THE DYNAMICS OF GLUCOSE-INSULIN ENDOCRINE METABOLIC
REGULATORY SYSTEM
by
Jiaxu Li
A Dissertation Presented in Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
ARIZONA STATE UNIVERSITY
December 2004
THE DYNAMICS OF GLUCOSE-INSULIN ENDOCRINE METABOLIC
REGULATORY SYSTEM
by
Jiaxu Li
has been approved
December 2004
APPROVED:
, Chair
Supervisory Committee
ACCEPTED:
Department Chair
Dean, Division of Graduate Studies
ABSTRACT
A model with two time delays is presented for modeling the insulin secretion ultradian oscillations in the glucose-insulin metabolic system. One delay is for the insulin
response time delay (around 6 minutes) to the glucose concentration level increase, and
the other is for the hepatic glucose production time delay (around 36 minutes). The
results of the analysis of this model are in agreement with the experimental observations
and exhibit intrinsic insulin secretion ultradian oscillations. The results show that both
these time delays are necessary for the insulin secretion ultradian oscillation sustainment and only the relative moderate glucose infusion rate and insulin degradation rate
can sustain the oscillations. The numerical simulations demonstrate that the insulin
concentration level peaks after the glucose concentration level. These results also indicate that the hepatic glucose production and its time delay are insignificant in modeling
intravenous glucose tolerance tests (IVGTT).
A generic dynamic IVGTT model and two models for special cases are developed to simulate the short time (30-120 minutes) dynamics. As expected, such models
frequently produce globally asymptotically stable steady state dynamics. The easy-tocheck conditions, which guarantee the steady state to be stable, are provided.
In the last model, we take the active β-cell mass into consideration and study the
effects of the β-cells in the glucose-insulin regulatory system. The numerical simulations
show that the insulin concentration peaks after the active β-cell mass peaks, which peaks
after the glucose concentration peaks. Other results are also in agreement with reported
data.
iii
In Memory of My Mother
To My Father
To My Wife and Daughters
To My Sisters
iv
ACKNOWLEDGMENTS
I would first of all like to thank Dr. Yang Kuang for his guidance during my doctoral study at Arizona State University. I am forever indebted to his advise, suggestions,
support, encouragement, understanding and, in particular, patience. I would like to
thank Dr. Steven Baer, Dr. Carlos Castillo-Chaves, Dr. Hal Smith and Dr. Horst Thieme
for their interest, carefully reading the manuscript, valuable input and suggestions for
improving this dissertation. It is my great pleasure to work with them and I feel so
lucky and proud that I have such a wonderful supervisory committee, one of the best
in the world. I would also like to thank the external reviewer for the valuable input.
My special thanks go to my master thesis advisor Prof. Xiudong Chen.
I would also like to extend my gratitude to Dr. Bingtuan Li for the various
broad discussions, to Dr. Athena Makroglou for her initiating the collaborate paper
[59] and providing references (for example, [65] and [64]), to Prof. Edoardo Beretta for
his providing the manuscript of [23], to Mr. Clint Mason for his providing reference [9]
and [85], to Ms. Debbie Olson and Ms. Joan Person for their administrative support,
to Dr. Jialong He for his IT support, and to Mr. Rafael Mendez for the proof-reading
of the most of this dissertation.
Last, but not the least, I would like to thank my wife, Dr. Guihua Li, for her
long lasting love and support.
Jiaxu Li
December 11, 2004
Arizona State University, Tempe, Arizona USA
v
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
CHAPTER 1 Introduction and Physiological Background . . . . . . . . . . . . .
1
1.
Diabetes Mellitus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2.
Glucose-Insulin Endocrine Metabolic Regulatory System . . . . . . . . .
4
3.
The pancreas and Its Endocrine Hormones . . . . . . . . . . . . . . . . .
6
3.1.
The pancreas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.2.
Glucose Transporters . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.3.
Secretion and Actions of Insulin . . . . . . . . . . . . . . . . . . . 10
3.4.
Insulin Receptors . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.5.
Insulin Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.6.
Insulin Degradation and Clearance . . . . . . . . . . . . . . . . . 16
3.7.
Production and Consumption of Glucose . . . . . . . . . . . . . . 17
4.
Glucose Tolerance Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.
The Organization of This Dissertation . . . . . . . . . . . . . . . . . . . 21
CHAPTER 2 The Ultradian Oscillations of Insulin Secretion . . . . . . . . . . . 23
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.
Sturis-Tolic ODE Model and Current Research Status . . . . . . . . . . . 25
3.
Two Time Delay DDE Model . . . . . . . . . . . . . . . . . . . . . . . . 34
4.
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
vi
5.
Global Stability of Steady State . . . . . . . . . . . . . . . . . . . . . . . 43
6.
Linearization and Local Analysis . . . . . . . . . . . . . . . . . . . . . . 44
7.
Numerical Analysis of Stability Switches and Bifurcations . . . . . . . . . 56
8.
7.1.
Insulin Response Time Delay τ1 . . . . . . . . . . . . . . . . . . . 59
7.2.
Glucose Infusion Rate Gin . . . . . . . . . . . . . . . . . . . . . . 60
7.3.
Insulin Degradation Rate di . . . . . . . . . . . . . . . . . . . . . 63
7.4.
Hepatic Glucose Production τ2 . . . . . . . . . . . . . . . . . . . . 63
7.5.
Parameter τ1 vs. Gin . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.6.
Parameter τ1 vs. di . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.7.
Parameter Gin vs. di . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.8.
Insulin Concentration Peaks after Glucose Concentration Peaks . 68
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
CHAPTER 3 Modeling Intra-Venus Glucose Tolerance Test
. . . . . . . . . . . 75
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.
Current Research Status . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.
More Generic IVGTT Model . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.
Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.
Global Stability of Steady State . . . . . . . . . . . . . . . . . . . . . . . 87
6.
Local Stability of Steady State and Stability Switch . . . . . . . . . . . . 93
7.
Delay Independent Stability Results for Discrete Delay Model . . . . . . 95
8.
Delay Dependent Stability Conditions . . . . . . . . . . . . . . . . . . . . 99
8.1.
The case of discrete delay . . . . . . . . . . . . . . . . . . . . . . 100
8.2.
The case of distributed delay . . . . . . . . . . . . . . . . . . . . . 101
8.3.
Expression of H(α) . . . . . . . . . . . . . . . . . . . . . . . . . . 102
vii
9.
Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
CHAPTER 4 The Effects of Active β-Cells: A Preliminary Study . . . . . . . . 108
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
2.
Current Research Status . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.
Active β-Cell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.
Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.1.
Insulin Response Delay and Hepatic Glucose Production Are Critical for Sustain Insulin Secretion Oscillations . . . . . . . . . . . . 115
4.2.
Insulin Response Time Delay τ1 as a Bifurcation Parameter . . . 115
4.3.
Glucose Infusion Rate Gin as a Bifurcation Parameter . . . . . . . 116
4.4.
Peaks of Oscillations in One Cycle
4.5.
β-cell Deactivation Rate k ∈ [0.01, 2] as a Bifurcation Parameter . 119
4.6.
Parameter σ as a Bifurcation Parameter . . . . . . . . . . . . . . 120
4.7.
The Changes of Insulin Degradation Rate di ∈ [0.025, 0.1] Do Not
. . . . . . . . . . . . . . . . . 118
Affect the Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 122
5.
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
viii
LIST OF TABLES
Table
Page
1.4.1.
Fasting Glucose Tolerance Test
1.4.2.
Oral Glucose Tolerance Test . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.3.
Gestational Diabetes Glucose Tolerance Test . . . . . . . . . . . . . . 21
2.2.1.
Parameters in the Sturis-Tolic Model (2.2.1). . . . . . . . . . . . . . . 28
2.2.2.
Parameters of the functions in the Sturis-Tolic Model (2.2.1). . . . . . 28
2.7.1.
Parameters of the functions in Two Time Delay Model (2.3.1). . . . . 57
3.9.1.
Parameters for subjects 6 and 7 in IVGTT Models (b5 = 23min.) . . . 104
4.2.1.
Parameters of the Model 4.2.1 . . . . . . . . . . . . . . . . . . . . . . 111
ix
. . . . . . . . . . . . . . . . . . . . . 20
LIST OF FIGURES
Figure
Page
1.2.1.
Glucose-Insulin Regulatory System . . . . . . . . . . . . . . . . . . . . . . .
7
1.3.1.
Langerhans islets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3.2.
The β cells secrete insulin when glucose concentration level elevated . . . . . . . 12
1.3.3.
Insulin signals cells to utilize glucose . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1.
Insulin Secretion Ultradian Oscillations . . . . . . . . . . . . . . . . . . . . . 24
2.2.1.
Physiological Glucose-Insulin Regulatory System . . . . . . . . . . . . . . . . 26
2.2.2.
Functions fi (I), i = 1, 2, 4, 5.
2.3.1.
Two Time Delay Glucose-Insulin Regulatory Model . . . . . . . . . . . . . . . 35
2.7.1.
Bifurcation diagram of τ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.7.2.
Periods of periodic solutions when τ1 ∈ [0, 20] and bifurcation diagram of di
2.7.3.
Bifurcation diagram of Gin ∈ [0, 2.16]
2.7.4.
Limit cycles in (Gin , G, I)-space when Gin ∈ [0, 2.16] . . . . . . . . . . . . . . 62
2.7.5.
Periods of periodic solutions when Gin ∈ [0, 2.16] . . . . . . . . . . . . . . . . 62
2.7.6.
Periods and peak time differences when di changes in [0.001, 0.7]
2.7.7.
Hepatic production delay has no impact to sustained oscillations . . . . . . . . . 64
2.7.8.
Stability Region in (τ1 , Gin )-plane . . . . . . . . . . . . . . . . . . . . . . . 65
2.7.9.
Bifurcation diagrams and stability regions in (τ1 , Gin )-space . . . . . . . . . . . 66
. . . . . . . . . . . . . . . . . . . . . . . . . 29
. . . 60
. . . . . . . . . . . . . . . . . . . . . 61
. . . . . . . . 63
2.7.10. Stability Regions in (τ1 , di )-plane and (Gin , di )-plane . . . . . . . . . . . . . . 67
2.7.11. Glucose concentrations peak before insulin does . . . . . . . . . . . . . . . . . 69
3.9.1.
Periodic solutions for the discrete delay model (3.3.2) for subject 6 and 7 . . . . . 105
4.3.1.
Glucose-Insulin with Active β-cell Interaction Diagram . . . . . . . . . . . . . 113
4.3.2.
Function g(G) in GIβ-Model
. . . . . . . . . . . . . . . . . . . . . . . . . 115
x
4.4.1.
Orbits of (G, I, β) of GIβ model . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4.2.
Bifurcation diagram of τ1 ∈ [0, 20] . . . . . . . . . . . . . . . . . . . . . . . 117
4.4.3.
Bifurcation diagram of Gin ∈ [0, 3.0] . . . . . . . . . . . . . . . . . . . . . . 118
4.4.4.
Periodic solutions and periods when Gin ∈ [0, 3.0] . . . . . . . . . . . . . . . . 119
4.4.5.
Peaks of Oscillations in One Cycle . . . . . . . . . . . . . . . . . . . . . . . 120
4.4.6.
Bifurcation diagram of k ∈ [0.01, 2] . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.7.
Bifurcation diagram of σ ∈ [0.0001, 0.1] . . . . . . . . . . . . . . . . . . . . . 121
4.4.8.
Limit Cycles when σ ∈ [0.0001, 0.1] . . . . . . . . . . . . . . . . . . . . . . . 122
4.4.9.
There is no bifurcation when di ∈ [0.005, 0.01]
4.5.1.
Possible β-cell pulsatile oscillation?
. . . . . . . . . . . . . . . . . 122
. . . . . . . . . . . . . . . . . . . . . . 124
xi
CHAPTER 1
Introduction and Physiological Background
1. Diabetes Mellitus
Human bodies need to maintain a glucose concentration level in a narrow range
(70 - 109 ml/dl or 3.9 - 6.04 mmol/l). If one’s glucose concentration level is significantly
out of the normal range (70 - 110 ml/dl), this person is considered to have a the plasma
glucose problem: hyperglycemia (≥140 mg/dl or 7.8 mmol/l after an Oral Glucose
Tolerance Test, or ≥100 mg/dl or 5.5 mmol/l after a Fasting Glucose Tolerance Test)
or hypoglycemia (less than 40 mg/dl or 2.2 mmol/l) ([89], [96]).
Diabetes mellitus is a disease in the glucose-insulin endocrine metabolic regulatory system, in which the pancreas either does not release insulin or does not properly
use insulin to uptake glucose in the plasma ([9], [85]), which is referred as hyperglycemia.
The consequences are that the body does not metabolize the glucose and builds
up hyperglycemia which eventually damages the regulatory system. Complications of
diabetes mellitus include retinopathy, nephropathy, peripheral neuropathy and blindness ([25]).
Diabetes mellitus is one of the worst diseases with respect to size of the affected
population. According to the data published by American Diabetes Association ([94]),
in the United States in 2002, 18.2 million people - 6.3% of the total population - had
2
diabetes. The direct and indirect cost of diabetes in 2002 was $132 billions. The world
wide diabetics population is much higher, especially in underdeveloped countries.
Diabetes mellitus is currently classified as type 1 diabetes or type 2 diabetes
([9], [85]). Type 1 diabetes was previously called insulin-dependent diabetes mellitus
(IDDM) or juvenile-onset diabetes. It develops when the body’s immune system destroys pancreatic beta cells, the only cells in the body that make the hormone insulin,
which regulates blood glucose. This form of diabetes usually strikes children and young
adults, although disease onset can occur at any age. Type 1 diabetes may account
for 5% to 10% of all diagnosed cases of diabetes. Risk factors for type 1 diabetes include autoimmune, genetic, and environmental factors. Type 2 diabetes is adult onset
or non-insulin-dependent diabetes mellitus (NIDDM) as this is due to a deficit in the
mass of β cells, reduced insulin secretion [53], and resistance to the action of insulin
[32]. The relative contribution and interaction of these defects in the pathogenesis of
this disease remains to be clarified [17]. About 90% to 95% of all diabetics diagnose
type 2 diabetes. Type 2 diabetes is associated with older age, obesity, family history
of diabetes, prior history of gestational diabetes, impaired glucose tolerance, physical
inactivity, and race/ethnicity. African Americans, Hispanic/Latino Americans, Native
Americans, some Asian Americans, Native Hawaiian, and other Pacific Islanders are
at particularly high risk for type 2 diabetes. Type 2 diabetes is increasingly being
diagnosed in children and adolescents ([93]).
In addition to Type 1 and Type 2 diabetes, gestational diabetes is a form of
glucose intolerance that is diagnosed in some women during pregnancy ([9], [85], [97]).
Gestational diabetes occurs more frequently among African Americans, Hispanic/Latino
Americans, and Native Americans. It is also more common among obese women and
3
women with a family history of diabetes. During pregnancy, gestational diabetes requires treatment to normalize maternal blood glucose levels to avoid complications in
the infant. After pregnancy, 5% to 10% of women with gestational diabetes are found
to have type 2 diabetes. Women who have had gestational diabetes have a 20% to
50% chance of developing diabetes in the next 5-10 years. Other specific types of diabetes result from specific genetic conditions (such as maturity-onset diabetes of youth),
surgery, drugs, malnutrition, infections, and other illnesses. Such types of diabetes may
account for 1% to 5% of all diagnosed cases of diabetes ([97]).
The relative contribution and interaction of these defects in the pathogenesis of
this disease remains to be clarified ([17]).
Due to the large population of diabetes patients in the world and the big health
expenses, many researchers are motivated to study the glucose-insulin endocrine metabolic
regulatory system so that we can better understand how the mechanism functions ([79],
[84], [85], [67], [74], [31], [85], [4] and their references), what cause the dysfunctions of
the system ([9] and its rich references), how to detect the onset of the either type of
diabetes including the so called prediabetes ([10], [83], [8], [97], [23], [57], [6], [63] and
their references), and eventually provide more reasonable, more effective, more efficient
and more economic treatments to diabetics. For example, according to Bergman ([6],
2002), there are now approximately 50 major studies published per year and more than
500 can be found in literature related to the so called minimal model ([10], [83], [8]) for
modeling the intra-venous glucose tolerance test.
4
2. Glucose-Insulin Endocrine Metabolic Regulatory System
Metabolism is the process of extracting useful energy from chemical bounds. A
metabolic pathway is a sequence of enzymatic reactions that take place in order to
transfer chemical energy from one form to another. The chemical adenosine triphosphate (ATP) is a common carrier of energy in a cell. There are two different ways to
form ATP:
1. adding one inorganic phosphate group (HP O42− ) to the adenosine diphosphate
(ADP), or
2. adding two inorganic phosphate groups to the adenosine monophosphate (AMP).
The process of inorganic phosphate group addition is referred to phosphorylation. Due
to the fact that the three phosphate groups in ATP carry negative charges, it requires
lots of energy to overcome the natural repulsion of like-charged phosphates when additional groups are added to AMP. So considerable amount of energy is released during
the hydrolysis of ATP to ADP ([51], [89] and [91]).
In the glucose-insulin endocrine metabolic regulatory system, the two pancreatic endocrine hormones, insulin and glucagon, are the primary dynamic factors that
regulate the system.
When the plasma glucose concentration rises, the elevation in the ratio of ATP/ADP
in a β cell in the pancreas causes ATP-sensitive K+ channels (KATP channels) in the
plasma membrane to close. The decreased K+ permeability leads to membrane depolarization, opening of voltage-dependent Ca2+ channels, Ca2+ influx, and eventual rise
of the cytosolic Ca2+ concentration ([Ca2+ ]c ) that triggers exocytosis ([91]).
5
When the serum insulin concentration increases, more insulin receptors of cells
are bound by insulin. The binding of insulin to its receptors on the surfaces of cell
membranes leads to an increase in glucose transporter (GLUT4) molecules in the outer
membrane of muscle cells and adipocytes, and therefore to an increase in the uptake
of glucose from blood into muscle and adipose tissue. Thus, the intracellular glucose is
consumed and energy is released ([91]).
After some amount of the plasma glucose is utilized by the cells and the concentration level is low, the β cells are signaled not to release insulin. Then the amount of
extracellular glucose transported into intracellular by the glucose transporters is significantly reduced or even stopped due to the decreased number of insulin receptors bound
by insulin. Therefore, the consumption of glucose is tremendously decreased.
When the glucose concentration level is low, the α cells in the pancreas will
release glucagon to the liver and the liver will convert glucagon into glucose. The liver
also converts glycogen into glucose.
In short, when human’s the plasma glucose concentration level is high, the following processes will occur:
1. the pancreas is signaled to release insulin from β cells;
2. serum insulin (including newly secreted insulin) binds to the cells’ insulin receptors,
3. the insulin receptors bound by insulin cause the glucose transporters (GLUT4)
transport glucose molecules into the cells;
4. the cells consume the glucose and convert to energy.
6
These processes decrease the glucose concentrations in the plasma. Almost all the cells
in human body have insulin receptors, including fat cells and muscle cells. Glucose
is also utilized by other cells without insulin involvement. The brain cell is a typical
example.
When a human’s the plasma glucose concentration level is low, a different series
of processes will occur:
1. the pancreas is signaled to release glucagon from α cells;
2. glucagon is transported to the liver;
3. the liver converts the glucagon to glucose.
These processes increase the glucose concentration level in human plasma.
Exogenous glucose infusion also increases glucose concentration. The typical exogenous glucose infusions include meal ingestion, oral glucose intake, continuous enteral
nutrition, and constant glucose infusion.
The liver plays a key role in keeping the glucose and insulin amount in human
plasma oscillating smoothly ([96]). Figure 1.2.1, which is adapted from [96], illustrates
the plasma glucose-insulin endocrine metabolic regulatory system.
3. The pancreas and Its Endocrine Hormones
3.1. The pancreas. The pancreas lies interior to a human’s stomach, in the
size of a human’s fist and is in the bend of the duodenum. Scattered through out
inside of the pancreas, there are about a million Langerhans islets. Each Langerhans
islet contains about three hundred β cells and each β cell contains about one thousand
granules. Approximately 5% of the total pancreatic mass is comprised of endocrine
7
High
Plasma Glucose
Level
Low
Plasma Glucose
Level
Pancreas
α-cells release
ß-cells release insulin
glucagon
Glucagon
Insulin
Exercises,
fasting
and others
Glucose Infusion,
meal, enteral,
oral intake
and others
Liver
Insulin helps
to consume
plasma glucose
Liver converts partial
glucagon released
from α-cells and partial
glycogen stored in liver
to glucose
Normal
Plasma Glucose
Level
Figure 1.2.1.
Glucose-Insulin Regulatory System
The dashed lines indicate that exercises and fasting consume glucose and lower the glucose concentration, which signals the pancreas to release glucagon and the liver converts the glucagon and glycogen
to glucose. The solid lines indicate that the glucose infusion elevate the plasma glucose concentration
level which signals the pancreas to secrete insulin and consume the glucose. (This figure is adapted
from [96].)
cells. These endocrine cells are clustered in groups within the pancreas, which look
like little islands of cells when examined under a microscope. The pancreas is both an
endocrine and an exocrine gland. The exocrine functions are concerned with digestion.
The endocrine function consists primarily for the secretion of the two major hormones,
insulin and glucagon, which participate in the regulation of carbohydrate metabolism.
Five types of cells in a Langerhans islet are identified: β cells, which occupy
65-80% of the islet and make insulin; α cells, which occupy 15-20% and make glucagon;
δ cells, which occupy 3-10% and make somatostatin ([87]); and pancreatic polypeptidecontaining P P cells and D1 cells comprise 1% ([2]), about which little is known. Figure
8
3.1 shows a Langerhans islet and the cells it contains. The β cells in the pancreas are
the only cells in which the growth hormone insulin is synthesized and secreted. The
insulin is synthesized in β cells as part of a larger preprohormone - preproinsulin - which
includes a 23 amino acid leader sequence attached to proinsulin; this leader sequence
is lost upon entrance of the molecule into the endoplasmic reticulum leaving the proinsulin molecule. Kallikrein, an enzyme present in the islets, aids in the conversion
of proinsulin to insulin. In this conversion, a C-peptide chain is removed from the
proinsulin molecule producing the disulfide-connected α and β chains that are insulin.
Insulin is an anabolic hormone, that is, it increases the storage of glucose, fatty acids
and amino acids in cells and tissues.
Figure 1.3.1.
Langerhans islets
A Langerhans islet contains α-cells, β-cells, δ-cells and others. This image contains three islets of a
house. (The image is from [89].)
It has been believed that the β cells do not replicate and neogenerate after one’s
birth. If one’s β cells are damaged in a large amount, he/she might have to suffer
diabetes due to β cell dysfunction. However, this hypothesis was recently challenged
by S. Bonner-Weir ([12], 2000). The new perspective assumes that the β-cells can be
replicated and neogenerated. Several in − vivo and in − vitra experiments support this
9
new perspective ([13], [14], [34], [82] and [49]).
The α cells release glucagon, a protein hormone that has important effects in the
regulation of carbohydrate metabolism. Glucagon is a catabolic hormone, that is, it
mobilizes glucose, fatty acids and amino acids from storage into the blood. When the
glucose concentration level in the plasma is low, the liver will convert the glucagon to
glucose.
Both insulin and glucagon are important in the regulation of carbohydrate, protein and lipid metabolism.
Somatostatin is secreted from the δ cells in the Langerhans islets in the pancreas and is a hormone inhibiting the secretion of many other hormones. Somatostatin
acts through both endocrine and paracrine pathways to affect its target cells. In the
pancreas, somatostatin appears to act primarily in a paracrine manner to inhibit the secretion of both insulin and glucagon. In the brain (hypothalamus) and the spinal cord
it may act as a neurohormone and neurotransmitter. The effects of somatostatin to
glucose-insulin regulatory system is small, indirect and negligible. Its paracrine manner
makes the secretion of insulin and glucagon smoother.
3.2. Glucose Transporters. Glucose is transported by its transporters. There
are total five transporters in the family, that is, GLUT1 to GLUT5 ([91]).
• GLUT1 is ubiquitously distributed in various tissues.
• GLUT2 is found primarily in intestine, kidney and liver.
• GLUT3 is found in the intestine.
• GLUT4 is primarily contained in insulin-sensitive tissues such as skeletal muscle
and adipose tissue.
10
• GLUT5 is found in the brain and testis. GLUT5 is also the major glucose transporter present in the membrane of the endoplasmic reticulum (ER) and serves the
