Scale-free avalanches in arrays of FitzHugh-Nagumo oscillators.

The activity in the brain cortex remarkably shows a simultaneous presence of robust collective oscillations and neuronal avalanches, where intermittent bursts of pseudo-synchronous spiking are interspersed with long periods of quiescence. The mechanisms allowing for such coexistence are still a matter of an intensive debate. Here, we demonstrate that avalanche activity patterns can emerge in a rather simple model of an array of diffusively coupled neural oscillators with multiple timescale local dynamics in the vicinity of a canard transition. The avalanches coexist with the fully synchronous state where the units perform relaxation oscillations. We show that the mechanism behind the avalanches is based on an inhibitory effect of interactions, which may quench the spiking of units due to an interplay with the maximal canard. The avalanche activity bears certain heralds of criticality, including scale-invariant distributions of event sizes. Furthermore, the system shows increased sensitivity to perturbations, manifested as critical slowing down and reduced resilience.

Perspectives on adaptive dynamical systems.

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems, such as the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges and give perspectives on future research directions, looking to inspire interdisciplinary approaches.

Transitional cluster dynamics in a model for delay-coupled chemical oscillators.

Cluster synchronization is a fundamental phenomenon in systems of coupled oscillators. Here, we investigate clustering patterns that emerge in a unidirectional ring of four delay-coupled electrochemical oscillators. A voltage parameter in the experimental setup controls the onset of oscillations via a Hopf bifurcation. For a smaller voltage, the oscillators exhibit simple, so-called primary, clustering patterns, where all phase differences between each set of coupled oscillators are identical. However, upon increasing the voltage, secondary states, where phase differences differ, are detected, in addition to the primary states. Previous work on this system saw the development of a mathematical model that explained how the existence, stability, and common frequency of the experimentally observed cluster states could be accurately controlled by the delay time of the coupling. In this study, we revisit the mathematical model of the electrochemical oscillators in order to address open questions by means of bifurcation analysis. Our analysis reveals how the stable cluster states, corresponding to experimental observations, lose their stability via an assortment of bifurcation types. The analysis further reveals complex interconnectedness between branches of different cluster types. We find that each secondary state provides a continuous transition between certain primary states. These connections are explained by studying the phase space and parameter symmetries of the respective states. Furthermore, we show that it is only for a larger value of the voltage parameter that the branches of secondary states develop intervals of stability. For a smaller voltage, all the branches of secondary states are completely unstable and are, therefore, hidden to experimentalists.

Emergence of synergistic and competitive pathogens in a coevolutionary spreading model.

Cooperation and competition between pathogens can alter the amount of individuals affected by a coinfection. Nonetheless, the evolution of the pathogens' behavior has been overlooked. Here, we consider a coevolutionary model where the simultaneous spreading is described by a two-pathogen susceptible-infected-recovered model in an either synergistic or competitive manner. At the end of each epidemic season, the pathogens species reproduce according to their fitness that, in turn, depends on the payoff accumulated during the spreading season in a hawk-and-dove game. This coevolutionary model displays a rich set of features. Specifically, the evolution of the pathogens' strategy induces abrupt transitions in the epidemic prevalence. Furthermore, we observe that the long-term dynamics results in a single, surviving pathogen species, and that the cooperative behavior of pathogens can emerge even under unfavorable conditions.

Deterministic and stochastic effects in spreading dynamics: A case study of bovine viral diarrhea.

Bovine viral diarrhea (BVD) is a disease in cattle with complex transmission dynamics that causes substantial economic losses and affects animal welfare. The infection can be transient or persistent. The mostly asymptomatic persistently infected hosts are the main source for transmission of the virus. This characteristic makes it difficult to control the spreading of BVD. We develop a deterministic compartmental model for the spreading dynamics of BVD within a herd and derive the basic reproduction number. This epidemiological quantity indicates that identification and removal of persistently infected animals is a successful control strategy if the transmission rate of transiently infected animals is small. Removing persistently infected animals from the herd at birth results in recurrent outbreaks with decreasing peak prevalence. We propose a stochastic version of the compartmental model that includes stochasticity in the transmission parameters. This stochasticity leads to sustained oscillations in cases where the deterministic model predicts oscillations with decreasing amplitude. The results provide useful information for the design of control strategies.

Individual-based model for the control of Bovine Viral Diarrhea spread in livestock trade networks.

