Epidemics on Static and Adaptive Networks

Abstract

The COVID-19 pandemic has had a disruptive impact on healthcare systems and everyday life of the majority of the people around the globe. Despite many years of research on network epidemiology, many key aspects of disease transmission and in particular the response of people to the spread of a disease, remain poorly understood. On the basis of epidemiological modelling lie the Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR) models. In this dissertation, we aim to improve the understanding of the spread of contagious diseases, with an emphasis on the interplay between disease spread and personal behaviour, applied to the SIS and SIR models. The first part starts with the analysis of the eigenvalue spectrum of the infinitesimal generator of the Markovian SIS model with self-infections (Chapter 2). Based on the eigenvalue spectrum, which we believe encodes the majority of the dynamics, we derive an alternative definition of the epidemic threshold. We show that the epidemic threshold approximately coincides with the effective infection rate for which the third-largest eigenvalue is minimal. Contrary to the SIS process, where only an eigenvalue analysis is possible, the SIR process is completely solved on an arbitrary, heterogeneous network (Chapter 3). The benefit of the exact solution is demonstrated by analytically computing the time when the number of infections is maximal. The second part concerns the interplay between the spread of a disease and the response of people to the disease spread. We develop the Generalised Adaptive SIS (GASIS) model to describe how individuals break and create links in the contact graph. The decisions for breaking or creating links are based on the viral state of the nodes attached to that link. For all 36 instances in the G-ASIS model, we analyse the relation between the epidemic threshold and the effective link-breaking rate (Chapter 4). We derive the first-order and second-order mean-field approximation of the G-ASIS model (Chapter 5) and illustrate that the second-order approximation is able to qualitatively approximate the Markovian model more accurately than the first-order approximation. The G-ASIS mean-field model is extended to arbitrary link-breaking and link-creation responses, which are not only related to the number of susceptible and infectious neighbours of a node, but may also depend on the presence of the virus in the whole population (Chapter 6). For all possible link-breaking and link-creation responses, epidemic waves cannot occur in the mean-field adaptive SIS process. In the final part,we develop theNetwork-Inference-based Prediction Algorithm(NIPA) for forecasting the spread of contagious diseases on heterogeneous networks (Chapter 7). The contact graph is assumed to be unknown and is inferred by NIPA from the number of reported cases. NIPA is a hybrid method, combining epidemiological knowledge, machine-learning and networks. Network-based forecasting, and NIPA in particular, seems favourable for predicting epidemic outbreaks, which is demonstrated by showing that NIPA outperforms many other forecasting algorithms for estimating the spread of COVID-19.