Concurrency-Induced Transitions in Epidemic Dynamics on Temporal Networks

Social contact networks underlying epidemic processes in humans and animals are highly dynamic. The spreading of infections on such temporal networks can differ dramatically from spreading on static networks. We theoretically investigate the effects of concurrency, the number of neighbors that a node has at a given time point, on the epidemic threshold in the stochastic susceptible-infected-susceptible dynamics on temporal network models. We show that network dynamics can suppress epidemics (i.e., yield a higher epidemic threshold) when the node’s concurrency is low, but can also enhance epidemics when the concurrency is high. We analytically determine different phases of this concurrency-induced transition, and confirm our results with numerical simulations.

Figure 1

Schematic of an activity-driven network with $m=3$.

Figure 2

Epidemic threshold and the numerically-simulated prevalence when $m=1$ (a),(c) and $m=10$ (b),(d). In (a) and (b), all nodes have the same activity potential value $a$. The solid lines represent the analytical estimate of the epidemic threshold [Eq. (8); we plot Eq. (9) instead in (a)]. The dashed lines represent the epidemic threshold obtained from the individual-based approximation (Supplemental Material [32]). The color indicates the prevalence. In (c) and (d), the activity potential ($\epsilon \le a_i\le 0.9$, $1\le i\le N$) obeys a power-law distribution with exponent 3. In (a)–(d), we set $N=2000$ and adjust the values of $a$ and $\epsilon$ such that the mean degree is the same ($⟨k⟩=0.1$) in the four cases. We simulate the stochastic SIS dynamics using the quasistationary state method [36], as in [31], and calculate the prevalence averaged over 100 realizations after discarding the first 15 000 time steps. We set the step size $\mathrm{\Delta }t=0.002$. Qualitatively similar results are obtained for the variant of the activity-driven model with a reinforcement mechanism of link creation [37] (Fig. S3 [32]).

Figure 3

Phase diagrams for the epidemic threshold, ${\beta }_c$, when the activity potential is (a) equal to $a$ for all nodes, or (b) obeys a power-law distribution with exponent 3 ($\epsilon \le a_i\le 0.9$). We set $⟨k⟩=0.1$ at $m=1$ and adjust the value of $a$ and $\epsilon$ such that ${\beta }_c$ takes the same value for all $m$ at $\tau =0$. In the “die out” phase, infection eventually dies out for any finite $\beta$. In the “suppressed” phase, ${\beta }_c$ is larger than the ${\beta }_c$ value at $\tau =0$. In the “enhanced” phase, ${\beta }_c$ is smaller than the ${\beta }_c$ value at $\tau =0$. The solid and dashed lines represent ${\tau }_{*}$ [Eq. (10)] and ${\tau }_c$, respectively. The color bar indicates the ${\beta }_c$ values. In the gray regions, ${\beta }_c>100$.