Introduction
The pandemic of 2019, due to SARS-CoV-2, named COVID-19 and referred to as coronavirus, has been responsible for nearly two million of deaths worldwide in 2020 alone. It is well-established that this virus primarily spreads from person-to-person contact by respiratory droplets produced when an infected person coughs or sneezes. Subsequently, the droplets come into contact with the eyes, nose or mouth of a nearby person or when a person touches an infected surface, then makes contact with their eyes, nose or mouth. Since the virus is small, 0.06–0.14 microns in diameter, it can be contained in or attached to such emitted droplets. Droplets as small as one micron can carry enough viral load to cause an infection. A particular concern is the interaction of droplets with ventilation systems, which can capture emitted pathogens, but can also potentially could enhance their propagation. This has implications on situation-specific safe distancing and the design of building filtration systems, air distribution, heating, air-conditioning and decontamination systems, for example using UV-c and related technologies. In order to facilitate such system redesigns, fundamental analysis tools are needed that are easy to use. Accordingly, this work develops a combined Digital-Twin and Machine-Learning framework to optimize ventilation systems by building on rapidly computable respiratory emission models developed in Zohdi. This framework ascertains the placement and flow rates of multiple mobile ventilation units, in order to optimally sequester particles released from respiratory emissions such as coughs, sneezes, etc. There have been dramatic advances in technologies associated with automation across many industries, which have the potential to drastically improve system efficiency, quality and safety. Some approaches are methodical and systematic, while some are ad-hoc and haphazard. In the world of systems engineering, increasingly sophisticated and integrated approaches for digital systems are appearing at a rapid rate. One key component to advancements in this areas are Digital-Twins, which are digital replicas of complex physical systems that can be safely manipulated and optimized in a virtual world and deployed in the physical world afterwards, reducing costs of experiments and accelerating development of new technologies. Digital-Twins blend AI, machine learning, software analytics and data to create living digital computer models that can update and change in tandem with their physical counterparts-and ultimately control them in real time. Updates to the Digital-Twin are made continuously in real-time, which necessitates technologies like rapid wireless communication, hyperspectral cameras, sensor fusion and fast simulations of process behavior. Today, there is no shortage of general simulation software; however, the fundamental limitations are ease-of-use and integration of models with in-field data and easy deployment for researchers. A core issue across all domains is the ability of a system to adapt to rapid changes in the environment and system capabilities by autonomously modifying operation-with humans in the loop. By developing these tools for food systems researchers, we believe that we can enhance the ability of industry to conduct research that inevitably involve Digital Twins, as well as benefiting society and industry.
The presented work also involves blending Machine-learning with Digital Twins, whereby simulations iteratively learn from their mistakes and constantly evolve to improve and optimize the model, safely in a virtual environment. This rapid simulation should be designed to run rapidly in tandem with a real system. Accordingly, the Digital-Twin and Machine-Learning in this framework combines these rapidly computable models with genomic Machine-Learning methods to ascertain the placement and flow rates of multiple mobile ventilation units, in order to optimally sequester particles released from respiratory emissions such as coughs, sneezes, etc. Specifically:
A model is developed for the propagation of emitted particles that is rapidly computable,
A qualitative analysis of the model is discussed,
A numerical discretization for the flow of the distribution of particles is constructed,
A genomic-based Machine-Learning algorithm to optimize the mobile ventilation systems is developed and
Numerical examples are provided.
