When asked to calculate the total volume of SARS-CoV-2 in the world for the BBC Radio 4 show More or lessI admit I had no idea what the answer would be. My wife suggested that it would be the size of an Olympic swimming pool. “Either that or a teaspoon,” she said. “Usually it is one or the other with such questions.”

So how do you get an approximation of the total volume? Fortunately, after doing some of these for my book, I have a mold with these large-scale estimates on the back of the cover The math of life and death. However, before we embark on this particular numerical journey, I should be clear that this is an approximation based on the most reasonable of assumptions, but I am happy to admit that there may be places where it can be improved.

Where should I start? We should first calculate how many SARS-CoV-2 particles there are in the world. To do this, we need to know how many people are infected. (We assume that humans, rather than animals, are the primary reservoir for the virus.)

According to statistics website Our world in dataHalf a million people test positive for COVID every day. We do know, however, that many people are not included in this count because they are asymptomatic or choose not to be tested – or because widespread testing is not readily available in their country.

Using statistical and epidemiological modelingThe Institute of Health Metrics and Ratings has estimated that the actual number of people infected every day is more like 3 million.

The amount of virus each currently infected carries around (their viral load) depends on how long it has been since they were infected. On average, the viral load is believed to increase and peak six days after infection, then they steadily decrease.

Of all the people infected now, those who got infected yesterday will add a little to the total. Those who got infected a few days ago will add a little more. The infected three days ago. On average, people infected six days ago have the highest viral load. That 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 humans are harboring at any point during their infection. Since we know roughly how viral load changes over time, it is enough to have an estimate of the maximum viral load. On unpublished study took data on the number of virus particles per Grams of a number of different tissues in infected monkeys and increased the size of the tissue to be Representative of man. Their rough estimates for maximum viral load range from 1 to 100 billion virus particles.

Let’s work with the upper end of the estimates so that we end up with an overestimate of the total volume. If you add up all of the viral load contributions from each of the 3 million people infected on each of the previous days (assuming that 3 million rate is roughly constant), we find there are roughly two trillion (2×10¹⁸ or two) Billions of billions) virus particles in the world at any time.

That sounds like a really big number and it is. It’s about the same as that Number of grains of sand on the planet. When calculating the total volume, however, we have to take into account that SARS-CoV-2 particles are extremely small. Estimates of the diameter range from 80 to 120 nanometers. A nanometer is a billionth of a meter. The radius of SARS-CoV-2 is approximately 1000 times thinner than that of a human hair. In our calculation below, let’s use the average value for a diameter of 100 nanometers.

To calculate the volume of an individual spherical Virus particles we need to use the formula for the volume of a sphere, which will undoubtedly be on everyone’s tongue:

V = 4π r³ / 3

Assume a radius of 50 nanometers (in the middle of the estimated range) of SARS-CoV-2 for the value of *r*The volume of a single virus particle is 523,000 nanometers³.

Multiply that *tiny* Volume from the *very large* The number of particles that we calculated earlier and the conversion into meaningful units gives a total volume of about 120 milliliters (ml). If we want to put all of these virus particles together in one place, we need to remember that spheres do not fit together perfectly.

## Close the ball packing

When you think about the orange pyramid you might see in the grocery store, you’ll remember that a significant portion of the space it takes up is empty. The best thing you can do to minimize the empty space is in a configuration called a “tight sphere packing” where the empty space is about 26% of the total volume. This increases the total *collected volume* of SARS-CoV-2 particles to approx. 160 ml – lightly small enough to fit into approx. six shot glasses. Even take the top end of the diameter estimate and take that into account Size of the spike proteins All SARS-CoV-2 would still not fill a Coke can.

It turns out that the total volume of SARS-CoV-2 was between my wife’s rough estimates for the teaspoon and the swimming pool. It is amazing to believe that all of the trouble, disruption, hardship, and loss of life that has arisen over the past year could only represent a few sips of arguably the worst drink in history.

Christian Yates, Lecturer in Mathematical Biology, *Bath University*

This article is republished by The conversation under a Creative Commons license. read this original article.

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