The spreading of viruses by airborne aerosols: lessons from a first-passage-time problem for tracers in turbulent flows

We study the spreading of viruses, such as SARS-CoV-2, by airborne aerosols, via a new first-passage-time problem for Lagrangian tracers that are advected by a turbulent flow: By direct numerical simulations of the three-dimensional (3D) incompressible, Navier-Stokes equation, we obtain the time $t_R$ at which a tracer, initially at the origin of a sphere of radius $R$, crosses the surface of the sphere \\textit{for the first time}. We obtain the probability distribution function $\\mathcal{P}(R,t_R)$ and show that it displays two qualitatively different behaviors: (a) for $R \\ll L_{\\rm I}$, $\\mathcal{P}(R,t_R)$ has a power-law tail $\\sim t_R^{-\\alpha}$, with the exponent $\\alpha = 4$ and $L_{\\rm I}$ the integral scale of the turbulent flow; (b) for $l_{\\rm I} \\lesssim R $, the tail of $\\mathcal{P}(R,t_R)$ decays exponentially. We develop models that allow us to obtain these asymptotic behaviors analytically. We show how to use $\\mathcal{P}(R,t_R)$ to develop social-distancing guidelines for the mitigation of the spreading of airborne aerosols with viruses such as SARS-CoV-2.