A discrete fractional-order mathematical model for in vitro fertilization dynamics

Infertility affects millions of couples worldwide, and in vitro fertilization (IVF) is the foremost assisted reproductive technology. However, IVF success rates vary considerably due to the memory-dependent nature of hormonal regulation, embryo development, and repeated treatment cycles. Clinical data are typically collected at discrete monthly intervals, yet few mathematical models simultaneously capture both the discrete time structure and the memory effects inherent in IVF. This work develops and rigorously analyses a discrete fractional-order (FO) IVF model using the dual Caputo nabla fractional difference (CNFD) operator, which naturally incorporates long-range memory while aligning with the cycle-based format of medical records. The model stratifies the IVF process into six compartments: infertile couples, patients under treatment, high-, medium-, and low-quality embryos, and positive pregnancy outcomes. Existence, uniqueness, and non-negativity of solutions are proved. Equilibrium analysis yields a unique positive endemic equilibrium, and sufficient conditions for global asymptotic stability (GAS) and Mittag-Leffler stability (MLS) are established via a novel Volterra-type discrete Lyapunov function (LF). Numerical simulations, performed with representative parameter values, confirm the theoretical results and show that lower FOs introduce stronger memory effects and slower convergence, thereby reproducing realistic, protracted IVF dynamics. The theoretical findings are not tied to a specific experimental dataset and thus provide a general framework that could assist clinicians in optimizing treatment decisions and personalizing patient management.