Mathematics of the pandemic

The May issue of the Spanish magazine on popular science Investigación y Ciencia, associated to Scientific American, contains an article authored by Bartolo Luque, Fernando Ballesteros and Octavio Miramontes, in which they apply to the current pandemic a mathematical model that dates back over a century. This model describes the first phase of the pandemic, the exponential rise of the first part of the logistics curve that I referred to in a previous post in this blog, and makes it possible to compare the evolution of the disease in various countries, depending on the virus containment measures that have been taken in each of them. It could also serve to explain why Spain has become the country with the most cases in Europe and with the most deaths per 100,000 inhabitants in the world.

The attached figure, modified from one that appears in the article mentioned above, shows the evolution of the pandemic in Spain during the month of March. Since the vertical axis is marked with a logarithmic scale, an exponential ascent appears in the figure as a straight line. The slope of this line defines the speed of ascent of the number of accumulated cases. It can be seen that the ascent was very steep in our country during the first half of the month, until the state of alarm was declared on March 14 and measures were taken to stop the explosive expansion of the pandemic. Starting a few days later, the slope of the line decreases considerably, until at the end of March the real data begins to fall below the theoretical line, which means that the exponential phase of the logistics curve ends and little by little it moves to the linear phase, during which the semi-logarithmic curve flattens.

The previous figure allows us to estimate what would have happened if the alarm state (and the subsequent measures) had been taken a week before it was, towards March 7. In such a case, the change in the slope of the exponential ascent line (the orange curve) would have been reached earlier (around March 11) and the ascent of the logistic curve would have been considerably reduced; remember that the vertical axis is logarithmic, which means that a small decrease is much more important than it appears.

The attached figure shows what would have happened, compared to what actually happened. Here the vertical axis is no longer logarithmic, but linear, which makes it simpler to compare the values ​​ appearing in the figure. The red curve represents the real data; the green curve shows what would have happened if the precautionary measures had been taken a week earlier. To estimate the concrete values ​​that would have been reached, I have calculated that part of the curve by multiplying each value by the quotient of the two corresponding values ​​on the actual curve one week later. Due to daily variations in the number of new cases, the green curve appears at some points slightly above the red curve.

The conclusions are clear. If these measures had been taken a week earlier, the maximum number of cases of the disease would have dropped from more than 200,000 to less than 100,000, suggesting that the number of deaths would also have been reduced to less than half. Instead of more than 25,000, we’d have at most 12,000. Or if we consider that the official figures are probably lower than the real ones (between 25 and 76%, according to Mortality Monitoring data and other studies), the decrease in the number of deaths would have been even more spectacular.

It is clear that the decision of the Spanish government to delay control measures (contrary to what they did in other countries, such as Japan, South Korea, Greece and Portugal) to allow the demonstrations to take place on March 8, had consequences disastrous and should be considered, at the very least, as a tragic mistake that has cost us more than 100,000 patients and over 10,000 deaths.

This at least is what mathematics says.

The same post in Spanish

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Manuel Alfonseca