Impact of temporary lockdown on disease extinction in assortative networks

TL;DR

This paper addresses how temporary lockdowns influence extinction risk in the susceptible-infected-susceptible (SIS) model on assortative networks. It combines a theoretical heterogeneous mean-field/WKB framework with kinetic Monte Carlo simulations to quantify how lockdown duration $T$ and strength $\xi$ interact with network heterogeneity (degree variance $\epsilon$) and degree-degree correlations ($\alpha$) to shape the extinction probability $\mathcal{P}$ and mean time to extinction (MTE). The main finding is that larger $T$ or $\xi$ increase $\mathcal{P}$, and that higher heterogeneity or assortativity amplifies this effect, enabling substantially milder lockdowns to achieve a target $\mathcal{P}$, with $\mathcal{P}$ scaling as $\mathcal{P}\sim e^{-N\Delta\mathcal{S}}$ where $\Delta\mathcal{S}$ is the action barrier. These results have practical implications for designing topology-aware interventions in realistic populations and suggest that targeted or structurally informed lockdowns can be markedly more efficient than homogeneous-model predictions.

Abstract

Changing environmental conditions can significantly affect the dynamics of disease spread. These changes may arise naturally or result from human interventions; in the latter case, lockdown measures that lead to abrupt but temporary reductions in transmission rates are used to combat disease spread. However, the impact of these measures on rare events in realistic populations has not been studied so far. Here, we analyze the susceptible-infected-susceptible (SIS) model in a stochastic setting where disease extinction -- a sudden clearance of the infection -- occurs via a rare, large fluctuation. We use a semiclassical approximation and extensive numerical simulations to show how the extinction risk of the disease depends on both the duration and magnitude of the lockdown, in heterogeneous assortative networks, with degree-degree correlations between neighboring nodes.

Figures (5)

• Figure 1: EP for random networks with $N = 1000$, $\langle k \rangle = 50$, and $R_0 = 1.2$. Shown are results for homogeneous (squares), bimodal (circles), and gamma (triangles) networks. For the bimodal and gamma distributions we took a COV of $\epsilon = 0.3$. Panel (a) shows results for varying $T$ with $\xi = 1.0$, while panel (b) corresponds to $T = 2.0$ with varying $\xi$. The solid lines show the analytical results for the homogeneous case.
• Figure 2: EP for networks with $N = 1000$, $\langle k \rangle = 50$, $T = 2.0$, and $\xi = 1.0$: shown are bimodal (circles) and gamma (triangles) networks. Panel (a) shows random networks ($\alpha=0$) with $R_0 = 1.47$ and varying $\epsilon$, while panel (b) shows assortative networks with $R_0 = 1.6$, $\epsilon = 0.5$ and varying $\alpha$.
• Figure 3: Shown are contour lines of equal EP, $\mathcal{P}_T=\mathcal{F}\mathcal{P}_0$. Here, triangles ($\mathcal{F}=10^4$) and circles ($\mathcal{F}=10^6$) represent KMC simulations for different values of $T$, $\xi$, and $\epsilon$, on random gamma-distributed networks with $N = 1000$, $R_0 = 1.24$, and $\langle k \rangle = 50$. The reference value $\mathcal{P}_0=2.96\times10^{-8}$ corresponds to the EP without lockdown in a homogeneous network, which was obtained by performing $10^{10}$ realizations, whereas $\mathcal{P}_T$ was calculated using $10^8$ realizations. In panel (a), the lockdown duration is fixed at $T = 2$ while $\epsilon$ and $\xi$ vary, whereas panel (b) shows results for varying $\epsilon$ and $T$ with fixed $\xi = 1.0$.
• Figure 4: Shown are contour lines of equal EP, $\mathcal{P}_T=\mathcal{F}\mathcal{P}_0$. Here, triangles ($\mathcal{F}=10^4$) and circles ($\mathcal{F}=10^6$) represent KMC simulations for different values of $T$, $\xi$, and $\alpha$, on gamma-distributed networks with $N = 1000$, $R_0 = 1.24$, $\epsilon=0.5$ and $\langle k \rangle = 50$. The reference value $\mathcal{P}_0=2.96\times10^{-8}$ corresponds to the EP without lockdown in a homogeneous network. Here, $\mathcal{P}_0$ and $\mathcal{P}_T$ were obtained by performing $10^{10}$ and $10^8$ realizations, respectively. In panel (a), the lockdown duration is fixed at $T = 2$ while $\alpha$ and $\xi$ vary, whereas panel (b) shows results for varying $\alpha$ and $T$ with fixed $\xi = 1.0$.
• Figure 5: Extinction path $p$ versus $x$, for $R_0 = 1.5$ and $\xi = 0.5$. Solid lines show the unperturbed path [Eq. \ref{['eq:path0']}], while dashed lines show the paths during lockdown [Eq. \ref{['eq:pathenergy1d']}] for durations $T = 1.5, 2.5, 4$ (bottom to top) for which $E_0\simeq 0.01584$, $0.01106$, and $0.00615$. Circles mark the intersections between the perturbed and unperturbed segments. Inset shows a log-log plot of $E_0$ versus $T$ for $R_0=1.5$ and $\xi=0.5$.