How the Wells‑Riley Equation Predicts Flu Spread in Enclosed Spaces
This article explains the Wells‑Riley mathematical model for estimating airborne infection risk, outlines its key assumptions and parameters, walks through a classroom case study, and shows how ventilation and exposure time influence the predicted number of flu infections.
Airborne disease transmission is a key research area in infectious disease epidemiology. In a closed room, the risk that others become infected when someone has influenza can be quantified using the Wells‑Riley equation.
Wells‑Riley Equation
The Wells‑Riley equation is a mathematical model that estimates infection risk by considering ventilation, the emission rate of infectious quanta from the source, exposure time, breathing rate, and the number of susceptible individuals.
The equation is expressed as:
(Equation omitted in source)
where:
I : Expected number of new infections.
S : Number of susceptible individuals in the room.
q : Quanta generation rate of the infected source.
p : Average breathing rate per person (m³/h).
t : Exposure time (h).
Q : Room ventilation rate (m³/h).
Case Study: Classroom Scenario
Assume a classroom with 50 students (susceptible) and one influenza patient. The conditions are:
Ventilation rate: 600 m³/h.
Average breathing rate per student: 0.5 m³/h.
Quanta generation rate of the infected person: 20 quanta/h.
Exposure time: 2 hours.
Using the Wells‑Riley equation, the total exposure to infectious quanta is calculated, followed by the infection probability and the expected number of infections. The result predicts approximately 1 to 2 students becoming infected during this period.
Key parameters significantly affect the outcome, as illustrated in the following figures:
Modeling Process
1. Basic Assumptions
The air in the room is uniformly mixed.
The ventilation system works uniformly throughout the space.
The infected source emits infectious quanta at a constant rate.
Susceptible individuals breathe at a constant rate and inhale the quanta.
2. Key Parameters
Number of susceptible individuals (S).
Quanta generation rate (q).
Average breathing rate (p).
Exposure time (t).
Ventilation rate (Q).
3. Derivation of the Equation
The derivation considers the accumulation of infectious quanta, the exposure of susceptibles, and the resulting infection probability, leading to the final expression for the expected number of infections.
By adjusting ventilation, exposure time, and other parameters, the Wells‑Riley equation provides a scientific method to assess airborne infection risk and informs public health policies and indoor environment management.
Practical Recommendations
Increase the number and efficiency of indoor ventilation devices to ensure rapid air exchange.
Adjust work and study schedules to reduce prolonged stays in enclosed spaces.
Strengthen isolation and management of infected individuals to lower the emission of infectious quanta.
The Wells‑Riley equation offers a valuable theoretical basis for evaluating and mitigating the spread of airborne diseases.
Model Perspective
Insights, knowledge, and enjoyment from a mathematical modeling researcher and educator. Hosted by Haihua Wang, a modeling instructor and author of "Clever Use of Chat for Mathematical Modeling", "Modeling: The Mathematics of Thinking", "Mathematical Modeling Practice: A Hands‑On Guide to Competitions", and co‑author of "Mathematical Modeling: Teaching Design and Cases".
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