Lumping of Degree-Based Mean Field and Pair Approximation Equations for Multi-State Contact Processes

Contact processes form a large and highly interesting class of dynamic processes on networks, including epidemic and information spreading. While devising stochastic models of such processes is relatively easy, analyzing them is very challenging from a computational point of view, particularly for large networks appearing in real applications. One strategy to reduce the complexity of their analysis is to rely on approximations, often in terms of a set of differential equations capturing the evolution of a random node, distinguishing nodes with different topological contexts (i.e., different degrees of different neighborhoods), like degree-based mean field (DBMF), approximate master equation (AME), or pair approximation (PA). The number of differential equations so obtained is typically proportional to the maximum degree kmax of the network, which is much smaller than the size of the master equation of the underlying stochastic model, yet numerically solving these equations can still be problematic for large kmax. In this paper, we extend AME and PA, which has been proposed only for the binary state case, to a multi-state setting and provide an aggregation procedure that clusters together nodes having similar degrees, treating those in the same cluster as indistinguishable, thus reducing the number of equations while preserving an accurate description of global observables of interest. We also provide an automatic way to build such equations and to identify a small number of degree clusters that give accurate results. The method is tested on several case studies, where it shows a high level of compression and a reduction of computational time of several orders of magnitude for large networks, with minimal loss in accuracy.Comment: 16 pages with the Appendi

Lumping the approximate master equation for multistate processes on complex networks

Complex networks play an important role in human society and in nature. Stochastic multistate processes provide a powerful framework to model a variety of emerging phenomena such as the dynamics of an epidemic or the spreading of information on complex networks. In recent years, mean-field type approximations gained widespread attention as a tool to analyze and understand complex network dynamics. They reduce the model\u2019s complexity by assuming that all nodes with a similar local structure behave identically. Among these methods the approximate master equation (AME) provides the most accurate description of complex networks\u2019 dynamics by considering the whole neighborhood of a node. The size of a typical network though renders the numerical solution of multistate AME infeasible. Here, we propose an efficient approach for the numerical solution of the AME that exploits similarities between the differential equations of structurally similar groups of nodes. We cluster a large number of similar equations together and solve only a single lumped equation per cluster. Our method allows the application of the AME to real-world networks, while preserving its accuracy in computing estimates of global network properties, such as the fraction of nodes in a state at a given time

Moment Closure - A Brief Review

Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one "moment", a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing "higher-order moments" in terms of "lower-order moments". In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in mathematics, physics, chemistry and quantitative biolog

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

Approximations and inference of dynamics on networks

The study of dynamics on networks is a subject area that has wide-reaching applications in areas such as epidemic outbreaks, rumour spreading, and innovation diffusion. In this thesis I look at how to both approximate and infer these dynamics. Specifically, I first explore mean-field approximations for SIS epidemic dynamics. I outline several established approximations of varying complexity, before investigating how their accuracy depends on the network and dynamical parameters. Next, I use a method called approximate lumping to coarse-grain SIS dynamics, and I show how this method allows us to derive mean-field approximations directly from the full master equation description, rather than via ad hoc moment closures, as is common. Finally, I consider inference of network dynamic parameters on multilayer networks. I focus on a case study of SIS dynamics occurring on a two-layer network, where the dynamics on one of the layers is unobserved or “hidden”. My goal is to estimate the SIS parameters, assuming I only have data about the events occurring on the visible layer. To do this I develop several simpler approximate models of the dynamics which have tractable likelihoods, and then use Markov chain Monte Carlo routines to infer the most likely parameters for these approximate dynamics

esys-Escript User’s Guide: Solving Partial Differential Equations with Escript and Finley Release - 3.3.1 (r4302)

esys.escript is a python-based environment for implementing mathematical models, in particular those based on coupled, non-linear, time-dependent partial differential equations. It consists of five major components • esys.escript core library • finite element solver esys.finley (which uses fast vendor-supplied solvers or our paso linear solver library) • the meshing interface esys.pycad • a model library. • an inversion library. The current version supports parallelization through both MPI for distributed memory and OpenMP for distributed shared memory. In this release there are a number of small changes which are not backwards compatible. Please see Appendix B to see if your scripts will be affected. If you use this software in your research, then we would appreciate (but do not require) a citation. Some relevant references can be found in Appendix D. For Python3 support, see Appendix E

esys-Escript User’s Guide: Solving Partial Differential Equations with Escript and Finley Release - 3.2.1 (r3613)

esys.escript is a python-based environment for implementing mathematical models, in particular those based on coupled, non-linear, time-dependent partial differential equations. It consists of four major components • esys.escript core library • finite element solver esys.finley (which uses fast vendor-supplied solvers or our paso linear solver library) • the meshing interface esys.pycad • a model library. The current version supports parallelization through both MPI for distributed memory and OpenMP for distributed shared memory. Please see Chapter 2 for changes to the way to launch esys.escript scripts. For more info on this and other changes from previous releases see Appendix B. If you use this software in your research, then we would appreciate (but do not require) a citation. Some relevant references can be found in Appendix D

Exact and approximate epidemic models on networks: theory and applications

This thesis is concerned with modelling the spread of diseases amongst host populations and the epidemics that result from this process. We are primarily interested in how networks can be used to model the various heterogeneities observable in real-world populations. Firstly, we start with the full system of Kolmogorov/master equations for a simple Susceptible-Infected-Susceptible (SIS) type epidemic on an arbitrary contact network. From this general framework, we rigorously derive sets of ODEs that describe the exact dynamics of the expected number of individuals and pairs of individuals. We proceed to use moment closure techniques to close these hierarchical systems of ODEs, by approximating higher order moments in terms of lower order moments. We prove that the simple first order mean-field approximation becomes exact in the limit of a large, fully-connected network. We then investigate how well two different pairwise approximations capture the topological features of theoretical networks generated using different algorithms. We then introduce the effective degree modelling framework and propose a model for SIS epidemics on dynamic contact networks by accounting for random link activation and deletion. We show that results from the resulting set of ODEs agrees well with results from stochastic simulations, both in describing the evolution of the network and the disease. Furthermore, we derive an analytic calculation of the stability of the disease-free steady state and explore the validity of such a measure in the context of a dynamically evolving contact network. Finally, we move on to derive a system of ODEs that describes the interacting dynamics of a disease and information relating to the disease. We allow individuals to become responsive in light of received information and, thus, reduce the rate at which they become infected. We consider the effectiveness of different routes of information transmission (such as peer-to-peer communication or mass media campaigns) in slowing or preventing the spread of a disease. Finally, we use a range of modelling techniques to investigate the spread of disease within sheep flocks. We use field data to construct weighted contact networks for flocks of sheep to account for seasonal changes of the flock structure as lambs are born and eventually become weaned. We construct a range of network and ODE models that are designed to investigate the effect of link-weight heterogeneity on the spread of disease