Towards improved social distancing guidelines: Space and time dependence of virus transmission from speech-driven aerosol transport between two individuals

It is now recognized that aerosol transport contributes to the transmission of the SARS-CoV-2 virus. Here, we improve existing social distancing guidelines for airborne pathogens, which are typically given in terms of distance with vague statements about contact times. Also, estimates of inhalation of virus in a contaminated space usually assume a well-mixed environment, which is realistic for some, but not all, situations. In particular, we consider a local casual interaction of an infected individual and a susceptible individual, both maskless, account for the air flow and aerosol transport characteristics of speaking and breathing in a poorly ventilated space, and propose social distancing guidelines that involve both space and contact time, based on a conservative model of fluid dynamics of the interactions.

Figure 1

Conical flows from breathing and speech. (a) Flow visualization of the conical jet produced by breathing from the mouth, adapted from Ref. [18]. Speaking can produce similar conical jets. The cone angle is $2\alpha$. A (susceptible) receiver, shown with a head sketched on the right, is facing the asymptomatic (infected) breather/speaker at a distance $\ell$. (b) Simulated transport of exhaled material during repeated speech. Results of numerical simulations from Ref. [18]: Example of the instantaneous field of a passive scalar used to visualize the dilution of the air exhaled from an infected speaker, displayed over a vertical symmetry plane after 9.5 cycles of periodic speaking-inhaling (each cycle lasts 4 s), with a volume per breath of 0.75 liters. The color map shows the concentration field $c$ of the passive scalar with $c=1.0$ at the exit of the mouth and $c=0$ in the ambient far field (the scale is saturated here to better visualize the field close to the receiver). An isoconcentration line at $c=0.05$ is displayed to help quantify the dilution levels far from the mouth of the speaker to the left. Both scale bars are 10 cm.

Figure 2

Sketch of ideal respiratory flow regions. The expiratory flow region is a cone with half angle $\alpha$ and apparent origin at $O$. The mouth is modeled as a circle of radius $a$, located at $x=acot\alpha$. The inhalatory flow region is a hemisphere with radius $R$ (determined by the inhaled volume, see text) and centered at $x=acot\alpha$ (the exit of the mouth). In the inhaled volume, only the shaded region is from the previous exhalation.

Figure 3

Dependence of infection probability $p$ on (a) distance $\ell$ and (b) speaking time $t$ (for interaction with a single maskless asymptomatic speaker). Values of $\ell$ or $t$ for each curve are given in the legends. The spatial distance is truncated at 3 m, beyond which we expect the ambient flows to become important even in a poorly ventilated space [18]. Two horizontal lines $p=0.63$ (or equivalently $N=N_{\text{inf}}$) and $p=0.2$ are plotted. Reported values in these figures are based on typical numbers listed in Table 2, but large variations of $c_v$ are possible [34], with larger values implying a shorter contact time for the same risk of infection.

Figure 4

Space-time diagram of infection risks. The color map (a) shows the infection risk $p$ as a function of distance $\ell$ and contact time $t$ with ${\varphi }_0=6\times {10}^{-9}$, in which the straight line indicates $p=0.63$. (b) shows the effects of ${\varphi }_0$. The infection risk is considered lower (higher) in the region below (above) the straight lines, where $p=0.63$ or, equivalently, $N=N_{\text{inf}}$ is proposed as a boundary for approximating relative risk. In (b) $t^{*}$ indicates the time for the jet to reach $\ell$ in the initial spreading stage. Below $t^{*}$ is the no-risk region. The plot is for a susceptible individual interacting with a single maskless asymptomatic speaker in a poorly ventilated space. Reported values in these figures are based on typical numbers listed in the legend and Table 2, but large variations of $c_v$ are possible [34], with larger values implying shorter contact time for the same risk of infection.