A stage-structured diabetes progression model with control-relapse dynamics: well-posedness, persistence, and stability

Abstract

We develop and analyze a nonlinear stage-structured model for diabetes progression that distinguishes uncontrolled and controlled diabetes and incorporates bidirectional management–relapse transitions together with bounded prevalence-weighted deterioration effects. We establish local and global well-posedness, positivity, boundedness, and the existence of a positively invariant feasible region. We also characterize the equilibrium structure, derive conditions for the existence of a positive equilibrium, and prove persistence of the downstream disease classes. A central analytical feature is the decomposition of the system into an upstream linear subsystem and a downstream nonlinear subsystem, which clarifies the mechanism governing the long-term dynamics. Local asymptotic stability of the positive equilibrium is established through Jacobian analysis and the Routh–Hurwitz criterion, while a Lyapunov–LaSalle argument yields a sufficient condition for global asymptotic stability. As an extension, we formulate an optimal-control version of the model with prevention, management, and relapse-prevention interventions and solve the resulting system numerically over a 10-year horizon. The simulations show that combining all three controls yields the largest reduction in uncontrolled diabetes and severe-complication burden, while sensitivity analysis identifies the progression-related parameters with the strongest influence on severe outcomes. These results show that the proposed framework is mathematically tractable and biologically relevant for studying long-term diabetes complication dynamics.