(374j) Mathematical Analysis of Percolation Processes with Application to Epidemics and Filtration

Percolation is the basis of numerous processes in chemical engineering, such as gelation of polymers, growth of epidemics, flow through porous media, and filtration, to give a few examples. We have developed efficient computer algorithms and mathematical analysis to find exact and approximate solutions to properties including thresholds, critical exponents, and geometric properties. For the problem of filtration, we have studied flow through a packed bed of spheres, and determined the size of the particles that can make it though large system. For epidemiology, the S (Susceptible) -- I (Infected) -- R (Recovered) model can be analyzed in a mean-field differential equation approach, and the solution can also be shown to be identical in many ways to percolation on complete graphs or branched (Bethe) lattices. This ties in percolation to network and graph theory. Some new results include exact thresholds to certain two-dimensional lattices, percolation for systems with long-range connections, and results for continuum percolation. There are also connections between percolation and aggregation theory. Some related models of practical importance include drilling percolation, strongly connected omponents in directed percolation, and hole percolation. Overall, the subjects have a broad application to many problems of interest to chemical engineers.