Modeling the role of respiratory droplets in Covid-19 type pandemics

Abstract

In this paper, we develop a first principles model that connects respiratory droplet physics with the evolution of a pandemic such as the ongoing Covid-19. The model has two parts. First, we model the growth rate of the infected population based on a reaction mechanism. The advantage of modeling the pandemic using the reaction mechanism is that the rate constants have sound physical interpretation. The infection rate constant is derived using collision rate theory and shown to be a function of the respiratory droplet lifetime. In the second part, we have emulated the respiratory droplets responsible for disease transmission as salt solution droplets and computed their evaporation time, accounting for droplet cooling, heat and mass transfer, and finally, crystallization of the dissolved salt. The model output favourably compares with the experimentally obtained evaporation characteristics of levitated droplets of pure water and salt solution, respectively, ensuring fidelity of the model. The droplet evaporation/desiccation time is, indeed, dependent on ambient temperature and is also a strong function of relative humidity. The multi-scale model thus developed and the firm theoretical underpinning that connects the two scales-macro-scale pandemic dynamics and micro-scale droplet physics-thus could emerge as a powerful tool in elucidating the role of environmental factors on infection spread through respiratory droplets.

FIG. 1.

Experimental setup showing the acoustic levitation of a droplet illuminated by a cold LED source. A diffuser plate is used for uniform imaging of the droplet. A CCD camera fitted with the zoom lens assembly is used for illumination. The schematic is not to scale.

FIG. 2.

A schematic of the collision rate model for the infection to occur. Infected person P ejects a cloud of infectious droplets D denoted by small red dots, and the cloud approaches a healthy person H with a relative velocity ${\overrightarrow{V}}_{DH}$ to infect them. The figure also shows the collision volume swept by the droplet cloud D and H with their respective effective diameters.

FIG. 3.

Instantaneous droplet images taken by a CCD camera (top left panel) and dark field micrograph of the final salt precipitate (top right panel). Comparison of experiments and simulations in the bottom left and right panels. Evolution of the normalized droplet diameter as a function of time for pure water (left panel) and the salt water solution droplet with 1% NaCl (right panel).

FIG. 4.

(a) D_crit, (b) τ, (c) σ_D, and (d) X_p as a function of T_∞ and RH_∞.The black dots A and B denote two typical conditions. Case A represents (T_∞, RH_∞) = (8, 55), while case B represents (T_∞, RH_∞) = (28, 77). Color bars have been clipped at reasonable values to show the respective variations over the wider region of interest.

FIG. 5.

Evolution of the normalized mass of water in the droplet (left panel) and droplet temperature (right panel) as a function of time for cases A and B.

FIG. 6.

Contours of calculated eigenvalues (a) λ₁ and (b) λ₂ as a function of T_∞ and RH_∞. (c) Infection rate constant, k₁, and (d) rate ratio over seven days as a function of T_∞ and RH_∞. Case A represents (T_∞, RH_∞) = (8, 55), while case B represents (T_∞, RH_∞) = (28, 77). The rate ratio for A and B are 16.60 and 10.33, respectively. Color bars have been clipped at reasonable values to show the respective variations over the wider region of interest.

FIG. 7.

Flow-diagram outlining the interconnections of the model developed.