Epidemic spreading and aging in temporal networks with memory

Time-varying network topologies can deeply influence dynamical processes mediated by them. Memory effects in the pattern of interactions among individuals are also known to affect how diffusive and spreading phenomena take place. In this paper we analyze the combined effect of these two ingredients on epidemic dynamics on networks. We study the susceptible-infected-susceptible (SIS) and the susceptible-infected-recovered (SIR) models on the recently introduced activity-driven networks with memory. By means of an activity-based mean-field approach, we derive, in the long-time limit, analytical predictions for the epidemic threshold as a function of the parameters describing the distribution of activities and the strength of the memory effects. Our results show that memory reduces the threshold, which is the same for SIS and SIR dynamics, therefore favoring epidemic spreading. The theoretical approach perfectly agrees with numerical simulations in the long-time asymptotic regime. Strong aging effects are present in the preasymptotic regime and the epidemic threshold is deeply affected by the starting time of the epidemics. We discuss in detail the origin of the model-dependent preasymptotic corrections, whose understanding could potentially allow for epidemic control on correlated temporal networks.

Figure 1

Plot of the ratio ${\lambda }_c/{\lambda }_{\text{ML}}$ between the epidemic thresholds in the memory (WM) and in the memoryless (ML) cases, for different values of the exponent $\nu$ of the distribution $F\left(a\right)=B{a}^{-(\nu +1)}$. The dashed line is the mean-field memoryless results, while the solid lines are the outcomes of the ABMF equations in the presence of memory.

Figure 2

The SIS model on the effective static network for $\nu =2.4, \epsilon =0.01, \mu =1.5\times {10}^{-2}$, and $N={10}^4$. The ratio between the epidemic threshold found in simulations and the result of the memoryless model in Eq. (7) is shown as a function of $\langle k\rangle /N$. For $\langle k\rangle /N>0.01$, we observe practically no dependence on $\langle k\rangle$.

Figure 3

The SIS model on the effective static network. The ratio between the epidemic threshold with memory and the epidemic threshold of the memoryless case is shown as a function of the reinforcement parameter $\beta =[0.01,0.2,0.4,0.6,1,1.4,1.8,2.2]$, with $\nu =2.4, \epsilon =0.01, \mu =1.5\times {10}^{-2}$, and $N=5\times {10}^4$. The points are averages of different realizations of the network with different sequences of activity $a_1,a_2,...,a_N$: 32 realizations for $\langle k\rangle =6$, 16 realizations for $\langle k\rangle =20$, and 4 realizations for $\langle k\rangle =100$.

Figure 4

The SIS epidemic process on the activity-driven network MA. The lifespan $L$ is shown as a function of the ratio between the epidemic threshold with memory and the epidemic threshold of the memoryless for different values of ${\langle k\rangle }_0$, with $N=5\times {10}^4, \nu =2.4$, and $a\in [{10}^{-2},1]$. We consider $4\times {10}^3$ epidemic realizations for each value of $\lambda$. The dynamics starts from the most active site and at small ${\langle k\rangle }_0$ backtracking effects are dominant, favoring the epidemic spreading; this on one side lowers the value of the threshold (value of $\lambda$ corresponding to the peak) but also increases the lifespan of the system at small $\lambda$.

Figure 5

The SIS epidemic process on the activity-driven network RA. The ratio between the epidemic threshold with memory and the epidemic threshold of the memoryless case is shown as a function of the reinforcement parameter $\beta =[0.01,0.2,0.4,0.6,1,1.4,1.8,2.2]$, with $N=5\times {10}^4, \nu =2.4$, and $a\in [{10}^{-2},1]$. The dotted line represents the memoryless result (ML) and the solid line is the analytical prediction obtained from Eq. (15) in the memory case (WM). We consider $4\times {10}^3$ epidemic realizations for each value of $\lambda$.

Figure 6

The SIS epidemic process on the activity-driven network MA. The ratio between the epidemic threshold with memory and the epidemic threshold of the memoryless case is shown as a function of the reinforcement parameter $\beta =[0.01,0.2,0.4,0.6,1,1.4,1.8,2.2]$, with $N=5\times {10}^4, \nu =2.4$, and $a\in [{10}^{-2},1]$. The dotted line represents the memoryless result (ML) and the solid line is the analytical prediction obtained from Eq. (15) (WM). We consider $4\times {10}^3$ epidemic realizations for each value of $\lambda$.

Figure 7

The SIR epidemic process on the activity-driven network RA. The ratio between the epidemic threshold with memory and the epidemic threshold of the memoryless case is shown as a function of the reinforcement parameter $\beta =[0.01,0.2,0.4,0.6,1,1.4,1.8]$, with $N=5\times {10}^4, \nu =2.4$, and $a\in [{10}^{-2},1]$. The dotted line represents the memoryless result (ML) and the solid line is the analytical prediction obtained from Eq. (15) in the memory case (WM). We consider $2\times {10}^3$ epidemic realizations for each value of $\lambda$.

Figure 8

The SIR epidemic process on the activity-driven network MA. The ratio between the epidemic threshold with memory and the epidemic threshold of the memoryless case is shown as a function of the reinforcement parameter $\beta =[0.01,0.2,0.4,0.6,1,1.4,1.8]$, with $N=5\times {10}^4, \nu =2.4$, and $a\in [{10}^{-2},1]$. The dotted line represents the memoryless result (ML) and the solid line is the analytical prediction obtained from Eq. (15) in the memory case (WM). We consider $4\times {10}^3$ epidemic realizations for each value of $\lambda$.