Epidemics and percolation in small-world networks

How Behaviour and the Environment Influence Transmission in Mobile Groups

The movement of individuals living in groups leads to the formation of physical interaction networks over which signals such as information or disease can be transmitted. Direct contacts represent the most obvious opportunities for a signal to be transmitted. However, because signals that persist after being deposited into the environment may later be acquired by other group members, indirect environmentally-mediated transmission is also possible. To date, studies of signal transmission within groups have focused on direct physical interactions and ignored the role of indirect pathways. Here, we use an agent-based model to study how the movement of individuals and characteristics of the signal being transmitted modulate transmission. By analysing the dynamic interaction networks generated from these simulations, we show that the addition of indirect pathways speeds up signal transmission, while the addition of physically-realistic collisions between individuals in densely packed environments hampers it. Furthermore, the inclusion of spatial biases that induce the formation of individual territories, reveals the existence of a trade-off such that optimal signal transmission at the group level is only achieved when territories are of intermediate sizes. Our findings provide insight into the selective pressures guiding the evolution of behavioural traits in natural groups, and offer a means by which multi-agent systems can be engineered to achieve desired transmission capabilities.

Building Ventilation as an Effective Disease Intervention Strategy in a Dense Indoor Contact Network in an Ideal City

Emerging diseases may spread rapidly through dense and large urban contact networks, especially they are transmitted by the airborne route, before new vaccines can be made available. Airborne diseases may spread rapidly as people visit different indoor environments and are in frequent contact with others. We constructed a simple indoor contact model for an ideal city with 7 million people and 3 million indoor spaces, and estimated the probability and duration of contact between any two individuals during one day. To do this, we used data from actual censuses, social behavior surveys, building surveys, and ventilation measurements in Hong Kong to define eight population groups and seven indoor location groups. Our indoor contact model was integrated with an existing epidemiological Susceptible, Exposed, Infectious, and Recovered (SEIR) model to estimate disease spread and with the Wells-Riley equation to calculate local infection risks, resulting in an integrated indoor transmission network model. This model was used to estimate the probability of an infected individual infecting others in the city and to study the disease transmission dynamics. We predicted the infection probability of each sub-population under different ventilation systems in each location type in the case of a hypothetical airborne disease outbreak, which is assumed to have the same natural history and infectiousness as smallpox. We compared the effectiveness of controlling ventilation in each location type with other intervention strategies. We conclude that increasing building ventilation rates using methods such as natural ventilation in classrooms, offices, and homes is a relatively effective strategy for airborne diseases in a large city.

Equivalence of several generalized percolation models on networks

In recent years, many variants of percolation have been used to study network structure and the behavior of processes spreading on networks. These include bond percolation, site percolation, k-core percolation, bootstrap percolation, the generalized epidemic process, and the Watts threshold model (WTM). We show that-except for bond percolation-each of these processes arises as a special case of the WTM, and bond percolation arises from a small modification. In fact "heterogeneous k-core percolation," a corresponding "heterogeneous bootstrap percolation" model, and the generalized epidemic process are completely equivalent to one another and the WTM. We further show that a natural generalization of the WTM in which individuals "transmit" or "send a message" to their neighbors with some probability less than 1 can be reformulated in terms of the WTM, and so this apparent generalization is in fact not more general. Finally, we show that in bond percolation, finding the set of nodes in the component containing a given node is equivalent to finding the set of nodes activated if that node is initially activated and the node thresholds are chosen from the appropriate distribution. A consequence of these results is that mathematical techniques developed for the WTM apply to these other models as well, and techniques that were developed for some particular case may in fact apply much more generally.

Crossover phenomena of percolation transition in evolution networks with hybrid attachment

A first-order percolation transition, called explosive percolation, was recently discovered in evolution networks with random edge selection under a certain restriction. For many real world networks, the mechanism of preferential attachment plays a significant role in the formation of heterogeneous structures, but the network percolation in evolution process with preferential attachment has not yet been concerned. We propose a tunable network percolation model by introducing a hybrid mechanism of edge selection into the Bohman-Frieze-Wormald model, in which a parameter adjusts the relative weights between random and preferential selections. A large number of simulations indicate that there exist crossover phenomena of percolation transition by adjusting the parameter in the evolution processes. When the strategy of selecting a candidate edge is dominated by random selection, a single discontinuous percolation transition occurs. When a candidate edge is selected more preferentially based on nodes degree, the size of the largest component undergoes multiple discontinuous jumps, which exhibits a peculiar difference from the network percolation of random selection with a certain restriction. Besides, the percolation transition becomes continuous when the candidate edge is selected completely preferentially.