function of transporting glucose to the cytosol following its dephosphorylation by
the ER enzyme glucose-6-phosphatase.
When the concentration of blood glucose increases in response to food intake,
pancreatic GLUT2 molecules mediate an increase in glucose uptake which leads to
increased insulin secretion. Recent evidence has shown that the cell surface receptor
for the human T cell leukemia virus (HTLV) is the ubiquitous GLUT1. ([91])
3.3. Secretion and Actions of Insulin. Insulin secretion is pulsatile and is
regulated primarily by the glucose metabolism ([67], [74]). Numerous in-vivo and invitro experiments have shown that insulin concentration oscillates in two different time
scales: rapid oscillation with a period of 5-15 minutes and ultradian oscillation with a
range of 50-140 minutes ([67], [74] and their cited references). The rapid oscillations
are caused by coordinating periodic secretory bursting of insulin from β cells contained
in millions of the Langerhans islets in the pancreas. These bursts are the dominant
mechanism of insulin release at basal level ([67]). Ultradian oscillations of insulin concentration are believed to be mainly due to glucose interaction in the plasma ([79], [84],
[74]). These ultradian oscillations are best seen after meal ingestion, oral glucose intake,
continuous enteral nutrition or intravenous glucose infusion ([79]). In addition, muscle,
the brain, nerve and others utilize the plasma glucose to complete the regulatory system
feedback loop. So, insulin production, glucose infusion and production (for example,
meal and continuous enteral nutrition in daily life) and glucose utilization (for example,
in daily life, exercise) are the three major variables of this intricate regulatory system
([74], [79]).
11
P. Gilon, M. A. Ravier, J.-C. Jonas, and J.-C. Henquin summarized the mechanism of insulin secretion control in 2002 ([39]). Glucose stimulates insulin secretion
from β-cells by activating two pathways that require metabolism of the sugar as follows
([47]).
• Triggering Pathway The GLUT2 transports the glucose into the β cell. It
causes the rise in the ratio of ATP/ADP which causes ATP-sensitive K+ channels
(KATP channels) in the plasma membrane to close. The decreased K+ permeability
leads to membrane depolarization, opening of voltage-dependent Ca2+ channels,
Ca2+ influx, and the eventual rise of the cytosolic Ca2+ concentration ([Ca2+ ]c )
that triggers exocytosis. This pathway is also called KATP channel-dependent
pathway. See Figure 1.3.2 for an illustration.
• Amplifying Pathway The KATP channel-independent pathway simply increases
the efficiency of the Ca2+ on exocytosis when the concentration of Ca2+ has been
elevated.
The pulsatility of insulin secretion might result from oscillations in either of these
transduction pathways. Because metabolism and [Ca2+ ]c play key roles in the control
of insulin secretion and have been reported to oscillate, many efforts have been spent
to investigate which of these two mechanisms is the primary factor of pulsatile insulin
secretion ([39]). The essential role of Ca2+ influx in the generation of [Ca2+ ]c oscillations
by glucose, in either whole islets or single β-cells, is demonstrated by their abrogation
upon omission of extracellular Ca2+ ([44], [38]) or blockade of voltage-dependent Ca2+
channels ([26]). [Ca2+ ]c oscillations are linked to oscillations of the membrane potential
in β-cells ([72], [38]), and it is assumed that mixed [Ca2+ ]c oscillations result from an
irregular (so-called “periodic”) electrical activity ([3], [46], [20]). Synchronization of the
12
β-cell electrical activity ([62]) by gap junctions is likely to underlie the synchronization
of [Ca2+ ]c oscillations between β-cells within the islet ([44], [50] and [43]). See Figure
Gl
uc
os
e
1.3.2 for an illustration.
GL
UT
2
Glucose-6-phosphate
Glucokinase
Glucose Metabolism
NAD(P)H
H+
ATP
Close K+
channels
Elevate
ADP
K+
Protein
Phosphorylations
Elevated Ca2+
Gr
Cell Depolarization
an
ule
s
Ca2+
influx
Open Ca2+
channeles
In
su
lin
Figure 1.3.2.
The β cells secrete insulin when glucose concentration level elevated
The facilitated GLUT2 transport the glucose into the β cell and the glucose is phosphorylated by
glucokinase. The ratio of ATP:ADP is elevated. The glucose metabolism causes ATP-sensitive K+
channels to close, the membrane to depolarize and the Ca2+ channels to open. This triggers a cascade
of protein phosphorylations and leads to insulin exocytosis [68]. (The figure is partially adapted from
[68].)
The insulin has five major actions. These include:
• facilitation of glucose transport through certain membranes (e.g. adipose and
muscle cells);
• stimulation of the enzyme system for conversion of glucose to glycogen (liver and
muscle cells);
• slow-down of gluconeogenesis (liver and muscle cells);
13
• regulation of lipogenesis (liver and adipose cells); and
• promotion of protein synthesis and growth (general effect).
These actions of insulin are mediated by the binding of the hormone to membrane receptors to trigger several simultaneous actions. A major effect of insulin is to promote
the entrance of glucose and amino acids in cells of muscle tissues, adipose tissue and
connective tissue. Glucose enters the cell by facilitated diffusion along an inward gradient created by low intracellular free glucose and by the availability of a specific carrier
called transporter. In the presence of insulin, the rate of movement of glucose into the
cell is greatly stimulated in a selective fashion. ([89].)
In the liver, insulin does not affect the movement of glucose across membranes
directly but facilitates glycogen deposition and decreases glucose output. Consequently,
there is a net increase in glucose uptake. Insulin induces or represses the activity of
many enzymes; however, it is not known whether these actions are direct or indirect. For
example, insulin suppresses the synthesis of key gluconeogenic enzymes and induces the
synthesis of key glycolytic enzymes such as glucokinase. Glycogen synthetase activity is
also increased. Insulin likewise increases the activity of enzymes involved in lipogenesis
.
3.4. Insulin Receptors. In molecular biology, the insulin receptor is a transmembrane glycoprotein that is activated by insulin. It belongs to the large class of
tyrosine kinase receptors. Two α subunits and two β subunits make up the insulin receptor. The β subunits pass through the cellular membrane and are linked by disulfide
bonds ([90]).
The insulin receptors are embedded in the plasma membrane of hepatocytes
and myocytes. The binding of insulin to the receptors is the initial step in a signal
14
transduction pathway, triggering the consumption and metabolism of glucose ([89], [86]).
Bound by insulin, the insulin receptor phosphorylates from ATP to several proteins
in the cytoplasm, including insulin receptor substrates (IRS-1 and IRS-2) containing
signaling molecules, activates Phosphatidylinositol 3-kinase (PI-3-K) and leads to an
increase in glucose transporter (GLUT4) molecules ([98]) in the outer membrane of
muscle cells and adipocytes, and therefore to an increase in the uptake of glucose from
blood into muscle and adipose tissue ([89]). GLUT4 will transport the glucose to the
cells efficiently. Figure 1.3.3 elucidates this signaling pathway.
Intracellular phosphorylation of glucose is rapid and efficient and therefore the
glucose concentration is low. Thus, a certain amount of glucose moves into the cell
regardless of the existence of insulin. With insulin, however, the rate of glucose entry is
much increased due to the facilitated diffusion as mediated by the glucose transporters
([89]). Refer to Figure 1.3.3.
However, the kinetics of insulin receptor binding are complex. The number of
insulin receptors of each cell changes opposite to the circulating insulin concentration
level. Increased insulin circulating level reduces the number of insulin receptors per cell
and the decreased circulating level of insulin triggers the number of receptors to increase.
The number of receptors is increased during starvation and decreased in obesity and
acromegaly. But, the receptor affinity is decreased by excess glucocorticoids. The
affinity of the receptor for the second insulin molecule is significantly lower than for the
first bound molecule. This may explain the negative cooperative interactions observed
at high insulin concentrations. That is, as the concentration of insulin increases and
more receptors become occupied, the affinity of the receptors for insulin decreases.
Conversely, at low insulin concentrations, positive cooperation has been recorded. That
15
Insulin
Glucose
I
I
G
I
G
G
-S-SααUnit Unit
Insulin receptor
Cell membrane
G
G
G
-S-S-
-S-S-
ßUnit
G
G
G
PI-3 Kinase
G
G
ßUnit
ATP
G
G
G
G
GLUT4
IRS-1
IRS-2
Phosphorylations
Other activities
Figure 1.3.3.
Insulin signals cells to utilize glucose
Insulin binds to its receptors on the membrane of the cells and phosphorylates several proteins in the
cytoplasm, including insulin receptor substrates (IRS-1 and IRS-2) containing signaling molecules, activates Phosphatidylinositol 3-kinase (PI-3-K) and leads to an increase in glucose transporter (GLUT4)
molecules. This leads to an increase in glucose transporter (GLUT4) molecules. GLUT4 will transport
the glucose to the cells efficiently.
is, the binding of insulin to its receptor at low insulin concentrations seems to enhance
further binding (([89]), [86]).
3.5. Insulin Resistance. Insulin resistance is defined as when insulin is inefficient in causing the plasma glucose to enter the cells of a body and to be utilized by
the cells for energy, even if there is enough insulin in serum. That is, the cells resist
the insulin. In addition, the liver may continue to secrete glucose into the bloodstream
even when the glucose is not needed.
The reasons for insulin resistance occurring are still uncertain. Certain genes
predispose certain people to develop insulin resistance. Some factors are, for example,
lack of exercise, obesity, and chronically high blood sugar levels may cause insulin
resistance in susceptible individuals. [95]
16
Previously, the perspective was that the abnormal binding to the insulin receptors
of the cells was the major reason of insulin resistance. This is no longer believed to be
the case. [95]
Currently, many researchers are active in determining the cause of insulin resistance at the cellular and molecular levels. “Postbinding abnormalities”, believed by
most researchers, is the cause of insulin resistance. Several chemical pathways and
genes causing the abnormalities have been identified. A typical example is that the
glucose transporter GLUT4 is deficient in some individuals showing insulin resistance.
The activity of GLUT4 is to transport the glucose into the body cells after the insulin
is bound to the insulin receptors. [95]
3.6. Insulin Degradation and Clearance. Insulin degradation is a broad
and rich research area and this is not the major focus of this dissertation. We will only
discuss this briefly. (For more information, refer to [5], [27], [33], [42] and their cited
references.)
Insulin is cleared mainly by the liver and kidney, but most other tissues also
degrade the hormone ([33]). Insulin-degrading enzyme (IDE) is the major enzyme in
the proteolysis of insulin in addition to several peptides ([27]). It resides in a region
of chromosome 10q that is linked to Type 2 diabetes ([42]). IDE is the major enzyme
responsible for insulin degradation in vitro, but the extent to which it mediates insulin
catabolism in vivo has been controversial, with doubts expressed that IDE has any
physiological role in insulin catabolism ([33] and cited references). Insulin is degraded
by enzymes in the subcutaneous tissue ([64]) and interstitial fluid as well ([7]). The
insulin is degraded by insulin receptors as well as when the insulin is bound to its
receptors ([85]).
17
3.7. Production and Consumption of Glucose. Glucose is liberated from
dietary carbohydrates such as starch or sucrose by hydrolysis within the small intestine,
and then is absorbed into the blood. The most often ways of glucose infusion are through
meal ingestion; oral glucose intake; continuous enteral nutrition; and constant glucose
infusion ([79] and [84]).
Insulin controls the hepatic glucose production (conversion from glucagon) and
release rate by the liver ([89]). When the blood glucose level drops, the liver converts
glycogen to glucose and releases it into the bloodstream. When there is enough glucose
in the bloodstream, insulin secreted by the pancreas signals the liver to shut down
glucose production. In healthy people, the pancreas continually measures blood glucose
levels and responds by secreting just the right amount of insulin. The liver converts the
glycogen to glucose as well as when the plasma glucose concentration level is low.
The insulin receptor leads that the glucose molecules go into the muscle cells,
fat cells and others. These cells utilize the glucose. Elevated concentrations of glucose
in the blood stimulate the release of insulin. Insulin acts on cells throughout the body
to stimulate uptake, utilization and storage of glucose. Within seconds to minutes the
rate of glucose entry into tissue cells increases 15 to 20 times. Once glucose enters
the tissue cells, insulin enhances its oxidation, stimulates its conversion to glycogen,
activates transport of amino acids into cells, promotes protein synthesis and inhibits
virtually all liver enzymes that promote gluconeogenesis. The effects of insulin on
glucose metabolism vary depending on the target tissue. Two important effects are
([89]) (see also Figure 1.3.3 for an illustration.):
• Higher Insulin Concentration Leads to More Glucose Uptake
Insulin
facilitates entry of glucose into muscle, adipose and several other tissues. The only
18
mechanism by which cells can take up glucose is by facilitated diffusion through a
family of hexose transporters. In many tissues, e.g., muscle, the major transporter
used for uptake of glucose (GLUT4) is made available in the plasma membrane
through the action of insulin.
• Lower Insulin Concentration Leads to Less Glucose Uptake
In the
absence of insulin, GLUT4 glucose transporters are present in cytoplasmic vesicles,
where they are useless for transporting glucose. Binding of insulin to receptors
on such cells leads rapidly to fusion of those vesicles with the plasma membrane
and insertion of the glucose transporters, thereby giving the cell the ability to
efficiently take up glucose. When blood levels of insulin decrease and insulin
receptors are no longer occupied, the glucose transporters are recycled back into
the cytoplasm. Therefore, the glucose uptake is significantly decreased.
Insulin stimulates the liver to store glucose in the form of glycogen. A large
fraction (50%) of glucose absorbed from the small intestine is immediately taken up by
hepatocytes, which convert it into the storage polymer glycogen ([89]).
Insulin has several effects in the liver that stimulate glycogen synthesis. First, it
activates the enzyme hexokinase, which phosphorylates glucose, trapping it within the
cell. Coincidentally, insulin acts to inhibit the activity of glucose-6-phosphatase. Insulin
also activates several of the enzymes that are directly involved in glycogen synthesis,
including phosphofructokinase and glycogen synthase. The net effect is clear: when the
supply of glucose is abundant, insulin signals the liver to store as much of it as possible
for use later ([89]).
Many cells consume the glucose without involvement of the insulin receptor effect.
The brain and the liver do not use GLUT4 to transport glucose. Instead, a type of
19
insulin-independent transport is used. This constitutes the insulin-independent glucose
utilizations ([89]).
4. Glucose Tolerance Test
A series of glucose tolerance tests have been developed over the year and applied
in clinics and experiments ([93], [10], [8], [41], [76], [16] and [61]). Each of the glucose
tolerance tests is to diagnose if an individual has diabetes or has potential to have
diabetes. The basic idea is to test one’s glucose-insulin endocrine metabolic system
after a large amount of glucose infusion.
The glucose tolerance tests include Fasting Glucose Tolerance Test (FGTT),
Oral Glucose Tolerance Test (OGTT), Intra Venous Glucose Tolerance Test (IVGTT),
frequently sampled Intra Venous Glucose Tolerance Test (fsIVGTT) ([93], [60] and [59]).
The Fasting Glucose Tolerance Test (FGTT) needs the individual to fast for 8-10 hours
before his/her the plasma glucose is sampled. The meanings of the test results are
summarized in Table 1.4.1. The Oral Glucose Tolerance Test (OGTT) is another type
of glucose tolerance test. The individual is given a glass of glucose liquid (75mg) to
intake and his/her the plasma glucose level will be sampled. The test result meanings
are defined in Table 1.4.2. To diagnose gestational diabetes, a pregnant woman is
required to drink a glass of glucose water containing 50mg glucose. Her the plasma
glucose is sampled one hour later. The meanings of the test results are listed in Table
1.4.3. The American Diabetes Association suggests two tests need to be performed to
determine if an individual has diabetes or pre-diabetes ([93]).
The Intra-venous Glucose Tolerance Test (IVGTT) and the frequently sampled
Intra-venous Glucose Tolerance Test (fsIVGTT) are to test the insulin sensitivity or
20
Table 1.4.1.
Fasting Glucose Tolerance Test
The plasma Glucose
Meaning
70-99 mg/dl (3.9-5.4 mmol/l)
normal glucose tolerance
100-125 mg/dl (5.5-6.9 mmol/l) impaired fasting glucose (pre-diabetes)
Over 126 mg/dl (7.0 mmol/l) and above
probable diabetes
Table 1.4.2.
Oral Glucose Tolerance Test
The plasma Glucose
Meaning
Below 140 mg/dl (7.8 mmol/l)
normal glucose tolerance
140-200 mg/dl (7.8-11.1 mmol/l) impaired fasting glucose (pre-diabetes)
Over 200 mg/dl (11.1 mmol/l)
probable diabetes
response to high the plasma glucose concentration. The procedure of IVGTT is similar
to other glucose tolerance tests but the plasma glucose and serum insulin are sampled
more frequently. In the test, the individual to be tested needs to fast 8-10 hours and
is then given a bolus of glucose infusion, for example, 0.33 g/kg body weight [23]
or 0.5 g/kg body weight of a 50% solution and is administered into an antecubital
vein in approximately 2.5 minutes. Within the next 180 minutes, the individual’s
the plasma glucose and serum insulin are sampled frequently. According to the rich
information in the sampled data, the insulin sensitivity can be accurately determined.
Many models study the Intravenous Glucose Tolerance Test (IVGTT), which focuses
on the metabolism of glucose in a short time period starting from the infusion of big
bolus (0.33 g/kg) of glucose at time t = 0. As pointed out in Chapter 2, due to the
large amount of intravenous glucose infusion, the insulin response time delay of the
small amount of hepatic glucose production is insignificant and thus negligible and
furthermore is assumed at a small constant infusion rate in the models ([10], [8], [23],
[57] and [63]). The most noticeable model is the so called “Minimal Model” which
21
Table 1.4.3.
Gestational Diabetes Glucose Tolerance Test
The plasma Glucose
Meaning
Below 140 mg/dl (7.8 mmol/l)
normal glucose tolerance
Over 140 mg/dl (7.8 mmol/l) abnormal, needs oral glucose tolerance test
contains minimal number of parameters ([10], [8]) and it is widely used in physiological
research work to estimate metabolic indices of glucose effectiveness (SG ) and insulin
sensitivity (SI ) from the intravenous glucose tolerance test (IVGTT) data by sampling
over certain periods (usually 180 minutes) ([41]). Also a few are on the control through
meals and exercise ([25]). See also a review paper by Mari ([60]) for a classification of
models.
5. The Organization of This Dissertation
In this dissertation, we propose a more realistic DDE model for the insulin secretion ultradian oscillations in Chapter 2. This model (Model (2.3.1)) contains two time
delays: the first mimic the hepatic glucose production time delay and the other reflects
the insulin response time delay to increased glucose concentration. Both analytical and
numerical analysis are performed. The results obtained include global and local stability analysis of steady state, persistence of solutions and numerical simulation with
insightful results.
In Chapter 3, we propose three models (Model (3.3.1), (3.3.2) and (3.3.3)) for
modeling the effective and powerful intravenous glucose tolerance test. We performed
global and local stability analysis of the steady state and numerical simulations based
on clinic data from diabetics.
22
In Chapter 4, we present another DDE model to investigate the effects of the
mass of the active β cells. Our numerical analysis shows that we simulated the glucoseinsulin endocrine metabolic system taking active β cell mass into account. Due to the
fact that this area is relatively new, our study is still preliminary. More thorough studies
are needed.
CHAPTER 2
The Ultradian Oscillations of Insulin Secretion
1. Introduction
Endocrine systems often secrete hormones in pulses [21] [56]. Examples include
the release of growth hormone and gonadotropins, and also the secretion of insulin from
the pancreas, which are secreted over intervals of 1-3 hours and 80-150 minutes, respectively. It has been suggested that relative to constant or stochastic signals, oscillatory
signals are more effective at producing a sustained response in the target cells [40] [58].
Numerous in-vivo and in-vitro experiments have shown that insulin concentration
oscillates in two different time scales: rapid oscillation with a period of 5-15 minutes
and ultradian oscillation with a range of 80-150 minutes ([79], [67], [74] and [73]).
The mechanisms underlying both types of oscillations are not fully understood.
The rapid oscillations may arise from an intra-pancreatic pacemaker mechanism [77]
and caused by coordinating periodic secretory bursting of insulin from β cells contained
in the millions of the Langerhans islets in the pancreas. These bursts are the dominant mechanism of insulin release at basal level ([67]). Often, the rapid oscillation is
superimposed on the slow (ultradian) oscillation ([79]).
Ultradian oscillations of insulin concentration are believed to be mainly due to
glucose interaction in the plasma and an instability in the insulin-glucose feedback sys-
24
B
A
100
40
Insulin (µU/ml)
Insulin (µU/ml)
50
30
20
10
40
20
0
160
Glucose (gm/dl)
Glucose (gm/dl)
0
140
80
60
120
100
140
120
100
80
80
60
240
480
720
960
1200
60
1440
C
160
240
40
40
Insulin (µU/ml)
Insulin (µU/ml)
50
30
20
10
0
30
20
10
140
180
120
Glucose (gm/dl)
Glucose (gm/dl)
120
D
100
80
60
40
160
140
120
100
240
480
720
960
Figure 2.1.1.
1200
1440
240
480
720
840
1200
Insulin Secretion Ultradian Oscillations
These figures illustrate the insulin secretion ultradian oscillations. The glucose infusion rate are A.
meal ingestion; B. oral glucose intake; C. continuous enteral nutrition; D. constant glucose infusion,
respectively. (The figures are adapted from [79].)
tem ([79], [84], [74] and [60]). These ultradian oscillations are best seen after meal
ingestion, oral glucose intake, continuous enteral nutrition or intravenous glucose infusion (Figure 2.1.1). In addition, muscles, the brain, nerves and others utilize the
plasma glucose to complete the regulatory system feedback loop ([79], [84]). So, insulin
production, glucose infusion and production (for example, meal and continuous enteral
nutrition in daily life) and glucose utilization (for example, in daily life, exercise) are
the three major factors of this intricate regulatory system ([74], [79] and [59]).
The hypothesis that the ultradian insulin secretion is an instability in the insulinglucose feedback system has been the subject of a number of studies, including some
which have developed a mathematical model of the insulin-glucose feedback system
([51], [79], [84], [31] and [4]).
This chapter is organized as follows. Section 2 summarizes the current study
status with focus on the Sturis-Tolic Model. Section 3 presents our two time delay
25
model for the insulin secretion ultradian oscillations. Preliminary results are presented
in Section 4. Two global stability results of the steady state are given in Section 5 and
followed by Section 6 for the local stability study. Intensive numerical simulations are
presented in Section 7 and then Section 8 is for discussion.
2. Sturis-Tolic ODE Model and Current Research Status
To determine whether the ultradian oscillations could result from the interaction
between insulin and glucose, a parsimonious nonlinear mathematical model consisting the six ordinary differential equations including the major mechanisms involved
in glucose regulation was developed by J. Sturis, K. S. Polonsky, E. Mosekilde and
E. Van Cauter ([79]) in 1991 and recently simplified by I. M. Tolic, E. Mosekilde and
J. Sturis ([84]) in 2000. The purpose of these two models was to provide a possible
mechanism for the origin of the ultradian insulin secretion oscillations. Included in this
model is the feedback loop (refer to Figure 2.2.1): glucose stimulates pancreatic insulin
secretion, insulin stimulates glucose uptake and inhibits hepatic glucose production,
and glucose enhances its own uptake. The model takes following form.
26
Insulin
(-)
Glucose
Production
(-)
Insulin
Secretion
(-)
Glucose
Utilization
(-)
Glucose
Figure 2.2.1.
Physiological Glucose-Insulin Regulatory System
These four negative feedback loop show the glucose stimulating pancreatic beta cells to secrete insulin,
insulin stimulating glucose uptake and inhibiting hepatic glucose production, and also positive feedback
as glucose enhances its own uptake ([79]). (This figure is adapted from [79].)


