Multi-Objective Identification and Optimal Control for Generalized SIR Epidemic Models with Time-Dependent Parameters

This paper investigates a multi-objective optimization framework for the simultaneous parameter identification and optimal vaccination control of a generalized non-linear SIR epidemic model. In contrast to traditional single-objective approaches, we formulate a vector optimization problem to concurrently minimize deviations from observed trajectories, disease prevalence, and public health operational costs, unlocking the identification of Pareto-optimal strategies. We establish the global existence, uniqueness, and physical invariance of the state solutions within a classical Sobolev space framework by employing a contraction mapping argument under the Bielecki weighted norm. Via the direct method of the calculus of variations and compact Sobolev embeddings, the existence of weak Pareto-optimal solutions is rigorously proved. First-order necessary optimality conditions are subsequently derived using the geometric Dubovitskii–Milyutin functional formalism, which yields a well-posed backward linear adjoint system and a precise characterization of the coupled variational inequalities. Finally, we implement a robust numerical architecture driven by an adjoint-based Augmented Lagrangian Method coupled with a quasi-Newton solver over a piecewise linear continuous subspace for the approximation and simulation of the Pareto-optimal frontier, which is successfully validated through three distinct numerical examples.