Description of model[edit]
Wells-Riley assumes that the air contains doses of the infectious bacterium or virus, and that you become infected if you breathe one dose in, i.e., the probability a person becomes infected, , is given by[1][2]
This dose is not a single bacterium or virus, but however many are needed to start an infection. These infectious doses are sometimes called 'quanta' - no relation to quantum physics. The doses are breathed out or otherwise emitted by the infectious person into the air of the room, such that in the room there is a concentration of these doses per unit volume of the air. If you are breathing in air at volume rate , then after a time in the room, the
The Wells-Riley then relies on standard Poisson statistics which predicts for the probability of infection
after a time in the room. This is just the Poisson statistics expression for the probability of one or more doses being inhaled, once we know the mean number.
So the prediction is that you are more likely to become infected if the concentration of infectious doses in the room air is high, or if you spend longer in the room. The concentration of doses will tend to be high in small, poorly ventilated rooms, and smaller in larger, better ventilated rooms.
Relation of the Wells-Riley model to epidemiology[edit]
Note that the Wells-Riley model approaches transmission of an airborne diseases as a physical transport problem, i.e., as the problem of how a virus or bacterium gets from one human body to another.
For transmission of COVID-19, for example, this would be how a virus breathed out by an infected person, can cross a room and be breathed in by a susceptible person. This is a different approach from that taken in the epidemiology of infectious diseases, which may gather information about who (e.g., nurses, factory workers) becomes infected, in what situations (e.g., the home, factories), and understand the spread of a disease in those terms - without considering how a virus or bacterium actually gets from one person to another.
However, the probability of infection predicted by the Wells-Riley model is close to the attack rate (also called secondary attack rate, and note that this 'rate' is a probability not a rate) in epidemiology. Compare the definition of in this page with the definition of the attack rate.
Mechanism of transmission of infectious diseases through the air[edit]
Wells-Riley is only applicable for transmission directly via the air, not via the susceptible person picking up the infectious agent from a surface (fomite transmission).[1] Because the model assumes the air is well mixed it does not account for the region within one or two metres of an infected person, having a higher concentration of the infectious agent. Breathing and speaking produce a cone of warm (body temperature), humid air that moves out into and dissipates into the room air over a distance of about one to two metres,[8] while a sneeze, with its much faster moving air, produces air movement up to metres away.[9] If the person breathing/speaking/sneezing is infected then an infectious agent such as tuberculosis bacterium or a respiratory virus is expected to be more concentrated in this cone of air, but the infectious agent can also (at least in some cases) spread into the room air.[10]
Estimating the number of inhaled doses requires more assumptions. The assumptions made by the model are essentially:[2][1]
Assumptions 4 to 6 mean that the concentration of doses in the room air, , is
Doses can be removed in three ways:
Assuming we can add the rates of these processes
for the lifetime of the infectious agent in air, the lifetime of a dose in the air before settling onto a surface or the floor, and the lifetime of the dose before it is removed by room ventilation or filtration. Then the concentration of doses is
If the susceptible person spends a time inside the room and inhales air at a rate (volume per unit time) then they inhale a volume and so a number of infectious doses
or
Putting all this together, the Wells-Riley prediction for the probability of infection is
where:
The lifetime of room air is just one over the air changes per hour - one measure of how well ventilated a room is. Building standards recommend several air changes per hour, in which case will be tens of minutes.
Relation to carbon dioxide concentration in a room[edit]
The Wells-Riley model assumes that an infected person continuously breathes out infectious virus. They will also continuously breathe out carbon dioxide, and so excess carbon dioxide concentration has been proposed as a proxy infection risk.[11][12][13] In other words, the higher the carbon dioxide concentration in a room, the higher the risk of infection by an airborne disease. The excess concentration of carbon dioxide is that over the background level in the Earth's atmosphere, which is assumed to come from human respiration (in the absence of another source such as fire).[11][12][13] Then the excess concentration of carbon dioxide is[13]
for people each exhaling carbon dioxide at a rate . Carbon dioxide neither sediments out (it is a molecule) nor decays, leaving ventilation as the only process that removes it. In the second equality we used , i.e., the rate of production of carbon dioxide is the breathing rate (volume of air exhaled per second = volume of air inhaled per second) times the concentration of carbon dioxide in exhaled breath .[13] Note that this implies that we can estimate how well ventilated a room is[13] if we know how many people are in the room, and the room's volume, from
If for the virus ventilation is the dominant route for removal of the virus, i.e., , the Wells-Riley prediction for the infection probability is then
which predicts that the higher the room concentration of carbon dioxide, the higher the infection risk.
Application to COVID-19[edit]
Although originally developed for other diseases such as tuberculosis, Wells-Riley has been applied[5][14] to try and understand (the still poorly understood[15][16]) transmission of COVID-19, notably for a superspreading event in a chorale rehearsal in Skagit Valley (USA).[5]
The Wells-Riley model is implemented as an interactive Google Sheets spreadsheet,[6] and interactive apps showing estimates of the probability of infection.[17] Even for the simple Wells-Riley model, the infection probability, , depends on seven parameters. The probability of becoming infected is predicted to increase with how infectious the person is ( - which may peak around the time of the onset of symptoms and is likely to vary hugely from one infectious person to another,[2][15] how rapidly they are breathing (which for example will increase with exercise), the length of the time they are in the room, as well as the lifetime of the virus in the room air.
This lifetime can be reduced by both ventilation and by removing the virus by filtration. Large rooms also dilute the infectious agent and so reduce risk - although this assumes that the air is well mixed - a highly approximate assumption.[2][8][10] A study of a COVID-19 transmission event in a restaurant in Guangzhou, went beyond this well-mixed approximation, to show that a group of three tables shared air with each other, to a greater extent than with the remainder of the (poorly ventilated) restaurant.[18] One infected person on one of these tables (a few metres apart) infected people on the other two tables.
The COVID-19 pandemic has led to work on improving the Wells-Riley model to account for factors such as the virus being in droplets of varying size which have varying lifetimes,[19] and an improved model[19] also has an interactive app.[20]