Coronavirus: all the SARS-CoV-2 in the world could fit in a soda can (and how that mathematical conclusion was reached)

Christian Yates*

The Conversation

When asked to estimate the total volume of SARS-CoV-2 in the world for the BBC’s More or Less program on Radio 4, I will admit that I had no idea what the answer would be. My wife suggested it would be the size of a lap pool. “That or a teaspoon,” he said. “Usually it’s one or the other with these kinds of questions.”

So how do you begin to calculate what the total volume really is? Fortunately, I am used to making these types of estimates, having carried out several of them for my book “The mathematics of life and death.”

However, before embarking on this particular numerical journey, I must make it clear that this is a approximation based on the most reasonable assumptions, but I’ll happily admit that there may be room for improvement.

So where do you start? It is best if we first calculate how many SARS-CoV-2 particles there are in the world. For this, we will need to know how many people are infected. (We will assume that humans, rather than animals, are the most important reservoir for the virus.)

Complex sum

According to the statistics website Our World in Data, half a million people test positive for COVID-19 every day. However, we know that many people will not be included in this count because they are asymptomatic or choose not to be tested, or because generalized tests are not readily available in your country.

Using statistical and epidemiological models, the Institute for Health Metrics and Evaluations, in the United States, has estimated that the real number of people infected each day is closer to 3 millions.

The amount of virus that each currently infected person carries with them (their viral load) it depends on how long ago they were infected. On average, viral loads are thought to increase and peak about six days after infection, after which they steadily decline.

Of all the people who are infected now, those who were infected yesterday will contribute a little to the total count. Those who were infected a couple of days ago will contribute a little more. Those infected three days ago a little longer. On average, people infected six days ago have the highest viral load. This contribution will then decrease for people who were infected seven, eight, or nine days ago, and so on.

The last thing we need to know is the number of virus particles that people harbor at any one time during infection. Since we know roughly how viral load changes over time, it is sufficient to have an estimate of the viral load. maximum viral load.

An unpublished study took data on the number of virus particles per gram from a variety of different tissues in infected monkeys and increased the tissue size to be representative of humans. Your rough estimates of peak viral loads range between 1 billion and 100 billion of virus particles.

Let’s work with a value in the middle of this range (the geometric mean) at 10,000 million. When you add up all the viral load contributions of each of the 3 million people who were infected on each of the previous days (assuming this rate of 3 million is roughly constant), we find that there are roughly two hundred quadrillion (2×10¹⁷ or two hundred million trillion) virus particles in the world at any one time.

It sounds like a really big number, and it is. It is approximately the same as the number of grains of sand on the planet. But when we calculate the total volume, we must remember that the SARS-CoV-2 particles are extremely small.

Estimates of its diameter range from 80 and 129 nanometers. A nanometer is one billionth of a meter. To put it in perspective, the radius of SARS-CoV-2 is about 1,000 times thinner than a human hair.

Let’s use the average value for the 100 nanometer diameter in our further calculation. To calculate the volume of a single spherical virus particle, we need to use the formula for the volume of a sphere which is undoubtedly on the tip of everyone’s tongue:

Assuming a radius of 50 nanometers (in the center of the estimated range) of SARS-CoV-2 for the value of r, the volume of a single virus particle turns out to be 523,000 nanometers³. Multiplying this very small volume by the large number of particles we calculated earlier and converting it to significant units gives us a total volume of approximately 120 milliliters (ml).

If we wanted to collect all these virus particles in one place, we would have to remember that the spheres do not pack perfectly.

Free space inside a can

If you think about the pyramid of oranges that you can see in the warehouse, you will remember that an important part of the space it occupies is empty. In fact, the best thing you can do to minimize void space is a configuration called “closed sphere packaging” in which the void space occupies approximately 26% of the total volume.

This increases the total accumulated volume of SARS-CoV-2 particles to approximately 160 ml, small enough to fit inside about six shot glasses. Even taking the upper end of the diameter estimate and taking into account the size of the spike proteins, all the SARS-CoV-2 wouldn’t fill a can of Coke.

It turns out that the total volume of SARS-CoV-2 was between my wife’s rough estimates for the spoon and the pool.

It’s amazing to think that all the troubles, disruptions, hardships, and loss of life in the past year could constitute just a few drinks of what would undoubtedly be the worst drink ever.

Christian Yates is Professor of Mathematical Biology at the University of Bath, UK.

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