To illustrate the effect of contact structure on the resulting phylogenetic tree, we perform simulations of epidemic outbreaks for three different network models: (a) the Erdös-Rényi (ER) random graph [29], (b) the Barabási-Albert (BA) graph [30] and (c) the Watts-Strogatz (WS) graph [31] with a low rewiring probability, (see Methods). Both the BA and the WS are representative for two important aspects of contact heterogeneity. Networks generated by the BA model have both a large variance in the degree distribution as well as short mean path lengths. Networks generated by the WS model have low degree distribution variance and long mean path lengths. We compare these network models to a full graph (FG), which corresponds to a model with random mixing.

Each of the three network models can be tuned with different parameters. The ER model generates networks with Poisson degree distributions where the mean can be varied. The BA model produces scale-free networks with a power-law degree distribution. The WS model produces networks that have a high degree of local clustering, but a degree distribution that lies between a Dirac distribution (all nodes have the same degree) and a Poisson distribution.

We track the exact spreading pattern (i.e. who infects whom) of a susceptible-infected-removed (SIR) epidemic for different networks generated by each model to obtain the infection tree for each of these networks. The parameters for the network models are chosen such that all networks have the same mean degree, , yet different degree distributions, path length distributions and clustering coefficients. Figure 1A shows the imbalance measured by the Sackin index of the resulting infection trees for the three network models and the full graph at different values of , captured by the transmissibility (see Methods for detailed definitions of the Sackin index and the transmissibility).

For large values of (large ) the whole network is infected, independent of the contact structure. Not surprisingly, the epidemic size is similar for all network types in this parameter range, since almost all individuals in the population eventually become infected before the epidemic dies out. The balancedness of the resulting trees, however, differ significantly for the three networks types. The ER model is virtually indistinguishable from the random mixing model (FG). For sufficiently large the BA model has higher imbalance than the ER and FG. Finally, the most striking difference in imbalance is observed for the WS.

For low (low ) the imbalance vanishes for all networks for the simple reason that no epidemic outbreak occurs (see Figure 1B). Interestingly, the imbalance is generally largest at , where the transmissibility is just large enough for an epidemic outbreak to occur. In this case each individual infects just one other individual on average, which results in an infection tree that continuously mostly branches off to one side and thus is maximally unbalanced (Figure S2).

For all network types except the BA model, the imbalance of the transmission tree is maximal for values of right around , but then converges to a smaller value as approaches unity. This can be explained by the fact that the SIR infection process is equivalent to a birth-death process. When , the death rate is de facto zero (birth rate death rate) and thus all lineages survive to the end. If the death rate vanishes the expected imbalance of the resulting transmission trees is minimal and is given by the Yule model (see Methods).

The level of imbalance of the transmission trees for the different network types shown in Figure 1 obviously depends on the choice of the network model parameters. In the following we will investigate how imbalance depends on the average number of neighbors and on local connectivity. Moreover, we henceforth use the expected Sackin index given by the Yule model to define a normalized Sackin index (see Methods), which has an expected value of zero for infection trees based on an SIR model with death rate zero.

We focus on the ER graph because in the limit of a large number of neighbors this model is expected to converge to the random mixing model. Furthermore, to eliminate contributions to imbalance resulting from a non-zero death rate we show the results for . Figure 2 shows the imbalance for an ER model with nodes and an average number of neighbors . The effect of on networks generated by the WS and BA model are reported in the supporting text S2.

Increasing the mean number of neighbors essentially increases the number of infections caused by a single individual and therefore the imbalance is expected to decreases with increasing number of neighbors. This is confirmed by the results presented in Figure 2. A small average number of neighbors results in more unbalanced transmission trees for a reason that is similar to why nonzero death rates increase imbalance. Once a node has infected all of its neighbors, it can no longer infect anyone else and is essentially removed from the system despite remaining infectious.

In the WS model, the mean path length is directly related to the rewiring probability [31]. The WS model with rewiring probability essentially generates the same type of network as the ER model. Therefore the imbalance of the transmission trees resulting from epidemics spreading on such networks should converge with increasing rewiring probability to the same value as for ER random graphs. Figure 3 shows imbalance as a function of the rewiring probability and transmissibility.

