Epidemic thresholds of the susceptible-infected-susceptible model on networks: A comparison of numerical and theoretical results

Recent work has shown that different theoretical approaches to the dynamics of the susceptible-infected-susceptible (SIS) model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks. Here we present large-scale numerical simulations of the SIS dynamics on various types of networks, allowing the precise determination of the effective threshold for systems of finite size $N$. We compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean-field theory and of the quenched mean-field theory. We show that the latter is in general more accurate, scaling with $N$ with the correct exponent, but often failing to capture the correct prefactor.

Figure 1

(a) Plot of $\chi =N(\langle {\rho }^2\rangle -{\langle \rho \rangle }^2)/\langle \rho \rangle$ and ${\chi }_N=N(\langle {\rho }^2\rangle -{\langle \rho \rangle }^2)$ for the SIS process on annealed SF networks with degree exponent $\gamma =2.25$ and different network sizes. The susceptibility $\chi$ is more efficient to determine the effective size-dependent threshold. (b) Effective threshold from the susceptibility peak ${\lambda }_p\left(N\right)$ as a function of $N$, for annealed SF networks with $\gamma =2.25$ and $\gamma =3.5$, compared with HMF predictions. The effective threshold shows a very good agreement with the numerically evaluated threshold ${\lambda }_c^{\mathrm{HMF}}$ [Eq. (1)]. Inset: Differences between ${\lambda }_p\left(N\right)$ and the theoretical predictions ${\lambda }_c^{\mathrm{HMF}}\left(N\right)$ for $\gamma =2.25$ and $\gamma =3.5$.

Figure 2

Susceptibility as a function of $\lambda$ for RRN of increasing size (from bottom to top) and degree $k=10$. The susceptibility peak is closer to the theoretical prediction of the pair approximation than to the HMF and QMF results.

Figure 3

Susceptibility $\chi$ as a function of $\lambda \sqrt{k_{\max }}$ computed in star graphs with different values of $k_{\max }$ (increasing from bottom to top).

Figure 4

Effective threshold ${\lambda }_p\left(N\right)$ from the susceptibility peak as a function of network size $N$ for uncorrelated SF networks with $\gamma =2.25$, compared with the HMF and QMF predictions. Inset: Susceptibility $\chi$ as a function of $\lambda$ for different network sizes (increasing from right to left).

Figure 5

(a) Numerical values of ${\rho }_s$, $\chi$, and ${\chi }_N$ as a function of $N$ evaluated at the susceptibility peak in SF networks with $\gamma =2.25$. (b) Stationary density of infected nodes as a function of the distance from the threshold ${\lambda }_p$ in SF networks with $\gamma =2.25$ and $N={10}^7$. Lower points represent the subcritical phase.

Figure 6

Effective threshold ${\lambda }_p\left(N\right)$ as a function of network size $N$ for uncorrelated SF networks with $\gamma =2.75$, compared with the HMF and QMF predictions. Inset: Susceptibility $\chi$ as a function of $\lambda$ for different network sizes (increasing from right to left).

Figure 7

(a) Susceptibility $\chi$ as a function of $\lambda$ for uncorrelated networks with $\gamma =3.5$ and different sizes. (b) Effective thresholds ${\lambda }_p\left(N\right)$ determined by the rightmost and leftmost (when present) susceptibility peak as a function of network size $N$ for uncorrelated networks with $\gamma =3.5$, compared with the HMF and QMF predictions.

Figure 8

Quasistationary distribution of the number of active nodes $P_n$ for an uncorrelated network of size $N=3\times {10}^6$ and $\gamma =3.5$. Different curves are for decreasing values of $\lambda$ (bottom to top).