Given these assumptions, the total absorbed dose by the susceptible individual can be calculated as follows:
where
and
,
d0 is the initial wet particle size on exhalation by the infectious,
is the shrinkage factor defined as the ratio of the particle initial wet diameter
d0 to the equilibrium diameter
de after it is exposed to the typically subsaturated conditions of the room and has lost its volatile components,
and
are the minimum and maximum particle size that can be aerosolized and contain
k copies of the pathogen,
texp is the exposure duration of the susceptible individual,
ρp is the pathogen number concentration, that is, the viral load, in the infectious respiratory tract fluid,
is the number concentration of exhaled particles at the mouth/nose of the infectious,
is the fractional ratio at which the particle concentration of the exhaled air by the infectious individual decreases until it reaches the breathing zone of the susceptible individual due to (turbulent and/or molecular) mixing with the room air or particle deposition losses,
is the outward filter penetration of the face mask fabric worn by the infectious,
is the outward face seal leakage of the face mask worn by the infected individual during exhalation,
and
are, respectively, the ratios of the exhale flow rate through the filter and face seal leaks to the total exhale flow rate of the infectious,
is the inward filter penetration of the face mask fabric worn by the susceptible,
is the inward face seal leakage of the face mask worn by the susceptible,
and
are, respectively, the ratios of the inhale flow rate through the filter and face seal leaks to the total inhale flow rate of the susceptible,
is the intake/deposition efficiency of the inhaled particles within the respiratory of the susceptible individual, and
and
are the volumetric inhalation rate (also called ventilation rate) of the infectious and susceptible, respectively. It should be noted that many of the parameters present in
Eq. 3 are also functions of the room conditions, for example, RH, temperature, ventilation type, air velocity, which are neglected here.
We assume that the
ρp is constant and independent of the particle size, even though it has been shown that particles of different sizes have different production sites within the respiratory tract (
5) and particles of different origins might have different viral loads (
74). The SARS-CoV-2 viral load,
ρp, is in the very broad range of 10
2mL
−1 to 10
11 mL
−1 (
23). Mean values for the currently measured SARS-CoV-2 variants are 10
8.2 mL
−1 to 10
8.5 mL
−1 (
75). Here we use 10
8.5 mL
−1 to obtain an upper estimate on risk of infection, which should be more applicable to the new variants of SARS-CoV-2. The increase in viral load with the new variants currently circulating globally is constant with findings in other studies (e.g., see ref.
76, and references therein). The SARS-CoV-2 ID
is not known very well, and, in the literature, a range of values between 100 and 1,000 is used, that is,
(
3,
21) and 100 (
26). In this investigation, we assume ID
, which, for a fixed pathogen dose, gives a risk of infection that is, at most, half (or 2 times) the values calculated with ID
(or 400).
values are calculated based on the multimodal fits found by ref.
5, which is obtained based on measurements from more than 130 subjects aged 5 y to 80 y, using aerosol size spectrometers and in-line holography covering wet particle sizes, that is,
d0, from 50 nm up to 1 mm. The multimodal fits presented by ref.
5 provide an average estimation of
for an adult (gender plays no role). The smallest particle size considered for infection risk analyses, that is,
, is 0.2
m, which is about 2 times the size of the SARS-CoV-2 virus (e.g., see refs.
3 and
4). As for the upper limit, we considered
and assumed larger particles deposit to the ground very quickly and in the vicinity of the infectious person. However, it should be noted that there is an ongoing debate regarding the advection distance of exhaled particles in different respiratory activities and room conditions (e.g., see refs.
9 and
29, and references therein, for more details). Particles exhaled by the infectious are moist and, depending on the RH, may decrease considerably in size by evaporation until they reach the breathing zone of the susceptible. Unless otherwise stated, we have assumed all the particles shrink by a factor of 4, that is,
w = 4.0, which is the expected shrinkage factor for RH
(
5), which is a conservative estimate for RH encountered in typical indoor environments (
4). The values published in table 15 of ref.
