EMPA researchers improve epidemic forecasts with the reproduction matrix

EMPA researchers improve epidemic forecasts with the reproduction matrix

The reproductive number R is often used as an indicator to predict how quickly an infectious disease will spread. EMPA researchers have developed a mathematical model that is just as easy to use but provides more accurate predictions than R. Their model is based on a reproduction matrix that takes into account the heterogeneity of society.

“Your friends have more friends than you,” wrote US sociologist Scott Feld in 1991. Feld's so-called friendship paradox says that a given person's friends have, on average, more friends than the person himself. This is based on a simple probability calculation: well-connected people are more likely to appear in other people's social circles.

If you look at a person's circle of friends, it is very likely that that circle contains very well-connected people with an above-average number of friends. “

Ivan Lunati, EMPA researcher, head of the Computational Engineering Laboratory

A similar principle served Lunati and his team as the basis for a new mathematical model that can be used to more accurately predict the development of case numbers during an epidemic.

But what do social circles and infectious diseases have in common? “The more contacts a person has, the more people they can infect in an epidemic,” Lunati explains. However, conventional epidemiological models assume that, on average, each infectious person infects the same number of other people over the course of the epidemic. This number is called the reproduction number (R). When R is greater than one, the number of cases increases exponentially; When R is less than one, it decreases.

Of course, this model is simplified: “The number of cases cannot increase indefinitely because the population has a finite size,” says Lunati. According to researchers, the rapid exponential growth mainly occurs at the beginning of a wave. However, as time goes on, there are fewer and fewer people who can still be infected, as is likely to be the case after the Covid pandemic.

No infinite number of “superspreaders”

This infection curve can be calculated using mathematical methods to predict its peak. By assuming that each infectious person infects the same number of other people, the model deviates from the empirically measured waves of infection. Although it can reproduce the beginning of the wave well, the number of new infections later is faster than predicted, so the peak ultimately turns out to be slightly lower than calculated - even if no new protective measures affect the course of the infection.

Together with EMPA researchers Hossein Gorji and Noé Stauffer, who is also a doctoral student at EPFL, Lunati asked the question: How do we make such predictions more accurate? Your answer has parallels to the Friendship Paradox. “People with a lot of social contacts become infected particularly quickly and in turn infect many others,” explains Lunati. The researchers also refer to people like hubs or superspreaders. At the beginning of a wave of infections, they are the ones who drive the increase in the number of cases. However, the number of such transgressors in society is relatively small. Once they're all infected - which happens fairly quickly given their high levels of contact - the spread of the disease slows. Traditional models based on the R reproduction number do not take this slowdown into account.

In a recent study published in the Journal of the Royal Society Interface, Gorji, Stauffer and Lunati propose using a reproduction matrix instead of the reproduction number. This matrix shows how quickly people belonging to different population groups become infected by other groups, thus taking into account the heterogeneity of contacts.

“We wanted to go beyond the simplified interpretation of the reproductive number r and better capture the complexity of real epidemic waves,” says Hossein Gorji. “The propagation matrix allows us to more accurately predict disease spread by accounting for both nonlinearity and heterogeneity, which are often overlooked in traditional models.”

The research project was supported by the Swiss National Science Foundation (SNSF).

Beyond epidemics

When defining this reproductive matrix, the researchers relied on data from other studies. For their model, they divided society into groups based on age. On average, people between the ages of 10 and 25 have the most contacts. “Grouping by age is of course a generalization, as interpersonal contact is much more complex,” explains Lunati. "Furthermore, our model assumes that both superspreaders and the number of cases are evenly distributed throughout the country. This assumption is not very problematic for small countries with highly interconnected regions and relatively uniform social structures. However, it is for large countries. We would also need to take into account the geographical distribution of the population and contacts between regions."

The researchers tested their new model with Covid data from Switzerland and Scotland - both relatively small countries. They were able to show that the matrix enables much more accurate predictions of infection peaks. “Of course, our model is also very simplified,” says Lunati. However, the strength of the matrix model lies precisely in its simplicity: “It is very easy to use, but at the same time much more realistic than the R value alone.”

The usefulness of the new model is not limited to epidemics: it can be used in various systems - everywhere, everywhere, everywhere, the objects that spread across a network. In the future, researchers would like to use it to simulate the spread of views, opinions and behavior in a society - for example when it comes to the introduction of new technologies or sustainable living.

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