## Safe Distance: Quantitative Analysis on the Distance A Covid-19 Virus Spreads During Its Lifespan

#### Abstract:

Most of the diseases caused by virus mainly spread through droplets in the air. The pathogen bearing droplets go deep into people’s lungs and cause infection. In this paper, we analyze the safe distance, the minimum range to keep droplets containing virus particles from entering lungs, and thereby carrying the virus inside the lung. Einstein equation for diffusivity of a particle and the wide of the Gaussian distribution of the particles in Brownian movement are used in the calculation of the range a virus-containing mucosalivary droplet can reach. Moreover, we used datas recorded in a previous paper named “Visualization of sneeze ejecta: steps of fluid fragmentation leading to respiratory droplets” by B. E. Scharfman et. all to generate our results.#### Bibliography/Citations:

(1) https://en.wikipedia.org/wiki/Coronavirus_disease_2019#cite_note-JHU_ticker-8

(2) https://www.cdc.gov/coronavirus/2019-ncov/symptoms-testing/symptoms.html

(3) https://www.cdc.gov/coronavirus/2019-ncov/prevent-getting-sick/how-covid-spreads.html

(4) "Coronavirus disease (COVID-19): How is it transmitted?". www.who.int. Retrieved 6 December 2020.

(5) CDC (11 February 2020). "Coronavirus Disease 2019 (COVID-19)". Centers for Disease Control and Prevention. Retrieved 6 December 2020

(6) Visualization of sneeze ejecta: steps of fluid fragmentation leading to respiratory droplets. B.E.Scharfman, A.H.Techet, J.W.M.Bush, and L.Bourouiba. Exp Fluids (2016) 57:24. DOI 10.1007/s00348-015-2078-4.

(7) https://baike.baidu.com/item/2019新型冠状病毒/24267858?fr=aladdin#3

(8) Hirt, C.; Claessens, S.; Fecher, T.; Kuhn, M.; Pail, R.; Rexer, M. (2013). "New ultrahigh- resolution picture of Earth's gravity field". Geophysical Research Letters. 40 (16): 4279–4283. Bibcode:2013GeoRL..40.4279H. doi:10.1002/grl.50838

(9) Ball, Philip (2008). "Water: Water—an enduring mystery". Nature. 452 (7185): 291–2. Bibcode:2008Natur.452..291B. doi:10.1038/452291a. PMID 18354466. S2CID 4365814. Archived

(10) Symon, Keith R. (1971). Mechanics (3rd ed.). Addison-Wesley. ISBN 978-0-201-07392-8.

(11) 考虑空气阻力的斜抛运动研究. Xuepeng Guo. Physics Teaching. 39 (4): 14-16. DOI 1002-0748(2017)4 - 0014

(12) Collins Concise Dictionary, p. 328

(13) Collins Concise Dictionary, p. 1520

(14) Roser, Max; Appel, Cameron; Ritchie, Hannah (8 October 2013). "Human Height". Our World in Data.

(15) https://web.archive.org/web/20070715171817/http://aerodyn.org/Drag/tables.html

#### Additional Project Information

#### Research Plan:

I did a research on finding out safe distance of COVID 19 virus in real life, with the consideration of air resistance, viscoelasticity of the mucosal vary liquid contaminated with the virus, and turbulence effect on the virus entity. Professor Gerald Fuller, Professor in Chemistry Engineering from Stanford University, guided me during the research. I applied knowledge of fluid dynamics, drag force, and advanced calculus as foundations of my work. Based on physical equations, I applied calculus to calculating safe distance on theory, which is, the maximized distance a COVID 19 virus could travel after being ejected from a patient’s nose.

First, I divided the path of the virus surrounded by mucus into two sections: when it moves inside the bronchus and after it is emitted from the nose. Then, I came up with two methods investigating the hypothesis. The first method is to create mathematical models for each part and combine Maxwell model of viscoelastic fluid, Einstein equation for diffusivity, and Gaussian distribution in Brownian movement to generate a trajectory equation for the pathogen-carrying particles. The second method is to only create mathematical models for the second portion of the path (movement outside the body and in the air) using equations for projectile motion under the influence of air resistance, and use the result from “Visualization of sneeze ejecta: steps of fluid fragmentation leading to respiratory droplets” by B. E. Scharfman et. all to generate the initial velocity of this projectile motion

I spent a month building up the mathematical model of the projectile of a COVID 19 virus after emitted from nasal cavity. After that, I spent another two weeks reading supplementary materials and essays studying the feature of viscoelastic mucosal vary mucus such as snivel that is potentially a pathogen carrier. Finally, I spent one more month completing and proliferating my essay.

#### Questions and Answers

1. What was the major objective of your project and what was your plan to achieve it?

a. Was that goal the result of any specific situation, experience, or problem you encountered?

