Corona Daily 181: Virus Math: Step-by-Step

Dr Christian Yates, a Mathematician at UK’s Bath university, was asked to calculate the total volume of the Covid-19 virus in the world. If it was possible to collect in one place all virus particles from all the infected humans, how much space would that collection occupy? A small room? A soccer field? A stadium?  His step-by-step calculations and the result they led to are fascinating.

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The first step is to estimate the total number of covid infected people in the world. (Viruses in animals and those lingering in air or on surfaces are ignored). Recently, the number of daily cases has declined from a high of 800,000 to 400,000. Dr Yates takes half a million people testing positive every day as the base estimate. This number, however, doesn’t include the asymptomatic cases, and those unable or unwilling to test. The models of IHME (Institute for Health Metrics and Evaluations) estimate the true number to be 3 million cases every day.

When 3 million cases are added every day for seven days (average time the virus lasts in a human body), it gives 21 million people who at any given point of time are likely to carry some amount of virus.

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Having ascertained the 21 million infected humans, the volume of virus (viral load) in them should be estimated. Studies have shown people infected six days ago carry the highest viral load. Before and after that day, the virus quantum declines. A study measured the virus particles in grammes in monkeys, and extrapolated to humans. The viral load ranges from 1 billion to 100 billion virus particles. Dr Yates has taken 10 billion particles as the geometric (not arithmetic) mean.

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The next step is to multiply the number of infected people (21 million) by the volume (10 billion particles). Rounding it off gives a large figure of two hundred quadrillion, or two hundred million billion. (2x10¹⁷). This number may be intimidating. It is almost as large as the number of grains of sand on earth. But then the SARS-CoV-2 particles are much smaller than a grain of sand. To calculate the space they will occupy, the size of the virus must be taken into the equation.

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The diameter of the covid virus is estimated to be about 100 nanometres. Billion nanometres make a meter. By way of perspective, Dr Yates compares it to a human hair which is a 1000 times thicker than the virus.

The virus being spherical, we use the sphere volume equation we learnt (or didn’t) in school days (V = 4 π r³/3). With a radius of 50 nanometre (If nothing else, each of us probably remembers radius being half of diameter), the volume of a single virus particle is 523,000 cubic nanometres.

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Having estimated the total number of viruses existing at any given time (2x10¹⁷), and the size or volume of each (523,000 cubic nanometres), Dr Yates uses more formulas and adjustments to get the final result. The total volume of the covid virus in the world is about 120 ml (millilitres). Since the microscopic particles are sphere-shaped, they will leave gaps when packed together. (Like round oranges in a box). Statisticians estimate this empty space between round balls to be 26%. Taking that into account, the total collected volume of the Covid-19 virus is 160 ml. Dr Yates compares it to a standard Coke can which is more than double this size.

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The world’s entire covid virus population can be fitted in half of a Coke can. Just like an atom in an atom bomb, it shows that size doesn’t matter when it comes to massive destruction and devastation.

Ravi