Effect of individual behavior on epidemic spreading in activity-driven networks

In this work we study the effect of behavioral changes of individuals on the propagation of epidemic diseases. Specifically, we consider a susceptible-infected-susceptible model over a network of contacts that evolves in a time scale that is comparable to the individual disease dynamics. The phenomenon is modeled in the context of activity-driven networks, in which contacts occur on the basis of activity potentials. To offer insight into behavioral strategies targeting both susceptible and infected individuals, we consider two separate behaviors that may emerge in respiratory syndromes and sexually transmitted infections. The first is related to a reduction in the activity of infected individuals due to quarantine or illness. The second is instead associated with a selfish self-protective behavior of susceptible individuals, who tend to reduce contact with the rest of the population on the basis of a risk perception. Numerical and theoretical results suggest that behavioral changes could have a beneficial effect on the disease spreading, by increasing the epidemic threshold and decreasing the steady-state fraction of infected individuals.

Figure 1

Epidemic threshold ${\sigma }^{\text{AR}}$ as a function of the activity ratio ${\eta }_I/{\eta }_S$, from Eq. (2), for a network of $N=10000$ nodes, with $m=5$, ${\eta }_S=15$, $0\le {\eta }_I\le 15$, and $\mu =0.1$. A distribution $F\left(x\right)\propto x^{-\gamma }$, with $\gamma =2.1$ and a lower cutoff $\epsilon ={10}^{-3}$, is adopted.

Figure 2

Steady state fraction $I^{\infty }/N$ of infected individuals (color coded) as a function of $\lambda /\mu$ and ${\eta }_I/{\eta }_S$. The white solid line defines the theoretical threshold computed according to Eq. (2) and the white dashed line offers a conservative estimate of the epidemic threshold computed on the steady-state data by setting the fraction of infected individuals to 0.001. A distribution $F\left(x\right)\propto x^{-\gamma }$, with $\gamma =2.1$ and a lower cutoff $\epsilon ={10}^{-3}$, is adopted. Results are averaged over 50 independent trials with an initial infected number of $0.01N$ random individuals.

Figure 3

Trend of the steady-state fraction $I^{\infty }/N$ of infected individuals as a function of $\lambda /\mu$, for the same network as in Fig. 2, for two different pairs of activity rates: ${\eta }_I=1,{\eta }_S=15$ (diamonds) and ${\eta }_I=14,{\eta }_S=15$ (circles). The red (gray) closed diamond and the circle represent, respectively, theoretical epidemic thresholds computed through Eq. (2).

Figure 4

Thresholds' ratio ${\sigma }^{\text{SAR}}/{\sigma }^{\text{0}}$, computed as in Eq. (4), as a function of the exponent $\gamma$ of the activity distribution $F\left(x\right)\propto x^{-\gamma }$, with a lower cutoff $\epsilon ={10}^{-3}$. A network of $N=10000$ nodes is considered. Results are averaged over 1000 realizations and error bars depict plus or minus one standard deviation.

Figure 5

Steady state fraction $I^{\infty }/N$ of infected individuals (color coded) with a risk perception behavior as in Eq. (5), as a function of $1/\overline{I}$ and $\lambda /\mu$, for ${\overline{\eta }}_S={\eta }_I=15$, $\mu =0.1$, and $N=10000$. The white solid line defines the epidemic threshold for uniform and constant activity ${\eta }_S={\eta }_I=15$, computed according to Eq. (1), and the white dashed line offers a conservative estimate of the epidemic threshold with the inclusion of risk perception behavior, computed by setting the fraction of infected individuals to 0.001. A distribution $F\left(x\right)\propto x^{-\gamma }$, with $\gamma =2.1$ and a lower cutoff $\epsilon ={10}^{-3}$, is adopted. Results are averaged over 50 independent trials with an initial infected number of $0.01N$ random individuals.

Figure 6

Steady state fraction $I^{\infty }/N$ of infected individuals with a risk perception behavior as in Eq. (5), as a function of $\lambda /\mu$, for ${\overline{\eta }}_S={\eta }_I=15$, $\mu =0.1$, and $N=10000$. Shown in blue (solid) is the case with no risk perception behavior; in red (circles) and green (diamonds) are the cases $1/\overline{I}={10}^{-4}$ and 1, respectively. The network parameters are the same as in Fig. 5.

Figure 7

Steady state fraction $I^{\infty }/N$ of infected individuals (color coded) with a risk perception behavior as in Eq. (6), as a function of $1/\overline{\Delta }$ and $\lambda /\mu$, for ${\overline{\eta }}_S={\eta }_I=15$, $\mu =0.1$, and $N=10000$. The white solid line defines the epidemic threshold for uniform and constant activity ${\eta }_S={\eta }_I=15$, computed according to Eq. (1), and the white dashed line offers a conservative estimate of the epidemic threshold with the inclusion of risk perception behavior, computed by setting the fraction of infected individuals to 0.001. The network parameters are the same as in Fig. 5.

Figure 8

Steady state fraction $I^{\infty }/N$ of infected individuals with a risk perception behavior as in Eq. (6), as a function of $\lambda /\mu$, for ${\overline{\eta }}_S={\eta }_I=15$, $\mu =0.1$, and $N=10000$. Shown in blue (solid) is the case with no risk perception; in black (circles), green (squares), and red (crosses) are the cases with $1/\overline{\Delta }={10}^{-3},{10}^{-2}$, and 1, respectively. The network parameters are the same as in Fig. 5.