Statistical Inference on Stochastic Graphs

Abstract

This thesis considers modelling and applications of random graph processes. A brief review on contemporary random graph models and a general Birth-Death model with relevant maximum likelihood inference procedure are provided in chapter one. The main result in this thesis is the construction of an epidemic model by embedding a competing hazard model within a stochastic graph process (chapter 2). This model includes both individual characteristics and the population connectivity pattern in analyzing the infection propagation. The dynamic outdegrees and indegrees, estimated by the model, provide insight into important epidemiological concepts such as the reproductive number. A dynamic reproductive number based on the disease graph process is developed and applied in several simulated and actual epidemic outbreaks. In addition, graph-based statistical measures are proposed to quantify the effect of individual characteristics on the disease propagation. The epidemic model is applied to two real outbreaks: the 2001 foot-and-mouth epidemic in the United Kingdom (chapter 3) and the 1861 measles outbreak in Hagelloch, Germany (chapter 4). Both applications provide valuable insight into the behaviour of infectious disease propagation with di erent connectivity patterns and human interventions.