Epidemic spreading on modular networks: The fear to declare a pandemic

In the past few decades, the frequency of pandemics has been increased due to the growth of urbanization and mobility among countries. Since a disease spreading in one country could become a pandemic with a potential worldwide humanitarian and economic impact, it is important to develop models to estimate the probability of a worldwide pandemic. In this paper, we propose a model of disease spreading in a structural modular complex network (having communities) and study how the number of bridge nodes $n$ that connect communities affects disease spread. We find that our model can be described at a global scale as an infectious transmission process between communities with global infectious and recovery time distributions that depend on the internal structure of each community and $n$. We find that near the critical point as $n$ increases, the disease reaches most of the communities, but each community has only a small fraction of recovered nodes. In addition, we obtain that in the limit $n\to \infty$, the probability of a pandemic increases abruptly at the critical point. This scenario could make the decision on whether to launch a pandemic alert or not more difficult. Finally, we show that link percolation theory can be used at a global scale to estimate the probability of a pandemic since the global transmissibility between communities has a weak dependence on the global recovery time.

Figure 1

Schematic illustration of four communities ($C_1$, $C_2$, $C_3$, and $C_4$, represented by circles) in which each row corresponds to a different global structure. The first column shows the global structure in which connections between communities are superlinks, and successive columns show different numbers of bridge nodes $n$ (stars) for the same global structure: $n=1$ (second column), $n=2$ (third column), and $n=3$ (fourth column). In all configurations for the first row $C_2$ and $C_3$ have two superlinks ($k^g=2$), while $C_1$ and $C_4$ have only one ($k^g=1$); for the second row $C_1$ has $k^g=1$, and $C_2$, $C_3$, and $C_4$ have $k^g=2$.

Figure 2

Schematic illustration of the SIR model at the final state in a network with communities and $n=1$. On the left, each large circle represents a community, stars denote bridge nodes, and small circles are the internal nodes or individuals of each community. The pink nodes correspond to recovered individuals, while the black ones are susceptible. On the right, we show the network of communities at a global scale where each circle is a supernode or community. The pink supernodes are the communities where the epidemic developed, and are white otherwise, i.e., the disease did not reach the community or only developed as an outbreak. The area enclosed with a dotted line corresponds to the figure on the left.

Figure 3

Schematic illustration of the definition of ${\tau }_R$ (a) and ${\tau }_I$ when a community $A$ infects $B$ (b). The figures illustrate the time evolution of the number of infected nodes $I\times N$ in community $A$ (dark blue) and community $B$ (light blue). The horizontal dotted line corresponds to the threshold $s_c$ above which a community is regarded as infected. In panel (a), the time ${\tau }_R$ (red dashed interval) corresponds to the time interval between the moment at which community $A$ becomes infected, and the moment it recovers. In panel (b), the time ${\tau }_I$ (red dashed interval) corresponds to the time interval between the times in which the two communities $A$ and $B$ get infected.

Figure 4

Normalized time distribution of ${\tau }_R$ and ${\tau }_I$ for different values of $n$ for two connected ER communities with $\langle k\rangle =3$. (a) Distribution of ${\tau }_R$. (b) The conditional distribution $P\left({\tau }_I\right|{\tau }_R)$ for $n=1$ and ${\tau }_R=15$ (black), ${\tau }_R=16$ (red), and ${\tau }_R=17$ (light blue). (c) $T_{{\tau }_R}^g$ as a function of ${\tau }_R$ for $n=1$ (black), $n=3$ (red), and $n=20$ (light blue). (d) The conditional distribution $P\left({\tau }_I\right|{\tau }_R)$ for ${\tau }_R=15$ and $n=1$ (black), $n=3$ (red), and $n=20$ (light blue). The results were obtained with over ${10}^6$ realizations for $T=0.70$, $N={10}^4$, and $s_c=100$.

Figure 5

Time evolution of the fraction of infected communities for different values of $n$: 1 (black), 3 (red), and 20 (light blue). For each value of $n$, we show the average value of $I^g$ obtained from 100 realizations of the aggregated network (symbols) and the network with communities (line). Panel (a) corresponds to $T=0.70$ for an ER network composed of ER communities with $\langle k^g\rangle =\langle k\rangle =3$. Note that for $n=1$, the disease reaches a macroscopic fraction of communities only for $T\gtrsim 0.6$ (see Fig. 7). Panel (b) corresponds to $T=0.60$ for SF networks at a global scale with $\lambda =3$ and $2\le k^g\le 200$, with SF communities in which $\lambda =2.5$ and $2\le k\le 200$. For the simulations, we use $N={10}^4$, $N^g=5\times {10}^3$, and $s_c=100$. We set the time $t=0$ as the moment at which $I^g{N}^g=s_c$ [30, 31]. The insets show $I^g\langle t\rangle$ as a function of $t/\langle t\rangle$ for different values of $n$, where $\langle t\rangle$ is the time at which the fraction of infected communities is maximum. These results were obtained from the aggregated network.

Figure 6

(a) Heat-map of the critical microscopic transmissibility for a pandemic, $T_{c,\mathrm{pand}}$, in the plane $\langle k^g\rangle$ and $\langle k\rangle$ for an ER network of ER communities and $n=1$. The black region indicates that there is no pandemic phase in the network for any value of the microscopic transmissibility. (b) Critical microscopic transmissibility for a pandemic $T_{c,\mathrm{pand}}$ as a function of the global mean degree $\langle k^g\rangle$ for ER network of ER communities with $\langle k\rangle =3$ and different values of $n$: 1 (black), 3 (red), and 20 (light blue). For each value of $n$, the system is in a pandemic phase above the curves, while below it is free of a pandemic. The vertical dotted line indicates the limit $\langle k^g\rangle =1$ and the horizontal dotted line corresponds to $T_{c,\mathrm{pand}}=T_c=1/\langle k\rangle$. The inset shows $T_{c,\mathrm{pand}}-1/\langle k\rangle$ as a function of $\langle k^g\rangle$ in log-log scale for the curves shown in the main plot. The curves and the surface are obtained from Eq. (11). Note that the $\mathrm{slope}=-1/2$ is predicted in Eq. (12).

