https://www.lemonde.fr/sciences/article/2020/02/05/faites-plutot-vacciner-vos-amis-c-est-mathematique_6028500_1650684.html

Mathematician Etienne Ghys evokes the implications that the « paradox of friendship » could have in strategies to fight pandemics.

Carte blanche. To understand how a virus spreads in a population, biology is of course very important, but it is not enough: mathematics is needed. Once a number of parameters – the transmission rate, incubation time, etc. – are known, the virus can be transmitted to the population. Once a certain number of parameters are known – transmission rate, incubation time, etc. – formidable mathematical problems still need to be solved. In the simplest epidemiological model, the population is broken down into three compartments: healthy people, infected people and people who are immunized after the disease. Healthy people can be infected with a certain probability when they meet a person who is already infected. An infected person becomes immune after a certain period of time. This leads to relatively simple differential equations.
It is clear that this model (developed a century ago) is very naïve. Many others, increasingly complex, have been imagined and work in many situations. The major difficulty is that most of these models are based on an assumption of population homogeneity, whereby individuals come into contact at random and the probability of infection does not depend on the individuals who meet. The population would have to be broken down into a multitude of compartments, taking into account, for example, their age, where they live, etc. The main difficulty is that most of these models are based on an assumption of population homogeneity, whereby individuals come into contact randomly and the probability of infection does not depend on which individuals meet. This becomes extremely complicated.
The problem is to understand the « network of contacts ». Draw 7 billion dots on a sheet of paper, one per human being, and join 2 dots with a line every time the 2 corresponding individuals met last week. Since this « drawing » is impossible to do in practice, we try instead to describe its global properties. For example, it is thought to be a « small world »: any two human beings can be connected by a very short series of individuals such that each is a friend of the next. It is even said that a string of length 6 should be enough, which can be worrisome if the virus is transmitted between friends.

Large network theory

On a much smaller scale, a group of researchers carried out an experiment in a high school in the United States: for one day, a thousand students wore small detectors around their necks, and it was possible to obtain a complete list of all the encounters between them (within three meters, for at least one minute). The researchers were then able to analyze in detail the properties of this network of encounters and then how an infectious disease could spread in this high school.
The theory of very large networks is currently in full expansion, both in mathematics and computer science. Here is a very simple but surprising theorem: « A majority of individuals have fewer friends than their friends ». Let’s take the following example: Mr. X has 100 friends who are friends only with him. So, of these 101 people, all but one of them have only one friend, but their (only) friend has 100 friends. It turns out that this phenomenon always happens, regardless of the nature of the friendship network.
As an application, let’s imagine that there are only a small number of vaccines available, and that it is a matter of choosing which people should be vaccinated. We could vaccinate randomly selected people, but a much better idea would be to randomly select one person and ask them to name one of their friends, and vaccinate that friend. If the friend has more friends, more people are likely to become infected and it would be better to vaccinate that friend. In the previous example, it is Mr. X.
The paradox of friendship goes further. Not only do your friends (in general) have more friends than you, but they are said to be happier than you!