In network theory, multidimensional networks, a special type of multilayer network, are networks with multiple kinds of relations. Increasingly sophisticated attempts to model real-world systems as multidimensional networks have yielded valuable insight in the fields of social network analysis, economics, urban and international transport, ecology, psychology, medicine, biology, Commerce, climatology, physics, computational neuroscience, operations management, infrastructures and finance.
The rapid exploration of complex networks in recent years has been dogged by a lack of standardized naming conventions, as various groups use overlapping and contradictory terminology to describe specific network configurations (e.g., multiplex, multilayer, multilevel, multidimensional, multirelational, interconnected). Formally, multidimensional networks are edge-labeled multigraphs. The term “fully multidimensional” has also been used to refer to a multipartite edge-labeled multigraph. Multidimensional networks have also recently been reframed as specific instances of multilayer networks. In this case, there are as many layers as there are dimensions, and the links between nodes within each layer are simply all the links for a given dimension.
Different Routes of Infection
Multilayer networks can be usefully applied in contexts where a pathogen can be transmitted through multiple modes or pathways of infection, as the multiplex approach provides a framework to account for multiple transmission probabilities. Considering the presence of multiple transmission modes can influence the efficacy of targeted interventions, particularly if nodes were traditionally targeted according to their degree in only one layer. This has implications for situations where data, networks, and resultant optimal control strategies are only available for one mode of transmission, leading to overconfidence in the efficacy of control.
In the context of veterinary epidemiology, animal movements are typically considered the most effective transmission mode between farms (direct contacts). However, other infection mechanisms might play an important role such as wind-borne spread and fomites disseminated through contaminated clothes, equipment, and vehicles by personnel (indirect contacts). Ignoring one mode of transmission could lead to inaccurate farm risk predictions and ineffective targeted surveillance. This has been demonstrated in a network analysis that considered both direct (cattle movements) and indirect (veterinarian movements) contacts to reveal that indirect contact, despite being less efficient in transmission, can play a major role in spread of a pathogen within a network.
In another example, Stella used an “ecomultiplex model” to study the spread of Trypanosoma cruzi (cause of Chagas disease in humans) across different mammal species. This pathogen can be transmitted either through invertebrate vectors (Triatominae or kissing bugs) or through predation when a susceptible predator feeds on infected prey or vectors. Thus, their model included two ecological/transmission layers: the food-web and vector layers. Their results showed that studying the multiplex network structure offered insights on which host species facilitate parasite spread, and thus which would be more effective to immunize in order to control the spread. At the same time, they showed how, in this system, when parasites spread occurs primarily through the trophic layer, immunizing predators hampers parasite transmission more than immunizing prey.
Furthermore, multilayer network analysis can help differentiate between different types of social interactions that may lead to disease transmission. For example, sex-related dynamics of contact networks can have important implications for disease spread in animal populations, as seen in the spread of Mycobacterium bovis in European badgers (Meles meles). The authors constructed an interconnected network that distinguished male-male, female-female, and between-sex contacts recorded during proximity loggers. Inter-layer between-sex edges and edges in the male-male layer were more important in connecting groups into wider social communities, and contacts between different social communities were also more likely in these layers.
Dynamics of Coupled Processes: The Spread of Two Pathogens
Another application of multilayer networks in epidemiology is to model the concurrent propagation of two entities through a network, such as two different pathogens co-occurring in the same population or the spread of disease awareness alongside the spread of infection. In both scenarios, the spread of one entity within the network interacts with the spread of the other, creating a coupled dynamical system. A multiplex approach can allow for each coupled process to spread through a network that is based on the appropriate type of contact for propagation (i.e., contact networks involved in pathogen transmission vs. interaction or association networks that allow information to spread). In the case of two infectious diseases concurrently spreading through a network, a multiplex approach can be particularly useful if infection of a node by pathogen A alters the susceptibility to pathogen B, or if coinfection of a node influences its ability to transmit either pathogen. For example, when infection by one pathogen increases the likelihood of becoming infected by another pathogen, it could theoretically facilitate the spread of a second pathogen and thus alter epidemic dynamics. This type of dynamic is likely to widespread in wild and domestic animals due to the importance of co-infection in affecting infectious disease dynamics by influencing the replication of pathogens within hosts. However, when there is competition or cross-immunity, the spread of one pathogen could reduce the spread of a second pathogen. For example, this type of dynamic could be expected for pathogens strains characterized by partial cross-immunity, such as avian influenza, or microparasite-macroparasite coinfections in which infection with one parasite reduces transmission of a second, such as infection with gastrointestinal helminths reducing the transmission of bovine tuberculosis in African buffalo (Syncerus caffer). Similar “within-node” dynamics could be important at a farm-level in livestock movement networks. For example, the detection of a given pathogen infection in a farm might cause it to be quarantined, thus reduce its susceptibility and ability to transmit other pathogen infections.
