Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping

We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to here as single-vertex-transition (SVT) processes, lead to the analysis of quasi-birth-and-death (QBD) processes in the theory of continuous-time Markov chains. We illustrate how to analyse a number of summary statistics for these processes, such as absorption probabilities and first-passage times. We extend the graph-automorphism lumping approach [Kiss, Miller, Simon, Mathematics of Epidemics on Networks, 2017; Simon, Taylor, Kiss, J. Math. Bio. 62(4), 2011], by providing a matrix-oriented representation of this technique, and show how it can be applied to a very wide range of dynamical processes on networks. This approach can be used not only to solve the master equation of the system, but also to analyse the summary statistics of interest. We also show the interplay between the graph-automorphism lumping approach and the QBD structures when dealing with SVT processes. Finally, we illustrate our theoretical results with examples from the areas of opinion dynamics and mathematical epidemiology

Networks and the epidemiology of infectious disease

The science of networks has revolutionised research into the dynamics of interacting elements. It could be argued that epidemiology in particular has embraced the potential of network theory more than any other discipline. Here we review the growing body of research concerning the spread of infectious diseases on networks, focusing on the interplay between network theory and epidemiology. The review is split into four main sections, which examine: the types of network relevant to epidemiology; the multitude of ways these networks can be characterised; the statistical methods that can be applied to infer the epidemiological parameters on a realised network; and finally simulation and analytical methods to determine epidemic dynamics on a given network. Given the breadth of areas covered and the ever-expanding number of publications, a comprehensive review of all work is impossible. Instead, we provide a personalised overview into the areas of network epidemiology that have seen the greatest progress in recent years or have the greatest potential to provide novel insights. As such, considerable importance is placed on analytical approaches and statistical methods which are both rapidly expanding fields. Throughout this review we restrict our attention to epidemiological issues

Stochastic spreading on complex networks

Complex interacting systems are ubiquitous in nature and society. Computational modeling of these systems is, therefore, of great relevance for science and engineering. Complex networks are common representations of these systems (e.g., friendship networks or road networks). Dynamical processes (e.g., virus spreading, traffic jams) that evolve on these networks are shaped and constrained by the underlying connectivity. This thesis provides numerical methods to study stochastic spreading processes on complex networks. We consider the processes as inherently probabilistic and analyze their behavior through the lens of probability theory. While powerful theoretical frameworks (like the SIS-epidemic model and continuous-time Markov chains) already exist, their analysis is computationally challenging. A key contribution of the thesis is to ease the computational burden of these methods. Particularly, we provide novel methods for the efficient stochastic simulation of these processes. Based on different simulation studies, we investigate techniques for optimal vaccine distribution and critically address the usage of mathematical models during the Covid-19 pandemic. We also provide model-reduction techniques that translate complicated models into simpler ones that can be solved without resorting to simulations. Lastly, we show how to infer the underlying contact data from node-level observations.Komplexe, interagierende Systeme sind in Natur und Gesellschaft allgegenwärtig. Die computergestützte Modellierung dieser Systeme ist daher von immenser Bedeutung für Wissenschaft und Technik. Netzwerke sind eine gängige Art, diese Systeme zu repräsentieren (z. B. Freundschaftsnetzwerke, Straßennetze). Dynamische Prozesse (z. B. Epidemien, Staus), die sich auf diesen Netzwerken ausbreiten, werden durch die spezifische Konnektivität geformt. In dieser Arbeit werden numerische Methoden zur Untersuchung stochastischer Ausbreitungsprozesse in komplexen Netzwerken entwickelt. Wir betrachten die Prozesse als inhärent probabilistisch und analysieren ihr Verhalten nach wahrscheinlichkeitstheoretischen Fragestellungen. Zwar gibt es bereits theoretische Grundlagen und Paradigmen (wie das SIS-Epidemiemodell und zeitkontinuierliche Markov-Ketten), aber ihre Analyse ist rechnerisch aufwändig. Ein wesentlicher Beitrag dieser Arbeit besteht darin, die Rechenlast dieser Methoden zu verringern. Wir erforschen Methoden zur effizienten Simulation dieser Prozesse. Anhand von Simulationsstudien untersuchen wir außerdem Techniken für optimale Impfstoffverteilung und setzen uns kritisch mit der Verwendung mathematischer Modelle bei der Covid-19-Pandemie auseinander. Des Weiteren führen wir Modellreduktionen ein, mit denen komplizierte Modelle in einfachere umgewandelt werden können. Abschließend zeigen wir, wie man von Beobachtungen einzelner Knoten auf die zugrunde liegende Netzwerkstruktur schließt

