Stability and Hopf bifurcation analysis for the hypothalamic-pituitary-adrenal axis model with memory

Kaslik, Eva; Neamţu, Mihaela

doi:10.1093/imammb/dqw020

Cited by 8 publications

(10 citation statements)

References 33 publications

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“…Remark In the case of the minimal model of the HPA axis, it has been show that there exists a unique equilibrium state. For the extended 4‐dimensional model , Proposition only shows the existence of at least 1 equilibrium state.…”

Section: Positively Invariant Sets and Equilibrium Statesmentioning

confidence: 99%

“…This implies

w_{1} w_{2} w_{3} \overset{w_{4}}{true} < a w_{1} w_{4} + b \overset{w_{4}}{true}

, which in turn, is equivalent to

false(\overset{I 3}{true} false)

, and b. is proved. Point c. follows from . □…”

Section: Bifurcation Analysismentioning

confidence: 99%

“…For simplicity, let η = μ =1 and hence, the considered feedback functions are

f_{1} (x) = \frac{c_{1}^{α}}{c_{1}^{α} + x^{α}}, f_{2} (x) = \frac{c_{2}^{α}}{c_{2}^{α} + x^{α}}, f_{3} (x) = \frac{x^{β}}{c_{3}^{β} + x^{β}}

with α =4 and β =5 as in Sriram et al, c 1 =2 ng/mL as in Kaslik and Neamtu, and c 2 = c 3 =0.8 ng/mL.…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…Point c. follows from. 12 For the bifurcation analysis, due to the complexity of the problem, we restrict our attention to Dirac kernels and Gamma kernels.…”

Section: Bifurcation Analysismentioning

confidence: 99%

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Stability and Hopf bifurcation analysis of a 4‐dimensional hypothalamic‐pituitary‐adrenal axis model with distributed delays

Kaslik

Neamţu

2018

Math Methods in App Sciences

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A 4‐dimensional mathematical model of the hypothalamus‐pituitary‐adrenal axis is investigated, incorporating the influence of the glucocorticoid receptors concentration and general feedback functions. The inclusion of distributed time delays provides a more realistic modeling approach, since the whole past history of the variables is taken into account. The positivity of the solutions and the existence of a positively invariant bounded region are proved. It is shown that the considered 4‐dimensional system has at least 1 equilibrium state, and a detailed local stability and Hopf bifurcation analysis is given. Numerical results reveal the fact that an appropriate choice of the system's parameters leads to the coexistence of 2 asymptotically stable equilibria in the nondelayed case. When the total average time delay of the system is large enough, the coexistence of 2 stable limit cycles is revealed, which successfully model the ultradian rhythm of the hypothalamus‐pituitary‐adrenal axis both in a normal disease‐free situation and in a diseased hypocortisolism state, respectively. Numerical simulations reflect the importance of the theoretical results.

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Section: Positively Invariant Sets and Equilibrium Statesmentioning

confidence: 99%

“…This implies

w_{1} w_{2} w_{3} \overset{w_{4}}{true} < a w_{1} w_{4} + b \overset{w_{4}}{true}

, which in turn, is equivalent to

false(\overset{I 3}{true} false)

, and b. is proved. Point c. follows from . □…”

Section: Bifurcation Analysismentioning

confidence: 99%

“…For simplicity, let η = μ =1 and hence, the considered feedback functions are

f_{1} (x) = \frac{c_{1}^{α}}{c_{1}^{α} + x^{α}}, f_{2} (x) = \frac{c_{2}^{α}}{c_{2}^{α} + x^{α}}, f_{3} (x) = \frac{x^{β}}{c_{3}^{β} + x^{β}}

with α =4 and β =5 as in Sriram et al, c 1 =2 ng/mL as in Kaslik and Neamtu, and c 2 = c 3 =0.8 ng/mL.…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…Point c. follows from. 12 For the bifurcation analysis, due to the complexity of the problem, we restrict our attention to Dirac kernels and Gamma kernels.…”

Section: Bifurcation Analysismentioning

confidence: 99%

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Stability and Hopf bifurcation analysis of a 4‐dimensional hypothalamic‐pituitary‐adrenal axis model with distributed delays

Kaslik

Neamţu

2018

Math Methods in App Sciences

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“…In 2015, Kaslik and Neamtu [13] proved the occurrence of oscillating solutions in a neighborhood of the unique equilibrium point, by performing a Hopf bifurcation analysis, considering several distributed time delays and fractional derivatives in the minimal three-dimensional mathematical model previously analyzed in [12].…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a hypothalamic-pituitary-adrenal axis model with inclusion of glucocorticoid receptor and memory

Kaslik¹,

Navolan²,

Neamţu³

2017

AIP Conference Proceedings

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Abstract. This paper analyzes a four-dimensional model of the hypothalamic-pituitary-adrenal (HPA) axis that includes the influence of the glucocorticoid receptor in the pituitary. Due to the spatial separation between the hypothalamus, pituitary and adrenal glands, distributed time delays are introduced in the mathematical model. The existence of the positive equilibrium point is proved and a local stability and bifurcation analysis is provided, considering several types of delay kernels. The fractional-order model with discrete time delays is also taken into account. Numerical simulations are provided to illustrate the effectiveness of the theoretical findings.

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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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Stability and Hopf bifurcation analysis for the hypothalamic-pituitary-adrenal axis model with memory

Cited by 8 publications

References 33 publications

Stability and Hopf bifurcation analysis of a 4‐dimensional hypothalamic‐pituitary‐adrenal axis model with distributed delays

Stability and Hopf bifurcation analysis of a 4‐dimensional hypothalamic‐pituitary‐adrenal axis model with distributed delays

Stability analysis of a hypothalamic-pituitary-adrenal axis model with inclusion of glucocorticoid receptor and memory

Mathematical modeling of tumor surface growth with necrotic kernels

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