An Eight-Dimensional Nonlinear Dynamical Systems Model for Polycystic Ovary Syndrome: A Theoretical Control-Theoretic Framework with Verified Stability, Global Sensitivity Analysis, and an Explicit Falsifiability Roadmap

Polycystic ovary syndrome (PCOS) is a heterogeneous endocrine-metabolic-reproductive disorder affecting an estimated 8–13% of reproductive-age women worldwide, with mechanistic contributions from hyperandrogenism, insulin resistance, ovulatory dysfunction, chronic low-grade inflammation, hypothalamic-pituitary-ovarian axis dysregulation, and gut-microbiome dysbiosis [1, 10, 13, 15–17]. This paper develops an eight-dimensional nonlinear ordinary-differential-equation (ODE) model, x(t) = [A, I, E, O, M, H, C, D]⊤, representing normalized androgen, insulin, estrogen, ovulation index, gut-microbiome diversity, hypothalamic GnRH pulse frequency, systemic inflammation, and adiposity, respectively, coupled to an eight-channel therapeutic control input u(t) ∈ [0, 1]⁸. Control enters each equation in a mechanistically motivated, clearance-enhancing or set-point-restorative form, and the microbiome equation is specified as a globally stable linear relaxation; together these choices guarantee that the healthy attractor x* is not postulated but numerically verified as an exact fixed point of the closed-loop vector field (residual norm ∥f(x*) + Bu∥ ≈ 8 × 10⁻¹⁷), lying inside the physiological reference domain. Local exponential stability is confirmed by the Jacobian spectrum at x (maximum real eigenvalue part ≈ −0.104 across the swept genomic-risk range), and global convergence toward x* under sustained treatment is supported empirically (strict monotonic Lyapunov decay along the simulated closed-loop trajectory) while being explicitly not claimed to hold for the untreated system, which is shown numerically to leave the normalized reference domain. A fully reproducible global Sobol sensitivity analysis (Saltelli estimator, Nbase = 256, implemented from first principles without third-party sensitivity packages) identifies basal insulin production (S1 = 0.410) and insulin-androgen coupling (S1 = 0.338) as the dominant drivers of the model's disease-state output, with the illustrative genomic-risk covariate contributing S1 = 0.187; one-at-a-time perturbation analysis independently reproduces the same ranking. Monte Carlo forward propagation of literature-informed parameter uncertainty (N = 2000) yields a 95% interval for the final normalized androgen level of [0.60, 1.29] at 12 months under the full protocol. This is a purely theoretical, in-silico framework. No patient data, no fitted clinical parameters, and no clinical outcome (remission, live-birth, or otherwise) is claimed, predicted, or estimated anywhere in this manuscript; all reported quantities are properties of the simulated dynamical system itself, are reproduced verbatim by the embedded Python 3 code (Appendix B) from a fixed random seed, and are explicitly bounded by the falsifiability criteria set out in Section 10.