Spatiotemporal patterns and optimal control in a diffusive Alzheimer’s disease model with pharmacokinetics

A diffusive Alzheimer’s disease (AD) model that incorporates pharmacokinetics is presented, focusing on the interactions between β-amyloid (Aβ), fiber and pharmacokinetics, with pharmacodynamics serving as a connecting framework. Our primary interest lies in understanding how the combined effects of diffusion, fiber growth, fiber nucleation, and pharmacokinetics influence the spatiotemporal distribution of Aβ and fiber in the brain. In particular, an extensive investigation is conducted into the existence of the Turing instability of constant equilibrium solutions, Turing instability of periodic solutions, and Turing–Hopf bifurcation. Interestingly, our results indicate that the type and existence of Turing instability depend on the ratio of the diffusion rates of Aβ and fiber. This suggests that varying intensities of intervention measures should be applied at different stages of AD. Furthermore, we find that even small variations in the fiber growth rate and nucleation rate can lead to a range of system behaviors, including uniform steady state, spatially inhomogeneous steady state, spatially homogeneous periodic solutions, and spatially inhomogeneous periodic solutions. This implies that controlling the fiber growth rate and nucleation rate can serve as potential therapeutic targets. Last but not least, the optimal control problem is introduced to minimize plaque concentration, treatment side effects and costs as much as possible. Numerical simulations have been implemented to support our theoretical findings and offer valuable insights for the management of AD.