Dynamical behaviour of a tumour-immune model focusing on the dosage of targeted chemotherapeutic drug

In this paper, we have considered a tumour-immune model with the dosage effect of the targeted chemotherapeutic drug and its efficacy. The chemotherapeutic drug efficacy is characterized via way of means of its impact on the abnormal size of beginning tumours. In pharmacology, the effectiveness of a drug is a very important factor in developing a dosage for any disease. With these points of view regarding drug efficacy, we present a four compartmental mathematical model, viz, the population of tumour cells, T(t), CD8 T-killer cells, E-k(t), CD4 T-helper cells, E-h(t), and the amount of targeted chemotherapeutic drug, C(t), in the bloodstream, at any time t. A qualitative analysis of the proposed model, including the existence and boundedness of the model has been described here. The dynamics of the system are presented by examining the admissibility and stability of the system at trivial, dead, tumour-free, and co-existing equilibrium points with their respective biological significance. The conditions for the local stability of all equilibrium points are derived by using the Jacobian matrix and Routh-Hurwitz's criterion. Numerical calculations have been performed using the Multi-step Differential Transformation Method (Ms-DTM), and are presented and discussed briefly in terms of graphs. We have expressed that the model is suitable for the tumour removal with the passage of time, regardless of the initial population or tumour size. Also, the amount the dosage of the drug with an assigned numerical value for its efficacy is one of the important factors in the treatment. The efficacy term is included to see the effect of drug dosages in CD8 T-killer cells and CD4 T-helper cells compartment.