Shielding of viruses such as SARS-Cov-2 from ultraviolet radiation in particles generated by sneezing or coughing: Numerical simulations of survival fractions

Abstract

SARS-CoV-2 and other microbes within aerosol particles can be partially shielded from UV radiation. The particles refract and absorb light, and thereby reduce the UV intensity at various locations within the particle. Previously, we demonstrated shielding in calculations of UV intensities within spherical approximations of SARS-CoV-2 virions within spherical particles approximating dried-to-equilibrium respiratory fluids. The purpose of this paper is to extend that work to survival fractions of virions (i.e., fractions of virions that can infect cells) within spherical particles approximating dried respiratory fluids, and to investigate the implications of these calculations for using UV light for disinfection. The particles may be on a surface or in air. Here, the survival fraction (S) of a set of individual virions illuminated with a UV fluence (F, in J/m²) is assumed described by S(kF) = exp(-kF), where k is the UV inactivation rate constant (m²/J). The average survival fraction (S_p) of the simulated virions in a group of particles is calculated using the energy absorbed by each virion in the particles. The results show that virions within particles of dried respiratory fluids can have larger S_p than do individual virions. For individual virions, and virions within 1-, 5-, and 9-µm particles illuminated (normal incidence) on a surface with 260-nm UV light, the S_p = 0.00005, 0.0155, 0.22, and 0.28, respectively, when kF = 10. The S_p decrease to <10⁻⁷, <10⁻⁷, 0.077, and 0.15, respectively, for kF = 100. Results also show that illuminating particles with UV beams from widely separated directions can strongly reduce the S_p. These results suggest that the size distributions and optical properties of the dried particles of virion-containing respiratory fluids are likely important to effectively designing and using UV germicidal irradiation systems for microbes in particles. The results suggest the use of reflective surfaces to increase the angles of illumination and decrease the S_p. The results suggest the need for measurements of the S_p of SARS-CoV-2 in particles having compositions and sizes relevant to the modes of disease transmission.

Keywords:

• Aerosol
• Covid-19
• disinfection
• respiratory droplets
• saliva
• UV inactivation

Introduction

Some diseases caused by viruses and bacteria can be transmitted by sneezed or coughed aerosols, droplets (of any size), or by particles remaining after droplets of respiratory or other fluids have dried-to-equilibrium (termed “dried particles” here) (Huang et al. 2021; Lu et al. 2020; Miller et al. 2020). SARS-CoV-2 (Severe Acute Respiratory Syndrome coronavirus 2), the causative agent of COVID-19 (coronavirus disease 2019), is an enveloped, positive-sense, and single-stranded RNA virus, as are other viruses in the coronavirus subfamily (SARS, MERS, and types causing the common cold). SARS-CoV-2 is found in fluids of the nasopharynx (Landry et al. 2020; Yilmaz et al. 2021), nose (Péré et al. 2020), throat (Zou et al. 2020), and lung (Wolfel et al. 2020). It occurs in saliva (Azzi et al. 2020; Wyllie et al. 2020) and sputum (Pan et al. 2020).

Respiratory fluids may be aerosolized by coughing, sneezing, talking or breathing. Microbes in particles on surfaces may be transferred to humans by direct contact, via fomites, or may be reaerosolized (Fisher et al. 2012; Kesavan et al. 2017; Krauter and Biermann 2007; Paton et al. 2015; Qian et al. 2014). An approach to reducing transmission of diseases associated with microbes in the respiratory tract is to reduce the exposure of persons to viable microbes in the air (e.g., Cheng et al. (2020) for SARS-CoV-2), and/or on surfaces/fomites (Boon and Gerba 2007; Rutala and Weber 2019). Face mask mandates and usage lead to lower rates of COVID-19 cases and deaths (Guy et al. 2021; Joo et al. 2021; Mitze et al. 2020).

There is a need for improved designs of buildings and vehicles, and the airflows within them, to reduce the transmission of airborne diseases while allowing human interactions to the extent possible (Augenbraun et al. 2020; Morawska et al. 2020). There is a need for decontamination of face masks, other personal protective equipment, filters (Fischer et al. 2020; Woo et al. 2012), and surfaces in many locations.

UV germicidal irradiation (UVGI), i.e., UV light at wavelengths from 200- to 320-nm, has been used for over 100 years to inactivate microbes. Low-pressure mercury lamps with a dominant emission peak at 254 nm have been the most common UVGI source. The use of UVGI for air and surface decontamination (including sources, biophysics, and inactivation systems) has been described in detail (Kowalski 2009). Improvements in UVGI sources and application techniques continue to be described as do studies of effectiveness (Anderson et al. 2018; Lindsley et al. 2018; Morawska et al. 2020; Rutala and Weber 2019).

A problem with using UVGI to inactivate microbes is that in some cases viruses and bacteria can be partially shielded from UVGI by clumping of microbes (e.g., Coohill and Sagripanti 2008; Kesavan et al. 2014; Handler and Edmonds 2015) or being on or within particles (Doughty et al. 2021; Osman et al. 2008). This problem is important, in part because inactivation rates obtained with measurements of individual, separated microbes may be higher than rates of inactivation of microbes within clumps or particles. A UV inactivation system may be designed for, and tested with, individual microbes, but then in actual use the microbes may occur within larger particles. In such a case, users and designers may have a false sense of security. Also, system designers may wonder how the problem of shielding from UV could be reduced or circumvented. Removing particles using filters or inactivating particles by increasing the UV intensity are possible solutions. But, questions remain about other possible solutions and tradeoffs.

