Distinction and Connection between Contact Network, Social Network, and Disease Transmission Network

Abstract

In this paper we discuss the distinction and connection between three closely related networks in animal ecology and epidemiology studies: the contact, social, and disease transmission networks. We provide a robust theoretical definition and interpretation of these three networks, demonstrate that social and disease transmission networks can be derived as spanning subgraphs of contact network, and show examples based on real-world high-resolution cattle contact structure data. Furthermore, we establish a modeling framework to track potential disease transmission dynamics and construct transmission network based on the observed animal contact network.

Networks are fundamental concepts for current domestic animal and wildlife studies, as many ecological and epidemiological processes take place on a network structure (Krause et al., 2014). Among various networks, contact network, social network, and disease transmission network are three important, closely related, but different types of networks. However, current studies sometimes use them without clear definitions or appreciation of the interrelationship or distinction among these three types of networks (Eubank et al., 2004; Harmede et al., 2009; Eames et al., 2015). In this paper, we compare the relationship and difference between these networks from a graph theory perspective using empirical data from a cattle study as an example, and specifically discuss how the edges in these three networks are formed (i.e., the dyadic interaction between a pair of individual animals).

Recent technology advances have facilitated researchers to construct highly precise and accurate animal contact networks by using radio-frequency tags and/or proximity loggers, and have revealed the dynamic nature (e.g., changing intensity of dyadic interaction throughout a day) of the animal contact network (Chen et al., 2013; 2014; 2015). These devices have been widely used in many ecological and epidemiological studies to facilitate network analysis (Krause et al., 2014). From a graph theory perspective, a network (or equivalently a graph, G) comprises two major components: the set of vertices (V) and the set of edges (E) which represent the dyadic interaction between vertices: G={V, E} (Newmann, 2010).

In general, we propose that both social and disease transmission networks can be regarded as different spanning subgraphs of the contact network (assuming the set of vertices remain the same, i.e., a constant population size in the network without demography such as birth/death/immigration/emigration, etc). Spanning subgraph are subgraphs (i.e., a subset of both V and E in the original graph) obtained by only edge deletion (V remains the same in the spanning subgraph, and E is a subset of original network/graph). We argue that social networks focus on social interactions and not every contact warrants a social interaction. For disease transmission network, the same contact network can give rise to multiple potential transmission networks, depending on individual’s epidemiological status (i.e., the contact network alone is not sufficient to infer “Who Acquire Infection From Whom”, WAIFW). Furthermore, certain social interactions are also able to facilitate disease transmission (Hens et al., 2009; Pellis et al., 2015). Thus, contact network, social network, and disease transmission network cannot be used interchangeably but they are tightly related, and the details of such connections and distinctions are discussed as follows:

Contact network (CN)

Contact network captures quantitative information of position, proximity and contacts (e.g., frequency and duration in weighted network) between pairs of individual animals in a certain period of time. Before the development and deployment of automated positioning devices, contact information were usually captured by direct observation from researchers, and was time consuming and labor intensive. Thanks to the technology advances, nowadays proximity loggers, radio-frequency tags, and GPS collars have been widely implemented in both confined and open areas for human beings, domestic animals, and wildlife to provide precise and accurate positioning and contact information for extended periods of time. There are some differences in using proximity loggers and radio-frequency tags/GPS collars to construct the contact networks. Proximity loggers send out and receive signals simultaneously within a short distance range (acting as transceiver, i.e., both transmitter and receiver together in one device), record the contact information using a pre-defined threshold distance to determine whether two individuals are in close proximity and consequently in contact (e.g., contacting with whom for how long), and do not record individual’s absolute position (e.g., in x-y-z coordinates). This system is highly efficient (it only records data whenever there is a contact), but the trade-off is that only relative position information is available. On the other hand, radio-frequency tags and GPS collars only emit signals out and rely on external receivers (e.g., radio-frequency receiver for tags, and satellite for GPS) to record the real-time absolute position of the animals (e.g., x-y-z coordinates for radio-frequency tags, and latitude/longitude data for GPS are available in real-time), and then use trigonometry to determine the distance between individuals to further construct the contact network (by using a pre-defined threshold distance for a contact, too). This is a more tedious framework, but more useful when the absolute position information is required in certain studies, such as investigating both host-host contact and host-environment contact at the same time.

