Interacting epidemics on overlay networks

The interaction between multiple pathogens spreading on networks connecting a given set of nodes presents an ongoing theoretical challenge. Here, we aim to understand such interactions by studying bond percolation of two different processes on overlay networks of arbitrary joint degree distribution. We find that an outbreak of a first pathogen providing immunity to another one spreading subsequently on a second network connecting the same set of nodes does so most effectively if the degrees on the two networks are positively correlated. In that case, the protection is stronger the more heterogeneous the degree distributions of the two networks are. If, on the other hand, the degrees are uncorrelated or negatively correlated, increasing heterogeneity reduces the potential of the first process to prevent the second one from reaching epidemic proportions. We generalize these results to cases where the edges of the two networks overlap to arbitrary amount, or where the immunity granted is only partial. If both processes grant immunity to each other, we find a wide range of possible situations of coexistence or mutual exclusion, depending on the joint degree distribution of the underlying networks and the amount of immunity granted mutually. These results generalize the concept of a coexistence threshold and illustrate the impact of large-scale network structure on the interaction between multiple spreading agents.

Figure 1

The method we use to study interacting epidemics on overlay networks. (a) We start with a set of nodes connected by two random networks ${\Gamma }_1$ (solid lines) and ${\Gamma }_2$ (dashed lines). (b) We then select or “occupy” edges on ${\Gamma }_1$ independently with probability $T_1$. If $T_1$ is large enough, a giant connected component of occupied edges emerges (thick solid lines), representing a large outbreak and occupying a fraction $S_1\left(T_1\right)$ of the nodes (white dots). (c) The nodes in that component (gray dots) are considered immune and removed from ${\Gamma }_2$, leaving the residual network of ${\Gamma }_1$ on ${\Gamma }_2$ (black dashed lines). (d) Edges on that network are then occupied with probability $T_2$, with a giant connected component of occupied edges (thick dashed lines) emerging if $T_2$ is larger than a threshold which depends on $T_1$ and the structure of ${\Gamma }_1$ and ${\Gamma }_2$. If the immunity granted by the first epidemic is only partial the gray nodes are not removed in step c, but edges connecting to them on ${\Gamma }_2$ (gray dashed lines) are assumed to be occupied with a reduced probability.

Figure 2

The critical transmission probability $T_{\text{C},2}$ as a function of heterogeneity for $T_{\text{C},1}=0.25$, expressed as the standard deviation in the degree distribution for an average of $⟨k⟩=6$, for positive (squares), negative (circles), and no (triangles) correlation between degrees of nodes on both networks. The critical transmission as determined from simulations of the dynamic SIR model on a finite network of $N=10000$ nodes is shown for comparison (open symbols). Where only a filled dot can be seen, the open one is in the same position. Details on how the networks were generated, and how the critical transmission probabilities estimated from simulations, can be found in the appendix.

Figure 3

The critical transmission probability (solid line) and fraction of nodes in the GCOC (long dashes) for a process running on a random network with $T_2=0.66$ after a previously running process with $T_1=0.33$ has granted partial immunity in the sense that nodes in the GCOC of the first process have their edges occupied by the second with probability $\sigma T_2$.

Figure 4

(a) and (b) Regions of possible coexistence for two epidemics spreading with transmission probabilities $T_1$ and $T_2$ and complete mutual immunity on nonoverlapping overlay networks with independently distributed degrees of average degree $⟨k_1⟩=⟨k_2⟩=6$ with standard deviation (a) $s=2.5$ and $s=8$ (b). Regions where none of the two processes can form a GCOC are black and ones where only one can form a GCOC are white, with the number inside the area indicating which of the two processes is able to do so. In regions of dark gray, there can be coexistence, but also exclusion of either process by the other, whereas in areas or light gray, there is always coexistence. (c) Regions of possible coexistence for $T_1=T_2$ as a function of standard deviation in node degree for complete mutual immunity on nonoverlapping overlay networks with independently distributed degrees of average degree $⟨k_1⟩=⟨k_2⟩=6$.

Figure 5

(a) and (b) The effect of partial immunity $\sigma =0.5$ (a) and mutual facilitation $\sigma =1.5$ (b) on the regions of possible coexistence for the network used in Fig. 4a (average degree $⟨k_1⟩=⟨k_2⟩=6$ with standard deviation $s=2.5$). (c) Regions of possible coexistence for $T_1=T_2$ as a function of partial immunity $\sigma$ in the same network.