Dynamical properties of a stochastic tumor–immune model with comprehensive pulsed therapy

IF 3.4 2区 数学 Q1 MATHEMATICS, APPLIED
Wei Li , Bingshuo Wang , Dongmei Huang , Vesna Rajic , Junfeng Zhao
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Abstract

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.

具有综合脉冲疗法的随机肿瘤免疫模型的动态特性
本文通过考虑随机扰动和脉冲效应,建立了一个具有综合脉冲疗法的随机肿瘤-免疫模型。以定理的形式给出了模型解的一些性质。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,通过构建李亚普诺夫函数证明了系统解在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了解的期望有界性。接着,结合伊托ˆ微分法则和强大数定律,分别得到了肿瘤细胞、狩猎 T 细胞和辅助 T 细胞消亡和非均值持久性的充分条件,以及肿瘤的弱持久性和随机持久性。结果通过数值 Milsteins 方法得到证实。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,说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化的主导因素,杀伤率越大,肿瘤细胞的数量就会迅速减少。增加脉冲治疗的频率对肿瘤消退也有显著效果。这一结论与肿瘤治疗的临床观察结果一致。
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来源期刊
Communications in Nonlinear Science and Numerical Simulation
Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
CiteScore
6.80
自引率
7.70%
发文量
378
审稿时长
78 days
期刊介绍: The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.
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