Modeling the UVGI within microbes in particles can help in understanding shielding quantitatively, and in designing systems which mitigate the effects of shielding from UVGI and thereby achieve a desired level of inactivation. A key objective of modeling is to predict wavelength-dependent UV intensity within particles/droplets in a way that is (1) based on fundamental physical first principles, (2) approximates the detailed geometry of multiple particle systems and variable illumination directions, and yet is (3) computationally tractable. The fundamental physical principles in this case are represented by Maxwell's wave equations which here (with non-magnetic media and no free charges) reduce to the 3D vector partial differential equation (Bohren and Huffman 1983): (1) $$\nabla \mathrm{ }\times \mathrm{ }\nabla \mathrm{ }\times \mathbf{E}\mathrm{ }–\mathrm{ }{(2\pi m/\lambda )}^2\mathbf{E}=\mathrm{ }0,$$(1) where ∇ × is the curl operator, E is the complex electric field vector for time-harmonic fields, λ is the wavelength in free space (for UVGI applications, typically close enough to the wavelength in air), and m = m_r + i m_i is the complex refractive index of the material, where m_r (or m_i) is referred to as the real (or imaginary) part of the complex refractive index. (In another common notation, n is used for m_r, and k is used for m_i, but here k is used for the UV inactivation rate constant.) Note that m is a material property and thus is position dependent. Quantities of interest, such as fluence and absorption rate, are typically obtained from the squared magnitude of E.

In general, it is possible to solve Equation (1)(1) $$\nabla \mathrm{ }\times \mathrm{ }\nabla \mathrm{ }\times \mathbf{E}\mathrm{ }–\mathrm{ }{(2\pi m/\lambda )}^2\mathbf{E}=\mathrm{ }0,$$(1) , for a given distribution of m and exciting electric field(s) (typically monochromatic plane wave(s)), using purely numerical means (e.g., a discrete dipole method or finite-difference time-domain method). Such approaches, applied to the systems of interest here (which, overall, are on the order of 4–100 wavelengths in diameter), would be computationally demanding.

Methods based upon analytical (as opposed to numerical) solutions to Equation (1)(1) $$\nabla \mathrm{ }\times \mathrm{ }\nabla \mathrm{ }\times \mathbf{E}\mathrm{ }–\mathrm{ }{(2\pi m/\lambda )}^2\mathbf{E}=\mathrm{ }0,$$(1) can be less computationally intensive. For examples relevant to modeling UV in microbes in particles, analytical solutions are possible when m is piecewise-constant, and when the boundaries separating one value of m from another are in the form of (1) spherical surfaces and/or (2) infinite plane surfaces. For such conditions E can be represented exactly in the piecewise-constant regions by fundamental basis functions (which are solutions to Equation [1](1) $$\nabla \mathrm{ }\times \mathrm{ }\nabla \mathrm{ }\times \mathbf{E}\mathrm{ }–\mathrm{ }{(2\pi m/\lambda )}^2\mathbf{E}=\mathrm{ }0,$$(1) ), and the mathematical problem reduces to matching the continuity conditions on the boundary surfaces. The most basic case of such solutions is Lorenz-Mie theory, which provides the exact E (derived using separation-of-variables methods to Equation [1]) for a single sphere in an unbounded medium and excited by a plane wave. The extension of this approach to multiple spheres, and to spheres adjacent to plane boundaries is termed the Multi-Sphere T-Matrix (MSTM) method which has been described in detail (Mackowski 2008; Mackowski and Mishchenko 2011, 2013). The method requires the application of non-trivial mathematical techniques and cannot be simply described in a few equations or sentences. The implementation of the solution also requires numerical methods for solving linear equations. The overall computational effort, however, is considerably less than that associated with a direct numerical solution to Equation (1)(1) $$\nabla \mathrm{ }\times \mathrm{ }\nabla \mathrm{ }\times \mathbf{E}\mathrm{ }–\mathrm{ }{(2\pi m/\lambda )}^2\mathbf{E}=\mathrm{ }0,$$(1) for the system under consideration.

Some previous attempts to approximate the UV intensities in clusters or agglomerates have assumed that the UV light travels in a straight line as it passes through the surfaces of spores (as if it were gamma rays), but attenuated by UV absorbing molecules in the material (Handler 2016; Handler and Edmonds 2015). No reflection or refraction was included in the model. This approach does not suggest the regions within a sphere that are well shielded because the rays refract (bend) away from these regions as they enter the sphere, regions which are clear in exact solutions for spheres illuminated with 266-nm light (Hill et al. 2015, Fig. 7), and in intensities calculated using geometrical optics (Chowdhury et al. 1992). Kowalski et al. (2019) used Mie Theory to calculate scattering and extinction cross sections for an individual sphere. However, they assumed that the scattered light does not interact with any other spheres in the cluster even if the spheres are in contact. Even for a particle immediately downstream from a given sphere Kowalski et al. (2019) assume that “any forward scattering that reaches the second particle is minor,” and thus they ignore it. These assumptions may be useful for approximations of UVGI in clusters in water in cases where the difference between the refractive index of bacteria and water is small (and so the refraction at the spore surface and the scattering by the individual bacterium tend to be small). However, Kowalski et al. (2019) claimed incorrectly that such an approach was applicable to spores and other microbes in air. In Kowalski et al. (2019, Fig. 2), rays traveling from the first to the second sphere are drawn parallel to the rays incident upon the sphere. In reality, a solid sphere in air, having a typical refractive index (1.5), illuminated by a plane wave, focuses a large fraction of the light impinging upon it into a small region just outside the sphere (Barber and Hill 1990, Figs. 4.32 and 4.34). Short-focal-length ball lenses are used in microscopy (Agbana et al. 2018). Microbes in air refract and reflect light at their surface.

