Research Article

# A Model Showing the Relative Risk of Viral Aerosol Infection from Breathing and the Benefit of Wearing Masks in Different Settings with Implications for Covid-19

### Gerald D Barr*

Independent Health Research, Peat Inn, UK

*Address for Correspondence: Gerald D Barr, Independent Health Research, Peat Inn, UK, Tel: +44-796-342-9784; ORCID ID: orcid.org/0000-0002-9472-1903; E-mail: gdbarr@outlook.com

Submitted: 19 June 2020; Approved: 09 July 2020; Published: 09 July 2020

Citation this article: Barr GD. A Model Showing the Relative Risk of Viral Aerosol Infection from Breathing and the Benefit of Wearing Masks in Different Settings with Implications for Covid-19. American J Epidemiol Public Health. 2020;4(3): 062-069. https://dx.doi.org/10.37871/ajeph.id32

Copyright: © 2020 Barr GD. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Keywords: Covid-19; SARS-CoV-2; Aerosol; Masks; Respirator

Background: Widespread use of masks in the general population is being used in many countries for control of Covid-19. There has been reluctance on the part of the WHO and some governments to recommend this.

Methodology: A basic model has been constructed to show the relative risk of aerosol from normal breathing in various situations together with the relative benefit from use of different masks.

Results: The benefit from mask use between individuals is multiplicative not additive and although social distancing at 2 meters appears beneficial with regards to aerosol infectivity, in confined areas this is time limited requiring additional measures such as masks. The model shows the relative benefit of masks when social distancing is not possible at all times, or when in confined areas which can also be aided by efficient ventilation. Where a person is in one place for a prolonged period there is more risk requiring protection.

Conclusion: Masks should be used in the above situations especially at an early stage of an outbreak. Public health planning requires stockpiling of masks and encouraging everyone to have suitable masks in their household when supplies are normalised. In the absence of widely available good quality masks the use of a cloth mask will be better than no protection at all.

## Introduction

Aerosol spread occurs much more from coughing and sneezing, compared to speaking or normal breathing, however, some individuals produce excessively high amounts of aerosol particles > 10,000 per litre in normal breathing [11] which may be one mechanism for super spreading. From an observation in influenza [12] causing 100 per cent infection in a confined space it was postulated that for airborne infection there must be a relationship between the amount of virus expelled into the air, the ventilation of the area and the time exposed. The objective of this study was to construct a simple mathematical model to show the relative benefit of mask wearing in relation to the infectious dose of a virus in aerosol. A Health and Safety review found a six times reduction in bioaerosol using a surgical mask with an estimated 100 times reduction for an FFP3 mask, with much of the inefficiency of surgical masks being due to leakage or poor fit [13]. Cloth masks are acknowledged to give some protection which may only have an improvement factor of two [14].

## Method

The model was formulated, based on the formulae described below, on an Excel spread sheet which allows input values to be changed easily and is available for general use on request. Input values available from the literature were used.

Infectious Dose ID = P(av) EvF1 (Bv /Ev) F2 t where t is time in minutes. (Bv cannot exceed Ev i.e. max for Bv/Ev is 1)

Infectious Dose (ID) is the amount of virus needed to be inhaled to cause infection. Many factors could affect this, and the bioavailability of the virus once inhaled together with mucosal defences. A review by Nikitin suggested a figure of 1.9 x 103 for influenza [15]. Regarding fine aerosol that can penetrate more deeply into the lungs it is quite possible that a much lower dose is required. For this comparative study an ID at 1000 was used especially as this fitted best with scenario 2 (Table 2) and the original observation. Not all aerosol particles contain virus which may only be 10% of aerosol but can be more [16]. For this study 30% of particles being infectious was taken at av = 0.3.

For an infected person breathing out aerosol particles P = 500/liter is used which is the cut off value between a low and high producer of aerosol [11].

Ev is the volume of air expired by the infectious person per minute [tidal volume x breaths/min] for this study taken as normal breathing 500 ml x 12 breaths per minute = 6 litres/min.

Bv is the volume of air inspired by a non-infected person per minute [tidal volume x breaths/min] for this study taken as normal breathing 500 ml x 12 = 6 litres/min.

F1 is the expiratory filtration factor of a mask for an infectious person breathing out, F2 is the filtration factor for non-infected person breathing in. For no mask F is 1, for x 6 reduction is 0.167, x100 reduction 0.01. Using this data and that from a study [17] on the reduction factor, inspiratory and expiratory at 30, 50 and 80 l/min extrapolating to 6 l/min cloth mask F1 is 0.75 F2 0.5, surgical mask F1 is 0.333 F2 is 0.167, FFP3 F1 is 0.333 F2 0.01 .

At a distance under one meter the volume is not applicable due to risk of directly breathing the same air e.g. one person breathes in as the other breathes out. Greater than 1 meter the exhaled infectious air is dispersed into a volume V.

Ve is the amount of ventilation, given by the amount of original air remaining after replacement with fresh air per minute.