The goal of the result is to generate a theoretical average safe distance for the COVID 19 virus. It is practical, especially under the current situation when people around the world are troubled by the pandemic. This result is influenced by my experience and problems I encountered in daily life as well. Walking on the street, I am always wondering how far should I keep from other people so that our safe can be guaranteed. With the safe distance calculated, it is much more convenient and less embarrassing when keeping the most suitable distance from other people.

b. Were you trying to solve a problem, answer a question, or test a hypothesis?

I was trying to solve a problem, which is to calculate the average safe distance (minimum distance that prevent infection) of COVID 19 virus. By investigating the theoretical maximum distance of a COVID 19 pathogen containing mucosalivary particle could reach, I get the result of the expected safe distance.

2. What were the major tasks you had to perform in order to complete your project?

a. For teams, describe what each member worked on.

This is an individual research program. I am the one who did all of the job, including: creating a mathematical model for the investigated particle, applying calculus and specific physical equations to do the calculation, analyzing results (graphs) generated from a previous experiment and concluding new results from it, as well as composing the research report and essay.

3. What is new or novel about your project?

a. Is there some aspect of your project's objective, or how you achieved it that you haven't done before?

My project is mostly innovative and objective. Even though the topic of my project, investigating the safe distance of COVID-19 virus, was done by many countries as soon as the pandemic was identified, I analyzed the problem in a novel approach. I consider the path of the virus particles inside the lung and bronchus before they are emitted from nasal channels, making the estimation more accurate. This is the part of my achieving that I haven’t done this project before.

b. Is your project's objective, or the way you implemented it, different from anything you have seen?

Using two different solutions is the way my project different from anything I have seen. Usually, as most of the researchers studying the safe distance of virus did, the estimated average value of safe distance would be calculated. However, in my research, I not only give an average value but also generate an equation solving the accurate value for different cases. Therefore, the accuracy and practicalness of my project increase enormously.

c. If you believe your work to be unique in some way, what research have you done to confirm that it is?

I combine Einstein equation for diffusivity of a particle and the the Gaussian distribution of the particles in Brownian movement in order to calculate the range a virus-containing mucosalivary droplet can reach. This is a novel investigating method, because I combine the fluid dynamic of viscoelastic material with the traditional air-resistance considerate model of projectile motion of the particle. This research can confirm that my work is unique in method of calculation and problem analyzation.

4. What was the most challenging part of completing your project?

a. What problems did you encounter, and how did you overcome them?

It takes a precise modeling as considerate as possible to calculate the accurate result. These elements include resistances the virus would face in the snot and air. These seemingly easy factors actually require knowledge in aerodynamics and hydrodynamics, not to say, differential and integral calculus calculation in this project beyond my capability, making the whole work come to a halt for some time.

b. What did you learn from overcoming these problems?

In order to solve this problem, I specifically learned the physics and calculus knowledge required, and then practiced a lot. Through a hard work, I took command of some basic approach to untangle the major calculus puzzles, such as parameter method, integration by substitution, integration by parts, partial differential equations, differentiation of implicit function and etc.

With the equipment of these knowledge, I reexamined the calculus equation in my project, seeing that they were not longer that intricate as much. And this was exciting and I was inspired too. This project, challenging yet positive, made me understand the truth that mathematics is the fundamental instrument of all science. Without an exceptional capability, it would be almost impossible to solve inter-disciplinary issues.

5. If you were going to do this project again, are there any things you would you do differently the next time?

Currently, the first method of my research contains usage of data generated from my re-analysis of the result of a previous essay “Visualization of sneeze ejecta: steps of fluid fragmentation leading to respiratory droplets” by B. E. Scharfman et. all to get the average initial speed of sneeze ejecta edited from noses. If I was going to do this project again, without being limited by the COVID-19 situation, I would love to do the research studying this initial speed so that the abundance of my work can be expended, with an actual experiment introduced instead of only containing theoretical calculations.

6. Did working on this project give you any ideas for other projects?

Certainly, it does. While working on this problem, it inspires me that after investigating the safe distance of the Coronavirus Disease 2019, I can also design specially produced face-masks to prevent invasion of the virus from outside into the body even if people are not keeping proper safe distance. Regular face masks that are commonly seen in nowadays market are mostly filtering the virus-bearing particles to block it out of the mouth. The application of calculation of fluid dynamics of mucosalivary fluid decelerate the pathegon-carring particles reminds me that I can design a type of face mask with viscoelastic filters in the middle, so that even if some of the virus particles break into the face masks, the fluid in between could slow them down and prevent them from getting into human bodies.

7. How did COVID-19 affect the completion of your project?

COVID-19 influences my project a lot, but does affect the completion. It provides me the topic choice and study target. The entire research is built on the basis of solving the practical problems people may encounter during the COVID-19 pandemic. Fortunately, COVID-19 did not influence the completion or plan of my project, since my project is mostly theoretical and the major component of it is mathematical calculation, therefor it was easy for me to complete the project at home and online, which keeps it unaffected by the COVID-19 situation.