Numerical Solution of Tumor-immune Model including small Molecule Drug by Multi-step Differential Transform Method

Abstract

In this article, we have described a mathematical model for tumor-immune system with impact of small molecules drug, and solved by multi step differential transform method. The mathematical model has four compartments; population of tumor cell, CD8 T-killer cell, CD4 T-helper cell and amount of drug in the blood stream. Calculations are presented to verify the theoretical results so obtained for tumor free equilibrium point. The model is fit for removing tumor with passage of time irrespective of the initial population or size of the tumor.

Pseudo‐spectral method for controlling the drug dosage in cancer

“…These cells are also eliminated by interaction with cancer cells with the rate μ. The immune system is excited to grow due to the presence of cancer cells, which is shown by the term (ρx The effect of vaccine therapy is regarded in (4) and (5). Antiviral vaccines and anticancer vaccines have a significant difference.…”

“…The tumour–immune interaction model with the impact of small molecules drug is presented in [6 ]. The multi‐step differential transformation method is used to solve the tumour–immune interaction model in [5 ]. In [4 ], a fractional‐order model is presented.…”

“…In this study, we will analyse and expand a model of cancer during the treatment by adding immunotherapy and chemotherapy from a new perspective. Many models have been present to show the tumour-immune interaction considering chemoimmunotherapy [3][4][5][6]. The mixed chemo-immunotherapy model considering the innate and the adaptive immune response is presented in [3].…”

A numerical approach of tumor‐immune model with B cells and monoclonal antibody drug by multi‐step differential transformation method

“…However, the disease is a multi‐factorial sickness; the hazard elements of which incorporate utilization of tobacco items, unfortunate eating routine, air contamination, and much more. There are several kinds of treatment amongst which targeted chemotherapy is efficient 1‐5 . Targeted chemotherapeutic drugs are one of the drugs that block the growth and spread of malignant tumors by interfering with specific molecules that are involved in growth of malignant tumors.…”

“…But the drug compartment used in the model helps to kill tumor cells 91% faster than any regular chemotherapy. Recently, in the year 2019, we described a mathematical model for tumor‐immune system with the impact of targeted chemotherapeutic drug, where the role of efficacy of drug has been described in order to remove large tumor cells 1 . In the regime of mathematical biology, fractional calculus is now the epicenter to describe the cellular behavior with much preciseness, and several mathematical models for human disease have been depicted through various fractional‐order model 19‐23 …”