Top models

The contribution of mathematical epidemiology is to understand how an epidemic develops in order to help control it. A discipline that has been in demand since the early days of the Covid-19 epidemic. Mircea Sofonea, teacher-researcher at the Mivegec* laboratory, sheds light on this approach, which has perhaps never received so much media coverage.

"Basic reproduction number, R zero". A term that has been on everyone's lips since the start of the Covid-19 epidemic, and which you may never have heard before. This famous R0 is the key number in a discipline that has been taking center stage in recent months: mathematical epidemiology.

"The encounter between mathematical calculations and public health dates back to the 18th century," explains Mircea Sofonea, a teacher-researcher in the Infectious Diseases and Vectors: Ecology, Genetics, Evolution and Control laboratory. At the time, Europe was faced with smallpox. To combat this dreaded disease, a technique originating in Asia involved inoculating healthy people with virus taken from people who were only slightly ill. The aim: to protect patients from severe smallpox. It was a rather random method, since variolization - the forerunner of vaccination - was the cause of fatal smallpox in some patients.

Developing models

So how do we know whether this act should be encouraged to ensure collective protection despite its collateral victims? " It was difficult to envisage inoculating an entire village to compare mortality with another village that had not been inoculated, for example, as this would not have been ethically acceptable", explains the researcher in epidemiology and evolution of infectious diseases. The only alternative was to develop models. "In other words, simplify reality to answer a precise scientific question. In 1760, Swiss physician and mathematician Daniel Bernoulli developed one of these early models. Based on the study of differential equations, he estimated that collective variolization would increase life expectancy by 3 years. " This was the starting point of mathematical epidemiology ", says Mircea Sofonea.

"These models meet three objectives: to understand the past, to describe the present and to shed light on the future", explains the specialist, whose team was heavily involved with Covid-19. To better describe the epidemic, the researchers estimate the famous basic reproduction number R0, which reflects a disease's potential to spread. "Biologically speaking, it corresponds to the average number of people infected by a contagious individual. This figure characterizes the trajectory of the epidemic: when it's greater than 1, the epidemic is spreading, and when it's less than 1, it's under control", explains Mircea Sofonea. The researcher and his team estimated the R0 in France at the start of the epidemic at between 2.5 and 3.5. According to their model, the R0 would have fallen to 0.7 during the confinement period, which greatly reduced individual contact.

Behavioral leverage

"In the absence of pharmaceutical solutions, behavioral leverage, via physical distancing in particular, is the only weapon against infectious diseases", explains the epidemiologist. This is one of the reasons why the R0 estimated at 3 in Europe at the start of the epidemic was limited to 2 in Asia. "This discrepancy could be due to lifestyles and cultural differences: greeting habits, proximity, frequency of hand-washing, wearing of spontaneous masks.

Pending treatment, the focus is on prevention: "the delicate objective is to contain transmission while being as unrestrictive as possible". But to what extent? To find out, members of the Theoretical and Experimental Evolution team set out to determine the best control to apply to the epidemic during the first 100 weeks, "the estimated time needed to discover and implement a treatment or vaccine". They thus determined a theory of optimal control. Their strategy? Rapidly implement strong control, then progressively relax it. "Our models suggest that this strategy would be effective in containing the epidemic over the period in question. It would give better results than no control at all, but also than a strategy of constant strict control."

Consultation

Can these models dictate the measures to be implemented? You have to be careful when using them, and not rely on a single simulation," replies Mircea Sofonea. These models have no predictive value beyond the short term, but they do play a part in the decision-making process. And the choice of strategy to be adopted to contain an epidemic must be made in consultation with other disciplines: epidemiology, medicine and the human and social sciences must work together to propose the best measures to be implemented".

*Mivegec (UM-CNRS-IRD)