Advances in Rule-based Modeling: Compartments, Energy, and Hybrid Simulation, with Application to Sepsis and Cell Signaling

Biological systems are commonly modeled as reaction networks, which describe the system at the resolution of biochemical species. Cellular systems, however, are governed by events at a finer scale: local interactions among macromolecular domains. The multi-domain structure of macromolecules, combined with the local nature of interactions, can lead to a combinatorial explosion that pushes reaction network methods to their limits. As an alternative, rule-based models (RBMs) describe the domain-based structure and local interactions found in biological systems. Molecular complexes are represented by graphs: functional domains as vertices, macromolecules as groupings of vertices, and molecular bonding as edges. Reaction rules, which describe classes of reactions, govern local modifications to molecular graphs, such as binding, post-translational modification, and degradation. RBMs can be transformed to equivalent reaction networks and simulated by differential or stochastic methods, or simulated directly with a network-free approach that avoids the problem of combinatorial complexity. Although RBMs and network-free methods resolve many problems in systems modeling, challenges remain. I address three challenges here: (i) managing model complexity due to cooperative interactions, (ii) representing biochemical systems in the compartmental setting of cells and organisms, and (iii) reducing the memory burden of large-scale network-free simulations. First, I present a general theory of energy-based modeling within the BioNetGen framework. Free energy is computed under a pattern-based formalism, and contextual variations within reaction classes are enumerated automatically. Next, I extend the BioNetGen language to permit description of compartmentalized biochemical systems, with treatment of volumes, surfaces and transport. Finally, a hybrid particle/population method is developed to reduce memory requirements of network-free simulations. All methods are implemented and available as part of BioNetGen. The remainder of this work presents an application to sepsis and inflammation. A multi-organ model of peritoneal infection and systemic inflammation is constructed and calibrated to experiment. Extra-corporeal blood purification, a potential treatment for sepsis, is explored in silico. Model simulations demonstrate that removal of blood cytokines and chemokines is a sufficient mechanism for improved survival in sepsis. However, differences between model predictions and the latest experimental data suggest directions for further exploration

Lifted Bayesian filtering in multi-entity systems

This thesis focuses on Bayesian filtering for systems that consist of multiple, interacting entites (e.g. agents or objects), which can naturally be described by Multiset Rewriting Systems (MRSs). The main insight is that the state space that is underling an MRS exhibits a certain symmetry, which can be exploited to increase inference efficiency. We provide an efficient, lifted filtering algorithm, which is able to achieve a factorial reduction in space and time complexity, compared to conventional, ground filtering.Diese Arbeit betrachtet Bayes'sche Filter in Systemen, die aus mehreren, interagierenden Entitäten (z.B. Agenten oder Objekten) bestehen. Die Systemdynamik solcher Systeme kann auf natürliche Art durch Multiset Rewriting Systems (MRS) spezifiziert werden. Die wesentliche Erkenntnis ist, dass der Zustandraum Symmetrien aufweist, die ausgenutzt werden können, um die Effizienz der Inferenz zu erhöhen. Wir führen einen effizienten, gelifteten Filter-Algorithmus ein, dessen Zeit- und Platzkomplexität gegenüber dem grundierten Algorithmus um einen faktoriellen Faktor reduziert ist