Risk assessment for airborne disease transmission by poly-pathogen aerosols

Abstract

In the case of airborne diseases, pathogen copies are transmitted by droplets of respiratory tract fluid that are exhaled by the infectious that stay suspended in the air for some time and, after partial or full drying, inhaled as aerosols by the susceptible. The risk of infection in indoor environments is typically modelled using the Wells-Riley model or a Wells-Riley-like formulation, usually assuming the pathogen dose follows a Poisson distribution (mono-pathogen assumption). Aerosols that hold more than one pathogen copy, i.e. poly-pathogen aerosols, break this assumption even if the aerosol dose itself follows a Poisson distribution. For the largest aerosols where the number of pathogen in each aerosol can sometimes be several hundred or several thousand, the effect is non-negligible, especially in diseases where the risk of infection per pathogen is high. Here we report on a generalization of the Wells-Riley model and dose-response models for poly-pathogen aerosols by separately modeling each number of pathogen copies per aerosol, while the aerosol dose itself follows a Poisson distribution. This results in a model for computational risk assessment suitable for mono-/poly-pathogen aerosols. We show that the mono-pathogen assumption significantly overestimates the risk of infection for high pathogen concentrations in the respiratory tract fluid. The model also includes the aerosol removal due to filtering by the individuals which becomes significant for poorly ventilated environments with a high density of individuals, and systematically includes the effects of facemasks in the infectious aerosol source and sink terms and dose calculations.

Fig 1. Effect of ignoring multiplicity.

Ratio of the time required to reach a 50% infection risk when multiplicity is ignored τ_50,ignore to when it is fully accounted for τ_50,full for single pathogen infection probabilities r (an average dose of r⁻¹ Poisson distributed pathogen copies gives a mean infection risk of 63.21%) and different pathogen concentrations ρ_p in the respiratory tract fluid of the infectious individual as in the worked example later in the manuscript with a disease following the exponential model, but at steady-state with just the speaking mask-less infectious individual and the risk to a mask-less susceptible individual whose exposure starts after steady state is reached. This is a simplified version of Fig 5.

Fig 2. Required M_c based on pathogen concentration in infectious individuals.

M_c,I,j required to capture 99% of pathogen production for each diameter at aerosol production d₀ from an infectious individual, with each line being a different pathogen concentration in their respiratory tract fluid ρ_p,j (see legend).

Fig 3. Model solution for example.

Solution to the example case. (Top-Left) The total pathogen and infectious aerosol concentrations over time. (Top-Right) The infectious aerosol concentration densities in the room as a function of d₀ at t = 6 hr compared to the aerosol concentration densities being exhaled by speaking and coughing individuals from Johnson et al. [22] scaled by 10⁻⁴ to make them have comparable magnitudes. (Bottom-Left, Bottom-Right) The mean infection risk ${\mathfrak{R}}_E$ for the susceptible individuals based on the mask they are wearing (none, simple1, or simple2) using (Bottom-Left) r = 2.45 × 10⁻³ (Bottom-Right) r = 5.39 × 10⁻².

Fig 4. Sink strength by bin.

The strength of the sink terms for each bin with 80 bins, which is α without inactivation, α + γ for k = 1, and α + M_c γ for k = M_c (different values for Stage 1 and 2).

Fig 5. Effect of ignoring multiplicity, full version.

Full version of Fig 1 with more ρ_p and the effect of masks. Plot of the ratio of the time required to reach a 50% infection risk when multiplicity is ignored τ_50,ignore to when it is fully accounted for τ_50,full for different respiratory tract fluid pathogen concentrations ρ_p. We are considering the same situation as in the worked example, but at steady-state with just the speaking mask-less infectious individual and the risk to a susceptible individual whose exposure starts after steady state is reached. The ratio is shown for different combinations of mask on the susceptible individual (none and simple2) and for different r. The legend lists the r, mask combinations in the same order as the lines from top to bottom. We assumed a 100 nm diameter spherical pathogen and used 80 diameter bins and chose the M_c (maximum multiplicity considered) heuristic threshold to be T = 0.01 (include 99% of pathogen production).

Fig 6. Multiplicity’s impact on infection risk.

Plots of mean infection risk (${\mathfrak{R}}_E$) using the modified exponential dose-response model when all infectious aerosols have the same number of pathogen copies in them k. (Left) The infection risk as a function of k for fixed average dose 〈Δ〉 = r⁻¹ for different single pathogen infection probabilities r. (Right) The infection risk as a function of the dose scaled by r (〈Δ〉r) for different k and the same fixed r = 10⁻² (r⁻¹ = 100).

Fig 7. Effect of upper diameter limit d_M.

The example situation was calculated for different values of the upper diameter limit d_M (technically, calculated at the largest and then truncated down as needed). (Left) The total pathogen and infectious aerosol concentration densities over time for each d_M. Note that the differences in the total infectious aerosol concentration density are so small that the lines are right on top of each other. The mean infection risk for each combination of masks on a susceptible individual (none, simple1, simple2) for (Middle) r = 2.45 × 10⁻³ and (Right) r = 5.39 × 10⁻².