Graphs with specified degree distributions, simple epidemics, and local vaccination strategies

Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R0, the basic reproduction number, and τ0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine Rv, the reproduction number, and τv, the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

Graphs with specified degree distributions, simple epidemics, and local vaccination strategies

Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R 0, the basic reproduction number, and τ0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine R v , the reproduction number, and τ v , the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

From neurons to epidemics: How trophic coherence affects spreading processes

Trophic coherence, a measure of the extent to which the nodes of a directed network are organised in levels, has recently been shown to be closely related to many structural and dynamical aspects of complex systems, including graph eigenspectra, the prevalence or absence of feedback cycles, and linear stability. Furthermore, non-trivial trophic structures have been observed in networks of neurons, species, genes, metabolites, cellular signalling, concatenated words, P2P users, and world trade. Here, we consider two simple yet apparently quite different dynamical models-one a susceptible-infected-susceptible epidemic model adapted to include complex contagion and the other an Amari-Hopfield neural network-and show that in both cases the related spreading processes are modulated in similar ways by the trophic coherence of the underlying networks. To do this, we propose a network assembly model which can generate structures with tunable trophic coherence, limiting in either perfectly stratified networks or random graphs. We find that trophic coherence can exert a qualitative change in spreading behaviour, determining whether a pulse of activity will percolate through the entire network or remain confined to a subset of nodes, and whether such activity will quickly die out or endure indefinitely. These results could be important for our understanding of phenomena such as epidemics, rumours, shocks to ecosystems, neuronal avalanches, and many other spreading processes.

Correlated Observations, the Law of Small Numbers and Bank Runs

Empirical descriptions and studies suggest that generally depositors observe a sample of previous decisions before deciding if to keep their funds deposited or to withdraw them. These observed decisions may exhibit different degrees of correlation across depositors. In our model depositors decide sequentially and are assumed to follow the law of small numbers in the sense that they believe that a bank run is underway if the number of observed withdrawals in their sample is large. Theoretically, with highly correlated samples and infinite depositors runs occur with certainty, while with random samples it needs not be the case, as for many parameter settings the likelihood of bank runs is zero. We investigate the intermediate cases and find that i) decreasing the correlation and ii) increasing the sample size reduces the likelihood of bank runs, ceteris paribus. Interestingly, the multiplicity of equilibria, a feature of the canonical Diamond-Dybvig model that we use also, disappears almost completely in our setup. Our results have relevant policy implications.

How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application

Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging problems are still open. In this paper, we first investigate inhomogeneous site percolation on Bethe Lattices with two occupation probabilities, and then extend the result to percolation with m occupation probabilities. The critical behaviour of this inhomogeneous percolation is shown clearly by formulating the percolation probability P∞(p) with given occupation probability p, the critical occupation probability pc = sup{p|P∞(p) = o}, and the average cluster size χ(p) where p is subject to P∞(p) = o. Moreover, using the above theory, we discuss in detail the diffusion behaviour of an infectious disease (SARS) and present specific disease-control strategies in consideration of groups with different infection probabilities.

Meta-food-chains as a many-layer epidemic process on networks

Notable recent works have focused on the multilayer properties of coevolving diseases. We point out that very similar systems play an important role in population ecology. Specifically we study a meta-food-web model that was recently proposed by Pillai et al. [Theor. Ecol. 3, 223 (2009)]. This model describes a network of species connected by feeding interactions, which spread over a network of spatial patches. Focusing on the essential case, where the network of feeding interactions is a chain, we develop an analytical approach for the computation of the degree distributions of colonized spatial patches for the different species in the chain. This framework allows us to address ecologically relevant questions. Considering configuration model ensembles of spatial networks, we find that there is an upper bound for the fraction of patches that a given species can occupy, which depends only on the networks mean degree. For a given mean degree there is then an optimal degree distribution that comes closest to the upper bound. Notably scale-free degree distributions perform worse than more homogeneous degree distributions if the mean degree is sufficiently high. Because species experience the underlying network differently the optimal degree distribution for one particular species is generally not the optimal distribution for the other species in the same food web. These results are of interest for conservation ecology, where, for instance, the task of selecting areas of old-growth forest to preserve in an agricultural landscape, amounts to the design of a patch network.

The role of light in Chagas disease infection risk in Colombia

Background

Chagas disease is the most important vector-borne disease in Latin America and Rhodnius prolixus is the main vector in Colombia. Control strategies in this region have shown poor outcomes due to the insect’s ability to disperse between the sylvatic and the domestic habitat. Because insect migration to houses is responsible to sustain contact rates between vectors and humans, understanding the risk factors that promote migration could be important in designing control strategies. In this respect, it has been reported that adult triatomines have the ability to move over long ranges at night attracted by artificial light. Thus, light bulbs could be playing a critical role in house invasion. The main objective of this study is to understand the role of artificial light, or simply light, in house infestation by R. prolixus.

Methods

To investigate the role of light, we combined fieldwork in the village of Chavinave, Casanare, Colombia and a mathematical model of Rhodnius prolixus dynamics. The model allowed us to simulate insect mobility and distribution in the village based on field results. We created 11 scenarios representing different amounts of light in the village (from 0 to 100 %, with increments of 10 %) with 100 simulations each for a time of 1000 days (2.7 years) and compare the results between the scenarios.

