Indoor Air Quality, Bioaerosols & Hospital Epidemiology
Wells-Riley Airborne Infection & Ventilation Formula
Calculate Well-Mixed Indoor Inhaled Quanta Dose mu = (I * q * p * t) / Q, Wells-Riley Individual Infection Probability 1 - EXP(-mu), Expected Secondary Cases among S Susceptibles, and Outbreak Threshold.
Size classroom and hospital HEPA + outdoor airflow Q to suppress airborne R_event below 1!
During airborne respiratory outbreaks (measles, tuberculosis, influenza, SARS-CoV-2), William Firth Wells and Richard L. Riley’s (1978, American Journal of Epidemiology) Poisson hit model relates infection probability in a well-mixed room to Equivalent Clean-Air Delivery Rate Q (m^3/hr from outdoor HVAC makeup air + MERV-13/HEPA filtration + UVGI)! When I infectious occupants emit q infectious quanta/hr while susceptibles breathe at pulmonary rate p (m^3/hr) over t hours, Mean Inhaled Quanta Dose is mu = (I * q * p * t) / Q, Individual Infection Probability is P_inf = 1 - EXP(-mu) (bounded strictly below 100%, unlike naive linear proportions!), and Expected Secondary Infections among S susceptibles is E[C] = S * P_inf!
Verify on a Poorly Ventilated Conference Room—Infectors I = 1 (B2), Quanta_Emission_Rate q = 50 quanta/hr (speaking, B3), Pulmonary_Breathing_Rate p = 0.60 m^3/hr (B4), Meeting_Duration t = 2.0 hrs (B5 -> 60 quanta*m^3/hr numerator), Clean_Air_Flow Q = 120 m^3/hr (B6), and Susceptible_Attendees S = 20 (B7): Inhaled Dose mu is 60 / 120 = 0.5000 quanta (E2); Individual Infection Probability is 1 - EXP(-0.5) = 0.3935 (39.35% attack rate in E3); Expected Secondary Cases is 20 * 0.39347 = 7.87 cases (E4); triggering "Superspreading Risk (E[C] >= 1)" (E5)!
| Wells-Riley IAQ Metric (I=1, q=50/h, p=0.6 m3/h, t=2h, Q=120 m3/h, S=20) | Formula | Verified Output |
|---|---|---|
| Mean Inhaled Quanta Dose mu | (I * q * p * t) / Q_m3_hr | 0.5000 quanta (0.0400 @ Q=1,500) |
| Individual Attack Probability | 1 - EXP(-mu) | 0.3935 (39.35%; 3.92% @ Q=1,500) |
| Expected Secondary Cases E[C] | S_susceptibles * (1 - EXP(-mu)) | 7.87 cases (0.78 cases @ Q=1,500) |
| Room Outbreak Control Gate | Expected_Cases < 1.0 Target | Superspreading Risk (E[C] >= 1) |
Wells-Riley Airborne Infection Probability Formula
Validate Infectors I >= 1, Quanta q > 0, Breathing p > 0, Duration t > 0, Clean Airflow Q > 0, and Susceptibles S >= 1!
=LET(in_f, B2, q_em, B3, p_br, B4, tm_h, B5, q_vn, B6, sc_n, B7, IF(OR(in_f < 1, q_em <= 0, p_br <= 0, tm_h <= 0, q_vn <= 0, sc_n < 1), "Invalid Wells-Riley Input", LET(mu_q, (in_f * q_em * p_br * tm_h) / q_vn, ROUND(1 - EXP(-mu_q), 4))))ASHRAE 241 Clean-Air Upgrade Check (Boost Q + HEPA CADR to 1,500 m^3/hr)
Upgrade outdoor ventilation + portable HEPA CADR so B6 = 1,500 m^3/hr: Attack Probability E3 plunges 10x to 0.0392 (3.92%), Expected Cases E4 drops to 0.78 (< 1.0), and E5 confirms "Controlled Ventilation (E[C] < 1)"!
=IF(ISTEXT(E3), E3, IF(E4 >= 1, "Superspreading Risk (E[C] >= 1)", "Controlled Ventilation (E[C] < 1)"))Try the example
Use the original values or change them to test your understanding.