Viruses are microscopic parasites, generally much smaller than bacteria. They lack the capacity to thrive and reproduce outside of a host body.

Predominantly, viruses have a reputation for being the cause of contagion. Widespread events of disease and death have no doubt bolstered such a reputation. The 2014 outbreak of Ebola in West Africa, and the 2009 H1N1/swine flu pandemic (a widespread global outbreak) likely come to mind.

When I was asked to calculate the total volume of Sars-CoV-2 in the world for the BBC Radio 4 show More or Less, I will admit I had no idea what the answer would be. My wife suggested it would be the size of an Olympic swimming pool. “Either that or a teaspoon,” she said. “It’s usually one or the other with these sorts of questions.”

So how to set about calculating an approximation of what the total volume really is?

Fortunately, I have some form with these sorts of large-scale back-of-the-envelope estimations, having carried out a number of them for my book The Maths of Life and Death. Before we embark on this particular numerical journey, though, I should be clear that this is an approximation based on the most reasonable assumptions, but I will happily admit there may be places where it can be improved.

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So where to start? We’d better first calculate how many Sars-CoV-2 particles there are in the world. To do that, we’ll need to know how many people are infected. (We’ll assume humans rather than animals are the most significant reservoir for the virus.)

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

Of all the people who are infected now, those who got 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 more still. On average, people infected six days ago will have the highest viral load. This contribution will then decline for people who were infected seven or eight or nine days ago, and so on.

The final thing we need to know is the number of virus particles people harbour at any point during their infection. Since we know roughly how viral load changes over time, it’s enough to have an estimate of the peak viral load. An unpublished study took data on the number of virus particles per gram of a range of different tissues in infected monkeys and scaled up the size of tissue to be representative of humans. Their rough estimates for peak viral loads range from one billion to 100 billion virus particles.

Let’s work with a value in the middle of this range (the geometric mean) at 10 billion. When you add up all the contributions to the viral load of each of the 3 million people who became infected on each of the previous days (assuming this 3 million rate is roughly constant) then we find that there are roughly 200 quadrillion (2×10¹⁷ or two hundred million billion) virus particles in the world at any one time.