Cooperative epidemics on multiplex networks

Influence of Information Blocking on the Spread of Virus in Multilayer Networks

In this paper, we present the model of the interaction between the spread of disease and the spread of information about the disease in multilayer networks. Next, based on the characteristics of the SARS-CoV-2 virus pandemic, we evaluated the influence of information blocking on the virus spread. Our results show that blocking the spread of information affects the speed at which the epidemic peak appears in our society, and affects the number of infected individuals.

Effects of official information and rumor on resource-epidemic coevolution dynamics

Epidemic-related information and resources have proven to have a significant impact on the spread of the epidemic during the Corona Virus Disease 2019 (COVID-19) pandemic. The various orientation role of information has different effects on the epidemic spreading process, which will affect the individual’ awareness of resources allocation and epidemic spreading scale. Based on this, a three-layer network is established to describe the dynamic coevolution process among information dissemination, resource allocation, and epidemic spreading. In order to analyze dynamic coevolution process, the microscopic Markov chain (MMC) theory is used. Then, the threshold of epidemic spreading is deduced. Our results indicated that the official information orientation intensity inhibits the epidemics spreading, while rumor orientation intensity promotes epidemic spreading. At the same time, the efficiency of resource utilization restrains the expansion of the infection scale. The two kinds of information are combined with resources respectively. Official information will enhance the inhibitory effect of resources epidemics spreading, while rumor will do the opposite.

Metapopulation epidemic models with a universal mobility pattern on interconnected networks

The global pandemic of the coronavirus disease 2019 (COVID-19) exemplifies the influence of human mobility on epidemic spreading. A framework called the movement-interaction-return (MIR) model is a model to study the impact of human mobility on epidemic spreading. In this paper, we investigate epidemic spreading in interconnected metapopulation networks. Specifically, we incorporate the human mobility pattern called the radiation model into the MIR model. As a result, the proposed model is more realistic in comparison to the original MIR model. We use the tensorial framework to develop Markovian equations that describe the dynamics of the proposed model on interconnected metapopulation networks. Then we derive the corresponding epidemic thresholds by converting tensors into matrices. Comprehensive numerical simulations confirm our analysis.

Multilayer and Multiplex Networks: An Introduction to Their Use in Veterinary Epidemiology

Contact network analysis has become a vital tool for conceptualizing the spread of pathogens in animal populations and is particularly useful for understanding the implications of heterogeneity in contact patterns for transmission. However, the transmission of most pathogens cannot be simplified to a single mode of transmission and, thus, a single definition of contact. In addition, host-pathogen interactions occur in a community context, with many pathogens infecting multiple host species and most hosts being infected by multiple pathogens. Multilayer networks provide a formal framework for researching host-pathogen systems in which multiple types of transmission-relevant interactions, defined as network layers, can be analyzed jointly. Here, we provide an overview of multilayer network analysis and review applications of this novel method to epidemiological research questions. We then demonstrate the use of this technique to analyze heterogeneity in direct and indirect contact patterns amongst swine farms in the United States. When contact among nodes can be defined in multiple ways, a multilayer approach can advance our ability to use networks in epidemiological research by providing an improved approach for defining epidemiologically relevant groups of interacting nodes and changing the way we identify epidemiologically important individuals such as superspreaders.

Competition between vaccination and disease spreading

We study the interaction between epidemic spreading and a vaccination process. We assume that, similar to the disease spreading, the vaccination process also occurs through direct contact, i.e., it follows the standard susceptible-infected-susceptible (SIS) dynamics. The two competing processes are asymmetrically coupled as vaccinated nodes can directly become infected at a reduced rate with respect to susceptible ones. We study analytically the model in the framework of mean-field theory finding a rich phase diagram. When vaccination provides little protection toward infection, two continuous transitions separate a disease-free immunized state from vaccinated-free epidemic state, with an intermediate mixed state where susceptible, infected, and vaccinated individuals coexist. As vaccine efficiency increases, a tricritical point leads to a bistable regime, and discontinuous phase transitions emerge. Numerical simulations for homogeneous random networks agree very well with analytical predictions.

Spreading of two interacting diseases in multiplex networks

We consider the interacting processes between two diseases on multiplex networks, where each node can be infected by two interacting diseases with general interacting schemes. A discrete-time individual-based probability model is rigorously derived. By the bifurcation analysis of the equilibrium, we analyze the outbreak condition of one disease. The theoretical predictions are in good agreement with discrete-time stochastic simulations on scale-free networks. Furthermore, we discuss the influence of network overlap and dynamical parameters on the epidemic dynamical behaviors. The simulation results show that the network overlap has almost no effect on both epidemic threshold and prevalence. We also find that the epidemic threshold of one disease does not depend on all system parameters. Our method offers an analytical framework for the spreading dynamics of multiple processes in multiplex networks.

