Bifurcation analysis of a delayed mathematical model for tumor growth

Khajanchi, Subhas

doi:10.1016/j.chaos.2015.06.001

Cited by 50 publications

(11 citation statements)

References 31 publications

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“…A model system is called chaotic if the trajectory created by its equations satisfy the property known as sensitive dependence on initial conditions. 28 Such property can also be noticed in the tumor growth dynamics, 11,12,28,29,33 and is presumed to be a sign of the existence of chaotic scenario in this system. 28 The most important indicator of chaotic dynamics that exhibits this property is the maximum Lyapunov Table I.…”

Section: Results and Biological Interpretationsmentioning

confidence: 74%

“…The models have the ability to exhibit the existence of chaotic behaviors in the cancer-immune competitive system (delayed as well as non-delayed system), some examples 10,12,29,37 are in continuous models. Our model system (2) exhibits chaotic dynamics which has been verified by the most important indicator for chaotic dynamics is the maximum Lyapunov Characteristic Exponent(LCE).…”

Section: Discussionmentioning

confidence: 99%

“…[21][22][23][24][25] The theoretical investigation of tumor and immune interactive dynamics has a long history and a good precis can be observed in Adam and Bellomo. 3 The delayed responses 26,27 cannot be neglected for the cancer and immune system interplay, just as Villasana and Radunskaya, 19 Moghtadaei et al, 28 Khajanchi, 29 and Bi et al 10 revealed that discrete time lags should be considered to narrate the times required for the development of molecules, progression, differentiation of cell populations, transport, etc. In fact, dynamic relationships between cancer-immune system interplay with time lag have been investigated extensively; see Refs.…”

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confidence: 99%

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The influence of time delay in a chaotic cancer model

Khajanchi

Perc

Ghosh

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The tumor-immune interactive dynamics is an evergreen subject that continues to draw attention from applied mathematicians and oncologists, especially so due to the unpredictable growth of tumor cells. In this respect, mathematical modeling promises insights that might help us to better understand this harmful aspect of our biology. With this goal, we here present and study a mathematical model that describes how tumor cells evolve and survive the brief encounter with the immune system, mediated by effector cells and host cells. We focus on the distribution of eigenvalues of the resulting ordinary differential equations, the local stability of the biologically feasible singular points, and the existence of Hopf bifurcations, whereby the time lag is used as the bifurcation parameter. We estimate analytically the length of the time delay to preserve the stability of the period-1 limit cycle, which arises at the Hopf bifurcation point. We also perform numerical simulations, which reveal the rich dynamics of the studied system. We show that the delayed model exhibits periodic oscillations as well as chaotic behavior, which are often indicators of long-term tumor relapse.

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Section: Results and Biological Interpretationsmentioning

confidence: 74%

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

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The influence of time delay in a chaotic cancer model

Khajanchi

Perc

Ghosh

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Accounting for the complexity of immune responses to a tumor, various mathematical models were developed to understand its dynamical mechanism in the past few decades. Among these models, most of them are deterministic (e.g., [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and references therein), and only a few of them are stochastic ( e.g., [19,20,21,22,23] and references therein). In the investigation of deterministic models, some researchers focus on the estimation of parameters and the analysis of dynamical properties, such as stability, bifurcation and its stability and direction [4,5,6,7,8,9,10,11], while others further extend the applications of these models by exploring the optimal control of cancer treatment [12,13,14,15,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

A deterministic and stochastic model for the system dynamics of tumor–immune responses to chemotherapy

Liu

Pan

2018

Physica A: Statistical Mechanics and its Applications

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Modern medical studies show that chemotherapy can help most cancer patients, especially for those diagnosed early, to stabilize their disease conditions from months to years, which means the population of tumor cells remained nearly unchanged in quite a long time after fighting against immune system and drugs. In order to better understand the dynamics of tumor immune responses under chemotherapy, deterministic and stochastic differential equation models are constructed to characterize the dynamical change of tumor cells and immune cells in this paper. The basic dynamical properties, such as boundedness, existence and stability of equilibrium points, are investigated in the deterministic model. Extended stochastic models include stochastic differential equations (SDEs) model and continuous-time Markov chain (CTMC) model, which accounts for the variability in cellular reproduction, growth and death, interspecific competitions, and immune response to chemotherapy. The CTMC model is harnessed to estimate the extinction probability of tumor cells. Numerical simulations are performed, which confirms the obtained theoretical results.

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“…To understand the dynamics of tumor cells and their environment from the context of mathematical modeling have been studied by numerous researchers [1,3,4,[20][21][22][24][25][26]28,29]. The delay introduced to the biological system may cause the arising of phenomena such as existence of stability switches of the steady state [31], appearance of Hopf-bifurcation and periodic solutions or chaos [8,27,30].…”

Section: Introductionmentioning

confidence: 99%