# See beyond the top of the curve

| Photo: Charly Tribal Agence France-Presse

The Oracles of modern times, epidemiologists attempt to guess what the future holds for us. However, their projections are not used to precisely quantify the number of cases, hospitalizations and deaths to come.

Welcome to the epidemic with a policy of letting go would have been catastrophic. According to an epidemiological model developed at the University of Toronto, over 50,000 patients requiring intensive care would have flooded the hospitals of Ontario at the height of the crisis. The same model shows, however, that a campaign of large-scale screening and the physical separation of the individuals of the effects of life-saving. A combination of the two approaches, applied on and off depending on the occupancy rate of the hospitals, gives the best results. The number of cases remains low and hovers slightly over the month, as a wave.

“It is necessary to learn how to surf on this wave,” says Ashleigh Tuite, the epidemiologist who designed the model producing these projections.

All models are wrong but some are useful

— George Box

The Oracles of modern times, epidemiologists attempt to guess what the future holds for us. However, their projections are not used to precisely quantify the number of cases, hospitalizations and deaths to come.

“All models are wrong but some are useful “, said the statistician George Box there are more than four decades. The models give rather a general idea of the situation to decision-makers, who can test potential actions and foresee their consequences.

In Quebec, the research group of professor Marc Brisson, Laval University, is working these days to calibrate its epidemiological model to describe the progression of the COVID-19 in the province. Already, he has presented preliminary results to the government. In the next few weeks, his model will allow to evaluate different strategies of déconfinement.

“We must initiate the exit of the containment in may, or June ? Is it better to proceed gradually with the re-opening of schools, some industries ? With our model, we can explore these scenarios and determine which strategy has the least risk of producing a second wave, ” said Mr. Brisson. It develops a model decked out in the qualifier very baroque of ” dynamic stochastic compartmental “, which takes so much more sophisticated, the principles established there nearly a century ago.

A simple and effective approach

At the turn of the 1930s, the Scottish Anderson Gray McKendrick and William Ogilvy Kermack laid the foundations of the mathematical epidemiology. Their idea was to separate the population into three compartments : those “susceptible” (S) that can contract the virus ; people “infected” (I) which are contagious ; and those “withdrawn” (R), because dead or immunized. These three compartments interact constantly, and their value determines the flow from one box to the other.

To trace the evolution of the epidemic in the time thanks to ” SIR “,it is necessary to solve a system of differential equations. For an expert, it is a game of children : some concepts of mathematics taught at the college level are sufficient to frame the problem.

But before asking his computer to warm up the circuits to calculate the response, the modeller must choose a setting that dictates in large part the evolution of this simplified system : the famous number basic reproductive number, or R0. It is defined as the average number of individuals that a carrier will infect in a population of non-immune. It depends on the biological characteristics of the disease, but also public policies and cultural habits of the population.

In the framework of such a model, fix the value of R determines the rate of initial growth of the epidemic and the fraction-finals of the population that will be infected if not vaccinated.

A whole zoo of epidemiological models arises from variations of the model SIR. Note for example the model ” SIS ” (susceptible, infected, susceptible), which suits better to diseases like the seasonal flu, for which the population does not develop immunity in the long term.

Other architectures include compartments for exposed persons, deceased, in quarantine, with an immunity to the birth or asymptomatic carriers. The solution of these mathematical problems usually takes the form of a feature called ” logistics “.

It consists of three phases. First, a phase of exponential growth very fast. Then, a linear growth, where the number of cases (or deaths) is roughly constant from day to day. And finally, a phase of growth that will slowly stabilizing. “In terms of emotional, this is the stage of panic, the phase of hope and the phase of relief,” notes Pier-André Bouchard St-Amant, a professor at the national School of public administration who concocted his model house of the epidemic of COVID-19 in Quebec. “The models give a referrer is a theoretical structure that helps to think about the problem “, he adds.

“Zoom in” up to the individual

When the time comes to test hypotheses more subtle, the other compartmental models suffer, however, is a major problem : they assume a homogeneous population, and a virus well “mixed” in the space.

However, the spread of an epidemic depends on the initial locus of the outbreak, and every generation also maintains different contacts with the rest of the company. Thus, some researchers stratify their model according to the age groups. The model Ashleigh Tuite, for example, has 17 compartments, each divided into four age groups.

Other specialists, such as the team of Neil Ferguson, Imperial College, abandon altogether the notion of compartments and model each individual on a spatial grid at high resolution.

Antoine Allard, professor at the physics Department of the Université Laval and a specialist in mathematical epidemiology, explores another approach, based on the science of ” complex networks “. It is then to represent each of the individuals of a population as a point (or node) is connected to other individuals by lines (or edges).

This method allows you to make projections as a function of social structures much more realistic — representing the school and the workplace, for example. “The idea, he said, is to study how this structure then influences the spread of the disease and how it determines the final size of the epidemic. “

In February, Mr. Allard and american colleagues have signed a scientific article explaining how, regardless of R, the events “supercontagion” (where one person infects tens) affect the fate of an emerging infectious disease like the COVID-19.

One thing is for certain, the models allow scientists and decision-makers to refine their intuition about the behaviour of this new disease whose tentacles reach to the four corners of the planet. Even if the models are not perfect, they “shed light on the validity of our assumptions and the state of our knowledge,” remarks Ms. Tuite. They also show that, up to now, the measures adopted in Canada to have operated, ” she says.

“We are currently in a critical time because of our public policy work and we want to say that we have solved the problem, said for his part, Antoine Allard. But the potential for a combinatorial explosion in the number of cases is still very real. What we include in our models, it is that we should really not let go. “