Beyond Bifurcations: Predicting System Shifts From Dynamic Fluctuations
![Gillespie simulations-conducted across population sizes of 1,000, 10,000, and 100,000 with initial conditions of [latex]S=0.4[/latex], [latex]I=0.05[/latex], and [latex]R=0.595[/latex]-demonstrate that fluctuations in infectious individuals, when compared to mean-field solutions, exhibit empirically and theoretically derived variances, and that analyzing the quasi-stationary distribution for smaller populations ([latex]N=1,000[/latex]) provides insight into pathogen dynamics as defined in Table 1.](https://arxiv.org/html/2601.14869v1/Figures/high_quasi_sims.png)
New research reveals how inherent properties of changing systems can signal impending transitions, even without a clear tipping point, offering insights for forecasting in fields like epidemiology.



![Forecast accuracy is demonstrably linked to the distribution of out-of-sample volatility, suggesting that predictive models perform best when calibrated to the inherent uncertainty present in dynamic systems-a relationship quantified by [latex] \sigma^2 [/latex].](https://arxiv.org/html/2601.13014v1/x4.png)




