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 $\tau$ 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 $\alpha$ of tumor cell, deactivation rate $d_1$ of tumor cell, activation rate $c_2$ of effector cell and death rate $d_2$ 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.

在讨论中存在离散时间延迟的影响，我们提出并分析了肿瘤与免疫相互作用的概念性数学模型。所提出的模型由三个耦合的非线性常微分方程（ODE）的系统描绘，即肿瘤细胞，效应细胞和细胞因子白介素2（IL-2）。尽管简单，但是该模型可以具有复杂的动力学行为。我们讨论了数学模型的定性性质，包括解的存在性和正性。我们在我们的数学模型中找出无肿瘤的奇异点和内部稳态。我们研究了时滞和非时滞系统中生物可行稳态的局部稳定性分析。通过使用横向条件，我们通过使用时延分析了Hopf分叉$\tau$作为分叉参数。我们已经估计了应用拉普拉斯变换的时延参数的长度，以保持1期极限循环的稳定性，从而提供了控制肿瘤细胞生长振荡的作用方式的思想。我们进行了数值模拟，并探讨了其生物学意义，以验证我们的理论分析。我们还参考了内在增长率绘制了延迟模型的分叉图$\alpha$ 肿瘤细胞的失活率 $d_{\mathrm{1个}}$ 肿瘤细胞活化率 $C_{\mathrm{2个}}$ 细胞的效应和死亡率 $d_{\mathrm{2个}}$效应细胞。理论和数值分析表明，在存在IL-2的情况下，效应细胞可导致肿瘤细胞退化。