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 $\mathit{\text{for}}\mathit{\text{the}}\mathit{\text{first}}\mathit{\text{time}}$. We obtain the probability distribution function $\mathcal{P}(R,t_R)$ and show that it displays two qualitatively different behaviors: (a) for $R\ll L_{\mathrm{I}}$, $\mathcal{P}(R,t_R)$ has a power-law tail $\sim t_R^{-\alpha }$, with the exponent $\alpha =4$ and $L_{\mathrm{I}}$ the integral scale of the turbulent flow; (b) for $l_{\mathrm{I}}\lesssim R$, 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.