Results

None of the Gomez-Nuñez traps were positive at any stage of the study, suggesting that insects do not colonize houses. The model predicts that with current village connections the proportion of houses that have visiting insects should be around 98 %. Additionally we showed that an increase in light allows for insect spreading and migration to previously un-infested areas.

Conclusions

Increments in light could increase the contact rates between vectors and humans; a two-fold increase in human cases for a 30 % increase in the use and visibility of light on this particular village was estimated with the model.

Asymptotic Behaviour of Gossip Processes and Small-World Networks

Both small-world models of random networks with occasional long-range connections and gossip processes with occasional long-range transmission of information have similar characteristic behaviour. The long-range elements appreciably reduce the effective distances, measured in space or in time, between pairs of typical points. In this paper we show that their common behaviour can be interpreted as a product of the locally branching nature of the models. In particular, it is shown that both typical distances between points and the proportion of space that can be reached within a given distance or time can be approximated by formulae involving the limit random variable of the branching process.

Asymptotic Behaviour of Gossip Processes and Small-World Networks

Both small-world models of random networks with occasional long-range connections and gossip processes with occasional long-range transmission of information have similar characteristic behaviour. The long-range elements appreciably reduce the effective distances, measured in space or in time, between pairs of typical points. In this paper we show that their common behaviour can be interpreted as a product of the locally branching nature of the models. In particular, it is shown that both typical distances between points and the proportion of space that can be reached within a given distance or time can be approximated by formulae involving the limit random variable of the branching process.

Multiple Lattice Model for Influenza Spreading

Behavioral differences among age classes, together with the natural tendency of individuals to prefer contacts with individuals of similar age, naturally point to the existence of a community structure in the population network, in which each community can be identified with a different age class. Data on age-dependent contact patterns also reveal how relevant is the role of the population age structure in shaping the spreading of an infectious disease. In the present paper we propose a simple model for epidemic spreading, in which a contact network with an intrinsic community structure is coupled with a simple stochastic SIR model for the epidemic spreading. The population is divided in 4 different age-communities and hosted on a multiple lattice, each community occupying a specific age-lattice. Individuals are allowed to move freely to nearest neighbor empty sites on the age-lattice. Different communities are connected with each other by means of inter-lattices edges, with a different number of external links connecting different age class populations. The parameters of the contact network model are fixed by requiring the simulated data to fully reproduce the contact patterns matrices of the Polymod survey. The paper shows that adopting a topology which better implements the age-class community structure of the population, one gets a better agreement between experimental contact patterns and simulated data, and this also improves the accordance between simulated and experimental data on the epidemic spreading.

Coupled effects of local movement and global interaction on contagion

By incorporating segregated spatial domain and individual-based linkage into the SIS (susceptible-infected-susceptible) model, we propose a generalized epidemic model which can change from the territorial epidemic model to the networked epidemic model. The role of the individual-based linkage between different spatial domains is investigated. As we adjust the timescale parameter τ from 0 to unity, which represents the degree of activation of the individual-based linkage, three regions are found. Within the region of 0 < τ < 0.02 , the epidemic is determined by local movement and is sensitive to the timescale τ . Within the region of 0.02 < τ < 0.5 , the epidemic is insensitive to the timescale τ . Within the region of 0.5 < τ < 1 , the outbreak of the epidemic is determined by the structure of the individual-based linkage. As we keep an eye on the first region, the role of activating the individual-based linkage in the present model is similar to the role of the shortcuts in the two-dimensional small world network. Only activating a small number of the individual-based linkage can prompt the outbreak of the epidemic globally. The role of narrowing segregated spatial domain and reducing mobility in epidemic control is checked. These two measures are found to be conducive to curbing the spread of infectious disease only when the global interaction is suppressed. A log-log relation between the change in the number of infected individuals and the timescale τ is found. By calculating the epidemic threshold and the mean first encounter time, we heuristically analyze the microscopic characteristics of the propagation of the epidemic in the present model.

Development of a network based model to simulate the between-farm transmission of the porcine reproductive and respiratory syndrome virus

Contact structure within a population can significantly affect the outcomes of infectious disease spread models. The objective of this study was to develop a network based simulation model for the between-farm spread of porcine reproductive and respiratory syndrome virus to assess the impact of contact structure on between-farm transmission of PRRS virus. For these farm level models, a hypothetical population of 500 swine farms following a multistage production system was used. The contact rates between farms were based on a study analyzing movement of pigs in Canada, while disease spread parameters were extracted from published literature. Eighteen distinct scenarios were designed and simulated by varying the mode of transmission (direct versus direct and indirect contact), type of index herd (farrowing, nursery and finishing), and the presumed network structures among swine farms (random, scale-free and small-world). PRRS virus was seeded in a randomly selected farm and 500 iterations of each scenario were simulated for 52 weeks. The median epidemic size by the end of the simulated period and percentage die-out for each scenario, were the key outcomes captured. Scenarios with scale-free network models resulted in the largest epidemic sizes, while scenarios with random and small-world network models resulted in smaller and similar epidemic sizes. Similarly, stochastic die-out percentage was least for scenarios with scale-free networks followed by random and small-world networks. Findings of the study indicated that incorporating network structures among the swine farms had a considerable impact on the spread of PRRS virus, highlighting the importance of understanding and incorporating realistic contact structures when developing infectious disease spread models for similar populations.

