Dynamical Model of Mild Atherosclerosis: Applied Mathematical Aspects

The paper’s Theorem 2 claims that the reduced two-dimensional system (5)–(6) is locally asymptotically stable at the nonzero equilibrium E2 provided two inequalities hold (its items 1 and 2). From the paper, the reduced model is obtained via QSSA with Ls = d m / ((f+m)(1+e M)), leading to (5)–(6) and the Jacobian J2 in (12), with Theorem 2 stated explicitly on pp. 12–13 . At E2, the equilibrium identity a/(1+σ) g(L2) = ε + c, with g(L) = L/(1+L) and L2 = d m2/((f+m2)(1+e M2)), implies ε + c = a d m2/((1+σ)D) where D := (f+m2)(1+e M2) + d m2. The paper’s inequality (i) is algebraically equal to a d m2/((1+σ)D) + J11 < ε + c, where J11 > 0 at E2, hence (i) is impossible to satisfy at any nonzero equilibrium (contradiction). Its inequality (ii) is equivalent to J21 > 0, which, using c = b d M2/D at E2, reduces to D > f(1+e M2), automatically true for m2 > 0. Therefore Theorem 2’s sufficient conditions are vacuous: (i) cannot hold and (ii) is automatically true, so the theorem’s implication is incorrect. In contrast, the model’s solution derives the correct Jacobian at E2 from (5)–(6) and applies the 2×2 Routh–Hurwitz criteria (trace < 0, det > 0), giving necessary and sufficient stability conditions; it also proposes a corrected sufficient pair that exactly matches those criteria. This aligns with the standard Lyapunov indirect method and matrix theory. Key paper elements cited: the reduced system (5)–(6) and QSSA formula, the Jacobian (12), and Theorem 2 statement .