The COVID-19 pandemic: growth patterns, power law scaling, and saturation.

More and more countries are showing a significant slowdown in the number of new COVID-19 infections due to effective governmentally instituted lockdown and social distancing measures. We have analyzed the growth behavior of the top 25 most affected countries by means of a local slope analysis and found three distinct patterns that individual countries follow depending on the strictness of the lockdown protocols: rise and fall, power law, or logistic. For countries showing power law growth we have determined the scaling exponents. For countries that showed a strong slowdown in the rate of infections we have extrapolated the expected saturation of the total number of infections and the expected final date. Three different extrapolation methods (logistic, parabolic, and cutoff power law) were used. All methods agree on the order of magnitude of saturation and end dates. Global infection rates are analyzed with the same methods. The relevance and accuracy of these extrapolations is discussed.

Agent-based model for friendship in social networks.

A model is proposed to understand the structuring of social networks in a fixed setting such as, for example, inside a university. The friendship formation is based on the frequency of encounters and mutual interest. The model shows distinctive single-scale behavior and reproduces accurately the measurable experimental quantities such as clustering coefficients, degree distribution, degree correlation, and friendship distribution. The model produces self-organized community structures and can be described as a network of densely interconnected networks. For the friendships, we find that the mutual interest is the dominant factor, which optimizes the network and that the number of encounters determines the statistically relevant distributions.

Experimental observation and computer simulations of 3D triplet structures in diffusion limited growth of xenon crystals.

Changes of growth morphologies are induced by a perturbation of the thermal diffusion field in the surrounding melt of a growing xenon crystal. Apart from the dendritic morphology, seaweed and doublon morphologies and for the first time transitions from dendritic to triplet structures (first predicted by T. Abel, E. Brener, and H. Müller-Krumbhaar [Phys. Rev. E 55, 7789 (1997)10.1103/PhysRevE.55.7789] were observed experimentally. With 3D phase-field simulations it was possible to reproduce the experimental procedure and to verify that triplet structures can grow in a stable way even in the presence of anisotropic surface free energy as found for experimental substances.

Analysis and visualization of multiply oriented lattice structures by a two-dimensional continuous wavelet transform.

The phase-field-crystal model [K. R. Elder and M. Grant, Phys. Rev. E 70, 051605 (2004)] produces multigrain structures on atomistic length scale but on diffusive time scales. Since individual atoms are resolved but are treated identically it is difficult to distinguish the exact position of grain boundaries and defects within grains. In order to analyze and visualize the whole grains a two-dimensional wavelet transform has been developed, which is capable of extracting grain boundaries and the lattice orientation of a grain relative to a laboratory frame of reference. This transformation makes it possible not only to easily visualize the multigrain structure, but also to perform exact measurements on low- and high-angle boundaries, grain size distributions and boundary-angle distributions, which can then be compared to experimental data. The presented wavelet transform can also be applied to results of other atomistic simulations, e.g., molecular dynamics or granular materials.

Measurements of the three-dimensional shape of ice crystals in supercooled water.

Experimentally grown ice crystals from ultrapure supercooled water are imaged by means of Mach-Zehnder interferometry. By analyzing the fringe patterns the phase information and thus the three-dimensional shape of the ice crystals is recovered quantitatively. The integral parameters height of the basal plane, volume, and surface of the crystals are measured as a function of time and supercooling. It is found that all measured parameters follow a power law as a function of time and the exponents are found to be independent of the supercooling. The shape transition from the prismatic to the basal face along the main growth direction of the ice dendrites as a function of the distance from the tip is found to be a power law as well. Our findings support the validity of universal growth laws in pattern forming systems.

Quantitative description of morphological transitions in diffusion-limited growth of xenon crystals.

Changes of growth morphologies are induced by a perturbation of the temperature distribution in the surrounding of a growing xenon crystal. Apart from the dendritic morphology seaweed and doublon morphologies are found. We present a method which quantitatively describes growth morphologies by means of rotational, scale, and translational invariant transformations. Evolutions of growth morphologies are represented as paths in the morphology space. The presented method could be of some use for other fields of research where qualitative and quantitative information of different classes of images has to be identified.

Generalized length scales for three-dimensional dendritic growth.

The crucial length scale of dendritic growth is the tip radius. Usually it is determined by fitting the data to a theoretical function. We present a method of a generalized tip radius which is entirely based on geometric considerations and is not dependent on an underlying assumption of the shape of the tip. Furthermore the results are stable and the average change of the tip radius between successive images is less than 6%.

The relation between scabies and hypersensitivity to antigens of house dust mites and storage mites.

Twenty five non-atopic scabietic patients were examined to estimate their reaction to crude house dust mite Dermatophagoides farinae (D. farinae) and storage mite Tyrophagus putrescentia (T. putrescentiae) antigens. Skin prick testing (SPT) by extracts of both mites antigens showed significant higher positive results in scabietics when compared to non-scabietic control subjects. Moreover, 60% and 56% of scabietic patients showed positive levels of specific anti-D. farinae and T. putrescentia IgE respectively in comparison to 13.4% & 20% of control subjects. A significant difference has been obtained when the total number of positive results were compared to the total number of negative ones. The results revealed that there is an evidence of cross reactivity between Sarcoptes scabiei antigens and extracts of D. farinae and T. putrescentiae, and the hypersensitivity to house dust mite and storage mite antigens was significantly higher in scabietics than in controls. It could be concluded that there is some proof that other mites rather than Sarcoptes scabiei may have a role in the pathogenesis of scabies and the cross reactivity between S. scabiei and house dust mite and storage mite may explain the persistence of symptoms in some cases even after proper treatment of the disease.