53 are used to calculate
and
. However, since these rates are given for general physical activities, that is, sleeping, sitting, and light and heavy exercise, they are combined by optimal weighting factors that were found iteratively and that reproduced the rates found in the literature for different respiratory activities (
77–
79). Breathing and speaking ventilation rates assumed to be constant and equal to 0.57 m
3×h
−1 and 0.67 m
3×h
−1, respectively. While
Pex and
Lex are functions of particles diameter during exhalation,
d0,
Pin, and
Lin are dependent on particle diameter during inhalation,
. The penetration of mask fabric is also a function of breathing rates since it will influence the particle loss due to inertial impaction (important for larger than 1-
m particles) and the time required for capturing submicron particles due to Brownian diffusion. The penetration due to mask leakage is also a function of particle diameter and breathing rate; more details regarding these parameters can be found in
Mask Efficacy Measurements. The ICRP respiratory tract deposition (ICRP94) model (
53) is used to calculate
, The ICRP94 model can provide an estimate of particle inhalability and also the deposition efficiency in five different regions of the respiratory tract based on empirical and numerical models, namely, nasal, oral, thoracic bronchial, bronchioles, and alveolar regions. In order to capture the deposition of exhaled particles that have dried in the typically subsaturated air of a room, one also needs to consider that such particles will undergo hygroscopic growth as they enter the almost saturated environment within the respiratory tract, that is, with an RH of 99.5% (
4,
53,
80,
81). To take into account the hygroscopic growth of inhaled particles, the coupled equations describing rate of change in the particle size and its temperature are solved simultaneously, as explained well in section 13.2.1 of ref.
82, assuming fully dried particles consisting of pure NaCl crystals. This assumption is a good approximation for human aerosols, although a more detailed knowledge would be highly beneficial. The osmotic coefficient required for hygroscopic growth of the NaCl solution is calculated via formulations provided by ref.
83. The hygroscopic growth codes are verified against diffusional growth rate curves shown in figure 13.2 of ref.
82 and also those produced by the E-AIM web-app (
84). For all regions, the midresidence time in the region plus the time spent in all the previous regions is taken as the time duration for calculating the grown size of particles. The total time duration that the particles spend in the respiratory tract per each inhalation+exhalation maneuver is calculated as
, where
fR is the respiration frequency per minute provided by the ICRP94 model. The time that particles spend in each region is then calculated by the distribution of the total respiration time according to the time constants provided by the ICRP94 deposition model for thoracic bronchial, bronchioles, and alveolar regions. The particle residence times for the extrathoracic regions, which are not provided in the ICRP94 model, during inhalation or exhalation are assumed to be 0.1 s. The susceptible is assumed to be a 35-y-old nose-breather male. As mentioned above, the fractional ratio
is one of the most challenging parameters in
Eq. 3. Even the most detailed simulations to date are carried out by assuming the exhale flow behaves similarly to a turbulent jet in a room with quiescent air (e.g., see refs.
9 and
29, and references therein). Therefore, for situations where the infectious is not wearing a face mask, we use a simplified theoretical formulation recently proposed for particle-laden jet flows (
26,
27), that is,
, where
x is the distance between the source and the receptor,
a is the radius of the mouth (assuming a circular shape), and
α is the exhale jet half-angle. For
1 m,
1.2 cm, and
10
∘,
fd is ∼6.8%, which agrees well with the 4.9% experimentally measured for 0.77-
m particles by ref.
85. For nose breathing, ref.
79 found an average nose opening area of 0.56 cm
2 to 0.71 cm
2 (a = 0.42 cm to 0.48 cm) and
11.5
∘, where
2 to 3% at a distance of 1 m. For mouth breathing, ref.
79 found
0.61 cm to 0.75 cm and
17
∘, where
at a distance of 1 m. For speaking, ref.
79 found an average mouth opening of 1.8 cm
2, which corresponds to
0.76 cm; however, no information for
α is presented. In order to be on the conservative side when calculating infection risk, we assume
1.8 cm and
10
∘ to achieve
at a distance of 1 m. These values are used for all scenarios in which the infectious is not wearing a face mask, to calculate
fd. For scenarios in which the infectious is wearing a face mask,
fd = 1.