A Seven-Compartment PK/PD Model of Periodic Subcutaneous Tolerogenic Vaccination for Donor-Specific Transplant Tolerance: Ultimately Bounded Kinetics, Floquet Stability of the Vaccinated Periodic Orbit, and Exploratory Global Sensitivity Analysis

Background: Chronic immunosuppression in solid organ transplantation is associated with substantial long-term morbidity. Antigen-specific tolerogenic strategies, including a periodic subcutaneous tolerogenic immune training vaccine (LSTIV), represent a promising conceptual alternative, but quantitative frameworks that couple realistic depot pharmacokinetics, humoral (antibody-mediated) injury, and a numerically well-posed dynamical structure are absent from the literature.Methods: We present a mechanistic seven-compartment ordinary differential equation (ODE) model comprising alloreactive effector T cells (E), donor-specific regulatory T cells (R), tolerogenic dendritic cells (D), pro-tolerogenic cytokines (C), a normalized graft-health index (G), a subcutaneous nanoparticle pharmacokinetic (PK) depot (P), and donor-specific antibody (DSA) titer (A). The vaccine acts through a first-order depot with an Emax/Hill pharmacodynamic (PD) transfer function. Positivity of all seven compartments is guaranteed by non-negative inward flux at every boundary face, and ultimate boundedness of E, R, D is established rigorously by an explicit comparison-principle (super-solution) argument that accounts for the basal additive source terms rather than assuming the nested carrying-capacity factors alone act as impassable ceilings; the derived absorbing bounds are confirmed numerically from deliberately supra-capacity initial conditions. Because periodic dosing renders the governing equations non-autonomous, local asymptotic stability of the unforced (no-vaccine) equilibrium is assessed via the standard 6 by 6 Jacobian eigenvalue spectrum, while orbital stability of the periodically forced (vaccinated) states is assessed rigorously via Floquet theory, computing the monodromy matrix over one dosing period on the numerically converged periodic orbit. Global sensitivity is explored with Latin-hypercube partial rank correlation coefficients (PRCC, n equals 400) and a from-scratch Saltelli scheme delivering total-order and pairwise-interaction Sobol indices, explicitly framed as an exploratory, small-sample analysis rather than a converged variance decomposition. Demographic-noise robustness is assessed with a chemical Langevin equation (CLE) ensemble (n equals 150 trajectories per arm).Results: The no-vaccine model converges to a locally asymptotically stable rejection equilibrium (all Jacobian eigenvalues Re(lambda) less than 0; G(730) equals 0.059, Treg/Teff equals 0.21). Monthly LSTIV dosing converges instead to a 30-day periodic orbit that is locally asymptotically orbitally stable (all seven Floquet multipliers less than 1 in modulus; spectral radius approximately 0.127), with day-730 phase point G(730) equals 0.895, Treg/Teff equals 37.7. Quarterly dosing converges to a distinct, large-amplitude 90-day periodic orbit that is itself strongly orbitally stable (Floquet spectral radius approximately 0.002) despite its wide within-cycle excursion (G(730) equals 0.758, cycle range approximately 0.09 to 0.99); this oscillatory outcome is therefore a genuine stable periodic attractor of the forced system, not a sign of dynamical instability. PRCC identifies the graft-recovery rate gamma (rho equals 0.664, p less than 10 to the negative 50) and a baseline non-immune attrition rate delta zero (rho equals negative 0.625, p less than 10 to the negative 43) as the dominant drivers of long-term graft health; an exploratory total-order Sobol analysis is directionally consistent with this ranking (S sub T gamma equals 0.708, S sub T delta zero equals 0.617 at one representative seed) but repeated Saltelli replications at the same feasible sample size show substantial estimator spread (for example, S sub T gamma ranging approximately from 0.41 to 0.83 across five independent seeds), so these point estimates should be read only as corroborating, not independently establishing, the PRCC ranking. Under demographic (finite-population) noise, 100 percent of 150 monthly-dosed trajectories retain tolerance (G(730) greater than 0.5; mean 0.895, standard deviation 0.0003) while 0 percent of untreated trajectories spontaneously acquire it.Conclusions: This manuscript delivers a self-consistent, dimensionally sound, PK/PD-linked, antibody-coupled theoretical platform for periodic tolerogenic vaccination, in which a stable rejection equilibrium and distinct, dynamically stable dosing-interval-specific periodic orbits coexist as attractors realized under different forcing regimes of a single, ultimately bounded seven-compartment dynamical system. All results are strictly conditional on the stated model structure and parameters. A scientific and technical risk assessment, four quantitative falsifiability criteria, and a phased translational roadmap accompany a generalizable extension to major autoimmune diseases.