Dynamics of history-dependent epidemics in temporal networks

The structural properties of temporal networks often influence the dynamical processes that occur on these networks, e.g., bursty interaction patterns have been shown to slow down epidemics. In this paper, we investigate the effect of link lifetimes on the spread of history-dependent epidemics. We formulate an analytically tractable activity-driven temporal network model that explicitly incorporates link lifetimes. For Markovian link lifetimes, we use mean-field analysis for computing the epidemic threshold, while the effect of non-Markovian link lifetimes is studied using simulations. Furthermore, we also study the effect of negative correlation between the number of links spawned by an individual and the lifetimes of those links. Such negative correlations may arise due to the finite cognitive capacity of the individuals. Our investigations reveal that heavy-tailed link lifetimes slow down the epidemic, while negative correlations can reduce epidemic prevalence. We believe that our results help shed light on the role of link lifetimes in modulating diffusion processes on temporal networks.

Internal-external stimulus competition in a system of interacting moving particles: Persuasion versus propaganda

We propose a general nonlinear analytical framework to study the effect of an external stimulus in the internal state of a population of moving particles. This novel scheme allows us to study a broad range of excitation transport phenomena. In particular, considering social systems, it gives insight of the spatial dynamics influence in the competition between propaganda (mass media) and convincement. By extending the framework presented by Terranova et al. [Europhys. Lett. 105, 30007 (2014)], we now allow changes in individual's opinions due to a reflection induced by mass media. The equations of the model could be solved numerically, and, for some special cases, it is possible to derive analytical solutions for the steady states. We implement computational simulations for different social and dynamical systems to check the accuracy of our scheme and to study a broader variety of scenarios. In particular, we compare the numerical outcome with the analytical results for two possible real cases, finding a good agreement. From the results, we observe that mass media dominates the opinion state in slow dynamics communities; whereas, for higher agent active speeds, the rate of interactions increases and the opinion state is determined by a competition between propaganda and persuasion. This difference suggests that kinetics can not be neglected in the study of transport of any excitation over a particle system.

Integrated travel network model for studying epidemics: Interplay between journeys and epidemic

The ease of travelling between cities has contributed much to globalization. Yet, it poses a threat on epidemic outbreaks. It is of great importance for network science and health control to understand the impact of frequent journeys on epidemics. We stress that a new framework of modelling that takes a traveller's viewpoint is needed. Such integrated travel network (ITN) model should incorporate the diversity among links as dictated by the distances between cities and different speeds of different modes of transportation, diversity among nodes as dictated by the population and the ease of travelling due to infrastructures and economic development of a city, and round-trip journeys to targeted destinations via the paths of shortest travel times typical of human journeys. An example is constructed for 116 cities in China with populations over one million that are connected by high-speed train services and highways. Epidemic spread on the constructed network is studied. It is revealed both numerically and theoretically that the traveling speed and frequency are important factors of epidemic spreading. Depending on the infection rate, increasing the traveling speed would result in either an enhanced or suppressed epidemic, while increasing the traveling frequency enhances the epidemic spreading.

Reversible first-order transition in Pauli percolation

Percolation plays an important role in fields and phenomena as diverse as the study of social networks, the dynamics of epidemics, the robustness of electricity grids, conduction in disordered media, and geometric properties in statistical physics. We analyze a new percolation problem in which the first-order nature of an equilibrium percolation transition can be established analytically and verified numerically. The rules for this site percolation model are physical and very simple, requiring only the introduction of a weight W(n)=n+1 for a cluster of size n. This establishes that a discontinuous percolation transition can occur with qualitatively more local interactions than in all currently considered examples of explosive percolation; and that, unlike these, it can be reversible. This greatly extends both the applicability of such percolation models in principle and their reach in practice.

The role of the organization structure in the diffusion of innovations

Diffusion and adoption of innovations is a topic of increasing interest in economics, market research, and sociology. In this paper we investigate, through an agent based model, the dynamics of adoption of innovative proposals in different kinds of structures. We show that community structure plays an important role on the innovation diffusion, so that proposals are more likely to be accepted in homogeneous organizations. In addition, we show that the learning process of innovative technologies enhances their diffusion, thus resulting in an important ingredient when heterogeneous networks are considered. We also show that social pressure blocks the adoption process whatever the structure of the organization. These results may help to understand how different factors influence the diffusion and acceptance of innovative proposals in different communities and organizations.