Bovine Viral Diarrhea (BVD) is a cattle disease that causes substantial financial losses, in particular to the dairy industry. Hence, several countries including Germany introduced compulsory disease control programs. For the case of Germany in particular, all animals had to be tested and persistently infected animals (PI animals) were removed from the population. The program was successful in reducing the number of PI animals, but was overtly expensive. Alternative approaches were therefore discussed to eliminate the remaining PI animals and alter the testing system in order to reduce costs. Contributing to these efforts, we developed an agent-based model that aimed to cover all relevant aspects of the disease biology and would allow to evaluate different control strategies. For the biological part of the infection spread, the model includes horizontal and vertical transmission, transient and persistent infections. Moreover, several control strategies including import of animals, trade restrictions, vaccination, as well as various testing schemes were included. The model was furthermore defined to be stochastic, event-driven and hierarchical, with cattle movements as the main route of spreading between farms. For the spread within farms, we included susceptible-infected-recovered (SIR) dynamics with an additional permanently infectious class. The interaction between the farms was described by a supply and demand farm manager mechanism governing the network structure and dynamics. Additionally, we carried out a sensitivity analysis of the input parameters to study the impact of extreme values on the model. Since the population size in the model is limited, we tested the influence of the initial population size on the model results. Our results showed that the model could accurately describe the dynamics of the disease in the presence and absence of disease control. Although we developed the model for the spread of BVD, it may be adapted to similar diseases of cattle and swine.

A metapopulation network model for the spreading of SARS-CoV-2: Case study for Ireland.

We present preliminary results on an all-Ireland network modelling approach to simulate the spreading the Severe Acute Respiratory Syndrome CoronaVirus 2 (SARS-CoV-2), commonly known as the coronavirus. In the model, nodes correspond to locations or communities that are connected by links indicating travel and commuting between different locations. While this proposed modelling framework can be applied on all levels of spatial granularity and different countries, we consider Ireland as a case study. The network comprises 3440 electoral divisions (EDs) of the Republic of Ireland and 890 superoutput areas (SOAs) for Northern Ireland, which corresponds to local administrative units below the NUTS 3 regions. The local dynamics within each node follows a phenomenological SIRX compartmental model including classes of Susceptibles, Infected, Recovered and Quarantined (X) inspired from Science 368, 742 (2020). For better comparison to empirical data, we extended that model by a class of Deaths. We consider various scenarios including the 5-phase roadmap for Ireland. In addition, as proof of concept, we investigate the effect of dynamic interventions that aim to keep the number of infected below a given threshold. This is achieved by dynamically adjusting containment measures on a national scale, which could also be implemented at a regional (county) or local (ED/SOA) level. We find that - in principle - dynamic interventions are capable to limit the impact of future waves of outbreaks, but on the downside, in the absence of a vaccine, such a strategy can last several years until herd immunity is reached.

Early warning of infectious disease outbreaks on cattle-transport networks.

Surveillance of infectious diseases in livestock is traditionally carried out at the farms, which are the typical units of epidemiological investigations and interventions. In Central and Western Europe, high-quality, long-term time series of animal transports have become available and this opens the possibility to new approaches like sentinel surveillance. By comparing a sentinel surveillance scheme based on markets to one based on farms, the primary aim of this paper is to identify the smallest set of sentinel holdings that would reliably and timely detect emergent disease outbreaks in Swiss cattle. Using a data-driven approach, we simulate the spread of infectious diseases according to the reported or available daily cattle transport data in Switzerland over a four year period. Investigating the efficiency of surveillance at either market or farm level, we find that the most efficient early warning surveillance system [the smallest set of sentinels that timely and reliably detect outbreaks (small outbreaks at detection, short detection delays)] would be based on the former, rather than the latter. We show that a detection probability of 86% can be achieved by monitoring all 137 markets in the network. Additional 250 farm sentinels-selected according to their risk-need to be placed under surveillance so that the probability of first hitting one of these farm sentinels is at least as high as the probability of first hitting a market. Combining all markets and 1000 farms with highest risk of infection, these two levels together will lead to a detection probability of 99%. We conclude that the design of animal surveillance systems greatly benefits from the use of the existing abundant and detailed animal transport data especially in the case of highly dynamic cattle transport networks. Sentinel surveillance approaches can be tailored to complement existing farm risk-based and syndromic surveillance approaches.

Interplay between competitive and cooperative interactions in a three-player pathogen system.

In ecological systems, heterogeneous interactions between pathogens take place simultaneously. This occurs, for instance, when two pathogens cooperate, while at the same time, multiple strains of these pathogens co-circulate and compete. Notable examples include the cooperation of human immunodeficiency virus with antibiotic-resistant and susceptible strains of tuberculosis or some respiratory infections with Streptococcus pneumoniae strains. Models focusing on competition or cooperation separately fail to describe how these concurrent interactions shape the epidemiology of such diseases. We studied this problem considering two cooperating pathogens, where one pathogen is further structured in two strains. The spreading follows a susceptible-infected-susceptible process and the strains differ in transmissibility and extent of cooperation with the other pathogen. We combined a mean-field stability analysis with stochastic simulations on networks considering both well-mixed and structured populations. We observed the emergence of a complex phase diagram, where the conditions for the less transmissible, but more cooperative strain to dominate are non-trivial, e.g. non-monotonic boundaries and bistability. Coupled with community structure, the presence of the cooperative pathogen enables the coexistence between strains by breaking the spatial symmetry and dynamically creating different ecological niches. These results shed light on ecological mechanisms that may impact the epidemiology of diseases of public health concern.