dG(t)



= G′ = Gin − f2 (G(t)) − f3 (G(t))f4 (Ii (t)) + f5 (x3 ),



dt













dIp (t)
Ip (t)
Ip (t) Ii (t)


−
)−
,
= Ip′ = f1 (G(t)) − E(



dt
Vp
Vi
tp














Ip (t) Ii (t)
Ii (t)
dIi (t)

′


=
E(
=
I
−
)
−
,

i

dt
Vp
Vi
ti









dx1 (t)
3



= x′1 = (Ip − x1 ),



dt
td













dx2 (t)
3



= x′2 = (x1 − x2 ),



dt
td













3
dx3 (t)



= x′3 = (x2 − x3 ),



dt
td







(2.2.1)
27
where G(t) is the amount of glucose, Ip (t) and Ii (t) are the amount of insulin in the
plasma and the intercellular space, respectively, Vp is the plasma insulin distribution
volume, Vi is the effective volume of the intercellular space, E is the diffusion transfer
rate, tp and ti are insulin degradation time constants in the plasma and intercellular space, respectively, Gin indicates (exogenous) glucose supply rate to plasma, and
x1 (t), x2 (t) and x3 (t) are three auxiliary variables associated with certain delays of the
insulin effect on the hepatic glucose production with total time td . f1 (G) is a function
modeling the pancreatic insulin production as controlled by the glucose concentration,
f2 (G) and f3 (G)f4 (Ii ) are functions, respectively, for insulin-independent and insulindependent glucose utilization by various body parts (for example, brain and nerves (f2 ),
and muscle and fat cells (f3 f4 )) and f5 (x3 ) is a function modeling hepatic glucose production with time delay td collaborated with auxiliary variables x1 , x2 and x3 . Based
on experimental results ([79], [84]), all the parameters in the model are given in Table
(2.2.1) and fi , i = 1, 2, 3, 4, 5, take following forms and the parameters listed in Table
2.2.2.
Rm
,
1 + exp((C1 − G/Vg )/a1 )
(2.2.2)
f2 (G) = Ub (1 − exp(−G/(C2 Vg ))),
(2.2.3)
f1 (G) =
f3 (G) =
f4 (Ii ) = U0 +
G
,
C3 Vg
0.1(Um − U0 )
,
1 + exp(−β ln(Ii /C4 (1/Vi + 1/Eti )))
(2.2.4)
(2.2.5)
28
Table 2.2.1.
Parameters in the Sturis-Tolic Model (2.2.1).
Parameters Values
Units
Vp
3
l
Vi
11
l
E
0.2 l · min−1
tp
6
min
ti
100
min
Table 2.2.2.
Parameters of the functions in the Sturis-Tolic Model (2.2.1).
Parameters
Units Values
Vg
l
10
−1
Rm
µUmin
210
a1
mg · l−1
300
−1
C1
mg · l
2000
−1
Ub mg · min
72
−1
C2
mg · l
144
C3
mg · l−1
1000
f5 (x) =
Parameters
Units Values
U0 mgmin−1
40
−1
Um mgmin
940
β
1.77
C4
µUl−1
80
−1
Rg mgmin
180
−1
α
lµU
0.29
a1
µUl−1
26
Rg
,
1 + exp(α(x/Vp − C5 ))
(2.2.6)
Figure (2.2.2) display the graphs of the above functions, fi , i = 1, 2, 3, 4, 5. The
importance of these functions is their shapes rather than their forms [51].
This model comprised of two major negative feedback loops describing the effects
of insulin on glucose utilization and glucose production, respectively, and both loops
include the stimulatory effect of glucose on insulin secretion. The authors of [84] hoped
to identify a possible mechanism behind the efficiency of oscillatory insulin secretions.
Analysis of the original model revealed that the slow oscillations of insulin secretion
could arise from a Hopf bifurcation in the insulin-glucose feedback mechanism. The
model included several feedback loops (see Figure 2.2.1), including: glucose stimulating
29
70
800
60
50
600
40
400
30
20
200
10
0
0
10000
20000
30000
40000
0
100
200
300
400
30000
40000
G
I
f2 (G)
f4 (I)
200
160
150
120
100
80
50
40
0
0
0
50
100
150
200
0
10000
x
20000
G
f5 (I)
Figure 2.2.2.
f1 (G)
Functions fi (I), i = 1, 2, 4, 5.
pancreatic beta cells to secrete insulin, insulin stimulating glucose uptake and inhibiting
hepatic glucose production, and also positive feedback as glucose enhances its own
uptake.
The model includes two significant delays. One, 5-15 min., is sluggish effect of
insulin on glucose utilization, reflecting that the effect is dependent on the concentration of insulin in a slowly equilibrating intercellular compartment as opposed to the
concentration of the plasma insulin. The other delay, 25-50 min., is due to the time
lag between the appearance of insulin in the plasma and its inhibitory effect on hepatic
30
glucose production. This delay is simulated by introducing three auxiliary variables
x1 , x2 and x3 , which is called the third order delay. We demonstrate how the auxiliary
variables simulate time delay as follows. For simplicity, assume the first order delay,
that is, x′1 (t) = (Ip (t) − x1 (t))/td , where td > 0 is the time delay. Then
Ip (t − td ) = x1 (t − td ) + x′1 (t − td )td
Observe the Taylor’s expansion of x1 (t) at t − td ,
x1 (t) = x1 (t − td ) + x′1 (t − td )td + o(td ).
So x1 (t) ≈ Ip (t − td ). The occurrence of sustained insulin and glucose oscillations was
found numerically to be dependent on these two time delays.
Model simulations suggested that the interaction of the oscillatory insulin supply
with the glucose receptors of the glucose utilizing cells was of minimal importance. This
was because the oscillations in the concentration of the intercellular insulin were small,
and changes in the average glucose utilization only depend weakly on amplitude. However, with their model they were able to resolve conflicting results from clinical studies.
Different experimental conditions will influence hepatic glucose release. If hepatic glucose release is occurring near its maximum limit, an oscillatory insulin supply will be
more effective at lowering the blood glucose level than a constant supply. However, if
the insulin level is sufficiently high to cause the hepatic release of glucose to virtually
disappear, the opposite is observed. For insulin concentrations close to the point of
inflection of the insulin-glucose curves (f1 and f5 ), an oscillatory and a constant insulin
secretion produce similar effects. Under the assumption of constant glucose infusion,
the authors observed following numerical observations.
ST1 The ultradian insulin secretion oscillation is critically dependent on hepatic glucose production, that is, if there is no hepatic glucose production, then there is
31
no insulin secretion oscillation.
ST2 When the hepatic glucose production time delay τ2 ∈ (25, 50), the period ω of the
periodic solutions of both insulin and glucose is in interval (95, 140) (min.), that
is, ω ∈ (95, 140).
ST3 To obtain the ultradian oscillation (periodic solutions), it is necessary to break
the insulin into two separate compartments, the plasma and interstitial tissues.
ST4 The ultradian oscillation is sensitive to both the speed of insulin reaction to the
increased plasma glucose concentration level and the speed of the hepatic glucose
production triggered by insulin. Specifically, if the slope in the reflexive points of
function f1 and f5 is reduced by 10 − 20%, the oscillation becomes damped.
K. Engelborghs, V. Lemaire, J. Belair and D. Roose ([31], 2001) introduced a
single time delay in the Negative Feedback Loop Model and proposed following DDE
model.





G′ (t) = Eg − f2 (G(t)) − f3 (G(t))f4 (I(t)) + f5 (I(t − τ )),





(2.2.7)






I(t)

′

,
 I (t) = f1 (G(t)) −
t1
where the functions, fi , i = 1, 2, 3, 4, 5, and their parameters are assumed to be the
same as those in the Model (2.2.1). Eg stands for the glucose infusion rate and the
term 1/t1 is the insulin degradation rate. The positive constant delay τ mimics the
hepatic glucose production delay (5-15 min.). This model ignores the glucose stimulating insulin secretion time delay. Due to the complex chemical reactions on the β cells,
32
the insulin secretion occurs a few minutes after the plasma glucose concentration rises.
This significant time delay (5-15 min.) is not negligible in physiology.
The other DDE model proposed by K. Engelborghs, V. Lemaire, J. Belair and
D. Roose ([31], 2001) is trying to model the exogenous insulin infusion. The authors
assumed that the exogenous insulin infusion function takes the same form as internal
insulin production, which is, as the authors admitted, too artificial.





G′ (t) = Eg − f2 (G(t)) − f3 (G(t))f4 (I(t)) + f5 (I(t − τ2 )),





(2.2.8)






I(t)

′

+ (1 − α)f1 (G(t − τ1 )).
 I (t) = αf1 (G(t)) −
t1
Nevertheless, a noticeable addition to the work of [31] is the usage of DDEBifTool software package ([30]) to analyze and simulate the bifurcation diagram and
other numerical analysis.
Due to the lack of physiological meanings, we would not summarize the analytical
and numerical results presented in [31].
In 2004, D. L. Bennett and S. A. Gourley ([4]) modified the Sturis-Tolic ODE
Model ([79] and [84]) by removing the three auxiliary linear chain equations and their
associated artificial parameters and introducing a time delay into the model explicitly.
This time delay τ stands for the hepatic glucose production, which is the same as
proposed in [31]. Unlike [31] in which the sluggish effect of glucose on insulin is ignored,
D. L. Bennett and S. A. Gourley ([4]) kept the idea in [79] and [84] of breaking the
insulin in two compartments to simulate the time delay of insulin secretion triggered
by rising glucose concentration level. The DDE model takes following form. All the
33
parameters and functions are the same as that in model (2.2.1) given in (2.2.2) to (2.2.6)
and Table 2.2.1 and 2.2.2.





G′ (t) = Gin − f2 (G(t)) − f3 (G(t))f4 (Ii (t)) + f5 (Ip (t − τ )),















Ip (t) Ii (t)
Ip (t)
I ′ (t) = f1 (G(t)) − E(
−
p

Vp
Vi














Ip (t) Ii (t)
Ii (t)

′


−
)−
,
 Ii (t) = E(
Vp
Vi
)−
tp
,
(2.2.9)
ti
Their major analytical results are a sufficient condition of global asymptotical
stability induced by a Liapunov function for the case that the hepatic glucose production
time delay τ = 0 and one for the case τ > 0. This analytical result shows that if the
hepatic glucose production time delay τ and the insulin degradation time delay between
the plasma and interstitial compartments ti and td are sufficiently small, then solutions
converge globally to the steady state or the basel levels of glucose and insulin. In other
words, there are no sustained oscillations. For larger delay, whose range is not given in
[4], oscillatory solutions become possible and under these circumstances it seems that
likely candidates for having sustainable oscillatory insulin and glucose levels are those
subjects with low degradation rates of the two insulin compartments.
Two other observations in [4] are that large glucose infusion rate could cause
insulin secretion oscillations, and the insulin oscillations are sensitive to the values of
|f1′ (C1 Vg )| = Rm /(4a1 Vg ) or |f5′ (C5 Vp )| = Rg α/(4Vp ). This means if the β cells do not
release enough insulin into the bloodstream, or glucose production is not sensitive to
insulin and keeps at a constant moderate rate (Rg /2), then the insulin oscillation will
34
not sustain. Similarly, if the hepatic glucose production rate Rg is too small, regardless
of sensitivity to insulin, the oscillations of insulin and glucose disappear.
3. Two Time Delay DDE Model
Glucose molecules are in the bloodstream or the plasma. When the concentration
level rises, electronic signals are sent to the pancreas and the β cells secrete insulin.
The liver delivers the insulin into the plasma. This process takes about 5-15 minutes
depending different individuals. So, to more intuitively and precisely model the glucoseinsulin ultradian oscillations, we introduce two time delay parameters in to the glucose
and insulin regulatory system. The model diagram is shown in Figure 2.3.1. We remove
the insulin compartment split in the Sturis-Tolic Model ([79], [84]). The two time
delays are the hepatic glucose production time delay τ2 as in [4] and [31] and the effect
of glucose concentration level on insulin secretion time delay τ1 due to the complex
electro-chemical reactions when the rising glucose concentration level triggers the β
cells to release insulin. The delay τ1 can be referred as insulin response time delay. The
two time delay DDE model we propose is as follows.

dG(t)



= Gin − f2 (G(t)) − f3 (G(t))f4 (I(t)) + f5 (I(t − τ2 )),



dt


(2.3.1)






dI(t)


