be used as a vehicle to promote any specific policy. We also wish to point out that, as a model, it does not constitute proof that unvaccinated people pose a risk to vaccinated people, but rather conjecture on what might happen should the parameters of the model be correct.

Given the serious flaws of this modelling study, it is unfortunate that it has received wide media attention. The media narratives being sold based on this study are troubling, because they do not reflect reality but are causing further societal division in Canada.

The authors claim a VE against infection of 80% (but admit a possible lower bound of 40% for Omicron). Definitely 80% and even 40% are overly optimistic VE assumptions for the current vaccines. For example, a recent NEJM article showed double dose mRNA VE dropping to 15.4% after 15 to 19 weeks and dropping further to 8.8% after 25 or more weeks (https://www.nejm.org/doi/full/10.1056/NEJMoa2119451). Many other papers have confirmed these findings.

The authors also claim a baseline immunity in the unvaccinated of 20% based on an 'assumption' (i.e. no reference given). This is a very false assumption as this point in the pandemic. In the Omicron era, a large percentage of the unvaccinated (or vaccinated) population has already been infected. For example, the US CDC recently announced (https://www.cdc.gov/mmwr/volumes/71/wr/mm7117e3.htm) that as of February 2022, approximately 75% of children and adolescents had serologic evidence of previous infection with SARS-CoV-2, with approximately one third becoming newly seropositive since December 2021. Given these findings, a large percentage of the population has natural or hybrid immunity. Natural immunity alone has been shown to be as or even more effective than 2 doses of an mRNA vaccine. This is irrefutable and supported by >100 published studies, many in prestigious journals. For example, see this recent review paper: https://www.sciencedirect.com/science/article/pii/S0013935122002389

I hope the CMAJ retracts this article or at the very least publishes the critiques that are sure to follow.

Fisman et al. [1]’s main conclusion – that risk of infection among vaccinated people can be disproportionately attributed to unvaccinated people – does not follow from the model presented [2].

Their ad hoc parameter Ψ – defined as “the fraction of all infections among vaccinated people that derived from contact with unvaccinated people, divided by the fraction of all contacts [involving vaccinated people] that occurred with unvaccinated people” – is incorrectly asserted to represent “a normalized index of the degree to which risk in one group may be disproportionately driven by contact with another.”

The assertion is incorrect because the model as presented is blind as to whether the “contacts” in the normalizing denominator of Ψ are infectious or benign, irrespective of vaccination status.

In the model, most “contacts” are benign (not involving an infectious person and a susceptible person), whether vaccinated or unvaccinated. This means that the normalizing denominator of Ψ cannot be assumed to represent “contacts driving infection”, as advanced by Fisman et al.

It is easy to see that the ad hoc parameter Ψ is nonsensical, from figures in their paper:
a. Fig. 2A shows Ψ dropping dramatically with increasing reproduction number. This would mean that unvaccinated people threaten vaccinated people proportionately less when the presumed pathogen is more infectious. The state should not worry about unvaccinated people if the pandemic is sufficiently virulent?
b. Fig. 2B shows Ψ approaching large values as the mixing coefficient η approaches 1. This would mean that unvaccinated people are proportionately more of a threat to vaccinated people as the two groups are more and more isolated from each other, up to complete isolation. This is an absurd result.

The obvious parameter that Fisman et al. could have reported is the numerator of Ψ, which is “the fraction of all infections among vaccinated people that derived from contact with unvaccinated people”.

We plot this “numerator of Ψ”, for parameters used by Fisman et al., versus the mixing coefficient η, and for different population fractions of unvaccinated people, here: https://ocla.ca/wp-content/uploads/2022/04/plot-numer-Psi-v-eta-R0-6-1.jpg

We see that there is no indication of disproportionate infections caused by unvaccinated people, and that the “the fraction of all infections among vaccinated people that derived from contact with unvaccinated people” is bound by the relative populations of vaccinated and unvaccinated susceptible individuals for random mixing, and goes more and more quickly to a value of zero as isolation between the two groups increases, as it must.

These are trivial results. The only way to get the simple model to say what Fisman et al. have said is to concoct and misinterpret an ad hoc parameter (Ψ).

I offer the following comments regarding the recent model published in the Canadian Medical Association Journal.1  There are several important shortcomings and concerns with this paper.

First, this is a model.  It does not measure, observe or test anything.  It is a prediction based on a self-described "simple" model.  The predictions of this model have not been tested so, at most, this model should be regarded as an hypothesis.  Like any hypothesis it needs to be tested and validated before its predictions should be considered evidence.

Second, the output of any model is totally dependent on the quality and accuracy of its inputs.  This key input for this model is the vaccine effectiveness (VE) in preventing infection.  The model assumes that this VE is 40-80%.  