Message-passing theory for cooperative epidemics

The interaction among spreading processes on a complex network is a nontrivial phenomenon of great importance. It has recently been realized that cooperative effects among infective diseases can give rise to qualitative changes in the phenomenology of epidemic spreading, leading, for instance, to abrupt transitions and hysteresis. Here, we consider a simple model for two interacting pathogens on a network and we study it by using the message-passing approach. In this way, we are able to provide detailed predictions for the behavior of the model in the whole phase-diagram for any given network structure. Numerical simulations on synthetic networks (both homogeneous and heterogeneous) confirm the great accuracy of the theoretical results. We finally consider the issue of identifying the nodes where it is better to seed the infection in order to maximize the probability of observing an extensive outbreak. The message-passing approach provides an accurate solution also for this problem.

Markovian approach to tackle the interaction of simultaneous diseases

The simultaneous emergence of several abrupt disease outbreaks or the extinction of some serotypes of multistrain diseases are fingerprints of the interaction between pathogens spreading within the same population. Here, we propose a general and versatile benchmark to address the unfolding of both cooperative and competitive interacting diseases. We characterize the explosive transitions between the disease-free and the epidemic regimes arising from the cooperation between pathogens and show the critical degree of cooperation needed for the onset of such abrupt transitions. For the competing diseases, we characterize the mutually exclusive case and derive analytically the transition point between the full-dominance phase, in which only one pathogen propagates, and the coexistence regime. Finally, we use this framework to analyze the behavior of the former transition point as the competition between pathogens is relaxed.

Coevolution spreading in complex networks

The propagations of diseases, behaviors and information in real systems are rarely independent of each other, but they are coevolving with strong interactions. To uncover the dynamical mechanisms, the evolving spatiotemporal patterns and critical phenomena of networked coevolution spreading are extremely important, which provide theoretical foundations for us to control epidemic spreading, predict collective behaviors in social systems, and so on. The coevolution spreading dynamics in complex networks has thus attracted much attention in many disciplines. In this review, we introduce recent progress in the study of coevolution spreading dynamics, emphasizing the contributions from the perspectives of statistical mechanics and network science. The theoretical methods, critical phenomena, phase transitions, interacting mechanisms, and effects of network topology for four representative types of coevolution spreading mechanisms, including the coevolution of biological contagions, social contagions, epidemic-awareness, and epidemic-resources, are presented in detail, and the challenges in this field as well as open issues for future studies are also discussed.

Self-awareness control effect of cooperative epidemics on complex networks

Coinfection mechanism is a common interacting mode between multiple diseases in real spreading processes, where the diseases mutually increase their susceptibility, and has aroused widespread studies in network science. We use the bond percolation theory to characterize the coinfection model under two self-awareness control strategies, including immunization strategy and quarantine strategy, and to study the impacts of the synergy effect and control strategies on cooperative epidemics. We find that strengthening the synergy effect can reduce the epidemic threshold and enhance the outbreak size of coinfected networks. On Erdős-Rényi networks, the synergy effect will induce a crossover phenomenon of phase transition, i.e., make the type of phase transition from being continuous to discontinuous. Self-awareness control strategies play a non-negligible role in suppressing cooperative epidemics. In particular, increasing immunization or the quarantine rate can enhance the epidemic threshold and reduce the outbreak size of cooperative epidemics, and lead to a crossover phenomenon of transition from being discontinuous to continuous. The impact of quarantine strategy on cooperative epidemics is more significant than the immunization strategy, which is verified on scale-free networks.

Effect of network clustering on mutually cooperative coinfections

The spread of an infectious disease can be promoted by previous infections with other pathogens. This cooperative effect can give rise to violent outbreaks, reflecting the presence of an abrupt epidemic transition. As for other diffusive dynamics, the topology of the interaction pattern of the host population plays a crucial role. It was conjectured that a discontinuous transition arises when there are relatively few short loops and many long loops in the contact network. Here we focus on the role of local clustering in determining the nature of the transition. We consider two mutually cooperative pathogens diffusing in the same population: An individual already infected with one disease has an increased probability of getting infected by the other. We look at how a disease obeying the susceptible-infected-removed dynamics spreads on contact networks with tunable clustering. Using numerical simulations we show that for large cooperativity the epidemic transition is always abrupt, with the discontinuity decreasing as clustering is increased. For large clustering strong finite-size effects are present and the discontinuous nature of the transition is manifest only in large networks. We also investigate the problem of influential spreaders for cooperative infections, revealing that both cooperativity and clustering strongly enhance the dependence of the spreading influence on the degree of the initial seed.