= f1 (G(t − τ1 )) − di I(t),
dt
where the initial condition I(0) = I0 > 0, G(0) = G0 > 0, G(t) ≡ G0 for all t ∈ [−τ1 , 0]
and I(t) ≡ I0 for t ∈ [−τ2 , 0] with τ1 , τ2 > 0. In addition,
35
Liver
Delay
Glucose Infusion:
meal ingenstion,
oral intake,
enteral nutrition,
constant infusion
Liver converts
glucagon and
glycogen to
glucose
Insulin Controls
Hepatic
glucose production
Glucagon
secrete
Pancreas
α−cells
ß-cells
Delay
Glucose
production
Insulin secretion
Insulin production
Glucose Controls
glucagon secretion
Glucose Controls
insulin secretion
Glucose
utilization
Glucose
Insulin
Insulin independent:
brain cells, and
others
Insulin dependent:
fat cells, and
others
Figure 2.3.1.
Insulin clearance
Insulin helps cells consume glucose
Insulin degradation:
receptor, enzyme, and
others
Two Time Delay Glucose-Insulin Regulatory Model
The divide lines (dash-dot-dot) indicate insulin controlled hepatic glucose production with time delay;
the dash-dot lines indicate the insulin secretion from the β-cells stimulated by elevated glucose concentration level with time delay; the dashed lines indicate low glucose concentration level triggers α-cells
in pancreas to release glucagon; and the dot line indicates the insulin accelerates glucose utilization in
cells.
36
(i) Gin is due to glucose infusion, e.g., by meal ingestion, oral glucose intake, continuous enteral nutrition or intravenous glucose infusion;
(ii) f2 (G(t)) stands for insulin independent glucose consumption by the brain, nerve
cells and others. f2 (0) = 0, f2 (x) > 0 and f2′ (x) > 0 are bounded for x > 0.
Denote M2 := sup{f2 (x) : x ≥ 0} < ∞ and M2′ := sup{f2′ (x) : x > 0} < ∞.
(iii) f3 (G(t))f4 (I(t)) stands for insulin dependent utilization/uptake by muscle, fat
cells and others. f3 (x) = k3 x, where k3 > 0 is a constant. f4 (0) > 0, for
x > 0, f4 (x) > 0 and f4′ (x) > 0 are bounded above. f4 (I(t)) is in sigmoidal
shape. Denote M3′ := sup{f3′ (x) : x > 0} < ∞, m4 := inf{f4 (x) : x ≥ 0} > 0,
M4 := sup{f4 (x) : x ≥ 0} < ∞, and M4′ := sup{f4′ (x) : x > 0} < ∞.
(iv) f5 (I(t − τ2 )) indicates hepatic glucose production that is dependent on insulin in
the plasma with time delay τ2 > 0. The time delay τ2 > 0 reflects that the liver
does not convert the stored glucose and glycogen into glucose immediate when the
insulin concentration level decreases. When insulin concentration level increases,
the liver converts glucagon and glycogen to glucose decreasingly. f5 (0) > 0 and,
for x > 0, f5 (x) > 0 and f5′ (x) < 0. f5 (x) and |f5′ (x)| are bounded above for x > 0.
Denote M5 := sup{f5 (x) : x ≥ 0} < ∞ and M5′ := sup{|f5′ (x)| : x > 0} < ∞.
f5 (x) is in an inverse sigmoidal shape.
(v) f1 (G(t − τ1 )) stands for insulin secretion from the pancreas. Insulin is stored in
β-cell granules. Glucose is the primary stimuli of insulin secretion from β cells.
The delay is due to the complex electric processes inside of a islet. These processes
include that glucose molecules enter islets through GLUT2, elevate ATP and then
close the K+ channels. When K+ channels are closed, Ca2+ channels are open.
37
The influx of Ca2+ ions causes β cell granules to secrete insulin. f1 (0) > 0 and, for
x > 0, f1 (x) > 0, f1′ (x) > 0, f1′ (x) > 0 and bounded. Denote M1 := sup{f1 (x) :
x ≥ 0} < ∞ and M1′ := sup{f1′ (x) : x > 0} < ∞. f1 (x) is in sigmoidal shape.
(vi) di I(t) stands for insulin degradation and constant di is the degradation rate.
Insulin is cleared mainly by the liver and kidney ([33]). Insulin is degraded by
enzymes in the subcutaneous tissue ([64]) and interstitial fluid as well ([7]).
We will study this model analytically and numerically.
4. Preliminaries
We will give some preliminary analysis in this section. First we illustrate the
uniqueness of the steady state of the model (2.3.1).
Proposition 2.4.1 The Model (2.3.1) has unique positive steady state (G∗ , I ∗ ), where
G∗ is the unique solution of equation
−1
H(x) = Gin − f2 (x) − f3 (x)f4 (d−1
i f1 (x)) + f5 (di f1 (x)) = 0,
x > 0,
(2.4.1)
and
∗
I ∗ = d−1
i f1 (G ).
(2.4.2)
Proof All we have to show is that equation (2.4.1) has a unique root in (0, ∞). In fact,
observe that f1′ (x) > 0, f2′ (x) > 0, f4′ (x) > 0, f3′ (x) > 0, and f5′ (x) < 0, then H ′ (x) < 0.
Notice that
−1
H(0) = Gin − f2 (0) − f3 (0)f4 (d−1
i f1 (0)) + f5 (di f1 (0))
= Gin + f5 (d−1
i f1 (0)) > 0,
38
and
lim H(x) = Gin − x→∞
lim f2 (x) − x→∞
lim f3 (x)f4 (d−1
lim f1 (x))
i x→∞
x→∞
+f5 (d−1
lim f1 (x))
i x→∞
= Gin − M2 − f4 (d−1
lim (k3 x) + f5 (d−1
i M1 ) x→∞
i M1 )
< 0.
In addition, f1 (x) is strictly monotone increasing, so the proof is completed.
We show the positiveness and boundedness of the solutions of the model (2.3.1).
Proposition 2.4.2 All solutions of model (2.3.1) exist for all t > 0, are positive and
bounded. Furthermore,
Gin + M5
m4 k3
(2.4.3)
lim sup I(t) ≤ MI := d−1
i f1 (MG ).
(2.4.4)
lim sup G(t) ≤ MG :=
t→∞
and
t→∞
Proof. Observe that the |fi′ (x)|, i = 1, 2, 3, 4, 5, are bounded, thus fi (x), i = 2, 3, 4,
and fj (xt ), j = 1, 5, are Lipschitz and completely continuous in x ≥ 0 and xt ∈
C[− max{τ1 , τ2 }, 0], respectively. Then by Theorem 2.1, 2.2 and 2.4 on page 19 and 20
in [54], the solution of equation (2.3.1) with given initial condition exists and unique
for all t ≥ 0. If there exists a t0 > 0 such that G(t0 ) = 0 and G(t) > 0, for 0 < t < t0 ,
then G′ (t0 ) ≤ 0. So
0 ≥ G′ (t0 )
= Gin − f2 (G(t0 )) − f3 (G(t0 ))f4 (I(t0 )) + f5 (I(t0 − τ2 ))
= Gin − f2 (0) − f3 (0)f4 (I(t0 )) + f5 (I(t0 − τ2 ))
= Gin + f5 (I(t − τ2 )) > 0
39
This implies that G(t) > 0, for all t > 0. If ∃t′0 > 0 such that I(t′0 ) = 0 and I(t) > 0
′
′
for all 0 < t < t′0 , then I(t0 ) < 0. Therefore, 0 > I(t′0 ) = f1 (G(t0 ) − di I(t′0 − τ1 ) ≥
f1 (G(t′0 )) > 0 implies that I(t) > 0 for all t > 0.
Notice that m4 ≤ f4 (x) ≤ M4 and f5 (x) ≤ M5 and f3 (x) = k3 x, for x > 0. Thus
G′ (t) = Gin − f2 (G(t)) − f3 (G(t))f4 (I(t)) + f5 (I(t − τ2 ))
≤ Gin − m4 k3 G(t) + M5 .
Therefore, for any given t̄ > 0, if t > t̄, we have
d m4 k3 t
(e
G(t)) ≤ (Gin + M5 )em4 k3 t
dt
Z
em4 k3 t G(t) ≤ G(t̄) +
G(t) ≤ G(t̄)e−m4 k3 t +
= G(t̄)e−m4 k3 t +
t̄
Z
t
t̄
t
(Gin + M5 )em4 k3 s ds
e−m4 k3 s ds
Gin + M5 −m4 k3 t̄
(e
− e−m4 k3 t )
m4 k3
Thus
lim sup G(t) ≤
t→∞
Gin + M5
:= MG
m4 k3
Since |f1 (x)| ≤ M1 , given ǫ > 0, I ′ (t) ≤ f1 (MG + ǫ) − di I(t) for sufficiently large t > 0.
Then we have
lim sup I(t) ≤ d−1
i f1 (MG + ǫ).
t→∞
Notice that ǫ > 0 is arbitrary, so
lim sup I(t) ≤ d−1
i f1 (MG ) := MI .
t→∞
The following lemma is elementary. See [48] for a proof.
40
Lemma A Let f : R → R be a differentiable function. If l = lim inf t→∞ f (t) <
lim supt→∞ f (t) = L, then there are sequences {tk } ↑ ∞, {sk } ↑ ∞ such that for all
k, f ′ (tk ) = f ′ (sk ) = 0, limk→∞ f (tk ) = L and limk→∞ f (sk ) = l.
We will apply Lemma A in follows and prove a few preliminary results. Let
(G(t), I(t)) be a solution of (2.3.1). Throughout this paper, we define
G = lim sup G(t),
t→∞
G = lim inf G(t)
t→∞
and
I = lim sup I(t),
t→∞
I = lim inf I(t).
t→∞
Due to the Proposition 2.4.1 and 2.4.2, we see that these limits are finite.
The following lemma regarding the upper limits and lower limits of a solution of
the model (2.3.1).
Lemma 2.4.1 If (G(t), I(t)) is a solution of (2.3.1), then
f1 (G) ≤ di I ≤ di I¯ ≤ f1 (Ḡ),
(2.4.5)
f2 (Ḡ) + f3 (Ḡ)f4 (I) ≤ Gin + f5 (I),
(2.4.6)
¯
¯ ≤ f2 (G) + f3 (G)f4 (I),
Gin + f5 (I)
(2.4.7)
Proof. First we show (2.4.5) holds. Due to Fluctuation Lemma and Proposition 2.4.2,
there exist sequences {tk } ↑ ∞, such that I ′ (tk ) = 0, limk→∞ I(tk ) = I. Thus,
0 = I ′ (tk ) = f1 (G(tk − τ1 )) − di I(tk ) for all k.
Therefore,
f1 (G) − di I(tk ) ≤ f1 (G(tk − τ1 )) − di I(tk ) for k = 1, 2, 3, ...
41
Thus,
f1 (G) − di I ≥ 0.
On the other hand side, there exists a sequence {sk } ↑ ∞ such that limk→∞ I(sk ) =
I and I ′ (sk ) = 0 for all k > 0. So,
f1 (G) − di I(sk ) ≤ f1 (G(sk − τ1 )) − di I(sk ) for k = 1, 2, 3, ...
Thus,
f1 (G) − di I ≤ 0.
Now we show (2.4.6) holds. Again, due to Proposition 2.4.2 and Fluctuation
′
′
Lemma, there exists a sequence {tk } ↑ ∞ as k → ∞ such that limk→∞ G(tk ) = G and
′
0 = G′ (tk )
′
′
′
′
= Gin − f2 (G(tk )) − f3 (G(tk ))f4 (I(tk )) + f5 (I(tk − τ2 )),
k = 1, 2, 3, ....
Then, notice that f4 ↑ ∞ and f5 ↓ 0,
′
′
′
′
′
′
0 = Gin − f2 (G(tk )) − f3 (G(tk ))f4 (I(tk )) + f5 (I(tk − τ2 ))
≤ Gin − f2 (G(tk )) − f3 (G(tk ))f4 (I) + f5 (I),
k = 1, 2, 3, ...
and therefore
Gin − f2 (G) − f3 (G)f4 (I) + f5 (I) ≥ 0.
Similarly we can show (2.4.7) is true. According to Proposition 2.4.2 and Fluctu′
′
ation Lemma, there exists a sequence {sk } ↑ ∞ as k → ∞ such that limk→∞ G(sk ) = G
and
′
0 = G′ (sk )
′
′
′
′
= Gin − f2 (G(sk )) − f3 (G(sk ))f4 (I(sk )) + f5 (I(sk − τ2 )),
k = 1, 2, 3, ....
42
Then, notice that f4 ↑ ∞ and f5 ↓ 0,
′
′
′
′
′
′
0 = Gin − f2 (G(sk )) − f3 (G(sk ))f4 (I(sk )) + f5 (I(sk − τ2 ))
≥ Gin − f2 (G(sk )) − f3 (G(sk ))f4 (I) + f5 (I),
k = 1, 2, 3, ....
Thus,
0 ≥ Gin − f2 (G) − f3 (G)f4 (I) + f5 (I).
Apparently, Ḡ = G implies I¯ = I due to (2.4.5). If I¯ = I, then (2.4.6) and
¯ 3 (G) − f2 (Ḡ)) ≤ 0. That is, Ḡ = G.
(2.4.7) together lead to f2 (Ḡ) − f2 (G) ≤ f4 (I)(f
This complete the proof of following
Theorem 2.4.1 Let (G(t), I(t)) be a solution of the Model (2.3.1). Then Ḡ = G and
I¯ = I imply each other.
Following proposition proves the model (2.3.1) is persistent.
Proposition 2.4.3 Model (2.3.1) is persistent, that is, all solutions of Model (2.3.1)
are bounded by a pair of positive constants from above and below, respectively.
Proof. Notice that f2 (0) + f3 (0) = 0 and f4 (x) < M4 for all x ≥ 0. Then (2.4.7)
implies that
Gin ≤ f2 (G) + f3 (G)M4 ,
f or all t > 0.
(2.4.8)
Thus ∃δG > 0, tG > 0, such that G(t) > δG for t > tG > 0. Therefore (2.4.5)
implies that I(t) is bounded below.
43
On the other hand side, (2.4.3) and (2.4.5) imply that I(t) and G(t) are bounded
above.
5. Global Stability of Steady State
In this section, we will give one result of globally asymptotically stable equilibrium of this model using Lemma 2.4.1.
Theorem 2.5.1 Let
−1
F (x, y) = f3 (x)f4 (d−1
i f1 (y)) + f5 (di f1 (x)),
x, y ≥ 0.
(2.5.1)
If
F (x, y) ≥ F (y, x),
x ≥ y ≥ 0,
then the steady state (G∗ , I ∗ ) of (2.3.1) is globally asymptotically stable.
Proof Let (G(t), I(t)) be a solution of (2.3.1). Due to Lemma 2.4.1, we have
Gin − f2 (G) − f3 (G)f4 (I) + f5 (I) ≤ Gin − f2 (G) − f3 (G)f4 (I) + f5 (I)
that is,
0 ≥ [f2 (G) + f3 (G)f4 (I) − f5 (I)] − [f2 (G) + f3 (G)f4 (I) − f5 (I)]
= [f2 (G) + f3 (G)f4 (I) + f5 (I)] − [f2 (G) + f3 (G)f4 (I) + f5 (I)]
−1
≥ [f2 (G) − f2 (G)] + [(f3 (G)f4 (d−1
i f1 (G)) + f5 (di f1 (G)))
−1
−(f3 (G)f4 (d−1
i f1 (G)) + f5 (di f1 (G)))]
= [f2 (G) − f2 (G)] + [F (G, G) − F (G, G)]
≥ f2 (G) − f2 (G)
(2.5.2)
44
due to (2.5.2). Thus G = G.
−1
Remark Notice that f5 (d−1
i f1 (x)) ≤ f5 (di f1 (y)) for x ≥ y ≥ 0 means higher hep-
atic production of glucose helps to make oscillations happen (the case that (G∗ , I ∗ ) is
unstable).
Remark Notice that f3 (G) can be linear and f4 is bounded. If the glucose concentration G is big enough and there is no hepatic production (f5 ≡ 0), then the steady state
(G∗ , I ∗ ) will be globally stable and thus there is no oscillation.
6. Linearization and Local Analysis
We need following theorem for two special cases, where one of the two time
delays equals to zero. When both delays equal to zero, the linearized system of the
model (2.3.1) becomes a trivial 2-dimensional ODE. Now we state theorem here without
proof. For a proof, see Kuang ([54], 1993)(Theorem 3.1, page 77).
Theorem B In the following second order real scalar linear neutral delay equation
x′′ (t) + σx′′ (t − τ ) + ax′ (t) + bx′ (t − τ ) + cx(t) + dx(t − τ ) = 0,
(2.6.1)
where τ ≥ 0. Assume |σ| < 1, c + d 6= 0 and a2 + b2 + (d − σc)2 6= 0. Consider the
characteristic equation of (2.6.2)
λ2 + σλ2 e−λτ + aλ + bλe−λτ + c + de−λτ = 0.
(2.6.2)
The number of different imaginary roots with positive (negative) imaginary parts of
(2.6.2) can be zero, one, or two only.
(I) If there are no such roots, then the stability of the zero solution does not
change for any τ > 0.
45
(II) If there are any imaginary roots with positive imaginary part, an unstable
zero solution never becomes stable for any τ ≥ 0. If the zero solution is asymptotically
stable for τ = 0, then it is asymptotically stable for τ < τ0 , and it becomes unstable
for τ > τ0 where τ0 > 0 is a constant. It undergoes a supercritical Hopf bifurcation at
τ = τ0 .
(III) If there are two imaginary roots with positive imaginary part, iω+ and iω− ,
such that ω+ > ω− > 0, then the stability of the zero solution can change (when changes
from stable to unstable, the zero solution undergoes a supercritical Hopf bifurcation) a
finite number of times at most as τ is increased, and eventually it becomes unstable.
The number of such roots are determined by the following conditions.
If c2 ≤ d2 , then there is only one such root.
If c2 > d2 , then there are two such roots provided that
(A) b2 + 2c − a2 − 2dσ > 0, and
(B) (b2 + 2c − a2 − 2dσ)2 > 4(1 − σ 2 )(c2 − d2 ).
Otherwise, there is no such solution.
Now we try to linearize the model (2.3.1). Let G(t) = G1 (t) + G∗ and I(t) =
I1 (t) + I ∗ . Then system (2.3.1) becomes
G′1 (t) = Gin − f2 (G1 (t) + G∗ ) − f3 (G1 (t) + G∗ )f4 (I1 (t) + I ∗ ) + f5 (I1 (t − τ2 ) + I ∗ )
= −[f2′ (G∗ ) + f3′ (G∗ )f4 (I ∗ )]G1 (t) − f3 (G∗ )f4′ (I ∗ )I1 (t) + f5′ (I ∗ )I1 (t − τ2 )
I1′ (t) = f1 (G1 (t − τ1 ) + G∗ ) − di (I1 (t) + I ∗ )
= f1′ (G∗ )G1 (t − τ1 ) − di I1 (t).
We still use G(t) and I(t) to denote G1 (t) and I1 (t), respectively. Thus the linearized
46
system of (2.3.1) can be written as

dG(t)



= −AG(t) − BI(t) − CI(t − τ2 )



dt


(2.6.3)






dI(t)


= DG(t − τ1 ) − di I1 (t)
dt
where
A := f2′ (G∗ ) + f3′ (G∗ )f4 (I ∗ ) > 0,
B := f3 (G∗ )f4′ (I ∗ ) > 0,
C := −f5′ (I ∗ ) > 0,
D := f1′ (G∗ ) > 0.
Let




 G0 
 G(t) 




 = eλt 

,




I(t)
G0 , I0 > 0,
I0
λ ∈ C,
t>0
be a solution of (2.6.3). Then



λt
 −AG0 e
 G0 



 = 



λeλt 
λt
λ(t−τ2 )
− BI0 e − CI0 e
DG0 eλ(t−τ1 ) − di I0 eλt
I0


= eλt 


−A
−λτ2
−B − Ce
De−λτ1
−di







  G0 



.


