I have to admit that when I was asked to calculate the total volume of SARS-CoV-2 in the world for the program *More or less*on BBC Radio 4, I had no idea what the answer was going to be. My wife told me that it would be the equivalent of a lap pool. “That, or a teaspoon,” he told me. “In these types of questions, the answer is usually one or the other.”

So where do you start calculating the actual volume of the virus, even a guideline figure? Fortunately I have some experience with these types of large-scale rough estimates, as I have made a large number of them in my book *The Maths of Life and Death*.

However, before embarking on this particular numerical journey, I must make it clear that it is an estimate based on the most probable estimates; I therefore have no problem admitting that there might be parts of the calculation that could be improved.

So where do you start? What would interest us most, in the first place, would be to calculate the number of SARS-CoV-2 particles in the world. And to do so we have to know how many people are infected. (Here we will assume that the main host of the virus is the human being, and not any animal).

According to figures on the Our World in Data website, half a million people test positive for COVID every day. But we are aware that there are many people who do not enter into this calculation, either because they are asymptomatic, or because they choose not to take the test, or because they live in countries where there is no possibility of doing massive tests.

The Institute for Health Metrics and Evaluations has made estimates through statistical and epidemiological models according to which the true number of daily infections is closer to three million.

The amount of virus that each of the people who are infected today is carrying (that is, their viral load) depends on when they were infected. It is estimated that, on average, the viral load increases and reaches its peak around the sixth day after contracting the infection, and that it decreases steadily thereafter.

Of all the people currently infected, those who were infected yesterday will contribute little to the total. Those that were infected two days ago will contribute a little more. The ones that did it three ago, something else. On average, those who were infected six days ago have the highest viral load. However, the contribution will decrease with respect to those who were infected seven, eight or nine days ago. And so on.

The last piece of information we need to know is the number of virus particles that people have in their bodies throughout their infection. To the extent that we know, even roughly, what the variation in viral load is over time, that is enough for us to be able to make an estimate of the maximum figure for said viral load. An unpublished study has collected data on the number of virus particles per gram in a number of different tissues from COVID-19-infected monkeys, and calculated the proportion of such tissues a person would have. Rough estimates for peak viral load range from 1 billion to 100 billion virus particles.

We will work with an average value (the geometric mean) of this range, the figure of 10 billion. When the contributions of the viral loads of each of the three million people who were infected in the previous days are added (and assuming that this figure of three million remains more or less constant), we obtain that there are approximately 2×10¹⁷ o 200,000 trillion virus particles in the world at any given time.

It seems like a huge number, and it really is. This is a figure more or less similar to the number of grains of sand on the planet. But when we calculate the total space that the SARS-CoV-2 particles occupy we have to take into account that they are extremely small. Its diameter is estimated to be between 80 and 120 nanometers long, and a nanometer is one billionth of a meter. To get an idea, the radius of a SARS-CoV-2 particle is around a thousand times smaller than that of a human hair. For our later calculation, we will take as the mean value of the diameter of a virus particle the figure of 100 nanometers.

To obtain the volume of a single spherical particle of the virus we have to use the formula to calculate the volume of a sphere. Surely it is a formula that we all have on the tip of our tongue:

V = 4 π r³ / 3

If we take the radius value 50 nanometers (the average value of the estimate of its length) and incorporate it into the formula, we find that the volume of each virus particle is 523,000 cubic nanometers.

If we multiply this volume (very small) by the number of particles that we calculated before (very large) and we translate it to a unit of value that we understand, it gives us a result of 120 milliliters (ml). But if we wanted to put all these virus particles together in one place, we would have to take into account that, since they are spheres, they do not fit perfectly together.

## Compact packing of spheres

If we think of the pyramids of oranges that we can see in greengrocers, we will remember that an important part of the space they occupy is empty. In fact, if what we want is to minimize said empty space, the best way is to use an arrangement called “compact packing of spheres”, in which the unused space is reduced to approximately 26% of the total. By virtue of all this, the total (accumulated) volume of the SARS-CoV-2 particles would rise to approximately 160 ml (an amount that would easily fit in about six shot glasses). But even if we took the highest value of the diameter length estimate and added the length of the viral spicules to it, the sum of all the SARS-CoV-2 particles in the world would still fail to fill an entire can of Coke. -Line.

In the end, the total volume of SARS-CoV-2 in the world was halfway between the rough estimates made by my wife; between the teaspoon and the lap pool. It is surprising to think that all the problems, disturbances, difficulties and loss of life that we have suffered over the last year could be reduced to a few drinks of what would undoubtedly be the worst drink in history.

*Article translated thanks to the collaboration with Fundación Lilly*.