The Science Of Standing In Line

I was standing in line the other day…

No seriously, this is no normal situation where you would want to be standing in line. Not your run of the mill situation of frustration, waiting to get your driver’s license renewed; it also doesn’t have to do with queueing theory.

Let me set the scene.

Disclaimer: I am not a medical professional. I encourage you to get actual medical advice from established medical authorities, like the NICD and WHO. I am just some guy on the internet with a blog, and as the title suggests, take all of this with a few pinches of salt.

Disclaimer 2: I got the nice wording above from here.

Setting the Scene

It is the 17th of June, the middle of winter, the day saw a maximum of 8 °C. I was heading to an appointment with the ENT (ear, nose, and throat) for a consultation.

Before making my way to the doctor’s rooms, I was told that I had to head to the hospital across the road to get screened for covid symptoms. Apparently, there was a tent outside, in the parking lot of the hospital where I should wait in a line to get screened.

I arrive at the place at around 08h30; bear in mind that at this point in time, the temperature is around 5 °C, and see that there is a long line outside this large tent in the parking lot.

So I think that this must be the line I need to stand in.

The line moves very slowly, they appeared to be screening people at around a person every five minutes. I haven’t been here before and the doctor’s rooms tell me that this is just how it is now, so I understand the situation and just wait in line.

Time goes by, and the line still continues to move very slowly, at this point about forty minutes have gone by; eventually the line moves quite a bit forward, and what appears before me? Yes, a massive sign that reads: “COVID TESTING STATION”.

Ah, I now think that I understand why the line is moving so slowly…

So I get on the phone to see if everyone has to go through this process or not, the doctors rooms don’t know anything about this.

Five minutes go by, and someone from the hospital comes outside, he says that he is the head administrator for the hospital, he wants to know why so many people are standing in the line; he says that it’s unusual to have this many people standing in this line.

So I ask him what this line is, and he points me towards a small tent, hidden on the inside of the hospital entrance, with no signage which is meant to be the screening tent…

So, turns out I was standing in the wrong line all along…

To further solidify this, the guy standing in front of me tells the administrator that he tested positive two weeks prior to this — more on this later.

This situation however, got me thinking, being the engineer that I am, I thought I would have a little fun with understanding exactly how safe I was standing in line there.

Now, just a note to readers, I did not get infected from standing in line that day; I would like to use this post as an illustration of exactly how and why I didn’t get infected.

Risk factors

I thought that this could be used as a bit of a fun scientific investigation about the situation and the amount of risk that I put myself in while standing in line.

Now by this point in the year, everyone was wearing masks, so this adds an interesting feature to the investigation.

I am by no means going to try to fully calculate/examine/evaluate the situation, but rather do a back of the envelope discussion of the situation with the science as we currently know it. I thought that I would start by laying out the situation and get an idea of exactly how much risk I was in during the course of the line standing.

This great article by Wits gives an excellent outline of how we understand the spreading of the virus in different situations.

The article outlines three major ways that cause you to get infected: distance, dose, and dispersion; the explanation of these, taken from the article are (emphasis my own):

Distance: The further away you are from someone who is infected, the less likely you are to be infected by them or to breathe in particles they have breathed out.

Dose: To become infected you need to have contact with a minimum dose, which takes time and exposure to people with the virus. The longer you are exposed to an infectious person, the more people you are exposed to, and the fewer barriers (like cloth masks) between you, the more likely you are to be exposed to the virus. People who have symptoms or are about to develop symptoms, including mild illness, are generally more infectious – i.e., are able to produce larger doses of infected respiratory particles.

Dispersion: Because smaller particles hang around in the air, the movement of air makes a really big difference. The particles disperse quickly if you are outside, particularly if there is a breeze or wind. We also know that sunlight breaks down the virus. Small, enclosed spaces with closed windows are high risk, especially when they are crowded.

These three items give us some meaty material to cover, all of which are relevant to my experience in the line — there are so many handles on this luggage!

I am going to go over 8 different factors about the scenario of me standing in line, all of these changing the level of risk that I was absorbing throughout.

• The radiation analogy
• The date
• The weather
• The people geometry
• The people date
• The mathematics
• The miscellaneous
• All together now

The radiation analogy

Generally, I think of this similarly to how one can absorb a specified safe amount of radiation over a period of time without it becoming lethal, or giving side effects. In terms of radiation, the time scales are much larger, an example of this is that it is recommended that you only expose your body to a limited amount of radiation over your lifetime (approx. 100 mSv which is about 25 CT scans). However, relating to the virus, the time scales are very different because you can absorb a specific dosage of virus over a short period of time, but if you absorb below the safe threshold, and then stop absorbing virus, your chance of getting infected drops back down again.