According to graph theory, any network has a graph (G) representation as G={V, E}, meaning a network consists two components, the vertices (V) and edges (E). Structurally, contact networks are usually weighted networks (i.e., the edges of the network carries weight information). Here we focus on the set of edges, since edges contain the contact information while vertices are just the individuals being observed (assuming a closed population without birth, death, immigration, and emigration). The edge weight can be described as number (or frequency) of contacts between each pair of vertices (individuals) in a certain period of time, which reflects the intensity of the dyadic interaction. Here we present a real-world hourly cattle contact network that was constructed from a group of 21 cattle with radio-frequency tags from our previous study (Chen et al., 2014). Generally, the thicker the edge, the more contacts between that pair of individuals (Fig 1. upper left panel). While animal i contacting animal j simultaneously implies animal j contacting animal i, we can further differentiate whether one of the animal initiates the contact. And the latter is an example of directed network (e.g., directional information of contact is recorded in the network). Directed contact network can be generated by using accelerometers as supplement to the basic positioning devices (such as the radio-frequency tags). By knowing both the position and acceleration information together, we can explicitly infer which animal is approaching the other one (or whether they are embracing/charging each other at the same time).

Fig 1. Contact Network (CN), Social Network (SN), and Two Potential Transmission Networks (TN) derived and inferred from the same set of real-time animal position data.

Note: Fig 1 demonstrates the relationship and differences between CN (upper left panel), SN (upper right panel), and TN (two lower panels). CN is constructed from the real-time location system with a group of 21 individual calves wearing radio-frequency tags within one-hour period (8–9 AM, Aug 11, 2011). The width of the edges in CN indicates the intensity of interactions among individual calves. SN and TNs are both subgraphs of CN (they share the same set of vertices V, and the edges in SN and TN are subsets of the edges in CN). SN is derived by filtering out the potential random contacts (e.g., if two vertices have less than 5 contacts then their contacts are considered random) and converting the edges into binary (only non-random contacts lead to an edge). TN is derived by randomly selecting an initially infected individual, calculating transmission probability based on edge weights of CN, and forming a directed edge if transmission occurs. Because these networks focus on different aspects in zoological studies, their structures are different: CN is usually undirected and weighted, SN is usually undirected and unweighted, and TN is usually directed and unweighted. A certain CN is usually associated with one SN, but can give rise to multiple potential TNs (because of different initial number of infected individuals and system stochasticity associated with transmission probability).

Social network (SN)

Social network deals with social interactions and associations in a group (or groups) of animals. Generally speaking, a social interaction should occur within a certain distance to ensure physical interaction such as grooming and other intimate behaviors (exceptions include vocal communication, which can occur over long distance and does not necessarily require a physical contact, and hence is not discussed in this paper). Thus, accurate contact network also facilitates more accurate inference of social network. A close proximity contact is often a necessary condition for social interaction (i.e., usually no animal social interaction can occur without a physical contact, as discussed within this paper), but not sufficient condition: observing a physical contact between a pair of individual animals does not necessarily warrant a social interaction between them. A recorded contact must satisfy some set of criteria to be considered as a social interaction, such as longer than certain threshold duration. It is not surprising that a contact could be purely random, or triggered by some external factors. For example, recent studies have shown that the contact structure among cattle during feeding hours is substantially different from the other time periods especially resting hours in a day (Chen et al. 2015). While social networks record social interactions, which are relatively stable (especially should be stable during a short period of time, e.g. within a day), we can conclude that the contacts during feeding hours are not necessarily all social contacts, and the animals are competing for food or other resources.

Besides the difference in how the edges are formulated, social networks do not necessarily incorporate the information of interaction intensity (while contact networks usually do, as in the form of weighted network) and hence are commonly treated as binary network instead (i.e., focusing on whether or not a pair of individuals are connected, instead of how strongly they are connected; Fig 1. upper right panel). Furthermore, social interactions can be classified as “positive” (e.g., intimate behavior such as grooming) or “negative” (e.g., antagonistic behavior especially among reproductively mature male individuals). Such interaction cannot be directly obtained from the contact network, as we cannot directly differentiate whether or not a contact is friendly or aggressive. Thus, the more information we have besides contact network alone, the better we can characterize the animal social network.

Nevertheless, social network and contact network share the same set of vertices (i.e., same individuals in the network, assuming a constant population), and social network can be derived from contact network. From a graph theory perspective, a social network (SN) is a spanning subgraph (i.e., the subgraph shares the same set of vertices with the complete graph but only a subset of the edges) of contact network (CN): SN ⊆ CN. Note that any social interaction must rely on a physical contact ensures that the edges in SN (social interaction) must be a subset of CN (physical contact).

Transmission network (TN)

For directly transmitted diseases, pathogen transmission occurs mostly through close physical contact. Thus contact network also serves as a cornerstone for transmission network, and the more accurate the contact network we construct, the better we can evaluate pathogen transmission risk. The transmission probability between a pair of individuals is generally computed as a function of contact intensity (i.e., the weight of edges in the contact network), by a sigmoid or logistic function, for instance. Intuitively, it indicates that the more contacts, the higher probability of pathogen transmission. Nevertheless, knowing the probability of pathogen transmission is only the first step to construct the transmission network, or “Who Acquire Infection From Whom” network, WAIFW.

