A Hybrid Reciprocal Adaptive Ventilation with Passive Thermal Recovery (RAV-PTR) System: Rigorous Mathematical Modeling, Full Monte Carlo Uncertainty Quantification, Physiological Safety Assessment, Cost-Benefit Analysis, and Low-Cost Implementation for Eliminating Domestic Carbon Monoxide Poisoning Risk

Carbon monoxide (CO) poisoning from domestic heating causes approximately 28,900 deaths annually worldwide (GBD 2021). We present RAV-PTR, a low-cost (≤ $45 USD) sensor-driven hybrid system that integrates reciprocal counter-flow ventilation, passive porous-media heat recovery (η_rec = 0.68–0.82), and hysteresis control (activation at >9 ppm CO, deactivation at <7 ppm). The system is governed by a fully derived set of mass- and energy-balance ODEs solved via stiff BDF integration with terminal event detection. Full Monte Carlo uncertainty quantification (10,000 runs) with realistic parameter distributions (S ∼ Uniform(800,900), η_rec ∼ N(0.72,0.05)) confirms median peak CO reduction of ∼87% versus sealed conditions while limiting additional thermal loss to only 3–5% relative to constant ventilation. Physiological safety is demonstrated via Haldane equation and full Coburn-Forster-Kane differential model (COHb < 10% at all times). All results, including time-series curves and IQR statistics, are 100% reproducible from the provided open-source Python code. Grounded on real Taif winter climate data (ΔT = 11.5 K), the framework offers immediate deployability, explicit falsifiability, and a projected prevention of 1,500–3,000 deaths per year at 10% global adoption in high-risk households.