Vaccine effectiveness in four concepts

Pfizer’s vaccine efficacy is estimated at 95 percent. Does this mean, as is sometimes heard, that five out of every 100 people vaccinated will get sick from Covid? Thankfully, this is not the way this figure should be understood. A few definitions may be helpful to avoid such misunderstandings. Worldwide, the laboratory has selected 43,000 volunteers. Half of them, chosen at random, were vaccinated. The other half were “vaccinated” with a placebo: salt water. The volunteers could not know if they had really been vaccinated. It was then waited for 170 of them to experience symptoms of Covid and for their test results to be positive. Of these, eight had been vaccinated and 162 had received the placebo. Thus, the number of vaccinated patients was twenty times less than those who were not vaccinated. The risk of getting sick if you are vaccinated is therefore 5% of the risk of getting sick if you are not vaccinated. In other words, the risk of getting sick has been reduced by 95%, which is expressed as 95% clinical efficacy. This clinical trial has to be done before the vaccine is released, because an efficacy of more than 50 percent is required to obtain market authorization: 95 percent is therefore a very good score.
We are much more interested in real-world efficacy: the question now is how much the risk of disease is reduced in the real world for a vaccinated person. It’s quite different from a clinical trial, which mainly measures an individual’s degree of protection. The real effectiveness depends on the number of people vaccinated in the population: the more people vaccinated, the less the virus circulates, and the fewer infections and therefore the fewer people who get sick. Moreover, the duration of protection provided by the vaccine, which is still poorly known, is very important in reality, whereas it is of little importance in a clinical trial, which lasts only a short time. Actual efficacy can only be assessed after the vaccine has been released, through sensitive epidemiological investigations: it will take time to know it in the case of covid vaccines.

Benefits for all

There are two other kinds of efficiency to be added. Let’s not forget that vaccination is first and foremost a public health measure, which aims not only to limit the risk of disease for the vaccinated individual, but also for society as a whole, a significant proportion of which is not vaccinated (sometimes for good reasons). We can then estimate the indirect effectiveness, i.e. the reduction in risk that unvaccinated individuals benefit from those who are vaccinated and who do not contaminate them. Finally, there is overall effectiveness, perhaps the most important and most difficult to estimate: the decrease in average risk in the total population (vaccinated or not) compared to what that risk would be if no one were vaccinated. So these are four different notions of efficacy. In all cases, Covid vaccines will be extremely useful even if their
overall efficiency will likely be less than 95%. Even a value of 50% would prevent half of all diseases, lead to a significant decrease in the circulation of the virus in the population and save a large number of lives. As always, one must be careful with numbers. Let’s imagine that in a population there are ten times as many vaccinated people as unvaccinated people. Imagine that the risk of disease for a vaccinated person is five times less than for a non-vaccinated person. Since there are ten times as many people who are vaccinated, the number of vaccinated patients will be twice as many as the number of non-vaccinated patients. This does not mean that vaccination is ineffective.

Do not hesitate! As soon as you have the opportunity, vaccinate yourself!

Etienne Ghys

The Academy of Sciences opens its Comptes Rendus in free access

Paris, December 14, 2020

Historic publication of the Academy of Sciences, the journal Les Comptes Rendus de l’Académie des Sciences is now available online under the “free diamond access” formula. This publication model makes all articles permanently available worldwide, without any financial burden, neither for readers nor for authors. In addition, the Academy allows the deposit of preprints in open archives. Faithful to its mission of encouraging scientific life and transmitting knowledge, the Academy of Sciences is thus evolving the publishing of its scientific journals, in order to bring it in line with the principles of open science, in collaboration with the National Museum of Natural History, the CNRS and the University Grenoble Alpes.

In 2020, the Academy of Sciences has completely overhauled its scientific journals: the seven series of the Comptes Rendus de l’Académie des sciences are now available free of charge on the website.