Then again, this safe threshold shouldn’t be taken literally, from a podcast I enjoy: “It’s not a magic threshold.”

So, during the course of writing this, I thought it would be good to attempt to model the risk, which apparently has a unit. According to Wikipedia, a micromort is so defined: a unit of risk defined as a one-in-a-million chance of death.

So, although it’s quite difficult to define exactly how many micromorts I absorb in different situations with the virus, I am instead going to define this as a situation where if I absorb a minimum amount of my own fictional risk units over a period of time, then it’s almost a given that I get infected¹.

At a basic level, the level of risk absorbed over time will increase when exposed to someone who is infected, I think it is interesting to look at this where I am different distances away from someone who is infected (keeping their day of infection constant).

I cannot fathom why, but for some reason I think of this in terms of a sigmoid curve: the longer that you are around someone who is infected, the more risk units you absorb, essentially increasing the chance that you get infected. This is also related to the distance away from the individual, this is why I have the different curves which correspond to being different distances away from the individual.

The graph here has a dotted line, which due to the nature of how these things work, means that there is never a 100 % chance of getting infected, but it approaches this limit with increasing time.

Conveniently, this graph can be used to illustrate a number of different characteristics; for example, if instead of each curve representing the distance away from the person, but instead it represented three different types of protective equipment. The curve on the left would represent a situation where no masks were worn, the second situation where a simple material mask is worn, and the third situation being a high-grade surgical mask.

This is obviously a very simple scenario: two people facing one another with differing separation distances; however, the situation I spoke about above is far more complex and, I think, quite interesting to think about².

The date

One convenient fact about this analysis is the amount of data available. The many dashboards and testing data mean that it is sort of possible to figure out the statistics of the people who were standing in line at that point in time.

One of the interesting statistics to look at is the test positivity at that point in time on the 17th of June.

Now the first fact to consider here according to this report from week 25 (the week after I stood in line) is that the mean turnaround time for the tests at that point in time was approximately 9 days…

According to the report (on page 9), Gauteng had a test positivity of 17.7 %. Now, of course I am generalizing for the whole of the province, different testing stations (corresponding to different areas of the province) would have varying rates of test positivity; but I think it’s a decent method of getting an idea of the most likely level at that point in time³.

Drilling further into the document linked, page 21 provides a choropleth map (I know how to make those!) which represents the test positivity of different regions by making use of different colours. I can’t quite see where Bedfordview is within this map, so I have attempted to take the outline generated by Google maps of Gauteng and overlay this onto the Choropleth map from the NICD document.

The area directly above the label of “Apartheid Museum” is where Bedfordview is located… Directly in the middle of the second darkest red section which suggests a test positivity of between 20 - 30 % test positivity…

I think I can make out that the area is labelled by EKUN2 which corresponds to Ekurhuleni East 2 which according to the document (on page 18) which has a percentage testing positive (95% CI) of 0.157 (0.094 - 0.220) whereas the previous week had 0.026 (0.008 - 0.044). So what do these numbers mean?

Well, firstly it is clear that on this week, there appears to be a test positivity rate of approximately 15.7 % (I am going to simplify my analysis and say that this is 20 %, considering the error in their document.) So what does this number mean to me standing in the line? Well, basically it means that approximately every one in five people standing in the line that day were indeed infected.

How does this correspond to a risk level? Well this depends on how the line is organized and where the different people are standing in the line, for now, I am going to attempt to calculate the probability that the people in front of, and behind me were positive (a later section looks at how other people in the line affect the level of risk depending on where they are in relation to me.)

Secondly, I think that it is also interesting to note here that the previous week had a test positivity rate of 2.6 %; this is an increase of 13.1 % which is crazy… I guess that’s why this entry in the document is labelled in red.

This also explains something that the hospital administrator said when he came outside to speak to us in the line. He asked why there were so many people standing in the line, he said that it was unusual to have so many people standing in the line when compared to previous weeks. I guess the numbers now explain why the line was suddenly so long, obviously the virus was spreading very rapidly during this time period. I am beginning to believe that this period of time can be marked by the rapid increase in cases in Gauteng…

I find it rather interesting that, according to the headline fact on the NICD document, the test positivity rate for the country was 7.9 %, the situation for me was vastly different. The luxury of having access to this level of data detail meant that I could get a far better understanding of the risk I was being exposed to.