The MSTM Fortran program, details of which can be found in Mackowski (2008) and Mackowski and Mishchenko (2011, 2013) was used by Doughty et al. (2021) to calculate UV intensities in spherical particles which have a diameter (100 nm, in the range of the 60–140 nm reported for SARS-CoV-2 (Zhu et al. 2020)) and optical properties selected to approximate SARS-CoV-2 virions. In the rest of this paper, these spherical approximations of virions are termed “virions.” The virions were randomly positioned inside a “host” spherical particle (1-, 5, or 9-µm diameter) with optical properties approximating dried-to-equilibrium respiratory fluid. The 0.1-µm case is the lower-limit of host diameter. The host particles were in some cases resting on a surface. The calculated results showed that (1) virions in particles can be partially shielded from UVGI whether the host particle is on a surface or not; (2) virions can be protected both by absorption of UVGI within the host particle and by refraction at the air-particle surface; (3) the extent of shielding is affected by the wavelength of the UVGI and properties of the dried-droplet host particle; (4) shielding increases as the size of the host particle increases; (5) illumination with incident beams from widely separated directions tends to decrease the shielding from UVGI; and (6) shielding is decreased in particles illuminated uniformly from all directions as may occur for a particle in turbulent air illuminated for enough time to tumble and spin through all orientations relative to the incident beam(s).

In our previous modeling of shielding using the MSTM (Doughty et al. 2021), the key parameter calculated for each virion was (2) $$R_i=C_{\text{abs},\mathrm{i}}/C_{\text{abs},\text{single}}=Q_{\text{abs},\mathrm{i}}/Q_{\text{abs},\text{single}},$$(2) where C_abs,i is the absorption cross section of the i^th virion in a particle (either resting on the surface or not), defined so that C_abs,iI₀ is the net rate of energy transfer into the virion (i.e., the integration of the normal component of the Poynting vector over the surface of the i^th virion), where I₀ is the irradiance illuminating the host sphere. C_abs,single is the absorption cross section of an individual virion (either resting on the surface or not), i.e., defined so that C_abs,singleI₀ is the net rate energy transfer into an individual virion. The absorption efficiency of a sphere is Q_abs = C_abs/πa², where a is the radius. A complete derivation of the cross-section formulas for spheres-inside-spheres is in Mackowski (2014).

The R_i is especially useful for understanding the physics of shielding of microbes from UV and how and where it occurs. However, the survival fraction (S) is the more important quantity for modeling and designing systems and procedures for minimizing disease transmission. The simplest common expression for survival fraction (S) is the single-stage decay model (Kowalski 2009): (3) $$S\left(kF\right)\mathrm{ }=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-kF\right),$$(3) where k is the UV rate constant for inactivation (m²/J) and F is the UV-exposure dose (Fluence), energy per unit area (J/m²). Also, F = It, where I is the intensity of the irradiance (W/m²), and t is the time of illumination. For a given virus and set of experimental conditions (humidity, etc.) the S is measured for different F, and then k is determined using Equation (3)(3) $$S\left(kF\right)\mathrm{ }=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-kF\right),$$(3) . The k have been measured for SARS-CoV-2 (Heilingloh et al. 2020; Ratnesar-Shumate et al. 2020), and other viruses (Sagripanti and Lytle 2011; Schuit et al. 2020), bacteria, and fungi (Kowalski 2009). More complex expressions for S(F) of the survival fraction, such as those which incorporate an additional term or terms to account for shoulders, tails, or other features (Kowalski 2009, Ch. 3; Kowalski et al. 2019) are beyond the scope of this paper.

To account for shielding of spores within a cluster of spores, Handler and Edmonds (2015) and Handler (2016) estimated the F at each spore and used it to calculate an average S(F) for the spores in the cluster. They, and Kowalski et al. (2019), assumed that microbes not on the illuminated surface of a cluster or particle are illuminated by an attenuated plane wave that travels in the same direction as the incident wave. However, the UV light within particles does not take the form of a simple attenuating plane wave (e.g., a virion in a particle may be at a minimum or maximum of a standing wave within the particle) and so an S based on assuming attenuated plane waves is not an adequate approach in general (see “Methods”).

The objective of this study is to increase understanding of the survival fraction of SARS-CoV-2 virions (and other virions and microbes) when the virions (or other microbes) are within particles which partially shield them from UVGI. The hope is that increased understanding of shielding from UVGI, and the modeling capabilities described, will allow designers and users of germicidal- or solar-UVGI systems to better mitigate the effects of shielding from UVGI. Here, the survival fraction (S_i(kF)) for the i^th virion in a particle is calculated as (4) $$S_i\left(kF\right)\mathrm{ }=\mathrm{ }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-kF{R}_i\right),$$(4) (see “Methods”). We calculate and illustrate curves of average survival fractions vs. kF for many of the cases for which we calculated R_i previously (Doughty et al. 2021), for virions of SARS-CoV-2 (approximated as 100-nm diameter spherical particles) within otherwise homogeneous spherical particles having optical properties to approximate dried particles of sneezed or coughed respiratory fluids. As in Doughty et al. (2021), the host particles have diameters of 1-, 5- and 9-µm (and 0.1 µm for the individual virion); the UV wavelengths are 260 and 302 nm; the particle is in air or on a surface with m = 1.4 + i0.0001; and there are several combinations of incident beam directions. The UV wavelengths were used because 302-nm light is at the most germicidal end of the solar UV, and 260-nm light is highly damaging to nucleic acids and close to the 254-nm primary emission of low-pressure mercury lamps.

Methods

The problem of calculating the UV intensities and estimating the optical properties of the virions and dried respiratory fluids was described in the Introduction and in Doughty et al. (2021), with details on the MSTM given in Mackowski (2008, 2014) and Mackowski and Mishchenko (2011, 2013).

