Exploring the dynamics of a tumor-immune interplay with time delay

Abstract

With the effect of discrete time delay in deliberation, we propose and analyze a conceptual mathematical model for the tumor-immune interaction. The proposed model is delineated by a system of three coupled non-linear ordinary differential equations (ODEs), namely tumor cells, effector cells and cytokine Interleukin-2 (IL-2). Though simple, the model can have complicated dynamical behaviors. We have discussed the qualitative properties of the mathematical model including existence and positivity of the solutions. We find out tumor-free singular point and interior steady states in our mathematical model. We have studied the local stability analysis of the biological feasible steady states in both of the delayed and non-delayed system. By using transversality condition, we have analyzed Hopf bifurcation by using time delay τ as a bifurcation parameter. We have estimated the length of time delay parameter applying Laplace transformation for preserving the stability of period-1 limit cycle that provides the idea about the mode of action in controlling oscillations in the growth of tumor cells. We performed numerical simulations and explored their biological implications to validate our theoretical analysis. We have also drawn bifurcation diagram of delayed model with reference to the intrinsic growth rate α of tumor cell, deactivation rate d1 of tumor cell, activation rate c2 of effector cell and death rate d2 of effector cell. Theoretical and numerical analysis show that in presence of IL-2 the effector cells can cause the tumor cell population to regress.