Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response

Yafia, Radouane

doi:10.1016/j.nonrwa.2006.08.003

Cited by 20 publications

(16 citation statements)

References 13 publications

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“…In this century, cancer remains one of the most dangerous killers of humankind; every year millions of people suffer from cancer and die from this disease throughout the world; see Boyle et al [1]. Recently, there has been much interest in mathematical modeling of immune response with the intruder (see, e.g., Liu et al [2,3], Yafia [4], d'Onofrio et al [5,6], and the references cited therein). In fact, mathematical models are feasible to propose simple models which are capable of displaying some of the essential immunological phenomena.…”

Section: Introductionmentioning

confidence: 99%

Bifurcations of Tumor-Immune Competition Systems with Delay

Xiao

2014

Abstract and Applied Analysis

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A tumor-immune competition model with delay is considered, which consists of two-dimensional nonlinear differential equation. The conditions for the linear stability of the equilibria are obtained by analyzing the distribution of eigenvalues. General formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, steady-state bifurcation, and B-T bifurcation. Numerical examples and simulations are given to illustrate the bifurcations analysis and obtained results.

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Section: Introductionmentioning

confidence: 99%

Bifurcations of Tumor-Immune Competition Systems with Delay

Xiao

2014

Abstract and Applied Analysis

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“…Modelling tumour-immune interaction has attracted much attention in the last decades. This interaction is very complex and mathematical models can help to shape our understanding of dynamics this biological phenomenon [1,3,4,[6][7][8]15,22,23,27,29]. Much research has focussed on how to enhance the anti-tumour activity, by stimulating the immune system with vaccines or by direct injection of T cells or cytokines [24,25].…”

Section: Introductionmentioning

confidence: 99%

Analysis of tumour-immune evasion with chemo-immuno therapeutic treatment with quadratic optimal control

Ravindran

Sheriff

Krishnapriya

2017

Journal of Biological Dynamics

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A simple mathematical model for the growth of tumour with discrete time delay in the immune system is considered. The dynamical behaviour of our system by analysing the existence and stability of our system at various equilibria is discussed elaborately. We set up an optimal control problem relative to the model so as to minimize the number of tumour cells and the chemo-immunotherapeutic drug administration. Sensitivity analysis of tumour model reveals that parameter value has a major impact on the model dynamics. We numerically illustrate how does these delay can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumour. Finally, epidemiological implications of our analytical findings are addressed critically. ARTICLE HISTORY

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“…Mathematical models of tumor growth and tumor-immune system interaction are derived by mainly using Ordinary Differential Equations (ODEs) (see, for example, [7], [14], [17], [20] and references therein), Delay Differantial Equations (DDEs) (see, for example, [2], [3], [4], [8], [9], [11], [21], [22] and references therein) and Partial Differential Equations (PDEs) (see, for example, [10] and references therein). One of ODE models was introduced by Kuznetsov and Taylor [16] which has the following form:…”

Section: Introductionmentioning

confidence: 99%

“…Time delays in connection with tumor growth also appear in [2], [3], [4], [8]. Yafia ([21], [22]) first showed the existence and stability of periodic solutions of system (1.5) through Hopf bifurcation when the immune response negative (i.e., ω < 0) at which a biologically meaningful nonnegative solution exists. Later, he gave Hopf bifurcation analysis of system (1.5) by using σ as a bifurcation parameter in [23] and τ as a bifurcation parameter in [24].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis of a Modified Tumor-immune System Interaction Model Involving Time Delay

Kayan

Merdan

Yafia³

et al. 2017

Math. Model. Nat. Phenom.

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We study stability and Hopf bifurcation analysis of a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a reaction-diffusion system including time delay under the Neumann boundary conditions, and is based on Kuznetsov-Taylor's model. Choosing the delay parameter as a bifurcation parameter, we first show that Hopf bifurcation occurs. Second, we determine two properties of the periodic solution, namely its direction and stability, by applying the normal form theory and the center manifold reduction for partial functional differential equations. Furthermore, we discuss the effects of diffusion on the dynamics by analyzing a model with constant coefficients and perform some numerical simulations to support the analytical results. The results show that diffusion has an important effects on the dynamics of a mathematical model.

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Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response

Cited by 20 publications

References 13 publications

Bifurcations of Tumor-Immune Competition Systems with Delay

Bifurcations of Tumor-Immune Competition Systems with Delay

Analysis of tumour-immune evasion with chemo-immuno therapeutic treatment with quadratic optimal control

Bifurcation Analysis of a Modified Tumor-immune System Interaction Model Involving Time Delay

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