A way to obtain Equation (4)(4) $$S_i\left(kF\right)\mathrm{ }=\mathrm{ }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-kF{R}_i\right),$$(4) from Equation (3)(3) $$S\left(kF\right)\mathrm{ }=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-kF\right),$$(3) is as follows. In Equation (3)(3) $$S\left(kF\right)\mathrm{ }=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-kF\right),$$(3) , F is the same for each virion. We desire an expression for S_i(kF) based on the energy absorbed by the i^th virion in the host particle when the host particle is illuminated with fluence F. To achieve that expression, the numerator and denominator of the exponent in Equation (3)(3) $$S\left(kF\right)\mathrm{ }=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-kF\right),$$(3) are multiplied by C_abs,single to obtain (5) $$S\left(kF\right)\mathrm{ }=\mathrm{ }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\left(k/C_{\text{abs},\text{single}}\right)\mathrm{ }\left(F{C}_{\text{abs},\text{single}}\right)\right),$$(5) where FC_abs,single = energy absorbed by a single virion illuminated with F, and where k/C_abs,single is the yield for viral inactivation (inactivations/J if k is in J/m²), or the quantum yield for viral inactivation (inactivations/photon if k is in m²/photon and F is in photons/m² (Rauth 1965, Eq. 2(2) $$R_i=C_{\text{abs},\mathrm{i}}/C_{\text{abs},\text{single}}=Q_{\text{abs},\mathrm{i}}/Q_{\text{abs},\text{single}},$$(2) )). Then to obtain the equivalent S(kF) for the i^th virion, the FC_abs,single in Equation (5)(5) $$S\left(kF\right)\mathrm{ }=\mathrm{ }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\left(k/C_{\text{abs},\text{single}}\right)\mathrm{ }\left(F{C}_{\text{abs},\text{single}}\right)\right),$$(5) is replaced by the energy absorbed by the i^th virion when the particle is illuminated by F, that is, by FC_abs,i, to obtain (6) $$S_i\left(kF\right)\mathrm{ }=\mathrm{ }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\left(k/C_{\text{abs},\text{single}}\right)\mathrm{ }\left(F{C}_{\text{abs},\mathrm{i}}\right)\right),$$(6) which is equivalent to Equation (4)(4) $$S_i\left(kF\right)\mathrm{ }=\mathrm{ }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-kF{R}_i\right),$$(4) . Also in Equation (4)(4) $$S_i\left(kF\right)\mathrm{ }=\mathrm{ }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-kF{R}_i\right),$$(4) , the FR_i can be taken to define an F_i for each virion.

The key function illustrated in the figures is S_p(kF), the average of the survival fractions (S_i) of all the N modeled virions in all the particles with the same specified composition. It is defined as: (7) $$S_p\left(kF\right)=\frac{1}{N}\sum _{i=1}^N\exp \left(-\mathit{kF}{R}_i\right).\mathrm{ }$$(7)

The R_i can range over several orders of magnitude in absorbing particles with dimensions of several wavelengths. In such cases, large numbers of R_i are needed to adequately simulate the distribution of R_i. For 9-µm particles, 120 simulations of 100 virions/particle were calculated and combined to approximate the distribution of R_i for randomly positioned virions, and so N = 12,000 for this case. The values of N, and why these N were chosen were discussed in Doughty et al. (2021).

Additional parameters are defined as follows. The penetration depth, δ, is the distance z at which I(z)/I_o = 1/e = 0.368 for a UV planewave propagating in a material, where the intensity of the irradiance of the wave at position (z) is (8) $$\mathrm{I}\left(\mathrm{z}\right)\mathrm{ }={\mathrm{ }\mathrm{I}}_{\mathrm{o}}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\mathrm{z}/\delta \right),$$(8) where I_o is I at z = 0. The δ, m_i and UV wavelength λ are related as (9) $$\delta =\lambda /m_{i}4\pi .$$(9)

The notation for angles follows that used in the MSTM codes (Mackowski 2008). A line perpendicular to the surface and passing through the center of the particle and through the planar surface defines the z-axis. The zenith angle (β) is the angle from this axis. When there is more than one illumination beam with β ≠ 0˚, the azimuthal angles (α) between these beams are also specified.

Table 1 provides, for each curve in the figures, the wavelength (λ) in nm, diameter (d) of the host particle in µm, numbers of virions (N) used to obtain the S_p(kF), m_i, penetration depth (δ), incident angle(s) β (and α when needed), and S_p(kF) for specific values of kF: 1, 10, 100, 1000, 10,000. The S_p at specific kF are given because extracting accurate values from the Figures is tedious. The m_i for the virions is 0.0021 at 260 nm and 0.000092 at 302 nm.

Table 1. Figure numbers, parameters, and average survival fraction of virions (Sp) for virions having UV-inactivation rate constant k within host particles illuminated with fluence F, for selected values of the dimensionless parameter kF.