This fundamental evolution was made possible by the Academy through the establishment of two founding partnerships: 

– The Mersenne Center for Open Scientific Publishing (CNRS – Grenoble Alpes University), a pioneering publishing platform in open science, was chosen to publish the journals Mathematics, Physics, Mechanics, Chemistry, Geoscience and Biology.  This partnership is part of a memorandum of understanding signed with the CNRS on October 28, 2020, which aims to set up a mechanism for consultation and cooperation, particularly in the area of scientific publishing.

– The publication of the journal Palévol has been entrusted to the National Museum of Natural History, whose expertise is an undisputed reference in the international community of taxonomic and naturalist palaeontologists. This partnership between the Academy and the Museum reflects the historical convergence of the two institutions’ missions of general interest. 

“We are delighted with the fruitful collaborations established between the Academy and its prestigious partners. Thanks to them, this complex project, which was particularly close to our hearts, was able to see the light of day. In the perspective of the strategic objectives that the Academy has set itself for the coming years, it aims to lay the foundations for a renewal of French scientific publishing,” emphasizes Etienne Ghys, Permanent Secretary of the Academy of Sciences. 

“Thanks to the strong support, particularly financial, of the CNRS and the exceptional mobilization of its team, the Centre Mersenne has succeeded in enthusiastically taking up the challenge proposed by the Academy of Sciences. This prefigures a strengthened partnership that will make the Academy of Sciences, the CNRS and the University of Grenoble Alpes major players in open science,” said Evelyne Miot, scientific director of the Centre Mersenne.

“I can only salute with enthusiasm and pride the collaboration between the Academy of Sciences and the National Museum of Natural History,” said Bruno David, President of the National Museum of Natural History. “Paleontology has always been a discipline at the heart of the research conducted at the Museum, a discipline that has greatly contributed to its international reputation. The arrival of Palévol in this new partnership framework follows in the footsteps of such prestigious personalities as Lamarck, Cuvier, d’Orbigny, Gaudry and many others. I wish the same success to the beautiful magazine that is Palévol”.

The archives of the articles published between January 1, 2000 and December 31, 2019 in the 7 Comptes Rendus journals remain available for free access on the Elsevier website. Previous archives, up to 1835, are available on Gallica and soon on Persée.

The Comptes Rendus are a set of 7 peer-reviewed electronic journals. 

In 2020, the editorial line of some of them has been reoriented.

Comptes Rendus – Mathematics. A new impetus has just been brought by the enrichment of the editorial board and the broadening of the editorial objectives. CR-Mathématique now welcomes different types of publications, and particularly encourages: original and significant research work; articles presenting in a non-technical or synthetic way important or topical mathematical developments; texts presenting important mathematical works in a global way; thematic issues taking stock of various approaches to the same problem (for example to report on colloquia or working days); texts of historical, philosophical or didactic reflection closely related to mathematics. Editors: Jean-Michel Coron, Jean-Pierre Demailly, Étienne Ghys, Laure Saint-Raymond.

Comptes Rendus – Physique covers all areas of physics and astrophysics and mainly proposes dossiers. Thanks to this formula, which has become a reference in the field, readers will find in each issue the presentation of a particularly fast-developing subject. The authors are chosen among the most active researchers and the coordination of each thematic issue is ensured by a guest editor, guaranteeing that the most recent and significant results are taken into account. CR-Physique also allows space for new results (on the recommendation of an academician), editing, and presentation of the work of the Academy’s award winners. Editors: Denis Gratias, Jacques Villain.

Comptes Rendus – Mécanique publishes original research papers, journal articles, thematic issues and articles reflecting the history of the discipline. The journal covers all the fields of mechanics: dynamic systems / solid mechanics / fluid mechanics / acoustics, waves, vibrations / automation, signal processing. The articles are proposed in the form of original notes relating briefly an important discovery. The publication of the results is fast. The thematic issues present the most up to date dossiers in the treated fields. Editor: Jean-Baptiste Leblond.

Comptes Rendus – Chimie aims to maintain high-level scientific exchanges between the different sub-disciplines of chemistry. The journal publishes original research works (notes, short memoirs) and review articles (reviews, historical chronicles) in all fields of chemistry. Preliminary papers should describe new and important results, while full papers should provide a detailed view of new results. In all cases, the work must be of high general interest or exceptional specialized interest. The journal also places great emphasis on thematic issues, bringing together the best specialists in the field around a guest editor. Editor-in-chief: Pierre Braunstein.

