Size distribution of virus laden droplets from expiratory ejecta of infected subjects

The study addresses whether and how aerosol-mediated transmission of SARS-CoV-2 depends on the size distribution of expelled droplets and the viral load of infected individuals. It focuses on the probability that expelled droplets are virus-laden ("virusols"), especially in enclosed indoor environments where aerosol transport is relevant. Context is provided by recognized transmission modes (contact, droplet, aerosol) and heightened concern about airborne spread in poorly ventilated spaces with prolonged exposure. The paper notes key unknowns (e.g., minimum infectious dose, relationship between disease severity and viral load, fraction of infections via airborne route) and emphasizes that asymptomatic or mildly symptomatic individuals may still emit droplets that could leak through masks. Prior observations indicate that speech droplets can remain airborne and travel, motivating quantitative assessment of which droplet sizes are likely to carry virus as a function of viral load. The purpose is to quantify virus incorporation into droplets using Poisson statistics, derive size-dependent contamination probabilities, and infer implications for risk assessment and mitigation technologies.

The paper synthesizes literature on airborne transmission and droplet size distributions. Reviews indicate that a significant fraction of influenza virus-containing droplets are in respirable sizes, supporting potential airborne transmission. Prior works have used or discussed Poisson statistics for particle incorporation into droplets generated from suspensions, highlighting the probability of blank (virus-free) droplets. Studies by Fuchs & Sutugin and Raabe support Poisson-based models; others (e.g., Zuo et al.) report size-dependent viral loading skewed to larger particles. While some experimental work questioned strict Poisson applicability, it remains widely used. Recent COVID-19 studies reported broad viral load ranges (10²–10¹¹ RNA copies/mL) across specimen types and clinical severities, with severe cases generally showing higher loads than mild cases. Measured expiratory droplet size distributions (e.g., Lindsley et al., Johnson et al., Morawska et al.) are commonly modeled as lognormal, with modes spanning submicron to hundreds of microns and variable geometric standard deviations. Evaporation rapidly reduces droplet diameters (to roughly one-half or one-third), creating droplet nuclei that can carry virions.

The analysis models stochastic incorporation of discrete virions (RNA copies) into expiratory droplets using Poisson statistics. For an expelled droplet of diameter d_p and fluid viral concentration C_v (RNA copies/mL), the expected number of virions in the droplet is µ = (π/6) d_p^3 C_v. The number of virions per droplet follows a Poisson distribution P_n = (µ^n / n!) e^{-µ}, and the probability that a droplet carries at least one virion (i.e., is a virusol) is P_v = 1 − e^{−µ}. Viral load data compiled from literature span 10²–10¹¹ RNA copies/mL across patient categories, with severe cases typically exhibiting higher loads. The study classifies loads as: mild-to-moderate 10² < C_v < 2×10⁵ RNA copies/mL; severe C_v ≥ 2×10⁵ RNA copies/mL. Because evaporation rapidly reduces droplet size (to ~50% of initial diameter, sometimes one-third), the analysis relates pre-evaporation sizes to inhalable desiccated residues. For aerosol risk, focus is placed on droplets ≤20 µm at emission (~≤10 µm residue), which are likely to remain airborne and reach relevant regions in the respiratory tract (though virusols >10 µm residues are less likely to reach pulmonary regions). For polydisperse emissions commonly represented by lognormal distributions with volume median diameter d_G and geometric standard deviation σ_g, the fraction of virus-laden droplets F_v is computed by integrating the Poisson-based contamination probability over the size distribution: F_v = √(2π) σ_g ∫_0^∞ [1 − exp(−(π/6) d^3 C_v)] exp{−[ln(d/d_G)]² / (2 ln² σ_g)} (d/d_G) dd_G. A median propensity parameter µ_G = C_v d_G^3 is introduced to parametrize viral load and median size jointly. The authors evaluate virusol fractions and resultant virusol size distributions for σ_g between 1.5 and 4, ignoring unrealistically large dispersities (e.g., σ_g > 4) that imply unphysical mass content. Size distribution data from prior studies (e.g., Lindsley, Johnson, Morawska) are used to contextualize number concentrations and modes for expiratory activities (breathing, speaking, counting).