Figure	λ^A nm	d^B µm	N^C	mi^D	δ^E µm	Angle(s) Degrees	Sp @ kF = 1	Sp @ kF = 10	Sp @ kF = 100	Sp @ kF = 1000	Sp @ kF = 10000
1	260	0.1	1	0.00728	2.8	Β^F = 0	0.368	0.000045	0.000000	0.000000	0.000000
1	260	1	500	0.00728	2.8	β = 0	0.442	0.015551	0.000000	0.000000	0.000000
1	260	5	3000	0.00728	2.8	β = 0	0.671	0.217234	0.076803	0.004748	0.000000
1	260	9	12000	0.00728	2.8	β = 0	0.759	0.281645	0.148994	0.050528	0.002829
1	302	1	500	0.00241	10.	β = 0	0.377	0.003765	0.000000	0.000000	0.000000
1	302	5	3000	0.00241	10.	β = 0	0.569	0.163955	0.022992	0.000010	0.000000
1	302	9	12000	0.00241	10.	β = 0	0.617	0.225805	0.087981	0.008403	0.000000
2	302	9	12000	0.0004	59.9	β = 0	0.565	0.188023	0.034816	0.000059	0.000000
2	302	9	12000	0.00241	10.	β = 0	0.617	0.225805	0.087981	0.008403	0.000000
2	260	9	12000	0.00728	2.8	β = 0	0.759	0.281645	0.148994	0.050528	0.002829
2	260	9	12000	0.01456	1.4	β = 0	0.861	0.404283	0.179325	0.096010	0.019449
3	260	9	12000	0.00728	2.8	β = 0	0.759	0.281645	0.148994	0.050528	0.002829
3	260	9	12000	0.00728	2.8	β = 40	0.753	0.267206	0.125280	0.039222	0.003820
3	260	9	12000	0.00728	2.8	β = 80	0.710	0.291724	0.135020	0.049240	0.008897
3	260	9	12000	0.00728	2.8	β = 0,40	0.751	0.227918	0.075299	0.010311	0.000088
3	260	9	12000	0.00728	2.8	β = 0,80	0.718	0.185450	0.056947	0.012725	0.000289
3	260	9	12000	0.00728	2.8	β = 80, α^G = 0,180	0.683	0.123693	0.005391	0.000908	0.000067
3	260	9	12000	0.00728	2.8	β = 80, α = 0,90, 180,270	0.675	0.095058	0.003309	0.000716	0.000021
3	260	9	12000	0.00728	2.8	β = 0 + β = 80, α = 0,180	0.699	0.111898	0.004289	0.000004	0.000000
3	260	9	12000	0.00728	2.8	β = 0 + β = 80, α = 0,90, 180,270	0.686	0.087636	0.001282	0.000000	0.000000
4	260	5	3000	0.00728	2.8	All^H	0.589	0.017202	0.000000	0.000000	0.000000
4	260	9	4000	0.00728	2.8	All	0.751	0.080112	0.000000	0.000000	0.000000

^AWavelength in nm.

^BDiameter of host particle (µm).

^CNumber of virions used to obtain average.

^DImaginary part of refractive index of host particle.

^EPenetration depth in µm.

^FZenith angle of UV beam (degrees).

^GAzimuthal angle of illumination (degrees).

^H“All” indicates averaged over all orientations.

Each curve in the figures is calculated by selecting a set of kF that are evenly spaced in log space (sufficiently close that the curve is smooth) and then using Equation (7) with the N values of R_i calculated with the parameters given in the specific row in Table 1 (as in Doughty et al. 2021, Table 3), to calculate the S_p for each of the selected kF. S_p is plotted vs. the dimensionless parameter kF to emphasize the relation between k and F. If k = 1 m²/J and the abscissa is F in J/m², no change in the tick labels would be needed. If k = 0.1 m²/J and the abscissa is F in J/m², each labeled value for each tick would need to be increased by a factor of 10.

In the “Results and Discussion,” Figure 1 illustrates the effects of host particle diameter (0.1-, 1-, 5-, or 9-µm) on S_p(kF) for two λ, δ, and m_i(λ). Figure 2 illustrates the effects of δ on S_p(kF) by using a set of four m_i(λ), two at 260 nm and two at 302 nm, for 9-µm host particles. Figure 3 shows the effects of illumination from different incident angles and combinations of these on the shielding from UVGI, as seen in S_p(F), for 9-µm host particles. Figure 4 illustrates the effects on S_p(F) of illumination of particles over all orientations, as compared with illumination from one or a few angles.

Figure 1. Average survival fractions (S_p) vs. kF for 100 nm virions within spherical particles of dried respiratory fluids of the sized indicated and resting on a surface with real refractive index 1.4.

Figure 2. S_p vs. kF for different penetration depth (and m_i) for the dried respiratory fluids, and all d = 9 µm except for the 0.1 µm individual virion shown for comparison. The blue and green curves are at 302 nm and the red and purple are at 260 nm.

Figure 3. S_p vs. kF, for 9-µm particles, 260 nm UVGI, δ = 2.8 µm, with various combination of illumination directions: the three curves on the right are for a single illumination angle β = 0˚, 40˚ and 80˚; the next two curves have an illumination with β = 0˚ and a second wave, β = 40˚, or 80˚; and the next four curves have more widely separated beams. In each case, the total intensity illuminating the particle is the same.

Figure 4. S_p vs. kF for virions in particles where the orientation averaged curves for 5-µm (green) and 9-µm (blue) are in dashed lines and are for particles not on a surface, and the other curves are shown for comparison. All cases are with 260-nm UVGI.

Results and discussion

Average survival fractions are larger in larger particles

Figure 1 shows S_p (average of the S_i as in Eq. [7]) as a function of kF for virions within 1-, 5- and 9-µm diameter host particles, and for individual 0.1-µm-diameter virions. In Figure 1, the S_p increases with host particle size for both wavelengths used (260 and 302 nm). As seen in Figure 1 and the table, when λ = 260 nm and kF = 10 the S_p is: 0.000045 for the individual virion, 0.0156 for the 1-µm particles, 0.217 for 5-µm particles, and 0.28 for the 9-µm particles. When λ = 260 nm and kF = 100 the S_p is: < 0.0000005 for the 1-µm particles, 0.077 for the 5-µm particles, and 0.149 for the 9-µm particles.

In Figure 1, the S_p are larger in host particles when λ = 260 nm than when λ = 302 nm, but that does not mean that lower F are needed for inactivation. The m_i for the virions is 0.0021 at 260 nm and 0.000092 at 302 nm (Doughty et al. 2021, Table 2), and so virions modeled here tend to be less absorbing of UVGI and the k tend to be smaller at 302 nm than at 260 nm. Also, the quantum yields for inactivation are lower at 302 nm than at 260 nm for the viruses reported by Rauth (1965, Fig. 5) and that also tends to make the k smaller at 302 nm. Because k tends to be smaller at 302 nm than at 260 nm, a higher F is required to achieve the same kF and S_p.