Comptes Rendus – Géoscience, which traditionally covers all fields of Earth sciences (geophysics, geomaterials, geochemistry, surface geosciences, oceanography, stratigraphy, tectonics, geodynamics…), is now broadening its editorial policy by encouraging the publication of articles dealing with the “sciences of the Planet” in the broadest sense. The journal is more open to scientific themes at the heart of current societal and environmental issues: natural hazards, energy and metal-material supply, water resources, pollution, climate change, both in the continental and oceanic/atmospheric domains. The submission of interdisciplinary papers is encouraged, to better understand the global effects of human activities on the functioning of the “Earth system”. Editors: Ghislain de Marsily and François Chabaux.

Comptes Rendus – Biologies sees in 2020 its objectives profoundly modified. True to the spirit of its title, the journal focuses its articles on the scientific activities of members or winners of the Academy’s awards, which are very rich . It only receives submissions of research articles by invitation only, but solicits mostly the biggest names in biology for articles divided into several sections: “C’est paru dans la presse/ News and views”, “Articles et revues”, “Notices biographiques”, “Opinions et perspectives”. This last section allows discussions and hypotheses on various subjects. Thematic issues on topical issues will be regularly scheduled, such as the one on COVID 19, which is currently being prepared. The articles are fully bilingual (English/French) and publication is fast. Editors: Jean-François Bach, Pascale Cossart, Bernard Dujon, Jean-Dominique Lebreton.

Comptes Rendus – Palévol is a continuous flow journal, dedicated to research in paleontology, prehistory and evolutionary science. It publishes original research results in systematics, human paleotonology, prehistory, evolutionary biology, and macroevolution. The journal also publishes thematic issues under the responsibility of guest editors. The co-publishing partnership agreement with the Academy allows CR-Palévol to benefit from the rigorous publication standards in force for the Museum’s journals, from the respect of the different codes of nomenclature and from direct compatibility with the major international databases. Editors: Philippe Taquet and Michel Laurin.

Created by Colbert in 1666, the Academy of Sciences is an assembly of scientists, chosen from among the most eminent French and foreign specialists. The reflections and debates that it conducts have the role of providing everyone with a framework of expertise, advice and alert, with regard to the political, ethical and societal challenges posed by science. By virtue of this mission, it works for the sharing of science as a common good in order to inform citizens’ choices, and formulates recommendations on which government authorities can base their decisions. It also supports research, is committed to the quality of science education and promotes scientific life on an international level.

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To decide between two candidates, long live the simple majority!

In his carte blanche, the mathematician Etienne Ghys goes back over the different ways, from the most to the least fair, of electing a representative among two competitors.

By Etienne Ghys (perpetual secretary of the Academy of Sciences, director of research (CNRS) at ENS Lyon)

Carte blanche. Can mathematics shed some light on the American election soap opera? Let’s imagine a population voting for two candidates and assume that voters flip a coin to choose one or the other. At the end of the ballot, the ballots are counted and the candidate with the most votes is elected. Now suppose that, during the counting, the scrutineers make a few mistakes (or fraud), for example, by being wrong once out of 10,000. What is the likelihood that these small errors will distort the overall result and the other candidate will be elected? It turns out that this probability is of the order of 6 out of 1,000 (for the curious, it is 2/π times the square root of 1/10,000). Is this an acceptable risk in a democracy?

American elections are two-tiered. Each state elects its representatives by a majority and these in turn elect the president. Assuming one more reading error out of 10,000 (which is reasonable when looking at the American ballots), what is the probability of distorting the final result? The existence of this second level makes the probability much worse: one election in 20 would be distorted! This is far too much.

