arXiv subjects

Tyll Krüger

Publications and source records attributed to Tyll Krüger.

5 recordsLinked to original sources

arXiv2024

Chaos in opinion-driven disease dynamics

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.

nlin.CD↗

arXiv2023

An integro-differential model for the spread of diseases

In this study, we present an integro-differential model to simulate the local spread of infections. The model incorporates a standard susceptible-infected-recovered (\textit{SIR}-) model enhanced by an integral kernel, allowing for non-homogeneous mixing between susceptibles and infectives. We define requirements for the kernel function and derive analytical results for both the \textit{SIR}- and a reduced susceptible-infected-susceptible (\textit{SIS}-) model, especially the uniqueness of solutions. In order to optimize the balance between disease containment and the social and political costs associated with lockdown measures, we set up requirements for the implementation of control function, and show examples for three different formulations for the control: continuous and time-dependent, continuous and space- and time-dependent, and piecewise constant space- and time-dependent. Latter represent reality more closely as the control cannot be updated for every time and location. We found the optimal control values for all of those setups, which are by nature best for a continuous and space-and time dependent control, yet found reasonable results for the discrete setting as well. To validate the numerical results of the integro-differential model, we compare them to an established agent-based model that incorporates social and other microscopical factors more accurately and thus acts as a benchmark for the validity of the integro-differential approach. A close match between the results of both models validates the integro-differential model as an efficient macroscopic proxy. Since computing an optimal control strategy for agent-based models is computationally very expensive, yet comparatively cheap for the integro-differential model, using the proxy model might have interesting implications for future research.

math.DS↗

arXiv2021

A look into the future of the COVID-19 pandemic in Europe: an expert consultation

How will the coronavirus disease 2019 (COVID-19) pandemic develop in the coming months and years? Based on an expert survey, we examine key aspects that are likely to influence COVID-19 in Europe. The future challenges and developments will strongly depend on the progress of national and global vaccination programs, the emergence and spread of variants of concern, and public responses to nonpharmaceutical interventions (NPIs). In the short term, many people are still unvaccinated, VOCs continue to emerge and spread, and mobility and population mixing is expected to increase over the summer. Therefore, policies that lift restrictions too much and too early risk another damaging wave. This challenge remains despite the reduced opportunities for transmission due to vaccination progress and reduced indoor mixing in the summer. In autumn 2021, increased indoor activity might accelerate the spread again, but a necessary reintroduction of NPIs might be too slow. The incidence may strongly rise again, possibly filling intensive care units, if vaccination levels are not high enough. A moderate, adaptive level of NPIs will thus remain necessary. These epidemiological aspects are put into perspective with the economic, social, and health-related consequences and thereby provide a holistic perspective on the future of COVID-19.

q-bio.OT↗

arXiv1994

Some remarks on periodic billiard orbits in rational polygons

A polygon is called rational if the angle between each pair of sides is a rational multiple of $π.$ The main theorem we will prove is Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of the billiard flow. This is a strengthening of Masur's theorem, who has shown that any rational polygon has ``many'' periodic billiard trajectories; more precisely, the set of directions of the periodic trajectories are dense in the set of velocity directions $§^1.$ We will also prove some refinements of Theorem 1: the ``well distribution'' of periodic orbits in the polygon and the residuality of the points $q \in Q$ with a dense set of periodic directions.

math.DS↗

arXiv1994

Acceleration of bouncing balls in external fields

We introduce two models, the Fermi-Ulam model in an external field and a one dimensional system of bouncing balls in an external field above a periodically oscillating plate. For both models we investigate the possibility of unbounded motion. In a special case the two models are equivalent.

math.DS↗