Stochastic Modeling and Probabilistic Assessment of Polycystic Ovary Syndrome (PCOS): Symmetry and Asymmetry in Infertility and Treatment Dynamics

Polycystic Ovary Syndrome (PCOS) is a widespread hormonal disorder affecting women of reproductive age, often leading to infertility and associated complications. This study presents a comprehensive stochastic mathematical framework to analyze the dynamics of PCOS with a particular focus on infertility and treatment outcomes. Here, the transitions between compartments represent progression of women through clinical states of PCOS (risk, diagnosis, treatment, recovery) rather than infection or transmission, since PCOS is a non-communicable disorder. The model incorporates probabilistic elements to break the symmetric and predictable assumptions inherent in deterministic approaches. This allows it to reflect the randomness and asymmetry in hormonal regulation and ovulation cycles, enabling a more realistic representation of disease progression. By utilizing stochastic differential equations, the study evaluates the impact of treatment adherence on fertility restoration. We establish the conditions for disease extinction versus the existence of an ergodic stationary distribution, which represents a form of long-term statistical symmetry. The results emphasize the importance of early diagnosis and consistent treatment. Furthermore, the proposed approach provides a valuable tool for clinicians to predict patient-specific trajectories and optimize individualized treatment plans, accounting for the asymmetric nature of patient responses.