Noise Sensitivity

Of course, all of this depends on very unrealistic assumptions and does not in any way substantiate Donald Trump’s allegations of fraud! Assuming that voters flip a coin is obviously meaningless, even if one can be amazed by the near-equal results in Georgia, for example. However, this illustrates a phenomenon highlighted by mathematicians some twenty years ago: the “noise sensitivity” of various decision-making processes, which go far beyond elections. This concerns computer science, combinatorics, statistical physics and social sciences. When a large number of “agents”, who can be human beings or neurons for example, have “opinions”, what are the right processes that allow a global decision to be made in a stable manner? This stability means that we want the decision to be as insensitive as possible to noise, i.e. to small errors that we cannot control.

One can imagine many electoral processes. For example, each neighborhood could elect its representative who would then elect the city representative, who would elect its representative in the canton, then the department, and so on. It would be a sort of sports tournament, in successive stages, a bit like the American elections but with many more levels. This method happens to be extremely sensitive to noise, and it must absolutely be avoided. The slightest proportion of errors in the count would result in a very high probability of being wrong about the final result. This is unacceptable for a vote, but it is part of the charm of sports tournaments: it is not always the best who wins, and that’s just as well.

What is then the best method, the one that is the most stable? The answer is a bit distressing and shows that the question is badly asked. It is enough to ask a dictator to decide alone. This “method” is indeed very stable because, to change the result, you need an error on the only ballot that counts, which happens once out of 10,000. The question must therefore be rephrased by looking for equitable methods that give the same power to all voters. About ten years ago, three mathematicians demonstrated a difficult theorem in this context, which is ultimately only common sense. To decide between two candidates, simple majority voting is the most stable of all fair methods. Long live the majority!

Some references :

Vaughan Jones, knotter and ultra-creative mathematician

The 1990 Fields Medal winner, the New Zealander passed away on September 6, 2020. Etienne Ghys pays tribute to him in his column in “Le Monde”.

Carte blanche. The mathematician Vaughan Jones died on September 6, 2020 in Tennessee, USA. He had received the Fields Medal in Kyoto in 1990. Sometimes a mathematician builds bridges between fields that were thought to be completely independent. These are moments of grace in the development of mathematics, reserved for the most creative, like Vaughan. However, it should not be thought that it is eureka! that suddenly appears. It almost always takes a long maturation, hardly compatible with the demand for immediacy of our current university system. The University of Geneva allowed Vaughan Jones to blossom and give the best of himself.
Vaughan arrived in Switzerland in 1974 from New Zealand to do a doctorate in physics. One day, with his thesis almost finished, he passes the door of the mathematics department and is fascinated by André Haefliger’s course: he abandons physics to do a thesis in mathematics (although, of course, his training as a physicist will remain fundamental). He works on “von Neumann algebras”, a field so abstract that the spaces studied have non-integer dimensions. Imagine for example a space whose dimension is 3.14 ! Haefliger – his thesis supervisor – is not a specialist in this subject, which is a sign of the great originality of the student and the open-mindedness of his master.
The Swiss Pierre de la Harpe, who knows the subject well, will become a friend and a “big mathematical brother” of Vaughan. At that time, the small department of Geneva was a breeding ground animated by a few exceptional senior mathematicians who fought against any form of exaggerated specialization. Algebra, geometry and analysis were spoken about a lot, very often in the small Italian bistro on the first floor. On the day of Vaughan’s defense in 1979, he was dressed in a tuxedo, which contrasted with the way the jury was dressed. In 1990, during the Fields Medal ceremony, in the presence of very formal Japanese authorities, he had insisted on wearing the All Blacks jersey, out of attachment to his New Zealand origins.

Sideration of the specialists

After his thesis, he settled in the United States but he often returned to Geneva. One day, after one of his lectures, someone pointed out to him, perhaps at the Italian bistro, an analogy between a relationship he wrote on the board and what is called the “group of braids”, which Vaughan did not know. That was all it took to glimpse a link between the subject of his thesis and a theme that was new to him: the theory of knots. All this led to a major discovery in 1984: the “Jones polynomial” associated with a knot. Knots, in mathematics, are those we imagine, like those of sailors. The mathematical theory of knots dates back to the 19th century and had a priori nothing to do with von Neumann’s algebras. Vaughan’s announcement of an important application of these algebras in the field of nodes will generate a kind of astonishment among topology specialists. He was awarded the Fields Medal but was also elected Life Vice-President of the International Knotmakers Guild, something he was very proud of.
The rest of his career was admirable. For about twenty years, the Ecole normale supérieure de Lyon has organized a mathematics weekend for about fifty students and an experienced mathematician. In 2012, Vaughan Jones literally charmed the young students. We have not only lost a brilliant mathematician, but also a model of generosity and openness for young people.

