Periodic epidemic outbursts explained by local saturation of clusters.

Adding the notion of spatial locality to the susceptible-infected-recovered (or SIR) model, allows to capture local saturation of an epidemic. The resulting minimum model of an epidemic, consisting of five ordinary differential equations with constant model coefficients, reproduces slowly decaying periodic outbursts, as observed in the COVID-19 or Spanish flu epidemic. It is shown that if immunity decays, even slowly, the model yields a fully periodic dynamics.

Modeling the role of clusters and diffusion in the evolution of COVID-19 infections during lock-down.

The dynamics of the spread of epidemics, such as the recent outbreak of the SARS-CoV-2 virus, is highly nonlinear and therefore difficult to predict. As time evolves in the present pandemic, it appears more and more clearly that a clustered dynamics is a key element of the description. This means that the disease rapidly evolves within spatially localized networks, that diffuse and eventually create new clusters. We improve upon the simplest possible compartmental model, the SIR model, by adding an additional compartment associated with the clustered individuals. This sophistication is compatible with more advanced compartmental models and allows, at the lowest level of complexity, to leverage the well-mixedness assumption. The so-obtained SBIR model takes into account the effect of inhomogeneity on epidemic spreading, and compares satisfactorily with results on the pandemic propagation in a number of European countries, during and immediately after lock-down. Especially, the decay exponent of the number of new cases after the first peak of the epidemic is captured without the need to vary the coefficients of the model with time. We show that this decay exponent is directly determined by the diffusion of the ensemble of clustered individuals and can be related to a global reproduction number, that overrides the classical, local reproduction number.

Internal wave turbulence near a Texel beach.

A summer bather entering a calm sea from the beach may sense alternating warm and cold water. This can be felt when moving forward into the sea ('vertically homogeneous' and 'horizontally different'), but also when standing still between one's feet and body ('vertically different'). On a calm summer-day, an array of high-precision sensors has measured fast temperature-changes up to 1 °C near a Texel-island (NL) beach. The measurements show that sensed variations are in fact internal waves, fronts and turbulence, supported in part by vertical stable stratification in density (temperature). Such motions are common in the deep ocean, but generally not in shallow seas where turbulent mixing is expected strong enough to homogenize. The internal beach-waves have amplitudes ten-times larger than those of the small surface wind waves. Quantifying their turbulent mixing gives diffusivity estimates of 10(-4)-10(-3) m(2) s(-1), which are larger than found in open-ocean but smaller than wave breaking above deep sloping topography.