Systemic Risk in the Asymmetric Case: Theory and Experiments with Epidemiology using Semidefinite Programming

Abstract

This thesis approaches systemic risk from an epidemiology perspective, modeling the transfer of disease as a dynamical system and attempting to quantify the risk of asymptotic instability. This instability specifically means trending to a probability of infection for each member of the system that is nonzero. In this work, we (1) extend a susceptible-infected-susceptible model for epidemic spreading to allow for asymmetry and time-variance, (2) bound the epidemic threshold of the model with the joint spectral radius (JSR) of the relevant transition matrices, and (3) use a semide nite programming technique to compute an upper bound on the JSR. We also experiment with two real disease models{HIV and Zica{to validate our model as well as to test our the impact of the network parameters, transmission rates, and cure rates on the epidemic threshold. Our results indicate that allowing time variance in these asymmetric models requires a more complex computational tool for providing an upper bound because the interaction between matrices can lead to higher JSR's than repeating any individual transmission matrix would. Finally, we provide one example of an interventionist use for this model by analyzing the effect of removing one node from the graph and compare random selection vs. maximally connected selection for that node.