Wells-Riley model

The Wells-Riley model is a simple model of the airborne transmission of infectious diseases,^[1][2] developed by William F. Wells and Richard L. Riley for tuberculosis^[3] and measles.^[4]

Wells-Riley can also be applied to other diseases transmitted in the air, such as COVID-19.^[5][6] The model describes the situation where one or more infected people are sharing a room (and so the room air) with other people who are susceptible to infection. It makes predictions for the probability that a susceptible person becomes infected. The prediction is that infection is more likely for small poorly ventilated rooms, and if the infected person is highly infectious.

The Wells-Riley is a highly simplified model of a very complex process, but does at least make predictions for how the probability of infection varies with things within our control, such as room ventilation.

Description of model[]

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, ${\displaystyle P_{i}}$, is given by^[1][2]

${\displaystyle P_{i}={\mbox{probability one or more doses are inhaled}}}$

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. then relies on standard Poisson statistics to predict that when the concentration of infectious doses in the air is ${\displaystyle c_{DOSE}}$, and the susceptible person breathes in air (volume per unit time) at a rate ${\displaystyle B}$ then the probability of becoming infected after a time ${\displaystyle t_{R}}$ in the room is^{[citation needed]}

${\displaystyle P_{i}=1-\exp \left(-c_{DOSE}Bt_{R}\right)}$

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[]

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.

Mechanism of transmission of infectious diseases through the air[]

Determining the exact mechanism by which diseases that are transmitted via air is difficult.^[7][8] Even if the identity of the person who was responsible for the disease transmission is known (which is rarely the case ^[7]), and it is known they shared a room, this is not enough to reliably determine the mechanism. It still may not be known how close they were (within a metre, always at opposite ends of the room, ...), or if the person infected touched surfaces.^{[citation needed]}

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,^[9] while a sneeze, with its much faster moving air, produces air movement up to metres away.^[10] 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.^[11]

Assumptions made by the Wells-Riley model[]

Estimating the number of inhaled doses requires more assumptions. The assumptions made by the model are essentially:^[2][1]

• That the air contains the infectious agent in the form of doses.
• That infection occurs whenever a dose is breathed in.
• That an infected person breathes out doses at some constant rate ${\displaystyle r_{DOUT}}$.
• That the air inside the room is well-mixed, i.e., that when these doses are breathed out, they rapidly become uniformly distributed in the air.
• That the doses have some lifetime ${\displaystyle \tau }$, before being removed. This is due to a combination of the infectious agent leaving the air, and it decaying/dying.
• That the air is at steady state, i.e., concentration of doses in the air is not changing with time.

Assumptions 4 to 6 mean that the concentration of doses in the room air, ${\displaystyle c_{DOSE}}$, is

${\displaystyle c_{DOSE}={\frac {r_{DOUT}\tau }{V_{ROOM}}}}$

Doses can be removed in three ways:

• The infectious agent can decay or die. Viruses in particular are known to be fragile and often short-lived outside their host.
• The dose can fall from the air onto a surface such as the floor.
• Room ventilation or filtration can either remove the air containing the dose to the outside, or filter the dose from the air.

Assuming we can add the rates of these processes

${\displaystyle {\frac {1}{\tau }}={\frac {1}{\tau _{D}}}+{\frac {1}{\tau _{F}}}+{\frac {1}{\tau _{VF}}}}$

for ${\displaystyle \tau _{D}}$ the lifetime of the infectious agent in air, ${\displaystyle \tau _{F}}$ the lifetime of a dose in the air before settling onto a surface or the floor, and ${\displaystyle \tau _{VF}}$ the lifetime of the dose before it is removed by room ventilation or filtration. Then the concentration of doses is

${\displaystyle c_{DOSE}={\frac {r_{DOUT}}{V_{ROOM}\left(1/\tau _{D}+1/\tau _{F}+1/\tau _{VF}\right)}}}$

If the susceptible person spends a time ${\displaystyle t_{R}}$ inside the room and inhales air at a rate (volume per unit time) ${\displaystyle B}$ then they inhale a volume ${\displaystyle Bt_{R}}$ and so a number of infectious doses

${\displaystyle {\mbox{number of inhaled doses}}=c_{DOSE}Bt_{R}}$

or

${\displaystyle {\mbox{number of inhaled doses}}={\frac {r_{DOUT}Bt_{R}}{V_{ROOM}\left(1/\tau _{D}+1/\tau _{F}+1/\tau _{VF}\right)}}}$

Prediction equation[]

Putting all this together, the Wells-Riley prediction for the probability of infection is^{[citation needed]}

${\displaystyle P_{i}=1-\exp \left(-{\frac {r_{DOUT}Bt_{R}}{V_{ROOM}\left(1/\tau _{D}+1/\tau _{F}+1/\tau _{VF}\right)}}\right)}$

The lifetime of room air ${\displaystyle \tau _{VF}}$ 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 ${\displaystyle \tau _{VF}}$ will be tens of minutes.

Application to COVID-19[]

Although originally developed for other diseases such as tuberculosis, Wells-Riley has been applied^[5][12] to try and understand (the still poorly understood^[7][8]) 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.^[13][14] Even for the simple Wells-Riley model, the infection probability, ${\displaystyle P_{i}}$, depends on seven parameters. The probability of becoming infected is predicted to increase with how infectious the person is (${\displaystyle r_{DOUT}}$ - which may peak around the time of the onset of symptoms and is likely to vary hugely from one infectious person to another,^[2][7] 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][9][11] 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.^[15] 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,^[16] and an improved model^[16] also has an interactive app.^[17]

References[]