Small inter-event times govern epidemic spreading on networks

Many aspects of human and animal interaction, such as the frequency of contacts of an individual, the number of interaction partners, and the time between the contacts of two individuals, are characterized by heavy-tailed distributions. These distributions affect the spreading of, e.g., infectious diseases or rumors, often because of impacts of the right tail of the distributions (i.e., the large values). In this paper we show that when it comes to inter-event time distributions, it is not the tail but the small values that control spreading dynamics. We investigate this effect both analytically and numerically for different versions of the susceptible-infected-recovered model on different types of networks.

Figure 1

Final outbreak size of the node-centric SIR model in the infinite well-mixed population with $a=0.1$, obtained by direct numerical solutions of the master equation. The solid line shows the analytically obtained epidemic threshold [Eq. (5a)].

Figure 2

Final outbreak size $\mathrm{\Omega }$ of the node-centric SIR model on different networks as a function of the effective infection rate ${\lambda }_{\mathrm{eff}}$. (a) Complete graph with $N={10}^3$ nodes. (b) BA network having $N={10}^3$ nodes, the number of edges that each added node introduces $m=3$, and the size of the initial clique $m_0=3$. (c) Largest connected component of an email network having $N=1133$ nodes and 5451 edges [22]. For PL1, we set $\alpha =1.5$ or 2.5. Assuming the equilibrium, we drew the distribution of waiting times, i.e., the time to the first activation from ${\psi }^{\mathrm{w}}\left(\tau \right)=(\alpha -1)/{(1+\tau )}^{\alpha }$, independently for each node. For the Pareto distribution in (a), we set $\alpha =1.3$ and $\alpha =1+\sqrt{6/5}$ and drew the time to the first activation from ${\psi }^{\mathrm{w}}\left(\tau \right)=({\alpha }^{\prime }-1){\tau }_0^{{\alpha }^{\prime }-1}/{\tau }^{\alpha \prime }$ ($\tau \ge {\tau }_0$) for each node; we also set ${\tau }_0=1$ without loss of generality, because changing ${\tau }_0$ corresponds to rescaling the time. For EMM2, we set $a=0.3, \text{CV}=\sqrt{5}$, and drew the time to the first activation from ${\psi }^{\mathrm{w}}\left(\tau \right)=S_{\ell }^{*}{\lambda }_{\ell }exp(-{\lambda }_{\ell }\tau )+S_{\text{h}}^{*}{\lambda }_{\text{h}}exp(-{\lambda }_{\text{h}}\tau )$. For each of the six distributions of inter-event times, we scanned values of $\mu$ to span a range of ${\lambda }_{\mathrm{eff}}$. We measured $\mathrm{\Omega }$ as the average over $n=5000$ simulations. Error bars are omitted for clarity. They would be 3–5 times thicker than the lines.

Figure 3

Prevalence of the node-centric SIS model on different networks as a function of the effective infection rate ${\lambda }_{\mathrm{eff}}$. (a) Complete graph with $N={10}^3$ nodes. (b) BA network having $N={10}^3$ nodes, the number of edges that each added node introduces $m=3$, and the size of the initial clique $m_0=3$. (c) Largest connected component of an email network having $N=1133$ nodes and 5451 edges [22]. For EMM2, we set $a=0.3$ and $\text{CV}=\sqrt{5}$. Error bars are omitted for clarity. They would be 3–5 times thicker than the lines.

Figure 4

Edge-centric SIR model. (a) Final size. We assumed a regular random graph with $N={10}^3$ nodes and mean degree five. (b) Waiting-time distribution ${\psi }^{\mathrm{w}}\left(\tau \right)$ for the exponential distribution and the UMM2, both having $\langle \tau \rangle =1$ and $\langle {\tau }^2\rangle =2$. For the UMM2, we set $a=1$, which yields $\epsilon \approx 1.389$.

Figure 5

Final size of the variant of the node-centric SIR model in which an activated susceptible node can be infected by its infected neighbor, whereas an activated infected node does not infect its susceptible neighbor. (a) Complete graph with $N={10}^3$ nodes. (b) BA model with $N={10}^3$ and $m=m_0=3$. (c) Email network. We set $\alpha =1.5$ for PL1, and $a=0.3$ and $\text{CV}=\sqrt{5}$ for EMM2.

Figure 6

Final size of the variant of the node-centric SIR model in which an activated susceptible node is not infected by its infected neighbor, whereas an activated infected node can infect its susceptible neighbor. (a) Complete graph with $N={10}^3$ nodes. (b) BA model with $N={10}^3$ and $m=m_0=3$. (c) Email network. We set $\alpha =1.5$ for PL1, and $a=0.3$ and $\text{CV}=\sqrt{5}$ for EMM2.