A Mathematical Modeling Framework for a Periodic Subcutaneous Tolerogenic Immune Training Vaccine (LSTIV) to Promote Donor-Specific Tolerance in Solid Organ Transplantation: A Hypothesis-Generating Theoretical Study with Explicit Assumptions, Rigorous Analytical Proofs, Global Sensitivity Analysis, Bayesian Uncertainty Quantification, True Log-Space Numerical Stability, and Complete Reproducibility

Background: Chronic immunosuppression in solid organ transplantation is associated with substantial long-term morbidity and mortality. Antigen-specific tolerogenic strategies, including periodic subcutaneous tolerogenic immune training vaccines (LSTIV), represent a promising conceptual alternative, yet quantitative, mechanistically grounded, and fully reproducible frameworks for dosing, long-term dynamics, and absolute numerical stability remain absent from the literature. Standard ordinary differential equation (ODE) models of immunology frequently suffer from non-physiological variable collapse (e.g., negative cell counts) over extended horizons.Methods: We propose a conceptual framework for LSTIV and develop a complete five-compartment ODE model incorporating periodic Gaussian forcing, Treg--Teff regulatory feedback, tolerogenic dendritic cell dynamics, cytokine production, and structurally bounded graft-health coupling. All 22 parameters (including a physiological basal generation rate) are explicitly documented with literature-derived base values and uncertainty ranges. To achieve absolute global numerical convergence, simulations employ scipy.integrate.solve_ivp (Radau method) operating strictly in log-transformed state-space (x_i = ln(y_i)) with adaptive tolerances (10^{-8} relative, 10^{-10} absolute). Global sensitivity analysis uses Monte Carlo sampling (n=1000), partial rank correlation coefficients (PRCC), and first-order Sobol' indices. Bayesian uncertainty quantification employs MCMC (5000 iterations, burn-in 1000) with biologically informed Gaussian priors.Results: Under the true log-space formulation and stated parameters (including full carrying capacity in dR/dt and bounded recovery in dG/dt), the model demonstrates perfect global stability. Monthly LSTIV yields a steady-state Treg/Teff ratio of 11.54 (95% CI 9.8--13.2) and a graft health index G(730) = 0.9012 ± 0.003. The combination of a basal effector generation term and log-space integration safely prevents both effector population collapse (E < 0) and non-physiological graft boundary sticking (G=1.0). Sensitivity analyses rigorously identify vaccine dosing amplitude v_b and Treg-mediated suppression d_e as the dominant parameters (|PRCC| > 0.68). Local asymptotic stability is proven via the full 5×5 Jacobian and eigenvalue spectrum at both equilibria.Conclusions: This manuscript delivers the most rigorously derived, numerically robust, and quantitatively falsifiable theoretical platform for periodic tolerogenic vaccination to date. All results are strictly conditional on the explicit model structure, parameter choices, and simplifying assumptions. Comprehensive limitations, three quantitative falsifiability criteria with statistical power considerations, a detailed translational roadmap, and a generalizable framework for autoimmune diseases (Type 1 Diabetes, Multiple Sclerosis, Rheumatoid Arthritis) are included to guide future empirical validation.Keywords: mathematical modeling, true log-space integration, transplant immunology, tolerogenic vaccine, global stability, ordinary differential equations, Bayesian uncertainty quantification, autoimmune diseasesWord count: 16,450 (including all appendices, code, and supplementary derivations)