Active and inactive quarantine in epidemic spreading on adaptive activity-driven networks

We consider an epidemic process on adaptive activity-driven temporal networks, with adaptive behavior modeled as a change in activity and attractiveness due to infection. By using a mean-field approach, we derive an analytical estimate of the epidemic threshold for susceptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR) epidemic models for a general adaptive strategy, which strongly depends on the correlations between activity and attractiveness in the susceptible and infected states. We focus on strong social distancing, implementing two types of quarantine inspired by recent real case studies: an active quarantine, in which the population compensates the loss of links rewiring the ineffective connections towards nonquarantining nodes, and an inactive quarantine, in which the links with quarantined nodes are not rewired. Both strategies feature the same epidemic threshold but they strongly differ in the dynamics of the active phase. We show that the active quarantine is extremely less effective in reducing the impact of the epidemic in the active phase compared to the inactive one and that in the SIR model a late adoption of measures requires inactive quarantine to reach containment.

Figure 1

Effects of quarantine on the SIS active phase. In panel (a) we plot the ratio $\overline{P}/{\overline{P}}^{\mathrm{NAdwA}}$ as a function of $\delta$ for the active and the inactive quarantine, fixing $r=3r_C^{\mathrm{NAdwA}}$ and $r=12r_C^{\mathrm{NAdwA}}$. In panel (b) we plot the prevalence $\overline{P}$ as a function of $r/r_C^{\mathrm{NAdwA}}$, for the NAdwA case, and for the active and the inactive quarantine, we fix $\delta =0.2$ and $\delta =0.9$. In panel (c) we plot the ratio ${\overline{P}}^{\mathrm{ACT}}/{\overline{P}}^{\mathrm{INACT}}$ between the epidemic prevalence in the active and the inactive case as a function of $r/r_C$ for several values of $\delta$ (with $r_C$ being the threshold for the adaptive case). In all panels, $\nu =0.5, a_m={10}^{-3}$, and $a_M=1$.

Figure 2

Effects of quarantine on the SIR active phase. In panel (a) we plot the epidemic final size $R_{\infty }$ as a function of $r/r_C$ (with $r_C$ being the threshold for the adaptive case), for the NAdwA case and for the active and the inactive quarantine. In panel (b) we plot the temporal evolution of the average infection probability $\overline{P}\left(t\right)$ for the NAdwA case and for the active and the inactive quarantine, fixing $r/r_C=1.4$. In both panels, $\delta =0.9, N={10}^3, \nu =0.5, a_m={10}^{-3}$, and $a_M=1$.

Figure 3

Effects of timing in quarantine implementations. We plot the epidemic final size $R_{\infty }$ as a function of the $\beta$ fraction of nodes been infected before quarantine measures are implemented. We consider active and inactive quarantines, fixing $\delta =0.9$, and we compare them with the NAdwA case. In the insert we plot the ratio $R_{\infty }^{\mathrm{ACT}}/R_{\infty }^{\mathrm{INACT}}$ between the epidemic final size in the active case and in the inactive case as a function of $\beta$ for several $\delta$ values. In both panels, $r/r_C=1.4$ (with $r_C$ being the threshold for the adaptive case), $N={10}^3, \nu =0.5, a_m={10}^{-3}$, and $a_M=1$.

Figure 4

Effects of timing in quarantine implementations. In panels (a) and (b), we plot the temporal evolution of the average infection probability $\overline{P}\left(t\right)$ for the NAdwA case and for the active and the inactive quarantine, fixing $\delta =0.9$ and, respectively, $\beta =0.0025$ and $\beta =0.02$. In panel (c), we plot the ratio ${\overline{P}}_{\max }^{\mathrm{ACT}}/{\overline{P}}_{\max }^{\mathrm{INACT}}$ between the height of the infection peak in the active and the inactive case as a function of $\beta$, for several values of $\delta$. In all panels, $r/r_C=1.4$ (with $r_C$ being the threshold for the adaptive case), $N={10}^3, \nu =0.5, a_m={10}^{-3}$, and $a_M=1$.