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.

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