Persistent spatial patterns of interacting contagions

The spread of infectious diseases, rumors, fashions, and innovations are complex contagion processes, embedded in network and spatial contexts. While the studies in the former context are intensively expanded, the latter remains largely unexplored. In this paper, we investigate the pattern formation of an interacting contagion, where two infections, A and B, interact with each other and diffuse simultaneously in space. The contagion process for each follows the classical susceptible-infected-susceptible kinetics, and their interaction introduces a potential change in the secondary infection propensity compared to the baseline reproduction number R_{0}. We show that the nontrivial spatial infection patterns arise when the susceptible individuals move faster than the infected and the interaction between the two infections is neither too competitive nor too cooperative. Interestingly, the system exhibits pattern hysteresis phenomena, i.e., quite different parameter regions for patterns exist in the direction of increasing or decreasing R_{0}. Decreasing R_{0} reveals remarkable enhancement in contagion prevalence, meaning that the eradication becomes difficult compared to the single-infection or coinfection without space. Linearization analysis supports our observations, and we have identified the required elements and dynamical mechanism, which suggests that these patterns are essentially Turing patterns. Our work thus reveals new complexities in interacting contagions and paves the way for further investigation because of its relevance to both biological and social contexts.

Propagation of interacting diseases on multilayer networks

The study on the dynamics of interacting diseases has attracted considerable attention in recent years. This paper proposes a model for two interacting epidemics spreading concurrently on a two-layer network, where both the epidemic thresholds and the dynamics of disease outbreaks are investigated. The analytical expression of the epidemic threshold shows strong coupling between the two interacting epidemics. Moreover, two metrics, the maximum prevalence and the highest spreading speed, are proposed to describe the outbreak process. Theoretical analysis together with extensive simulations illustrate the functions of various factors, including the network topological parameters, percentage of overlapped network links, vulnerable individuals, and the reciprocity of the two diseases. It is found that the seemingly important factor, i.e., the percentage of overlapped links, possesses no effect on the propagation, while the frequently overlooked factor, i.e., the percentage of vulnerable individuals, has significant effects. For the interaction of the two diseases, the recovery state of one disease is more influential than the other in both the mutually enhanced and the mutually impaired situations.

Cooperative epidemics spreading under resource control

The input and allocation of public resources are of crucial importance to suppressing the outbreak of infectious diseases. However, in the research on multi-disease dynamics, the impact of resources has never been taken into account. Here, we propose a two-epidemic spreading model with resource control, in which the amount of resources is introduced into the recovery rates of diseases and the allocation of resources between two diseases is regulated by a parameter. Using the dynamical message passing method, we obtain resource thresholds of the two diseases and validate them on ER networks and scale-free networks. By comparing the results on scale-free networks with different power-law exponents, we find that the heterogeneity of the network promotes the spreading of both diseases. Especially, we find optimal allocation coefficients at different resource levels. And, we get a counterintuitive conclusion that when the available resources are limited, it is a better strategy to preferentially suppress the disease with lower infection rate. In addition, we investigate the effect of interaction strength and find that great interaction strength between diseases makes two diseases with different infectivity tend to be homogeneous.

Spreading Processes in Multiplex Metapopulations Containing Different Mobility Networks

We propose a theoretical framework for the study of spreading processes in structured metapopulations, with heterogeneous agents, subjected to different recurrent mobility patterns. We propose to represent the heterogeneity in the composition of the metapopulations as layers in a multiplex network, where nodes would correspond to geographical areas and layers account for the mobility patterns of agents of the same class. We analyze classical epidemic models within this framework and obtain an excellent agreement with extensive Monte Carlo simulations. This agreement allows us to derive analytical expressions of the epidemic threshold and to face the challenge of characterizing a real multiplex metapopulation, the city of Medellín in Colombia, where different recurrent mobility patterns are observed depending on the socioeconomic class of the agents. Our framework allows us to unveil the geographical location of those patches that trigger the epidemic state at the critical point. A careful exploration reveals that social mixing between classes and mobility crucially determines these critical patches and, more importantly, it can produce abrupt changes of the critical properties of the epidemic onset.

The spread of disease is strongly driven by how humans move and segregate across cities, regions, and countries. Understanding how epidemics arise from the interplay of these behaviors is crucial for implementing efficient containment and prevention policies. In this work, we propose a theoretical framework for the spread of pathogens that incorporates the real patterns of mobility, demographics, and human diversity observed in urban environments.

Our framework represents various populations in a community as layers in a multiplex network, where nodes correspond to geographical areas and layers account for movement patterns. With this framework, we analyze a real urban system, the city of Medellin in Colombia, considering the interplay between six socioeconomic classes displaying disparate demographic segregation and mobility habits. Our framework can identify those neighborhoods and classes that trigger epidemic outbreaks.

Remarkably, we observe that small changes to both mobility and social mixing among these subpopulations can trigger abrupt changes. As a consequence, containment strategies targeting a certain neighborhood can quickly change from efficient to useless. Finally, we provide a physical interpretation of the abrupt changes of those patches driving the unfolding of epidemics based on the effective number of contacts of agents.

