Figure 1

Examples of the functions $g_{k,M}\left(y\right)$ for $k=5$.Reuse & Permissions

Figure 2

(Color online). The nontrivial equilibria $y=Y\left({\beta }_0\right)$ of the HN and SFN mean-field models (1, 3), for $M=1$, $\overline{k}=12$, and ${\beta }_1$ set to $1$ (a), $1.25$ (b), and $1.5$ (c). Solid (dashed) curves denote stable (unstable) equilibria and have been numerically obtained by continuation [26]. For the SFN model we used a number $n=k_{\max }-k_{\min }+1=600$ of node degrees, which is consistent with a finite-size network of $N\cong (n∕k_{\min }{)}^2={10}^4$ nodes [3]. As for the stochastic simulations, at each value of ${\beta }_0$ we simulated the epidemiological process over a time horizon of $3\times {10}^4$ steps $(\Delta ={10}^{-2})$ starting from two different network states, i.e., by randomly infecting 2 and $40\%$ of nodes. In circles are the averages over the last ${10}^4$ steps.Reuse & Permissions

Figure 3

(Color online) The saddle-node bifurcation curves of the mean-field models (1, 3). For transmission parameter values in regions I and II, SFNs support the coexistence of two stable equilibria: the high endemic state and either a low endemic state (region I) or a totally susceptible population (region II). Above the HN curve, model (1) has also two attractors, namely, the high endemic state and the totally susceptible population. Unspecified parameters as in Fig. 2.Reuse & Permissions

Figure 4

(Color online) The value of ${\beta }_0$ at the saddle-node bifurcation in both HN and SFN models as a function of $n=k_{\max }-k_{\min }+1$. The value $n={10}^5$ corresponds to a finite-size SFN of $N\cong (n∕k_{\min }{)}^2\cong 2.8\times {10}^8$ nodes [3]. Other parameters set to $\overline{k}=2k_{\min }=12$, $M=1$, and ${\beta }_1=1.25$.Reuse & Permissions