Abrupt percolation in small equilibrated networks

Networks can exhibit an abrupt transition in the form of a spontaneous self-organization of a sizable fraction of the population into a giant component of connected members. This behavior has been demonstrated in random graphs under suppressive rules that passively or actively attempt to delay the formation of the giant cluster. We show that suppressive rules are not a necessary condition for a sharp transition at the percolation threshold. Rather, a finite system with aggressive tendency to form a giant cluster may exhibit an instability at the percolation threshold that is relieved through an abrupt and discontinuous transition to the stable branch. We develop the theory for a class of equilibrated networks that produce this behavior and find that the discontinuous jump is especially pronounced in small networks but disappears when the size of the system is infinite.

Ising model in clustered scale-free networks

The Ising model in clustered scale-free networks has been studied by Monte Carlo simulations. These networks are characterized by a degree distribution of the form P(k)∼k(-γ) for large k. Clustering is introduced in the networks by inserting triangles, i.e., triads of connected nodes. The transition from a ferromagnetic (FM) to a paramagnetic (PM) phase has been studied as a function of the exponent γ and the triangle density. For γ>3 our results are in line with earlier simulations, and a phase transition appears at a temperature T(c)(γ) in the thermodynamic limit (system size N→∞). For γ≤3, a FM-PM crossover appears at a size-dependent temperature T(co), so the system remains in a FM state at any finite temperature in the limit N→∞. Thus, for γ=3, T(co) scales as lnN, whereas for γ<3, we find T(co)∼JN(z), where the exponent z decreases for increasing γ. Adding motifs (triangles in our case) to the networks causes an increase in the transition (or crossover) temperature for exponent γ>3 (or ≤3). For γ>3, this increase is due to changes in the mean values 〈k〉 and 〈k(2)〉, i.e., the transition is controlled by the degree distribution (nearest-neighbor connectivities). For γ≤3, however, we find that clustered and unclustered networks with the same size and distribution P(k) have different crossover temperature, i.e., clustering favors FM correlations, thus increasing the temperature T(co). The effect of a degree cutoff k(cut) on the asymptotic behavior of T(co) is discussed.

Price of anarchy is maximized at the percolation threshold

When many independent users try to route traffic through a network, the flow can easily become suboptimal as a consequence of congestion of the most efficient paths. The degree of this suboptimality is quantified by the so-called price of anarchy (POA), but so far there are no general rules for when to expect a large POA in a random network. Here I address this question by introducing a simple model of flow through a network with randomly placed congestible and incongestible links. I show that the POA is maximized precisely when the fraction of congestible links matches the percolation threshold of the lattice. Both the POA and the total cost demonstrate critical scaling near the percolation threshold.

Epidemics in interconnected small-world networks

Networks can be used to describe the interconnections among individuals, which play an important role in the spread of disease. Although the small-world effect has been found to have a significant impact on epidemics in single networks, the small-world effect on epidemics in interconnected networks has rarely been considered. Here, we study the susceptible-infected-susceptible (SIS) model of epidemic spreading in a system comprising two interconnected small-world networks. We find that the epidemic threshold in such networks decreases when the rewiring probability of the component small-world networks increases. When the infection rate is low, the rewiring probability affects the global steady-state infection density, whereas when the infection rate is high, the infection density is insensitive to the rewiring probability. Moreover, epidemics in interconnected small-world networks are found to spread at different velocities that depend on the rewiring probability.

Link removal for the control of stochastically evolving epidemics over networks: a comparison of approaches

For many communicable diseases, knowledge of the underlying contact network through which the disease spreads is essential to determining appropriate control measures. When behavior change is the primary intervention for disease prevention, it is important to understand how to best modify network connectivity using the limited resources available to control disease spread. We describe and compare four algorithms for selecting a limited number of links to remove from a network: two "preventive" approaches (edge centrality, R0 minimization), where the decision of which links to remove is made prior to any disease outbreak and depends only on the network structure; and two "reactive" approaches (S-I edge centrality, optimal quarantining), where information about the initial disease states of the nodes is incorporated into the decision of which links to remove. We evaluate the performance of these algorithms in minimizing the total number of infections that occur over the course of an acute outbreak of disease. We consider different network structures, including both static and dynamic Erdös-Rényi random networks with varying levels of connectivity, a real-world network of residential hotels connected through injection drug use, and a network exhibiting community structure. We show that reactive approaches outperform preventive approaches in averting infections. Among reactive approaches, removing links in order of S-I edge centrality is favored when the link removal budget is small, while optimal quarantining performs best when the link removal budget is sufficiently large. The budget threshold above which optimal quarantining outperforms the S-I edge centrality algorithm is a function of both network structure (higher for unstructured Erdös-Rényi random networks compared to networks with community structure or the real-world network) and disease infectiousness (lower for highly infectious diseases). We conduct a value-of-information analysis of knowing which nodes are initially infected by comparing the performance improvement achieved by reactive over preventive strategies. We find that such information is most valuable for moderate budget levels, with increasing value as disease spread becomes more likely (due to either increased connectedness of the network or increased infectiousness of the disease).

