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A time delay model of tumour–immune system interactions: Global dynamics, parameter estimation, sensitivity analysis

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  • Rihan, F.A.
  • Abdel Rahman, D.H.
  • Lakshmanan, S.
  • Alkhajeh, A.S.

Abstract

Recently, a large number of mathematical models that are described by delay differential equations (DDEs) have appeared in the life sciences. In this paper, we present a delay differential model to describe the interactions between the effector and tumour cells. The existence of the possible steady states and their local stability and change of stability via Hopf bifurcation are theoretically and numerically investigated. Parameter estimation problem for given real observations, using least squares approach, is studied. The global stability and sensitivity analysis are also numerically proved for the model. The stability and periodicity of the solutions may depend on the time-lag parameter. The model is qualitatively consistent with the experimental observations of immune-induced tumour dormancy. The model also predicts dormancy as a transient period of growth which necessarily results in either tumour elimination or tumour escape.

Suggested Citation

  • Rihan, F.A. & Abdel Rahman, D.H. & Lakshmanan, S. & Alkhajeh, A.S., 2014. "A time delay model of tumour–immune system interactions: Global dynamics, parameter estimation, sensitivity analysis," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 606-623.
    • Handle: RePEc:eee:apmaco:v:232:y:2014:i:c:p:606-623
    DOI: 10.1016/j.amc.2014.01.111
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    References listed on IDEAS

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    1. F. A. Rihan & M. Safan & M. A. Abdeen & D. Abdel Rahman, 2012. "Qualitative and Computational Analysis of a Mathematical Model for Tumor-Immune Interactions," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-19, February.
    2. Park, Ju H., 2007. "Further results on passivity analysis of delayed cellular neural networks," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1546-1551.
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    Cited by:

    1. Liu, Xiangdong & Li, Qingze & Pan, Jianxin, 2018. "A deterministic and stochastic model for the system dynamics of tumor–immune responses to chemotherapy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 500(C), pages 162-176.
    2. George E. Chatzarakis & Tongxing Li, 2018. "Oscillation Criteria for Delay and Advanced Differential Equations with Nonmonotone Arguments," Complexity, Hindawi, vol. 2018, pages 1-18, April.
    3. Ruiqing Shi & Ting Lu & Cuihong Wang, 2019. "Dynamic Analysis of a Fractional-Order Model for Hepatitis B Virus with Holling II Functional Response," Complexity, Hindawi, vol. 2019, pages 1-13, August.
    4. Dzyubak, Larysa & Dzyubak, Oleksandr & Awrejcewicz, Jan, 2023. "Nonlinear multiscale diffusion cancer invasion model with memory of states," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    5. Rihan, F.A. & Lakshmanan, S. & Maurer, H., 2019. "Optimal control of tumour-immune model with time-delay and immuno-chemotherapy," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 147-165.
    6. Ghanizadeh, Mojtaba & Shariatpanahi, Seyed Peyman & Goliaei, Bahram & Rüegg, Curzio, 2021. "Mathematical modeling approach of cancer immunoediting reveals new insights in targeted-therapy and timing plan of cancer treatment," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    7. Zhou, Haihua & Song, Huijuan & Wang, Zejia, 2022. "The effect of time delay in regulatory apoptosis on a tumor model with angiogenesis," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    8. Jia, Yunfeng, 2020. "Bifurcation and pattern formation of a tumor–immune model with time-delay and diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 92-108.
    9. Fathalla A. Rihan & Chinnathambi Rajivganthi, 2021. "Dynamics of Tumor-Immune System with Random Noise," Mathematics, MDPI, vol. 9(21), pages 1-14, October.
    10. F. A. Rihan & C. Tunc & S. H. Saker & S. Lakshmanan & R. Rakkiyappan, 2018. "Applications of Delay Differential Equations in Biological Systems," Complexity, Hindawi, vol. 2018, pages 1-3, September.
    11. Huang, Chengdai & Cao, Jinde & Xiao, Min & Alsaedi, Ahmed & Alsaadi, Fuad E., 2017. "Controlling bifurcation in a delayed fractional predator–prey system with incommensurate orders," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 293-310.
    12. Dong, Yueping & Huang, Gang & Miyazaki, Rinko & Takeuchi, Yasuhiro, 2015. "Dynamics in a tumor immune system with time delays," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 99-113.
    13. Duan, Wei-Long & Lin, Ling, 2021. "Noise and delay enhanced stability in tumor-immune responses to chemotherapy system," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    14. Abraham J. Arenas & Gilberto González-Parra & Jhon J. Naranjo & Myladis Cogollo & Nicolás De La Espriella, 2021. "Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay," Mathematics, MDPI, vol. 9(3), pages 1-21, January.
    15. Dehingia, Kaushik & Das, Parthasakha & Upadhyay, Ranjit Kumar & Misra, Arvind Kumar & Rihan, Fathalla A. & Hosseini, Kamyar, 2023. "Modelling and analysis of delayed tumour–immune system with hunting T-cells," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 669-684.
    16. Rihan, F.A. & Velmurugan, G., 2020. "Dynamics of fractional-order delay differential model for tumor-immune system," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    17. Shyam Sundar Santra & Rami Ahmad El-Nabulsi & Khaled Mohamed Khedher, 2021. "Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator," Mathematics, MDPI, vol. 9(12), pages 1-9, June.
    18. Xu, Changjin & Farman, Muhammad & Akgül, Ali & Nisar, Kottakkaran Sooppy & Ahmad, Aqeel, 2022. "Modeling and analysis fractal order cancer model with effects of chemotherapy," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).

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