Remark 1
It is now widely accepted that masks are an integral part of controlling the spread of infectious diseases, such as COVID19. There are three general categories of face masks: (1) Cloth Masks (for the general public), which provides some protection to the wearer, but mainly serves to stop the spread of viruses (2) Surgical Masks (for healthcare workers and the general public), which provides partial protection and (3) N95 Respirators (primarily for healthcare workers), which provides high level of protection and use charged fibers (electrets) to attract particles of all sizes. N95 masks are excellent at trapping very small particles (below 0.1 micron), since their motion is random (Brownian), due to collisions with other particles, air molecules, etc., which will force them to collide with an electrically charged "sticky" fiber. It is also excellent in trapping large particles (above 0.3 microns) , since their motion is effectively linear across the mask(because of their inertia) and will collide with a fiber. It has more difficulty trapping medium-sized particles (0.1 microns-0.3 microns), which travel with airflow patterns. N95 means that it traps at least 95% of the particles. However, there are problems, due to mask ill-fitting, re-use due to shortages, use of alcohols and solvents for mask cleaning, etc., which may damage the mask or neutralize the fiber charge. Furthermore, the proper donning and doffing (removal from face) of a mask is nontrivial, due to strap elongation, facial hair, etc. Therefore, one must assume that some particles get through, regardless of the mask type. Accordingly, in workplaces, some type ventilation is needed, generally involving HEPA (High Efficiency Particulate Absorbing) grade filters, which remove 99.95% of the particles equal to 0.3 microns in size, with efficiency increasing for particles larger or smaller than this size. Such filters were started by the German military in WWII for gas masks, whereby increases in mask efficiency were noted by simply adding a piece of paper (cellulose) to the mask. Such systems started commercially in the 1950s for decontamination control in hospitals, nuclear facilities, homes vehicles, etc. They are comprised of randomly arranged fiberglass fibers between 0.5 and 2.0 microns. As with N95 masks, they are designed to capture a range of particles by similar mechanisms. Another common term is 'MERV' Minimum Efficiency Reporting Value (1987 ASHRAE), which is designed to report the worst case performance of a filter, ranging 0.3–10 microns, with a scale rating of from 1–16 (highest at 95%). It is essential, that ventilation systems be given much more attention than ever before.
Remark 2
We remark that some systems also employ UV-c light for further decontamination, in combination with the ventilation system. A dosage of 1 𝐽/𝑐𝑚2 is considered adequate to destroy COVID19. Many commercial mobile UV-c light systems now exist. UV-c light 254–265 nm is considered optimal. Wavelengths above this range are not effective, while wavelengths below produce ozone, which is dangerous (Zohdi). Heating contaminated at 70 °C for 60 min is effective, but impractical in a large workplace setting with sensitive equipment and people. For example, decontamination based on UV technology has become ubiquitous, with many variants now being proposed. UV light varies in wavelength from 10 to 400 nm, thus making it shorter that visible wavelengths and larger than X-rays. Short wave UV light (UV-c) can damage DNA and sterilize surfaces making in useful in the medical industry. This was first noted in 1878 (Downes and Blunt) when the effect of short-wavelength light killing bacteria was discovered. By 1903 it was known the most effective wavelengths were around 250 nm (UV-c), for which Niels Finsen won a Nobel Prize (for skin-based tuberculosis eradication using UV light). Contaminants in the indoor environment are almost entirely organic carbon-based compounds, which break down when exposed to high-intensity UV at 240–280 nm. Despite the attractiveness of using UV-c light, the literature has shown that it is difficult to ensure that all surfaces are completely decontaminated due to shadowing effects. Thus, the use of ultraviolet germicidal irradiation (UVGI) is effective only as a component in a multistage process-it alone carries the risk of residual contamination. Thus, purely UV-c protocols should be adopted if there is no other choice. However, they can be an integral part of a multistage process involving a combination of (a) gas vapors and (b) heat and humidity. The topic of decontamination technologies is of paramount interest (see references Anderson et al., Battelle, Boyce et. al., Card et al., Heimbuch and Harish, Heimbuch et al., Ito and Ito, Lin et al., Kanemitsu, Lindsley et al., Lore et al., Marra et al., Mills et al. and Nerandzic et al.), and the corresponding simulation of such processes has recently been undertaken in Zohdi and is a topic of ongoing research.