Does the Covid-19 pandemic herald the end of the mathematics laboratory concept?

In his column in “Le Monde”, the mathematician Etienne Ghys notes that the confinement has brutally accelerated, with the imposed teleconferences, a process of reduction of physical interactions between researchers.

Carte blanche. The months of confinement that we have just experienced will probably permanently change the working methods of scientific researchers, including those with no connection to biology. Mathematicians, for example, do not use experimental equipment, and their physical presence in the laboratory may not seem indispensable. They have been among those for whom teleworking has been the easiest to set up.
The site lists 739 mathematics presentations that can be participated in via the Internet, being able to interact live with the lecturers on all subjects, at any time of the day or night, taking advantage of the time difference. This opens up unprecedented possibilities for communication between researchers and abruptly accelerates a slowly evolving process. The consequences that this will have on the social life of the mathematical community are unknown.
Mathematicians usually work alone, but of course they need to exchange ideas with other colleagues. For a century, a major communication tool has been the laboratory seminar. These are meetings, usually weekly, during which a new result is presented to the members of a team. In France, the first seminar was created in 1920 by Jacques Hadamard, a professor at the Collège de France. At the beginning of the academic year, he would invite a few mathematicians to his home and distribute recently published research articles to be studied. He would then draw up an annual program.

The seminar, a Sunday mass

At the time, the Hadamard seminar was unique in France, but today, all the teams in the mathematics laboratories are organized around their seminars. Their role goes far beyond the transmission of knowledge: they are social events that unite the teams. They are sometimes compared to Sunday mass. Sometimes one attends out of obligation, or to see friends and colleagues. It must be said that it is not always easy to follow a mathematics conference and that one is often lost, sometimes from the very first sentences.
Over the last twenty years or so, the Internet has, of course, made these modes of communication evolve. First of all, all scientific journals are now available online. In the past, mathematicians used to go to their laboratories to be close to their library, which was their real working tool. This is still the case, but libraries have become virtual. E-mail, which is abused, has replaced the letters that were carefully written by thinking about each word. It’s not uncommon to see researchers, with a helmet on their head, collaborating via Skype with someone on the other side of the world, and forgetting to go and chat with their close colleagues in the common room of the laboratory.
This gradual evolution has great advantages, of course, but also obvious disadvantages. The weekly “face-to-face” seminars remained, however, and made it possible to preserve the human link within the teams. The pandemic suddenly accelerated this evolution: the seminars had to meet by videoconference, and participants no longer had to be members of the same laboratory. Lists of “global web-seminars” emerged, offering impressive amounts of live conferencing, each more appealing than the last. This evolution is probably irreversible. Does it herald the end of the mathematics laboratory concept? That would be a pity.
This summer I’m going to take part in a conference in Russia… without leaving home.

Percolation theory or the art of modeling a pandemic

The mathematician Etienne Ghys details the theory established by two British researchers in 1957 to understand the propagation of a fluid in a random environment. Like any modeling, it requires juggling with a lot of unknowns.

Carte blanche. Many articles have described the development of an epidemic over time, with an exponential growth in the number of new cases at first, then the famous peak, and finally the long-awaited decrease. There has been less discussion of contagion across a territory.
The mathematical theory of percolation is interested in this kind of problem. The word comes from the Latin percolatio meaning “filtration” and of course it evokes the coffee percolator: boiling water under pressure finds its way through the ground coffee particles, just as a virus finds its way into a population.
The theory originated in 1957 in an article by two British researchers, John Michael Hammersley and Simon Ralph Broadbent. Their initial motivation was for the much-talked-about breathing masks. In their case, these were protective masks for coal miners. The porous filter is likened to a regular network of very fine interconnected tubes, a number of which are randomly plugged, and the question is to understand whether a gas can pass through such a maze.

