Geometric threshold of endemicity in curvature-weighted contact networks

Let $\overline{\kappa}$ denote the mean curvature of the network, let $\kappa^\star>0$ be a constant, and let $R_0$ denote the basic reproduction number of the unweighted system. The disease-free equilibrium is the equilibrium with no infection, namely $y_i=0$ for all $i$. Geometric threshold of endemicity. There exists a constant $\kappa^\star>0$ such that, if

$$\overline{\kappa} > \kappa^\star,$$

then the disease-free equilibrium is globally stable, even when the unweighted system satisfies $R_0>1$. Sufficiently positive curvature prevents endemic persistence by reducing the effective connectivity of the network. The claim proposes that sufficiently large mean curvature can suppress endemicity despite supercritical transmission in the unweighted network; its resolution is not established in the supplied text.