Maths in a minute: The Wells-Riley model

What is the risk of catching an airborne disease? There are many mathematical models that describe the behaviour of gases and contaminants within them. An interesting example is the Wells-Riley equation, which estimates the expected number $I$ of people who become infected by sharing a room with people who have an airborne disease:

  \[ I=S\left(1-exp\left(-\frac{q \Gamma t}{Q}\right) \right) \]    
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More ventilation means less infections

Here $S$ is the number of people in the room who are susceptible to the disease. And, loosely speaking, $\Gamma $ describes the total rate at which already infected occupants in the room emit infectious material, $q$ measures the average breathing rate per person, and $t$ the time period people are sharing the room. Finally the number $Q$ measures the ventilation rate of the room, that is, the rate at which fresh air enters it.

A close look at the equation reveals that more people will become infected (that is $I$ will increase) the longer the time people spend in the room (as time $t$ increases) or the more infectious material that is emitted (as $\Gamma $ increases). More infectious material might be emitted because more infected people are in the room, or the people themselves might be more infectious.

More happily, the better the ventilation of the room (the larger the $Q$) the smaller the number of people who become infected. The Wells-Riley equation assumes that the ventilation $Q$ is uniform across the space which isn’t usually the case in reality, as air flows created by people and ventilation systems also matter. However, the Wells-Riley equation (along with many other relevant mathematical expressions) can form part of more complex models that describe real life more accurately.

You can read about how the Wells-Riley model is used in Clearing the air: Making indoor spaces COVID safe and A breath of fresh air.

This article was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from the collaboration here.

The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit to find out more.

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