Parameterizing Airborne Infection in SEIR Models: Ventilation, Aerosols, and ExposureAirborne transmission plays a central role in the spread of many respiratory infections (e.g., influenza, measles, SARS-CoV-2). Traditional SEIR (Susceptible–Exposed–Infectious–Recovered) compartmental models capture population-level dynamics but often treat transmission as a simple mass-action term (βSI/N). For airborne pathogens, this simplification can miss critical drivers: aerosol generation and decay, room ventilation, deposition, infectious quanta, and heterogeneity in exposure. This article explains how to parameterize airborne infection within SEIR frameworks by linking mechanistic aerosol physics and exposure processes to epidemiological parameters, illustrates practical model formulations, discusses data sources and parameter estimation, and examines implications for control measures such as ventilation and masking.
1. Why explicit parameterization matters
- Mechanistic realism: Airborne transmission depends on environmental factors (ventilation rate, room volume), human behavior (speaking, shouting), and pathogen-specific aerosol properties (aerosol size distribution, viability decay). Folding these into a single β can obscure which interventions will be effective.
- Setting-specific risk: Infection risk differs dramatically between outdoor, well-ventilated indoor, and poorly ventilated crowded spaces. Explicit parameters enable scenario-specific predictions.
- Intervention assessment: Quantifying effects of ventilation upgrades, air cleaning, occupancy limits, or masking requires models that respond to changes in aerosol concentration dynamics rather than only to population contact rates.
- Heterogeneity and superspreading: Aerosol concentrations and individual emission rates drive overdispersion in transmission. Mechanistic parameterization captures potential superspreading events better than homogeneous-mixing β terms.
2. Conceptual link: From quanta to β
A convenient bridge between aerosol physics and epidemiological transmission is the concept of infectious quanta. First introduced by Wells and Riley, a quantum is defined as the dose of airborne infectious material that will cause infection in 63% (1 – e^−1) of susceptible persons under steady exposure. Using quanta, the probability p of a susceptible person in a well-mixed indoor space becoming infected during an exposure time T with a time-varying airborne concentration Cq(t) (quanta per volume) is:
p = 1 − exp(− ∫_0^T Cq(t) · IIR · dt )
where IIR is the inhalation rate (volume per time) of the susceptible (m^3/h). In many applications, assuming steady-state concentration Cq and constant inhalation rate gives p = 1 − exp(− Cq · IIR · T).
To integrate this into an SEIR model, express the force of infection λ (per susceptible per time) in terms of quanta concentration and exposure:
λ(t) = IIR · Cq(t)
If Cq(t) scales linearly with the number of infectious individuals I(t) in the population (e.g., in a homogeneously mixing building or community), Cq(t) = κ · I(t) where κ aggregates emission, dilution, and removal processes, then λ(t) = (IIR · κ) · I(t). Comparing with the classic mass-action λ = β I(t)/N, one can identify β = N · IIR · κ (units adjusted as needed). However, κ depends on environmental parameters (ventilation, filtration, deposition, decay), emission heterogeneity, and occupancy patterns.
3. Modeling airborne concentration dynamics
Instead of collapsing all factors into β, explicitly model the airborne quanta concentration Cq(t) in an environment (single room or multiple connected spaces). A common well-mixed box model describes the time evolution:
dCq/dt = (∑_j q_j(t)) / V − (λ_v + λ_f + λ_d + λ_i) · Cq(t)
where:
- q_j(t) is the quanta emission rate (quanta/time) from infectious person j (depends on activity: breathing, speaking, singing);
- V is room volume (m^3);
- λ_v is ventilation air exchange rate (h^−1) (outdoor air exchange);
- λ_f is effective removal rate due to filtration/air cleaning (h^−1);
- λ_d is deposition onto surfaces (h^−1) (size-dependent);
- λ_i is biological decay/inactivation rate of the pathogen in aerosols (h^−1).
At steady-state with constant emission Q = ∑ q_j, Cq_ss = Q / (V · Λ) where Λ = λ_v + λ_f + λ_d + λ_i. For k infectious occupants emitting on average q̄ quanta/time, Q = k q̄ and Cq_ss = k q̄ / (V Λ). The per-susceptible force of infection is then λ = IIR · Cq_ss.
Key practical points:
- Ventilation λ_v is often reported in air changes per hour (ACH). Converting ACH to h^−1 is direct.
