Quantifying the preventive effect of wearing face masks

An important task in combating the current Covid-19 pandemic lies in estimating the effect of different preventive measures. Here, we focus on the preventive effect of enforcing the use of face masks. Several publications study this effect, however, often using different measures such as: the relative attack rate in case-control studies, the effect on incidence growth/decline in a specific time frame and the effect on the number of infected in a given time frame. These measures all depend on community-specific features and are hence not easily transferred to other community settings. We argue that a more universal measure is the relative reduction in the reproduction number, which we call the face mask effect, EFM. It is shown how to convert the other measures to EFM. We also apply the methodology to four empirical studies using different effect-measures. When converted to estimates of EFM, all estimates lie between 15 and 40%, suggesting that mandatory face masks reduce the reproduction number by an amount in this range, when compared with no individuals wearing face masks.


Comments to the Author(s)
This paper looks at the topical problem of estimating the effect of wearing face masks and, specifically, the reduction in the (current) reproduction number, R, that results from the mandatory wearing of face masks as compared with no use of masks. While the theory set out in Sections 2 and 3 is straightforward and well-known, the application to observational data and interpretation of the results described in Sections 4 and 5 is of interest. The paper shows signs of having (understandably perhaps) been written in haste, and would benefit from careful proofreading as there are many typographical errors. In places the English lacks fluency and in general the descriptions could be expressed much more concisely and irrelevant details omitted (for example, why introduce notation for different sorts of vaccine efficacy in the Introduction?).
The effect of face mask wearing is to reduce the rate of effective contacts by lowering both susceptibility and transmission, and hence to reduce the reproduction number R by the same factor. The face mask effect EFM is defined in Section 2 to be the proportionate decrease in R. Most of the notation introduced in this section is not required. The equations in Section 3 express R in terms of various observable quantities such as the growth rate and increase in cases, thus enabling the estimation of EFM. These expressions could well be stated in an Appendix. There is no new theory here and the paper could itself be reduced to a short note, concentrating on Sections 4 and 5, without loss. The beginning of Section 5 would be a suitable introduction.
The estimation and discussion sections (Section 4 and 5) demonstrate the calculation of EFM using various data sets, and have some interesting and potentially useful results, although these are necessarily qualified by all sorts of caveats. Mandatory face-covering does not guarantee compliance and the latter may vary substantially between communities. More importantly for the work presented here, the connections between R and the various observable quantities taken from Section 3 are for an emerging epidemic in a homogeneously mixing population and even, in 3.3 for a Markov SIR model, and it is assumed that everything in terms of mixing behaviour and other interventions remains fixed during the period over which the effect of face masks is to be estimated. The sensitivity of the results to all these assumptions needs to be discussed. There is some discussion of confounding effects but it seems unlikely that individual behaviour would have remained unchanged in the face of a rapidly growing pandemic during the time periods covered by these studies. How is population heterogeneity likely to affect the results? Is it reasonable to apply theory derived for an emerging epidemic to data from Kansas in July 2020?
Minor point: Equation 2 would look simpler in terms of the index of dispersion of the generation time distribution rather than its coefficient of variation Review form: Referee 2 Is the manuscript an original and important contribution to its field? Good

Is the paper of sufficient general interest? Good
Is the overall quality of the paper suitable? Good Can the paper be shortened without overall detriment to the main message? No Do you think some of the material would be more appropriate as an electronic appendix? Yes

Recommendation?
Accept with minor revision (please list in comments) Comments to the Author(s) Major 1. I may be wrong but much of the theory in section 3 seems applicable to any intervention that may impact on both infectivity and susceptibility, including vaccination.
2. Section 2 is important, showing how face mask effects impinge on reproduction number. It is less clear what section 3 adds, which is largely standard theory on how the reproduction number relates to other epidemiological parameters.
3. A limitation of the data from the Kansas study is that compliance would not have been 100% in counties with mandatory face masks, nor 0% in counties without mandatory face masks. It is therefore difficult to assess individual transmission parameters.
4. The description of the paper in section 4.4 is very brief, in contrast to 4.1, 4.2 and 4.3. This makes the paper feel unbalanced. Is this study reliable? 5. Is it possible to derive confidence intervals for the studies in 4.2 and 4.3? How dependable are these findings? 6. The authors might like to mention the possibility of publication bias i.e. only positive findings making it to publication. Minor 1. Some of the nomeclature is questionable. 4.1 is not a case-control study. 4.2 and 4.3 not retrospective studies.
2. In section 4.3 the incidence of 42 should be 142?