The curves of S_p(kF) vs kF in Figures 1–4 are not simple exponentials (except for the individual-virion curves) because they are averages of many (N) simple exponentials. For the i^th exponential curve, S_i = 1/e when kF = 1/R_i. As kF increases, the S_i of the virions with the largest R_i decrease faster than do the S_i of virions with smaller R_i. Consequently, the relative contributions of the different virions to the S_p vs kF curve changes with kF. For large kF, the virions with the smallest R_i dominate the S_p.

Average survival fractions vs. kF are larger in particles with higher absorptivity

Figure 2 shows the S_p vs. kF curves for virions in host particles having different UV absorptivities for the material of the dried droplet, as indicated by the m_i stated in the caption and Table 1. The curves are labeled by the penetration depth δ, which is useful for understanding the shielding. In Figure 2, the m_i was chosen to span a wider range than the estimated “typical” values used in the other figures. In Figure 2, at 260 nm, in addition to the δ = 2.8 and 10 µm in Figure 1, there is also a more strongly absorbing particle with δ = 1.4 µm (m_i = 0.01456). At 302 nm, there is also a more weak-absorbing particle with δ = 59.9 µm (m_i = 0.00040). This larger range of δ is used to better account for the large variations in some of the primary light-absorbing components of saliva or nasal fluids, such as, at 260 nm, nucleic acids (e.g., Poehls et al. 2018), and at 302 nm, uric acid (Hawkins et al. 1963; Riis et al. 2018, Figure 2).

In Figure 2, the S_p of even the least absorbing particle is many times greater than that of the individual virion. For example, even when the δ is 6.7 particle diameters (i.e., the 9-µm particle at 302 nm with δ = 59.9 µm), when kF = 10, the S_p for the virions in the particle is 0.188, which is 4200 times greater than the S_p = 0.000045 for the individual virion. Because the estimated absorption is so low, the increase in S_p occurs primarily because of refractive shielding. For solar wavelengths longer than the 302-nm UVGI used here, the absorption tends to be even lower, with larger δ. However, because of refractive shielding the large increase in S_p remains (as compared to that of the individual virions).

This increase in S_p for higher absorption by the host particle is also seen in Figure 1, where the S_p are smaller at λ = 302 nm than at λ = 260 nm because at 302 nm the particles are less absorbing of UVGI. For example, when kF = 10, the S_p for 1-µm particles are 0.0156 at 260 nm, but 0.00377 at 302 nm. Also when kF = 10, the S_p for 5-µm particles are 0.217 at 260 nm but 0.164 at 302 nm.

Survival fractions are smaller in particles illuminated from widely separated directions

Figures 3 and 4 illustrate S_p vs. kF for illumination with beams with β = 0˚ or β ≠ 0˚, or with beams from multiple incident directions (with F split equally between the beams). These figures illustrate how illumination with beams from multiple angles can result in reductions in S_p. In Figure 3, when kF = 10, the S_p in the 9-µm particles range from 0.0876 with 5 widely separated UV beams, to 0.282 with one UV beam at normal incidence. For the individual virion, S_p = 0.000045. When kF = 100, the S_p range from 0.149 with one UV beam with β = 0˚, to 0.00128 with 5 widely separated UV beams, i.e., 116 times smaller than the single-beam case.

In Figure 3, the curves can be grouped into three main categories: (a) illumination with one beam only, with β = 0˚, 40˚, or 80˚; (b) illumination with multiple beams, but the differences in illumination angles are less than 90˚; and (c) illumination with multiple beams where at least one of the angular differences between two of the angles of incidence is greater than 90˚.

With only one illumination beam the differences in S_p distributions are relatively small, but not negligible. The set of beams providing the lowest S_p includes five beams: one beam heading perpendicular to the surface (β = 0˚), and four beams, each with β = 80˚ from the vertical but approaching from different azimuthal angles, α = 0˚, 90˚, 180˚, 270˚. At kF = 100, the S_p = 0.00128 for this set. A set of four beams with β = 80˚ and α = 0˚, 90˚, 180˚, or 270˚ (similar to the 5-beam case but with no β = 0˚ beam), has S_p = 0.00331.

Highly UV-reflective surfaces (Krishnamoorthy and Tande 2016; Lindsley et al. 2018; Rutala et al. 2013; Ryan et al. 2010) can increase overall UVGI and can help in directing UVGI into shaded regions of a room. The results in Figure 3 suggest an additional benefit of highly UV-reflective surfaces, i.e., helping to achieve illumination with multiple beams where the angular differences between the angles of incidence with respect to the surface is relatively large (see Doughty et al. (2021) for the R_i distributions).

In Figures 3 and 4, the UV illumination energy is split between the beams so that the total intensity in each of the cases in Figure 3 can be directly compared. Potential increases in the overall UV-illumination intensity that could result from generating additional beams using reflective surfaces could very likely decrease the S_p. However, such calculations depend upon factors such as the positions, angles and reflectivities of the relevant surfaces, and are beyond the scope of this paper.

Figure 4 shows the orientation-averaged S_p vs kF for virions in 5- and 9-µm particles (dashed lines). Such orientation averaged results might be obtained with a fixed-orientation particle illuminated equally from all directions or with a particle that rotates (tumbles and spins) through all orientations with respect to one or more UV beams.

