2025-007

Question

Consider the propagation of a virus modeled by the SIS (Susceptible-Infected-Susceptible) system in two distinct networks:

• A Random Network (Erdős-Rényi).
• A Scale-Free Network with a degree exponent \(2 < \gamma < 3\).

Both networks have the same number of nodes \(N\) (where \(N \to \infty\)) and the same average degree \(\langle k \rangle\).

Given that the epidemic threshold is defined by the critical spreading rate \(\lambda_c\), below which the virus dies out exponentially, select the correct statement regarding the behavior of this threshold and immunization strategies:

A) Since both networks have the same average degree \(\langle k \rangle\), the epidemic threshold \(\lambda_c\) will be identical for both, as the average connection density determines the initial spreading speed in any topology.

B) In the Scale-Free Network, the epidemic threshold is given by \(\lambda_c = \frac{\langle k \rangle}{\langle k^2 \rangle}\). Since the second moment \(\langle k^2 \rangle\) diverges to infinity in large networks with \(\gamma < 3\), the threshold tends to zero (\(\lambda_c \to 0\)). This implies that even viruses with very low transmission rates can become endemic.

C) The Random Immunization strategy is equally effective in both types of networks. To eradicate the disease, it suffices to vaccinate a fraction of the population \(g\) such that the effective transmission rate is reduced, regardless of the variance of node degrees.

D) In the SIS model on Scale-Free Networks, the absence of an epidemic threshold occurs only if the network is infinite and has \(\gamma > 3\). For \(2 < \gamma < 3\), the hubs are less relevant, and the network behaves similarly to a random network.

E) None of the above.

Original idea by: Giancarlo Maldonado Cárdenas