Our framework provides a reliable, time-saving platform for analyzing the spread of pathogens among various populations, allowing researchers to identify areas critical to the unfolding of diseases. With further improvements, our formalism could be extended to accommodate more sophisticated commuting patterns and epidemic models.

Interactive social contagions and co-infections on complex networks

What we are learning about the ubiquitous interactions among multiple social contagion processes on complex networks challenges existing theoretical methods. We propose an interactive social behavior spreading model, in which two behaviors sequentially spread on a complex network, one following the other. Adopting the first behavior has either a synergistic or an inhibiting effect on the spread of the second behavior. We find that the inhibiting effect of the first behavior can cause the continuous phase transition of the second behavior spreading to become discontinuous. This discontinuous phase transition of the second behavior can also become a continuous one when the effect of adopting the first behavior becomes synergistic. This synergy allows the second behavior to be more easily adopted and enlarges the co-existence region of both behaviors. We establish an edge-based compartmental method, and our theoretical predictions match well with the simulation results. Our findings provide helpful insights into better understanding the spread of interactive social behavior in human society.

Advances from the nexus of animal behaviour and pathogen transmission: new directions and opportunities using contact networks

Contact network models have enabled significant advances in understanding the influence of behaviour on parasite and pathogen transmission. They are an important tool that links variation in individual behaviour, to epidemiological consequences at the population level. Here, in our introduction to this special issue, we highlight the importance of applying network approaches to disease ecological and epidemiological questions, and how this has provided a much deeper understanding of these research areas. Recent advances in tracking host behaviour (bio-logging: e.g., GPS tracking, barcoding) and tracking pathogens (high-resolution sequencing), as well as methodological advances (multi-layer networks, computational techniques) started producing exciting new insights into disease transmission through contact networks. We discuss some of the exciting directions that the field is taking, some of the challenges, and importantly the opportunities that lie ahead. For instance, we suggest to integrate multiple transmission pathways, multiple pathogens, and in some systems, multiple host species, into the next generation of network models. Corresponding opportunities exist in utilising molecular techniques, such as high-resolution sequencing, to establish causality in network connectivity and disease outcomes. Such novel developments and the continued integration of network tools offers a more complete understanding of pathogen transmission processes, their underlying mechanisms and their evolutionary consequences.

Mutually cooperative epidemics on power-law networks

The spread of an infectious disease can, in some cases, promote the propagation of other pathogens favoring violent outbreaks, which cause a discontinuous transition to an endemic state. The topology of the contact network plays a crucial role in these cooperative dynamics. We consider a susceptible-infected-removed-type model with two mutually cooperative pathogens: An individual already infected with one disease has an increased probability of getting infected by the other. We present a heterogeneous mean-field theoretical approach to the coinfection dynamics on generic uncorrelated power-law degree-distributed networks and validate its results by means of numerical simulations. We show that, when the second moment of the degree distribution is finite, the epidemic transition is continuous for low cooperativity, while it is discontinuous when cooperativity is sufficiently high. For scale-free networks, i.e., topologies with diverging second moment, the transition is instead always continuous. In this way we clarify the effect of heterogeneity and system size on the nature of the transition, and we validate the physical interpretation about the origin of the discontinuity.

Message passing theory for percolation models on multiplex networks with link overlap

Multiplex networks describe a large variety of complex systems, including infrastructures, transportation networks, and biological systems. Most of these networks feature a significant link overlap. It is therefore of particular importance to characterize the mutually connected giant component in these networks. Here we provide a message passing theory for characterizing the percolation transition in multiplex networks with link overlap and an arbitrary number of layers M. Specifically we propose and compare two message passing algorithms that generalize the algorithm widely used to study the percolation transition in multiplex networks without link overlap. The first algorithm describes a directed percolation transition and admits an epidemic spreading interpretation. The second algorithm describes the emergence of the mutually connected giant component, that is the percolation transition, but does not preserve the epidemic spreading interpretation. We obtain the phase diagrams for the percolation and directed percolation transition in simple representative cases. We demonstrate that for the same multiplex network structure, in which the directed percolation transition has nontrivial tricritical points, the percolation transition has a discontinuous phase transition, with the exception of the trivial case in which all the layers completely overlap.

Correlated edge overlaps in multiplex networks

We develop the theory of sparse multiplex networks with partially overlapping links based on their local treelikeness. This theory enables us to find the giant mutually connected component in a two-layer multiplex network with arbitrary correlations between connections of different types. We find that correlations between the overlapping and nonoverlapping links markedly change the phase diagram of the system, leading to multiple hybrid phase transitions. For assortative correlations we observe recurrent hybrid phase transitions.