Robust Weak Chimeras in Oscillator Networks with Delayed Linear and Quadratic Interactions

This is the author's accepted version. The final version is available from APS via the DOI in this recordWe present an approach to generate chimera dynamics (localized frequency synchrony) in oscillator networks with two groups of (at least) two elements using a general method based on delayed interaction with linear and quadratic terms. The coupling design yields robust chimeras through a phase-model based design of the delay and the ratio of linear and quadratic components of the interactions. We demonstrate the method in the Brusselator model and experiments with electrochemical oscillators. The technique opens the way to directly bridge theory for phase models and the dynamics of real-world oscillator networks.CB has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement no. 626111. IZK acknowledges support from National Science Foundation CHE-1465013 grant

Weak Chimeras in Modular Electrochemical Oscillator Networks

This is the final version. Available on open access from Frontiers Media via the DOI in this recordWe investigate the formation of weak chimera states in modular networks of electrochemical oscillations during the electrodissulution of nickel in sulfuric acid. In experiment and simulation, we consider two globally coupled populations of highly non-linear oscillators which are weakly coupled through a collective resistance. Without cross coupling, the system exhibits bistability between a one- and a two-cluster state, whose frequencies are distinct. For weak cross coupling and initial conditions for the one- and two-cluster states for populations 1 and 2, respectively, weak chimera dynamics are generated. The weak chimera state exhibits localized frequency synchrony: The oscillators in each population are frequency-synchronized while the two populations are not. The chimera state is very robust: The behavior is maintained for hundreds of cycles for the rather heterogeneous natural frequencies of the oscillators. The experimental results are confirmed with numerical simulations of a kinetic model for the chemical process. The features of the chimera states are compared to other previously observed chimeras with oscillators close to Hopf bifurcation, coupled with parallel resistances and capacitances or with a non-linear delayed feedback. The experimentally observed synchronization patterns could provide a mechanism for generation of chimeras in biological systems, where robust response is essential

Beyond clustering: mean-field dynamics on networks with arbitrary subgraph composition

Clustering is the propensity of nodes that share a common neighbour to be connected. It is ubiquitous in many networks but poses many modelling challenges. Clustering typically manifests itself by a higher than expected frequency of triangles, and this has led to the principle of constructing networks from such building blocks. This approach has been generalised to networks being constructed from a set of more exotic subgraphs. As long as these are fully connected, it is then possible to derive mean-field models that approximate epidemic dynamics well. However, there are virtually no results for non-fully connected subgraphs. In this paper, we provide a general and automated approach to deriving a set of ordinary differential equations, or mean-field model, that describes, to a high degree of accuracy, the expected values of system-level quantities, such as the prevalence of infection. Our approach offers a previously unattainable degree of control over the arrangement of subgraphs and network characteristics such as classical node degree, variance and clustering. The combination of these features makes it possible to generate families of networks with different subgraph compositions while keeping classical network metrics constant. Using our approach, we show that higher-order structure realised either through the introduction of loops of different sizes or by generating networks based on different subgraphs but with identical degree distribution and clustering, leads to non-negligible differences in epidemic dynamics

Exact deterministic representation of Markovian SIR epidemics on networks with and without loops

In a previous paper Sharkey et al. (Bull Math Biol doi: 10.1007/s11538-013-9923-5, 2012) proved the exactness of closures at the level of triples for Markovian SIR (susceptible-infected-removed) dynamics on tree-like networks. This resulted in a deterministic representation of the epidemic dynamics on the network that can be numerically evaluated. In this paper, we extend this modelling framework to certain classes of networks exhibiting loops. We show that closures where the loops are kept intact are exact, and lead to a simplified and numerically solvable system of ODEs (ordinary-differential-equations). The findings of the paper lead us to a generalisation of closures that are based on partitioning the network around nodes that are cut-vertices (i.e. the removal of such a node leads to the network breaking down into at least two disjointed components or subnetworks). Exploiting this structural property of the network yields some natural closures, where the evolution of a particular state can typically be exactly given in terms of the corresponding or projected states on the subnetworks and the cut-vertex. A byproduct of this analysis is an alternative probabilistic proof of the exactness of the closures for tree-like networks presented in Sharkey et al. (Bull Math Biol doi: 10.1007/s11538-013-9923-5, 2012). In this paper we also elaborate on how the main result can be applied to more realistic networks, for which we write down the ODEs explicitly and compare output from these to results from simulation. Furthermore, we give a general, recipe-like method of how to apply the reduction by closures technique for arbitrary networks, and give an upper bound on the maximum number of equations needed for an exact representation

