Memory effects in disease modelling through kernel estimates with oscillatory time history.

We design a linear chain trick algorithm for dynamical systems for which we have oscillatory time histories in the distributed time delay. We make use of this algorithmic framework to analyse memory effects in disease evolution in a population. The modelling is based on a susceptible-infected-recovered SIR-model and on a susceptible-exposed-infected-recovered SEIR-model through a kernel that dampens the activity based on the recent history of infectious individuals. This corresponds to adaptive behavior in the population or through governmental non-pharmaceutical interventions. We use the linear chain trick to show that such a model may be written in a Markovian way, and we analyze the stability of the system. We find that the adaptive behavior gives rise to either a stable equilibrium point or a stable limit cycle for a close to constant number of susceptibles, i.e. locally in time. We also show that the attack rate for this model is lower than it would be without the dampening, although the adaptive behavior disappears as time goes to infinity and the number of infected goes to zero.

Estimating true prevalence through questionnaire data.

We present a general analytical method for obtaining unbiased prevalence estimates based on data from regional or national testing programs, where individual participation in the testing program is voluntary but where additional questionnaire data is collected regarding the individual-level reason/motivation for being tested. The approach is based on re-writing the conditional probabilities for being tested, being infected, and having symptoms, so that a series of equations can be defined that relate estimable quantities (from test data and questionnaire data) to the result of interest (an unbiased estimate of prevalence). The final estimates appear to be robust based on prima-facie examination of the temporal dynamics estimated, as well as agreement with an independent estimate of prevalence. Our approach demonstrates the potential strength of incorporating questionnaires when testing a population during an outbreak, and can be used to help obtain unbiased estimates of prevalence in similar settings.

Spacing distribution in the two-dimensional Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at noninteger ß.

A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature ß. For 2×2 matrices with Gaussian distribution we analytically compute the nearest-neighbor spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective ß_{eff}(ß) in our analytic formula that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of ß. It reproduces the 2D Poisson distribution at ß=0 exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by noninteger values ß=1.4 and ß=2.6 from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.

Territorial behaviour of buzzards versus random matrix spacing distributions.

A deeper understanding of the processes underlying the distribution of animals in space is crucial for both basic and applied ecology. The Common buzzard (Buteo buteo) is a highly aggressive, territorial bird of prey that interacts strongly with its intra- and interspecific competitors. We propose and use random matrix theory to quantify the strength and range of repulsion as a function of the buzzard population density, thus providing a novel approach to model density dependence. As an indicator of territorial behaviour, we perform a large-scale analysis of the distribution of buzzard nests in an area of 300 square kilometres around the Teutoburger Wald, Germany, as gathered over a period of 20 years. The nearest and next-to-nearest neighbour spacing distribution between nests is compared to the two-dimensional Poisson distribution, originating from uncorrelated random variables, to the complex eigenvalues of random matrices, which are strongly correlated, and to a two-dimensional Coulomb gas interpolating between these two. A one-parameter fit to a time-moving average reveals a significant increase of repulsion between neighbouring nests, as a function of the observed increase in absolute population density over the monitored period of time, thereby proving an unexpected yet simple model for density-dependent spacing of predator territories. A similar effect is obtained for next-to-nearest neighbours, albeit with weaker repulsion, indicating a short-range interaction. Our results show that random matrix theory might be useful in the context of population ecology.

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems.

We study the transition between integrable and chaotic behavior in dissipative open quantum systems, exemplified by a boundary driven quantum spin chain. The repulsion between the complex eigenvalues of the corresponding Liouville operator in radial distance s is used as a universal measure. The corresponding level spacing distribution is well fitted by that of a static two-dimensional Coulomb gas with harmonic potential at inverse temperature ß∈[0,2]. Here, ß=0 yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and ß=2 equals the distribution obtained from the complex Ginibre ensemble, describing the fully chaotic limit. Our findings generalize the results of Grobe, Haake, and Sommers, who derived a universal cubic level repulsion for small spacings s. We collect mathematical evidence for the universality of the full level spacing distribution in the fully chaotic limit at ß=2. It holds for all three Ginibre ensembles of random matrices with independent real, complex, or quaternion matrix elements.

Engineered nanoparticles induced brush border disruption in a human model of the intestinal epithelium.

Nanoparticles hold great promise in cell biology and medicine due to the inherent physico-chemical properties when these materials are synthesized on the nanoscale. Moreover, their small size, and the ability to functionalize the outer nanoparticle surface makes them an ideal vector suited to traverse a number of physical barriers in the human body. While nanoparticles hold great promise for applications in cell biology and medicine, their downfall is the toxicity that accompanies exposure to biological systems. This chapter focuses on exposure via the oral route since nanomaterials are being engineered to act as carriers for drugs, contrast agents for specialized imaging techniques, as well as ingested pigments approved by regulatory agencies for human food products. After these nanomaterials are ingested they have the potential to interact with a number of biologically significant tissues, one of which is the epithelium of the small intestine. Within the small intestine exists enterocytes whose principal function is nutrient absorption. The absorptive process is aided by microvilli that act to increase the surface area of the epithelium. Dense arrays of microvilli, referred to as the brush border, have recently been shown to undergo disruption as a consequence of exposure to nanomaterials. This chapter aims to set the stage for detailed mechanistic studies at the cell biology level concerning this newly emerging nanotoxicity research paradigm, as the underlying structural characterization responsible for the existence of microvilli have been elucidated.