Robust oscillations in SIS epidemics on adaptive networks: Coarse-graining by automated moment closure

We investigate the dynamics of an epidemiological susceptible-infected-susceptible (SIS) model on an adaptive network. This model combines epidemic spreading (dynamics on the network) with rewiring of network connections (topological evolution of the network). We propose and implement a computational approach that enables us to study the dynamics of the network directly on an emergent, coarse-grained level. The approach sidesteps the derivation of closed low-dimensional approximations. Our investigations reveal that global coupling, which enters through the awareness of the population to the disease, can result in robust large-amplitude oscillations of the state and topology of the network.Comment: revised version 6 pages, 4 figure

A new ghost cell/level set method for moving boundary problems:application to tumor growth

In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth

Numerical Simulation of Grain Boundary Grooving By Level Set Method

A numerical investigation of grain-boundary grooving by means of a Level Set method is carried out. An idealized polygranular interconnect which consists of grains separated by parallel grain boundaries aligned normal to the average orientation of the surface is considered. The surface diffusion is the only physical mechanism assumed. The surface diffusion is driven by surface curvature gradients, and a fixed surface slope and zero atomic flux are assumed at the groove root. The corresponding mathematical system is an initial boundary value problem for a two-dimensional Hamilton-Jacobi type equation. The results obtained are in good agreement with both Mullins' analytical "small slope" solution of the linearized problem (W.W. Mullins, 1957) (for the case of an isolated grain boundary) and with solution for the periodic array of grain boundaries (S.A. Hackney, 1988).Comment: Submitted to the Journal of Computational Physics (19 pages, 8 Postscript figures, 3 tables, 29 references