Determining the critical probability

More generally, these researchers study the propagation of a fluid in a random environment. One of their examples is a very simple model of an epidemic. It involves a huge orchard in which fruit trees are planted regularly in a square network. It is assumed that at some point in time one of the trees has a disease that it can potentially transmit to its neighbors. Each diseased tree can infect each of its four neighbors with a certain probability p (the lower the probability, the lower the trees respect the “social distancing”).
How will the epidemic spread? Hammersley and Broadbent show that if p does not exceed a certain critical value, the epidemic remains localized: these are clusters in which the contamination reaches only a small group of trees. When this critical value is exceeded, the disease suddenly invades a large part of the orchard (infinite if the orchard is infinite) and it is the pandemic.
Of course, this theorem is of interest only if this critical probability can be determined. Numerical simulations suggested that the cluster-pandemic transition occurs for p = 0.5, and it was not until 1980 that this was rigorously established. Unfortunately, this kind of precise result is only known in very simple cases, such as that of a regularly planted orchard. As soon as the trees are more or less in disorder, the phenomenon is less well understood.

Very partial information

In this case, the trees are flesh and blood individuals that fortunately are not planted regularly and are moving around. Moreover, the number of contacts of an individual, i.e. the number of people he meets in a day, and that he can potentially contaminate, is extremely variable from one individual to another. It depends on where he lives, his age, and many other parameters.
Only very partial information is available on the statistics of these contacts. A final problem arises: when a sick person meets a healthy person, the probability of contamination is also variable, and not well known.
In order to do this properly, a large number of parameters should be precisely known, many of which are inaccessible. The modeler must select a small number of them that seem most relevant to him, and of which he has a reasonable knowledge. He must then determine whether the other parameters – which he knows little about – could have a significant influence on the outcome of his predictions. This is not an easy task. Mathematical modeling is an art.

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.

Get your friends vaccinated instead, it’s mathematical

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!

The mutual attractions of the abbot Sigorgne

This ecclesiastic, who popularized Newton’s ideas, is an example of the spirit of the Enlightenment that deserves to be brought out of oblivion, according to the mathematician Etienne Ghys.

Carte blanche. You probably don’t know Abbot Sigorgne. However, he was the subject of a fascinating symposium on October 4 and 5 in Mâcon, bringing together specialists in the history of science and literature. Born in 1719 and died in 1809, in Mâcon, it is difficult to classify him: mathematician, physicist, writer, man of the church? In our society of immediacy, we must always remember the importance of historical research to better understand our contemporary world, which owes so much to the Enlightenment.
In the 18th century, the battle raged between the English, supporters of Newton’s theory of gravitation, and the French, supporters of Descartes’ theory. According to Descartes, space is filled with an unknown fluid, forming whirlpools of all sizes that drag the planets in their course. According to Newton, space is empty and the bodies are subjected to mysterious forces of mutual attraction that act instantaneously, even if the distances between them are considerable.

As we know, Newtonians will win the battle against the Cartesians (while waiting for Einstein’s arrival with his theory of general relativity). Voltaire will play an important role by writing his wonderful Elements of Newton’s Philosophy (1738) in an almost journalistic tone. Newton will penetrate scientific France thanks to the translations and commentaries of Emilie du Châtelet. But it was Abbot Sigorgne who allowed Newton to enter university teaching by writing his Newtonian Institutions in 1747. Of course, Sigorgne is not as well known as Condorcet, d’Alembert, Voltaire or Rousseau, but history is not reduced to celebrities, and it is important to look at a less well-known Mâconnais than Antoine Griezmann.

