Rapid decay in the relative efficiency of quarantine to halt epidemics in networks

Several recent studies have tackled the issue of optimal network immunization by providing efficient criteria to identify key nodes to be removed in order to break apart a network, thus preventing the occurrence of extensive epidemic outbreaks. Yet, although the efficiency of those criteria has been demonstrated also in empirical networks, preventive immunization is rarely applied to real-world scenarios, where the usual approach is the a posteriori attempt to contain epidemic outbreaks using quarantine measures. Here we compare the efficiency of prevention with that of quarantine in terms of the tradeoff between the number of removed and saved nodes on both synthetic and empirical topologies. We show how, consistent with common sense, but contrary to common practice, in many cases preventing is better than curing: depending on network structure, rescuing an infected network by quarantine could become inefficient soon after the first infection.

Figure 1

(a) Ratio between the fraction of nodes $n_Q$ saved by quarantine and the fraction of nodes $n_{\text{NI}}$ not reached by the infection in the absence of any intervention as a function of the fraction $R$ of nodes recovered at the time the quarantine is implemented. The network is an Erdös-Rényi graph with $N={10}^5$ nodes. The curves are for growing values (bottom to top) of the spreading probability $p$. The dashed line indicates when the ratio is 1, i.e., the value discriminating between convenient quarantine (ratio $>1)$ and inconvenient quarantine (ratio $<1)$. (b) The same plot for a scale-free network with $\gamma =3$. (c) The same plot for a random geometric graph. For this figure (and in all the others) we performed 10 000 realizations of the process. Solid lines are average values computed over 100 intervals (of width 0.01), while the shaded areas around them are the variations between the minimum and the maximum in 1000 intervals.

Figure 2

(a) Ratio between the fraction $n_Q$ of nodes saved by quarantine and the fraction of nodes $n_{\text{OP}}$ saved by preventive immunization as a function of the fraction $R$ of nodes in state R when quarantine is applied. The network is an Erdös-Rényi graph with $N={10}^5$ nodes. The curves are for growing values (top to bottom) of the spreading probability $p$. The dashed line indicates when the ratio is 1, i.e., the value discriminating between convenient quarantine (ratio $>1$) and inconvenient quarantine (ratio $<1$). (b) The same plot for a scale-free network with $\gamma =3$. (c) The same plot for a random geometric graph.

Figure 3

Comparison between analytical predictions [Eq. (4)] and simulation results for the fraction $R^{*}$ of recovered nodes after which quarantine becomes less convenient than preventive immunization. The dashed line indicates perfect equality. Simulations are performed on Erdös-Rényi (ER) networks with various values of the average degree $〈k〉$ and on a scale-free (SF) network. Error bars are not shown when smaller than symbols size. System size is $N={10}^5$.

Figure 4

Ratio between the fraction $n_Q$ of nodes saved by quarantine and the fraction of nodes $n_{\text{OP}}$ saved by “informed” preventive immunization (i.e., with $p$ known a priori) as a function of the fraction of nodes removed when quarantine is applied. The network is an Erdös-Rényi graph with $N={10}^5$ nodes. The curves are for growing values (top to bottom) of the spreading probability $p$.

Figure 5

Ratio between the fraction $n_Q$ of nodes saved by quarantine and the fraction of nodes $n_{\text{OP}}$ saved by preventive immunization as a function of the fraction of nodes removed. The topology is (a) WAN and (b) U.S. power grid. In the WAN topology (a) we report both the results of the simulations implementing either the complete quarantine (dashed lines) or the improved one (solid lines). The curves are for growing values (top to bottom) of the spreading probability $p$.

Figure 6

Comparison of the number of nodes to be quarantined $Q$ as a function of $R$ in the case of complete quarantine (dashed lines) or improved quarantine (solid lines). Different panels refer to different values of the spreading probability $p$. The topology is the WAN.

Figure 7

Scatter plot of $t^{*}$ vs $\overline{t}$ for ER graphs with various $〈k〉$ and scale-free with $\gamma =3$ and $k_{\text{min}}=3$ ($〈k〉\approx 5.01$) showing that the two quantities are almost identical. The solid line indicates perfect equality.