Modelling means building a bridge between aspects of the real world and fictions: matching a fragment of reality with models—mathematical ones here—that can explain or predict it. Thus, empirical laws concerning large numbers are explained by theorems such as the law of large numbers and the central limit theorem (see Tangente 149, 2012). These results lead to PAC ("probably approximately correct") predictions, with both "probably" and "approximately" quantified: in opinion polling, every prediction comes with a range, which quantifies the approximation, and a confidence level for that range, which quantifies the "probably."
Models and toy models -------------------------
Modelling also means venturing into possible worlds, where simulation plays the role of experimentation. In the midst of the coronavirus pandemic, biological experiments are being conducted in vitro, along with therapeutic trials—experiments on human subjects within the framework of evidence-based medicine—and a great deal of modelling and simulation. In such an experimental context, these are known as toy models (or toy models). We might have toy models of how epidemics evolve over time in each country, and use these fictional worlds to examine the effects of various forms of lockdown or the lifting of lockdown restrictions. Because reliable, precise information is lacking for certain crucial parameters, particularly the numbers infected and immune, simulations will be run not on a single model but with various classes of models and several values for the parameters involved.
In the case of the Covid-19 pandemic, statistical methods for dynamically estimating parameters ultimately enabled these models—mere products of the imagination—to produce forecasts that, at the beginning of lockdown, proved relevant over the very short term—two or three days—at the scale of a country or region. Longer-term forecasting, however, was entirely illusory. After a few weeks of lockdown, we are working with other models, for example at the Institut Pasteur de Paris and Inserm. Provided nothing changes in how the country is organized, these can reasonably be used to forecast a few weeks ahead. In a sense, we are trying to "clothe" aspects of the pandemic in models that take account of the available information—but an outfit that fits one day will no longer fit the following month.
Since we need to move forward, decisions will have to be made without any guarantee that they are the right ones, and we will observe the results: are events unfolding as the model predicts, must new elements be incorporated into it, or must it be replaced altogether? Models, and toy models in particular, are not meant to tell "the truth." They allow us to compare possible courses of action, making them tools for thinking and decision-making.