Epidemic spreading in networks with nonrandom long-range interactions

An “infection,” understood here in a very broad sense, can be propagated through the network of social contacts among individuals. These social contacts include both “close” contacts and “casual” encounters among individuals in transport, leisure, shopping, etc. Knowing the first through the study of the social networks is not a difficult task, but having a clear picture of the network of casual contacts is a very hard problem in a society of increasing mobility. Here we assume, on the basis of several pieces of empirical evidence, that the casual contacts between two individuals are a function of their social distance in the network of close contacts. Then, we assume that we know the network of close contacts and infer the casual encounters by means of nonrandom long-range (LR) interactions determined by the social proximity of the two individuals. This approach is then implemented in a susceptible-infected-susceptible (SIS) model accounting for the spread of infections in complex networks. A parameter called “conductance” controls the feasibility of those casual encounters. In a zero conductance network only contagion through close contacts is allowed. As the conductance increases the probability of having casual encounters also increases. We show here that as the conductance parameter increases, the rate of propagation increases dramatically and the infection is less likely to die out. This increment is particularly marked in networks with scale-free degree distributions, where infections easily become epidemics. Our model provides a general framework for studying epidemic spreading in networks with arbitrary topology with and without casual contacts accounted for by means of LR interactions.

Figure 1

Average age of the nearest neighbor nodes in different age groups (see text) by using the WS model with node ages and a random rewiring of links. The ages are organized from top to bottom at probability 0.0 in the following groups: 0–5, 5–10, 10–15, 15–20, 20–30, 30–40, 40–50, 50–60, 60–70, and 70+.

Figure 2

Average age of the nearest neighbor nodes in different age groups (see text) by using the WS model with node ages in which the rewiring probability depends explicitly on the internode distance. The ages are organized in groups as in Fig. 1.

Figure 3

Results of the simulations (dashed lines) and the exact GNLDS-MMCA (solid lines) for (left) ER and (right) BA networks having $n=1000$ nodes, $\overline{k}\approx 6$, with $\beta =0.02$ and $\delta =0.12$, and for different values of the parameter $x$. The results are the average of 250 realizations. The values of conductivity parameters are, from bottom to top, 0.0, 0.03, 0.06, 0.09, and 0.13.

Figure 4

(a) Evolution of the epidemic threshold for a star graph with $n=1000$ nodes as a function of $x$. (b). Results of the simulations (dashed lines) and the exact GNLDS-MMCA (solid lines) for a star graph with $n=1000$ nodes, where $\tau =1/\sqrt{999}\approx 0.0316.$ Parameters in the left plot are $\beta =0.002$, $\delta =0.01$ and starting with about 20% nodes infected; since $\beta /\delta =0.01>\tau$, the infection will become epidemic for any value of the conductance $x$. Parameters in the right plot are $\beta =0.0002$, $\delta =0.024$; thus $\beta /\delta =0.0083<\tau$, and even starting with all nodes infected, the epidemic dies out for all values of the parameter $x$. The results are the average of 100 realizations. The values of conductivity parameter are, from bottom to top, 0.0, 0.03, 0.06, 0.09, 0.13, 0.20, 0.30, and 0.50.

Figure 5

Results of the simulations (dashed lines) and the exact GNLDS-MMCA (solid lines) for networks with power-law degree distributions $p\left(k\right)\sim k^{-\gamma }$ with (left) $\gamma =1.89$ and (right) $\gamma =1.98$. The results are the average of 100 realizations for networks with $n=1000$ nodes, $\beta =0.02$, and $\delta =0.12$ and for different values of the parameter $x$. The values of conductivity parameter are, from bottom to top, 0.0, 0.03, 0.06, 0.09, 0.13, 0.15, and 0.20.

Figure 6

Rate of epidemic spreading measured by $t_{1/2}$, i.e., the time needed to infect 50% of the whole population, for different values of the power-law exponent $\gamma$ and for different values of the conductance $x$.

Figure 7

Probability of infection $\overline{d}\left(k\right)x^{\overline{d}\left(k\right)-1}$ for different values of the conductance $x$ for BA and ER networks having the same number of nodes, $n=1000$, and the same average degree.

Figure 8

Illustration of the evolution of the degree of the nodes in a network created with the BA model as the values of the conductance change. Here for a BA network with $n=1000$ and $\overline{k}\approx 6$ we represent the values of the generalized degree $k\left(x\right)$ for every node for different values of $x$. Notice that there is an inversion of the centrality of the nodes as the values of the conductance increases.

Figure 9

Illustration of the evolution of the cumulative degree distribution of the nodes for the network studied in Fig. 8 for (a) $x=0$, (b) $x=0.3$, (c) $x=0.5$, and (d) $x=1$. Notice that there is change in the distribution from a power law at $x=0$ to Poissonian-like distributions as $x$ increases.

Figure 10

(a) Change of the rank correlation between $k_i\left(x=0\right)$ and $k_i\left(x\ne 0\right)$ as a function of the conductance $x$ for scale-free (SF) networks with different exponents of the power law. (b) Scaling of the conductance at which an inversion in the rank correlation occurs as a function of the exponent of the power law in SF networks (see text for explanations).

Figure 11

Results of the simulations and GNLDS-MMCA for the networks of collaboration among (left) jazz musicians and (right) for the corporate directors of the top 500 corporations in the United States. The results are the average of 250 realizations with $\beta =0.02$, $\delta =0.12$ and for different values of the conductance parameter $x$. The values of conductivity parameter are, from bottom to top, 0.0, 0.03, 0.06, 0.09, 0.13, and 0.20.