- Poisson statistical barrier: For most practical airborne-relevant droplet sizes, a large fraction of droplets are virus-free unless the propensity parameter µ = (π/6) d^3 C_v approaches or exceeds unity. - Mild-to-moderate viral loads: For C_v < 2×10⁵ RNA copies/mL (typical of mild-to-moderate cases), droplets with emission diameter <20 µm (~10 µm residue) are unlikely to carry virus; virusol formation is largely inhibited in this size range. - Very low viral loads: For C_v < 10⁴ RNA copies/mL, virusol fraction is <0.1% for droplets below 60 µm; thus >99.9% of droplets below 60 µm are virus-free. - Severe viral loads: Even for high loads (e.g., C_v ~10⁸ RNA copies/mL), droplets <2 µm (pre-evaporation) are unlikely to be contaminated. - Cut-off diameters: The smallest droplet diameter likely to be contaminated increases as viral load decreases. Figure 2 delineates cut-off diameters below which the contaminated fraction is <0.01%, 0.1%, or 1%, across C_v from 10⁴–10¹⁰ RNA copies/mL. - Exposure implication: For expiratory activities dominated by 0.8–1.8 µm droplets (e.g., speaking, breathing) with number concentrations up to ~1 cm⁻³, even at C_v ~10⁸ RNA copies/mL, the expected inhaled virus-laden droplets can be <1 over an hour in a typical indoor setting. - Polydisperse emissions and virusol shift: Virusol size distributions are shifted toward larger diameters relative to the original droplet distribution due to the size dependence of Poisson incorporation. The median virusol diameter can be 1.5–20× larger than the original median, depending on σ_g (greater shifts for larger σ_g). For σ_g = 2, clear upward shifts are shown; virusol σ_g tends to be smaller than the original droplet σ_g. - Fraction contaminated in polydisperse case: For µ_G < 0.005, <10% of the droplet spectrum is contaminated (covers droplets <20 µm from mild-to-moderate cases). There is a crossover near µ_G ≈ 0.6 where the virusol fraction is ~50% largely independent of σ_g (0.58–0.62 range). - Practical mitigation insight: Because virusols are biased toward larger sizes, filtration requirements for ultrafine particles may be relaxed for controlling aerosolized infection risk; coarser filters can be effective with lower pressure drops and higher clean air delivery rates.

By incorporating Poisson statistics at the droplet formation stage, the study demonstrates that most small expiratory droplets are statistically unlikely to carry virions unless viral loads are very high and/or droplets are sufficiently large. This directly addresses the central question of which droplet sizes present aerosolized infection risk: mild-to-moderate cases (C_v < 2×10⁵ RNA copies/mL) contribute minimally via droplets <20 µm, suggesting aerosol transmission risk is dominated by severe cases and larger droplets. The analysis also shows that even with high viral loads, sub-2 µm droplets are largely virus-free, aligning with observations that virion incorporation probability grows rapidly with droplet volume. For polydisperse emissions, virusol distributions shift to larger sizes and slightly narrower dispersity, implying shorter airborne residence times and increased capture by standard filtration technologies. These findings refine risk assessment by indicating that control strategies focusing on larger aerosol fractions are likely to be effective, and they help contextualize superspreading events as scenarios involving high viral loads, prolonged exposure, or emissions skewed to larger droplets.

The work quantifies a Poisson-driven statistical barrier to virion incorporation into expiratory droplets and introduces the concept of virusols (virus-laden droplets). It shows that for mild-to-moderate viral loads, formation of inhalable virusols (≤20 µm at emission, ~≤10 µm residue) is strongly suppressed, while aerosolized risk is more pertinent for severe cases and larger droplets (>2 µm pre-evaporation). Virusol size distributions are shifted to larger diameters than the original droplets, reducing airborne residence times and enabling effective control using relatively coarser filtration with lower pressure drops. These insights support more realistic airborne risk assessments and inform design and evaluation of mitigation technologies in indoor environments. Future research should integrate more comprehensive, high-resolution droplet size data from current pandemics, refine viral load–severity relationships, and incorporate environmental dynamics (ventilation, humidity) and infectious dose quantification into probabilistic exposure models.

- Data sparsity and variability: Available droplet size distributions are limited and heterogeneous, especially for submicron sizes and contemporary COVID-19 contexts. Extremely large reported dispersities (e.g., σ_g > 4) were excluded as potentially unphysical. - Assumptions on evaporation: The model assumes rapid size reduction to ~50% (or 1/3 in some studies) to relate emission sizes to residue sizes; real-world evaporation depends on humidity, temperature, and composition. - Poisson incorporation assumption: While widely used and supported by several studies, strict Poisson behavior may not hold in all atomization contexts; some experimental reports question applicability. - Simplified risk translation: The study focuses on probability of droplet contamination, not the full chain to infection (e.g., viable virion counts, dose-response, deposition efficiency, environmental removal, and ventilation dynamics). - Infectious dose uncertainty: Minimum infectious dose for SARS-CoV-2 is not established; translating virusol counts to infection probabilities relies on uncertain thresholds. - Patient heterogeneity: Viral loads vary temporally and across individuals; categorical thresholds (mild vs severe) are approximate and may not capture outliers (e.g., high-load asymptomatic cases).