Interaction tumor-immune model with time-delay and immuno-chemotherapy protocol

This research explores a delay differential model to describe the dynamics of tumour-immune interactions in presence of immuno-chemotherapy. The model includes a constant delay in the recruitment term of the immune cells to illustrate the time lag between the stimulated accumulations of immune cells in the vicinity of cancer cells. The efficiency of solutions and the boudness have been supported by the findings. Also, the conditions for local stability and the existence of Hopf bifurcation are investigated. In particular, sufficient conditions dependent on the delay parameter under which the interior equilibrium is asymptotically stable are constructed. Afterwards, we established the length of delay to preserve stability. The numerical simulations show that the combination of immuno-chemotherapy protocol reduces the tumor load in a few months of therapy. The numerical simulations are presented to illustrate our theoretical results and show that the blend of immuno-chemotherapy convention decreases the tumour load in a couple of long periods of treatment.