Virus loading as a queue

Science could use more interdisciplinarity. Models from disparate disciplines can be transferred and may give insights, at least by analogy, if not direct mathematical application.

Queueing models are common in transport. A queue forms when the inputs exceed the outputs. For instance think of cars at a bottleneck. Imagine cars arrive at 1 per second, but can be served (move through the bottleneck) at 2 per second, in this case there is no queue. Instead imagine cars arrive at 4 per second, but can be served at 2 per second, here the queue grows in length by 2 cars per second. At the end of an hour, the queue is 7200 cars in length.

This same logic could be applied to viruses, though the math is a bit more complicated since viruses and antibodies have doubling rates rather than arrival and server capacity or departure rates. Your body is exposed to a single copy of a virus, it doubles at some rate and causes damage as discussed below. If all is going well, your body produces antibodies [and cytotoxic T lymphocytes (CTLs)] in response, which serve (kill) the virus and its production system.

Lev Osherovich, biologist friend of mine, notes:

Covid-19 virus. Source wikipedia
Covid-19 virus. Source wikipedia

There’s a lot more to a successful immune response against SARS-CoV-2 than just antibody production — for example, there is a strong component of cellular (cytotoxic) immunity that goes after infected cells before they can churn out more virus. However, it’s a matter of time and chance for the immune system to figure out how to make the right combination of antibodies and cytotoxic T lymphocytes (CTLs) that can quell the virus — in some patients that happens quickly, before the virus penetrates deep into the lungs to cause acute respiratory distress syndrome, but in many patients (particularlly elderly and immunocompromised) the virus gets there first. In general, once an effective immune response is launched, the immune system can rapidly scale up production of the right plasma cells (antibody factories) and CTLs — the bottleneck is in finding a winning combination through random chance. We hope (but are not certain) that once an effective immune response occurs, it will rapidly clear the virus and will prevent re-infection — at its best, anti-viral immunity is an all-or-nothing process.


Antibody and CTL production varies by individual, older individuals and those who are immune-compromised may produce less effectively than young adults.

For someone newly exposed, the virus has a head start, so the antibody and CTL doubling rate has to be shorter (it doubles more quickly) to catch up. If the virus lead is too great, your body is overwhelmed not just by viremia (high viral titres) but cytokine release syndrome (CRS) — “when large numbers of white blood cells are activated and release inflammatory cytokines, which in turn activate yet more white blood cells.” There is a maximum queue length (number of un-dealt with viruses and cytokenes ) in the body, at which point you die. As you approach this maximum, you get sicker.

Lev writes:

The typical cause of death is not viremia (high viral titres) but cytokine release syndrome (CRS), an inflammatory over-reaction triggered by too much virus in the lung but exacerbated by pre-exisiting conditions and susceptibilitiies. There are evidently many patients who experience high viral titre (and are very ill) but do not progress to CRS, ARDS and death, and likewise there are some patients who progress rapidly to CRS and death without high viral titres. In the context of your queuing model, in the some cases the virus (or the immune system) jumps the queue and kills the patient. While presumably having high viral titre increases the odds of such an event, there seems to be some element of bad luck as well as predisposition.


In the figure below, viruses start doubling at time 1 (first exposure), while antibodies (and CTLs) don’t start until time 16, but they double twice as fast. By time period 28, they start to noticeably slow the growth of viruses, and due to the power of compound interest, effectively destroy them by time period 30. Assuming the patient can survive a load of 10 million viruses in their body, everything is fine. But there is a threshold, and if the antibodies aren’t fast enough (start too late, double too slowly), the patient won’t make it.

Virus input (demand or arrival rate) varies by location, e.g. cruise ships or nursing homes with recirculated air and exposure to many other infected people increases loading, so it is not a single virus that infects the subject, but many, thereby giving the virus an even longer head start, and making the job of eradication that much more difficult.

Antibody production (capacity) can be stimulated with vaccines or with previous exposure.


While obviously the body is more complex than a queueing model, so is traffic. The aim of a model is to give us a way of thinking, which might suggest solutions (reducing viral intake (defense), speeding response (offense), and so on).

Your daily commute won’t ever be the same | National Geographic

Emily Sohn at National Geographic wrote: Your daily commute won’t ever be the same.

The title is a bit overwrought [for most Americans, things will be largely indistinguishable, since most Americans drive in the suburbs], but the content identifies areas where things might differ from today. Lots of us were interviewed, my contributions below:

An empty platform with a monitor indicating with the green train icon that a largely empty train is coming

Facial sensors like ones that have been deployed on public transportation in China may also measure your temperature, rejecting entry to a bus or subway station if you have a fever. Once antibody testing is accurate and easily available, you might even show an immunity card before you can board, though there will be a fine line between safety and hassle, adds David Levinson, a transport engineer at the University of Sydney, Australia.

“All of those invasions of privacy are things that are not going to make people happy, and are going to make them more likely to use other modes than public transit,” he says. “People will find alternatives if they can.”

For some people, the future of commuting might be no commute at all. About half of adults with jobs in the U.S. are working from home during the pandemic, according to a report published by the Brookings Institution in April. That’s more than double the percentage who did some telecommuting two years ago. Close to 20 percent of chief financial officers surveyed by Brookings said they planned to permanently retain remote work for at least 20 percent of their workers.

This kind of societal shift could further reduce crowding and the spread of disease on mass transit, Levinson says, especially if people go into offices only occasionally, if they bike when they can, and if they get better about staying home when sick.

How public transport will change as lockdowns are eased

Josh Dutton at Yahoo News (which is a relatively bigger deal in Australia than the US due to a tie-up with one of the TV networks (7)) wrote: ‘Think outside the box’: How public transport will change as lockdowns are eased‘. I was quoted:

These shops in Waterloo are a bit of ghost town (2010-04-25)
These shops in Waterloo are a bit of ghost town (2020-04-25)

University of Sydney transport analyst Professor David Levinson also believes people will walk or cycle to work.

“This means more space needs to be available for footpaths, and more protected and dedicated cycle lanes need to be deployed rapidly,” he said.

“This can be taken from the excessive space given over to cars, especially parked cars, now.”

As for catching taxis and rideshares to work, Professor Levinson believes this only alleviates issues for people who don’t already own cars “and that gets expensive pretty quickly”
Issues might arise from parking, particularly in Sydney’s CBD, Professor Levinson said as there is “only a small and finite” number of spaces.

“If everyone were to go back to work at the same rates as pre-COVID, they could not all possibly drive. However, if work-at-home remains in place, say for half the workforce, then twice as many people could drive [to] the CBD,” he said.

Professor Levinson added congestion “tends to be self-limiting”

“As traffic increases, travel times rise, and as travel time rises demand (traffic) drops, so it finds a level,” he said.

“The equilibrium might be more congested than before the virus for a period if people are going back to work in high numbers but still transit averse, especially on bad weather days.

“But if the virus were effectively extinguished, I suspect transit ridership would go back to near normal levels, rather than stay at ghost town levels.”

He added he believes people will be working from home “at much higher levels post-COVID than pre-COVID” leading to “a lot less” travel.