“All models are wrong, but some are useful.” This is the summary given by the statistician George Box in 1976. The sentence is of course intended to be provocative, but it holds a measure of truth: the role of the model is not to hold fully to reality, but to describe it, a lot of simplification. The form should allow a comparison, and the Latin origin of the word indicates the idea of \u200b\u200bmeasurement. If the paradigms were omnipresent, they did not generate much confusion: in economics, they promise a return to growth and stagnation; In the field of health, the pandemic crisis shows us how difficult it is to predict the evolution of a pandemic; While climate models are constantly being revised based on new data. Should we conclude from this that the model is good only from the point of view of its designer?
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Let’s take the example of an astrophysics model with astrophysicist Aurélie Guilbert-Lepoutre. His work at the Lyon Geology Laboratory consists of modeling a comet, as part of a future European mission called Comet Interceptor.
Its mission will be to wait in orbit around the Earth for a new comet to pass within its reach, to join it and study it. So astrophysicists hope to analyze a comet devoid of any interaction with the Sun, because those studied so far have been modified as they pass close to our star. The only problem: we will have to choose the best filter with the least amount of information. So Aurélie Guilbert-Lepoutre must design a guilty model to guide her choice. “Concretely, making a model is first and foremost writing physical equations, which involves a bit of mathematics, explains the astrophysicist. And then you have to define it,” that is, transforming abstract and connected data into numbers understood by a computer. “It then becomes a collection of thousands of lines that give commands to the computer,” she says. Here the mathematics takes over: “A layer of mathematics must be added to make separate approximations for continuous phenomena. For example, integrals that turn into a series.” pong” between the model and the simulation itself. Aurélie Guilbert-Lepoutre explains: “We enter the initial conditions, run the model, look at what it presents and compare it with the real data. And if it’s not good, we restart a simulation by changing this parameter or such and such, which requires experience… “In his case, fifteen years of practice let him know which parameters to change. Finally, I got a guilty model” spherical”, which would become a “numerical cube”. We are far from the image of a “perfect” model, which would accurately reproduce a comet. Because the important thing here is not to model its appearance, but to predict its temperature distribution, the presence of freezing dust … In short, Therefore, no high-resolution model is needed.” Nobody uses a hiking map with all the pedestrian paths when they take the highway, the designer compares. As for the model, it’s the same thing, you have to build it according to your needs.
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Last May, the Institute for Mathematics for Planet Earth (IMPT) was born. “The idea is to identify what mathematics can do for major societal issues.”, explains mathematician Laure Saint-Raymond, director of the institute. Climate, physics, geology, biology, social sciences… all disciplines can benefit from it. “It is important to note that not all of them are at the same level of modeling.”advances the world of mathematics. Thus physics will be the most developed, due to its history that allows it to obtain complex mathematical models.
“Biology is already less than that, she completes, Because it is the latest interface. The idea of quantifying things is still starting to grow, like the spread of tumors. “
The field of social sciences lags behind, whose equations are four or five centuries old.Laureate Laure Saint Raymond. “Be careful, this does not mean that it is easier, but on the contrary, you have to learn how to build new ones. It is like a Lego game: you start with one building, then two, and so on. A sequel until the formation of a new complex pattern.” His colleague Arnaud Guillin, who is also a mathematician and executive director of the institute, adds an important point: the distinction between the mentality of a mathematician and that of other scientists, especially when it comes to modeling. “ The mathematician will try to manipulate the model to understand it, he explains, To circumvent it, make it take into account quantities that do not necessarily physical or biological interest. “ Laure Saint-Raymond adds: “We must not forget that the model is not a reality, whereas in mathematics, it is the axiom, and from there we begin. When you’re a physicist, you’re writing a model that will only be relevant until the observation puts it in from the point of view of mathematics, that’s the starting point, and you get her to say what she can.”
This forms a loop: the physicist tries to describe what he observes, and the mathematics explains it. “We take it to its limits.”, summarizes Arnaud Guillin. So the role of the mathematician will not be to determine whether a model is good or not, but only to determine its intrinsic properties. This is why the two mathematicians, within their new institute, require every project “Powered by both mathematicians and non-mathematicians, in order to always reset the interface between disciplines”. Hopefully, one day every discipline will be equipped with solid mathematical models like those used in physics. “We have the idea that it is possible for all disciplines”says Laure Saint-Raymond.
Even in economics: “I was looking at papers from economists this week, and it’s clear that their models aren’t able to explain certain things like spatial heterogeneity, and the fact that people don’t behave the same way everywhere. So yeah, there’s room for improvement, and it’s not just because of climate change.” .
Here we touch on the key feature of the model: it should “answer a specific question that should be as general as possible,” as defined by the model’s author of Infectious Disease Epidemiology Elisabetta Virgo. She is an Inrae researcher and part of the scientific committee of a group called ModCov19, dedicated to coordinating efforts to model the spread of covid-19. As everyone knows, the difficulties are great to successfully predict the development of the epidemic. “One of the simplest models of epidemic dynamics is the SIR model, which consists in dividing the population into three categories, the epidemiologist explains. First, healthy individuals, S, in any condition of the disease studied; then the infected, I, in feeling infected and contaminated; and finally, those who have already been infected and are no longer infectious, R.” Next, we define “transitions” between each category, such as the proportionality relationship between the number of S and the fraction I in group N. This results in the present case in a nonlinear system of equations – because each parameter can vary According to time – in the form: dS / dt = – ß (S x I) / N (ß is the transmission rate constant, and dS / dt is the derivative of S according to time t). This formal form makes it possible to obtain “mechanical” modeling. [du processus de contamination]that is, we translate into equations of mechanisms capable of explaining the phenomenon.” Elisabetta Verge summarizes a position contrary to the so-called “phenomenal models, which describe relationships without trying to explain”.
Other models exist, for example those “so-called person-centred, or meta-populations, where many connected populations are considered”. The first is particularly effective in assessing follow-up contact cases. But all these models have inherent flaws in their construction. “We often take the hypothesis of a homogeneous population, admits Elisabeta Vergu, whether from the point of view of age, city scale, or individual household,” which constitutes a bias rooted in the mathematical difficulties still encountered in medicine. Globalism.
“There is still a challenge to be faced: confronting these models with data, which calls for the development of new methods for estimating standards. Fortunately, progress has been made since the beginning of the pandemic crisis. There is a joint effort, the specialist rejoices, and even if there are not always immediate effects, the science of Epidemiology and mathematics are communicating more and more closely.” Epidemiology, but also climatology, criminology, cancer science, economics … every discipline needs mathematics.
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