We identify two limiting cases for the imbalance of the epidemic. For values of there is essentially no epidemic outbreak and the imbalance remains small. For values of close to but larger than an epidemic can occur and the imbalance is maximal. As increases further, the number of shortcuts in the network increases and the mean path length decreases, as does the imbalance. For values substantially larger than the network converges to something similar to an ER graph and the hence normalized imbalance converges to zero (for and ) or to a fixed value for finite populations and small mean degree.

In the supporting text S1 we derive an analytical approximation for the normalized Sackin index given the transmission network (see Figure S1),(1)Here, is the average number of infections caused by an infected individual until that individual is removed (i.e. the excess degree in the transmission network) and is the mean shortest path length in the transmission network. This equation shows that assuming the transmission network were known, imbalance depends on one hand on the mean path length, , and on the other hand on the average excess degree . For networks generated by the configuration model, depends on the first and second moment of the degree distribution. BA networks are characterized by a large degree distribution variance, as well as a short mean path length. For low rewiring probabilities, WS networks have small degree distribution variances and large mean path lengths. These observations together with the analytical approximation in equation (1) can help explain why it is not always possible to distinguish between the BA and WS models when considering the Sackin index as a measure of tree topology (see Figure 1). This ambiguity is most pronounced when considering two idealized networks: a chain and a star. These two topologically very different networks would result in identical transmission trees (see Figure S2) and therefore be indistinguishable using tree imbalance alone.

Note that and in equation (1) refer to the transmission network rather than the actual contact network. The connection between contact networks and transmission networks has recently been studied in the context of epidemic percolation networks [32]. Unfortunately, the exact relationship between the quantities and in the transmission network and the contact network has not yet been described. However, since the transmission network is a subgraph of the contact network, it is feasible to assume that contact networks that display long or short mean shortest paths also result in transmission networks with long or short mean shortest paths, respectively, and contact networks that have large or small mean excess degrees result in transmission networks with large or small mean excesss degrees, respectively.

Up to this point we have only considered the case where the full transmission network is known and we can thus infer the average phylogenetic tree of the disease outbreak. It is clear, however, that in the real world we only have access to a limited subset of leaves from a phylogenetic tree. It is thus necessary to study the robustness of the tree shape under random sampling of leaves.

Figure 4 shows the imbalance of the tree as a function of the number of sampled lineages. All non sampled branches are pruned from the tree and the sampled branches are joined together at their last common ancestor to create the sample tree (see Figure 5A). For small enough sampling sizes (around 1%) the ER and WS graphs become indistinguishable, indicating that the imbalance is driven by the finer structures of the tree, rather than the backbone. The imbalance of the BA network converges much slower to that of the ER network.

Figures 5B and 5C show two schemes of time sampling for which we study the effect on tree imbalance. In the first scheme we truncate the tree at a time point before the end of the epidemic (Figure 5B). This corresponds to the situation where samples of all individuals in an ongoing epidemic are available. In the second scheme, we use only those sequences from individuals that are infectious at a time point and exclude sequences from individuals who are no longer infectious or have died before (Figure 5C). This corresponds to a snapshot of an epidemic.

In Figure 6A, we observe that tree balancedness saturates at a certain value for ER and BA models, even before the epidemic has stopped. In the case of the WS model, tree imbalance continues to grow exponentially until the last individual has been infected. This indicates that in the ER and BA models, the early stages of the epidemic contribute more strongly to tree imbalance. In contrast, in the WS model the late stage infections contribute more strongly than the early stage infections. This is consistent with the observations made in the case of random sampling, since random sampling tends to destroy the tree structure towards the tips of the tree, while conserving the structure towards the root of the tree. This differentiation can no longer be observed when a snapshot of the epidemic is used to create the tree (Figure 6B).

The two schemes of time sampling are studied here because they are characteristic for data sampling in different biological contexts. The first scheme reflects the typical situation for real epidemics for which sequence information is sampled over a broad time window. The second scheme is more applicable to phylogenetic trees based on pathogen populations from within an individual host. While we have concentrated so far on the inference of epidemiological contact structure from phylogenetic trees, we note that our approach can also be used to study the imbalance of within-host trees, which may result from spatial structure or compartmentalization. Both these schemes are idealizations of available real data. In most situations the sample structure will in fact be a combination of one of the two time sampling schemes and random sampling as discussed in the previous section.

