EQUILIBRIA ANALYSIS OF A NETWORKED BIVIRUS EPIDEMIC MODEL USING POINCARÉ–HOPF AND MANIFOLD THEORY

The paper proves the counting formula ∑k(−1)^{n_k}=1 by (i) perturbing the region near any stable boundary equilibria so they become interior, (ii) diffeomorphically mapping the interior to S^{2n} minus a point, and (iii) adding the north pole as a source so Poincaré–Hopf applies on the compact boundaryless sphere; this yields Equation (3.5) and Theorem 3.3 correctly, with explicit handling of boundary tangencies and corners . By contrast, the model’s argument tries to apply Poincaré–Hopf directly to a smoothed collar M “diffeomorphic to” Ξ_{2n} and asserts the vector field −F is outward along ∂M. This misses two critical issues highlighted by the paper: Ξ_{2n} is a manifold-with-corners (not a manifold-with-boundary), and trajectories are confined to boundary faces (e.g., x^2=0), so the boundary condition for Poincaré–Hopf fails; zeros also occur on the boundary, which the model does not resolve . The model also invokes a collar theorem for manifolds-with-boundary without justifying smoothing of corners and outward-pointing everywhere on the new boundary. Hence the model’s proof is flawed/incomplete even though it reaches the correct formula; the paper’s proof is sound.