Effect of network clustering on mutually cooperative coinfections

The spread of an infectious disease can be promoted by previous infections with other pathogens. This cooperative effect can give rise to violent outbreaks, reflecting the presence of an abrupt epidemic transition. As for other diffusive dynamics, the topology of the interaction pattern of the host population plays a crucial role. It was conjectured that a discontinuous transition arises when there are relatively few short loops and many long loops in the contact network. Here we focus on the role of local clustering in determining the nature of the transition. We consider two mutually cooperative pathogens diffusing in the same population: An individual already infected with one disease has an increased probability of getting infected by the other. We look at how a disease obeying the susceptible-infected-removed dynamics spreads on contact networks with tunable clustering. Using numerical simulations we show that for large cooperativity the epidemic transition is always abrupt, with the discontinuity decreasing as clustering is increased. For large clustering strong finite-size effects are present and the discontinuous nature of the transition is manifest only in large networks. We also investigate the problem of influential spreaders for cooperative infections, revealing that both cooperativity and clustering strongly enhance the dependence of the spreading influence on the degree of the initial seed.

Figure 1

(a) Degree distribution generated by the Serrano-Boguñá algorithm using a Poisson degree distribution with average degree $z=4$ (red circles). The red solid line represents the preassigned distribution. (b) Clustering distributions of the Poisson networks generated by the Serrano-Boguñá algorithm (symbols). Solid lines represent the preassigned clustering distributions which are $\overline{c}\left(k\right)=c_0{(k-1)}^{-{\alpha }_0}$, with ${\alpha }_0=0.6$. From bottom to top, the points represent clustering for networks with $c_0=0.2$ (red circles), $c_0=0.4$ (blue squares), $c_0=0.6$ (magenta diamonds), and $c_0=0.8$ (black triangles), respectively. Data are obtained by averaging over 200 realizations. The network size is $N={10}^5$. The parameter ${\beta }_0$ regulating the assortativity is fixed to ${\beta }_0=0.1$.

Figure 2

Results for a single doubly infected seed. (a) Final fraction of population in the doubly recovered state (ab) versus $p$ for networks of size $N={10}^4$ and different values of $c_0$. Each point is a single realization. There are 200 realizations for each value of $p$. Values of $p$ are increased by intervals $\mathrm{\Delta }p=0.002$. (b) Probability $P_{ab}$ that ${\rho }_{ab}>T=0.005$. Initially, all nodes are susceptible except for a randomly selected node which is in the doubly infected state.

Figure 3

Results for two distinct singly infected seeds. (a) Final fraction of population in the doubly recovered state (ab) versus $p$ for networks of size $N={10}^4$ and different values of $c_0$. Each point is a single realization. There are 200 realizations for each value of $p$. Values of $p$ are increased by intervals $\mathrm{\Delta }p=0.002$. (b) Probability $P_{ab}$ that ${\rho }_{ab}>T=0.005$. Initially all nodes are susceptible except for two singly infected nodes randomly selected provided the distance between them is larger than 8.

Figure 4

Temporal evolution of densities of singly or doubly recovered nodes (states a, b, or ab, respectively) for cooperative SIR dynamics around the threshold ($p=0.25$). The network has size $N={10}^5$. Initially all nodes are susceptible except for a single randomly-selected node which is in the doubly infected (AB) state.

Figure 5

Results of simulations for zero ($c_0=0$) and high ($c_0=0.8$) connectivity and different system sizes. Panels (a) and (b) are the results for $c_0=0$ for networks of sizes $N=3\times {10}^4,{10}^5,3\times {10}^5,{10}^6$. For each value of $N$ the data are averaged over ${10}^5$ realizations: (a) shows the average final fraction $\langle {\rho }_{ab}\rangle$ of the population in the doubly infected state (AB) for large outbreaks versus $p$; (b) shows the probability $P_{ab}$ that ${\rho }_{ab}>T=0.05$. Panels (c) and (d) show the same quantities for the case of large clustering, $c_0=0.8$. Values of $\langle {\rho }_{ab}\rangle$ are not plotted if the number of realizations to be averaged is less than 10.

Figure 6

Results for $c=0.8$ and $q=0.4$ for networks of sizes $N={10}^5,3\times {10}^5,{10}^6$. For each value of $N$ the data are averaged over ${10}^5$ realizations. (a) Average final fraction $\langle {\rho }_{ab}\rangle$ in the doubly recovered state (ab) for large outbreaks versus $p$; (b) probability $P_{ab}$ that ${\rho }_{ab}>T=0.002$. Values of $\langle {\rho }_{ab}\rangle$ are not plotted if the number of realizations with ${\rho }_{ab}>T$ is less than 10.

Figure 7

Histograms of ${\rho }_{ab}$ in the doubly recovered state (ab) for $N={10}^5$ (dotted lines) and $N={10}^6$ (solid lines) and $N_r={10}^5$ realizations. Top: $c=0.8$ and $q=1.0$. Each peak corresponds to a value of $p$ starting with $p=0.27$ for the rightmost one and decreasing of $\mathrm{\Delta }p=0.005$ at each curve. The main is in lin-lin scale, and the inset is the same plot in lin-log scale. Sharper peaks on the right correspond to the larger system, while on the left they are suppressed as the system size increases, indicating a critical value of $p$ around 0.25. Bottom: $c=0.8$ and $q=0.4$. Each peak corresponds to a value of $p$ starting from right with $p=0.35$ and decreasing of $\mathrm{\Delta }p=0.01$ at each curve. The main is in lin-lin scale, and the inset is the same plot in lin-log scale. Sharper peaks correspond to the larger system. In this case, peaks corresponding to the larger system are never suppressed, consistently with a continuous transition.

Figure 8

Results from simulations on networks with $c_0=0$ (red) and $c_0=0.8$ (black), with $p=0.255$. System size is $N={10}^5$ (filled symbols) and $N={10}^6$ (empty symbols). Main: probability $P_{ab}$ that a double-infection started in a node of degree $k$ originates a macroscopic outbreak. Inset: Average relative size of the macroscopic outbreaks when the infection is seeded in a node of degree $k$.