After the “R0”, here is the “k”

Après le «R0», voici le «k»

Après le «R0», voici le «k»

The higher the “k” is close to 1, the more the propagation is homogeneous and steady : most of the sick are going to infect roughly the same number of people. To the contrary, when the “k” near 0, with a minority of smaller and smaller will be responsible for the majority of the spread.


May 29, 2020

Updated on may 30, 2020 at 16h18


After the “R0”, here is the “k”

Après le «R0», voici le «k»

Après le «R0», voici le «k»

Jean-François Cliche

The Sun

Simplifying a bit, one could say that the viruses have two main strategies for spread. A bit like the decisions that one takes in finance, some viruses choose to spread regular and predictable, but often by “small steps”, while others have a more risky and often nothing, but who “pays” more when it works. The more one discovers about the COVID-19, the more she seems to belong to the second category — and that might explain some of its “mysteries”.

At the beginning of the pandemic, the media have talked a lot about the famous “basic reproductive number” of the virus, or “R0”, which is the average number of people that each sick person will infect. In general, the studies arrive at an R0 between 2 and 3 for the COVID-19, which means that each person who contracts the disease will pass to 2 or 3 other people on average. By comparison, the R0 of seasonal influenza is usually about 1.3 to 1.5.

But, like all averages, this R0 eraser a part of the reality. “Certain segments of the population have more contacts than others [and] as the contacts play a big role in the R0, you can observe a lot of variation in the R0. […] At the limit, the same, each individual has its own R0,” explains Benoît Mâsse, a researcher in epidemiology at the University of Montreal.

It is for this reason that the epidemiologists also calculate a “factor of dispersion”, or “k”, which roughly indicates the extent to which R0 varies from one group or one individual to another. There are several ways to calculate it, but in general it varies between 0 and 1. The closer he is to 1, the more the propagation is homogeneous and steady : most of the sick are going to infect roughly the same number of people. Conversely, the larger the k approach 0, and the more the disease progresses by “suddenly”, by leaps and unpredictable : a large part of the sick do not infect person or almost, but a small minority will be responsible for the majority of the spread (this is the famous “super-spreaders”).

Gold although some studies have placed the COVID-19 among the diseases which are progressing on a regular and predictable in the whole “there seems to be a trend for k relatively small, around 0.1,” says Mâsse. Concretely, a k as low means that 80 % of infections are caused by approximately 10 % of the patients : “If you have 100 people who are infecting 300 others, there are 10 who will be responsible for 240 infections while the other 90 will be responsible for 60 infections” shows there.

In short, if this coronavirus was an animal, we would not have to deal with an ant that moves in a being always not the same length, but rather is a frog ahead in doing good more or less erratic.

This may help to explain some features a bit amazing of the pandemic. If the COVID-19 propagates well by to-shot, then the virus typically has a need for multiple inputs in a same country before the epidemic did not take root, since most of the sick do not infect the person. This could in principle explain, by example, why the first case to have been detected in Italy, appeared in Rome at the end of January, while it is the north of the country which has come to be the most affected, and only from the end of February. Or why some individual cases appear to have “spun-off” to a degree yet unheard, like the famous “patient 31” that has contaminated dozens of people alone in a church in South Korea (the people who then went on to infect others), or even as this German businessman who seems to have been the source of the epidemic in the north of Italy.

The good news, however, indicates Marc Dionne, physician at the national Institute of public health of Quebec, it is only from the moment when one knows that one has to deal with a virus that “depends on” the events of super-spreading to spreading, then this gives us tracks that are more targeted.

We talked a lot about the case of the meat factories, where a high proportion of employees contract the disease, but this is not the kind of situation that most worries Dr. Dionne. In cases like these, the risk of transmission is high among workers and their families because they are all people who have come together on a daily basis, but the virus will not come out necessarily of this social circle.

“If we think of celebrations such as weddings for example, there are elements in these situations which will lead to a high probability of transmission in the group at the same time a high probability of scattering out of the group,” he says. This kind of gatherings held indoors, where the risk of contagion is higher than outside, which brings together people who do not see often, is especially likely to cause episodes of “super-spread”, says Dr. Dionne.

As it emits more of the droplets, and (if applicable) of virus when shouting or singing), one can also think of places such as bars, where many people have to speak loudly to be understood, or singing in a choir or some places of worship — a few cases of super-spreading have been documented in a number of choirs.

“The epidemic is going to continue, but, if one manages to avoid these situations, we should draw not too bad in the meantime that herd immunity develops slowly, or that a vaccine arrives,” says Dr. Dionne.

Le Soleil

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