Reduced Order Particle-Fluid Interaction Model
In its most basic form, a cough can be considered as a high-velocity release of a random distribution of particles of various sizes, into an ambient atmosphere (Fig. 1). We refer the reader to Wei and Li, Duguid, Papineni and Rosenthal, Wei and Li, Zhu et al., Chao et al., Morawska et al., VanSciver et al., Kwon et al., Tang et al., Xie et al., Gupta et al., Wan et al., Villafruela et al., Nielson Zhang and Li, Lindsley et. al. and Burridge et al. for extensive reviews of coughs and other respiratory emissions. We will consider the model problem in Fig. 1.
Assumptions
Following formulations for physically similar problems associated with particulate dynamics from the fields of blasts, explosions and fire embers (Zohdi), we make the following assumptions (Zohdi):
We assume the same initial velocity magnitude for all particles under consideration, with a random distribution of outward directions away from the source of the cough. This implies that a particle non-interaction approximation is appropriate. Thus, the inter-particle collisions are negligible. This has been repeatedly verified by "brute-force" collision calculations using formulations found in Zohdi.
We assume that the particles are spherical with a random distribution of radii 𝑅𝑖, 𝑖=1,2,3...𝑃𝑛, where 𝑃𝑛 is the total number of particles. The masses are given by 𝑚𝑖=𝜌𝑖43𝜋𝑅3𝑖, where 𝜌𝑖 is the density of the particles.
We assume that the cough particles are quite small and that the amount of rotation, if any, contributes negligibly to the overall trajectory of the particles. The equation of motion for the 𝑖𝑡ℎ particle in the system is equation (2.1) with initial velocity 𝑣𝑣𝑖(0) and initial position 𝑟𝑟𝑖(0). The gravitational force is ΨΨ𝑔𝑟𝑎𝑣𝑖=𝑚𝑖𝑔𝑔, where 𝑔𝑔=(𝑔𝑥,𝑔𝑦,𝑔𝑧)=(0,0,−9.81)𝑚/𝑠2.
For the drag, we will employ a general phenomenological model equation (2.2) where 𝐶𝐷 is the drag coefficient, 𝐴𝑖 is the reference area, which for a sphere is 𝐴𝑖=𝜋𝑅2𝑖, 𝜌𝑎 is the density of the ambient fluid environment and 𝑣𝑣𝑓 is the velocity of the surrounding medium which, in the case of interest, is air. We will assume that the velocity of the surrounding fluid medium (𝑣𝑣𝑓) is given, implicitly assuming that the dynamics of the surrounding medium are unaffected by the particles. We will discuss these assumptions further, later in the work.
Qualitative Behavior of the Simplified Model
A simplified Stokesian model, which can be solved analytically provides some physical insight, is discussed in Zohdi. A summary of that analysis is as follows. For a (low Reynolds number) Stokesian model, the differential equation for each particle is equation (2.3) where 𝑐𝑖=𝜇𝑓6𝜋𝑅𝑖, where 𝜇𝑓 is the viscosity of the surrounding fluid (air) and the local Reynolds number for a particle is 𝑅𝑒=def2𝑅𝑖𝜌𝑎||𝑣𝑣𝑓−𝑣𝑣𝑖||𝜇𝑓 and 𝜇𝑓 is the fluid viscosity. The trends are equation equation.
In summary
Large particles travel far and settle quickly and
Small particles do not travel far and settle slowly.
Remark 3
The ratio of the Stokesian drag force to gravity is equation, which indicates that for very small particles, drag will dominate the settling process and for larger particles, gravity will dominate.
Simulation of the Model: Cough Particle Generation and Propagation
For the general model, in order to more accurately model the effects of drag, one can take into account that the empirical drag coefficient varies with Reynolds number.
For example, consider the following piecewise relation (Chow): equation, equation, equation, equation and equation, where, as in the previous section, the local Reynolds number for a particle is 𝑅𝑒=def2𝑅𝑖𝜌𝑎||𝑣𝑣𝑓−𝑣𝑣𝑖||𝜇𝑓 and 𝜇𝑓 is the fluid viscosity.Footnote 1We note that in the zero Reynolds number limit, the drag is Stokesian. In order to solve the governing equation, equation (3.1).