Above we demonstrated that contact structure can result in strongly unbalanced trees. Here we investigate whether real epidemics also result in unbalanced trees. To this end we examine the imbalance of a phylogenetic tree constructed from 5961 patient sequences of the Swiss HIV cohort study [33] (see Figure 7).

Since SIR dynamics with low (i.e. small mean degree or transmissibility close to the critical value ) can potentially also generate strongly unbalanced trees, we compare the imbalance of the HIV tree to an SIR epidemic with random mixing and an , corresponding to the range of realistic that has been estimated for the HIV epidemic in Switzerland [34]. The sampled individuals cover 30–40% of all Swiss HIV infected individuals and we therefore restrict the total epidemic size to the range . It has been argued that the HIV epidemic is still in the exponential stage in developed countries [35]. However, because saturation of an epidemic also causes increased imbalance, we make the conservative assumption that the total population is finite and can be equal to the current epidemic size. We take the range of possible population sizes to be . As a null model, we use a likelihood-free test of departure from random mixing based on [36]. We repeatedly sample parameters uniformly from the intervals above and simulate an epidemic outbreak using these parameters under the assumption of random mixing. We then randomly sample between and individuals from the simulated tree and calculate the normalized Sackin index of the resulting subtree (blue line and shaded areas in Figure 7). We compare this to subtrees with identical number of sampled individuals from the HIV tree from [33] (red line and shaded areas in Figure 7; see Figures S3, S4 and S6 for an analysis using an alternative imbalance measure, as well as a more detailed view of the effect of individual parameters on tree imbalance).

Comparing the HIV tree with an SIR epidemic with equal number of individuals connected by random mixing shows that the HIV tree exhibits strong imbalance. The normalized Sackin index of the HIV tree is with a minimum/maximum of / based on 100 bootstrap trees constructed from sequences with the amino acid positions resampled. The range of values of the normalized Sackin index of the HIV tree as well as the bootstrap trees is outside the 95% confidence interval for the SIR model, implying that the imbalance of the HIV tree is statistically highly significant.

One important component of contact structure in the HIV epidemic is the preferential transmission within transmission groups (such as heterosexuals, intravenous drug users, and men having sex with men) [33]. Subepidemics occurring within these transmission groups are therefore expected to show decreased levels of imbalance. Indeed, calculating the Sackin index for the three largest transmission clusters [33] reveals much more balanced trees in these subepidemics (see Figure 7). However, the observed level of imbalance is still significant, suggesting that contact structure is present even within these transmission groups. As we pointed out above, the imbalance in the SIR model increases with approaching . Therefore, the significance of the imbalance of the subepidemics depends on the choice of and thus .

In summary, our analysis of the HIV tree reveals substantial imbalance in the entire epidemic, possibly extending to the subepidemics, which is consistent with what would be expected from our knowledge of HIV transmission.

The Swiss HIV cohort study was approved by individual local institutional review boards of all participating centers (www.shcs.ch). Written informed consent was obtained for each SHCS study participant.

We consider a disease spreading amongst a susceptible population that displays susceptible-infected-removed (SIR) type dynamics [1], [40]. In the limit of large population size and random mixing the model can be described by the simple system of differential equations(2)(3)(4), and are the number of susceptible, infected, and removed individuals in each compartment at time . Here, is the rate of transmission per contact between a susceptible and infected individual and is the removal rate of infected individuals. In the context of a network the transmissibility is the probability that an individual will transmit the disease across a single contact over the whole duration of the epidemic. This can be calculated from and by averaging over the distribution of waiting times for transmission and recovery. For a given recovery time , the probability that transmission occurs before the individuals recovers is given by . Thus, if the recovery times are exponentially distributed [5],(5)The basic reproductive ratio is the number of secondary infections caused by an infected individual placed into a wholly susceptible population () [1]. In fully mixed populations, an epidemic can occur when . Here, . In non-homogeneous populations this threshold also depends on the contact structure. For networks generated by the configuration model [41], [42], i.e. random contact networks with a given degree distribution, the expected total number of second neighbors (neighbors of my neighbors) is given by , where and are the first and second moments of the degree distribution [43]. Then is the average number of nodes two steps away per neighbor. Thus the expected number of secondary infections per infected individual is . For an epidemic to occur must be greater than 1 or [2], [5]. When the population additionally displays community structure (such as clustering and modularity) this threshold changes again. For example, the Watts-Strogatz model incorporates local connectedness by starting with a regular network where every node is connected to a fixed number of close neighbors. Then, each connection is rewired to a randomly chosen node with a certain probability, thus creating shortcuts in the contact network [31]. In this case the threshold for an epidemic outbreak also depends on this rewiring probability [37].