I0
So the characteristic equation of (2.6.3) is given as
 


−λτ2
  λ 0   −A
−B − Ce

 
 
det 
−
 
 
0 λ
De−λτ1
−di



= det 

λ+A
−λτ1
−De

B + Ce−λτ2 

λ + di


 








47
= (λ + A)(λ + di ) + De−λτ1 (B + Ce−λτ2 )
= λ2 + (A + di )λ + di A + DBe−λτ1 + DCe−λ(τ1 +τ2 )
= 0.
We denote the characteristic equation as
∆(λ) = λ2 + (A + di )λ + di A + DBe−λτ1 + DCe−λ(τ1 +τ2 ) = 0.
(2.6.4)
Note ∆(0) = di A + DB + DC > 0. So λ = 0 is not a solution of the characteristic
equation (2.6.4). So if there is any stability switch of the trivial solution of the linearized
system (2.6.3), there must exist a pair of pure imaginary roots of the characteristic
equation (2.6.4).
If τ1 = 0 and τ2 = 0, the original model (2.3.1) is an ODE model. The characteristic equation of its linearized equation is given by
∆(λ) = λ2 + (A + di )λ + di A + DB + DC = 0.
Then due to A + di > 0 and di A + DB + DC > 0, the steady state (G∗ , I ∗ ) is stable.
If τ2 = 0 but τ1 > 0, the characteristic equation of the linearized system takes
the following form.
∆(λ) = λ2 + (A + di )λ + di A + (DB + DC)e−λτ1 = 0.
Then due to Theorem B ([54]), di ≤
D(B+C)
A
(2.6.5)
means there exists only one positive root
of (2.6.5). That is, there exists an τ10 > 0 such that the trivial solution of the linearized
system (2.6.3) is stable when τ1 ∈ (0, τ10 ) and unstable when τ1 ≥ τ10 .
Similarly if τ1 = 0 and τ2 > 0, then d2i ≥ 2DB − A2 implies the trivial solution
of the linearized system (2.6.3) is stable. If d2i < 2DB − A2 and 2DB + D2 C 2 >
A2 + d2i + (di A + DB)2 , then the trivial solution of the linearized system (2.6.3) has at
most finite number of stability switches and eventually is unstable.
48
Now assume both τ1 > 0 and τ2 > 0. Let λ = ωi, ω > 0, be such an eigenvalue
in (2.6.4), then we have
∆(ωi) = −ω 2 + (A + di )ωi + di A + DB(cos ωτ1
−i sin ωτ1 ) + DC(cos ω(τ1 + τ2 ) − i sin ω(τ1 + τ2 ))
= [−ω 2 + di A + DB cos ωτ1 + DC cos ω(τ1 + τ2 )]
+i[(A + di )ω − DB sin ωτ1 − DC sin ω(τ1 + τ2 )]
= 0.
That is,




 −ω 2 + di A + DB cos ωτ1 + DC cos ω(τ1 + τ2 ) = 0,
This leads to
(2.6.6)



 (A + di )ω − DB sin ωτ1 − DC sin ω(τ1 + τ2 ) = 0.
(−ω 2 + di A)2 + ((A + di )ω)2
= (DB cos ωτ1 + DC cos ω(τ1 + τ2 ))2 + (DB sin ωτ1 + DC sin ω(τ1 + τ2 ))2 ,
that is,
ω 4 − 2di Aω 2 + d2i A2 + (A2 + 2di A + d2i )ω 2
= D2 (B 2 + C 2 + 2BC cos ωτ2 ) ≤ D2 (B 2 + C 2 + 2BC) = D2 (B + C)2 .
So
ω 4 + (A2 + d2i )ω 2 + [d2i A2 − D2 (B + C)2 ] ≤ 0.
This is impossible in the case that di A ≥ D(B +C). We summarize the results obtained
in this section as follows.
Proposition 2.6.1 In the linearized system (2.6.3),
49
(a) When τ1 = 0 and τ2 = 0, the steady state of (2.6.3) is a stable spiral point.
(b) When τ1 > 0 and τ2 = 0, if di ≥ D(B + C)/A, then there exists a stability switch,
i.e., there exists an τ10 > 0 such that the trivial solution of the linearized system
(2.6.3) is stable when τ1 ∈ (0, τ10 ) and unstable when τ1 ≥ τ10 .
(c) When τ1 = 0 and τ2 > 0,
(c.1) if d2i ≥ 2DB − A2 , then the trivial solution of the linearized system (2.6.3)
is stable.
(c.2) If d2i < 2DB − A2 and 2DB + D2 C 2 > A2 + d2i + (di A + DB)2 , then the
trivial solution of the linearized system (2.6.3) has at most finite number of
stability switches and eventually is unstable.
(d) When τ1 > 0 and τ2 > 0, if di ≥ D(B + C)/A, then the steady state of the
linearized system (2.6.3) is stable.
Thus Proposition 2.6.1 leads to the following trivial result for the model (2.3.1).
Theorem 2.6.1 In Model (2.3.1),
(a) when τ1 = 0 and τ2 = 0, the steady state (G∗ , I ∗ ) is a stable spiral point.
(b) When τ1 > 0 and τ2 = 0, if di ≥ f1 (G∗ )(B − f5′ (I ∗ ))/(f2′ (G∗ ) + f3′ (G∗ )f4 (I ∗ )), then
there exists a stability switch, i.e., there exists an τ10 > 0 such that the trivial
solution of the steady state (G∗ , I ∗ ) is stable when τ1 ∈ (0, τ10 ) and unstable when
τ1 ≥ τ10 .
(c)
When τ1 = 0 and τ2 > 0,
50
(c.1)
if d2i ≥ 2f1′ (G∗ )B − (f2′ (G∗ ) + f3′ (G∗ )f4 (I ∗ ))2 , then the steady state (G∗ , I ∗ )
is stable.
(c.2)
if
d2i < 2f1 (G∗ )f3 (G∗ )f4′ (I ∗ ) − (f2′ (G∗ ) + f3′ (G∗ )f4 (I ∗ ))2 ,
and
2f1 (G∗ )f3 (G∗ )f4′ (I ∗ ) + (f1 (G∗ )f5′ (I ∗ ))2
> (f2′ (G∗ ) + f3′ (G∗ )f4 (I ∗ ))2 + d2i +
+(di (f2′ (G∗ ) + f3′ (G∗ )f4 (I ∗ )) + f1′ (G∗ )(f3 (G∗ )f4′ (I ∗ )))2
then there are at most a finite number of stability switch and eventually steady
state (G∗ , I ∗ ) is unstable.
(d) When τ1 > 0 and τ2 > 0, if the insulin degradation rate
di ≥
f1′ (G∗ )(f3 (G∗ )f4′ (I ∗ ) − f5′ (I ∗ ))
:= d0 ,
f2′ (G∗ ) + f3′ (G∗ )f4 (I ∗ )
(2.6.7)
the steady state (G∗ , I ∗ ) is stable.
Remark If the parameters and functions fi , i = 1, 2, 3, 4, 5, take the values in (2.7.1)
to (2.7.5) and Table (2.2.1) and (2.2.2), then the threshold value d0 = 0.6669 when
Gin = 0.54. So when di = 1/26 = 0.03849 < d0 . So, (2.6.7) does not hold. In fact, the
insulin and glucose oscillation is sustained provided that τ2 = 36 and τ1 is sufficiently
large (greater than 5.2).
To further analyze the stability of the steady state of the model (2.3.1) and the
cases of the oscillations to be sustained, we will apply Rouchè′ s T heorem to analyze
51
when τ1 > 0 and τ2 > 0 that the steady state (G∗ , I ∗ ) is unstable. Recall following
Rouchè′ s T heorem ([19], p.125-126).
Rouchè’s Theorem Given two functions f (z) and g(z) analytic in a simple connected
region A ⊂ C with boundary γ, a simple loop homotopic to a point in A. If |f (z)| >
|g(z)| on γ, then f (z) and f (z) + g(z) have the same number of roots in A.
We start from a more generic equation and leave the system (2.6.3) as a special
case.
Let
S1 = {
2m
: m, n ∈ Z+ , m, n ≥ 1}
2n − 1
and
S2 = {
2m − 1
: m, n ∈ Z+ , m, n ≥ 1}.
2n
Clearly Q+ = S1 ∪ S2 and S1 ∩ S2 = ∅. Further we have
Lemma 2.6.1 S1 and S2 are dense in Q+ thus in R+ .
Proof.
∀r ∈ Q+ \ S1 , ∃p, q ∈ Z+ such that r =
2p−1
.
2q
Thus
2
2p − 1 − 2k
(4kp − 2k − 2)/2k
=
1
(4kq − 1)/2k
2q − 2k
2(2kp − 2k − 1)
∈ S1 ∀k = 1, 2, 3, ...
=
2(2kq) − 1
rk =
and limk→∞ rk = (2p − 1)/2q = r. That is, S 1 = Q+ . Similarly, S 2 = Q+ .
Proposition 2.6.2 For characteristic equation
λk +
k−1
X
j=1
aj λj + b + ce−λσ1 + de−λσ2 = 0,
k ≥ 2, σ1 , σ2 > 0,
(2.6.8)
where b, c, d > 0, aj ∈ R, j = 1, 2, 3, ..., k,, if b < d − c or b < c − d, then ∃σ10 > 0 and
σ20 > 0 such that the characteristic equation (2.6.8) has at least one root with positive
real part for σ1 > σ10 and σ2 > σ20 and σ1 /σ2 ∈ S1 or σ1 /σ2 ∈ S2 .
52
We need following lemmas to prove Proposition 2.6.2.
Lemma 2.6.2 For the equation
k k
ǫ z +
k−1
X
aj ǫj z j + b + ce−p1 z + de−p2 z = 0,
j=1
k ≥ 2,
p1 , p2 > 0,
z∈C
(2.6.9)
where b, c, d > 0, aj ∈ R, j = 1, 2, 3, ..., k, assume
(i) b < d − c, and p1 /p2 ∈ S1 , or
(ii) b < c − d, and p1 /p2 ∈ S2 .
Then, ∃ǫ0 > 0 such that for all ǫ, 0 < ǫ < ǫ0 , the equation (2.6.9) has at least one root
with positive real part.
Proof. Let
f (z) = b + ce−p1 z + de−p2 z .
We show that f (z) has a zero with positive real part. Since p1 and p2 are S1 related
in case (i) or S2 related in case (ii), there exist integer m, n ≥ 1 such that
for case (i), or
p1
p2
=
2m−1
2n
p1
p2
=
2m
2n−1
for case (ii). Let z = x + qπi, where q = 2m/p1 = (2n −
1)/p2 for case (i) or q = (2m − 1)/p1 = 2n/p2 for case (ii). Then
f (z) = b + ce−p1 x e−p1 qπi + de−p2 x e−p2 qπi
= b + ce−p1 x cos p1 qπ + de−p2 x cos p2 qπ − i(ce−p1 x sin p1 qπ + de−p2 x sin p2 qπ)
= b + ce−p1 x cos 2mπ + de−p2 x cos (2n − 1)π
(= b + ce−p1 x cos (2m − 1)π + de−p2 x cos 2nπ for case (ii))
= b + ce−p1 x − de−p2 x
:= H(x).
(= b − ce−p1 x + de−p2 x for case (ii))
53
Notice that H(0) = b + c − d < 0 (H(0) = b − c + d < 0 for case (ii)) and limx→∞ H(x) =
b > 0, therefore H(x) has at least one zero x0 ∈ (0, ∞). So f (z) has at least one zero
z0 = x0 + qπi with x0 > 0.
We perturb f (z) by gǫ (z) given by
gǫ (z) = ǫk z k +
k−1
X
aj ǫj z j ,
ǫ > 0,
(2.6.10)
j=1
with small ǫ > 0 and show that f (z) + gǫ (z) has the same number of zeros as f (z) if ǫ
is small. To this end, we first construct a simple loop γ homotopic to a point and then
show |f (z)| > |gǫ (z)| on γ.
Let z = x, x ∈ (−∞, ∞), then
|f (z)| = b + ce−p1 x + de−p2 x > b.
Let z = x + 2qπi, x ∈ (−∞, ∞), then
|f (z)| = |b + ce−p1 x e2qp1 πi + de−p2 x e2qp2 πi |
= |b + ce−p1 x cos 2qp1 π + de−p2 x cos 2qp2 π
−i(ce−p1 x sin 2qp1 π + de−p2 x sin 2qp2 π)|
= |b + ce−p1 x cos 4mπ + de−p2 x cos 2(2n − 1)π|
(= |b + ce−p1 x cos 2(2m − 1)π + de−p2 x cos 4nπ| for case (ii))
= b + ce−p1 x + de−p2 x > b.
Let z = Kx0 + yi, y ∈ [0, 2qπ], where K > 1 such that b − ce−p1 Kx0 − de−p2 Kx0 > b/2.
Then
|f (z)| = |b + ce−p1 Kx0 e−p1 yi + de−p2 Kx0 e−p2 yi |
≥ b − ce−p1 Kx0 − de−p2 Kx0 > b/2.
54
Let z = yi, y ∈ [0, 2qπ], then
|f (z)| = |b + ce−p1 yi + de−p2 yi |




