Sharp-threshold global stability of the rumor-free equilibrium

Prove global asymptotic stability of the rumor-free equilibrium under the sharp threshold condition \(\mathcal R_0<1\), rather than only under the stronger sufficient conditions \(\bar{\mathcal R}_0<1\) and \(\lambda\rho(\mathbf A)<\delta\).

Background

The paper establishes global asymptotic stability of the rumor-free equilibrium only under sufficient conditions involving both the upper-bound reproduction number $\bar{\mathcal R}_0=\beta^U\rho(\mathbf B)/\mu$ and the awareness-layer condition $\lambda\rho(\mathbf A)<\delta$.

The basic reproduction number $\mathcal R_0$, which incorporates the equilibrium awareness distribution and is therefore sharper, is shown to govern local stability and endemic-equilibrium existence. However, the authors do not prove that $\mathcal R_0<1$ alone guarantees global stability of the rumor-free equilibrium. They explicitly identify this sharp-threshold result as unresolved, while numerical experiments provide evidence for the expected global convergence.