The impact of contact tracing in clustered populations

The tracing of potentially infectious contacts has become an important part of the control strategy for many infectious diseases, from early cases of novel infections to endemic sexually transmitted infections. Here, we make use of mathematical models to consider the case of partner notification for sexually transmitted infection, however these models are sufficiently simple to allow more general conclusions to be drawn. We show that, when contact network structure is considered in addition to contact tracing, standard “mass action” models are generally inadequate. To consider the impact of mutual contacts (specifically clustering) we develop an improvement to existing pairwise network models, which we use to demonstrate that ceteris paribus, clustering improves the efficacy of contact tracing for a large region of parameter space. This result is sometimes reversed, however, for the case of highly effective contact tracing. We also develop stochastic simulations for comparison, using simple re-wiring methods that allow the generation of appropriate comparator networks. In this way we contribute to the general theory of network-based interventions against infectious disease

Probabilistic predictions of SIS epidemics on networks based on population-level observations

We predict the future course of ongoing susceptible-infected-susceptible (SIS) epidemics on regular, Erdős-Rényi and Barabási-Albert networks. It is known that the contact network influences the spread of an epidemic within a population. Therefore, observations of an epidemic, in this case at the population-level, contain information about the underlying network. This information, in turn, is useful for predicting the future course of an ongoing epidemic. To exploit this in a prediction framework, the exact high-dimensional stochastic model of an SIS epidemic on a network is approximated by a lower-dimensional surrogate model. The surrogate model is based on a birth-and-death process; the effect of the underlying network is described by a parametric model for the birth rates. We demonstrate empirically that the surrogate model captures the intrinsic stochasticity of the epidemic once it reaches a point from which it will not die out. Bayesian parameter inference allows for uncertainty about the model parameters and the class of the underlying network to be incorporated directly into probabilistic predictions. An evaluation of a number of scenarios shows that in most cases the resulting prediction intervals adequately quantify the prediction uncertainty. As long as the population-level data is available over a long-enough period, even if not sampled frequently, the model leads to excellent predictions where the underlying network is correctly identified and prediction uncertainty mainly reflects the intrinsic stochasticity of the spreading epidemic. For predictions inferred from shorter observational periods, uncertainty about parameters and network class dominate prediction uncertainty. The proposed method relies on minimal data at population-level, which is always likely to be available. This, combined with its numerical efficiency, makes the proposed method attractive to be used either as a standalone inference and prediction scheme or in conjunction with other inference and/or predictive models

Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis

An adaptive network model using SIS epidemic propagation with link-type-dependent link activation and deletion is considered. Bifurcation analysis of the pairwise ODE approximation and the network-based stochastic simulation is carried out, showing that three typical behaviours may occur; namely, oscillations can be observed besides disease-free or endemic steady states. The oscillatory behaviour in the stochastic simulations is studied using Fourier analysis, as well as through analysing the exact master equations of the stochastic model. By going beyond simply comparing simulation results to mean-field models, our approach yields deeper insights into the observed phenomena and help better understand and map out the limitations of mean-field models

Phase coherence and attractor geometry of chaotic electrochemical oscillators

2012-2013 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

A monotonic relationship between the variability of the infectious period and final size in pairwise epidemic modelling

For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks