Mathematical and Dynamic Modelling of the Interplay Between Serum Vitamin D, Ferritin Levels, and Thyroid Function in the Tobruk Population: A Differential Equation and Confounding Analysis Approach
EMAD Awadh
Source abstract
Vitamin D and iron are involved in thyroid hormone synthesis and metabolic regulation, yet observational studies often ignore feedback kinetics and demographic confounders. We develop a two-variable ODE model of the hypothalamic–pituitary–thyroid (HPT) axis with ferritin- and vitamin D-dependent terms, prove that it has a unique, globally attracting steady state whose TSH level decreases with ferritin and saturates at high ferritin, and pair it with gender-adjusted regression to test whether serum 25-hydroxyvitamin D [25(OH)D] and ferritin are independently associated with TSH in Tobruk, Libya. In a cross-sectional study (October 2025 – March 2026) of 650 adults (108 males, 542 females; 39 with thyroid dysfunction: 34 females and 5 males), we measured TSH, 25(OH)D and ferritin. Analysis used Spearman correlation, Mann–Whitney U tests and multiple linear regression, together with analytical (nondimensionalization, stability, omitted-variable bias) and numerical (global sensitivity, PRCC) mathematical analysis. Vitamin D deficiency (<20 ng/mL) affected 63.4% (95% CI 59.6–67.0) and low ferritin 49.2% (45.4–53.1); 30.5% had both. TSH was not correlated with 25(OH)D (ρ = −0.013, p = 0.741). A weak unadjusted TSH–ferritin correlation (ρ = 0.098, p = 0.013) disappeared after gender adjustment (B = −0.10, 95% CI −2.54 to 2.34, p = 0.936), while gender strongly predicted ferritin (B = 52.70, p < 0.001). Among females, cases (n = 34) and controls (n = 508) did not differ in vitamin D or ferritin; the 5 male cases were too few for a separate comparison. After gender adjustment, we found no evidence of an independent association of TSH with ferritin or 25(OH)D. The model predicts only a weak, saturating dependence of TSH on ferritin above the range in which thyroid peroxidase is limited, which is compatible with the empirical null result. The study is small for thyroid-dysfunction comparisons (n = 39: 34 females, 5 males), so modest effects cannot be excluded.
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