Influence of multiple delays in brain tumor and immune system interaction with T11 target structure as a potent stimulator

We present a mathematical model which describes the growth of malignant gliomas in presence of immune responses by considering the role of immunotherapeutic agent T11 target structure (T11TS). The model consider five populations, namely, glioma cells, macrophages, cytotoxic T-lymphocytes, TGF - beta and IFN - gamma. The model system has highly nonlinear terms with four discrete time lags, but remains tractable. The goal of this work is to better understand the effect of multiple delays on the interaction between gliomas and immune components in conjunction with an administration of T11 target structure. Analytically, we investigate the conditions for the asymptotic stability of equilibrium points, the existence of Hopf bifurcations and the maximum value of the delay to preserve the stability of limit cycle. For the set of parameter values estimated from experimental data, time delays have hardly any influence on the system behavior. Numerical simulations are carried out to investigate the dynamics of the model with different values for delays with and without administration of T11 target structure.