14-Apr-2021
Dear Dr Britton The Editor of Proceedings A has now received comments from referees on the above paper and would like you to revise it in accordance with their suggestions which can be found below (not including confidential reports to the Editor).
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Yours sincerely Raminder Shergill proceedingsa@royalsociety.org on behalf of Professor Vincenzo Capasso Board Member Proceedings A Reviewer(s)' Comments to Author: Referee: 1 Comments to the Author(s) This paper looks at the topical problem of estimating the effect of wearing face masks and, specifically, the reduction in the (current) reproduction number, R, that results from the mandatory wearing of face masks as compared with no use of masks. While the theory set out in Sections 2 and 3 is straightforward and well-known, the application to observational data and interpretation of the results described in Sections 4 and 5 is of interest. The paper shows signs of having (understandably perhaps) been written in haste, and would benefit from careful proofreading as there are many typographical errors. In places the English lacks fluency and in general the descriptions could be expressed much more concisely and irrelevant details omitted (for example, why introduce notation for different sorts of vaccine efficacy in the In troduction?).
The effect of face mask wearing is to reduce the rate of effective contacts by lowering both susceptibility and transmission, and hence to reduce the reproduction number R by the same factor. The face mask effect EFM is defined in Section 2 to be the proportionate decrease in R. Most of the notation introduced in this section is not required. The equations in Section 3 express R in terms of various observable quantities such as the growth rate and increase in cases, thus enabling the estimation of EFM. These expressions could well be stated in an Appendix. There is no new theory here and the paper could itself be reduced to a short note, concentrating on Sections 4 and 5, without loss. The beginning of Section 5 would be a suitable introduction.
The estimation and discussion sections (Section 4 and 5) demonstrate the calculation of EFM using various data sets, and have some interesting and potentially useful results, although these are necessarily qualified by all sorts of caveats. Mandatory face-covering does not guarantee compliance and the latter may vary substantially between communities. More importantly for the work presented here, the connections between R and the various observable quantities taken from Section 3 are for an emerging epidemic in a homogeneously mixing population and even, in 3.3 for a Markov SIR model, and it is assumed that everything in terms of mixing behaviour and other interventions remains fixed during the period over which the effect of face masks is to be estimated. The sensitivity of the results to all these assumptions needs to be discussed. There is some discussion of confounding effects but it seems unlikely that individual behaviour would have remained unchanged in the face of a rapidly growing pandemic during the time periods covered by these studies. How is population heterogeneity likely to affect the results? Is it reasonable to apply theory derived for an emerging epidemic to data from Kansas in July 2020?
Minor point: Equation 2 would look simpler in terms of the index of dispersion of the generation time distribution rather than its coefficient of variation Referee: 2 Comments to the Author(s) Major 1. I may be wrong but much of the theory in section 3 seems applicable to any intervention that may impact on both infectivity and susceptibility, including vaccination.
4. The description of the paper in section 4.4 is very brief, in contrast to 4.1, 4.2 and 4.3. This makes the paper feel unbalanced. Is this study reliable? 5. Is it possible to derive confidence intervals for the studies in 4.2 and 4.3? How dependable are these findings?
6. The authors might like to mention the possibility of publication bias i.e. only pos itive findings making it to publication.

Is the paper of sufficient general interest? Good
Is the overall quality of the paper suitable? Good Can the paper be shortened without overall detriment to the main message? No Do you think some of the material would be more appropriate as an electronic appendix? Yes

Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
The extra comments and discussion have usefully addressed the issues that were raised by the reviewers. I have no further comments.

Review form: Referee 2
Is the manuscript an original and important contribution to its field? Good

Is the paper of sufficient general interest? Good
Is the overall quality of the paper suitable? Good

Can the paper be shortened without overall detriment to the main message? Yes
Do you think some of the material would be more appropriate as an electronic appendix? No

Recommendation?
Accept as is

13-May-2021
Dear Dr Britton, On behalf of the Editor, I am pleased to inform you that your Manuscript RSPA-2021-0151.R1 entitled "Quantifying the preventive effect of wearing face masks" has been accepted for publication subject to minor revisions in Proceedings A. Please find the referees' comments below.
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10-Jun-2021
Dear Dr Britton I am pleased to inform you that your manuscript entitled "Quantifying the preventive effect of wearing face masks" has been accepted in its final form for publication in Proceedings A.
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Response to comments of the referees and Associate editor
First I would like to thank the two referees and the Associate for many useful comments which helped improve the manuscript. The manuscript has been revised accordingly, also correcting a few typos and other errors.
Please find below all comments by the referees and my response to each of them. In order to distinguish between referee's comments and my response, my responses are written in italics and start with ">>".

