An optimal control problem for anti-inflammatory treatments of Alzheimer's disease.

We present and analyze an optimal control problem to model anti-inflammatory treatment strategies for Alzheimer's disease, using a system of differential equations that captures interactions between A $\beta$ -peptides, microglial cells, interleukins, and neurons. These interactions operate through mechanisms such as protein polymerization, inflammation processes, and neural stress responses. In particular, inflammation is highlighted as a key factor in the onset and progression of Alzheimer's disease, driven by a hysteresis effect related to the degradation rate d of monomers and the initial concentration of interleukins. This implies a critical inflammation threshold that determines whether the disease persists over the long term. The optimal control problem we propose seeks to minimize the concentration of toxic oligomers by modulating interleukin production and degradation rates, representing potential anti-inflammatory treatment effects. Under natural constraints on treatment dose efficacy and cumulative exposure, our goal is to assess whether it is possible to shift the system from a persistent disease state to a disease-free equilibrium. We provide the necessary conditions of the optimal solution and supplement our theoretical findings with numerical simulations, which illustrate the system's behavior under different parameter settings and the imposed constraints of the optimal control problem.