Inference for epidemics on networks

Abstract

One of the motivating questions for many epidemiologists is “how quickly or widely will a particular infection spread?" To answer this question, often epidemic models are used to model the spread of a disease, with different epidemic models making different assumptions about the development of the disease; for example, two similar epidemic models might differ in whether they assume that people develop immunity after recovering from the disease. The advantage of these epidemic models is that they can be used to quickly estimate the epidemic model parameters, given an observed outbreak of a disease. One assumption of most standard epidemic models is that an infectious individual has an equal probability of spreading the disease to any susceptible person in a population. The classical Susceptible-Infective-Recovered (SIR) model is an epidemic model with this assumption, which says that any pair of individuals has an equal probability of having an interaction [1]; when this interaction is between a susceptible and infectious individual, we call this interaction adequate contact [1] if the interaction results in a susceptible individual contracting the disease. This assumption of equal probability of interaction (called homogeneous-mixing) can be a restrictive and unreasonable assumption in situations where within a population there are some pairs of individuals that never interact or some pairs of individuals that interact with higher probability. A more general model considers networks, in which nodes represent people and edges represent a possible path of infection; that is, if Node A (infectious) and Node B (susceptible) don’t share an edge then Node A cannot directly infect Node B. However without this homogeneous-mixing assumption, inference for the epidemic model parameters can be computationally intensive. This thesis will answer two questions: 1. given observed properties of the network and the final epidemic size(s), can we efficiently estimate the epidemic model parameters; and, 2. given observed properties of the network and known times and types of events, can we efficiently estimate the epidemic model parameters? We will answer these questions by assuming the data came from the homogeneous-mixing SIR model, and then estimating the epidemic model parameters. We then use a linear model to adjust these estimates, which provides a fast way to estimate epidemic model parameters. We will also show that the error of the models we present are, on average, never greater than 0.2; that is, the estimated epidemic model parameters are on average within 0.2 of the true epidemic model parameters.