Reviewer(s)' Comments to Author:
Referee: 1 Comments to the Author(s) This paper looks at the topical problem of estimating the effect of wearing face masks and, specifically, the reduction in the (current) reproduction number, R, that results from the mandatory wearing of face masks as compared with no use of masks. While the theory set out in Sections 2 and 3 is straightforward and well-known, the application to observational data and interpretation of the results described in Sections 4 and 5 is of interest. The paper shows signs of having (understandably perhaps) been written in haste, and would benefit from careful proofreading as there are many typographical errors. In places the English lacks fluency and in general the descriptions could be expressed much more concisely and irrelevant details omitted (for example, why introduce notation for different sorts of vaccine efficacy in the Introduction?).
>> The manuscript has now been carefully edited removing several typos and other inconsistencies. Further, unnecessary technical parts in Section 1 and 3 have been removed.
The effect of face mask wearing is to reduce the rate of effective contacts by lowering both susceptibility and transmission, and hence to reduce the reproduction number R by the same factor. The face mask effect EFM is defined in Section 2 to be the proportionate decrease in R. Most of the notation introduced in this section is not required. The equations in Section 3 express R in terms of various observable quantities such as the growth rate and increase in cases, thus enabling the estimation of EFM. These expressions could well be stated in an Appendix. There is no new theory here and the paper could itself be reduced to a short note, concentrating on Sections 4 and 5, without loss. The beginning of Section 5 would be a suitable introduction.
>> Many good points! Unnecessary notation in Section 2 is dropped. Some notation and equations in Section 3 have also been dropped. Instead the initial discussion on under what assumptions the modelling results are valid has been extended upon request (new Section 3.1). It is true that the theory presented in Section 3 does not contain any new theory. However, finding the different results in the literature requires quite a lot of searching and I think it is worth gathering them in one place. I checked several other manuscripts in Roy Soc Proc A and found that many of them contained a section called "Mathematical background" (which the section now is renamed to, plus "assumptions"). I did not find any papers having an appendix in the main text and would prefer not putting this theory in a Supplementary Material which readers may not find. If allowed by the journal I can move Section 3 into an Appendix at the end of the main text, or if requested to a separate Supplementary Material.
The estimation and discussion sections (Section 4 and 5) demonstrate the calculation of EFM using various data sets, and have some interesting and potentially useful results, although these are necessarily qualified by all sorts of caveats. Mandatory face-covering does not guarantee compliance and the latter may vary substantially between communities. More importantly for the work presented here, the connections between R and the various observable quantities taken from Section 3 are for an emerging epidemic in a homogeneously mixing population and even, in 3.3 for a Markov SIR model, and it is assumed that everything in terms of mixing behaviour and other interventions remains fixed during the period over which the effect of face masks is to be estimated. The sensitivity of the results to all these assumptions needs to be discussed. There is some discussion of confounding effects but it seems unlikely that individual behaviour would have remained unchanged in the face of a rapidly growing pandemic during the time periods covered by these studies. How is population heterogeneity likely to affect the results? Is it reasonable to apply theory derived for an emerging epidemic to data from Kansas in July 2020? >> Many important points! I have added a new Section 3.1 detailing the underlying assumptions. The Conclusion/Discussion section has been expanded with additional paragraphs. In 2nd full paragraph of p10 I discuss effects of confounding errors due to differences and changes during the study period. In the next paragraph effects of possible publication bias is brought up. In the paragraph thereafter I discuss effects of adherence and also how changed behaviour during the study period may affect estimation.
Minor point: Equation 2 would look simpler in terms of the index of dispersion of the generation time distribution rather than its coefficient of variation >> Index of dispersion is defined by \sigma^2/\mu. I agree with the comment if it would have been defined as \sigma^2/\mu^2, but not now. Further, I prefer coefficient of variation since it lacks measurement unit. For this reason I did not change to dispersion index.