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Digital contact tracing is a relevant tool to control infectious disease outbreaks, including the COVID-19 epidemic. Early work evaluating digital contact tracing omitted important features and heterogeneities of real-world contact patterns influencing contagion dynamics. We fill this gap with a modeling framework informed by empirical high-resolution contact data to analyze the impact of digital contact tracing in the COVID-19 pandemic. We investigate how well contact tracing apps, coupled with the quarantine of identified contacts, can mitigate the spread in real environments. We find that restrictive policies are more effective in containing the epidemic but come at the cost of unnecessary large-scale quarantines. Policy evaluation through their efficiency and cost results in optimized solutions which only consider contacts longer than 15–20 minutes and closer than 2–3 meters to be at risk. Our results show that isolation and tracing can help control re-emerging outbreaks when some conditions are met: (i) a reduction of the reproductive number through masks and physical distance; (ii) a low-delay isolation of infected individuals; (iii) a high compliance. Finally, we observe the inefficacy of a less privacy-preserving tracing involving second order contacts. Our results may inform digital contact tracing efforts currently being implemented across several countries worldwide.

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Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.

Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the "native" Hilbert space H in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both L 2 -orthonormal and orthogonal in H (Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of H. These results have strong connections to n-widths in Approximation Theory, and they establish that errors of optimal approximations are closely related to the decay of the eigenvalues.Though the eigenspaces and eigenvalues are not readily available, they can be well approximated using the standard n-dimensional subspaces spanned by translates of the kernel with respect to n nodes or centers. We give error bounds for the numerical approximation of the eigensystem via such subspaces. A series of examples shows that our numerical technique via a greedy point selection strategy allows to calculate the eigensystems with good accuracy. *

Digital contact tracing is increasingly considered as a tool to control infectious disease outbreaks. As part of a broader test, trace, isolate, and quarantine strategy, digital contract tracing apps have been proposed to alleviate lock-downs, and to return societies to a more normal situation in the ongoing COVID-19 crisis. Early work evaluating digital contact tracing did not consider important features and heterogeneities present in real-world contact patterns which impact epidemic dynamics. Here, we fill this gap by considering a modeling framework informed by empirical high-resolution contact data to analyze the impact of digital contact tracing apps in the COVID-19 pandemic. We investigate how well contact tracing apps, coupled with the quarantine of identified contacts, can mitigate the spread of COVID-19 in realistic scenarios such as a university campus, a workplace, or a high school. We find that restrictive policies are more effective in confining the epidemics but come at the cost of quarantining a large part of the population. It is possible to avoid this effect by considering less strict policies, which only consider contacts with longer exposure and at shorter distance to be at risk. Our results also show that isolation and tracing can help keep re-emerging outbreaks under control provided that hygiene and social distancing measures limit the reproductive number to 1.5. Moreover, we confirm that a high level of app adoption is crucial to make digital contact tracing an effective measure. Our results may inform app-based contact tracing efforts currently being implemented across several countries worldwide.

It is well-known that radial basis function interpolants suffer of bad conditioning if the basis of translates is used. In the recent work [12], the authors gave a quite general way to build stable and orthonormal bases for the native space N Φ (Ω) associated to a kernel Φ on a domain Ω ⊂ R s . The method is simply based on the factorization of the corresponding kernel matrix. Starting from that setting we describe a particular basis which turns out to be orthonormal in N Φ (Ω) and in 2,w (X), where X is a set of data sites of the domain Ω. The basis arises from a weighted singular value decomposition of the kernel matrix. This basis is also related to a discretization of the compact operatorand provides a connection with the continuous basis that arises from an eigen-decomposition of T Φ . Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.

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