一类具有抗原性的肿瘤-免疫系统的定性分析

Abstract

Abstract:

In this paper, we propose and investigate a tumor-immune system interaction model with antigenicity. The existence of equilibria of the model is determined, and the local dynamics of each feasible equilibrium is analyzed. The global dynamics of the model is obtained by excluding the existence of periodic solutions. It is found that, under certain conditions, the saddle-node bifurcation and the bi-stability of strong equilibrium with tumor and equilibrium without tumor may occur for the model, which imply that the growth and development of the tumor will depend on its initial state. The obtained theoretical analysis results are verified by numerical simulations.

Key words: Tumor-immune system, Antigenicity, Stability, Saddle-node bifurcation

CLC Number:

O175.1

Xin Xie,Jianquan Li,Yuping Wang,Dian Zhang. A Qualitative Analysis of a Tumor-Immune System with Antigenicity[J].Acta mathematica scientia,Series A, 2021, 41(6): 1969-1979.

Trendmd

Figures/Tables 2

References 20

1	Subhas K , Nieto J J . Mathematical modeling of tumor-immune competitive system, considering the role of time delay. Applied Mathematics and Computation, 2019, 340, 180- 205 doi: 10.1016/j.amc.2018.08.018
2	Kirschner D , Panetta J C . Modeling immunotherapy of the tumor-immune interaction. Journal of Mathematical Biology, 1998, 37, 235- 252 doi: 10.1007/s002850050127
3	Khajanchi S , Banerjee S . Stability and bifurcation analysis of delay induced tumor immune interaction model. Applied Mathematics and Computation, 2014, 248, 652- 671 doi: 10.1016/j.amc.2014.10.009
4	Khajanchi S , Nieto J . Mathematical modeling of tumor-immune competitive system, considering the role of time delay. Applied Mathematics and Computation, 2019, 340, 180- 205 doi: 10.1016/j.amc.2018.08.018
5	Letellier C , Denis F , Aguirre L A . What can be learned from chaotic cancer model?. Journal of Theoretical Biology, 2013, 322, 7- 16 doi: 10.1016/j.jtbi.2013.01.003
6	Kuznetsov V A , Makalkin I A , Taylor M A , et al. Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bulletin of Mathematical Biology, 1994, 56, 295- 321 doi: 10.1016/S0092-8240(05)80260-5
7	Robert D , Schreiber , Lloyd J , et al. Cancer immunoediting: integrating immunity's roles in cancer suppression and promotion. Science, 2011, 331 (6024): 1565- 1570 doi: 10.1126/science.1203486
8	Matthew D V , Michael H K , Robert D S , et al. Natural innate and adaptive immunity to cancer. Annual Review of Immunology, 2011, 29 (1): 235- 71 doi: 10.1146/annurev-immunol-031210-101324
9	Araujo R P , McElwain D . A history of the study of solid tumour growth: The contribution of mathematical modelling. Bull Math Biol, 2004, 66, 1039- 1091 doi: 10.1016/j.bulm.2003.11.002
10	黄佩, 蔺小林, 李建全, 等. 一类肿瘤-免疫系统动力学性态的全局分析. 高校应用数学学报A辑, 2019, 34 (2): 181- 189
	Huang P , Lin X L , Li J Q , et al. Global analysis of a kind of tumor-immune system dynamics. Journal of Applied Mathematics Series A, 2019, 34 (2): 181- 189
11	陶有山, 边保军. 抑制剂作用下肿瘤生长模型的参数识别. 数学物理学报, 2009, 29 (5): 1175- 1186
	Tao Y S , Bian B J . Parameter identification of tumor growth model under the action of inhibitor. Acta Math Sci, 2009, 29 (5): 1175- 1186
12	卫雪梅, 崔尚斌. 一个肿瘤生长自由边界问题解的渐近性态. 数学物理学报, 2007, 27 (4): 648- 659
	Wei X M , Cui S B . The Asymptotic behavior of the solution of a tumor growth free boundary problem. Acta Math Sci, 2007, 27 (4): 648- 659
13	Adam J A , Bellomo N . A Survey of models for tumor-immune system dynamics. Modeling & Simulation in Science Engineering & Technology, 1996, 59 (5): 1023- 1024
14	马知恩, 周义仓, 李承治. 常微分方程定性与稳定性方法. 北京: 科学出版社, 2015
	Ma Z E , Zhou Y C , Li C Z . Qualitative and Stable Methods of Ordinary Differential Equations. Beijing: Science Press, 2015
15	Adams J F . Stable Homotopy and Generalised Homology. Chicago: University of Chicago Press, 1974
16	Kirschner D , Panetta J C . Modeling immunotherapy of the tumor-immune interaction. Journal of Mathematical Biology, 1998, 37 (3): 235- 252 doi: 10.1007/s002850050127
17	Bodnar M , Piotrowska M J , Urszula Foryś , et al. Model of tumour angiogenesis-analysis of stability with respect to delays. Mathematical Bioences and Engineering, 2013, 10 (1): 19- 35
18	Pillis L D , Fister K R , Gu W , et al. Mathematical model creation for cancer chemo-immunotherapy. Computational and Mathematical Methods in Medicine, 2009, 10 (3): 165- 184 doi: 10.1080/17486700802216301
19	Chaplain M, Matzavinos A. Mathematical Modelling of Spatio-Temporal Phenomena in Tumour Immunology. Berlin: Springer, 2006
20	Galach M . Dynamics of the tumor-immune system competition: the effect of time delay. Applied Mathematics and Computer Science, 2003, 13, 395- 406