The authors cite two references to support the lower bound (40%) estimate.  The first is a surveillance report from the United Kingdom at a time (December 2021) when Omicron Variant was just emerging.2  The data in this report are based on Delta Variant but the report makes it clear that  lower VE with Omicron is anticipated.  The second reference is simply another unvalidated model.3

The authors cite only a single reference to support an upper bound (80%) estimate of VE.4

The authors' use of this single reference is highly problematic for three reasons.  First, the reference only covers data up until October 20, 2021 - six months ago!  It does not take into account the impact of new Variants (Omicron and B.2) or continuing waning immunity.  Second, the reference study does not support a VE estimate of 80%.  The reference study measured VE for three vaccines between July 1, 2021 and October 20, 2021 at 49%, 52% and 70%.  Third, the authors have failed to acknowledge abundant new evidence5,6, including some from Ontario,7,8, showing little or no persistent (and perhaps even negative) VE against infection.

In summary, this paper describes an untested hypothesis based on flawed assumptions, invalidating its conclusions.

Richard Schabas MD MHSc FRCPC

1.  https://www.cmaj.ca/content/194/16/E573
2.  https://assets.publishing.service.gov.uk/government/uploads/system/uploa...
4.  https://academic.oup.com/cid/advance-article/doi/10.1093/cid/ciac106/652...
5. https://www.thelancet.com/journals/laninf/article/PIIS14733099(21)007684/fulltext
6. https://www.nejm.org/doi/full/10.1056/NEJMoa2201570
7.  https://www.medrxiv.org/content/10.1101/2021.06.28.21259420v2
8.  https://www.medrxiv.org/content/10.1101/2021.06.28.21259420v3

Fisman and colleagues[1] present a oversimplification of a complex epidemiological, social, and bioethical issue. The findings are predetermined by the authors’ own model design choices; something that should never occur in science. That the authors make strong ethical and political claims that feed existing social polarization makes the flawed design even more problematic.

The authors use a compartmental SIR model to compute the infection burden in vaccinated and unvaccinated population subgroups and assess contribution of the unvaccinated group to the cumulative rate of infection among the vaccinated. The study’s main conclusion—that mixing with unvaccinated people increases the risk of infection among the vaccinated—is predetermined by the authors choice of model and parameters. By ignoring waning immunity (from both vaccination and prior infection), the authors have constructed a model in which herd immunity always occurs, leaving some residual proportion of the population uninfected indefinitely. In this hypothetical scenario, it is a foregone conclusion that if one group with high baseline immunity is mixed with another group of lower baseline immunity then a greater proportion of the high-immunity group will become infected before herd immunity is achieved, than if they had not mixed. This is nothing more than dilution. The model[2] contains two crucial parameters: ‘vaccine efficacy’ and ‘baseline immunity in unvaccinated’. If these are set to any combination where the latter is higher, then the findings are reversed; the vaccinated increase risk for the unvaccinated. Obviously, both conclusions are similarly flawed. In the context of observed waning of vaccine efficacy against infection[3], even the authors acknowledge that “it is unlikely that SARS-CoV-2 will be eliminated”. Why then is their analysis based on an assumption that it will be?

It is especially problematic that a modelling paper so detached from reality contains such explicit and strong condemnation of ‘the unvaccinated’. The authors discuss the theoretical risk that the unvaccinated pose to the vaccinated via their disproportionate demand for healthcare resources–something not considered in their model–without any acknowledgement of the vast difference in healthcare demands of, say, a healthy 18-year-old compared with a comorbid 80-year-old. The potential for this work to foster social division and misplaced anger and blame is at odds with public health ethics. The authors claim support for Canada’s vaccine mandates without any acknowledgment that these policies helped to ignite nation-wide protest. The combination of deeply flawed modelling, moral condemnation and politicisation should be sufficient to retract a paper published in Canada’s preeminent medical journal. Unfortunately, the damage has already been done, with many media outlets added fuel to the fire through uncritical reporting of this study. Trust in public health and science has been further eroded.

Dear Sir,

The article by Fisman et al.1 has garnered much medical and societal attention. The authors used a mathematical model to simulate COVID-19 infection risk across various patterns of interactions amongst vaccinated and vaccine-free individuals. The authors conclude that individuals who avoid vaccination contribute to negative health consequences for others. Such an assertion is incorrect and biased for the following reasons:

1. Uses problematic mathematical modeling as a surrogate for real-world data. Mathematical modeling has been used throughout the COVID-19 response to justify lockdown measures while promoting unscientific public health edicts. As there is abundant real world data, why they would choose a mathematical model is unclear.

2. Overestimates vaccine effectiveness against symptomatic infection. The range of vaccine effectiveness against symptomatic infection was 40-80%. The upper bound limit of 80% may apply to the old Delta variant but the lower bound limit of 40% does not apply to the Omicron variant. Current data shows that vaccine effectiveness against Omicron symptomatic infection ranges from 0% to 75% independent of vaccine type, duration since primary series, or duration since booster(s).2

3. Overestimates the risk of transmission (secondary attack rate). The authors overstate the ability of vaccines to reduce the risk of transmission. Kampf has shown that the proportional rate of symptomatic Covid-19 cases among fully vaccinated patients >60 years old has been increasing since July 2021. In the week of October 27, 2021, it was 58.9%.3 If anything, COVID-19 vaccines do a poor job at reducing transmission of disease.