- Filtration has fractional removal f per pass; if recirculated air rate is r (h^−1), λ_f ≈ f · r.
- Deposition λ_d depends on aerosol size distribution; for respirable aerosols (0.5–5 µm) it’s small (≈0.1 h^−1) compared to ventilation in many settings.
- Biological decay λ_i varies with pathogen, humidity, UV; values often come from chamber or laboratory aerosol studies.
4. From room-level exposure to population-level SEIR
There are two common approaches to incorporate the concentration model into SEIR:
- Population-averaged compartment linking via an effective β(t)
- Compute an effective time-dependent transmission rate β(t) from ensemble-average exposure environments: β(t) = N · IIR · κ(t) where κ(t) reflects average per-infectious airborne concentration per infectious person, accounting for typical settings and occupancy patterns.
- Use standard SEIR: dS/dt = −β(t) S I / N dE/dt = β(t) S I / N − σ E dI/dt = σ E − γ I dR/dt = γ I
This hybrid keeps SEIR structure but makes β interpretable and responsive to interventions (e.g., increasing ventilation reduces κ and hence β).
- Coupled micro-environment SEIR (metapopulation or event-based)
- Partition population into locations (homes, offices, classrooms, transport). For each location m model Cq_m(t) with its own V_m and Λ_m, and let occupancy in location m at time t be S_m, E_m, I_m, R_m.
- Force of infection for susceptiblest in location m: λ_m(t) = IIR · Cq_m(t)
- Transmission occurs within each location; individuals move between locations according to schedules.
- This captures heterogeneity and superspreading events (e.g., a single infectious person in a choir rehearsal with high q̄ and low Λ produces high attack rates).
The choice depends on data availability and computational resources. Metapopulation models are more realistic for targeted interventions but require detailed occupancy and environment data.
5. Parameterizing emission rates (q)
Quanta emission q is highly variable:
- Activity: breathing (low), speaking (moderate), loud speaking/singing (high), heavy exercise (high).
- Individual variability: “high emitters” can produce orders of magnitude more aerosols than low emitters.
- Pathogen load and infectiousness: viral concentration in respiratory fluid and fraction that becomes aerosolized and remains infectious drive q.
Typical approaches:
- Use published quanta emission estimates from outbreak reconstructions (e.g., Wells–Riley fits) and experimental aerosol studies. For SARS-CoV-2, literature estimates range widely from <1 to >100 quanta/h depending on activity and variant.
- Model q as q = q0 · a · v, where q0 is a baseline per-person emission (breathing), a is activity multiplier (speaking, singing), and v represents individual infectiousness heterogeneity sampled from a distribution (log-normal, gamma).
- Calibrate q (or its distribution) by fitting model outputs to observed outbreak sizes, secondary attack rates, or household studies.
Include uncertainty: use distributions, not single point estimates, and propagate through the model.
6. Ventilation, filtration, and mitigation parameters
Key controllable parameters and how to represent them:
- Ventilation rate (λ_v, ACH): measured or estimated from building HVAC specs or tracer-gas experiments. Increase in λ_v reduces steady-state concentration linearly.
- Filtration (λ_f): incorporate MERV or HEPA filter efficiency and recirculation flow. Effective removal = recirculated flow rate × filter efficiency / room volume.
- Portable air cleaners: treat as added λ_f based on CADR (clean air delivery rate): λ_cleaner = CADR / V.
- Masking: reduce emission (source control) and inhalation (protection). Represent masks by reducing q_j by factor (1 − e_s) and/or reducing inhalation IIR by factor (1 − e_r), or simply applying overall multiplicative reduction in λ. For example, if masks reduce emission by 50% and inhalation by 30%, net reduction ≈ 1 − (1 − 0.5)(1 − 0.3) = 0.65 (i.e., 65% reduction).
- UV germicidal irradiation: add λ_uv to Λ based on UV dose and room air circulation patterns.
- Occupancy scheduling and time spent: account for time-weighted exposures across multiple spaces.
Provide sensitivity analyses to rank interventions: often increasing ventilation or adding high-efficiency filtration yields predictable reductions; masking offers strong reductions especially when universal; reducing occupancy and exposure time also helps linearly in quanta-dose frameworks.
7. Data sources and estimation approaches
- Empirical outbreak studies: reconstruct emission rates and effective quanta by fitting the Wells–Riley framework to documented events (choir, restaurants, buses).