The fact that this level of data is available to the public, being transparent, is great. I just hope that this information is used to make calculated and informed decisions with regards to policy and laws; hopefully it is used to get an understanding of what is happening in close to real-time so that lives (and livelihoods) can be saved by the decision makers. It’s amazing to see that the information that is recorded, is then made available to the public, which allows them to also make decisions based on their analysis and understanding.

The weather

The weather conveniently saved me greatly in the situation; firstly, being outside meant that there was wind and sun. These two in combination greatly reduced the risk per unit time.

I am not so sure how to evaluate how much the effect of the sun affected the risk units, but the wind is vastly different.

The winds direction, coupled with the geometry of the line, will change the risk units. For some reason, I have in my mind that the simple presence of wind greatly reduces the risk, but different wind directions will result in different risk units.

Consider the figure below where the wind is blowing in the Easterly direction.

Considering I am in the middle here, the only risk that I am absorbing is the people to the West/left of me; however, the person on the far right is absorbing the most risk… Sucks to be them…

So in general, the further to the right you are in the line, the more risk units will be absorbed as every extra person down the line will increase the probability of a person being infected and spreading virus. This is greatly simplified, completely ignoring the effects of the movement of particles around objects…

Okay, considering the figure above, where the wind direction, Easterly, is the 0 °C rotation, then different wind directions will change the amount of risk units.

The following figure illustrates the risk units that would be absorbed at differing wind directions.

So generally, if the wind is blowing perpendicular to the direction of the line, that will result in the lowest risk units for everyone involved.

The moral of this story is… Umm… Make the line direction dynamically shift with the wind direction?

The people geometry

Okay, now before I forget, I want to get an idea, based on these numbers (every fifth person is infected) what the probabilities are of me standing next to (or near to) someone who was infected. For the purposes of this part of the analysis, I am going to ignore the fact that the dude standing in front of me already had it and was now no longer infectious.

Let me just think about the simplest scenario here, the person in front of, and behind me. If the probability of being infected for each person in the line sits at 20 %, then the people in front of and behind me each have this probability, thus there is a 40 % chance that I am standing next to someone who is infected.

What if I now think about two people in front of and behind me? Well, this would then be 20 % for every person, this can easily be modelled with the use of a Binomial distribution.

So, in the situation where I am looking at the person in front of, and behind me, it looks like the following.

This means that of the two people that surround me, both of them have a 20 % chance of being infected, what about the two people next to those two?

Again, what about the next two people surrounding them?

Is it possible to understand the situation the further out of the line that I look? What is the probability that someone in each of these scenarios is actually infected? I guess that, using the Binomial distribution, it is possible to get an idea of finding the probability of at least one of these people being infected.

What does this mean? Well, if I look at the two people next to me in line (n = 2), what are the chances of at least one of those people being infected? Note that this includes the probability for one person being infected in addition to the probability of both being infected. Expanding this to two people on either side is simple (n = 4) and onwards on either side of the line (n = x).

So, in general, as I look further and further out into more people on either side of me, the probability of at least a single person being infected approaches 1. Thinking back to the length of the line on that day, I would think that there were definitely people who were infected.

The people date

There is a further interesting fact to consider about the people that are standing in this line - why are the people standing in the line? We know why I was standing in line… So I am an exception here, but I think that there are three different reasons that people will be standing in this line:

1. The individual is experiencing related symptoms and believes they have the virus,
2. The individual has been in contact with someone who tested positive and is now checking whether they got infected,
3. The individual is a hypochondriac.

So I think that the first point to consider is that if you are infected, then over the course of time from when your body is first exposed, until you have completely recovered, it is believed that different amounts of virus are shed from your body.

The following three quotes from Harvard give a general idea of the course of the virus and the infectious periods:

The time from exposure to symptom onset (known as the incubation period) is thought to be three to 14 days, though symptoms typically appear within four or five days after exposure.

A person with COVID-19 may be contagious 48 to 72 hours before starting to experience symptoms. Emerging research suggests that people may actually be most likely to spread the virus to others during the 48 hours before they start to experience symptoms.

