Epidemics on Networks: Analysis, Network Reconstruction and Prediction

The field of epidemiology encompasses a broad class of spreading phenomena, ranging from the seasonal influenza and the dissemination of fake news on online social media to the spread of neural activity over a synaptic network. The propagation of viruses, fake news and neural activity relies on the contact between individuals, social media accounts and brain regions, respectively. The contact patterns of the whole population result in a network. Due to the complexity of such contact networks, the understanding of epidemics is still unsatisfactory. In this dissertation, we advance the theory of epidemics and its applications, with a particular emphasis on the impact of the contact network. Our first contribution focusses on the analysis of the N-Intertwined Mean-Field Approximation (NIMFA) of the Susceptible-Infected-Susceptible (SIS) epidemic process on networks. We propose a geometric approach to clustering for epidemics on networks, which reduces the number of NIMFA differential equations from the network size N to the number m << N of clusters (Chapter 2). Specifically, we show that exact clustering is possible if and only if the contact network has an equitable partition, and we propose an approximate clustering method for arbitrary networks. Furthermore, for arbitrary contact networks, we derive the closed-form solution of the nonlinear NIMFA differential equations around the epidemic threshold (Chapter 3). Our solution reveals that the topology of the contact network is practically irrelevant for the epidemic outbreak around the epidemic threshold. Lastly, we study a discrete-time version of the NIMFA epidemic model (Chapter 4). We derive that the viral state is (almost always) monotonically increasing, the steady state is exponentially stable, and the viral dynamics is bounded by linear time-invariant systems. In the second part, we consider the reconstruction of the contact network and the prediction of epidemic outbreaks. We show that, for the stochastic SIS epidemic process on an individual level, the exact reconstruction of the contact network is impractical. Specifically, the maximum-likelihood SIS network reconstruction is NP-hard, and an accurate reconstruction requires a tremendous number of observations of the epidemic outbreak (Chapter 5). For epidemic models between groups of individuals, we argue that, in the presence of model errors, accurate long-term predictions of epidemic outbreaks are not possible, due to a severely ill-conditioned problem (Chapter 6). Nonetheless, short-term forecasts of epidemics are valuable, and we propose a prediction method which is applicable to a plethora of epidemic models on networks (Chapter 7). As an intermediate step, our prediction method infers the contact network from observations of the epidemic outbreak. Our key result is paradoxical: even though an accurate network reconstruction is impossible, the epidemic outbreak can be predicted accurately. Lastly, we apply our network-inference-based prediction method to the outbreak of COVID-19 (Chapter 8). The third part focusses on spreading phenomena in the human brain. We study the relation between two prominent methods for relating structure and function in the brain: the eigenmode approach and the series expansion approach (Chapter 9). More specifically, we derive closed-form expressions for the optimal coefficients of both approaches, and we demonstrate that the eigenmode approach is preferable to the series expansion approach. Furthermore, we study cross-frequency coupling in magnetoencephalography (MEG) brain networks (Chapter 10). By employing a multilayer network reconstruction method, we show that there are strong one-to-one interactions between the alpha and beta band, and the theta and gamma band. Furthermore, our results show that there are many cross-frequency connections between distant brain regions for theta-gamma coupling.