Chaotic and Stochastic Components in an Influenza Surveillance Series: Nonlinear Dynamics and Predictive Modeling...

Chaotic dynamics has been the subject of both theoretical and empirical research in epidemiology, with the most recent research strongly focusing on SARS-CoV-2. However, few empirical studies have been undertaken with respect to influenza, even though evidence of chaos has also been found in influenza surveillance data. Furthermore, empirical studies on chaos are focused on reconstructing hidden attractors in epidemiological time series to filter out noise; however, dynamical noise affecting chaotic dynamics can have relevant epidemiological features that are, in this way, left unresearched and that can be used for epidemiological surveillance and risk analysis by capturing the main underlying nonlinear processes associated with epidemiological dynamics.

This study aimed to reinforce empirical research on chaotic dynamics in influenza surveillance and the study of the dynamical noise affecting that chaotic dynamics, addressing the consequences for epidemiological risk analysis and surveillance.

Working with the weekly share of positive influenza tests for the Northern Hemisphere from January 2009 to March 2025 compiled by Our World in Data using FluNet data from the World Health Organization, we applied a recent method based on topological data analysis for reconstructing underlying attractors from time series and decomposing the dynamics down to independent and identically distributed noise. We adapted the method to the epidemiological context so that it can be used for predictive decomposition with direct application to epidemiological risk analysis and surveillance.

We found evidence of a low-dimensional chaotic attractor in the researched surveillance data, with a fractal dimension between 1 and 2, and a positive statistically significant largest Lyapunov exponent. The chaotic dynamics had power law scaling associated with long-wave influenza outbreaks, and it is affected by a stochastic component that is nonstationary in variance, leading to turbulent bursts in the outbreak dynamics. Testing different machine learning algorithms using the attractor as input for prediction and a 10-week rolling window, we found the following largest R2 scores for the prediction of the target series: 92.11% (1 week ahead), 85.95% (2 weeks ahead), 81.75% (3 weeks ahead), 77.59% (4 weeks ahead), and 73.35% (5 weeks ahead).

The main results reinforce previous theoretical and empirical studies on chaos in epidemiology. Our findings showed that there is a 2-dimensional chaotic attractor that can support up to a 1-month prediction of the target surveillance series with high prediction scores and that the attractor plus noise can be modeled in a way that supports uncertainty quantification and epidemiological risk analysis.