  We develop a spatially dependent generalisation to the Wells-Riley model and
its extensions applied to COVID-19, that determines the infection risk due to
airborne transmission of viruses. We assume that the concentration of
infectious particles is governed by an advection-diffusion-reaction equation
with the particles advected by airflow, diffused due to turbulence, emitted by
infected people and removed due to the room ventilation, inactivation of the
virus and gravitational settling. We consider one asymptomatic or
presymptomatic infectious person who breathes or talks, with or without a mask
and model a quasi-3D setup that incorporates a recirculating air-conditioning
flow. A semi-analytic solution is available and this enables fast simulations.
We quantify the effect of ventilation and particle emission rate on the
particle concentration, infection risk and the `time to probable infection'
(TTPI). Good agreement with CFD models is achieved. Furthermore, we derive
power laws that quantify the effect of ventilation, emission rate and
infectiousness of the virus on the TTPI. The model can be easily updated to
take into account modified parameter values. This work paves the way for
establishing `safe occupancy times' at any location and has direct
applicability in mitigating the spread of the COVID-19 pandemic.
