Mathematical Modeling, Bio-Dynamic Analysis, and Optimal Control of Periodontal Disease Progression Incorporating Time Delays, Host-Modulation Therapy, and Realistic Clinical Scenarios

Vollständiger Abstract

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Periodontitis is a chronic multi-factorial inflammatory condition characterized by subgingival dysbiosis and immunopathological tissue breakdown, ultimately leading to connective-tissue attachment loss and alveolar bone resorption. This paper extends a previously formulated four-dimensional ordinary differential equation (ODE) model of subgingival bacterial biofilm, neutrophil/cytokine dynamics, tissue damage, and local antimicrobial pharmacokinetics into a five-dimensional delay-differential-equation (DDE) framework that (i) incorporates a physiologically realistic lag τ1 in immune-cell activation and a lag τ2 in the onset of tissue-repair signalling, and (ii) introduces an explicit host-modulation therapy (HMT) compartment v(t) — representing, e.g., a sub-antimicrobial-dose MMP-inhibitor — that suppresses the bystander immunopathological damage coefficient β2 through a saturating inhibition law. Beyond the original stability theory (disease-free/endemic equilibria, Lyapunov/LaSalle global stability, R0 via the Next-Generation-Matrix method), we perform an equilibrium-continuation bifurcation analysis of the (B, I) subsystem across the R0 = 1 threshold and a closed-form delay-induced Hopf bifurcation analysis of the immune-activation lag τ1, identifying the critical delay τ1* beyond which sustained inflammatory oscillations emerge — a candidate mechanistic explanation for cyclical, flare-and-remission patterns observed clinically in chronic periodontitis. A dual optimal-control problem (local antimicrobial S1(t) and host-modulation S2(t)) is solved via Pontryagin's Minimum Principle and a Forward-Backward Sweep numerical scheme. Two new clinically motivated scenarios — a smoker phenotype with delayed immune kinetics, and a diabetic host under optimal dual therapy — are added to the original three scenarios. An extended Latin-Hypercube/PRCC sensitivity analysis (now spanning nine parameters, including τ1, τ2 and the HMT potency κ) confirms host susceptibility (σ) and bystander immune damage (β2) as dominant drivers of bone loss, while showing that HMT potency (κ) is itself a strong protective factor (PRCC ≈ −0.74). Numerical results show that combining antimicrobial and host-modulation control lowers day-60 cumulative tissue damage from D(60) ≈ 0.96 (antimicrobial alone) to D(60) ≈ 0.66 in a poorly controlled diabetic host — a clinically meaningful improvement that is preserved once realistic immune/repair delays are reintroduced.