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The Wells-Riley model is a simple model of the airborne transmission of infectious diseases, developed by Wells and RIley for tuberculosis and measles. It can also be applied to other diseases transmitted in the air, such as COVID-19,  and measles. 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, that depend on properties of the room such as size, ventilation rate, how infectious the infected person so, and so on. The Wells-Riley model 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.

It assumes that the air contains doses of the infectious bacterium or virus, and that you become infected if you breathe one dose in. 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. Wells-Riley then relies on standard Poisson statistics to predict that the probability a person becomes infected, $$P_i$$, is given by

$$P_i= \mbox{probability one or more doses are inhaled}$$

When the concentration of infectious doses in the air is $$c_{DOSE}$$, and the susceptible person breathes in air (volume per unit time) at a rate $$B$$ then the probability of becoming infected after a time $$t_R$$ in the room is

$$P_i=1-\exp\left(-c_{DOSE}Bt_R\right)$$

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.

Finding out the exact mechanism by which diseases that are transmitted across the air, actually go from an infected person into a susceptible person, is notoriously difficult. Even if we know the identity of the infected person who was responsible for a person becoming infected (which we rarely do ) then we do not know how far apart they were, if they touched or not, or if the person who became infected touched any surfaces. The Wells-Riley model is only for transmission directly via the air, not via the susceptible person picking up the infectious agent from a surface. And because it 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.

Assumptions made by the model and predictions for the how transmission depends on room ventilation, size and other factors
Estimating the number of inhaled doses requires more assumptions. The assumptions made by the model are essentially :


 * 1) That the air contains the infectious agent in the form of doses.
 * 2) That infection occurs whenever a dose is breathed in.
 * 3) That an infected person breathes out doses at some constant rate $$r_{DOSE}$$.
 * 4) 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.
 * 5) That the doses have some lifetime $$\tau$$, before being removed. This is due to a combination of the infectious agent leaving the air, and it decaying/dying.
 * 6) 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, $$c_{DOSE}$$, is

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

Doses can be removed in three ways:


 * 1) The infectious agent can decay or die. Viruses in particular are known to be fragile and often short-lived outside their host.
 * 2) The dose can fall from the air onto a surface such as the floor.
 * 3) 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

$$\frac{1}{\tau}=\frac{1}{\tau_D}+\frac{1}{\tau_F}+\frac{1}{\tau_{VF}}$$

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

$$c_{DOSE}=\frac{r_{DOUT}}{V_{ROOM}\left(1/\tau_D+1/\tau_F+1/\tau_{VF}\right)}$$

If the susceptible person spends a time $$t_R$$ inside the room and inhales air at a rate (volume per unit time) $$r_{BIN}$$ then they inhale a volume $$r_{BIN}t_R$$ and so a number of infectious doses

$$\mbox{number of inhaled doses}=c_{DOSE}Bt_R$$

or

$$\mbox{number of inhaled doses}=\frac{r_{DOUT}Bt_R}{V_{ROOM}\left(1/\tau_D+1/\tau_F+1/\tau_{VF}\right)}$$

Putting all this together, the Wells-Riley prediction for the probability of infection is

$$P_i=1-\exp\left(-\frac{r_{DOUT}Bt_R}{V_{ROOM}\left(1/\tau_D+1/\tau_F+1/\tau_{VF}\right)}\right)$$

The Wells-Riley model and COVID-19
Although originally developed for other diseases such as tuberculosis, the Wells-Riley model has been applied to try and understand (the still poorly understood ) COVID-19 transmission, notably for a superspreading event event in a chorale rehearsal in Skagit Valley (USA). The model is implemented as an interactive Google Sheets, and interactive apps showing estimates of the probability of infection. Even for the simple Wells-Riley model, the infection probability, $$P_i$$, depends on seven parameters. The probability of becoming infected is predicted to increase with how infectious the person is ($$r_{DOUT}$$ - which may peak around the time of the onset of symptoms and is likely to vary hugely from one infectious person to another ), 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.