Complexities and bifurcations induced by drug responses in a pulsed tumour-immune model

Drug response plays a key role in exploring how drug toxicity affects the evolution of tumour cells. Depending on the tumour size in tissue, we initiate pulsed comprehensive therapies, and then construct an impulsive tumour-immune model, in which whether comprehensive therapies applied or not only relies on the threshold of tumour size. By employing the definitions and properties of the Poincare map, we show that the effector cell eradication periodic solution is globally stable under threshold conditions. In the light of the bifurcation theorems, it is shown that transcritical bifurcations can occur with respect to many treatment parameters including depletion rate, chemotherapeutic drug concentration, medicine toxicity coefficient and accumulation rate of effector cells. Then we provide conditions for the existence of order−k(k ≥ 1) periodic solutions. The results indicate that threshold R0 is sensitive to treatment parameters and the proposed system exists with very complex dynamics when treatment parameters are chosen as bifurcation parameters. Moreover, the therapeutic protocol with a smaller chemotherapy drug dosage and more frequent intakes is more effective to maintain higher tumour cell depletion rate