Epidemic spreading with awareness diffusion on activity-driven networks

Acknowledging the significance of awareness diffusion and behavioral response in contagion outbreaks has been regarded as an indispensable prerequisite for a complete understanding of epidemic spreading. Recent studies from the research community have accumulated overwhelming evidence for the incessantly evolving structure of the underlying networks. Thus there is an impelling need to capture the interplay between the epidemic spreading and awareness diffusion on time-varying networks. In this paper, we consider a behavioral model in which susceptible individuals become alert and adopt a preventive behavior under the local risk perception characterized by a decision-making threshold. The impact of awareness diffusion on the epidemic threshold is investigated under the framework of activity-driven network. Results show that the local epidemic situation in risk perception and the duration of preventive effect are crucial for raising the epidemic threshold. The analytical results are corroborated by Monte Carlo simulations.

Figure 1

The probability of the nodes with activity $a_1$ connecting to the nodes with activity $a$ in the instantaneous network $G_t$, obtained from numerical simulation and theoretical analysis. We assume there are $n$ different activity classes, in which the nodes' activities are uniformly sampled in (0,1] with the sampling interval $\frac{1}{n}$ and the activity distribution takes the form $F\left(a\right)\propto a^{-2}$. Here we choose $a_1$ as the minimum activity in the network. For the example of $n=5$, the activity is selected from $a\in \{0.2,0.4,0.6,0.8,1\}$ with $a_1=0.2$. The symbols (circle for $n=5$, square for $n=10$, diamond for $n=30$) and dotted lines represent the results of numerical simulations and theoretical analysis, respectively. The size of network is set to be $N=10000$ in each realization, and the simulation results are averaged over 100 realizations.

Figure 2

Time evolution of $S^t$, $A^t$ and $I^t$ under different cases of risk perception with the infection rate $\lambda =0.2$. The results are obtained when (a) $T_i=1,T_a=\infty ,h=1$; (b) $T_i=1,T_a=\infty ,h=\infty$; (c) $T_i=3,T_a=3,h=1$; (d) $T_i=3,T_a=3,h=\infty$. The symbols and dotted lines represent the results of Monte Carlo simulation and the SAIS model, respectively. Other parameters are set to be $N=5000$, $I^0=0.1$, $c=0.5$, $\mu =1$.

Figure 3

Final fraction of infected nodes as a function of the infection rate $\lambda$, for different $T_i$ when the alert threshold is set to be $T_a=\infty$. In the figure, (a) $h=1$ and (b) $h=\infty$. The epidemic threshold suggested by Monte Carlo simulations and the SAIS model are recorded as soon as $I^{\infty }$ is slightly larger than 0.015 (the critical value above which the realization is considered as endemic). In (a), the epidemic threshold given by theoretical analysis is labeled by red triangle. Other parameters are set to be the same as those in Fig. 2.

Figure 4

Epidemic threshold under different infection rates of alert nodes $\tilde{\lambda }=c\lambda$. The epidemic thresholds suggested by Monte Carlo simulations and the SAIS model (31, 32, 33) are recorded following the same steps in Fig. 3. Other parameters are set to be the same as those in Fig. 2.

Figure 5

Final fraction of infected nodes $I^{\infty }$ as a function of infection rate $\lambda$, for different $T_i\mathrm{s}$ and $T_a\mathrm{s}$ with duration $h=1$. From (a) to (f), $T_i$ is respectively fixed at 1,2,3,4,5,6, and $T_a=\{1,2,3,4,5,6\}$. The epidemic thresholds suggested by Monte Carlo simulations and the SAIS model are recorded following the same steps in Fig. 3. Other parameters are set to be the same as those in Fig. 2.

Figure 6

Coupled dynamics of infection and awareness for the risk perception with $h=1,T_i=4,T_a=2$. In the figure, (a) final fraction of infected nodes $I^{\infty }$, (b) time evolution of infected nodes $I^t$, (c) final fraction of alert nodes $A^{\infty }$, and (d) time evolution of alert nodes $A^t$. In (b) and (d), the symbols and dotted lines represent the results of Monte Carlo simulation and the SAIS model, respectively. Other parameters are set to be the same as those in Fig. 2.

Figure 7

Final fraction of infected nodes $I^{\infty }$ as a function of infection rate $\lambda$, for different $T_i\mathrm{s}$ and $T_a\mathrm{s}$ with duration $h=\infty$. From (a) to (f), $T_i$ is respectively fixed at 1,2,3,4,5,6 and $T_a=\{1,2,3,4,5,6\}$. The epidemic thresholds suggested by Monte Carlo simulations, the SAIS model and theoretical analysis are recorded and labeled following the same steps in Fig. 3. Other parameters are set to be the same as those in Fig. 2.

Figure 8

Final fraction of infected nodes $I^{\infty }$ with time-varying activity as a function of infection rate $\lambda$ for three cases of threshold conditions. In the figure, (a), (b), and (c) are related to the duration $h=1$, and (d), (e), and (f) are related to the duration $h=\infty$. Other parameters are set to be the same as those in Fig. 2.