Epidemics: flattening the exponentials


Carte blanche. These last days will have at least allowed the French to understand in their flesh what an exponential is. We have all become aware that the powers of 2 grow really fast: 1, 2, 4, 8, 16, 32, 64, etc., to exceed one billion in just 30 steps. What is less well known is that while the number of new infections in an epidemic doubles every three days, half of those infected since the beginning of the epidemic have been infected for less than three days. The exponential function has terrifying aspects.
The first scientist to highlight this type of growth was Leonhard Euler, in 1760, in an important article entitled “General Research on the Mortality and Multiplication of the Human Race”. In 1798, Thomas Malthus understood that exponential growth is a threat to humanity. Fortunately, in 1840, Pierre-François Verhulst discovered “logistic growth”, which allowed him to understand why the exponential growth must eventually calm down. This is the curve that was presented so clearly on a television set by our Minister of Health.
In a purely exponential growth, the number of new cases of contamination is proportional to the number of people contaminated. In formula, the derivative y’ of the number of cases y is proportional to y, which translates into a diabolically simple equation y’ = ay, whose exponential solution y = exp (at) may bring back memories to the reader. The coefficient ‘a’ depends on the average number of contacts we have: the larger it is, the faster the exponential explodes.

Bell curve

In a logistic growth, the number of new cases of contamination is proportional to the number of people already contaminated, but also to the number of people who are contaminable, i.e. who have not already been contaminated. Fortunately, the number of contagious people decreases as the epidemic progresses, and the evolution is reversed.
In the formula, y’ = ay (1-y/b) where b denotes the total population. In this model, the number of new cases follows the bell curve drawn by the minister. There is an exponential growth at the beginning (when the number of cases is still small), then a maximum, and finally a decrease. The only parameter we can act on is this seemingly innocuous coefficient “a”, which is related to the average number of our contacts. When we decrease “a”, the curve keeps the same speed, but it flattens. Certainly the peak comes later, but it will be lower. The epidemic lasts longer, but it is less deadly. That’s why you have to stay home!
In the 18th century, the question was raised as to the value of inoculation in the fight against smallpox, which had decimated nearly half of Europeans. It was a very primitive version of vaccination, but one that presented dangers for inoculated patients (unlike vaccination). Mathematician Daniel Bernoulli will write an article entitled “Testing a new analysis of smallpox mortality and the benefits of inoculation to prevent it” which mathematically demonstrates that inoculation is beneficial. Alas, it will not be listened to.
A few years later, the article “Inoculation” in Diderot and d’Alembert’s encyclopedia stated: “When it is a question of the public good, it is the duty of the thinking part of the nation to enlighten those who are susceptible to light, and to drag along by the weight of authority this crowd over whom the evidence has no hold. »
This may be true, but it is even truer when “the thinking party” clearly explains its choices by drawing a curve on a TV set.