A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man

David Greenhalgh, Qamar Khan

Research output: Contribution to conferenceAbstract

Abstract

In this paper we develop previously studied mathematical models of
the regulation of testosterone by luteinizing hormone and
luteinizing hormone release hormone in the human body. We propose a
delay differential equation mathematical model which improves on
earlier simpler models by taking into account observed experimental
facts. We show that our model has four possible equilibria, but only
one unique equilibrium where all three hormones are present. We
perform stability and Hopf bifurcation analyses on the equilibrium
where all three hormones are present. With no time delay this
equilibrium is unstable, but as the time delay increases through an
infinite sequence of positive values Hopf bifurcation occurs
repeatedly. This is of practical interest as biological evidence
shows that the levels of these hormones in the body oscillate
periodically. We next discuss stability of the other equilibria heuristically using analytical methods. Then we describe simulations with realistic parameter values and show that our model can mimic the regular fluctuations of the three hormones in the body and explore numerically some of our heuristic conjectures. A brief discussion concludes the paper.

Conference

ConferenceWorkshop on multiscale modelling of biological systems
CountryUnited Kingdom
CityStirling
Period5/07/106/07/10

Fingerprint

Hormones
Delay Differential Equations
Mathematical Model
Hopf Bifurcation
Time Delay
Analytical Methods
Unstable
Model
Heuristics
Fluctuations
Differential equation
Simulation

Keywords

  • human hormonal systems
  • testosterone
  • luteinizing hormone
  • luteinizing hormone release hormone
  • time delay
  • differential equations
  • equilibrium and stability
  • limit cycles
  • Hopf bifurcation

Cite this

Greenhalgh, D., & Khan, Q. (2010). A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man. Abstract from Workshop on multiscale modelling of biological systems, Stirling, United Kingdom.
Greenhalgh, David ; Khan, Qamar. / A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man. Abstract from Workshop on multiscale modelling of biological systems, Stirling, United Kingdom.
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title = "A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man",
abstract = "In this paper we develop previously studied mathematical models ofthe regulation of testosterone by luteinizing hormone andluteinizing hormone release hormone in the human body. We propose adelay differential equation mathematical model which improves onearlier simpler models by taking into account observed experimentalfacts. We show that our model has four possible equilibria, but onlyone unique equilibrium where all three hormones are present. Weperform stability and Hopf bifurcation analyses on the equilibriumwhere all three hormones are present. With no time delay thisequilibrium is unstable, but as the time delay increases through aninfinite sequence of positive values Hopf bifurcation occursrepeatedly. This is of practical interest as biological evidenceshows that the levels of these hormones in the body oscillateperiodically. We next discuss stability of the other equilibria heuristically using analytical methods. Then we describe simulations with realistic parameter values and show that our model can mimic the regular fluctuations of the three hormones in the body and explore numerically some of our heuristic conjectures. A brief discussion concludes the paper.",
keywords = "human hormonal systems, testosterone, luteinizing hormone, luteinizing hormone release hormone, time delay, differential equations, equilibrium and stability, limit cycles, Hopf bifurcation",
author = "David Greenhalgh and Qamar Khan",
year = "2010",
language = "English",
note = "Workshop on multiscale modelling of biological systems ; Conference date: 05-07-2010 Through 06-07-2010",

}

Greenhalgh, D & Khan, Q 2010, 'A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man' Workshop on multiscale modelling of biological systems, Stirling, United Kingdom, 5/07/10 - 6/07/10, .

A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man. / Greenhalgh, David; Khan, Qamar.

2010. Abstract from Workshop on multiscale modelling of biological systems, Stirling, United Kingdom.

Research output: Contribution to conferenceAbstract

TY - CONF

T1 - A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man

AU - Greenhalgh, David

AU - Khan, Qamar

PY - 2010

Y1 - 2010

N2 - In this paper we develop previously studied mathematical models ofthe regulation of testosterone by luteinizing hormone andluteinizing hormone release hormone in the human body. We propose adelay differential equation mathematical model which improves onearlier simpler models by taking into account observed experimentalfacts. We show that our model has four possible equilibria, but onlyone unique equilibrium where all three hormones are present. Weperform stability and Hopf bifurcation analyses on the equilibriumwhere all three hormones are present. With no time delay thisequilibrium is unstable, but as the time delay increases through aninfinite sequence of positive values Hopf bifurcation occursrepeatedly. This is of practical interest as biological evidenceshows that the levels of these hormones in the body oscillateperiodically. We next discuss stability of the other equilibria heuristically using analytical methods. Then we describe simulations with realistic parameter values and show that our model can mimic the regular fluctuations of the three hormones in the body and explore numerically some of our heuristic conjectures. A brief discussion concludes the paper.

AB - In this paper we develop previously studied mathematical models ofthe regulation of testosterone by luteinizing hormone andluteinizing hormone release hormone in the human body. We propose adelay differential equation mathematical model which improves onearlier simpler models by taking into account observed experimentalfacts. We show that our model has four possible equilibria, but onlyone unique equilibrium where all three hormones are present. Weperform stability and Hopf bifurcation analyses on the equilibriumwhere all three hormones are present. With no time delay thisequilibrium is unstable, but as the time delay increases through aninfinite sequence of positive values Hopf bifurcation occursrepeatedly. This is of practical interest as biological evidenceshows that the levels of these hormones in the body oscillateperiodically. We next discuss stability of the other equilibria heuristically using analytical methods. Then we describe simulations with realistic parameter values and show that our model can mimic the regular fluctuations of the three hormones in the body and explore numerically some of our heuristic conjectures. A brief discussion concludes the paper.

KW - human hormonal systems

KW - testosterone

KW - luteinizing hormone

KW - luteinizing hormone release hormone

KW - time delay

KW - differential equations

KW - equilibrium and stability

KW - limit cycles

KW - Hopf bifurcation

UR - http://dx.doi.org/10.1016/j.na.2009.01.031

M3 - Abstract

ER -

Greenhalgh D, Khan Q. A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man. 2010. Abstract from Workshop on multiscale modelling of biological systems, Stirling, United Kingdom.