A Polya Urn Stochastic Model for the Analysis and Control of Epidemics on Networks by Mikhail Hayhoe

This thesis introduces a model for epidemics on networks based on the classical Polya process. Temporal contagion processes are generated on the network nodes using a modified Polya sampling scheme that accounts for spatial infection among neighbouring nodes. The stochastic properties and asymptotic behaviour of the resulting network Polya contagion process are analyzed. Given the complicated nature of this process, three classical Polya processes, one computational and two analytical, are proposed to statistically approximate the contagion process of each node, demonstrating a good fit for a range of system parameters. An optimal control problem is formulated for minimizing the average infection using a limited curing budget, and a number of different curing strategies are presented, including a proven convergent gradient descent algorithm. The feasibility of the problem is proven under high curing budgets by deriving conservative lower bounds that turn some processes into supermartingales. Extensive simulations run on large-scale networks demonstrate the effectiveness of our proposed strategies.