On star-convex volumes in 2-D hydrodynamical flows and their relevance for coherent transport.

Oceanic surface flows are dominated by finite-time mesoscale structures that separate two-dimensional flows into volumes of qualitatively different dynamical behavior. Among these, the transport boundaries around eddies are of particular interest since the enclosed volumes show a notable stability with respect to filamentation while being transported over significant distances with consequences for a multitude of different oceanic phenomena. In this paper, we present a novel method to analyze coherent transport in oceanic flows. The presented approach is purely based on convexity and aims to uncover maximal persistently star-convex (MPSC) volumes, volumes that remain star-convex with respect to a chosen reference point during a predefined time window. Since these volumes do not generate filaments, they constitute a sub-class of finite-time coherent volumes. The new perspective yields definitions for filaments, which enables the study of MPSC volume formation and dissipation. We discuss the underlying theory and present an algorithm, the material star-convex structure search, that yields comprehensible and intuitive results. In addition, we apply our method to different velocity fields and illustrate the usefulness of the method for interdisciplinary research by studying the generation of filaments in a real-world example.

An extended transfer operator approach to identify separatrices in open flows.

Vortices of coherent fluid volume are considered to have a substantial impact on transport processes in turbulent media. Yet, due to their Lagrangian nature, detecting these structures is highly nontrivial. In this respect, transfer operator approaches have been proven to provide useful tools: Approximating a possibly time-dependent flow as a discrete Markov process in space and time, information about coherent structures is contained in the operator's eigenvectors, which is usually extracted by employing clustering methods. Here, we propose an extended approach that couples surrounding filaments using "mixing boundary conditions" and focuses on the separation of the inner coherent set and embedding outer flow. The approach refrains from using unsupervised machine learning techniques such as clustering and uses physical arguments by maximizing a coherence ratio instead. We show that this technique improves the reconstruction of separatrices in stationary open flows and succeeds in finding almost-invariant sets in periodically perturbed flows.

Quantifying transient spreading dynamics on networks.

Spreading phenomena on networks are essential for the collective dynamics of various natural and technological systems, from information spreading in gene regulatory networks to neural circuits and from epidemics to supply networks experiencing perturbations. Still, how local disturbances spread across networks is not yet quantitatively understood. Here, we analyze generic spreading dynamics in deterministic network dynamical systems close to a given operating point. Standard dynamical systems' theory does not explicitly provide measures for arrival times and amplitudes of a transient spreading signal because it focuses on invariant sets, invariant measures, and other quantities less relevant for transient behavior. We here change the perspective and introduce formal expectation values for deterministic dynamics to work out a theory explicitly quantifying when and how strongly a perturbation initiated at one unit of a network impacts any other. The theory provides explicit timing and amplitude information as a function of the relative position of initially perturbed and responding unit as well as depending on the entire network topology.

Regional responsibility and coordination of appropriate inpatient care capacities for patients with COVID-19 - the German DISPENSE model.

As of late 2019, the COVID-19 pandemic has been a challenge to health care systems worldwide. Rapidly rising local COVID-19 incidence rates, result in demand for high hospital and intensive care bed capacities on short notice. A detailed up-to-date regional surveillance of the dynamics of the pandemic, precise prediction of required inpatient capacities of care as well as a centralized coordination of the distribution of regional patient fluxes is needed to ensure optimal patient care. In March 2020, the German federal state of Saxony established three COVID-19 coordination centers located at each of its maximum care hospitals, namely the University Hospitals Dresden and Leipzig and the hospital Chemnitz. Each center has coordinated inpatient care facilities for the three regions East, Northwest and Southwest Saxony with 36, 18 and 29 hospital sites, respectively. Fed by daily data flows from local public health authorities capturing the dynamics of the pandemic as well as daily reports on regional inpatient care capacities, we established the information and prognosis tool DISPENSE. It provides a regional overview of the current pandemic situation combined with daily prognoses for up to seven days as well as outlooks for up to 14 days of bed requirements. The prognosis precision varies from 21% and 38% to 12% and 15% relative errors in normal ward and ICU bed demand, respectively, depending on the considered time period. The deployment of DISPENSE has had a major positive impact to stay alert for the second wave of the COVID-19 pandemic and to allocate resources as needed. The application of a mathematical model to forecast required bed capacities enabled concerted actions for patient allocation and strategic planning. The ad-hoc implementation of these tools substantiates the need of a detailed data basis that enables appropriate responses, both on regional scales in terms of clinic resource planning and on larger scales concerning political reactions to pandemic situations.

Transition to reconstructibility in weakly coupled networks.

Across scientific disciplines, thresholded pairwise measures of statistical dependence between time series are taken as proxies for the interactions between the dynamical units of a network. Yet such correlation measures often fail to reflect the underlying physical interactions accurately. Here we systematically study the problem of reconstructing direct physical interaction networks from thresholding correlations. We explicate how local common cause and relay structures, heterogeneous in-degrees and non-local structural properties of the network generally hinder reconstructibility. However, in the limit of weak coupling strengths we prove that stationary systems with dynamics close to a given operating point transition to universal reconstructiblity across all network topologies.