 d − c − b,
≥ 


 c − d − b,
for case (i),
for case (ii)
:= η0 > 0.
Let η0′ := min{η0 , b/2}. Denote
γ := {z = x + yi ∈ C :
z = x or z = x ± 2qπi,
x ∈ [0, Kx0 ]
or z = yi or z = Kx0 + yi y ∈ [0, 2qπ].
γ ◦ := {z = x + yi ∈ C : 0 < x < Kx0 , 0π < y < 2qπ}.
Clearly, γ is a simple loop homotopic to the original, z0 = x0 + qπi ∈ γ ◦ and |f (z)| >
η0′ on γ. Choose r0 > 0 such that γ ⊂ A := {z ∈ C : |z| < r0 }. Denote ∂A := {z ∈ C :
|z| = r0 }. Thus ∀z ∈ ∂A, z = r0 eθi , θ ∈ [0, 2π], we have
k k
|gǫ (z)| = |ǫ z +
k−1
X
j=1
j j
ǫk r0k
aj ǫ z | ≤
+
k−1
X
j=1
|aj |ǫj r0j .
(2.6.11)
Obviously ∃ǫ0 > 0 such that ∀ǫ, 0 < ǫ < ǫ0 ,
|gǫ (z)| < η0′ ,
z ∈ ∂A.
∀z ∈ A, z = reθi , then r < r0 , and
|gǫ (z)| = |ǫk z k +
k−1
X
j=1
aj ǫj z j | ≤ ǫk r k +
k−1
X
j=1
|aj |ǫj rj < ǫk r0k +
Thus
|gǫ (z)| < η0′
for all z ∈ γ.
k−1
X
j=1
|aj |ǫj r0j .
55
Therefore |f (z)| > |gǫ (z)| on γ. By Rouchè′ s T heorem ([19], p125-126), f (z) and
f (z) + gǫ (z) have the same number of zeros in γ ◦ . That is, f (z) + gǫ (z) = 0 has at least
one root ẑǫ ∈ γ ◦ .
Proof of Proposition 2.6.2.
Assume b < d − c, and σ1 /σ2 ∈ S1 (or b < c − d,
and σ1 /σ2 ∈ S2 ). In Lemma 2.6.2, choose p10 and p20 such that p10 /p20 ∈ S1 (or
p10 /p20 ∈ S2 ). Suppose ǫ0 is given by (2.6.11) in the proof of Lemma 2.6.2. Let σ10 =
p10 /ǫ0 and σ20 = p20 /ǫ0 . Then ∀σ1 > σ10 , σ2 > σ20 and σ1 /σ2 ∈ S1 (or σ1 /σ2 ∈ S2 ), ∃ǫ,
0 < ǫ < ǫ0 such that
σ1 = p1 /ǫ > σ10 and σ2 = p2 /ǫ > σ20 .
Let λ = ǫz. Then (2.6.8) becomes (2.6.9) in Lemma 2.6.2 and thus the conclusion
follows.
Remark In Lemma 2.6.2, given p1 and p2 that are S1 or S2 related, if we carefully
choose ǫ0 in the proof of Lemma 2.6.2, an estimate of unstable region of σ1 and σ2 can
be given. For the special case k = 2, r0 and ǫ0 can be chosen as
r0 =
q
K 2 x20 + q 2 π 2
and
q
ǫ0 = ( a21 + 4η0′ − a1 )/2r0 .
Let k = 2 and we apply the Proposition 2.6.2 to the linearized system (2.6.3) and we
have
Proposition 2.6.3 If di A < D|C − B|, then there exist τ10 > 0 and τ20 > 0 such that
the characteristic equation of the system (2.6.3) has at least one root with positive real
part if
(i) di A < D(C − B), τ1 > τ10 , τ1 + τ2 > τ20 and τ1 /(τ1 + τ2 ) ∈ S1 , or
(ii) di A < D(B − C), τ1 > τ10 , τ1 + τ2 > τ20 and τ1 /(τ1 + τ2 ) ∈ S2
56
This is straight forward if in Proposition 2.6.2 choose k = 2, a1 = A + di , b =
Proof.
di A, c = DB, d = DC, σ1 = τ1 , σ2 = τ1 + τ2 .
Therefore, we have
Theorem 2.6.2 In model (2.3.1), if
di <
f1′ (G∗ )|f3 (G∗ )f4′ (I ∗ ) + f5′ (I ∗ )|
f2′ (G∗ ) + f3′ (G∗ )f4 (I ∗ )
(2.6.12)
then there exist τ10 > 0 and τ20 > 0 such that if τ1 > τ10 and τ2 > τ20 , the steady state
(G∗ , I ∗ ) is unstable if
(i) di <
f1′ (G∗ )(f3 (G∗ )f4′ (I ∗ )+f5′ (I ∗ ))
,
f2′ (G∗ )+f3′ (G∗ )f4 (I ∗ )
(ii) di < −
τ1 > τ10 , τ1 + τ2 > τ20 and τ1 /(τ1 + τ2 ) ∈ S1 , or
f1′ (G∗ )(f3 (G∗ )f4′ (I ∗ )+f5′ (I ∗ ))
,
f2′ (G∗ )+f3′ (G∗ )f4 (I ∗ )
τ1 > τ10 , τ1 + τ2 > τ20 and τ1 /(τ1 + τ2 ) ∈ S2
Remark Theorem 2.6.2 indicates the insulin concentration oscillation in Model (2.3.1)
is sustained when the (2.6.12) holds. In next section, intensive simulations are performed
with the functions given in (2.2.2) to (2.2.6). When the di = 0.03849 and Gin = 0.54,
(2.6.12) does hold. If τ1 = 6 and τ2 ∈ [25, 45], the insulin concentration oscillation
is sustained (see Figure 2.7.7). When the di = 0.03849 and Gin = 2.54, (2.6.12) also
holds. But, in this case, the delay parameters are required to be very large, for example,
τ1 = 70 and τ2 = 90, and the steady state is unstable.
7. Numerical Analysis of Stability Switches and Bifurcations
In this section, we further explore numerical analysis on a group of particular
functions and parameters (Table 2.7.1) developed from experiments ([79] and [84]) and
also used in [31] and [4]. The functions fi , i = 1, 2, 3, 4, 5, take the following forms
57
Table 2.7.1.
Parameters of the functions in Two Time Delay Model (2.3.1).
Parameters
Units Values
Vg
l
10
−1
Rm
µU min
210
−1
a1
mg · l
300
−1
C1
mg · l
2000
Ub mg · min−1
72
−1
C2
mg · l
144
−1
C3
mg · l
1000
f1 (G) =
Parameters
Units
U0 mg · min−1
Um mg · min−1
β
C4
µUl−1
Rg mg · min−1
α
lµU−1
a1
µUl−1
0.1Rm
,
1 + exp((C1 − G/Vg )/a1 )
f2 (G) = 0.1Ub (1 − exp(−G/(C2 Vg ))),
f3 (G) =
f4 (Ii ) = 0.1U0 +
G
,
10C3 Vg
0.1(Um − U0 )
,
1 + exp(−βln(Ii /C4 (1/Vi + 1/Eti )))
f5 (x) =
0.01Rg
,
1 + exp(α(x/Vp − C5 ))
Values
40
940
1.77
80
180
0.29
26
(2.7.1)
(2.7.2)
(2.7.3)
(2.7.4)
(2.7.5)
and the parameters are defined in table (2.7.1) based on [79] and [84]. The glucose
G and insulin I in Sturis-Tolic models (2.2.1)([79], [84]) and later the models studied
by K. Engelborghs, V. Lemaire, J. Belair and D. Roose ([31]) and D. L. Bennett and
S. A. Gourley ([4]) are in the unit mg and µU, respectively. We divide G and I by 100
and 10, respectively, in the Model (2.3.1) (therefore the functions fi , i = 1, 2, 3, 4, 5, in
(2.7.1), (2.7.2), (2.7.3), (2.7.4) and (2.7.5)) so that the glucose G is in the unit of mg/dl
58
and insulin I is in the unit of µU/ml. But, some of the simulations are done simply
using the original fi , i = 1, 2, 3, 4, 5.
We analyze the dynamics of the glucose and insulin numerically in following
cases.
I Take the insulin response time delay τ1 as the bifurcation parameter and let other
parameters are fixed;
II Take the glucose infusion rate Gin as the bifurcation parameter and let other
parameters are fixed;
III Take the insulin degradation di as the bifurcation parameter and let other parameters are fixed;
IV Take the hepatic glucose time delay τ2 as the bifurcation parameter and let other
parameters are fixed;
V Take both insulin response time delay τ1 and glucose infusion rate Gin as bifurcation parameters;
VI Take both insulin response time delay τ1 and the insulin degradation di as bifurcation parameters;
VII Take both the insulin degradation di and the glucose infusion rate Gin as bifurcation parameters.
Let di = 0.03849 be fixed, then d0 = 0.6669 in Theorem 2.6.1. So Theorem 2.6.1
does not apply and lead to a stable steady state (G∗ , I ∗ ).
59
Periodic Solutions in (tau1, G, I) when (Gin = 0.54 (mg/dl/min), tau2 = 36 (min))
140
30
25
100
20
80
I (mU/ml)
Glucose (red, mg/dl), Insulin (blue, mU/ml)
120
60
15
10
40
5
20
0
0
20
0
2
4
6
8
10
12
tau1 (min) (Gin = 0.54, tau 2 = 36)
14
16
Figure 2.7.1.
18
0
5
40
60
10
80
100
120
20
15
140
20
tau1 (min)
G (mg/dl)
Bifurcation diagram of τ1
where τ1 ∈ [0, 20], di = 0.03849, τ2 = 36 and Gin = 0.54. Left: the upper bifurcation diagram is for
glucose G and the lower one is for insulin I; right: Periodic Solutions.
7.1. Insulin Response Time Delay τ1 . In this subsection we take the insulin
response time delay τ1 as a bifurcation parameter and study the behavior of the system.
We let parameter τ1 change from 0 to 20 and Gin = 0.54, di = 0.03849 and
τ2 = 36 are fixed. Figures 2.7.1 and 2.7.2 show the stability regions, limit cycles and
period of periodic solutions in this case. To summarize the observations of these three
figures, we have
Numerical Observation 2.7.1 In model (2.3.1), assume di = 0.03849, Gin = 0.54,
τ2 = 36 and τ1 changes in [0, 20]. Then there exists a τ10 , 5.1 < τ10 < 5.2, such that
the model (2.3.1) has a stability switch at τ10 , that is
(1) The unique steady state (G∗ , I ∗ ) is stable when τ1 ∈ [0, τ10 ) and unstable when
τ1 ∈ [τ10 , 20].
(2) When τ1 ∈ [τ10 , 20], there exists a periodic solution. While τ1 changes from
τ10 to 20, the amplitudes and periods of periodic solutions increase significantly.
60
200
Period of Periodic Solutions of G and I (overlap)
Glucose concentration (mg/dl)
200
180
160
140
150
100
120
50
0
0
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.7
100
12
Insulin concentration (mU/l)
80
60
40
20
0
0
5
Figure 2.7.2.
10
15
tau1 (min) (Gin = 0.54, tau2 = 36)
20
25
10
8
6
4
0.2
0.3
0.4
0.5
di (mU/l/min), Gin=54 (mg/dl/min), tau =7.5 (min), tau =36 (min)
1
2
Periods of periodic solutions when τ1 ∈ [0, 20] and bifurcation diagram of di
Left: where di = 0.03849, τ2 = 36 and Gin = 0.54 are fixed This curve indicates the period of the
periodic solutions increase significantly when τ1 increases from 0 to 20 (min). Right: bifurcation
diagram of di ∈ [0.001, 0.7] (µU/(l·min)).
7.2. Glucose Infusion Rate Gin . We take the glucose infusion rate Gin as a
bifurcation parameter. We consider the case τ1 = 6.0 and the case τ1 = 7.5, respectively.
All other parameters are fixed.
Let parameter Gin vary from 0 to 2.16, di = 0.03849, τ2 = 36, τ1 = 6.0 or
τ1 = 7.5 are fixed. Figures 2.7.3, 2.7.4 and 2.7.5 show the stability regions, limit cycles
and period of periodic solutions in this case. To summarize the observations of these
three figures, we have, for the case τ1 = 6.0,
Numerical Observation 2.7.2 In model (2.3.1), assume τ1 = 6, τ2 = 36 and Gin
02
01
02
changes in [0, 2.16]. Then there exist G01
in and Gin , 0.05 < Gin < 0.075, 0.7 < Gin <
02
0.8, such that the model (2.3.1) has stability switches at G01
in and Gin , that is
(1) The unique steady state (G∗ , I ∗ ) is stable when Gin ∈ [0, G01
in ), unstable when
02
02
Gin ∈ [G01
in , Gin ] and stable when Gin ∈ (Gin , 2.16].
02
(2) When Gin ∈ [G01
in , Gin ], there exists a periodic solution. There exists an
02
01
02
a0 ∈ [G01
in , Gin ], while Gin changes from Gin to Gin , the amplitudes increase when
100
100
90
90
80
80
Glucose (red, mg/dl), Insulin (blue, mU/ml)
Glucose (red, mg/dl), Insulin (blue, mU/ml)
61
70
60
50
40
30
70
60
50
40
30
20
20
10
10
0
0
0.5
1
1.5
Gin (mg/dl/min) ( tau 1 = 6 (min), tau 2 = 36 (min))
Figure 2.7.3.
2
2.5
0
0
0.5
1
1.5
Gin (mg/dl/min) ( tau 1 = 7.5 (min), tau
2
2
2.5
= 36 (min))
Bifurcation diagram of Gin ∈ [0, 2.16]
These two figures indicate when τ1 = 6.0, there are two bifurcation points when Gin changes from 0
to 2.16 (mg/(dl·min), and there is only one bifurcation point when τ1 = 7.5. Left: τ2 = 36, τ1 = 6.0.
Right: τ2 = 36, τ1 = 7.5 .
Gin < a0 and decreases when Gin > a0 . The periods of periodic solutions change
insignificantly in the range (102, 108).
For the case τ1 = 7.5, we have
Numerical Observation 2.7.3 In model (2.3.1), assume τ1 = 7.5, τ2 = 36 and Gin
03
changes in [0, 2.16]. Then there exist G03
in , 0.8 < Gin < 1.0 such that the model (2.3.1)
has a stability switch at G03
in , that is
(1) The unique steady state (G∗ , I ∗ ) is unstable when Gin ∈ [0, G03
in ], stable when
Gin ∈ (G03
in , 2.16].
′
(2) When Gin ∈ [0, G03
in ], there exists a periodic solution. There exists an a0 ∈
03
′
[0, G03
in ], while Gin changes from 0 to Gin , the amplitudes increase when Gin < a0 and
decreases when Gin > a′0 . The periods of periodic solutions vary insignificantly in the
range (115, 122).
62
Periodic Solutions in (tau , G, I) when (tau = 7.5 (min), tau = 36 (min))
Periodic Solutions in (tau 1, G, I) when (tau 1 = 6 (min), tau 2 = 36 (min))
1
1
2
20
20
18
16
15
12
I (mU/ml)
I (mU/ml)
14
10
8
10
6
100
90
80
5
100
70
G (mg/dl)
60
0
1
0.5
2
1.5
80
2.5
60
40
0.5
G (mg/dl)
Gin (mg/dl/min)
Figure 2.7.4.
0
2
1.5
1
2.5
Gin (mg/dl/min)
Limit cycles in (Gin , G, I)-space when Gin ∈ [0, 2.16]
Left: τ2 = 36, τ1 = 6.0. Right: τ2 = 36, τ1 = 7.5.
Period of Periodic Solutions
Period of Periodic Solutions
120
140
120
100
G(red, mg/dl), I(blue, mU/ml)
G(red, mg/dl), I(blue, mU/ml)
100
80
60
40
80
60
40
20
0
20
0
0.5
1
1.5
Gin (mg/dl/min) ( tau = 6 (min), tau = 36 (min))
1
Figure 2.7.5.
2
2
2.5
0
0
0.5
1
1.5
Gin (mg/dl/min) ( tau = 7.5 (min), tau = 36 (min))
1
2
2.5
2
Periods of periodic solutions when Gin ∈ [0, 2.16]
These two figures show clearly that the periods of periodic solutions change insignificantly when Gin
varies in [0, 2.16] (mg/(dl·min) and τ2 = 36, τ1 = 6.0 (left figure) and τ1 = 7.5 (right figure).
63
140
Time (min) shift between peaks of G and I of periodic solutions
30
Period (min) of periodic solutions
120
100
80
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
0.6
di (mU/l/min), Gin=54 (mg/dl/min), tau1=7.5 (min), tau2=36 (min)
Figure 2.7.6.
0.7
0.8
25
20
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
di (mU/l/min), Gin=54 (mg/dl/min), tau1=7.5 (min), tau2=36 (min)
0.7
0.8
Periods and peak time differences when di changes in [0.001, 0.7]
Left: periods of the periodic solutions decreases along with di increases. The decrease is dramatic when
di is small and insignificant when di is larger. Right: the time shift that insulin peaks after glucose
peaks. The changes along with di is in the pace with the periods.
7.3. Insulin Degradation Rate di . We take the insulin degradation rate di
as a bifurcation parameter and let di vary from 0.001 to 0.07 (µU/(l·min)) while other
parameters are fixed.
The right hand side figure in Figures 2.7.2 shows the stability region when di
changes from 0.001 to 0.7 (µU/(l·min)) and Gin = 0.54, τ2 = 36 and τ1 = 7.5 are
fixed. The dotted line is the steady state when it is unstable and the solid lines are
the amplitude of the periodic solution bifurcated when the steady state switches its
stability. The left figure in Figure 2.7.6 shows the periods of the periodic solutions
when di changes in [0.001, 0.7]. It is noticeable that the period of the oscillations
decreases dramatically when di increases and tend to be smaller. The right figure shows
in each cycle, how long it takes the insulin concentration to peak after the glucose
concentration level peaks.
7.4. Hepatic Glucose Production τ2 . Let the hepatic glucose production τ2
be the bifurcation parameter and others are fixed.
64
Periodic Solutions in (tau , I, G) when (tau = 6 (min), Gin = 0.54 (mg/dl/min))
Glucose (red, mg)
8500
2
1
8000
8500
7500
7000
8000
6500
25
35
40
45
G (mU/ml)
30
100
7500
Insulin (blue, mU)
95
90
7000
85
80
45
6500
100
75
70
25
30
35
40
tau (min) when tau = 6 (min), Gin = 0.54 (mg/dl/min)
2
Figure 2.7.7.
40
95
90
35
85
45
1
I (mg/dl)
80
30
75
70
25
tau2 (min)
Hepatic production delay has no impact to sustained oscillations
Periodic solutions (right) and Amplitude of periodic solutions (left) when τ2 ∈ [25, 45] and τ1 =
6.0, Gin = 0.54, di = 0.03846 fixed. Theses two figures indicates the hepatic time delay τ2 does not
have impact to the oscillations when τ2 ∈ [25, 45]
Figure 2.7.7 shows the stability regions, limit cycles and period of periodic solutions when τ2 ∈ (25, 45). To summarize the observations of these three figures, we
have
Numerical Observation 2.7.4 In this case we observe that the hepatic delay τ2 is
not sensitive in its physiologically meaningful range.
7.5. Parameter τ1 vs. Gin . In this subsection, we take both the insulin response delay τ1 and the glucose infusion rate Gin as bifurcation parameters. We try to
identify the stability regions in the (τ1 , Gin ) space.
Let di = 0.03849 and τ2 = 36 are fixed. We compute the bifurcation points for
the parameters τ1 and Gin varying in τ1 , Gin ∈ [4.9, 20] × [0, 2.16]. Figure 2.7.8 shows
the stability regions and stability region in the (τ1 , Gin )-plane. Summarize all above,
we have
Numerical Observation 2.7.5 In model (2.3.1), assume τ2 = 36, di = 0.03849, then
there exists a curve composed of bifurcation points and divides the rectangular [4.9, 20]×
65
220
90
200
80
180
70
Stable Region
160
G_in
Stable Region
140
60
120
50
Unstable Region
100
40
80
30
Unstable Region
60
20
40
20
0
10
0
2
4
6
8
10
12
14
16
tau_1
Figure 2.7.8.
18
20
0
5
5.5
6
Stability Region in (τ1 , Gin )-plane
Left: Stability Region in (τ1 , Gin )-plane (di = 0.03846, τ2 = 36). This figures demonstrates that the
oscillations will not be sustained if the glucose infusion rate Gin is very large or the insulin response
time delay is very small. Right: Zoomed in left figure.
[0, 2.16] in (τ1 , Gin )-plane into two regions. The steady state of the model (2.3.1) is stable
in one of these two regions and unstable in the other region (refer to Figure 2.7.8). In
more details,
(1) There exists G0in > 1.40 such that if Gin < G0in , there is a unique stability
switch when τ1 varies from 0 to 20. That is, there exists a τ0 , 0 < τ0 < 20 depending
on Gin such that the steady state (G∗ , I ∗ ) is stable for τ1 ∈ [0, τ0 ) and unstable for
τ1 ∈ [τ0 , 20].
(2) There exists a τ01 , 5.1 < τ01 < 5.2, such that the steady state is always stable
if 0 < τ1 < τ01 .
(3) There exists a τ02 , τ01 < τ02 < 6.2, such that when τ1 ∈ (τ01 , τ02 ), stability
02
switches twice while Gin increase from 0 to 2.16. That is, there exist G01
in and Gin
02
01
02
depending on τ1 , 0 < G01
in < Gin < 90, such that when Gin ≤ Gin or Gin ≥ Gin , the
02
steady state (G∗ , I ∗ ) of Model (2.3.1) is stable; when Gin ∈ (G01
in , Gin ), the steady state
(G∗ , I ∗ ) of Model (2.3.1) is unstable.
66
100
G (mg/dl)
140
80
120
70
100
60
80
50
60
40
40
30
20
I (µU/l)
20
Period (min)
Period of periodic solutions (min)
90
Stable region
0
2.5
10
Unstable region
2
0
4
5
6
τ 1 (min)
7
8
9
0
0.5
1
1.5
2
2.5
Gin (mg/dl/min)
9
1.5
Gin (mg/dl/min
)
8
7
1
6
0.5
5
0
Figure 2.7.9.
τ 1 (min)
4
Bifurcation diagrams and stability regions in (τ1 , Gin )-space
Left: The stability regions in 3D-mesh show for G (top) and I (bottom) in (τ1 , Gin ) ∈ [4.9, 9] × [0, 2.16]
while τ2 = 36. Each plane τ1 = c ∈ [4.9, 9] and each plane Gin = c ∈ [0, 2.16] intersect the 3D-mesh
at the bifurcation diagrams in Gin ∈ [0, 2.16] and τ1 ∈ [4.9, 9], respectively. Right: Periods of Periodic
Solutions. This figure shows clearly that the periods jump out at bifurcation points.
(4) When τ1 > τ02 , there exists a Ḡin (τ1 ) > 80 such that (G∗ , I ∗ ) is stable if
Gin < Ḡin (τ1 ) and unstable if Gin ≥ Ḡin (τ1 ).
The right figure in Figure 2.7.9 shows the periods of the periodic solutions for
each point (τ1 , Gin ) ∈ [4.9, 9] × [0, 2.16]. This figure shows clearly that the periods jump
out at bifurcation points. The left figure shows the stability region in 3D mesh for G
(top) and I (bottom) in (τ1 , Gin ) ∈ [4.9, 9] × [0, 2.16]. Each plane τ1 = c ∈ [4.9, 9]
and each plane Gin = c ∈ [0, 2.16] intersect the 3D mesh at the stability region in
Gin ∈ [0, 2.16] and τ1 ∈ [4.9, 9], respectively.
7.6. Parameter τ1 vs. di . In this subsection, we take both the insulin response
delay τ1 and the insulin degradation rate di as bifurcation parameters. We try to identify
the stability regions in the (τ1 , di ) space.
To study the relationship between the parameter τ1 vs. di , we compute the
bifurcation points for the parameters τ1 and di varying in [4.9, 20] × [0.0025, 0.7]. Figure
67
0.6
0.6
0.5
Stable Region
di (muU/l/min)
di (mU/l/min): from 0.0025 to 0.7
0.5
0.4
Unstable Region
0.3
0.2
0.3
0.2
0.1
0
Stable Region
0.4
Unstable Region
0.1
5
6
7
8
tau1 (min) (Gin=54, tau 2=36)
Figure 2.7.10.
9
10
11
0
0.1
0.2
0.3
0.4
0.5
Gin (mg/dl/min)
0.6
0.7
0.8
Stability Regions in (τ1 , di )-plane and (Gin , di )-plane
Left: stability region in (τ1 , di )-space, Gin = 0.54, τ2 = 36. Right: stability region in (Gin , di )-plane,
τ1 = 6.0, τ2 = 36
2.7.10 shows the stability region in this case. To summarize the observations of these
three figures, we have
Numerical Observation 2.7.6 In model (2.3.1), assume τ2 = 36, Gin = 0.54, then
there exists a curve composed of bifurcation points and divides the rectangular [4.9, 20]×
[0.0025, 0.7] in (τ1 , di )-plane into two regions. The steady state of the model (2.3.1) is
stable in one of these two regions and unstable in the other region (refer to Figure
2.7.10).
7.7. Parameter Gin vs. di . In this subsection, we take both the insulin degradation rate di and the glucose infusion rate Gin as bifurcation parameters. We try to
identify the stability regions in the (di , Gin ) space.
The stability region in (Gin , di )-plane is shown in the right hand side figure
of Figure 2.7.10. We computed the bifurcation points in the rectangular (Gin , di ) ∈
[0, 85] × [0.075, 0.7] while τ1 = 6, τ2 = 36 are fixed.
Numerical Observation 2.7.7 In model (2.3.1), assume τ1 = 6, τ2 = 36, then a
68
curve composed with bifurcation points and divides the rectangular [0.0075, 0.7]×[0, 0.85]
in (τ1 , di )-plane into two regions. The steady state of the model (2.3.1) is stable in one
of these two regions and unstable in the other region (refer to Figure 2.7.10).
7.8. Insulin Concentration Peaks after Glucose Concentration Peaks.
We observed that in one cycle of the insulin secretion ultradian oscillation, the insulin
concentration peaks after glucose concentration peaks in a 15-35 min range depending
on the parameters τ1 , τ2 , Gin and di . Figure 2.7.11 shows four cases indicating the
differences of the peaks. From the physiological point of view, the insulin secretion
is triggered by the increased glucose concentration level and, in turn, the increased
insulin concentration level helps the cells to consume the plasma glucose and thus
bring the glucose concentration level down. The left hand side figure in Figure 2.7.11
demonstrates that the offset of the peaks between the glucose concentration level and
the insulin concentration level increase when the insulin response time delay τ1 increases
from its bifurcation point τ10 (≈ 5.15) to 20 minutes.
8. Discussion
Let’s recall the physiological background first. The experiments have revealed,
both in-vivo and in-vitro, that the insulin secretion ultradian oscillations sustain in the
daily life of healthy subjects in the interval 80-150 minutes ([73], [79], [60], [74], [75],
[67] and their cited references). Due to the complex chemical reactions in the glucoseinsulin endocrine metabolism, the secretion of insulin stimulated by glucose contains two
significant time delays: one is so called hepatic glucose production time delay caused
by the time lag between the appearance of insulin in the plasma and its inhibitory
effect on liver’s converting glucagon into glucose ([15], [69]); the other one is due to
69
Insulin Concentration Peaks 22.0761 min after Glucose Concentration
30
79
77
76
75
74
73
72
71
7800
7810
7820
7830
7840
7850
7860
7870
7880
7890
7900
Insulin (mU/l)
8.8
8.6
8.4
8.2
Time (min) shift between peaks of G and I of periodic solutions
Glucose (mg/dl)
78
25
20
15
10
5
8
7.8
7800
7810
7820
7830
7840
7850
7860
7870
7880
t (min) (tau = 5.5, tau = 36, Gin = 0.54, di = 0.03849)
1
2
Figure 2.7.11.
7890
7900
0
0
0.1
0.2
0.3
0.4
0.5
0.6
di (mU/l/min), Gin=54 (mg/dl/min), tau1=7.5 (min), tau2=36 (min)
0.7
0.8
Glucose concentrations peak before insulin does
Left: in one cycle of the insulin secretion ultradian oscillation, insulin concentration level peaks after
glucose concentration. Right: the difference of time (min.) between the insulin concentration and the
glucose concentration level peak changes from 20 to 35 min: di ∈ (0.001, 0.7) while other parameters
are fixed
chemical reactions and physiological actions of insulin on the uptakes of glucose ([88],
[66]). This oscillation has been modelled by J. Sturis, K. S. Polonsky, E. Mosekilde
and E. Van Cauter ([79], 1991) and I. M. Tolic, E. Mosekilde and J. Sturis ([84], 2000)
(model (2.2.1)), K. Engelborghs, V. Lemaire, J. Belair and D. Roose ([31], 2001) (model
(2.2.8)) and D. L. Bennett and S. A. Gourley ([4]) (model (2.2.9)).
In this chapter, we introduced a more realistic model (Model (2.3.1)) with two
time delays. One delay, τ2 , is the same as introduced in [4] and the other one is for
the delay, τ1 , caused by the chemical reactions in Langerhans islets. So the effort of
breaking insulin into two different compartment is unnecessary and thus the number of
equations is reduced by one. Our analytical and numerical analysis yield the following
observations and exhibit intrinsic insulin secretion ultradian oscillations.
Let’s recall the results obtained in [79] [84] and [4] and compare with our results.
We list the major results in [79] and [84] as [STx], the major results in [4] as [BGx], our
results as [Ax], where x is a number.
70
ST1 The ultradian insulin secretion oscillation is critically dependent on hepatic glucose production, that is, if there is no hepatic glucose production, then there is
no insulin secretion oscillation. ([79] and [84])
−1
A1 From Theorem 2.5.1, we can notice that f5 (d−1
i f1 (x)) ≤ f5 (di f1 (y)) for x ≥ y ≥ 0
means higher hepatic production of glucose helps to make oscillations happen (the
case that (G∗ , I ∗ ) is unstable). Comparing with [ST1], notice that f3 (G) can be
linear and f4 is bounded, which is the case used by [79] and [84]. If G is big enough
and there is no hepatic production (f5 ≡ 0), then the steady state (G∗ , I ∗ ) will be
globally stable and thus the insulin oscillation will not occur.
A2 If the hepatic glucose production τ2 = 0, then no insulin oscillation is observed
due to Theorem 2.6.1. This also confirms above [ST1]. This seems a surprising
observation. But, from physiological point of view, the hepatic glucose production is much smaller relative to the glucose infusion thus its impact should be
relatively small, but is not ignorable. Neither is the time delay, for the normal situation (refer to IVGTT models in Chapter 3 for the case that the hepatic glucose
production triggered by insulin can be completely ignored.)
ST2 When the hepatic glucose production time delay τ2 ∈ (25, 50), the period ω of the
periodic solutions of both insulin and glucose is in interval (95, 140) (min.), that
is, ω ∈ (95, 140). ([79] and [84])
A3 Figure 2.7.7 indicates that the hepatic glucose production time delay τ2 ∈ [25, 45]
has no impact to the insulin oscillation if the glucose infusion rate, the insulin
degradation rate and the insulin secretion time delay τ1 are in their “right ranges”,
respectively (Numerical Observation 2.7.4).
71
A4 Figure 2.7.11 indicates that during one cycle of the insulin and glucose concentration oscillations (95-135 min depending on τ1 , Gin and di ), the insulin concentration level peaks after the glucose concentration level does. This reflects the
physiological fact that glucose stimulates insulin secretion. On the other hand,
the glucose concentration level bottoms before the insulin concentration level bottoms. This reflects that higher insulin concentration helps the glucose uptake by
cells. Our [A2] and [A3] confirms [ST2].
ST3 To obtain the ultradian oscillation (periodic solutions), it is necessary to break
the insulin into two separate compartments, the plasma and interstitial tissues.
([79] and [84])
A5 Theorem 2.6.1 reveals that if there is no insulin response to glucose stimulation
time delay (τ1 = 0), insulin secretion oscillation will not be sustained. Comparing
with [ST3], the effort of compartment split on insulin is overcome by introducing
the insulin response time delay.
BG2 If the hepatic glucose production time delay τ and the insulin degradation rates
between the plasma and interstitial compartments ti and td are sufficiently small,
then there is no sustained oscillations. For larger delay or large glucose infusion
rate, oscillatory solutions become possible [4].
A6 Theorem 2.6.2 and Theorem 2.6.1 help to understand the role of insulin degradation rate. For a subject, if one’s insulin degradation rate is sufficiently small
or sufficiently large (di ≥
f1′ (G∗ )(f3 (G∗ )f4′ (I ∗ )−f5′ (I ∗ ))
),
f2′ (G∗ )+f3′ (G∗ )f4 (I ∗ )
the insulin secretion ultradian
oscillation does not occur. The oscillation can be sustained if the degradation rate
is moderate (di <
f1′ (G∗ )|f3 (G∗ )f4′ (I ∗ )+f5′ (I ∗ )|
).
f2′ (G∗ )+f3′ (G∗ )f4 (I ∗ )
The numerical observation indicates
72
that if the insulin degradation is too small, the glucose maintains in a low basel
level (see Figure 2.7.10, Figure 2.7.10). This is considered to be hypoglycemic.
On the other hand, if the insulin degradation is too large, the glucose basel level
is high which may lead to hyperglycemia (see Figure 2.7.10, Figure 2.7.10). This
provides more insightful information than the general statements in [BG2].
A7 When the glucose infusion rate and insulin degradation rate are moderate (not
too high and not too low), the insulin oscillation is sustained if the time delay
τ1 > 5.15 and τ2 ∈ (25, 45) (Figure 2.7.1). In this case, along with the increase of
time delay τ1 from 5.15 to 10 minutes, the amplitude of oscillation increases and
the period of the oscillation increases significantly from about 98 to 140 minutes
(Figure 2.7.2). All these are consistent to the experiments ([73], [79], [60], [74],
[75], [67] and their cited references). Comparing with [BG2], the bifurcation
diagram in Figure 2.7.1 quantifies the behaviors of the dynamics in the particular
case.
Besides, we also have following observations from our model (2.3.1).
A8 Theorem 2.4.1 guarantees that there is no such case for a healthy subject, he/she
neither experiences a stable glucose concentration at basel level and at the same
time have his/her insulin concentration oscillates nor does he/she have his/her
insulin concentration stabilizes at its basel level and the glucose concentration
oscillates. In other words, the insulin concentration and glucose concentration
levels vary simultaneously with an observed variation (see Figure 2.7.11).
A9 From Proposition 2.4.1, we know that the glucose concentration and insulin concentration basel levels are unique.
73
A10 Considering the small glucose infusion rate v.s. the insulin response time delay,
Figure 2.7.8 reveals that when the insulin response to glucose stimulation time
delay τ1 is in the range of 5.15 minutes to 6.15 minutes, small glucose infusion
rate Gin < 0.135 (mg/dl/min) would not make insulin oscillations to be sustained.
(See also left figures in Figure 2.7.3 and Figure 2.7.4.) This indicates if the glucose
infusion is small but the insulin response time is quick, then the small amount
of glucose added into the bloodstream would be consumed quickly and no more
insulin would need to be secreted.
On the other hand, Figure 2.7.8 also reveals that when the insulin response to
glucose stimulation time delay τ1 is approximately greater than 6.15 minutes, the
insulin oscillations sustain at small, even zero, glucose infusion rate (refer to the
right figures in Figure 2.7.3 and Figure 2.7.4.) This is possibly due to the slow
response of insulin secretion to the glucose stimulation and thus the glucose uptake
by cells are not that fast.
So only moderate glucose infusion rate 0.135 < Gin < 0.675 (mg/dl/min) can
make insulin secretion ultradian oscillations to be sustained when the insulin
response to glucose stimulation is fast.
A11 When considering the large glucose infusion rate v.s. the insulin response time
delay. Again, Figure 2.7.8 shows when the glucose infusion rate is high, Gin > 1.40
(mg/dl/min), no matter how fast or slow the insulin responds to the glucose
stimulation, the insulin oscillation will not be sustained. This indicates that
the insulin can not be produced and released from β-cells to uptake large amount
glucose in the plasma and thus sustains oscillation. This explains why the IVGTT
models in the Chapter 3 focus only on the study of asymptotically stable steady
74
state (the basel levels of glucose and insulin concentrations).
A12 From Figure 2.7.5 and Figure 2.7.2, when di = 0.03849, τ2 = 36, the larger the
glucose stimulation delay τ1 ∈ (5.15, 15), the period is significantly larger; the
larger the glucose infusion rate Gin , the smaller the period, but not significantly.
So, our analytical results and numerical observations have confirmed the existing research results or observations and also provided more insightful and quantitative
information.
CHAPTER 3
Modeling Intra-Venus Glucose Tolerance Test
1. Introduction
Due to the increased occurrence of pathological conditions such as diabetes (8% of
American people in 2002 ([94])) and obesity (one third, or currently 18 million American
adults ([92]), the quantification of insulin sensitivity from some relatively non-invasive
tests has gained increased interest and importance in physiological research. This lead
to some new studies on the existing models ([10], [8], [60]) and the introduction of some
alternative ones ([23], [63], [60]).
To detect the onset of diabetes or to diagnose the potential of having diabetes,
several tests or protocols are currently in use. These include Oral Glucose Tolerance Test
(OGTT), Fasting Glucose Tolerance Test (FGTT), Intra-venous Glucose Tolerance Test
(IVGTT) and the frequently sampled Intra-venous Glucose Tolerance Test (fsIVGTT)
([93]). They all test the insulin sensitivity or response to high the plasma glucose
concentration with a big bolus of glucose infusion. The most effective and accurate test
is the Intra-venous Glucose Tolerance Test (IVGTT) and the frequently sampled Intravenous Glucose Tolerance Test (fsIVGTT). During IVGTT, with designated frequently
sampled the plasma glucose concentration levels after a big bolus of glucose intra-venous
infusion, for example, 0.33g/kg body weight [23], the rich data can reveal how efficient
76
the insulin sensitivity of the subject’s glucose-insulin metabolic regulatory system is.
Many mathematical models study the Intravenous Glucose Tolerance Test (IVGTT)
([10], [8], [23], [57], [60], [63] and [61]). The most widely used is the so called “Minimal Model” ([10], [8]), which is challenged by [23]. [23] proposed an alternative model
called “Dynamic Model”. Besides the exogenous glucose intake, the liver produces a
small amount of glucose and converts glycogen into glucose when the plasma glucose
concentration level is low, with a delay of about 30 to 50 minutes ([79], [84]). As pointed
out in the numerical observation 2.7.5 in Chapter 2, due to the insulin response delay
(about 5 to 15 minutes), although the huge glucose intravenous infusion, the small
amount of hepatic glucose production is still sustained until a large amount of insulin
is released from the β cells. But, in a study of dynamics in a short period (about 30
min), the insulin response time delay of the liver glucose production is usually ignored
(the term f5 (I(t − τ2 )) in model (2.2.1), (2.2.9), (2.2.7), (2.2.8) and (2.3.1). This is in
agreement with the absence of this term in these models ([10], [8], [23], [57] and [63]).
As shown in Chapter 2, the absence of this term or the time delay often admits global
asymptotically stable steady state (Section 8 in Chapter 2, [79] and [84]).
This chapter is organized as follows. In Section 2, we summarize the current
status with focus on the minimal model ([10], [8]) and dynamic model ([23], [63]). Then
we present our genetic model and two other less general ones (still more general than the
dynamic model [23]) in Section 2, and present our generic models in Section 3. The basic
properties of the models we present in Section 3 will be discussed in Section 4. Section 5
contains some general global stability results. Section 6 presents a formal derivation of
linearization and characteristic equations. Section 7 provides some delay independent of
local stability results and Section 8 gives delay dependent stability conditions. Guided
77
by the results of Section 8, we performed extensive computer simulation (using clinic
data) and succeeded in finding periodic solutions in our discrete delay model, which is
shown in Section 9. Discussions on the implications of our results are summarized in
Section 10.
2. Current Research Status
The dynamic relationship between glucose and its controlling hormone insulin
has been mathematically modelled and studied by many researchers since the sixties
([11], [45], [55], [70], [79], [84], [85], [31], and the references cited in [29]). Most of
these models consist of several ordinary differential equations, the number of equations
is often proportional to that of factors considered. Some of these equations are simply
linear and were judged unacceptable for various reasons ([10], [8]), such as parameters
are not identifiable or have poor fits to experimental data. Nevertheless, the most
noticeable model, the so called “Minimal Model” which contains minimal number of
parameters ([10], [8]), is widely used in physiological research work to estimate glucose
effectiveness (SG) and insulin sensitivity (SI) from intravenous glucose tolerance test
(IVGTT) data by sampling over certain periods. Also a few are on the control through
meals and exercise ([25]). According to ([6]), 2002), there are now approximately 50
major studies published per year and more than 500 can be found in the literature,
according to the same author, which involve the minimal model.
Currently, the most widely used model in physiological research on the metabolism
of glucose is the so-called “minimal model”, which describes intra-venous glucose tolerance test (IVGTT) experimental data well using the smallest set of identifiable and
meaningful parameters ( [10], [65]). After incorporating the insulin dynamics, it takes
78
the form of [23])


