First-passage-time problem for tracers in turbulent flows applied to virus spreading

We study the spreading of viruses, such as SARS-CoV-2, by airborne aerosols, via a 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 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 $R\ll L_{\mathrm{I}}, \mathcal{P}(R,t_R)$ 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.

Figure 1

A schematic diagram illustrating how aerosols with viruses (red points) may be advected by a turbulent flow, from the person at the center (the source) to other persons at different distances from the center.

Figure 2

Plots of the complementary cumulative probability distribution functions (CPDFs) $\mathcal{Q}$ vs the scaled first-passage time $t_R$ (see text): (a) Log-log plots of $\mathcal{Q}\left(\frac{t_R/T_{\mathrm{eddy}}}{R/L_{\mathrm{I}}}\right)$ for $R/L_{\mathrm{I}}=0.06$ (blue), $R/L_{\mathrm{I}}=0.10$ (purple), $R/L_{\mathrm{I}}=0.14$ (green), and ${\left(\frac{t_R/T_{\mathrm{eddy}}}{R/L_{\mathrm{I}}}\right)}^{-3}$ (black dashed line); the inset shows log-log plots of $\mathcal{Q}(t_R/T_{\mathrm{eddy}})$ for the same values of $R/L_{\mathrm{I}}$. (b) Semilog plots of $\mathcal{Q}(t_R/T_{\mathrm{eddy}})$ for $R/L_{\mathrm{I}}=1.02$ (pink), $R/L_{\mathrm{I}}=1.43$ (blue), $R/L_{\mathrm{I}}=1.84$ (purple), and $R/L_{\mathrm{I}}=2.04$ (green).

Figure 3

(a) Log-log plots of the complementary CPDFs $\mathcal{Q}\left(\frac{\gamma t_R}{R/L}\right)$ of the scaled first-passage time $\frac{\gamma t_R}{R/L}$, for $R\ll L$ and $\gamma =0.01$; the complementary CPDFs, for $R/L=0.0002$ (green), $R/L=0.00035$ (blue), and $R/L=0.0005$ (orange), collapse onto one curve; (b) semilog plots of the complementary CPDFs of the scaled first-passage time $t_R/R^2$, for $L\lesssim R$ and $\gamma =30$. The complementary CPDF of $t_R/R^2$, for $R/L=10$ (purple), $R/L=14$ (green), $R/L=18$ (blue), and $R/L=20$ (orange), collapse onto one curve. Plots of the complementary CPDFs $\mathcal{Q}\left(\gamma t_R\right)$ vs $\gamma t_R$ are shown in the insets.

Figure 4

Surface plots of $W(R,t)$, the probability of virus-laden aerosol particles not reaching a person (at a distance $R$ from an infected person), up until time $t$ (the first-passage time) in (a) vs $R$ and $t$, and (b) vs $R/L_I$ and $t/T$, in the diffusive region, for the representative values $K=0.1{\mathrm{m}}^2/\mathrm{s}, u_{\mathrm{rms}}=0.05$ m/s, and $L_{\mathrm{I}}=2$ m.