[1]	Lv Dongting. Globally Asymptotic Stability of 2-Species Reaction-Diffusion Systems of Spatially Inhomogeneous Models [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1171-1183.
[2]	Liang Qing. Existence, Uniqueness and Stability of the Global Solutions to Two Classes of Pantogragh Stochastic Functional Differential Equations with Markovian Switching [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1184-1205.
[3]	Liu Yu, Chen Guanggan, Li Shuyong. Nonlinear Stability of Traveling Waves for Stochastic Kuramoto-Sivashinsky Equation [J]. Acta mathematica scientia,Series A, 2025, 45(3): 790-806.
[4]	Liu Jia, Bao Xiongxiong. Asymptotic Stability of Pyramidal Traveling Front for Nonlocal Delayed Diffusion Equation [J]. Acta mathematica scientia,Series A, 2025, 45(1): 44-53.
[5]	Xiao Jianglong, Song Yongli, Xia Yonghui. Spatiotemporal Dynamics Induced by the Interaction Between Fear and Schooling Behavior in a Diffusive Model [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1577-1594.
[6]	Tang Jun, Wu Ailong. Event-Triggered Control for a Class of Stochastic Time-Delay Nonlinear Systems [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1607-1616.
[7]	Zhang Yongxue, Jia Wensheng. Stability of Solutions for Controlled Systems of Generalized Multiobjective Multi-Leader-Follower Games Under Bounded Rationality [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1689-1702.
[8]	Yang Hong, Zhang Xiaoguang. A Stochastic SIS Epidemic Model on Simplex Complexes [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1392-1399.
[9]	Fan Tianjiao, Feng Lichao, Yang Yanmei. Generalization of Inequality and Its Application in Additive Time-Varying Delay Systems [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1335-1351.
[10]	Guo Yan, Xu Xiaochuan. Local Solvability and Stability of the Inverse Spectral Problems for the Discontinuous Sturm-Liouville Problem with the Mixed Given Data [J]. Acta mathematica scientia,Series A, 2024, 44(4): 859-870.
[11]	Pan Lijun, Lv Shun, Weng Shasha. Riemann Solution and Stability of Coupled Aw-Rascle-Zhang Model [J]. Acta mathematica scientia,Series A, 2024, 44(4): 885-895.
[12]	Zhang Chunguo, Sun Baonan, Fu Yuzhi, Yu Xin. Stabilization of 2-D Mindlin Timoshenko Plate Systems with Local Damping [J]. Acta mathematica scientia,Series A, 2024, 44(4): 946-959.
[13]	Gao Caixia, Zhao Dongxia. The Delayed Control and Input-to-State Stability of ARZ Traffic Flow Model with Disturbances [J]. Acta mathematica scientia,Series A, 2024, 44(4): 960-977.
[14]	Xu Rui, Zhou Kaijuan, Bai Ning. A with-in Host HIV-1 Infection Dynamics Model Based on Virus-to-cell Infection and Cell-to-cell Transmission [J]. Acta mathematica scientia,Series A, 2024, 44(3): 771-782.
[15]	Qiu Yangcong, Wang Qiru. Hyers-Ulam-Rassias Stability of First-Order Nonlinear Dynamic Equations on Time Scales [J]. Acta mathematica scientia,Series A, 2024, 44(2): 465-475.

A Qualitative Analysis of a Tumor-Immune System with Antigenicity

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 2

References 20

Related Articles 15

Metrics

Comments

Recommended 10