Two-pathogen model with competition on clustered networks

Networks provide a mathematically rich framework to represent social contacts sufficient for the transmission of disease. Social networks are often highly clustered and fail to be locally treelike. In this paper, we study the effects of clustering on the spread of sequential strains of a pathogen using the generating function formulation under a complete cross-immunity coupling, deriving conditions for the threshold of coexistence of the second strain. We show that clustering reduces the coexistence threshold of the second strain and its outbreak size in Poisson networks, while exhibiting the opposite effects on uniform-degree models. We conclude that clustering within a population must increase the ability of the second wave of an epidemic to spread over a network. We apply our model to the study of multilayer clustered networks and observe the fracturing of the residual graph at two distinct transmissibilities.

Figure 1

The three triangles that a focal node may be connected to. (A) The focal node has two uninfected neighbors (green), neither of which are capable of transmitting infection. (B) Both nodes are infected (red), but each direct edge fails to infect the focal node. (C) Only one neighbor is infected; however, it can infect the focal node by first infecting the susceptible neighbor and then a further transmission to the focal node.

Figure 2

The percolation properties of the 2-strain model over clustered doubly-Poisson networks with clustering coefficient, $\mathcal{C}$, and fixed average degree $\mu +2\nu =2$ of treelike and triangles, respectively. (a) The epidemic threshold of strain-1 (solid) as a function of $\mathcal{C}$. The critical thresholds for a GC to exist solely among treelike edges (small dashed lines) or triangle edges (long dash lines from Eq. (19) are plotted in (a). Similar analysis in plot (b) shows the coexistence threshold, $T^{*}$, as a function of increasing clustering coefficient from Eq. (20). Also plotted in (b) is the difference $T_{\delta }=T_c-T^{*}$ between the epidemic and coexistence thresholds. Plot (c) shows the expected epidemic size of each strain. Scatter points indicate experimental results of bond percolation on a network of size $N=40000$ with 70 repetitions. Solid lines represent the theoretical predictions of Eqs. (11) and (17) for each strain.

Figure 3

The residual network of the three networks described in Sec. 2b. In this model [9, 17], clustering can be shown to increase both the size of the GRCC and also the coexistence threshold relative to the configuration model. We also observe dichotomous results depending on the nature of the degree assortativity among the clustered edges. When clustering is assortatively confined to low-degree nodes, the results of the Poisson experiment are reproduced.

Figure 4

An example of the multilayer network used to in the numerical example. The green layer consists solely of triangles while the orange layer is treelike. Each layer is connected via a few tree-like edges to allow the GC to span the network.

Figure 5

The expected epidemic size of each strain on a Poisson distributed clustered multilayer network with 2-layers. In this experiment, the orange layer has a clustering coefficient of $C=0$ while the green layer is set to $C=1/3$. Interlayer treelike edges have been added to allow the GC to span the entire network. Scatter points indicate experimental results of bond percolation on a network of size $N=20000$ with 25 repeats. Solid lines represent the theoretical predictions of Eqs. Also plotted is the SLCC and the SLRCC, peaks in which indicate a phase transition. From this plot we can see that peaks in the SLCC and the SLRCC do not align with each other, their separation defines the region of coexistence of both strains.