Analysis of Optimal Control in an Age–Structured Epidemic Model with Spatial Diffusion

In the present paper, we analyze an age and space-structured SVIRD epidemic model that couples age-transport dynamics with spatial diffusion and a nonlocal infection operator. Two distributed, bounded controls-vaccination U_1 (t,a,x) and treatment U_2 (t,a,x) are introduced to augment baseline vaccination and recovery rates and an optimal control problem is posed to minimize a finite-horizon cost functional balancing infection burden and quadratic control costs. Under standard regularity, boundedness and Lipschitz assumptions on model coefficients and the contact kernel, we establish well-posedness of the controlled state system. Existence of an optimal control pair is proved by the direct method, using uniform a-priori estimates, Aubin–Lions compactness, and weak lower semicontinuity of the cost. We derive first-order necessary optimality conditions via Pontryagin’s Maximum Principle (PMP) for distributed systems: the adjoint PDEs. The resulting coupled forward–backward optimality system is analyzed and interpreted, adjoint variables are shown to act as shadow prices that guide age and space varying vaccination and treatment intensities.