4. Underestimates the percentage of the unvaccinated population with natural immunity. The authors assume a baseline of previous infection of 20% in the unvaccinated population. Even before vaccinations were available, about 60% of Canadians had clear evidence of prior SARS-CoV-2 infection.4 The longevity of protection from natural immunity against symptomatic infection has been repeatedly proven superior to vaccination alone, meaning that the underestimation of those with natural immunity skews the model towards vaccination and away from current clinical evidence.

5. Does not account for waning vaccine immunity. The authors overestimate vaccine effectiveness and fail to account for the primary reason for the ongoing and relentless Omicron waves, namely waning vaccine immunity. Studies using real-world data demonstrate rapidly waning immunity in the fully vaccinated population.5

6. Conflict of interest. The lead author has multiple conflicts of interest and receives financial compensation from COVID-19 vaccine companies including Pfizer and AstraZeneca.

Promotion of poor research such as this leads to further stigmatization and division in society. We challenge the CMAJ to retract this study or issue a correction in their next publication.

Noting that once the Omicron variant started to dominate over Delta in late December 2021, infection rates started to increase faster among the vaccinated than the unvaccinated and the rate ratio (RR) of unvaccinated/vaccinated quickly dropped below 1.0, I’d suggest that the main finding of this study “We found that the risk of infection was markedly higher among unvaccinated people than among vaccinated people under all mixing assumptions” should be reviewed (See Ontario Public Health data available at https://covid-19.ontario.ca/data/case-numbers-and-spread for rates by vaccination status and https://data.ontario.ca/en/dataset/covid-19-vaccine-data-in-ontario to download data).

At this time vaccine passports granted the vaccinated access to high-risk venues such as restaurants bars and gyms and the vaccine effectiveness (VE) of 2 doses was much lower for Omicron than Delta. My interpretation of the diverging trends in rates at the time was that the privileges provided by vaccine passports increased the rate of high-risk contacts among the vaccinated more than the protection against infection offered by the vaccine – that is, the vaccinated rather than the unvaccinated had become the main drivers of Omicron wave. The policy decision to shut down the high-risk venues after Christmas seemed to have worked in reducing hospitalizations (however, case counts declined rapidly as access to PCR testing in Ontario became severely limited).

While the RR is typically used to calculate VE, trends in the RR could be due to waning immunity, differences in exposure risk, or differences in access to testing between the two groups, or some combination of these factors. Would monitoring the diverging trends in the rates and RR not be informative on which sub-group or sub-groups are likely driving the epidemic wave?

It would seem quite helpful to informing public health policy in a timely manor if the authors could explore informing their model based on near real time trends in rates or RRs. Perhaps monitoring trends in rates for divergence would be helpful to identify changes in drivers of pandemic growth!

Some are not though. A good model approximates the real world, and its assumptions reflect reality. The recent modeling by Fisman et al. published in the CMAJ is an example of a model presented in such a way that does not well represent the real world. Moreover, the interpretation of the model does not necessarily lead to the policy conclusions the authors present.

Of note, the authors make a major unrealistic assumption which is non-waning immunity among the cohort of individuals who are vaccinated. This means that these persons cannot transmit COVID19 to each other or to unvaccinated individuals. Where immunity wanes as it seems it does in the real world [2], or where there is not sterilizing immunity, the disproportionate impact that is reflected in the authors' models does not necessarily hold. This very seriously calls the authors' conclusion that the choices of the unvaccinated affect everyone “in a manner that is disproportionate to the portion of unvaccinated people in the population.” While not a perfect reflection of waning immunity, one suggestive result from the authors that reflects this is the sensitivity analysis that changes the effectiveness of the vaccine (fig 2 panel 4). Where the vaccine effectiveness drops to 40% the outcome is a much lower contribution of the unvaccinated to infection risk than their headline results suggest.

This issue aside, the authors then proceed to outline policy recommendations that do not necessarily follow from their results. In the parlance of economics what the authors have outlined is an externality. The unvaccinated impose costs on the vaccinated. The goal of any good policy is to “internalize” these externalities in whatever way is effective; if possible given medicines primary dictum of “first do no harm” this should be done in a non-coercive manner unless there are exigent circumstances. However, the authors leap directly to the idea that non-vaccinated individuals should be banned from public spaces and regulation should be enacted that may violate individual freedoms. Instead, of these coercive policy recommendations, the economics literature provides guidance on how to internalize these negative externalities via taxes and subsidies. In many cases the welfare implications of these policies are often superior to the complete bans and mandates that the authors seem to suggest [3]. These policies are much less coercive than this papers recommendations.

I am sympathetic to the idea that COVID19 is a disease that imposes externalities; vaccinated patients should not have to suffer because of the choices of unvaccinated persons. Moreover, vaccination continues to provide the best way to prevent bad outcomes from COVID19 and policy should attempt to ensure that the greatest number of people possible are vaccinated against it. However the model that the authors propose does not approximate the real world well and does not support the coercive measures that are suggested in this paper.