Large-deviations of disease spreading dynamics with vaccination.

We numerically simulated the spread of disease for a Susceptible-Infected-Recovered (SIR) model on contact networks drawn from a small-world ensemble. We investigated the impact of two types of vaccination strategies, namely random vaccination and high-degree heuristics, on the probability density function (pdf) of the cumulative number C of infected people over a large range of its support. To obtain the pdf even in the range of probabilities as small as 10-80, we applied a large-deviation approach, in particular the 1/t Wang-Landau algorithm. To study the size-dependence of the pdfs within the framework of large-deviation theory, we analyzed the empirical rate function. To find out how typical as well as extreme mild or extreme severe infection courses arise, we investigated the structures of the time series conditioned to the observed values of C.

Large deviations of a susceptible-infected-recovered model around the epidemic threshold.

We numerically study the dynamics of the SIR disease model on small-world networks by using a large-deviation approach. This allows us to obtain the probability density function of the total fraction of infected nodes and of the maximum fraction of simultaneously infected nodes down to very small probability densities like 10^{-2500}. We analyze the structure of the disease dynamics and observed three regimes in all probability density functions, which correspond to quick mild, quick extremely severe, and sustained severe dynamical evolutions, respectively. Furthermore, the mathematical rate functions of the densities are investigated. The results indicate that the so-called large-deviation property holds for the SIR model. Finally, we measured correlations with other quantities like the duration of an outbreak or the peak position of the fraction of infections, also in the rare regions which are not accessible by standard simulation techniques.

Large-deviations of the basin stability of power grids.

Energy grids play an important role in modern society. In recent years, there was a shift from using few central power sources to using many small power sources, due to efforts to increase the percentage of renewable energies. Therefore, the properties of extremely stable and unstable networks are of interest. In this paper, distributions of the basin stability, a nonlinear measure to quantify the ability of a power grid to recover from perturbations, and its correlations with other measurable quantities, namely, diameter, flow backup capacity, power-sign ratio, universal order parameter, biconnected component, clustering coefficient, two core, and leafs, are studied. The energy grids are modeled by an Erdos-Rényi random graph ensemble and a small-world graph ensemble, where the latter is defined in such a way that it does not exhibit dead ends. Using large-deviation techniques, we reach very improbable power grids that are extremely stable as well as ones that are extremely unstable. The 1/t-algorithm, a variation of Wang-Landau, which does not suffer from error saturation, and additional entropic sampling are used to achieve good precision even for very small probabilities ranging over eight decades.