Amid the ongoing COVID‐19 pandemic, public health authorities and the general population are striving to achieve a balance between safety and normalcy. Ever changing conditions call for the development of theory and simulation tools to finely describe multiple strata of society while supporting the evaluation of “what‐if” scenarios. Particularly important is to assess the effectiveness of potential testing approaches and vaccination strategies. Here, an agent‐based modeling platform is proposed to simulate the spreading of COVID‐19 in small towns and cities, with a single‐individual resolution. The platform is validated on real data from New Rochelle, NY—one of the first outbreaks registered in the United States. Supported by expert knowledge and informed by reported data, the model incorporates detailed elements of the spreading within a statistically realistic population. Along with pertinent functionality such as testing, treatment, and vaccination options, the model accounts for the burden of other illnesses with symptoms similar to COVID‐19. Unique to the model is the possibility to explore different testing approaches—in hospitals or drive‐through facilities—and vaccination strategies that could prioritize vulnerable groups. Decision‐making by public authorities could benefit from the model, for its fine‐grain resolution, open‐source nature, and wide range of features.

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We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as 'leapfrogging' orbits. These solutions, which consists of two pairs of identical yet oppositely-signed vortices, were known to W. Gröbli (1877) and A. E. H. Love (1883), and can be parameterized by a dimensionless parameter α related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Tophøj and Aref (2012) both indicate, to many digits, that the bifurcation occurs when 1/α = φ 2 , where φ is the golden ratio. This study aims to explain the origin of this remarkable value. Using a trick from the gravitational two-body problem, we change variables to render the Floquet problem in an explicit form that is more amenable to analysis. We then implement G. W. Hill's method of harmonic balance to high order using computer algebra to construct a rapidly-converging sequence of asymptotic approximations to the bifurcation value, confirming the value found earlier.

The COVID-19 pandemic has laid bare the importance of non-pharmaceutical interventions in the containment of airborne infectious diseases. Social distancing and mask-wearing have been found to contain COVID-19 spreading across a number of observational studies, but a precise understanding of their combined effectiveness is lacking. An underdeveloped area of research entails the quantification of the specific role of each of these measures when they are differentially adopted by the population. Pursuing this research allows for answering several pressing questions like: how many people should follow public health measures for them to be effective for everybody? Is it sufficient to practice social distancing only or just wear a mask? Here, we make a first step in this direction, by establishing a susceptible–exposed–infected–removed epidemic model on a temporal network, evolving according to the activity-driven paradigm. Through analytical and numerical efforts, we study epidemic spreading as a function of the proportion of the population following public health measures, the extent of social distancing, and the efficacy of masks in protecting the wearer and others. Our model demonstrates that social distancing and mask-wearing can be effective in preventing COVID-19 outbreaks if adherence to both measures involves a substantial fraction of the population.

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The point vortex system is a system of longstanding interest in nonlinear dynamics, describing the motion of a two-dimensional inviscid fluid that is irrotational except at a discrete set of moving point vortices, at which the vorticity diverges. The leapfrogging orbit consists of two rotating pairs of like-signed vortices which, taken as a quartet, propagate at constant velocity. It is known that if the two pairs are initially widely separated, the motion is stable, while if they are closer together it becomes unstable, with this relation represented by a dimensionless parameter α defined in the text. We here demonstrate analytically that the transition from stability to instability happens at a critical value α = φ −2 , where φ is the golden ratio. This value had been hypothesized based on careful numerics by Tophøj and Aref, and by the present authors using a semi-analytic argument but not previously demonstrated through exact analysis.