The complexity of transmission network besides transmission probability lies on the individual epidemiological status. In a widely used susceptible-infected-susceptible (SIS) compartment modeling framework, transmission can only occur from infected individual I to susceptible individual S, thus requiring a contact specifically between S and I for transmission. Consequently, any contact between two Ss or between two Is does not contribute to pathogen transmission (i.e., pathogen cannot transmit if neither individual has it, nor is it important if both individuals already have the pathogen). That being said, a transmission network is a directed network (always from I to S) even if the contact is usually treated as non-directional. In fact, effective transmission networks (post-hoc) obtained from epidemiological studies only show the successful transmissions and do not incorporate the complete contact network structure. Furthermore, unlike contact network, transmission network is usually unweighted, and the edges indicate the flow of pathogen in the network.

In the absence of information regarding the epidemiological status of each individual over time, we demonstrate a modeling framework to infer potential transmission network from observed contact network. This modeling framework can be applied to either static contact network (where the network structure remains the same through time) or dynamic network (where the network structure varies, Chen et al. 2014, 2015). To simplify we demonstrate with static network here but it can be easily expanded to dynamic contact network scenario. Besides the contact structure information from the contact network, we still need some additional knowledge to infer the transmission network.

First, we need to know the epidemiological states of the given individuals in the population/contact network. For example, the number of initially infected individuals is an important factor for transmission network formulation, as no transmission shall occur if there is no infected individuals. In a widely adopted Susceptible-Infectious-Recovered (SIR) type of transmission model, each individual can be at one of the three epidemiological states: susceptible, infectious, or recovered. These states can be coded as numeric values (e.g., 0, 1, and 2). Then, we need to couple the contact structure information with epidemiological states. The only possible way for a new infection to occur between two animals i and j must satisfy the following conditions simultaneously:

1. There is positive number of contacts between i and j (i.e., no transmission shall occur if there is not any contact);

2. Either individual is infectious, the other one is susceptible (i.e., no transmission shall occur if both are susceptible, or both are infectious, or both are recovered, or one is susceptible while the other one is recovered);

3. The Bernoulli outcome of the total transmission probability (product of number of contacts, derived from the contact network, and the per-contact transmission probability) indicates a new infection outcome other than remaining susceptible for the susceptible individual (i.e., system stochasticity determines the outcome of a potential infectious contact).

At each time step, the simulated current individual’s epidemiological states are provided and current contact network is used to extract the contact number data. Then we identify the potential pairs of individuals (infectious and suscpetible) for new transmission, calculate the total transmission probability for each pair (combing the information of weights in the contact network, i.e., number of contacts), and update individual’s simulated epidemiological states based on the Bernoulli trial. Simulated recovery will be assumed to occur in a constant rate.

Thus we demonstrate that a given contact network is able to give rise to multiple potential transmission networks. First, even if we assume the same number of initially infected individual in the network (e.g., 1 initially infected), this individual could be randomly chosen in the contact network. Second, system stochasticity (as we see from the Bernoulli outcome) can also alter the potential transmission network. Consider a fixed initially infected individual and we can compute the total transmission probability for each of its connected susceptible neighbors in the contact network. Then whether or not a transmission occur depends on the outcome of the Bernoulli trial associated with the transmission probability. Thus, a given contact network can generate more than one potential transmission network because of such system stochasticity (Fig 1. two lower panels). In summary, a transmission network (TN) is also a spanning subgraph of contact network (CN): TN ⊆ CN, and we can safely conclude because transmission also relies on contact networks.

Now, we realize that both social network and transmission network can be considered as spanning subgraphs of the original contact network, thus we are able to apply similar network analysis techniques for them (e.g., finding the network density, modularity, and centrality, etc). The difference between social and transmission network is that social networks emphasize on the social interaction (more on behavior aspects), whereas transmission networks focus on the potential of pathogen transmission. The contacts between unfamiliar individuals i and j do not attribute into the social network, but there is still certain possibility of pathogen transmission for such random/unintentional contacts. On the other hand, a real social interaction may not necessarily trigger pathogen transmission, either (e.g., if neither individual is infected). Nevertheless, even if social network and transmission network are two different concepts, there are still connections between them and they are not completely independent. Many animal social behaviors, for example, grooming (e.g., licking among cattle) increase the risk of pathogen and parasite transmission – but that is still relevant to individual’s epidemiological status and system stochasticity, too.

To summarize, contact network is the foundation of both social network and transmission network. A given contact network usually associates with one certain true social network, but the same contact network is able to generate multiple potential simulated transmission networks.