Preserving system activity while controlling epidemic spreading in adaptive temporal networks

TL;DR

This work analytically derive the conditions for a widespread diffusion of epidemics in the presence of arbitrary adaptive behaviors, highlighting the crucial role of correlations between agents behavior in the infected and in the susceptible state.

Abstract

Human behaviour strongly influences the spread of infectious diseases: understanding the interplay between epidemic dynamics and adaptive behaviours is essential to improve response strategies to epidemics, with the goal of containing the epidemic while preserving a sufficient level of operativeness in the population. Through activity-driven temporal networks, we formulate a general framework which models a wide range of adaptive behaviours and mitigation strategies, observed in real populations. We analytically derive the conditions for a widespread diffusion of epidemics in the presence of arbitrary adaptive behaviours, highlighting the crucial role of correlations between agents behaviour in the infected and in the susceptible state. We focus on the effects of sick-leave, comparing the effectiveness of different strategies in reducing the impact of the epidemic and preserving the system operativeness. We show the critical relevance of heterogeneity in individual behavior: in homogeneous networks, all sick-leave strategies are equivalent and poorly effective, while in heterogeneous networks, strategies targeting the most vulnerable nodes are able to effectively mitigate the epidemic, also avoiding a deterioration in system activity and maintaining a low level of absenteeism. Interestingly, with targeted strategies both the minimum of population activity and the maximum of absenteeism anticipate the infection peak, which is effectively flattened and delayed, so that full operativeness is almost restored when the infection peak arrives. We also provide realistic estimates of the model parameters for influenza-like illness, thereby suggesting strategies for managing epidemics and absenteeism in realistic populations.

Figures (8)

• Figure 1: Epidemic threshold for sick-leave strategies. In each panel we show the ratio between the epidemic threshold in the NAD (non-adaptive) scenario, $r_C^{NAD}$, and when a sick-leave strategy is implemented, $r_C$. We consider: in panel (a) the uniform strategy, $r_C=r_C^U$; in panel (b) the targeted strategy, $r_C=r_C^T$; in panel (c) the $\varepsilon$-targeted strategy, $r_C=r_C^{\varepsilon}$. The ratio $r_C^{NAD}/r_C$ is plotted as a function of $\nu$, exponent of the $\rho_S(a_S)$ distribution, and $p$, fraction of sick-leaving nodes, through a heat-map. In all cases we fix the distribution of susceptible activity $\rho_S(a_S) \sim a_S^{-(\nu+1)}$ with lower cut-off $a_m=10^{-3}$ and upper cut-off $a_M=1$, moreover in panel (c) we fix $\varepsilon=0.1$, that is the fraction of high-activity nodes not performing sick-leave. Note that the gray zone in panel (c) corresponds to the region of prohibited $p>1-\varepsilon$ values. These results have been obtained analytically from Eqs. \ref{['eq:NAD']}, \ref{['eq:rcU']}-\ref{['eq:rcE']}.
• Figure 2: Epidemic threshold for $\varepsilon$-targeted strategy. In all panels we consider the ratio between the epidemic threshold in the NAD (non-adaptive) scenario, $r_C^{NAD}$, and when the $\varepsilon$-targeted strategy is implemented, $r_C^{\varepsilon}$. The ratio $r_C^{NAD}/r_C^{\varepsilon}$ is plotted as a function of $\nu$, exponent of the $\rho_S(a_S)$ distribution, and $p$, fraction of sick-leaving nodes, through a heat-map. In all cases we fix the distribution of susceptible activity $\rho_S(a_S) \sim a_S^{-(\nu+1)}$ with lower cut-off $a_m=10^{-3}$ and upper cut-off $a_M=1$. In each panel we consider a different fraction of high-activity nodes not performing sick-leave, $\varepsilon$, reported above each panel. Note that the gray zone in each panel corresponds to the region of prohibited $p>1-\varepsilon$ values. These results have been obtained analytically from Eqs. \ref{['eq:NAD']}, \ref{['eq:rcE']}.
• Figure 3: SIS Epidemic prevalence for sick-leave strategies. In both panels the SIS epidemic prevalence, $\overline{P}$, is plotted as a function of the control parameter, $r$, for different sick-leave strategies. We fix: the fraction of sick-leaving nodes $p=0.3$, the fraction of high-activity nodes not performing sick-leave $\varepsilon = 0.1$ and the distribution of susceptible activity $\rho_S(a_S) \sim a_S^{-(\nu+1)}$ with lower cut-off $a_m=10^{-3}$, upper cut-off $a_M=1$, $\nu=1$ in panel (a) and $\nu=3.5$ in panel (b). The results are obtained by iterating numerically Eqs. \ref{['eq:prevb']}-\ref{['eq:prev']}.
• Figure 4: SIR Epidemic final-size for sick-leave strategies. In both panels the SIR epidemic final-size, $R_{\infty}$, is plotted as a function of the control parameter, $r$, for different sick-leave strategies. We fix: the fraction of sick-leaving nodes $p=0.3$, the fraction of high-activity nodes not performing sick-leave $\varepsilon = 0.1$ and the distribution of susceptible activity $\rho_S(a_S) \sim a_S^{-(\nu+1)}$ with lower cut-off $a_m=10^{-3}$, upper cut-off $a_M=1$, $\nu=1$ in panel (a) and $\nu=3.5$ in panel (b). The results are obtained through numerical simulations over networks of $N=10^3$ nodes: each point is obtained by averaging over at least $10^3$ realizations of the dynamical evolution and of the underlying network, until the errors on $R_{\infty}$ and on the maximum of the infection peak, $\overline{P}_{max}$, are both lower than 1%.
• Figure 5: Infection peak for sick-leave strategies. In both panels we plot the fraction of infected nodes, $\overline{P}(t)$, as a function of time for several sick-leave strategies. We fix: the control parameter $r=3$, the fraction of sick-leaving nodes $p=0.3$, the fraction of high-activity nodes not performing sick-leave $\varepsilon = 0.1$ and the distribution of susceptible activity $\rho_S(a_S) \sim a_S^{-(\nu+1)}$ with lower cut-off $a_m=10^{-3}$, upper cut-off $a_M=1$, $\nu=1$ in panel (a) and $\nu=3.5$ in panel (b). The results are obtained through numerical simulations over networks of $N=10^3$ nodes: each curve is obtained by averaging over at least $10^3$ realizations of the dynamical evolution and of the underlying network, until the errors on the epidemic final-size, $R_{\infty}$, and on the maximum of the infection peak, $\overline{P}_{max}$, are both lower than 1%.