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In this chapter, we present a delay differential model with optimal control that describes the interactions of tumor cells (TCs) and immune response cells with external therapy. The intracellular delay is incorporated into the model to justify the time required to stimulate the effector cells (ECs). The optimal control variables are incorporated to identify the best treatment strategy with minimum side effects by blocking the production of new TCs and keeping the number of normal cells above \(75\%\) of its carrying capacity. The existence of the optimal control pair and optimality system has been established. Pontryagin’s maximum principle is applicable to characterize the optimal controls. The model displays a tumor-free steady state and up to three coexisting steady states.

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Delay Differential Equations of Tumor-Immune System with Treatment and Control

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  • 发表日期：

    2026年

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  • Fathalla A. Rihan

摘要

In this chapter, we present a delay differential model with optimal control that describes the interactions of tumor cells (TCs) and immune response cells with external therapy. The intracellular delay is incorporated into the model to justify the time required to stimulate the effector cells (ECs). The optimal control variables are incorporated to identify the best treatment strategy with minimum side effects by blocking the production of new TCs and keeping the number of normal cells above \(75\%\) of its carrying capacity. The existence of the optimal control pair and optimality system has been established. Pontryagin’s maximum principle is applicable to characterize the optimal controls. The model displays a tumor-free steady state and up to three coexisting steady states.