An Epidemiological and Economic Simulation Model to Evaluate Strategies for the Control of Bovine Virus Diarrhea in Germany.

Models can be used to plan, evaluate, and improve programs for animal disease control. In Germany, a nationwide compulsory program to eradicate Bovine viral diarrhea (BVD) is in force since January 2011. As it is associated with substantial expenditures, the program is currently under revision. To provide the basis for a science-based decision on the future course of BVD control in Germany, we evaluated 13 scenarios (sc1-13) with respect to the chance of reaching freedom from disease and their economic implications for a period of 20 years (2011-2030). To simulate the impact of different control strategies on disease dynamics, a disease spread model was developed. To estimate the effects of a transient infection (TI) on animal level, a gross margin analysis was performed. To assess the value of cattle that died prematurely, a valuation model was used. Finally, an economic model was developed to perform a cost-benefit analysis and to compare each control scenario with a baseline setting with no BVD control. Costs comprised the expenditures for diagnostics, vaccination, preventive culling, and trade restrictions. Benefits were animal and production losses avoided by having control measures in place. The results show that reducing the PI prevalence on animal level to 0% is only feasible in scenarios that combine antigen or antibody testing with compulsory vaccination. All other scenarios, i.e., those based exclusively on a "test and cull" approach, including the current control program, will, according to the model, not achieve freedom of BVD by 2030. On the other hand, none of the scenarios that may lead to complete BVD eradication is economically attractive [benefit-cost ratio (BCR) between 0.64 and 0.94]. The average direct costs of BVD in Germany are estimated at 113 million Euros per year (34-402 million Euros), corresponding to 28.3 million Euros per million animals. Only the concepts of the former and the current national BVD control program ("ear tag testing and culling") may reduce the BVD prevalence to 0.01% with an acceptable BCR (net present value of 222 and 238 million Euros, respectively, with a BCR of 1.22 and 1.24).

Accelerating dynamics of collective attention.

With news pushed to smart phones in real time and social media reactions spreading across the globe in seconds, the public discussion can appear accelerated and temporally fragmented. In longitudinal datasets across various domains, covering multiple decades, we find increasing gradients and shortened periods in the trajectories of how cultural items receive collective attention. Is this the inevitable conclusion of the way information is disseminated and consumed? Our findings support this hypothesis. Using a simple mathematical model of topics competing for finite collective attention, we are able to explain the empirical data remarkably well. Our modeling suggests that the accelerating ups and downs of popular content are driven by increasing production and consumption of content, resulting in a more rapid exhaustion of limited attention resources. In the interplay with competition for novelty, this causes growing turnover rates and individual topics receiving shorter intervals of collective attention.

Characterizing Complex Networks Using Entropy-Degree Diagrams: Unveiling Changes in Functional Brain Connectivity Induced by Ayahuasca.

With the aim of further advancing the understanding of the human brain's functional connectivity, we propose a network metric which we term the geodesic entropy. This metric quantifies the Shannon entropy of the distance distribution to a specific node from all other nodes. It allows to characterize the influence exerted on a specific node considering statistics of the overall network structure. The measurement and characterization of this structural information has the potential to greatly improve our understanding of sustained activity and other emergent behaviors in networks. We apply this method to study how the psychedelic infusion Ayahuasca affects the functional connectivity of the human brain in resting state. We show that the geodesic entropy is able to differentiate functional networks of the human brain associated with two different states of consciousness in the awaking resting state: (i) the ordinary state and (ii) a state altered by ingestion of the Ayahuasca. The functional brain networks from subjects in the altered state have, on average, a larger geodesic entropy compared to the ordinary state. Finally, we discuss why the geodesic entropy may bring even further valuable insights into the study of the human brain and other empirical networks.

Erratum: Chimera patterns in two-dimensional networks of coupled neurons [Phys. Rev. E 95, 032224 (2017)].

This corrects the article DOI: 10.1103/PhysRevE.95.032224.

Tracking online topics over time: understanding dynamic hashtag communities.