dG(t)



= G′ = −[b1 + X(t)]G(t) + b1 Gb ,



dt











dX(t)
′
= X = −b2 X(t) + b3 [I(t) − Ib ],


dt













dI(t)



= I ′ = b4 [G(t) − b5 ]+ t − b6 [I(t) − Ib ].
(3.2.1)
dt
The initial conditions are:
G(0) = b0 ,
X(0) = 0,
I(0) = b7 + Ib .
Here G(t) [mg/dl],I(t) [µUI/ml] is the plasma glucose, insulin concentration at time t
[min], respectively. X(t) [min−1 ] is an auxiliary function representing insulin-excitable
tissue glucose uptake activity, roughly proportional to insulin concentration in a “distant” compartment. Gb [mg/dl], Ib [µUI/ml] is the subject’s baseline glycemia, insulinemia, respectively. b0 [mg/dl] is theoretical glycemia at time 0 after the instantaneous
glucose bolus intake. b1 [min−1 ] is the insulin-independent constant of tissue glucose
uptake rate. b2 [min−1 ] is the rate constant describing the spontaneous decrease of
tissue glucose uptake ability. b3 [min−2 (µUI/ml)−1 ]is the insulin-dependent increase in
tissue glucose uptake ability, per unit of insulin concentration excess over the baseline.
b4 [(µUI/ml)(mg/dl)−1 min−1 ] is the rate of pancreatic release of insulin after the intake
of the glucose bolus, per minute per unit of glucose concentration above the “target”
glycemia b5 [mg/dl]. b6 [µUI/ml] is the first order decay rate for insulin in the plasma.
b7 [µUI/ml] is the plasma insulin concentration at time 0, above basal insulinemia,
immediately after the glucose bolus intake.
While the above minimal model has a minimal number of constants (b0 − b7 ),
and has been very useful in physiological research works, this minimal model has been
79
challenged by De Gaetano and Arino [23] from both physiological and modeling aspects
recently. [23] argues that it has the following three drawbacks associated with it. For
this model, the parameter fitting is to be divided into two separate parts: first, using
the recorded insulin concentration as given input data in order to derive the parameters
in the first two equations in the model, then using the recorded glucose concentration as
given input to derive the parameters in the third equation. However, the system is an
integrated physiological dynamic system and one should treat it as a whole and be able
to conduct a single-step parameter fitting process. Secondly, some of the mathematical
results produced by this model are not realistic. Specifically, it can be shown that the
minimal model dose not admit an equilibrium and the solutions may not be bounded.
Finally, the non-observable auxiliary variable X(t) is artificially introduced to delay the
action of insulin on glucose. An alternative and natural way is to explicitly introduce the
time delay in the model. To address these issues, De Gaetano and Arino [23] introduced
the following aggregated delay differential model which they named as “dynamic model”
by use of certain simple and specific functions and introduces a time delay in a particular
way. It takes the form of


dG(t)



= G′ = −b1 G(t) − b4 I(t)G(t) + b7 ,



dt








dI(t)
b6 Z


= I ′ = −b2 I(t) +

dt
The initial condition now takes the form of
G(0) = Gb + b0 , I(0) = Ib + b3 b0 ,
b5
t
t−b5
and for
(3.2.2)
G(s)ds.
t ∈ [−b5 , 0), G(t) = Gb .
As in the minimal model, G(t) [mg/dl],I(t) [µUI/ml] is the plasma glucose, insulin
concentration at time t [min], respectively. The Gb , Ib , b0 , b1 , b2 and b3 are the same
80
as, or similar to that in the minimal model with same units. b4 [min−1 pM−1 ] is the
constant measuring the insulin-dependent glucose disappearance rate per unit [pM]
of the plasma insulin concentration. b5 [min] is the number of minutes of the past
period whose the plasma glucose concentrations influence the current pancreatic insulin
secretion. b6 [min−1 pM/(mg/dl)] is the constant describing the second-phase pancreatic
insulin secretion rate per unit of average the plasma glucose concentration throughout
the previous b5 minutes. b7 [(mg/dl)min−1 ] is the constant increase in the plasma glucose
concentration due to constant baseline liver glucose release.
The outcome is that the model always admits a globally asymptotically stable
steady state.
One of the objectives of this chapter is to find out if and how this outcome
depends on the specific choice of functions and the way delay is incorporated. To
this end, we generalize the dynamical model to allow more general functions and an
alternative way of incorporating time delay. Our findings show that in theory, such
models can possess unstable positive steady states and produce oscillatory solutions.
However, for all the clinic data reported in [23], such unstable steady states do not
exist. Hence, our work indicates that the dynamic model does provide qualitatively
robust dynamics for the purpose of clinic application. We also perform simulations
based on data from a clinic study reported in [23] and point out some plausible but
important implications.
3. More Generic IVGTT Model
While the dynamic model solves the problems of minimal model, it implicitly or
explicitly made a few assumptions that may not be necessary or realistic. Specifically
81
some of the interaction terms are too special and thus too restrictive. For example, the
term b4 I(t)G(t) assumes mass action law applies here. A more popular, general and
realistic alternative is to replace this term by b4 I(t)G(t)/(αG(t) + 1). Since in a unit of
time, a unit of insulin can only process a limited amount of glucose. Also the way the
delay is introduced is somewhat restrictive, the justification of which consists of only
one subjective assumption “the delay term refers to the pancreatic secretion of insulin:
effective pancreatic secretion at time t is considered to be proportional to the average
value of glucose concentration in the b5 minutes preceding time t”. This naturally invites
other plausible ways of incorporating the time delay. The most noteworthy outcome of
the dynamical model is that it always admits a globally asymptotically stable steady
state.
We propose the following general and more realistic model for the interaction
of glucose and insulin. This model includes the “dynamic model” ( 3.2.2) as a special
case:

dG(t)



= G′ (t) = −f (G(t)) − g(G(t), I(t)) + b7 ,



dt


(3.3.1)






dI(t)


= I ′ (t) = −p(I(t)) + q(L(Gt )).
dt
The initial condition is
G(0) = Gb + b0 ,
I(0) = Ib + b3 b0 ,
and G(t) ≡ Gb ,
for t ∈ [−b5 , 0),
where Gt (θ) = G(t + θ), t > 0, θ ∈ [−b5 , 0]. Gb [mg/dl], Ib [µUI/ml] is the subject’s
baseline glycemia, insulinemia, respectively. The parameters b0 , b3 , b5 and b7 are the
same as in model (3.2.2). Functions f , g, p, q satisfy the following general conditions.
(i) f (0) = 0, f (∞) = ∞, 0 < f ′ (x) < ∞ for x > 0
82
(ii) g(0, 0) = 0, 0 < gx (x, y) < ∞, 0 < gy (x, y) < ∞ for x > 0
g(x, 0) = 0, g(0, y) = 0, g(∞, y) < ∞, and g(x, ∞) = ∞ when x 6= 0
(iii) p(0) = 0, p(∞) = ∞, 0 < p′ (x) < ∞ for x > 0
(iv) q(x) = 0, if and only if x = 0; L : C[−b5 , 0] → C[−b5 , 0] is a linear operator
defined as L(ϕ) =
R0
−b5
ϕ(s)d(µ(s)), where µ(s) is nondecreasing with
R0
−b5
d(µ(s)) = 1.
We will consider two cases for L(Gt ): the discrete and distributed cases. For discrete
delay, L(Gt ) = G(t−b5 ) and for distributed delay, L(Gt ) =
1
b5
R0
−b5
G(t+θ)dθ. We assume
q(L(Gt + ϕt )) > q(L(Gt )) for ϕt ∈ C[−b5 , 0] with ϕt (θ) > 0, θ ∈ [−b5 , 0]. Furthermore,
we assume that |q(ϕ1 ) − q(ϕ2 )| < kkϕ1 − ϕ2 k for ϕ1 , ϕ2 ∈ C[−b5 , 0], where k · k is C
norm. So q(L(·)) : C → R is Lipschitz.
We always assume that the model (3.3.1) has a unique equilibrium point (G∗ , I ∗ )
in R2+ = {(x, y) : x > 0, y > 0}.
We also present two other less general models (still more general than the dynamic model in [23]), for the convenience of analysis and applications. We first propose
the following specific model of glucose-insulin interaction.


b4 I(t)G(t)
dG(t)

′


=
G
(t)
=
−b
G(t)
−
+ b7
1


dt
αG(t) + 1



(3.3.2)






dI(t)



= I ′ (t) = −b2 I(t) + b6 G(t − b5 )
dt
with the same initial conditions of model (3.3.1). The parameters have the same meaning as those in “dynamic model” (3.2.2).
Comparing with the “dynamic model” (3.2.2), model (3.3.2) has two notable and
important differences: First, no mass action law is assumed for glucose concentration
change due to the insulin-dependent net glucose tissue uptake. We assume instead
that insulin-dependent net glucose tissue uptake takes the more general and realistic
83
Michaelis-Menten form G(t)/(αG(t) + 1) which has a maximum capacity b4 /α. The
parameter α in the response function G(t)/(αG(t) + 1) is non-negative. 1/α is the
half-saturation constant. The reason for this is simply due to the limit of time and the
capacity of insulin’s ability of digesting glucose. Second, we assume that the effective
pancreatic secretion (after the liver first-pass effect) at time t is affected by the value of
glucose concentration in the b5 minutes preceding time t instead of the average amount
in that period.
When
1 Z0
G(t + θ)dθ,
L(Gt ) =
b5 −b5
and the Michaelis-Menten kinetics is assumed, the model (3.3.1) becomes


b4 I(t)G(t)