Most people with coronavirus who have symptoms will no longer be contagious by 10 days after symptoms resolve. People who test positive for the virus but never develop symptoms over the following 10 days after testing are probably no longer contagious, but again there are documented exceptions.

So coupling these pieces of information with one last quote from the same document:

One of the main problems with general rules regarding contagion and transmission of this coronavirus is the marked differences in how it behaves in different individuals.

This suggests that there is a distribution on the levels of infectiousness from first infection until the virus has run its course; this is also coupled with the fact that different people will have differently shaped distributions. That means it’s possible to find a distribution for the average individual.

There’s a nice illustration of the different points of virus progression found here which I have adapted below.

Naturally, a lot of work has gone into modelling the virus shedding over this timeline, the papers from the WHO and this Nature article show how it can be modelled by a Gamma distribution, of which I have adapted here.

Interestingly enough, this curve (the simulated serial interval) appears to have been generated by the convolution of the infectiousness profile with the incubation period. This shows how the level of infectiousness can peak before the onset of symptoms which are illustrated in this figure by day zero.

What is so interesting to observe here, which is spoken about early on in the Nature paper, is the difference between the observed mean serial interval and the observed mean incubation period; particularly if the serial interval is shorter than the incubation period, transmission occurs before symptoms appear. This fact makes it difficult to contain by normal measures; the larger this difference between the two periods, the harder it is to contain and the longer time period infected individuals will go about their normal lives without knowing they are spreading the virus, increasing the number of people infected (is HIV a great example of this?).

I spoke to a friend about this a while back, as I thought that it would be possible to get an idea on the day of infection that different people would be standing in the line. I thought that there would be a single distribution, similar in shape to the Gamma distribution above; he suggested that there were instead two different peaks of the people in the line.

The first group would correspond to the people who are experiencing symptoms

Whereas the second group, the group who believe or know that they came into contact with someone who tested positive would likely be in the following space (and further to the left than this figure represents).

I believe that the distribution shape that he was talking about is called a Multimodal distribution, or in this case a bimodal distribution.

I would think it looks like the figure below, the placement of the two distributions along the day timeline is just a guess on my side.

The mathematics

Now, in a previous post, I spoke about the fact that I would be using the Navier-Stokes theorem to show how its dynamics assisted in keeping me virus free… I was listening to one of my favourite podcasts, Food Safety Talk where they spoke about how they believed that this equation can be used for certain modelling situations.

So, here’s the equation again…

However, I didn’t manage to find any evidence of this equation being applicable to this situation, I did however find a paper where the authors suggested that this set of equations doesn’t apply at all!

This paper appears to have noted the equation, but it is also interesting as it speaks about how different factors influence the risk of becoming infected, the following factors are looked at when considering risk levels⁴:

• Volume of space
• Air flow rate
• Air quality
• Duration of time spent in space
• Occupancy of space

What is interesting to note from this paper is that the infection risk increases linearly with increasing time spent in the different situations.

This is really important for situations such as going on a plane, or going into the office; you will spend large amounts of time in these spaces, which means that the other factors need to be modified to reduce the risk - better air flow rate, lower occupancy, social distancing distances. These are all variables that can be played with to ensure the lowest risk environment, it’s all a balancing act.

So fortunately, in my situation, there was a strong wind outside, in addition to the line being able to spread out and people social distancing well enough. All this adds up to me not getting infected.

The miscellaneous

When I started out writing this, I thought I would have a few miscellaneous things to include, but I think I managed to ramble sufficiently before I got to this section, so I will not put the reader through much more pain except to add one last point.

The person who was standing in front of me tested positive two weeks before, which meant that the virus had run its course on him, and thus he likely changed the entire scenario I spoke about above, but I ignored that because I had no idea how it would have changed my assumptions other than making the modelling more difficult! On the other hand, it did mean that my risk was reduced…

Altogether now

So what was the moral of this story? Uhhh… I guess, stay outdoors, and wear a mask?

1. I also don’t really know how to convert the micromorts into the chance that I die from covid, so it’s just easier to create a fictionalised version of the risk units instead. ↩

2. Obviously, I am generalising massively here, as you can see I haven’t included any units. Isn’t it lovely when you have a fictional unit, you can do anything you want with it!!! ↩

3. I am assuming here that the PCR tests are at this point are very reliable and are a good indicator of a positive/negative result. ↩

4. Some other interesting papers which speak about risk and how it is affected by wind, humidity, occupancy, masks, etc. can be seen here, here, here, and here. ↩