A Brief Theory of Epidemic Kinetics.

In the context of the COVID-19 epidemic, and on the basis of the Theory of Dynamical Systems, we propose a simple theoretical approach for the expansion of contagious diseases, with a particular focus on viral respiratory tracts. The infection develops through contacts between contagious and exposed people, with a rate proportional to the number of contagious and of non-immune individuals, to contact duration and turnover, inversely proportional to the efficiency of protection measures, and balanced by the average individual recovery response. The obvious initial exponential increase is readily hindered by the growing recovery rate, and also by the size reduction of the exposed population. The system converges towards a stable attractor whose value is expressed in terms of the "reproductive rate" R0, depending on contamination and recovery factors. Various properties of the attractor are examined, and particularly its relations with R0. Decreasing this ratio below a critical value leads to a tipping threshold beyond which the epidemic is over. By contrast, significant values of the above ratio may bring the system through a bifurcating hierarchy of stable cycles up to a chaotic behaviour.

Hall-petch law revisited in terms of collective dislocation dynamics.

The Hall-Petch (HP) law, that accounts for the effect of grain size on the plastic yield stress of polycrystals, is revisited in terms of the collective motion of interacting dislocations. Sudden relaxation of incompatibility stresses in a grain triggers aftershocks in the neighboring ones. The HP law results from a scaling argument based on the conservation of the elastic energy during such transfers. The Hall-Petch law breakdown for nanometric sized grains is shown to stem from the loss of such a collective behavior as grains start deforming by successive motion of individual dislocations.

Breakdown of avalanche critical behaviour in polycrystalline plasticity.

Acoustic emission experiments on creeping ice as well as numerical simulations argue for a self-organization of collective dislocation dynamics during plastic deformation of single crystals into a scale-free pattern of dislocation avalanches characterized by intermittency, power-law distributions of avalanche sizes, complex space-time correlations and aftershock triggering. Here, we address the question of whether such scale-free, close-to-critical dislocation dynamics will still apply to polycrystals. We show that polycrystalline plasticity is also characterized by intermittency and dislocation avalanches. However, grain boundaries hinder the propagation of avalanches, as revealed by a finite (grain)-size effect on avalanche size distributions. We propose that the restraint of large avalanches builds up internal stresses that push temporally the dynamical system into a supercritical state, off the scale-invariant critical regime, and trigger secondary avalanches in neighbouring grains. This modifies the statistical properties of the avalanche population. The results might also bring into question the classical ways of modelling plasticity in polycrystalline materials, based on homogenization procedures.

Two-threshold model for scaling laws of noninteracting snow avalanches.

The sizes of snow slab failure that trigger snow avalanches are power-law distributed. Such a power-law probability distribution function has also been proposed to characterize different landslide types. In order to understand this scaling for gravity-driven systems, we introduce a two-threshold 2D cellular automaton, in which failure occurs irreversibly. Taking snow slab avalanches as a model system, we find that the sizes of the largest avalanches just preceding the lattice system breakdown are power-law distributed. By tuning the maximum value of the ratio of the two failure thresholds our model reproduces the range of power-law exponents observed for land, rock, or snow avalanches. We suggest this control parameter represents the material cohesion anisotropy.