Dynamical Behaviour of a Tumor-Immune System with Chemotherapy and Optimal Control

Abstract

We have considered a tumor growth model with the effect of tumor-immune interaction and chemotherapeutic drug. We have considered two immune components-helper (resting) T-cells which stimulate CTLs and convert them into active (hunting) CTL cells and active (hunting) CTL cells which attack, destroy, or ingest the tumor cells. In our model there are four compartments, namely, tumor cells, active CTL cells, helper T-cells, and chemotherapeutic drug. We have discussed the behaviour of the solutions of our system. The dynamical behaviour of our system by analyzing the existence and stability of the system at various equilibrium points is discussed elaborately. We have set up an optimal control problem relative to the model so as to minimize the number of tumor cells and the chemotherapeutic drug administration. Here we used a quadratic control to quantify this goal and have considered the administration of chemotherapy drug as control to reduce the spread of the disease. The important mathematical findings for the dynamical behaviour of the tumor-immune model with control are also numerically verified using MATLAB. Finally, epidemiological implications of our analytical findings are addressed critically.

A multi‐regional epidemic model for controlling the spread of Ebola: awareness, treatment, and travel‐blocking optimal control approaches

“…Mathematical models play an important role in the investigation and the control of the propagation of human infectious diseases, allowing health policy-makers to predict the impact of particular vaccine and treatment programs or to derive more efficient strategies based on different mathematical methods [12][13][14][15][16][17][18][19][20][21][22][23][24] and references therein. Most of mathematical models conventional of the spread of an epidemic have often ignored or underestimated the fact that some infectious diseases can spread easily over large lands.…”

On a multiobjective optimal control of a tumor growth model with immune response and drug therapies

“…A simple prey–predator type model for the growth of tumor with discrete time delay in the immune system is considered in Saleem and Agrawal (). A tumor growth model with the effect of tumor–immune interaction and chemotherapeutic drug is also presented and studied in Sharma and Samanta (). In the immune system, both studies assume two immune components helper T cells.…”

Analysis of the Dynamics of a Tumor–Immune System with Chemotherapy and Immunotherapy and Quadratic Optimal Control

“…There are some other research works on tumor-immune dynamics [1,3,4,7,12,13,15,31,33,36,38,40]. Also there are some research works on the tumor growth models with optimal control strategies [8,10,11,[16][17][18][19]30,35]. These are very helpful to predict the most effective therapy and strategy to control the spread of diseases minimizing the total drug administered.…”