Evolution and emergence of infectious diseases in theoretical and real-world networks

One of the most important advancements in theoretical epidemiology has been the development of methods that account for realistic host population structure. The central finding is that heterogeneity in contact networks, such as the presence of 'superspreaders', accelerates infectious disease spread in real epidemics. Disease control is also complicated by the continuous evolution of pathogens in response to changing environments and medical interventions. It remains unclear, however, how population structure influences these adaptive processes. Here we examine the evolution of infectious disease in empirical and theoretical networks. We show that the heterogeneity in contact structure, which facilitates the spread of a single disease, surprisingly renders a resident strain more resilient to invasion by new variants. Our results suggest that many host contact structures suppress invasion of new strains and may slow disease adaptation. These findings are important to the natural history of disease evolution and the spread of drug-resistant strains.

Networks of neuroblastoma cells on porous silicon substrates reveal a small world topology

N2A cells on porous substrates create highly clustered, small world topology patterns.

Emergence of dynamical complexity related to human heart rate variability

We apply the refined composite multiscale entropy (MSE) method to a one-dimensional directed small-world network composed of nodes whose states are binary and whose dynamics obey the majority rule. We find that the resulting fluctuating signal becomes dynamically complex. This dynamical complexity is caused (i) by the presence of both short-range connections and long-range shortcuts and (ii) by how well the system can adapt to the noisy environment. By tuning the adaptability of the environment and the long-range shortcuts we can increase or decrease the dynamical complexity, thereby modeling trends found in the MSE of a healthy human heart rate in different physiological states. When the shortcut and adaptability values increase, the complexity in the system dynamics becomes uncorrelated.

Tight lower bound for percolation threshold on an infinite graph

We construct a tight lower bound for the site percolation threshold on an infinite graph, which becomes exact for an infinite tree. The bound is given by the inverse of the maximal eigenvalue of the Hashimoto matrix used to count nonbacktracking walks on the original graph. Our bound always exceeds the inverse spectral radius of the graph's adjacency matrix, and it is also generally tighter than the existing bound in terms of the maximum degree. We give a constructive proof for existence of such an eigenvalue in the case of a connected infinite quasitransitive graph, a graph-theoretic analog of a translationally invariant system.

Epidemic spreading and risk perception in multiplex networks: A self-organized percolation method

In this paper we study the interplay between epidemic spreading and risk perception on multiplex networks. The basic idea is that the effective infection probability is affected by the perception of the risk of being infected, which we assume to be related to the fraction of infected neighbors, as introduced by Bagnoli et al. [Phys. Rev. E 76, 061904 (2007)PLEEE81539-375510.1103/PhysRevE.76.061904]. We rederive previous results using a self-organized method that automatically gives the percolation threshold in just one simulation. We then extend the model to multiplex networks considering that people get infected by physical contacts in real life but often gather information from an information network, which may be quite different from the physical ones. The similarity between the physical and the information networks determines the possibility of stopping the infection for a sufficiently high precaution level: if the networks are too different, there is no means of avoiding the epidemics.

Exact results for a simple epidemic model on a directed network: explorations of a system in a nonequilibrium steady state

Motivated by fundamental issues in nonequilibrium statistical mechanics, we study the venerable susceptible-infected-susceptible (SIS) model of disease spreading in an idealized, simple setting. Using Monte Carlo and analytic techniques, we consider a fully connected, unidirectional network of odd number of nodes, each having an equal number of in- and out-degrees. With the standard SIS dynamics at high infection rates, this system settles into an active nonequilibrium steady state. We find the exact probability distribution and explore its implications for nonequilibrium statistical mechanics, such as the presence of persistent probability currents.

Percolation on fitness-dependent networks with heterogeneous resilience

The ability to understand the impact of adversarial processes on networks is crucial to various disciplines. The objects of study in this article are fitness-driven networks. Fitness-dependent networks are fully described by a probability distribution of fitness and an attachment kernel. Every node in the network is endowed with a fitness value and the attachment kernel translates the fitness of two nodes into the probability that these two nodes share an edge. This concept is also known as mutual attractiveness. In the present article, fitness does not only serve as a measure of attractiveness, but also as a measure of a node's robustness against failure. The probability that a node fails increases with the number of failures in its direct neighborhood and decreases with higher fitness. Both static and dynamic network models are considered. Analytical results for the percolation threshold and the occupied fraction are derived. One of the results is that the distinction between the dynamic and the static model has a profound impact on the way failures spread over the network. Additionally, we find that the introduction of mutual attractiveness stabilizes the network compared to a pure random attachment.

The evolutionary and ecological consequences of animal social networks: emerging issues

The first generation of research on animal social networks was primarily aimed at introducing the concept of social networks to the fields of animal behaviour and behavioural ecology. More recently, a diverse body of evidence has shown that social fine structure matters on a broader scale than initially expected, affecting many key ecological and evolutionary processes. Here, we review this development. We discuss the effects of social network structure on evolutionary dynamics (genetic drift, fixation probabilities, and frequency-dependent selection) and social evolution (cooperation and between-individual behavioural differences). We discuss how social network structure can affect important coevolutionary processes (host-pathogen interactions and mutualisms) and population stability. We also discuss the potentially important, but poorly studied, role of social network structure on dispersal and invasion. Throughout, we highlight important areas for future research.