BACKGROUND: Hashtags are widely used for communication in online media. As a condensed version of information, they characterize topics and discussions. For their analysis, we apply methods from network science and propose novel tools for tracing their dynamics in time-dependent data. The observations are characterized by bursty behaviors in the increases and decreases of hashtag usage. These features can be reproduced with a novel model of dynamic rankings. HASHTAG COMMUNITIES IN TIME: We build temporal and weighted co-occurrence networks from hashtags. On static snapshots, we infer the community structure using customized methods. On temporal networks, we solve the bipartite matching problem of detected communities at subsequent timesteps by taking into account higher-order memory. This results in a matching protocol that is robust toward temporal fluctuations and instabilities of the static community detection. The proposed methodology is broadly applicable and its outcomes reveal the temporal behavior of online topics. MODELING TOPIC-DYNAMICS: We consider the size of the communities in time as a proxy for online popularity dynamics. We find that the distributions of gains and losses, as well as the interevent times are fat-tailed indicating occasional, but large and sudden changes in the usage of hashtags. Inspired by typical website designs, we propose a stochastic model that incorporates a ranking with respect to a time-dependent prestige score. This causes occasional cascades of rank shift events and reproduces the observations with good agreement. This offers an explanation for the observed dynamics, based on characteristic elements of online media.

Robustness of chimera states in nonlocally coupled networks of nonidentical logistic maps.

We investigate the dynamics of nonlocally coupled time-discrete maps with emphasis on the occurrence and robustness of chimera states. These peculiar, hybrid states are characterized by a coexistence of coherent and incoherent regions. We consider logistic maps coupled on a one-dimensional ring with finite coupling radius. Domains of chimera existence form different tongues in the parameter space of coupling range and coupling strength. For a sufficiently large coupling strength, each tongue refers to a wave number describing the structure of the spatial profile. We also analyze the period-adding scheme within these tongues and multiplicity of period solutions. Furthermore, we study the robustness of chimeras with respect to parameter inhomogeneities and find that these states persist for different widths of the parameter distribution. Finally, we explore the spatial structure of the chimera using a spatial correlation function.

Chimera states in multi-strain epidemic models with temporary immunity.

We investigate a time-delayed epidemic model for multi-strain diseases with temporary immunity. In the absence of cross-immunity between strains, dynamics of each individual strain exhibit emergence and annihilation of limit cycles due to a Hopf bifurcation of the endemic equilibrium, and a saddle-node bifurcation of limit cycles depending on the time delay associated with duration of temporary immunity. Effects of all-to-all and non-local coupling topologies are systematically investigated by means of numerical simulations, and they suggest that cross-immunity is able to induce a diverse range of complex dynamical behaviors and synchronization patterns, including discrete traveling waves, solitary states, and amplitude chimeras. Interestingly, chimera states are observed for narrower cross-immunity kernels, which can have profound implications for understanding the dynamics of multi-strain diseases.

Identifying and modeling the structural discontinuities of human interactions.

The idea of a hierarchical spatial organization of society lies at the core of seminal theories in human geography that have strongly influenced our understanding of social organization. Along the same line, the recent availability of large-scale human mobility and communication data has offered novel quantitative insights hinting at a strong geographical confinement of human interactions within neighboring regions, extending to local levels within countries. However, models of human interaction largely ignore this effect. Here, we analyze several country-wide networks of telephone calls - both, mobile and landline - and in either case uncover a systematic decrease of communication induced by borders which we identify as the missing variable in state-of-the-art models. Using this empirical evidence, we propose an alternative modeling framework that naturally stylizes the damping effect of borders. We show that this new notion substantially improves the predictive power of widely used interaction models. This increases our ability to understand, model and predict social activities and to plan the development of infrastructures across multiple scales.

Chimera patterns in two-dimensional networks of coupled neurons.

We discuss synchronization patterns in networks of FitzHugh-Nagumo and leaky integrate-and-fire oscillators coupled in a two-dimensional toroidal geometry. A common feature between the two models is the presence of fast and slow dynamics, a typical characteristic of neurons. Earlier studies have demonstrated that both models when coupled nonlocally in one-dimensional ring networks produce chimera states for a large range of parameter values. In this study, we give evidence of a plethora of two-dimensional chimera patterns of various shapes, including spots, rings, stripes, and grids, observed in both models, as well as additional patterns found mainly in the FitzHugh-Nagumo system. Both systems exhibit multistability: For the same parameter values, different initial conditions give rise to different dynamical states. Transitions occur between various patterns when the parameters (coupling range, coupling strength, refractory period, and coupling phase) are varied. Many patterns observed in the two models follow similar rules. For example, the diameter of the rings grows linearly with the coupling radius.

Effective distances for epidemics spreading on complex networks.

We show that the recently introduced logarithmic metrics used to predict disease arrival times on complex networks are approximations of more general network-based measures derived from random walks theory. Using the daily air-traffic transportation data we perform numerical experiments to compare the infection arrival time with this alternative metric that is obtained by accounting for multiple walks instead of only the most probable path. The comparison with direct simulations reveals a higher correlation compared to the shortest-path approach used previously. In addition our method allows to connect fundamental observables in epidemic spreading with the cumulant-generating function of the hitting time for a Markov chain. Our results provides a general and computationally efficient approach using only algebraic methods.