Figure 7

Fraction of recovered individuals $R^{\mathrm{tot}}$ ($\square$, dashed line) and communities that developed an epidemic $R^g$ ($◯$, solid line) as a function of the microscopic transmissibility. Our results were obtained from the simulations (symbols) and Eqs. (9) and (10) (lines) for $n=1$ (black), $n=20$ (red), and $n={10}^4$ (light blue, only theory). The pink dotted line corresponds to the limit $n=\infty$ [see Eq. (13)]. Panel (a) corresponds to an ER network of ER communities with $\langle k^g\rangle =\langle k\rangle =3$. Panel (b) corresponds to SF networks at a global scale with $\lambda =3$ and $2\le k^g\le 200$, with SF communities in which $\lambda =2.5$ and $2\le k\le 200$. The simulations were performed over 100 network realizations with $N={10}^4$, $N^g=5\times {10}^3$, and (a) $s_c=600$ and (b) $s_c=100$.

Figure 8

Probability of a pandemic given that there is one community with an epidemic as a function of $T$ for $n=1$ (black), $n=3$ (red), $n=20$ (light blue). Panel (a): The results correspond to an ER network of ER communities with $\langle k^g\rangle =\langle k\rangle =3$, and different values of $n$. Panel (b) corresponds to SF networks at a global scale with $\lambda =3$ and $2\le k^g\le 200$, with SF communities in which $\lambda =2.5$ and $2\le k\le 200$. The simulations were performed over ${10}^3$ network realizations with $N={10}^4$, $N^g=5\times {10}^3$, and (a) $s_c=600$ and (b) $s_c=100$. The lines ($R^g$) correspond to the theory obtained from the Eqs. (9) and (10), and the symbols (${\mathrm{\Pi }}^g$) to the simulations.

Figure 9

Distribution of the number of recovered nodes at the final state, $P\left(s\right)$ for different values of $T$ for networks without community structure. Panel (a) corresponds to an ER network with $\langle k\rangle =3$. Panel (b) corresponds to a SF network with $\lambda =2.5$, $k_{\min }=2$, and $k_{\max }=200$. The simulations results were averaged over ${10}^6$ network realizations with $N={10}^4$.

Figure 10

The time $\langle t\rangle$ at which the fraction of infected communities is maximum, as a function of $n$ in linear-log scale. Panels (a) and (c) correspond to $T=0.50$ and $T=0.70$, respectively, for an ER network of ER communities with $\langle k^g\rangle =\langle k\rangle =3$. Panels (b) and (d) correspond to $T=0.40$ and $T=0.60$, respectively, for SF networks at a global scale with $\lambda =3$ and $2\le k^g\le 200$, with SF communities in which $\lambda =2.5$ and $2\le k\le 200$. The dashed line corresponds to a logarithmic fit $\langle t\rangle =A+Bln\left(n\right)$, where $A=92.8$ and $B=-10.4$ (a), $A=26.4$ and $B=-2.16$ (b), $A=52.2$ and $B=-5.7$ (c), and $A=21.7$ and $B=-1.6$ (d). For the simulations, we use $N={10}^4$, $N^g=5\times {10}^3$, and $s_c=100$. We set the time $t=0$ as the moment at which $I^g{N}^g=s_c$. The insets show $I^g\langle t\rangle$ as a function of $t/\langle t\rangle$ for different values of $n$.

Figure 11

Time evolution of the fraction of infected communities for different values of $n$: 1 (black), 3 (red), and 20 (light blue). Other parameters are the same as in Figs. 5 and 5. For each value of $n$, we show the average value of $I^g$ obtained from 100 realizations of the aggregated network (symbols) and the network with communities (line). Note that for panels (a) and (c), we do not show $I^g$ for $n=1$ because, in that case, the transmissibility is close or below $T_{c,\mathrm{pand}}$ [see Figs. 7 and 7].

Figure 12

Schematic figure of the backward (a) and forward branching (b). The direction of the branching process goes from bottom to top. Solid lines without any arrow represent undirected links, and arrows represent links with a direction. The blue area depicts the set of links used in the backward branching (a) and forward branching (b). For a backward (forward) branching, a node is reached through one of its outgoing (incoming) links with probability $k_{\mathrm{out}}P\left(k_{\mathrm{out}}\right)/\langle k_{\mathrm{out}}\rangle$ $[k_{\mathrm{in}}P\left(k_{\mathrm{in}}\right)/\langle k_{\mathrm{in}}\rangle ]$ and the in-component (out-component) continue to grow through its incoming (outgoing) and undirected links.

Figure 13

Mapping between link percolation and the SIR model with heterogeneous recovery distribution for ER networks with $\langle k\rangle =3$. The panels show $R$ (black) and $\mathrm{\Pi }$ (red) as a function of the total transmissibility $T$ for a heterogeneous recovery distribution given by Table 1 (a) and Table 2 (b). The lines correspond to the theoretical solutions of Eqs. (D2)–(D16), and the symbols to simulations. The simulations were performed over ${10}^4$ network realizations with $N={10}^4$ and $s_c=600$. The disagreement between the theoretical curves and the simulations around the critical point is due to finite size effects and the value of $s_c$ which cannot distinguish an epidemic from an outbreak near $T=T_c$ (see Appendix pp1).