Nonlinear dynamics in tumor-immune system interaction models with delays

Ruan, Shigui

doi:10.3934/dcdsb.2020282

Cited by 16 publications

(8 citation statements)

References 120 publications

(302 reference statements)

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“…Delay differential equations are widely used for tumor growth modelling, as the introduction of time delays in an ordinary tumor growth model significantly describes the complex cellular interaction and gives us a more realistic picture of the model [21][22][23][24][25][26][27][28]. The inclusion of time delays in tumor immune interaction also produces the nonlinear behavior of tumors [29]. Bi et al [30] incorporated the time delays regarding molecule production, progression, differentiation of cells, transport, etc., and observed rich, complex behavior for a tumor-immune interaction model.…”

Section: Introductionmentioning

confidence: 99%

A study on the dynamics of a breast cancer model with discrete-time delay

Das,

Dehingia,

Hinçal

et al. 2024

Phys. Scr.

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This study aims to discuss the impact of discrete-time delay on the anti-tumor immune response against tumor growth, excess levels of estrogen, and the source rate of immune cells in a breast cancer model. The non-negativity and boundedness of the solutions of the model are discussed. The existence of equilibria and their stability are examined. It is found that if the estrogen level is normal and the source rate of immune cells is low, the stability of the model around the co-existing equilibrium switches to instability via a Hopf bifurcation as the time delay increases. To validate the theoretical findings, a few numerical examples have been presented. The main result of this study is that the growth of tumors can be controlled if the immune system quickly generates an anti-tumor immune response. However, if the immune system takes a longer time to generate anti-tumor immune responses, the tumor growth cannot be controlled, and the system becomes unstable, which may result in the further spread of the disease.

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Section: Introductionmentioning

confidence: 99%

A study on the dynamics of a breast cancer model with discrete-time delay

Das,

Dehingia,

Hinçal

et al. 2024

Phys. Scr.

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“…In the same year, Niu et al 17 analyzed the stability of steady states in a diffusive tumor‐immune model with response and proliferation delay of immune system. In 2021, Ruan 18 reviewed some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effectors cells of the immune system.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of immunotherapy antitumor models with impulsive control strategy

Wang

Zhang

2021

Math Methods in App Sciences

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In this paper, using the methods of killing tumors and impulsive differential equations, two immunotherapy antitumor models for describing therapies of general tumors and advanced solid tumors are established. By using the theories of impulsive equations, small amplitude perturbation techniques, and the comparison technique, we obtain the conditions which guarantee the global asymptotical stability of the tumor‐eliminated periodic solution and system permanence, when immunotherapy alone is performed. The numerical results of the influences of the impulsive perturbation on the inherent oscillation show rich dynamics, such as period‐doubling bifurcation and chaos. Moreover, the effects of the combination of radiotherapy with immunotherapy on antitumor are obtained, including the threshold value of stability conditions of tumor‐eradication periodic solution when the mixed combination treatment of immunotherapy and radiotherapy is performed. Some numerical simulations for the effects of the timing of radiotherapy application and the timing of injection T cells on the threshold value are performed. Finally, we present some theoretical methods for suppressing the growth of tumors.

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“…They have calculated the range of delays for which immune cells hinder the tumor cells [13]. A review work of some recent delayed work on tumor-immune interactions was described by S. Ruan [14]. Moreover, a discussion on a three-delayed tumor-immune system interaction model have been carried out.…”

Section: Introductionmentioning

confidence: 99%

An Optimal Control Problem of Chemotherapy In Presence of Gene and Immunotherapy

Dehingia¹

2021

Preprint

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In this study, a cancer eradication model involving effector cells in the presence of gene therapy, immunotherapy, and chemotherapy has been proposed. Boundedness and positiveness of the solutions in the respective feasible domains of the proposed model are verified. Conditions for which the equilibrium points of the system are stable have been derived. Amounts of immunotherapy to be injected to a patient for eradication of cancerous tumor cells have been found. To minimize the chemotherapy drug-induced toxicity to the patient, an optimal control problem for the system has been constructed and solved. Numerical and graphical results have been presented. Through the investigation, it was seen that tumor cells can be eliminated in a specific time interval with the control of chemotherapeutic drug concentration.

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Nonlinear dynamics in tumor-immune system interaction models with delays

Cited by 16 publications

References 120 publications

A study on the dynamics of a breast cancer model with discrete-time delay

A study on the dynamics of a breast cancer model with discrete-time delay

Dynamics of immunotherapy antitumor models with impulsive control strategy

An Optimal Control Problem of Chemotherapy In Presence of Gene and Immunotherapy

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