The barrier method: a technique for calculating very long transition times.

In many dynamical systems, there is a large separation of time scales between typical events and "rare" events which can be the cases of interest. Rare-event rates are quite difficult to compute numerically, but they are of considerable practical importance in many fields, for example, transition times in chemical physics and extinction times in epidemiology can be very long, but are quite important. We present a very fast numerical technique that can be used to find long transition times (very small rates) in low-dimensional systems, even if they lack detailed balance. We illustrate the method for a bistable nonequilibrium system introduced by Maier and Stein and a two-dimensional (in parameter space) epidemiology model.

Harmonic measure for critical Potts clusters.

We present a technique, which we call "etching," which we use to study the harmonic measure of Fortuin-Kasteleyn clusters in the Q-state Potts model for Q=1-4 . The harmonic measure is the probability distribution of random walkers diffusing onto the perimeter of a cluster. We use etching to study regions of clusters which are extremely unlikely to be hit by random walkers, having hitting probabilities down to 10-4600. We find good agreement between the theoretical predictions of Duplantier and our numerical results for the generalized dimension D(q) including regions of small and negative q .

Design, fabrication, and preliminary results of a novel below-knee prosthesis for snowboarding: A case report.

Snowboarding with a below-knee prosthesis is compromised by the limited rotation capabilities of the existing below-knee prostheses, which are designed for use in normal walking. Based on snowboarding range of motion analyses, a novel below-knee prosthesis was designed with the aim to achieve similar range of motions like able-bodied snowboarders. The new prosthesis allows for passive inversion/eversion, passive plantarflexion/dorsiflexion and additional 'voluntary' plantarflexion/dorsiflexion initiated by lateral or medial rotation of the upper leg and knee. A prototype was built and was subsequently tested on a single subject, a highly professional snowboarder and candidate for the Olympic Winter Games. The movements of the subject were recorded on video, analyzed and compared to the recorded movements of an able-bodied snowboarder, and a snowboarder with a traditional below-knee prosthesis. The results indicated an increased similarity of inversion/eversion and plantarflexion/dorsiflexion between the snowboarder with the new below-knee prosthesis and the able-bodied snowboarder, whereas the snowboarder with the traditional below-knee prosthesis and the able-bodied snowboarder differed considerably. These results indicate that snowboarding with the new prosthesis is more comparable to able-bodied snowboarding. On a subjective basis this is confirmed by the test subject who stated that: "snowboarding with the new prosthesis is like it was before the amputation!".

SIS epidemics with household structure: the self-consistent field method.

We consider a stochastic SIS infection model for a population partitioned into m households assuming random mixing. We solve the model in the limit m --> infinity by using the self-consistent field method of statistical physics. We derive a number of explicit results, and give numerical illustrations. We then do numerical simulations of the model for finite m and without random mixing. We find in many of these cases that the self-consistent field method is a very good approximation.

Percolation on heterogeneous networks as a model for epidemics.

We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the heterogeneity. These results are qualitatively different from those of standard models in epidemiology, but correspond to real effects. We suggest that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms. We also investigate small world lattices, and show that the effects mentioned above are even stronger.

Geography in a scale-free network model.

We offer an example of a network model with a power-law degree distribution, P(k) approximately k(-alpha), for nodes, but which nevertheless has a well-defined geography and a nonzero threshold percolation probability for alpha>2, the range of real-world contact networks. This is different from p(c)=0 for alpha<3 results for the original well-mixed scale-free networks. In our lattice-based scale-free network, individuals link to nearby neighbors on a lattice. Even considerable additional small-world links do not change our conclusion of nonzero thresholds. When applied to disease propagation, these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering.

Fluctuation effects in an epidemic model.

We study a discrete epidemic model A+B-->2A in one and two dimensions (1D and 2D). In 1D for low concentration theta, we find that a depletion zone exists ahead of the front and the average velocity of the front approaches v=theta/2. In the 1D high concentration limit, we find that the velocity approaches v=1-e(-theta/2). In 2D, for low concentration we also find a depletion zone, and the velocity scales as v approximately theta(0.6), which is different from the scaling expected from the mean field approximation, v approximately theta(0.5). Analysis of the interface width scaling properties demonstrated that the front dynamics of this reaction are not governed by the Kardar-Parisi-Zhang equation.