AN INVESTIGATION OF VIRUS DYNAMICS ON STARLIKE GRAPHS

The paper’s Theorem 4.1 asserts a sharp threshold b* = (1−a)/√(n1+⋯+nk−1) for all k-level starlike graphs, claiming: below b* the only fixed point is 0 and trajectories converge to 0; above b* there is a unique positive fixed point and global convergence to it. This is explicitly stated for k=3 (Theorem 3.1) and extended to arbitrary k (Theorem 4.1) using concavity/convexity and an “analogous” argument for convergence, see the model definitions for 3-level (1.13) and the k-level extension and Theorem 4.1 statements in the PDF . However, for k≥4 the claimed threshold is generally incorrect: the correct sharp threshold is b* = (1−a)/ρ, where ρ is the Perron root of the level-quotient tridiagonal matrix similar to the symmetric tridiagonal with off-diagonals √(n_m). This specializes to √(n1+n2) when k=3, but is strictly smaller than √(n1+⋯+n_{k−1}) in general. A concrete counterexample (k=4, n1=n2=n3=1, a=1/2, b=0.30) lies above the paper’s threshold (0.30 > 0.5/√3) but still dies out because a + b ρ < 1 (with ρ = 2 cos(π/5) ≈ 1.618), contradicting Theorem 4.1 II(b) (existence/attractivity of a nontrivial fixed point) . The model’s Perron–Frobenius argument, global linear comparison F(d) ≤ (aI+bQ)d, and monotone-iteration under strict concavity along rays yield a complete and correct dichotomy at a + bρ = 1 (Seneta, 1981).