Proceedings of the Steklov Institute of Mathematics ( IF 0.5 ) Pub Date : 2021-07-26 , DOI: 10.1134/s0081543821030111 N. L. Grigorenko 1 , E. N. Khailov 1 , A. D. Klimenkova 1 , E. V. Grigorieva 2
The Lotka–Volterra competition model is applied to describe the interaction between the concentrations of healthy and cancerous cells in diseases associated with blood cancer. The model is supplemented with a differential equation characterizing the change in the concentration of a chemotherapeutic drug. The equation contains a scalar bounded control that specifies the rate of drug intake. We consider the problem of minimizing the weighted difference between the concentrations of cancerous and healthy cells at the end time of the treatment period. The Pontryagin maximum principle is used to establish analytically the properties of an optimal control. We describe situations in which the optimal control is a bang–bang function and situations in which the control may contain a singular arc in addition to bang–bang arcs. The results obtained are confirmed by corresponding numerical calculations.
中文翻译:
Lotka-Volterra 竞争数学模型中治疗癌症的最佳策略
Lotka-Volterra 竞争模型用于描述与血癌相关的疾病中健康细胞和癌细胞浓度之间的相互作用。该模型补充有表征化疗药物浓度变化的微分方程。该方程包含一个标量有界控制,用于指定药物摄入的速率。我们考虑在治疗期结束时最小化癌细胞和健康细胞浓度之间的加权差异的问题。庞特里亚金最大值原理用于通过分析建立最佳控制的属性。我们描述了最佳控制是 bang-bang 函数的情况以及控制可能包含除了 bang-bang 弧之外的奇异弧的情况。