Microtransition cascades to percolation

We report the discovery of a discrete hierarchy of microtransitions occurring in models of continuous and discontinuous percolation. The precursory microtransitions allow us to target almost deterministically the location of the transition point to global connectivity. This extends to the class of intrinsically stochastic processes the possibility to use warning signals anticipating phase transitions in complex systems.

Inferring Plasmodium vivax transmission networks from tempo-spatial surveillance data

BACKGROUND

The transmission networks of Plasmodium vivax characterize how the parasite transmits from one location to another, which are informative and insightful for public health policy makers to accurately predict the patterns of its geographical spread. However, such networks are not apparent from surveillance data because P. vivax transmission can be affected by many factors, such as the biological characteristics of mosquitoes and the mobility of human beings. Here, we pay special attention to the problem of how to infer the underlying transmission networks of P. vivax based on available tempo-spatial patterns of reported cases.

METHODOLOGY

We first define a spatial transmission model, which involves representing both the heterogeneous transmission potential of P. vivax at individual locations and the mobility of infected populations among different locations. Based on the proposed transmission model, we further introduce a recurrent neural network model to infer the transmission networks from surveillance data. Specifically, in this model, we take into account multiple real-world factors, including the length of P. vivax incubation period, the impact of malaria control at different locations, and the total number of imported cases.

PRINCIPAL FINDINGS

We implement our proposed models by focusing on the P. vivax transmission among 62 towns in Yunnan province, People's Republic China, which have been experiencing high malaria transmission in the past years. By conducting scenario analysis with respect to different numbers of imported cases, we can (i) infer the underlying P. vivax transmission networks, (ii) estimate the number of imported cases for each individual town, and (iii) quantify the roles of individual towns in the geographical spread of P. vivax.

CONCLUSION

The demonstrated models have presented a general means for inferring the underlying transmission networks from surveillance data. The inferred networks will offer new insights into how to improve the predictability of P. vivax transmission.

Message-passing approach for threshold models of behavior in networks

We study a simple model of how social behaviors, like trends and opinions, propagate in networks where individuals adopt the trend when they are informed by threshold T neighbors who are adopters. Using a dynamic message-passing algorithm, we develop a tractable and computationally efficient method that provides complete time evolution of each individual's probability of adopting the trend or of the frequency of adopters and nonadopters in any arbitrary networks. We validate the method by comparing it with Monte Carlo-based agent simulation in real and synthetic networks and provide an exact analytic scheme for large random networks, where simulation results match well. Our approach is general enough to incorporate non-Markovian processes and to include heterogeneous thresholds and thus can be applied to explore rich sets of complex heterogeneous agent-based models.

Signal transmission in a Y-shaped one-way chain

It has been found that noise plays a key role to improve signal transmission in a one-way chain of bistable systems [Zhang et al., Phys. Rev. E 58, 2952 (1998)]. We here show that the signal transmission can be sharply improved without the aid of noise, if the one-way chain with a single source node is changed with two source nodes becoming a Y-shaped one-way chain. We further reveal that the enhanced signal transmission in the Y-shaped one-way chain is regulated by coupling strength, and that it is robust to noise perturbation and input signal irregularity. We finally analyze the mechanism of the enhanced signal transmission by the Y-shaped structure.

Asymptotically inspired moment-closure approximation for adaptive networks

Adaptive social networks, in which nodes and network structure coevolve, are often described using a mean-field system of equations for the density of node and link types. These equations constitute an open system due to dependence on higher-order topological structures. We propose a new approach to moment closure based on the analytical description of the system in an asymptotic regime. We apply the proposed approach to two examples of adaptive networks: recruitment to a cause model and adaptive epidemic model. We show a good agreement between the improved mean-field prediction and simulations of the full network system.

Unstable supercritical discontinuous percolation transitions

The location and nature of the percolation transition in random networks is a subject of intense interest. Recently, a series of graph evolution processes have been introduced that lead to discontinuous percolation transitions where the addition of a single edge causes the size of the largest component to exhibit a significant macroscopic jump in the thermodynamic limit. These processes can have additional exotic behaviors, such as displaying a "Devil's staircase" of discrete jumps in the supercritical regime. Here we investigate whether the location of the largest jump coincides with the percolation threshold for a range of processes, such as Erdős-Rényipercolation, percolation via edge competition and via growth by overtaking. We find that the largest jump asymptotically occurs at the percolation transition for Erdős-Rényiand other processes exhibiting global continuity, including models exhibiting an "explosive" transition. However, for percolation processes exhibiting genuine discontinuities, the behavior is substantially richer. In percolation models where the order parameter exhibits a staircase, the largest discontinuity generically does not coincide with the percolation transition. For the generalized Bohman-Frieze-Wormald model, it depends on the model parameter. Distinct parameter regimes well in the supercritical regime feature unstable discontinuous transitions-a novel and unexpected phenomenon in percolation. We thus demonstrate that seemingly and genuinely discontinuous percolation transitions can involve a rich behavior in supercriticality, a regime that has been largely ignored in percolation.