- Laboratory aerosol studies: measure viral RNA or viable virus in aerosols under controlled activities; derive aerosol generation rates and decay constants.
- Environmental measurements: CO2 as a proxy for rebreathed air fraction; can be converted to relative exposure and used to scale κ. CO2 measurements enable real-time estimation of ventilation effectiveness and relative airborne risk.
- HVAC specifications and tracer-gas tests: obtain ACH and recirculation rates.
- Household transmission studies: estimate secondary attack rates and latent/infectious periods for model calibration.
- Genomic and contact-tracing data: inform overdispersion and identify superspreading contexts.
Parameter estimation methods:
- Bayesian calibration to observed case counts, outbreaks, or secondary attack rates—allows propagation of parameter uncertainty.
- Maximum likelihood or least squares fits for simpler aggregated models.
- Use hierarchical models to borrow strength across similar environments while allowing site-specific variation.
8. Representing heterogeneity and superspreading
Superspreading arises from a combination of high emitters, high-risk environments (low Λ, high occupancy), and long exposure duration. Model these by:
- Sampling individual infectiousness multipliers from a heavy-tailed distribution (e.g., log-normal or gamma with small dispersion parameter k).
- Modeling events explicitly (e.g., gatherings) with high activity multipliers (singing) and long durations.
- Using metapopulation/event frameworks to allow someone to attend a high-risk event once; this naturally produces overdispersed secondary case distributions.
Quantify overdispersion parameter k by fitting offspring distributions; this informs targeted control strategies (preventing high-risk events is disproportionately valuable).
9. Practical example: classroom scenario (summary)
Consider a classroom: V = 200 m^3, one infectious teacher speaking (q̄ = 50 quanta/h), 20 students, ventilation λ_v = 3 ACH (3 h^−1), negligible filtration, deposition+decay Λ_other = 0.5 h^−1, so Λ = 3.5 h^−1. Steady-state Cq = q̄ / (V Λ) = 50 / (200 × 3.5) ≈ 0.0714 quanta/m^3. If students inhale IIR = 0.5 m^3/h and spend T = 2 h:
Dose per student = Cq · IIR · T ≈ 0.0714 × 0.5 × 2 ≈ 0.0714 quanta → infection probability p ≈ 1 − e^−0.0714 ≈ 0.069 (6.9%). With 20 students expect ~1.4 secondary cases. Increasing ventilation to 6 ACH halves Cq and roughly halves risk; universal masking reducing emission+inhalation by 60% lowers p substantially.
10. Limitations and caveats
- Well-mixed assumption: Box models assume uniform concentration; in reality, near-field close contact can have higher exposure. Combine near-field (plume) models with room-scale models when proximity matters.
- Quanta concept: Quanta are a convenient epidemiological construct but collapse complex biology into a single parameter; quanta estimates vary by pathogen, variant, and study methods.
- Data gaps: Viable virus measurements in aerosols are sparse; many studies report RNA, not infectivity.
- Behavioral complexity: Occupancy patterns, adherence to masking, and activity levels change over time and are uncertain.
- External factors: Environmental conditions (temperature, humidity, UV) affect decay rates and aerosol behavior.
11. Recommendations for modelers
- Use a hybrid approach: link mechanistic room-scale concentration models to SEIR compartments to keep interpretability and responsiveness to interventions.
- Parameterize ventilation, filtration, and activity explicitly; avoid folding everything into an uninterpretable β when assessing engineering controls.
- Incorporate heterogeneity: sample emission rates and model events to capture superspreading potential.
- Use CO2 or tracer measurements when possible to validate ventilation and exposure proxies.
- Quantify uncertainty with probabilistic methods and report ranges, not only point estimates.
- When near-field transmission is important (close, prolonged contact), add a short-range exposure component in addition to the room-scale airborne concentration.
12. Conclusion
Parameterizing airborne infection in SEIR models requires bridging aerosol physics, human behavior, and epidemiology. Using quanta-based concentration models, explicit ventilation and removal rates, activity-dependent emission rates, and heterogeneity in infectiousness yields models that better predict setting-specific risk and the impact of interventions. While subject to assumptions (well-mixed air, quanta interpretation), these approaches allow more actionable guidance—showing how ventilation, filtration, masking, and occupancy reduction translate into reduced transmission in quantitative terms.
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