Stability Analysis of an Impulsive Differential Equation Model for Bone Metabolism under Estrogen Treatment during COVID-19 Infection

COVID-19-associated inflammation, immobilization, and altered endocrine signaling can disrupt bone remodeling. We formulate a reduced impulsive differential-equation model for the interactions among parathyroid hormone, active osteoclasts, and active osteoblasts under periodic estrogen intervention. After a quasi-steady reduction of the fast parathyroid-hormone variable, we establish positivity and dissipativity, derive the unique osteoclast-free periodic solution, and obtain its exact transverse Floquet multiplier, which is shown to be exact for the unreduced three-dimensional model as well. Sufficient conditions for global attraction and uniform persistence are proved, and a simple-eigenvalue bifurcation of positive periodic solutions is characterized through the Poincaré map. Analytical sensitivity and constrained pulse-design formulas are confirmed by numerical calculations.