About complex infection dynamics - and how we are the ones creating them

Seasonal influenza follows a clear, predictable pattern: there is one wave per year with infections rising in late autumn to early winter and peaking in late winter. The total number of influenza infections might change from year to year, but overall, the pattern is very stable across time and countries.

Now remember looking at the infection numbers during the COVID-19 pandemic, failing to find a clear, predictable pattern? And although predictability is quite big of a word, we can agree that in contrast to Influenza (of which we expect excactly one peak every year around late winter), COVID-19 showed no comparable stability or structure – so why is that?


In a recent research article, we took a deeper look at what drives such infection dynamics, including how societal reactions can influence the spreading of the disease. With the help of a model, we showed that society’s behaviour can actually induce complex dynamics. Not only that: Such complexity can even be optimal when considering expected costs for health and economy.

So let's dive into these findings.

If there's a virus, what drives infection waves?

Another driver of infections is society's preventative disease mitigation. By that we mean the different measures to fight infections, such as vaccinations, hygiene practices, and the reduction of social interactions.

The intensity of these mitigation measure depends on the number of infections. As more people get infected, society increases their mitigation efforts, until that eventually leads to the wave getting broken and infection numbers going down again.

When infection numbers are low, mitigation is accordingly lowered – which then leads to a new infection wave – and the cycle repeats. This leads to a "feedback loop": By adapting to the infection numbers, we correspondingly change them – forcing us to re-adapt.

Two driving forces interacting can create complexity

If seasonality and mitigation have an impact of comparable strength, a combination of both factors can result in complex infection dynamics. These can display varying numbers and heights of infection waves each year.

When does mitigation lead to complex infection dynamics?

To break it down, society’s influence on infections largely depends on how intensely and how fast people react to rising infection numbers. We measure this reaction using the maximal amount of mitigation that society is willing to adopt (maximal mitigation), and the time it lets pass before it reacts (mitigation delay).

Using the model, we can explore how specific mitigation behaviours (consisting of a pair of given maximal mitigation and mitigation delay) would shape infection dynamics. It allows us to experiment with a variety of mitigation measures. We portray them within a coordinate system: the higher up, the stronger the measures; the more to the right, the longer their delay. For each spot in this coordinate system, we can observe the infection dynamics in the model and assign the spot a color according to the exhibited number of infection waves per year. In the following, we illustrate this by showing the infection dynamics for some example spots in the coordinate system. (Here, mitigation delay is given in days. And a maximal mitigation of 1.0 would mean every infection is getting prevented.)

In a spot with mild mitigation measures and a large delay, the influence of seasonality dominates – as the yellow colour indicates, we witness one wave per year. Though mitigation does change the peaks’ heights, it is not strong enough to induce waves on its own.

With stronger and faster mitigation, multiple infection waves per year can appear. A recurring pattern can, nevertheless, still be made out.

However, in spots where shapes and colours appear to overlap, the dynamics becomes more complex and even this recurring pattern breaks down.

Now, we have located areas of complex-looking dynamics. But there remains a central question: Are infection numbers in these areas actually harder to predict? Mainly two effects limit forecasting efforts: A critical dependency on the exact societal reactions and on the initial number of infections.

Strong dependencies impede forecasts

At their core, forecasts of infection dynamics take in the current number of infections, and feed them in a model of the time evolution to get an estimate of the number of infections in the future. However, real life is not as set in stone as a simplified model – society’s reactions to infections will always slightly differ, both in intensity and delay. These variations can be imagined as a spot slightly “moving” within our figure. The effects of this can vary: A spot in the broad yellow area will remain in that area, even if movement is involved. However, where different colours are located close to each other, small variations can easily move the spot into a different "category". That further “complicates” the infection dynamics, as these jumps between colours can constantly change the dynamics. Consequently, in these areas, minor changes in mitigation behaviour can lead to big changes in the dynamics.

For instance, in the region of the last example spot, a slight movement is enough to alter the course of infection dynamics drastically. In this case, mitigating just slightly less can cut the infection wave's peak in half. (Note the change in scale on the axis!)

As we can see, in particular areas of the above figure, the infection dynamics strongly depends on the society. Without perfectly knowing how people react to high infection numbers right now and in the future, predicting infections is difficult.

In similar areas, an additional effect comes into play: a critical dependency on the initial number of infections, also called the inital conditions. In reality, one can never know these at perfect precision - not everyone infected is tested and reported. In the broad yellow area of the figure, a slight uncertainty in the initial conditions barely affects forecasts. However, in the more complex areas, a slight change in initial conditions can be enough to cause drastically different dynamics – even if the mitigation behaviour remains the same!

Changing the initial number of infections in our previous example, we observe vastly different infection dynamics - even leaving the mitigation behaviour unchanged. The largest infection wave is about double the size as before!

As we can see, infection dynamics can be highly complex, and hard to predict. Slight changes in how a society reacts to a disease and in the current number of infections both can vastly alter the number and height of infection waves observed in the future.

See the boxes below for more details on the different kinds of complexity.

So some areas are complex - but which ones are we likely to enter?

So, infection dynamics can be complex. But is this relevant outside of our abstract model? If societal reactions determine infection dynamics, we should ask ourselves how people are likely to act. This is a matter of balancing costs. Spreading infections come along with infection costs (people suffering, hospitals being overloaded, etc.) and mitigation costs (economic costs of limiting workforce, psychological costs of limiting contacts, etc.). Rationally, society will strive to act and mitigate in the most cost-optimal way. For this cost-optimal mitigation behaviour, total costs are minimal.

The relation of infection costs and mitigation costs resemble an inverse proportion: With rising mitigation measures, infection costs fall due to fewer people getting infected, whilst mitigation costs start to increase steeply as very high mitigation measures become harder to implement. It follows that the cost-optimal region will be found somewhere in the middle – where mitigation is strong enough to flatten infections but not yet immensely costly.

Complexity is cost-optimal

Knowing cost-optimal mitigation behaviour allows us to find the corresponding cost-optimal dynamics. For that, the optimal spot identified above can be transferred into the coordinate system of mitigation behaviour. Remember, it shows the number of infection waves for different mitigation behaviours.

Similar cost calculations can be done for the rest of the figure, revealing a new finding: The cost-optimal spots appear to almost perfectly align with “border areas” of the figure, where colors and shapes overlap, and thus with the complex region identified before. To quote the paper: “[The] cost-optimal region may coincide with the region of complex […] infection dynamics.”

This relation shows that not only is it possible for a society to end up with complex infection dynamics, but it is rational too. This can lead to us actually encountering these dynamics in real life. Coming back to our initial motivation: Why did COVID-19 display more complex dynamics than Influenza? COVID-19 infections brought an enormous burden, making it necessary to balance mitigation with infection costs. This, as demonstrated above, quickly leads to highly complex dynamics. Influenza, on the other hand, society does not mitigate to an extent anywhere close. Accordingly, we witness yearly recurring peaks of Influenza in winter.

This connection between societal behaviour, complexity, and cost-optimality is what you should take away from this article: That if an infectious disease is severe enough to be societally mitigated, complex infection dynamics become possible – and if mitigating cost-optimally, this possibility becomes reality.


See the paper for the full research:
Societal self-regulation induces complex infection dynamics and chaos, Phys. Rev. Research (2025)

Involved Scientists

Joel Wagner
Simon Bauer
Seba Contreras
Luk Fleddermann
Ulrich Parlitz
Viola Priesemann