Reconciling Descartes and Newton

Our abbot is a man of the Enlightenment, open to dialogue. He will exchange about a hundred letters with Georges-Louis Le Sage, a Geneva physicist, who will try to convince him that it is possible to reconcile Descartes and Newton. According to Le Sage’s theory, space is filled with microscopic particles that partially penetrate bodies by bouncing off atoms. That made it possible to explain the mysterious force of gravitation whose origin Newton himself admitted not to understand. However, this beautiful theory of The Wise Man was not successful.
Sigorgne is also a teacher. Several letters from Turgot show that he had not forgotten his teacher and that he could seriously discuss the Newtonian attraction and the geometry of ellipses or hyperbolas. Happy times when rulers knew geometry! On the other hand, fifty years later, it seems that Lamartine did not really benefit from his mathematics lessons.
Of course, all this is mixed with intense theological debates: how to reconcile Reason and Faith? The abbot, for example, violently attacked Rousseau’s Lettres écrites de la montagne (1764) by publishing the Lettres écrites de la plaine (1764), or the defense of miracles against the philosopher of Neuf-Châtel (1766).
At the end of his life, Sigorgne judged that “high scientific works no longer suited his age”, and wrote a collection containing a large number of fables, in the manner of La Fontaine. The manuscript was recently found in the archives of Mâcon. A literary historian made a detailed analysis of it and came up with a very nice idea: in collaboration with a school teacher, she worked on some of these fables in a class of CM1-CM2 in a neighboring village. A video maker staged the whole thing and produced a nice film. What an emotion to see in 2019 children declaiming forgotten texts, as if echoing the Age of Enlightenment!

In Shanghai, an obsession for the square root

The mathematician Etienne Ghys comes back on the strange formula which presides over the establishment of the Shanghai ranking.

Carte blanche. The famous Shanghai ranking list of universities was published as every year in August. We learn that the top trio is made up, as always, of Harvard, Stanford and Cambridge, and that the universities of Paris-Sud and the Sorbonne occupy the 37th and 44th positions. This ranking is criticized from all sides, except of course by the universities that are well placed. It is perhaps useful to explain how it is constructed, to show how little sense it makes.
First, the ARWU (Academic Ranking of World Universities) assesses five “indicators” for each university. These are the number of Nobel Prize or Fields Medal winners who work there, the number of alumni who have received these same honors, the total number of published articles, those published in the two journals Nature and Science, and finally the number of “highly cited” researchers.
Each of these indicators is problematic. For example, the list of most cited researchers includes 90 mathematicians, 16 of whom sign their papers … in Saudi Arabia. On the other hand, there are no French mathematicians in this list. Without being chauvinistic, this makes no sense.
Of course, these five indicators favor the big institutions and leave little chance to the small ones, even if they are excellent. To try to remedy this, a sixth indicator is used, which is a sort of average of the previous ones, divided by the total number of researchers in the university.

As in the decathlon

The icing on the cake is the formula used to aggregate all this and make an overall ranking. The “score” assigned to a university is an average of the square roots of the six indicators, assigned certain coefficients. You read that right: it is an average of the square roots. To understand the idea, we can refer to the decathlon. How do you aggregate a sportsman’s results in ten disciplines as different as high jump and shot put? The solution is to first transform each of the ten performances in a certain way, specific to each discipline, before calculating averages. An improvement of 1 cm in the high jump will earn you many more points if you jump 2.45 m (world record) than if you jump “only” 1.50 m. For a university that already has a lot of Nobel Prize winners, however, it is easier to recruit one more than it is for a university that has none. In order to take this into account, the ARWU did not look very far and decided to transform all indicators in the same way and to use the square root.
There are at least two differences between academics and decathletes. Firstly, there has been much debate among athletes in the past about what a good formula should be. Nothing of the sort has occurred among academics, and the arbitrary choice of the square root is puzzling. On the other hand, a decathlete participates in a competition that he has freely chosen and for which he knows the rules. This is not the case of universities, which do not have the mission of following rules imposed unilaterally by a Chinese institute that promotes square roots.
The ARWU also establishes world rankings by discipline. I have of course consulted the one concerning mathematics. There you learn that Princeton is first, the Sorbonne is second, Paris-Sud is in fifth place, and the French department of mathematics that follows, in a very honorable 27th place in the world, is my laboratory at the Ecole normale supérieure de Lyon. In the end, these rankings are not so bad…