a b s t r a c t Power law distributions, also known as heavy tail distributions, model distinct real life phenomena in the areas of biology, demography, computer science, economics, informa-tion theory, language, and astronomy, amongst others. In this paper, it is presented a review of the literature having in mind applications and possible explanations for the use of power laws in real phenomena. We also unravel some controversies around power laws.

We use symmetry to study two central pattern generator (CPG) models for biped locomotion. The first one is a coupled four-cell network, proposed by Golubitsky, Stewart, Buono, and Collins, that models rhythms associated to legs. A classification based on symmetry shows that this network can produce periodic solutions with rhythms corresponding to the standard bipedal gaits of run, walk, hop, gallop, and skip, among others. Moreover, the four-cell model can produce two types of hop, two types of gallop, and three additional symmetry types of periodic solutions that have yet to be identified with the rhythms of known bipedal gaits. The second locomotor CPG network models interlimb coordination in bipeds (arms+legs). It is obtained by breaking the symmetry between fore and hind legs in an eight-cell CPG network for quadruped gaits, also proposed by Golubitsky et al. We match the rhythms of perturbed periodic solutions found in this eight-cell network with legs rhythms produced by the four-cell CPG model. We also compare patterns of oscillation of gaits of the eight-cell model with results on bipedal interlimb coordination in the literature, showing that the eight-cell model is a plausible network for modeling human interlimb coordination.We show numerical simulations of periodic solutions corresponding to the bipedal gaits in the two CPG models. These simulations use clamped Hodgkin-Huxley equations to model cell internal dynamics and partial linear coupling (where only the electrical potentials of different cells are coupled). We use synaptic coupling in the four-cell model and diffusive coupling in the eight-cell model.

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We study a fractional order model for HIV infection where latent T helper cells are included. We compute the reproduction number of the model and study the stability of the disease free equilibrium. We observe that the reproduction number varies with the order of the fractional derivative α. In terms of epidemics, this suggests that varying α induces a change in the patients' epidemic status. Moreover, we simulate the variation of relevant parameters, such as the fraction of uninfected CD4 + T cells that become latently infected, and the CTLs proliferation rate due to infected CD4 + T cells. The model produces biologically reasonable results.