p-Modulus on Networks with Applications to the Study of Epidemics

Kansas State University, Manhattan KS

Investigators

Abstract

Modern approaches to data analysis often involve network structures. For example, the spread of an epidemic in a specific population can be described using a contact network representing the nature and strength of interpersonal interactions among individuals. Historically, these models were intractable due to the complexity of individual behavior and, therefore, many early disease models utilized simplified representations involving population averages and statistics. By differentiating individuals, network models carry a large amount of additional information that has become more accessible thanks to increasingly powerful computer processors and the advent of fast computational algorithms. This project develops new mathematical theories and computational algorithms to capture the essential features of networks and applies them to models in epidemiology. Results from this project will allow researchers to identify a number of valuable patterns in the data, including the subpopulations at highest risk, vulnerable transmission pathways, and effective mitigation strategies. The investigator and his colleagues study the mathematical concept of p-modulus on networks, focusing on the analysis of theoretical properties, the development of numerical algorithms, and the study of applications to the spread of diseases in contact networks. This is an interdisciplinary project intended to enhance both the theoretical understanding of the ways in which diseases spread in an interconnected network of individuals or sub-populations, and the computational tools available to researchers interested in modeling, simulating, and predicting the behavior of epidemics. The theory of p-modulus was originally developed in the field of complex analysis and has a connection to the concept of effective resistance in the context of electrical networks. The p-modulus provides a method for quantifying the richness of a family of walks: families of many short walks have a larger modulus than families of few long ones. Therefore, the flexibility of p-modulus provides a means for extracting fine structure characteristics of linked data.