Assessing the heterogeneity in the transmission of infectious diseases from time series of epidemiological data.

Structural features and the heterogeneity of disease transmissions play an essential role in the dynamics of epidemic spread. But these aspects can not completely be assessed from aggregate data or macroscopic indicators such as the effective reproduction number. We propose in this paper an index of effective aggregate dispersion (EffDI) that indicates the significance of infection clusters and superspreading events in the progression of outbreaks by carefully measuring the level of relative stochasticity in time series of reported case numbers using a specially crafted statistical model for reproduction. This allows to detect potential transitions from predominantly clustered spreading to a diffusive regime with diminishing significance of singular clusters, which can be a decisive turning point in the progression of outbreaks and relevant in the planning of containment measures. We evaluate EffDI for SARS-CoV-2 case data in different countries and compare the results with a quantifier for the socio-demographic heterogeneity in disease transmissions in a case study to substantiate that EffDI qualifies as a measure for the heterogeneity in transmission dynamics.

Solving the electronic Schrödinger equation for multiple nuclear geometries with weight-sharing deep neural networks.

The Schrödinger equation describes the quantum-mechanical behaviour of particles, making it the most fundamental equation in chemistry. A solution for a given molecule allows computation of any of its properties. Finding accurate solutions for many different molecules and geometries is thus crucial to the discovery of new materials such as drugs or catalysts. Despite its importance, the Schrödinger equation is notoriously difficult to solve even for single molecules, as established methods scale exponentially with the number of particles. Combining Monte Carlo techniques with unsupervised optimization of neural networks was recently discovered as a promising approach to overcome this curse of dimensionality, but the corresponding methods do not exploit synergies that arise when considering multiple geometries. Here we show that sharing the vast majority of weights across neural network models for different geometries substantially accelerates optimization. Furthermore, weight-sharing yields pretrained models that require only a small number of additional optimization steps to obtain high-accuracy solutions for new geometries.