Symbiotic and antagonistic disease dynamics on networks using bond percolation

In this paper we introduce a description of the equilibrium state of a bond percolation process on random graphs using the exact method of generating functions. This allows us to find the expected size of the giant connected component (GCC) of two sequential bond percolation processes in which the bond occupancy probability of the second process is modulated (increased or decreased) by a node being inside or outside of the GCC created by the first process. In the context of epidemic spreading this amounts to both an antagonistic partial immunity and a synergistic partial coinfection interaction between the two sequential diseases. We examine configuration model networks with tunable clustering. We find that the emergent evolutionary behavior of the second strain is highly dependent on the details of the coupling between the strains. Contact clustering generally reduces the outbreak size of the second strain relative to unclustered topologies; however, positive assortativity induced by clustered contacts inverts this conclusion for highly transmissible disease dynamics.

Figure 1

From top left: A primary disease has spread over a network of susceptible nodes (gray) to create a giant component of infected hosts (red) at its equilibrium. A seed node (dotted dark green) of degree $k=4$ has degree 2 in the RG (uninfected neighbors) and $k-l=2$ strain 1 infected neighbors. In the general case, the second strain then spreads both on the RG with transmissibility $T_2$ and over the GCC of strain 1 (hatched light green) with modulated transmissibility $T_2^{\prime }$. The light green nodes are coinfected with both strain 1 and strain 2, while the red (dark green) nodes only have strain 1 (2).

Figure 2

The 14 motifs that surround both focal nodes in the percolation model. In each case the lowest node in the motif is the focal node, for which there are two considerations, uninfected (yellow) and infected (red). There are three states a neighboring node could be in: uninfected (green), infected externally (gray), and infected directly (blue), although the latter occurs only when the focal node is infected. Motifs A–C are the triangle motifs surrounding an uninfected focal node; D and E are the two types of tree-like edges. Motifs F–H are the tree-like edges that could surround an infected focal node, and finally, motifs I–N are the triangles that an infected focal node can belong to. Among these motifs, there are 18 unique node sites in total. Symmetric triangles (about a vertical axis bisecting the focal node) contribute only one site type, while mixed triangles contribute two site types and tree-like edges contribute one site type each. The numbering convention for mixed triangles always proceeds from left to right; for instance, in mixed state triangle B the uninfected neighbor is B1, while the infected neighbor is B2. Also, we do not have to include the mirror image of mixed triangles since they occur with equal probability.

Figure 3

The outbreak fractions for several $(T_2,T_2^{\prime })$ combinations of the model all in the absence of clustering. (a) Complete cross-immunity $(T_2,T_2^{\prime })=(1,0)$, (b) partial cross-immunity $(T_2,T_2^{\prime })=(0.6,0.39)$, (c) partial coinfection $(T_2,T_2^{\prime })=(0.4,0.7)$, and (d) perfect coinfection $(T_2,T_2^{\prime })=(0,0.6)$. Markers are the average of 50 repeats of bond percolation over CCM networks of size $N=65000$, $\alpha =2.0$, and $\theta =0$; square markers are strain 1, while circles are strain 2. Solid lines are the theoretical results of Eq. (21) for strain 2.

Figure 4

The outbreak fractions for both strains for clustered [Eq. (23)] and unclustered networks as $T_1$ is varied under two disease couplings: (a) partial cross-immunity $(T_2,T_2^{\prime })=(0.6,0.4)$ and (b) partial coinfection $(T_2,T_2^{\prime })=(0.4,0.7)$. Simulations are the average of 50 repeats of bond percolation on networks with $N=35000$ and $\theta =0.0$, with $\alpha =2.0$ and 0.5 for the unclustered and clustered networks, respectively. Solid lines are the theoretical results of Eqs. (11) and (21). In general, clustering reduces the extent of plural infections in the network; however, degree assortativity within the contact topology causes a reversal of this at high (low) values of $T_1$ in (a) [(b)].

Figure 5

The outbreak fractions of the degree-$\delta$ model with clustering constrained to low degree assortativity for (a) partial immunity $(T_2,T_2^{\prime })=(0.8,0.65)$ and (b) partial coinfection $(T_2,T_2^{\prime })=(0.6,0.75)$ disease interactions. Simulations are the average of 50 repeats of bond percolation on networks with $N=35000$, $\alpha =2.0$; solid lines are the theoretical results of Eqs. (11) and (21). Clustering reduces the fraction of the network that becomes infected by both strains for all values of $T_1$.