Efficiency of prompt quarantine measures on a susceptible-infected-removed model in networks

This study focuses on investigating the manner in which a prompt quarantine measure suppresses epidemics in networks. A simple and ideal quarantine measure is considered in which an individual is detected with a probability immediately after it becomes infected and the detected one and its neighbors are promptly isolated. The efficiency of this quarantine in suppressing a susceptible-infected-removed (SIR) model is tested in random graphs and uncorrelated scale-free networks. Monte Carlo simulations are used to show that the prompt quarantine measure outperforms random and acquaintance preventive vaccination schemes in terms of reducing the number of infected individuals. The epidemic threshold for the SIR model is analytically derived under the quarantine measure, and the theoretical findings indicate that prompt executions of quarantines are highly effective in containing epidemics. Even if infected individuals are detected with a very low probability, the SIR model under a prompt quarantine measure has finite epidemic thresholds in fat-tailed scale-free networks in which an infected individual can always cause an outbreak of a finite relative size without any measure. The numerical simulations also demonstrate that the present quarantine measure is effective in suppressing epidemics in real networks.

Figure 1

Transition rules of the SIR model with quarantine measures. (a) A node $j$ is randomly selected among the susceptible neighbors of infected node $i$. (b) With probability $T$, disease is transmitted from $i$ to $j$. Immediately after (b), (c) this newly infected node $j$ is detected with probability $f$, and (d) node $j$ and its susceptible and infected neighbors (including node $i$) are promptly isolated. (e) If node $j$ is not detected, node $i$ tries to infect the next susceptible neighbor.

Figure 2

The order parameter $r$, as a function of $T$, for (a) the RGs and (b) the SFNs. Here $f$ is set to $f=0.01$ (open symbols) and 0.2 (solid symbols). The red circles, green triangles, and blue squares represent the results for the quarantine measure, the random vaccination, and the acquaintance vaccination, respectively. The black line represents the result of the original SIR model. In (a), the open green triangles (random vaccination with $f=0.01$) fall behind the open blue squares (acquaintance vaccination with $f=0.01$).

Figure 3

The order parameter $r$, as a function of $f$, for the SFNs. Here $T$ is set to $T=0.5$ (open symbols) and $T=1.0$ (solid symbols). The red circles, green triangles, and blue squares represent the results for the quarantine measure, the random vaccination, and the acquaintance vaccination, respectively.

Figure 4

Logarithmic plot of the order parameter, $r$, for (a) the RGs and (b) the SFNs with $N={10}^4$ (red circles), ${10}^5$ (green triangles), and ${10}^6$ (blue squares). The open symbols and solid symbols represent the results of $f=0.01$ and 0.2, respectively. Two vertical lines represent the epidemic threshold $T_c$ of $f=0.01$ (left) and $f=0.2$ (right) as given by Eq. (5).

Figure 5

The probability that a single infected node induces a global outbreak for (a) the RGs and (b) the SFNs with $N={10}^4$ (red circles), ${10}^5$ (green triangles), and ${10}^6$ (blue squares). The detection probability is set to $f=0.01$ (open symbols) and 0.2 (solid symbols), in which symbols denote the fraction of samples such that the fraction of R nodes exceeds 1%. The solid and dashed lines represent the probability of global outbreaks (8) for $f=0.01$ and 0.2, respectively. Two vertical lines represent the epidemic threshold $T_c$ of $f=0.01$ (left) and $f=0.2$ (right), as given by Eq. (5).

Figure 6

Phase boundary in the $(f,T)$ plane of the SIR model with the quarantine measure on (a) the RGs and (b) the SFNs. The solid lines represent epidemic threshold $T_c\left(f\right)$, which is evaluated from Eq. (5). Inset of (b): $T_c\left(f\right)$ for $f\ll 1$. The dotted line denotes $T_c\left(f\right)\propto f^{3/7}$.

Figure 7

The order parameter $r$, as a function of $T$, for real networks: (a) the sexual contact network between Brazilian prostitutes and sex buyers and (b) the friendship network of Gowalla users. Here $f$ is set to $f=0.01$ (open symbols) and 0.2 (solid symbols). The red circles, green triangles, and blue squares represent the results for the quarantine measure, the random vaccination, and the acquaintance vaccination, respectively. The black line represents the result of the original SIR model. The number of nodes is $N=16748$ for the sexual contact network and $N=196591$ for Gowalla network. The data at given $T$ and $f$ are averaged over ${10}^5$ trials.