Stability analysis of a fractional‐order delay dynamical model on oncolytic virotherapy

Singh, Hitesh K.; Pandey, Deepak

doi:10.1002/mma.6836

Cited by 4 publications

(2 citation statements)

References 35 publications

(56 reference statements)

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“…To appropriately model CTLs-mediated immune response to tumor cells, the time delay of the adaptive immune response should not be ignored. Besides, numerous results such as those of previous works [14][15][16] have demonstrated that time delays can produce rich dynamics in a system, such as the stability switches and periodic oscillations. Bi et al 14 considered three delays that, respectively, described tumor cells proliferation, effector cells growth, and the immune effector cells differentiation and studied the oscillation activity of tumor and immune cells.…”

Section: Introductionmentioning

confidence: 97%

Mathematical modeling of tumor surface growth with necrotic kernels

Zhang

Tian

Niu

et al. 2021

Math Methods in App Sciences

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A two‐dimensional tumor‐immune model with the time delay of the adaptive immune response is considered in this paper. The model is designed to account for the interaction between cytotoxic T lymphocytes (CTLs) and cancer cells on the surface of a solid tumor. The model considers the surface growth as a major growth pattern of solid tumors in order to describe the existence of necrotic kernels. The qualitative analysis shows that the immune‐free equilibrium is unstable, and the behavior of positive equilibrium is closely related to the ratio of the immune killing rate to tumor volume growth rate. The positive equilibrium is locally asymptotically stable when the ratio is smaller than a critical value. Otherwise, the occurrence of the delay‐driven Hopf bifurcation at the positive equilibrium is proved. Applying the center manifold reduction and normal form method, we obtain explicit formulas to determine the properties of the Hopf bifurcation. The global continuation of a local Hopf bifurcation is investigated based on the coincidence degree theory. The results reveal that the time of the adaptive immune system taken to response to tumors can lead to oscillation dynamics. We also carry out detailed numerical analysis for parameters and numerical simulations to illustrate our qualitative analysis. Numerically, we find that shorter immune response time can lead to longer patient survival time, and the period and amplitude of a stable periodic solution increase with the increasing immune response time. When CTLs recruitment rate and death rate vary, we show how the ratio of the immune killing rate to tumor volume growth rate and the first bifurcation value change numerically, which yields further insights to the tumor‐immune dynamics.

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Section: Introductionmentioning

confidence: 97%

Mathematical modeling of tumor surface growth with necrotic kernels

Zhang

Tian

Niu

et al. 2021

Math Methods in App Sciences

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“…A significant control objective in designing any controller is the stability of the closed loop system. Several stability conditions have been presented to analyze the stability of fractional order control systems 12–19 . Moreover, the stability of some classes of fractional order systems has been investigated in the works of He, 20 X. Zhang et al, 21 S. Zhang et al, 22 Li et al, 23 Li et al, 24 and Delavari et al 25 …”

Section: Introductionmentioning

confidence: 99%

Robust stability analysis of a general class of interval delayed fractional order plants by a general form of fractional order controllers

Ghorbani

Tavakoli-Kakhki

2021

Math Methods in App Sciences

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In this paper, the robust stability of interval fractional order plants with one time delay controlled by fractional order controllers is investigated in a general form. For robust stability analysis of the closed loop system by the zero exclusion principle, the distance between the origin and the value set of the characteristic function needs to be checked. It is known that the outer vertices of this value set may change at some switching frequencies and the repetitive calculation of these vertices at switching frequencies leads to additional calculations. In this study initially, new necessary and sufficient conditions are proposed to check the robust stability of a delayed fractional order closed loop system. Then, a novel robust stability testing function is presented based on some vertices, which are fixed for all positive frequencies. Therefore, no additional calculation is needed to obtain the outer vertices of the characteristic function value set for any pair of the switching frequencies. Also, a finite frequency range is presented to reduce the computational cost noticeably. Eventually, three numerical examples are given to verify the efficiency of the results of this paper.

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Stability analysis of a fractional‐order delay dynamical model on oncolytic virotherapy

Cited by 4 publications

References 35 publications

Mathematical modeling of tumor surface growth with necrotic kernels

Mathematical modeling of tumor surface growth with necrotic kernels

Robust stability analysis of a general class of interval delayed fractional order plants by a general form of fractional order controllers

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