Chaos in opinion-driven disease dynamics

Abstract

During the COVID-19 pandemic, it became evident that the effectiveness of applying intervention measures is significantly influenced by societal acceptance, which, in turn, is affected by the processes of opinion formation. This article explores one among the many possibilities of a coupled opinion-epidemic system. The findings reveal either intricate periodic patterns or chaotic dynamics, leading to substantial fluctuations in opinion distribution and, consequently, significant variations in the total number of infections over time. Interestingly, the model is exhibiting the protective pattern.

Figures (9)

• Figure 1: Example of a chaotic timelines of the evolution of the dynamics with parameters $n=10,\epsilon=0.25,\tau=0.55$ where MLE and autocorrelation indicate chaotic behavior. The timelines begin with 20,000+ time steps and finishes with 20,000+2,000. We show heatmaps of opinion $u(x, t)$, infected $z(x, t)$, sum of infected $Z(t)$ and entropies of $z(x, t)$ and $u(x, t)$ repectively. Entropies of $z$ and $u$ are computed with with base 10. In each plot there are fluctuations and irregularity of data, especially in $u$ and entropy of $u$.
• Figure 1: Cross sections along Epsilon and $\tau$ axes of Autocorrelation and MLE heatmaps in figure no. \ref{['fig:n10']}. Panel of 4 scatter plots includes: Maximum Autocorrelation along $\tau$ axis (top left), MLE along $\tau$ axis (bottom left), Maximum Autocorrelation along $\epsilon$ axis (top right), Maximum Autocorrelation along $\epsilon$ axis (bottom right). We can see large fluctuations in each of 4 plots. The low Autocorrelation values correlate with high MLEs values when compared between top and bottom row. It is expected behavior that increases confidence into chaoticity in these regions. \newlabelfig:crossSections0
• Figure 2: Example of a periodic timelines of the evolution of the dynamics with parameters $n=10,\epsilon=0.25,\tau=0.85$ where MLE and autocorrelation indicate periodic behavior. The timelines begin with 20,000+ time steps and finishes with 20,000+200. We show heatmaps of opinion $u(x, t)$, infected $z(x, t)$, sum of infected $Z(t)$ and entropies of $z(x, t)$ and $u(x, t)$ repectively. Entropies of $z$ and $u$ are computed with with base 10. In each plot there are fluctuations of data, especially in $u$ and entropy of $u$. One can clearly distinguish periodic behavior in each plot. \newlabelfig:periodicTimeline0
• Figure 3: Example of a chaotic timeline of sum of infected $Z(t)$ from figure no. \ref{['fig:chaoticTimeline']}. Simulation with parameters $n=10,\epsilon=0.25,\tau=0.55$ with extended time up to 20,000+5,000 steps is displayed. \newlabelfig:chaoticTimelineLong0
• Figure 4: Panel of heatmaps with measures of disorder of dynamics of system with $n=10$. The results are from the grid search simulations for $\epsilon$ and $\tau$ parameters that are on x and y axis respectively. Maximum Autocorrelation (top left), MLE (bottom left), Maximum Spectral Entropy (top right), Maximum Wavelet Entropy (bottom right) are displayed. In each plot there is large fluctuation of values. Missing tiles in Autocorrelation heatmap are NaN values that mark stationary solutions. The parameter spaces with chaotic simulations are easy to distinguish in Autocorrelation (low values), MLE (high values) and Spectral Entropy (high values). The low Autocorrelation values, high MLEs values and high Spectral Entropy overlap. It is expected behavior that increases confidence into chaoticity in these regions. Maximum Wavelet Entropy is an outlier that is less consistent with other measures. \newlabelfig:n100

Theorems & Definitions (4)

• Proposition 1
• Proof 1
• Proposition 2: Stability of the Equilibrium in the pure $2$-voter model
• Proof 2