In this paper, a stochastic tumor–immune model with comprehensive pulsed therapy is established by taking stochastic perturbation and pulsed effect into account. Some properties of the model solutions are given in the form of the Theorems. Firstly, we obtain the equivalent solutions of the tumor–immune system by through three auxiliary equations, and prove the system solutions are existent, positive and unique. Secondly, a Lyapunov function is constructed to prove the global attraction in the mean sense for the system solution, and the boundness of the solutions’ expectation is proved by the comparison theorem of the impulsive differential equations. Next, the sufficient conditions for the extinction and non-mean persistence of tumor cells, hunting T-cells and helper T-cells, as well as the weak persistence and stochastic persistence of the tumor, are obtained by way of combining Itoˆ’s differential rule and strong law of large numbers, respectively. The results pass the confirmation by numerical Milsteins method. The results show that when the noise intensity gradually increases, the tumor state changes from the weak persistence to the extinction, it demonstrates that the effect of stochastic perturbations on tumor cells is very prominent. In addition, by adjusting the value of a(nP) to simulate different medication doses, the results show that the killing rate of the medication to the tumor cells is the dominant factor in the long-term evolution of the tumor, and the bigger killing rate can lead to a rapid decrease in the number of tumor cells. Increasing the frequency of pulse therapy has also significant effects on tumor regression. The conclusion is consistent with the clinical observation of tumor treatment.

中文翻译：

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具有综合脉冲治疗的随机肿瘤-免疫模型的动力学特性


在本文中，通过考虑随机扰动和脉冲效应，建立了具有综合脉冲治疗的随机肿瘤-免疫模型。模型解的一些属性以定理的形式给出。首先，通过三个辅助方程得到肿瘤-免疫系统的等效解，并证明系统解是存在的、正的和唯一的。其次，构造 Lyapunov 函数来证明系统解的均值意义上的全局吸引力，并通过脉冲微分方程的比较定理证明解期望的边界性。接下来，通过分别结合 Itoˆ 微分法则和强数大数定律，获得了肿瘤细胞、狩猎 T 细胞和辅助 T 细胞灭绝和非均值持续存在的充分条件，以及肿瘤的弱持续性和随机持续性。结果通过数值 Milsteins 方法的确认。结果表明，当噪声强度逐渐增大时，肿瘤状态由弱持续变为消亡，表明随机扰动对肿瘤细胞的影响非常突出。此外，通过调整 a（nP） 的值来模拟不同的药物剂量，结果表明药物对肿瘤细胞的杀伤率是肿瘤长期演变的主导因素，较大的杀伤率会导致肿瘤细胞数量迅速减少。增加脉冲治疗的频率对肿瘤消退也有显着影响。结论与肿瘤治疗的临床观察一致。