In Figure 4 (see also the numbers in Table 1), the reductions in S_p with orientation averaging are remarkable. For 5-µm particles when kF = 100, the S_p is less than 5 × 10⁻⁷ for the orientation-averaged case and is 0.0768 for the single beam with β = 0˚. For kF > 9, the orientation-averaged case has a lower S_p than any of the other cases with 9-µm. For 9-µm particles when kF = 100, the S_p is less than 5 × 10⁻⁷ for the orientation-averaged case but is 0.00128 for the 5 beam case, and 0.149 for the single beam β = 0˚ case. Also in Figure 4, when kF > 11 the S_p for the orientation-averaged 5-µm particles is less than the S_p for 1-µm particles (β = 0˚). When kF > 33, the S_p for the orientation-averaged 9-µm particles is less than S_p for 1-µm particles (β = 0˚). Such results may appear confusing. How can orientation averaging be so important compared to a 9-fold difference in diameter when the smaller particle is only 1-µm diameter? The reason is that no virions are refractively shielded well in the orientation-averaged particles. But the wavelength is so short (260 nm), that even a 1-µm particle illuminated from one direction can have 100-nm regions that are refractively shielded (e.g., Hill et al. 2015, Figure 7a).

Summary: virions in particles tend to have larger UV survival fractions

Virions within UV-illuminated spherical particles (homogeneous except for the virions) tend to have larger S_p than do individual virions, as seen in the calculated values in the figures and table. Several key relations can be seen in the calculated results.

1. For host particles on a surface with a given m_r + i m_i, illuminated with normally incident UVGI (zenith angle β = 0˚) at a given wavelength, as the host particle size increases the S_p increases (except when kF<1 for 1-um particles at 302 nm) (Figure 1 and Table 1). This relation occurs for both wavelengths studied (260 and 302 nm). To achieve S_p = 0.001 in 1-µm particles with 260 nm light, kF must be approximately 4× higher for 1-µm particles, 230× higher for the 5-µm particles and 2000x greater for the 9-µm particles, all with respect to the individual virion. At 302 nm, the increases in kF required are smaller: 2×, 56× and 300×, respectively, because the absorption of UV by the particle is smaller.

2. For particles on a surface illuminated with normally incident UVGI (β = 0), as the penetration depth in the host particles decreases the S_p increases. This relation is most clear in Figure 2 where the curves are labeled with the four different δ, but is also seen in Figure 1.

3. UV illumination of particles from multiple directions, especially widely varying directions, can reduce the S_p (Figures 3 and 4, and Table 1). Particles illuminated equally from all directions tend to have especially low S_p (Figure 4). Particles in outdoor air for a sufficient time on a gusty and sunny day may be examples of particles illuminated from all directions. UVGI systems may be designed to illuminate particles somewhat uniformly from all orientations. Or, all-orientation S_p may be approximated by particles that tumble and spin sufficiently rapidly as they are carried within turbulent airflows through a UVGI system with illumination from one side.

Uncertainties

The compositions of airway fluids may vary with a person’s genetic makeup, disease state (if any), time since last drink or meal, degree of hydration, exertion level, region of the respiratory tract, and other factors. In the estimates of m_i from airway-fluid compositions used here, the molecules absorbing the most UVGI at 260-nm are DNA, RNA, and nucleosides/nucleotides. In using calculated S_p to improve the use of UVGI in reducing the transmission of disease, a sense of the ranges of measured concentrations of UV-absorbing molecules such as nucleic acids can be beneficial. Konečná et al. (2020) reported average dsDNA concentrations in whole saliva of 0.24 g/L in healthy subjects and 0.5 g/L in persons with periodontitis. In supernatants from centrifuged (1,600× g) samples they measured average concentrations of DNA of 0.054 g/L in healthy subjects and 0.07 g/L in persons with periodontitis. Fahy et al. (1995) measured a range of cell-free DNA concentrations from 0.06 to 5.8 g/L (average 0.5 g/L) in patients with asthma. Pandit et al. (2013) found the extracellular RNA in the saliva (supernatant after centrifugation) ranged from 0.17 to 0.76 g/L in healthy control subjects and from 0.09 to 0.36 g/L in cancer patients. SARS-CoV-2 and other microbes can induce neutrophils in saliva to release DNA, histones, and proteins to form Neutrophil Extracellular Traps (NETs) (Arcanjo et al., 2020; Lachowicz-Scroggins et al. 2019). Eosinophil Extracellular Traps (EETs) and Basophile Extracellular Traps (BETs) also contain DNA and contribute to airway defense and inflammation (Yousefi et al. 2020). Bacterial and viral DNA and RNA (Haro et al. 2020) also occur in airway fluids, but typically in lower concentrations than host DNA and RNA. To partially account for the large variability in DNA and RNA concentrations, a large range of m_i are used in the calculations here (Figure 2).

In this paper, the model host particles are spherical and homogeneous (except for the virions), and the positions of virions within each particle are random. The MSTM can be used to model UVGI in more-complex particles than those investigated here, e.g., particles with additional spherical regions inside and outside the particle, each with its own m_r and m_i. However, such modeling requires measurements or assumptions of the geometries of the particles and m_r and m_i of each of the various regions. The homogeneous-sphere-with-virions assumption used here is a useful case to investigate first, given the large numbers of relevant variables: m_r and m_i of the virion(s), host particle and surface; size(s) of the virions and the host particle; virion locations; wavelengths; number and angles of beams; presence of a surface or not. However, for more accurate modeling of the S_p of more complex particles, additional studies along the lines of those of Vejerano and Marr (2018) and Walker et al. (2021) would likely be needed.