Invited review: Epidemics on social networks

Since its first formulations almost a century ago, mathematical models fordisease spreading contributed to understand, evaluate and control the epidemic processes.They promoted a dramatic change in how epidemiologists  thought of the propagation of infectious diseases.In the last decade, when the traditional epidemiological models seemed to be exhausted,  new types of models were developed.These new models incorporated concepts from graph theory to describe and model the underlying social structure.Many of these works  merely produced a more detailed extension of the previous results,  but some otherstriggered a completely new paradigm  in the mathematical study of epidemic processes. In this review, we will introduce the basicconcepts of epidemiology, epidemic  modeling and networks, to finally provide a brief description of the mostrelevant results in the field. Received: 6 April 2013,  Accepted: 3 June 2013; Edited by: G. Mindlin; DOI: http://dx.doi.org/10.4279/PIP.050003Cite as: M N Kuperman, Papers in Physics 5, 050003 (2013) 

A lattice model for influenza spreading

We construct a stochastic SIR model for influenza spreading on a D-dimensional lattice, which represents the dynamic contact network of individuals. An age distributed population is placed on the lattice and moves on it. The displacement from a site to a nearest neighbor empty site, allows individuals to change the number and identities of their contacts. The dynamics on the lattice is governed by an attractive interaction between individuals belonging to the same age-class. The parameters, which regulate the pattern dynamics, are fixed fitting the data on the age-dependent daily contact numbers, furnished by the Polymod survey. A simple SIR transmission model with a nearest neighbors interaction and some very basic adaptive mobility restrictions complete the model. The model is validated against the age-distributed Italian epidemiological data for the influenza A(H1N1) during the [Formula: see text] season, with sensible predictions for the epidemiological parameters. For an appropriate topology of the lattice, we find that, whenever the accordance between the contact patterns of the model and the Polymod data is satisfactory, there is a good agreement between the numerical and the experimental epidemiological data. This result shows how rich is the information encoded in the average contact patterns of individuals, with respect to the analysis of the epidemic spreading of an infectious disease.

Phase transitions in supercritical explosive percolation

Percolation describes the sudden emergence of large-scale connectivity as edges are added to a lattice or random network. In the Bohman-Frieze-Wormald model (BFW) of percolation, edges sampled from a random graph are considered individually and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function with asymptotic value of α, a constant. The BFW process has been studied as a model system for investigating the underlying mechanisms leading to discontinuous phase transitions in percolation. Here we focus on the regime αε[0.6,0.95] where it is known that only one giant component, denoted C(1) , initially appears at the discontinuous phase transition. We show that at some point in the supercritical regime C(1) stops growing and eventually a second giant component, denoted C(2), emerges in a continuous percolation transition. The delay between the emergence of C(1) and C(2) and their asymptotic sizes both depend on the value of α and we establish by several techniques that there exists a bifurcation point α(c)=0.763±0.002. For αε[0.6,α(c)), C(1) stops growing the instant it emerges and the delay between the emergence of C(1) and C(2) decreases with increasing α. For αε(α(c),0.95], in contrast, C(1) continues growing into the supercritical regime and the delay between the emergence of C(1) and C(2) increases with increasing α. As we show, α(c) marks the minimal delay possible between the emergence of C(1) and C(2) (i.e., the smallest edge density for which C(2) can exist). We also establish many features of the continuous percolation of C(2) including scaling exponents and relations.

Topology-free design for amplitude death in time-delayed oscillators coupled by a delayed connection

This paper deals with amplitude death in time-delayed oscillators coupled by a delayed connection with topology uncertainty. A systematic procedure without trial-and-error testing for designing connection parameters is proposed from a robust control theory viewpoint. This procedure has the following two advantages: the designed connection parameters are valid for any network topology and the procedure is valid even for long-delay oscillators. These analytical results are verified by some numerical simulations.

Percolation centrality: quantifying graph-theoretic impact of nodes during percolation in networks

A number of centrality measures are available to determine the relative importance of a node in a complex network, and betweenness is prominent among them. However, the existing centrality measures are not adequate in network percolation scenarios (such as during infection transmission in a social network of individuals, spreading of computer viruses on computer networks, or transmission of disease over a network of towns) because they do not account for the changing percolation states of individual nodes. We propose a new measure, percolation centrality, that quantifies relative impact of nodes based on their topological connectivity, as well as their percolation states. The measure can be extended to include random walk based definitions, and its computational complexity is shown to be of the same order as that of betweenness centrality. We demonstrate the usage of percolation centrality by applying it to a canonical network as well as simulated and real world scale-free and random networks.