Solving the Dynamic Correlation Problem of the Susceptible-Infected-Susceptible Model on Networks

The susceptible-infected-susceptible (SIS) model is a canonical model for emerging disease outbreaks. Such outbreaks are naturally modeled as taking place on networks. A theoretical challenge in network epidemiology is the dynamic correlations coming from that if one node is infected, then its neighbors are likely to be infected. By combining two theoretical approaches—the heterogeneous mean-field theory and the effective degree method—we are able to include these correlations in an analytical solution of the SIS model. We derive accurate expressions for the average prevalence (fraction of infected) and epidemic threshold. We also discuss how to generalize the approach to a larger class of stochastic population models.

Figure 1

The prevalence $\rho$ is plotted as a function of the effective infection rate $\lambda$ in static networks. (a) Erdős-Rényi random networks with average degree 4 and 10, (b) scale-free network with minimum degree 3 and $\gamma =4.5$, 2.7, and 2.2. Lines are theoretical estimations of Eqs. (9) and (11), while the points are simulation results. We use networks with $N={10}^5$ nodes and 100 randomly chosen seeds for the infection. Each data point is an average over at least 100 independent epidemic outbreaks, performed on at least 10 different network realizations.

Figure 2

(a) and (c) Epidemic threshold ${\lambda }_c$, obtained by our method, the quenched mean-field method, and the quasistationary state method [10, 21], versus network size $N$. (b) and (d) Susceptibility $\chi$ as a function of the effective infection rate $\lambda$ for varying network size $N$, where the peak values determine the ${\lambda }_p(N)$ in (a) and (c). Panels (a) and (b) are for Erdős-Rényi random networks with average degree $⟨k⟩=4$; (c) and (d) are for scale-free networks with minimum degree 3 and $\gamma =2.7$. The results are obtained for the epidemic dynamics over at least ten different network realizations with initial infected seed number $I_0=1$.

Figure 3

(a) The epidemic prevalence $\rho$ is plotted as a function of the effective infection rate $\lambda$ in static random regular graphs with degree $k_0=4$ and 10. Lines are the theoretical predictions of Eq. (14), while the scatters are Monte Carlo simulation results. Other parameters are the same as those in Fig. 1. (b) Susceptibility $\chi$ as a function of the effective infection rate $\lambda$ for random regular graphs with different network sizes, where $k_0$ is fixed to 5.