Since we are not interested in the exact values of the parameters, we can choose by rescaling without loss of generality. Furthermore, it should be noted that as approaches 1, gets much larger than . Hence, the SIR model with large effectively reduces to an SI model.

In order to simulate transmission trees of epidemics occurring in heterogeneously connected populations, a C₊₊ implementation of Gillespie's Next-Reaction Method was used [44]. At the beginning of the simulation a single node is infected and a recovery time is sampled from the distribution of recovery times, . Each of the node's susceptible neighbors is then infected after a time chosen from the distribution of infections times, . If the infection time is shorter than the recovery time, the link is activated and the node is infected at time . The procedure is then repeated for each newly infected node. In case a node is scheduled to be infected by multiple neighbors, the earliest infection takes priority. By keeping track of who-infects-whom, each epidemic outbreak yields an infection tree.

We study three different network models: (a) the Erdös-Rényi (ER) random graph [29], (b) the Barabási-Albert (BA) graph [30] and (c) the Watts-Strogatz (WS) graph [31]. In the ER random graph every individual is connected to every other individual with a certain probability . This results in a graph with a Poissonian degree distribution with mean number of neighbors . The BA graph is constructed by preferential attachment. Each node is sequentially added to the graph and attached to neighbors, where nodes that already have many neighbors have a higher probability of being connected to the new node. This results in a degree distribution with a power-law tail. Such graphs are often referred to as scale-free [30]. Finally, WS graphs start out with a ring lattice, in which every node is connected to its nearest neighbors. Each link is then updated with probability in such a way that one end of the link is rewired to a randomly chosen node. Thus the node that loses the link decreases its degree by one and the node that the link is rewired to increases its degree by one. This process introduces shortcuts in the graph (i.e. decreases the mean shortest path) [31]. For the graph has strongly connected communities. For all links are randomly assigned and the graph is similar to the ER graph with the same mean number of neighbors (equal number of edges) [42]. For intermediate values of , the graphs often display both strong community structure and short path lengths, which are characteristics of small-world graphs [31].

The shape of a phylogenetic tree is described in part by its imbalance. Here, we use the Sackin index as a measure of imbalance [45], because of its analogy to path lengths in graph theory. The Sackin index is defined as follows: Let the distance of a leaf be the number of internal nodes that need to be traversed when following the path from the root of the tree to a leaf . Then the Sackin index is the sum of all such paths,(6)

When considering transmission trees, it is important to differentiate between two cases: The first case considers the complete transmission trees of an epidemic outbreak. This is essentially equivalent to a birth/death process. From the perspective of an individual, death corresponds to removal from the infectious class or the depletion of all its susceptible neighbors. In either case that individual can no longer infect anyone else. Thus the transmission trees have branches that do not all survive until the end of the epidemic.

In the second type of tree, all lineages are extant at the end of the epidemic. Such a transmission tree could be generated by an SI-type epidemic in an infinite size population where each individual can infect every other individual. These trees are generated by the Yule model.

The expected value of the Sackin index for a given number of leaves under the Yule model is given by [26],(7)with Euler's constant .

An exact expression for the expected value of the Sackin index is not known in the case where some lineages die before the end of the epidemic. However, it can be assumed that this will in general result in slightly more unbalanced trees.

Since the expected value of the Sackin index increases with tree size, we introduce a normalized Sackin index defined by(8) measures the relative deviation of the tree imbalance from what would be expected for an SI epidemic (or SIR with ).