Dynamics of Coupled Processes: Interactions Between Transmission Networks and Information/Social Networks
For coupled processes involving a disease alongside a social process (i.e., spread of information or disease awareness), we might expect that the spread of the pathogen will be associated with the spread of disease awareness or preventative behaviors such as mask-wearing, and in these cases theoretical models suggest that considering the spread of disease awareness can result in reduced disease spread. A model was presented by Granell, which represented two competing processes on the same network: infection spread (modeled using a Susceptible-Infected-Susceptible compartmental model) coupled with information spread through a social network (an Unaware-Aware-Unaware compartmental model). The authors used their model to show that the timing of self-awareness of infection had little effect on the epidemic dynamics. However, the degree of immunization (a parameter which regulates the probability of becoming infected when aware) and mass media information spread on the social layer did critically impact disease spread. A similar framework has been used to study the effect of the diffusion of vaccine opinion (pro or anti) across a social network with concurrent infectious disease spread. The study showed a clear regime shift from a vaccinated population and controlled outbreak to vaccine refusal and epidemic spread depending on the strength of opinion on the perceived risks of the vaccine. The shift in outcomes from a controlled to uncontrolled outbreak was accompanied by an increase in the spatial correlation of cases. While models in the veterinary literature have accounted for altered behavior of nodes (imposition of control measures) as a result of detection or awareness of disease, it is not common for awareness to be considered as a dynamic process that is influenced by how each node has interacted with the pathogen (i.e., contact with an infected neighbor). For example, the rate of adoption of biosecurity practices at a farm, such as enhanced surveillance, use of vaccination, or installation of air filtration systems, may be dependent on the presence of disease in neighboring farms or the farmers’ awareness of a pathogen through a professional network of colleagues.
There is also some evidence that nodes that are more connected in their “social support” networks (e.g., connections with family and close friends in humans) can alter network processes that result in negative outcomes, such as pathogen exposure or engagement in high-risk behavior. In a case based on users of injectable drugs, social connections with non-injectors can reduce drug-users connectivity in a network based on risky behavior with other drug injectors. In a model presented by Chen, a social-support layer of a multiplex network drove the allocation of resources for infection recovery, meaning that infected individuals recovered faster if they possessed more neighbors in the social support layer. In animal (both wild and domesticated) populations, this concept could be adapted to represent an individual’s likelihood of recovery from, or tolerance to, infection being influenced by the buffering effect of affiliative social relationships. For domestic animals, investment in certain resources at a farm level could influence a premise’s ability to recover (e.g., treatment) or onwards transmission of a pathogen (e.g., treatment or biosecurity practices). Sharing of these resources between farms could be modeled through a “social-support” layer in a multiplex, for example, where a farm’s transmissibility is impacted by access to shared truck-washing facilities.
Multilayer networks can be used to study the features of mixed species contact networks or model the spread of a pathogens in a host community, providing important insights into multi-host pathogens. Scenarios like this are commonplace at the livestock-wildlife interface and therefore the insights provided could be of real interest to veterinary epidemiology. In the case of multi-host pathogens, intralayer and interlayer edges represent the contacts between individuals of the same species and between individuals of different species, respectively. They can therefore be used to identify bottlenecks of transmission and provide a clearer idea of how spillover occurs. For example, Silk used an interconnected network with three layers to study potential routes of transmission in a multi-host system. One layer consisted of a wild European badger (Meles meles) contact network, the second a domesticated cattle contact network, and the third a layer containing badger latrine sites (potentially important sites of indirect environmental transmission). No intralayer edges were possible in the latrine layer. The authors demonstrated the importance of these environmental sites in shortening paths through the multilayer network (for both between- and within-species transmission routes) and showed that some latrine sites were more important than others in connecting together the different layers. Pilosof presented a theoretical model, labeling the species as focal (i.e., of interest) and non-focal, showing that the outbreak probability and outbreak size depend on which species originates the outbreak and on asymmetries in between-species transmission probabilities.
Similar applications of multilayer networks could easily be extended to systems where two or more species are domesticated animals, as well. Examples of these could be the study of a pathogen such as Bluetongue virus, which affects both cattle and sheep, or foot-and-mouth disease virus, which infects cattle, sheep, and pigs. In such cases, each species can be represented by a different level in the network, and interlayer edges are made possible as a result of mixed farms (i.e., cattle and sheep), different species from different farms grazing on the same pasture, or for other types of indirect contacts such as the sharing equipment or personnel.
Overall, multilayer approaches provide an elegant way to analyze cross-species transmission and spillover, including for zoonotic pathogens across the human-livestock-wildlife interface. They can be used to simultaneously model within-species transmission, identify heterogeneities among nodes in their tendency to engage in between-species contacts relevant for spillover and spillback, and better predict the dynamics of spread prior and subsequent to cross-species transmission events, which may contribute to forecasting outbreaks in target species. Measures of multilayer network centrality in this instance could be used to extend the superspreader concept into a community context; individuals that are influential in within-species contact networks and possess between-species connections might be predicted to have a more substantial influence on infectious disease dynamics in the wider community.