Optimal control of Alzheimer's disease model via multi-target combination therapy

Alzheimer's disease (AD) is a neurodegenerative disorder characterized by the accumulation of toxic protein plaques in the brain. Abundant evidence indicates that β-amyloid (Aβ) serves as an important pathological hallmark in AD, with Aβ oligomers (Aβo) exhibiting critical neurotoxicity. The imbalance of Ca²⁺ homeostasis has also been demonstrated to exhibit complex interplay with abnormal Aβ accumulation. Several Aβ-targeting drugs are currently employed in the treatment of AD. However, the multifactorial mechanisms driving AD contribute to the suboptimal efficacy of existing clinical therapies. Motivated by these factors, multi-target combination therapies emerge as the prominent trend in AD research. In this paper, we propose an optimal control model to describe Aβo dynamics under multi-target combined therapy. Also, the basic reproduction number R₀ is derived as a threshold parameter to characterize the AD process. We find that Aβo can be maintained at a low steady state when R₀ <  1, indicating the potential for disease control. The results of parameter sensitivity analysis based on R₀ further confirm the significant impact of control variables on the system evolution. Subsequently, we investigate the theoretical results of the optimality system, proving the existence and uniqueness of the optimal solution. Numerical simulations are conducted to demonstrate the control efficacy of different strategies, revealing that drug combination therapy exhibits superior control over Aβo steady-state compared to mono therapy. Moreover, cost-effectiveness analysis is employed to compare different control strategies, with the objective of identifying the optimal approach that balances drug costs with therapeutic efficacy. Notably, appropriately lowering the drug dosage in triple therapy retains the potential for maintaining therapeutic effects, thereby improving drug cost control. In summary, this work provides important insights into multi-target combination therapy for early treatment of AD.