Vejerano and Marr (2018) collected droplets of artificial saliva on a superhydrophobic surface and used optical microscopy to examine the particles as they dried as the RH was lowered. In their fluorescence images, particles that were less than roughly 20 µm appear to us as approximately spherical, but inhomogeneous. The nonsphericities appear to arise from the crystallization/precipitation of different materials in different locations. Walker et al. (2021, their Fig. 5c) collected dried particles of artificial saliva on a surface and examined them with scanning electron microscopy. The particles generally look smooth on the order of the apparent resolution, except for some unclear features. The smooth appearance suggests that particles remaining after drying in air (as in Walker et al. 2021) or after drying on a superhydrophobic surface can be quite different from the material remaining after a droplet of a liquid or liquid/solid mixture is deposited on a not-superhydrophobic surface and then dried (as in Papineni and Rosenthal 1997). Walker et al. (2021) also state that light scattering patterns indicate that the particles are inhomogeneous. Vejerano and Marr (2018) and Walker et al. (2021) are the only studies (to the authors' knowledge) with as much information related to morphologies of small particles from dried saliva or respiratory fluids with sizes in the range of interest (e.g., Fennelly 2020) and which dry in air or on a superhydrophobic surface. However, there are uncertainties in how to apply these studies to modeling the Sp of dried saliva or respiratory particles in the 0.5- to 9-µm size range because the sizes, positions and the optical properties (or compositions) of internal structures (if any) are not specified, and because respiratory fluids tend to be more complex than the artificial fluids used in some parts of the studies mentioned. Differences between human and model saliva, and between gastric and submaxillary mucin, are not insignificant (Sarkar et al. 2019). Because of the additional complexity of actual saliva and other respiratory fluids (containing DNA, RNA, mucins, and other proteins, carbohydrates, etc.), there is cause to wonder if crystallization/precipitation may be more likely to occur as smaller crystals in dried coughed/sneezed droplets, as compared with fewer larger crystals in the artificial salivas. For their fluorescence images, Vejerano and Marr (2018, Figs. 4 and 5) used an aqueous solution of gastric mucin, NaCl, and a phospholipid surfactant. Walker et al. (2021), following (Woo et al. 2012), used an aqueous solution of Na⁺, K⁺, Mg²⁺, Ca²⁺, Cl^-, HCO₃ ^-, H₂PO₄ ^-, SCN^-, amylase, mucin from porcine stomach, glucose and very small concentrations of glucose, amino acids, vitamins, cofactors, and sulfate. Neither includes DNA or RNA, which may affect the diffusion rates of various solutes (Dix and Verkman 2008) or particles (Linssen et al. 2021) within droplets.

Conclusions

Average survival fractions (S_p) of virions in particles of dried respiratory fluids exposed to UVGI can be much larger than survival fractions of virions not within such particles, as illustrated in Figures 1–4. The S_p were calculated using UV intensities within virions calculated with the MSTM, an exact method (within numerical error). The model particles are spherical and homogeneous, except for the virions. The particles may be on a surface and may be illuminated with any number of incident beams.

The key findings are:

1. The S_p increases as the particle size increases, where the smallest relevant particle is an individual virion (Figures 1 and 4).

2. The S_p increases as the UV-absorptivity of the material comprising the particle increases (Figures 1 and 2). However, even when the material of the particle absorbs very little UV light, the S_p increases as the particle size increases.

3. The S_p tends to decrease as more UV beams, from more widely separated directions, illuminate the particle, where the total UV energy is the same for each case (Figures 3 and 4). In the limiting case of equal illumination from all angles, the S_p tend to be especially low, e.g., S_p for the orientation-averaged 5-µm particles is smaller than the S_p for fixed orientation 1-µm particles unless kF<11 (Figure 4). The orientation-averaged case is equivalent to the one where particles rotate (spin and tumble) through all orientations with respect to the illumination beams during their time of exposure to UVGI.

4. Increases in S_p for virions in particles can be mitigated by using more intense UVGI sources, and/or more UVGI sources, and/or with highly UV-reflective surfaces (Lindsley et al. 2018; Rutala et al. 2013; Ryan et al. 2010). An advantage of using highly UV-reflective surfaces is that in addition to increasing the UV intensity, they can also be positioned to increase the numbers of directions from which particles are illuminated, and the angular separation between the various angles of illumination, and thus substantially reduce the S_p as compared to illumination with a single-beam or beams from similar directions.

Detailed, accurate modeling of the S_p of virions in particles requires knowledge of the morphologies of the virion-containing particles, i.e., the 3-D shapes, positions, and compositions or optical properties of each region of the particle.

Recommendations

In designing, testing, and using UVGI systems to inactivate viruses, recognize that virions within particles larger than a very few UV wavelengths tend to have larger survival fractions than do individual virions, and that the survival fractions increase with particle size.

In evaluating UVGI systems for efficacy of inactivation, test for inactivation with virion-containing particles having sizes and compositions representative of the particles that need to be inactivated. For example, if tests were done only with virion-containing particles smaller than 1 µm, but the system may be used with a significant fraction of particles as large as 4 to 5 µm, then recognize that the results shown here suggest that the measured survival fractions will very likely be higher for the 4- to 5-µm particles.

In designing and deploying UVGI systems for virions in particles sufficiently large- and/or absorptive to shield virions, consider, to the extent possible, systems that illuminate particles from many, widely separated, angles (see Figures 3 and 4). Consider employing systems with surfaces that are highly reflective at the relevant UV wavelengths, because such surfaces can increase both the UV intensities and the ranges of angles of incidence of the UV illumination, and thereby decrease the survival fractions. For systems in which particles of the relevant sizes spin and tumble through all orientations as they are illuminated, the need for illumination from many directions, including reflected beams, is not necessary, but it should be recognized that verification of adequate tumble/spin rates is not trivial for many rapid flow-through systems.

Code availability

Researchers interested in the MSTM codes to calculate the results Q_abs for virions may email D.W. Mackowski ([email protected]). Those interested in using the MSTM to generate the R_i for the cases studied here may email D. C. Doughty ([email protected]). Those interested in using the R_i to generate the Figures or Table 1 may email S. C. Hill.

Acknowledgments

Hill and Doughty acknowledge partial support from DEVCOM ARL. All authors acknowledge partial support from the Defense Threat Reduction Agency.