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Abstract. The dynamics of social networks are modeled with a system of continuous Stochastic Ordinary Differential Equations (SODE). With the proper amount of noise input, the SODE model captures dynamic features that are lacking in the corresponding deterministic ODE model. Therefore, sensitivity to noise levels is investigated by considering four different regimes: essentially deterministic, noise-enriched, noise-enlarged, and noise-dominated. Each regime is defined based on the behavior of solutions of the SODE, and geometry of the regimes are categorized with stochastic simulations.

A Rouse model for polymer chains is incorporated into the linear continuous stickslip molecular-based tube reptation ideas of Doi-Edwards and Johnson-Stacer. This treats the physically constrained (PC) molecular stretches as internal strain variables for the overall PC/chemically cross-linked (CC) system. It yields an explicit system of stress-strain equations for the system permitting simple calculations of complex stressstrain relations. The model that is developed here treats PC molecule as entrapped within a constraining tube, which is comprised of both CC and PC molecules. The model is compared with experimental data sets from the literature.

We report on two recent advances in the modeling of viscoelastic polymers: (i) a new constitutive model that combines the virtual stick-slip continuum "molecular-based" ideas of Johnson and Stacer with the Rouse bead chain ideas; (ii) a two-dimensional version of a model that accounts for stenosis-driven shear wave propagation in biotissue.