Continuous-Time Discrete-Distribution Theory for Activity-Driven Networks

Activity-driven networks are a powerful paradigm to study epidemic spreading over time-varying networks. Despite significant advances, most of the current understanding relies on discrete-time computer simulations, in which each node is assigned an activity potential from a continuous distribution. Here, we establish a continuous-time discrete-distribution framework toward an analytical treatment of the epidemic spreading, from its onset to the endemic equilibrium. In the thermodynamic limit, we derive a nonlinear dynamical system to accurately model the epidemic spreading and leverage techniques from the fields of differential inclusions and adaptive estimation to inform short- and long-term predictions. We demonstrate our framework through the analysis of two real-world case studies, exemplifying different physical phenomena and time scales.

Figure 1

Time evolution of the fraction of infected nodes for the flu (a) and Twitter (b) case studies. Comparison between the discrete-time continuous-distribution ADN process (blue, dashed), our continuous-time discrete-distribution approach (green, dotted) model, and theoretical predictions (red, solid) from Eq. (1).

Figure 2

Averaged Monte Carlo simulations of a discrete-time continuous-distribution ADN process (blue) and theoretical bounds on the endemic equilibrium state (computed for $k^{*}=1$, in red), for flu (a) and Twitter (b) case studies. From Table 2, ${\alpha }_1\lambda /\mu$ is equal to 0.988 in (a) and 1.785 in (b).

Figure 3

Averaged Monte Carlo simulations of a discrete-time continuous-distribution ADN process (blue) and theoretical bounds on the dynamics of the epidemic spreading (computed for $k^{*}=2$ with $ϵ={10}^{-3}$, in red), for flu (a) and Twitter (b) case studies.

Figure 4

Simulation of a discrete-time continuous-distribution ADN (blue), for the flu case study, and our predictions over a finite time horizon of one day and one week (red). Predictions in (a) and (b) are obtained with a constant estimate for $C$, while those in (c) and (d) are based on the on-line adaptive update in Eq. (7) with $\beta =1$. A similar result for the Twitter case study is presented in the Supplemental Material [32].