The time evolution of the infectious particle concentration C in the volume V is given by:

$\frac{dC}{dt}=\left(\frac{P{a}_{v}{E}_{v}{F}_{1}}{V}\right)-\lambda C$

The time evolution of the inspired dose N is given by:

$\frac{dN}{dt}=\left({B}_{v}{F}_{2}\right)C$

For zero decay constant λ the solutions are (where we consider that the infectious person has been present for a time Tstart before the uninfected person arrives):

$C\left(t\right)={C}_{start}+\left(\frac{P{a}_{v}{E}_{v}{F}_{1}}{V}\right)t$

$N\left(t\right)=\left({B}_{v}{F}_{2}\right)\left[{C}_{start}t+\left(\frac{P{a}_{v}{E}_{v}{F}_{1}}{V}\right)\left(\frac{1}{2}{t}^{2}\right)\right]$

${C}_{start}=\left(\frac{P{a}_{v}{E}_{v}{F}_{1}}{V}\right){T}_{start}$

For nonzero λ the solutions are (where we allow a different decay constant λpre during the warm-up phase before the uninfected person arrives):

$C\left(t\right)={C}_{final}+\left({C}_{start}-{C}_{final}\right)\mathrm{exp}\mathrm{exp}\left(-\lambda t\right)$

$N\left(t\right)=\left({B}_{v}{F}_{2}\right)\left[{C}_{final}t+\left({C}_{start}-{C}_{final}\right)\left(\frac{1-\mathrm{exp}\mathrm{exp}\left(-\lambda t\right)}{\lambda }\right)\right]$

${C}_{final}=\left(\frac{P{a}_{v}{E}_{v}{F}_{1}}{V\lambda }\right)$

${C}_{start}={C}_{pre-sat}\left(1-\mathrm{exp}\mathrm{exp}\left(-{\lambda }_{pre}{T}_{start}\right)\right)$

${C}_{pre-sat}=\left(\frac{P{a}_{v}{E}_{v}{F}_{1}}{V{\lambda }_{pre}}\right)$

${t}_{1/2}=\frac{\mathrm{ln}\mathrm{ln}\left(2\right)}{\lambda }$

The aerosol absorbed by each person inhaling would reduce the number of infectious particles also increased by use of a mask affecting exhalation too. Fx is the inhalation filtration factor for the infectious person and Fy is the filtration factor for the non-infected person exhaling. Using the calculated loss from a medical aerosol study of absorption giving an estimated 45% of particles absorbed [18].

The absorption loss during inhalation is given by:

$\lambda insp/\mathrm{exp}=\left(BvFloss\right)/V$

where Floss is the effective fraction of particles lost in one complete breath and accounts for the effect of any masks during breathing in or out. The contributions of multiple individuals could be added to this value.

Fabs is the fraction of particles absorbed with no mask then the effect of a mask can be estimated as:

Floss Bv = 1-(Fy F2) (1-Fabs)

Floss Ev = 1-(FxF1) (1-Fabs

The exponential decay constant λ represents all loss processes which can be combined as:

λ = λ1 + λ2 + …

λ (decay) = -ln(2)/(half-life for aerosol decay)

λ (Ve) = -ln(Volume retained after 1 min/V )

λ(insp/exp) = -ln( proportion inspiratory and expiratory loss)

This is a comparative analysis not an actual situation for any particular virus although the values used from the literature will give an approximation for Covid-19. Any of the chosen parameters could be less than the actual for instance the aerosol may only have 10% infectious particles, ID could be 3000 but equally particles exhaled by an individual could be 10 times more. The mask, no mask and ventilation analysis, however, is consistent and relative to these values.

The model assumes that the particles expired are instantly diluted to give a uniform concentration in the volume. In reality the distribution in space would be complex and so V must be regarded as an effective interaction volume rather than the precise volume of any particular region. Additionally, ventilation and convection or diffusion would create a complex spatial and temporal dependence of the concentration so the contribution to λ by these processes must be considered as an effective value to model the loss of the virus from the system and not a precise value. It does not take account of localised directional air flow which in theory could increase the infectious dose i.e. directing higher concentration aerosol towards a person.

Decay of aerosol will vary for virus, particle size and humidity. Adams et al. found a sharp decay at the start for Reo virus, although this was considerably reduced by humidity, with recovery over longer periods [19] which may be due to smaller particles surviving longer. Cowling found that 87% of particles [20] emitted are less than 1 micron, at this size there is less potential space for a virus, for instance an aerosol particle 1 micron in diameter is much more likely to have a maximum of one SARS-CoVid-2 particle measuring 0.12 microns in diameter. Smaller particles can stay airborne for hours in a humid environment with only a 10% loss of infectivity [21]. Yang found enough influenza virus in the air to cause infection in different public places and that a steady state will be reached [22]. For this study it was assumed one infectious particle releases only one viral particle and for the calculations the median aerosol half-life of 1.2 hours for SARS-CoVid-2 found by van Doremalen was used [23].

It is not certain what the minimum cumulative dose for infection is and over what period i.e. the minimum number of virus particles that nonspecific defences can cope with per hour or minute. It is assumed the ID is cumulative at least over a relatively short period for this study minutes to a few hours.

## Discussion

Compliance has been a problem in western countries even as far back as the Spanish Flu pandemic on the other hand masks have developed significantly since then. The health belief model [40] shows perception of risk of death and to health are important factors in compliance, also when livelihood becomes affected. In Hong Kong during the SARS epidemic the public mask usage was around 65%, but for asymptomatic individuals as low as 21.5% for H1N1 [41] rising to over 95% for Covid-19. A strong and consistent public health message is needed such as the campaign in Czechia, with visual media messages, “I protect you, you protect me” also introducing the compulsory public wearing of masks early on. This compares to the USA and UK where for weeks even after the exponential rise in cases and hospitals being overwhelmed the official message was that masks were not needed. Not surprisingly this has to led to a relatively low compliance and the need for mandatory mask use on public transport and in shops including fines for those non-compliant in Scotland. Generally, it is acknowledged that older people and females [31] will engage more with mask wearing. Compliance within households will be low in the absence of infection [42] but the best option is to keep households free from infection by general population measures.

## Acknowledgement

Timothy Heelis for providing assistance with the numerical calculations.

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