′

+ b7
 G (t) = −b1 G(t) −
αG(t) + 1

b6 Z 0


′

G(t + θ)dθ
I
(t)
=
−b
I(t)
+

2
b5
(3.3.3)
−b5
Clearly, both models ( 3.3.2) and (3.3.3) have a unique equilibrium point (G∗ , I ∗ )
in R2+ = {(x, y) : x ≥ 0, y ≥ 0}, where
µ
∗
G = 2b7 / (b1 − αb7 ) +
s
(b1 − αb7
)2
+ 4b7
µ
b4 b6
αb1 +
b2
¶¶
and
I∗ =
b6 ∗
G.
b2
Obviously, model (3.3.2)((3.3.3)) is a special case of model (3.3.1), where f (x) =
b1 x, g(x, y) = b4 xy/(αx + 1), p(x) = b2 x and q(L(xt )) = b6 x(t − b5 ) (q(L(xt )) =
(b6 /b5 )
R b6
t−b5
x(t−s)ds). For the same choice of f , g, p and q(L(xt )) = (b6 /b5 )
and α = 0, model (3.3.1) reduces to the dynamic model (3.2.2).
Rt
t−b5
x(s)ds
4. Preliminary Analysis
The following basic proposition is important for our study. Its proof is straightforward.
84
Proposition 3.4.1 All solutions of model (3.3.1) exist for all t > 0, and are positive
and bounded.
Proof Since the |f ′ (x)|, |gx (x, y)|, |gy (x, y)| and |p′ (x)| are bounded for x, y > 0, they
are Lipschitz and completely continuous for x > 0, y > 0. Notice that we assume q(x) is
Lipschitz in for x ≥ 0 and L : C[−b5 , 0] → C[−b5 , 0] is linear, then q(L(·)) is Lipschitz
in C[−b5 , 0]. By Theorem 2.1, 2.2 and 2.4 on page 19 and 20 in [54], the solution of
equation (3.3.1) with given initial condition exists and unique for all t ≥ 0.
Let (G(t), I(t)) be a solution of (3.3.1). If G(t0 ) = 0 for some t0 > 0 and
G(t) > 0 for 0 < t < t0 , then G′ (t0 ) ≤ 0. However, at t0 , due to the assumptions that
f (0) = g(0, y) = 0, we have G′ (t0 ) = −f (G(t0 )) − g(G(t0 ), I(t0 )) + b7 = b7 > 0. This
contradiction shows that G(t) > 0 for all t in the interval of existence. If I(t̄0 ) = 0
for some t̄0 > 0, then I ′ (t̄0 )) ≤ 0 and 0 ≥ I ′ (t̄0 ) = −p(I(t̄0 )) + q(L(Gt̄0 )) = q(L(Gt̄0 )).
Since Gt̄0 (θ) > 0 for θ ∈ [−b5 , 0], q(L(Gt̄0 )) > 0 by (iv) and thus I(t) > 0 for all t in
the interval of existence.
As for the boundedness of G(t), by the first equation of (3.3.1),
G′ (t) = −f (G(t)) − g(G(t), I(t)) + b7 ≤ −f (G(t)) + b7 .
Thus G(t) is bounded by MG = max{Gb + b0 , f −1 (b7 )}. And hence I(t) is bounded by
∆
MI = max{Ib + b3 b0 , p−1 (q(MG ))} due to
I ′ (t) = −p(I(t)) + q(Gt ) ≤ −p(I(t)) + q(MG ).
This completes the proof.
Let (G(t), I(t)) be a solution of (3.3.1). Throughout this paper, we define
G = lim sup G(t),
t→∞
G = lim inf G(t)
t→∞
I = lim sup I(t),
t→∞
I = lim inf I(t).
t→∞
85
Due to the Proposition 3.4.1, we see that these limits are finite.
As in Chapter 2, we apply following elementary lemma to obtain a few preliminary results. See [48] for a proof.
Lemma A Let f : R → R be a differentiable function. If l = lim inf t→∞ f (t) <
lim supt→∞ f (t) = L, then there are sequences {tk } ↑ ∞, {sk } ↑ ∞ such that for all
k, f ′ (tk ) = f ′ (sk ) = 0, limk→∞ f (tk ) = L and limk→∞ f (sk ) = l.
The following lemma is useful for establishing the fact that model (3.3.1) is
always persistent, which implies that both components of solutions of the model are
eventually bounded by positive constants from both above and below. Such bounds are
independent of initial data.
Lemma 3.4.1 Consider model (3.3.1). If I < I, then
p−1 (q(G)) ≤ I < I ≤ p−1 (q(G)).
If G < G, then
−f (G) − g(G, I) + b7 ≤ 0,
and
− f (G) − g(G, I) + b7 ≥ 0.
(3.4.1)
Proof Since I < I, by Lemma A, there exists {tk } ↑ ∞, {sk } ↑ ∞, such that I ′ (tk ) =
I ′ (sk ) = 0, limk→∞ I(tk ) = I and limk→∞ I(sk ) = I. Notice that p, q are continuous,
q ′ (·) > 0 and (G(t), I(t)) is a solution of (3.3.1). Hence, we have
0 = I ′ (tk ) = −p(I(tk )) + q(L(Gtk )) for all k.
For any ε > 0, there exists k0 > 0, such that
G + ε > Gtk (θ),
θ ∈ [−τ, 0] for all k > k0 .
86
Hence condition (iv) implies that q(L(Gtk )) ≤ q(G + ε) for k > k0 . Therefore,
0 = −p(I(tk )) + q(L(Gtk )) ≤ −p(I(tk )) + q(G + ε).
By letting k → ∞ and ε → 0, we have
p(I) ≤ q(G).
(3.4.2)
p(I) ≥ q(G).
(3.4.3)
Similarly, we have
(3.4.2) and (3.4.3) lead to
q(G) ≤ p(I) < p(I) ≤ q(G)
and then
p−1 (q(G)) ≤ I < I ≤ p−1 (q(G)).
If G < G, by Lemma A there exists {t′k } ↑ ∞, {s′k } ↑ ∞, such that G′ (t′k ) =
G′ (s′k ) = 0, limk→∞ G(t′k ) = G and limk→∞ G(s′k ) = G. Thus we have
0 = G′ (t′k ) = −f (G(t′k )) − g(G(t′k ), I(t′k )) + b7
and
0 = G′ (s′k ) = −f (G(s′k )) − g(G(s′k ), I(s′k )) + b7
for all k.
Since f, g are continuous and g2 (x, y) > 0 for all x > 0, without loss of generality,
assuming limk→∞ I(t′k ) and limk→∞ I(s′k ) exist, we have
0 =
lim (−f (G(t′k )) − g(G(t′k ), I(t′k ))) + b7
k→∞
= −f (G) − g(G, lim I(t′k )) + b7
k→∞
≤ −f (G) − g(G, I) + b7
87
and
0 = lim (−f (G(s′k )) − g(G(s′k ), I(s′k ))) + b7
t→∞
= −f (G) − g(G, lim I(s′k )) + b7
k→∞
≥ −f (G) − g(G, I) + b7 .
This completes the proof.
Proposition 3.4.2 The model (3.3.1) is persistent. That is, solutions are eventually
bounded by positive constants from both above and below.
Proof For a solution (G(t), I(t)) of (3.3.1), by Proposition 3.4.1,
G′ (t) = −f (G(t)) − g(G(t), I(t)) + b7
≤ −f (G(t)) + b7 .
∆
Using Lemma A, we can obtain that G ≤ f −1 (b7 ), where G = lim supt→∞ G(t).
Notice that (3.4.1) implies
f (G) + g(G, p−1 (q(f −1 (b7 )))) ≥ f (G) + g(G, p−1 (q(G))) ≥ b7 ,
which shows that G > 0 due to f (0) = 0 and g(0, y) = 0 for y ≥ 0. This together with
Lemma 3.4.1 shows that the model (3.3.1) is persistent.
5. Global Stability of Steady State
In this section, we provide several global stability results for the steady state
(G∗ , I ∗ ). The same method was used by [23] to establish the global stability of the positive steady state. As we shall see, for the general model (3.3.1), global asymptotically
stability of (G∗ , I ∗ ) is conditional.
Using mainly fluctuation type argument and Lemma 3.4.1, we can obtain
88
Theorem 3.5.1 For model (3.3.1), if
g(x, p−1 (q(y))) − g(y, p−1 (q(x))) ≥ 0,
(3.5.1)
for all x ≥ y > 0, then the unique equilibrium point (G∗ , I ∗ ) of (3.3.1) is globally
asymptotically stable.
Proof If I < I, then from Lemma 3.4.1,
p−1 (q(G)) ≤ I < I ≤ p−1 (q(G)).
Thus G < G and
−f (G) − g(G, p−1 (q(G))) + b7 ≤ −f (G) − g(G, I) + b7 ≤ 0,
−f (G) − g(G, p−1 (q(G))) + b7 ≥ −f (G) − g(G, I) + b7 ≥ 0.
Therefore
(f (G) − f (G)) + (g(G, p−1 (q(G))) − g(G, p−1 (q(G)))) ≤ 0.
Due to (3.5.1), we have
g(G, p−1 (q(G))) − g(G, p−1 (q(G))) ≥ 0.
Hence
f (G) − f (G) ≤ 0,
which indicates G = G and thus I = I. Since (G∗ , I ∗ ) is the only equilibrium point of
(3.3.1), we have
lim G(t) = G∗
t→∞
The proof is completed.
Theorem 3.5.2 For model (3.3.1), if
and
lim I(t) = I ∗ .
t→∞
89
q ′ (y)
f (x) + gx (x, p (q(y))) − gy (x, p (q(y))) ′ −1
>0
p (p (q(y)))
′
−1
−1
(3.5.2)
for all x, y > 0, then the unique equilibrium point (G∗ , I ∗ ) of (3.3.1) is globally asymptotically stable.
Proof If I < I, then from Lemma 3.4.1,
p−1 (q(G)) ≤ I < I ≤ p−1 (q(G))
we see that G < G, and
(f (G) − f (G)) + (g(G, p−1 (q(G))) − g(G, p−1 (q(G)))) ≤ 0.
(3.5.3)
Let
F (x, y) = f (x) + g(x, p−1 (q(y))),
(x, y) ∈ R2+ = {(x, y) : x > 0, y > 0}.
Then (3.5.3) is equivalent to
F (G, G) − F (G, G) ≤ 0.
(3.5.4)
By the mean value theorem, there exists a θ ∈ (0, 1) such that
F (G, G) − F (G, G) = (G − G)(Fx (ξ, η) − Fy (ξ, η))
∆
∆
where ξ = G + θ(G − G) and η = G − θ(G − G). Notice that
Fx (x, y) = f ′ (x) + gx (x, p−1 (q(y)))
and
Fy (x, y) = gy (x, p−1 (q(y)))
q ′ (y)
.
p′ (p−1 (q(y)))
(3.5.5)
90
¿From (3.5.2)
Fx (ξ, η) − Fy (ξ, η) = f ′ (ξ) + gx (ξ, p−1 (q(η)))
−gy (ξ, p−1 (q(η)))
q ′ (η)
p′ (p−1 (q(η)))
(3.5.6)
> 0.
On the other hand, (3.5.4) and (3.5.5) lead to
(G − G)(Fx (ξ, η) − Fy (ξ, η)) ≤ 0.
Clearly, (3.5.6) leads to G − G ≤ 0 and thus G = G and I = I. Since (G∗ , I ∗ ) is the
only equilibrium point of (3.3.1), we have
(G(t), I(t)) → (G∗ , I ∗ ) as t → ∞.
This completes the proof.
In model (3.3.1), if g(x, y) takes the special form
g(x, y) = g1 (x, y)/g2 (x)
where
(v) g1 (0, 0) = 0, g1 (x, 0) = g1 (0, y) = 0 for all (x, y) > 0,
(vi) g2 (x) ≥ c > 0 for some constant c. g2 (∞) = ∞.
(vii) (g1 )x (x, y) > 0, (g1 )y (x, y) > 0, for all x, y > 0.
(viii) g1 (x, ∞) = g1 (∞, y) = ∞ for x, y > 0, xy 6= 0.
(ix) g2′ (x) > 0 for all x > 0,
then we have
Theorem 3.5.3 For model (3.3.1), g(x, y) = g1 (x, y)/g2 (x), where g1 , g2 satisfy (v)–
(ix). Then (G∗ , I ∗ ) is globally asymptotically stable if
91
(a) (f (x) − b7 )g2 (x) is increasing for all x > 0
(b) g1 (x, p−1 (q(y))) − g1 (y, p−1 (q(x))) ≥ 0 for all x ≥ y > 0.
Proof We shall show that G = G and I = I. By Lemma 3.4.1, if I < I, then G < G
and
p−1 (q(G)) ≤ I < I ≤ p−1 (q(G)),
−f (G) − g1 (G, p−1 (q(G)))/g2 (G) + b7 ≤ 0
and
−f (G) − g1 (G, p−1 (q(G)))/g2 (G) + b7 ≥ 0.
Thus
−(f (G) − b7 )g2 (G) − g1 (G, p−1 (q(G))) ≤ 0
(3.5.7)
−(f (G) − b7 )g2 (G) − g1 (G, p−1 (q(G))) ≥ 0.
(3.5.8)
and
Hence, (3.5.7) and (3.5.8) imply
((f (G) − b7 )g2 (G) − (f (G) − b7 )g2 (G)) + (g1 (G, p−1 (q(G))) − g1 (G, p−1 (q(G)))) ≤ 0.
(3.5.9)
This together with the assumptions (a) and (b), we obtain
(f (G) − b7 )g2 (G) − (f (G) − b7 )g2 (G) = 0.
This implies that G = G, and therefore I = I. This completes the proof.
Corollary 3.5.1 For model (3.3.1), assume g(x, y) = g1 (x, y)/g2 (x), where g1 , g2 satisfy (v)–(ix). If (b) in Theorem 3.5.3 is replaced by
(b′ )
∂
q ′ (y)
∂
g1 (x, p−1 (q(y))) −
g1 (x, p−1 (q(y))) ′ −1
≥ 0 for all x, y > 0,
∂x
∂y
p (p (q(y)))
then (G∗ , I ∗ ) is globally asymptotically stable.
92
Proof We shall show that (b) in Theorem 3.5.3 is true when (b′ ) holds. Let
u(x, y) = g1 (x, p−1 (q(y))),
x, y > 0.
By the mean value theorem, for x ≥ y > 0, there exists θ ∈ (0, 1) such that
u(x, y) − u(y, x) = ux (ξ, η)(x − y) + uy (ξ, η)(x − y)
= (x − y)(ux (ξ, η) − uy (ξ, η)) ≥ 0,
where ξ = y + θ(x − y), η = x − θ(x − y). This completes the proof.
The following are direct results of the applications of the above theorems to the
specific models (3.3.2) and (3.3.3). For convenience, we define two new parameters.
.
a1 := b1 b7
and
.
γ := b4 b6 b1 b2
Corollary 3.5.2 For model (3.3.2) ((3.3.3)), if α ≥ γ, then the only equilibrium point
(G∗ , I ∗ ) of (3.3.2) ((3.3.3)) is globally asymptotically stable.
Proof We shall apply Theorem 3.5.2 to model (3.3.2) ((3.3.3)). For model (3.3.2), we
have
f ′ (x) + gx (x, p−1 (q(y))) − gy (x, p−1 (q(y)))
q ′ (y)
p′ (p−1 (q(y)))
b4 (b6 /b2 )y
b4 x b6
−
2
(αx + 1)
αx + 1 b2
¶
µ
γx
≥ b1 1 −
αx + 1
= b1 +
= b1 ((α − γ)x + 1)/(αx + 1) > 0
for all x > 0, if α ≥ γ.
Corollary 3.5.3 For model (3.3.2) ((3.3.3)), if α ≤ a1 = b1 /b7 , then the only equilibrium point (G∗ , I ∗ ) of (3.3.2)((3.3.3)) is globally asymptotically stable.
93
Proof We shall apply Theorem 3.5.3 to model (3.3.2). For (a) to hold, we need
d
d
((f (x) − b7 )g2 (x)) =
((b1 x − b7 )(αx + 1))
dx
dx
= 2b1 αx + (b1 − αb7 ) > 0
for all x > 0. This is true if b1 − αb7 ≥ 0, i.e., α ≤ b1 /b7 .
For (b) to hold, we need only to observe that
g1 (x, p−1 (q(y))) − g1 (y, p−1 (q(x))) ≡ 0 for all x, y > 0.
Combining the above two corollaries, we immediately arrive at the following
conclusion
Corollary 3.5.4 For model (3.3.2) ((3.3.3)), if a1 ≥ γ, i.e.,
.
b1 /b7 ≥ b4 b6 b1 b2 ,
then (G∗ , I ∗ ) is globally asymptotically stable for all α ≥ 0 and b5 > 0.
6. Local Stability of Steady State and Stability Switch
Although we have obtained several global stability results, we have yet to study
the local stability systematically. One of our motivations to study the general model
(3.3.1) is to see if an oscillatory solution can exist for certain parameter values. Recall
that both the minimal and dynamic models permit only globally asymptotically stable
positive steady states. We would like to know if and how this may change for more
realistic models. To this end, we need to obtain the characteristic equations associated
to our models.
Consider first model (3.3.1). Let
G1 (t) = G(t) − G∗ ,
I1 (t) = I(t) − I ∗ .
(3.6.1)
94
then model (3.3.1) is translated to




 G′1 (t) = −f (G1 (t) + G∗ ) − g(G1 (t) + G∗ , I1 (t) + I ∗ ) + b7
(3.6.2)



 I1′ (t) = −p(I1 (t) + I ∗ ) + q(L((G1 )t + G∗ ))
and has a unique equilibrium point at (0, 0).
Thus the linearized system of (3.6.2) is given by




 G′1 (t) = −(f ′ (G∗ ) + gx (G∗ , I ∗ ))G1 (t) − gy (G∗ , I ∗ )I1 (t),



 I1′ (t) = −p′ (I ∗ )I1 (t) + q ′ (G∗ )L((G1 )t ).
For convenience, we still use G(t) and I(t) to represent G1 (t) and I1 (t), respectively, and define
A = f ′ (G∗ ) + gx (G∗ , I ∗ ), B = gy (G∗ , I ∗ ), C = q ′ (G∗ ), D = p′ (I ∗ ).
Then A, B, C and D are positive. The linearized system of (3.6.2) can be rewritten as




 G′ (t) = −AG(t) − BI(t)
It is easy to see that
(3.6.3)



 I ′ (t) = −DI(t) + CL(Gt ).
G′′ (t) = −(A + D)G′ (t) − ADG(t) − BCL(Gt ).
(3.6.4)
Denote that
a = A + D,
c = AD,
d = BC
and τ = b5 .
Then (3.6.4) can be rewritten as
G′′ (t) + aG′ (t) + cG(t) + dL(Gt ) = 0.
(3.6.5)
If L(Gt ) takes the discrete delay form, i.e., L(Gt ) = G(t − τ ), t > 0, then the characteristic equation of (3.6.5) is given by
∆
D(λ) = λ2 + aλ + c + de−τ λ = 0.
(3.6.6)
95
If L(Gt ) takes the form of distributed delay, i.e., L(Gt ) = (1/τ )
the characteristic equation of (3.6.5) is given by
∆
D̃(λ) = λ2 + aλ + c +
Rt
t−τ
G(θ)dθ, t > 0, then
d Z 0 λθ
e dθ = 0.
τ −τ
(3.6.7)
7. Delay Independent Stability Results for Discrete Delay Model
In this section we consider only the case of discrete delay in model (3.3.1) and
therefore the results are applicable to model (3.3.2). We will need theorem B stated in
Section 6 of Chapter 2 (Theorem 3.1 on page 77 in [54], 1993). First, we have
Lemma 3.7.1 In (3.6.6), the stability of (0, 0) is determined as follows.
Case (1). If AD > BC, then the stability of (0, 0) does not change as τ ≥ 0 is
increasing;
Case (2). If AD ≤ BC, then the stability of (0, 0) can at most change once from
stable to unstable, i.e., if (0, 0) is stable, when τ = 0, then (0, 0) becomes unstable when
τ ≥ τ0 > 0 for some τ0 > 0; if (0, 0) is unstable when τ = 0, (0, 0) remains unstable for
all τ > 0.
Proof Compare (3.6.6) with (2.6.1). We have
σ = 0,
a = A + D,
b = 0,
c = AD,
d = BC.
Thus,
b2 + 2c − a2 − 2dσ = 2c − a2 = 2AD − (A + D)2
= −(A2 + D2 ) ≤ 0.
Hence, (A) is violated, and hence the case (III) in (Theorem B in Section 6 of Chapter
2). That is, (0, 0) cannot have multiple stability switches.
96
Since A, B, C, D > 0, we see that c2 > d2 (c2 ≤ d2 ) is equivalent to c > d (c ≤ d).
Thus we proved Case (1) and Case (2).
From Lemma 3.7.1, we have
Theorem 3.7.1 For (3.3.1), we have the following results on the local stability of (0, 0).
Case (i). If
(f ′ (G∗ ) + gx (G∗ , I ∗ ))p′ (I ∗ ) ≤ gy (G∗ , I ∗ )q ′ (G∗ ),
then (G∗ , I ∗ ) has at most one stability switch as τ ≥ 0 increases.
Case (ii). If
(f ′ (G∗ ) + gx (G∗ , I ∗ ))p′ (I ∗ ) > gy (G∗ , I ∗ )q ′ (G∗ ),
then stability of (G∗ , I ∗ ) does not change for any τ ≥ 0.
Proof Notice that (3.6.3) is the linearized system of (3.3.1) at (G∗ , I ∗ ), and A =
f ′ (G∗ ) + gx (G∗ , I ∗ ), D = p′ (I ∗ ), B = gy (G∗ , I ∗ ), C = q ′ (G∗ ).
Corollary 3.7.1 For model (3.3.2), the local stability of (G∗ , I ∗ ) can be determined as
follows.
√
Case (i). If γ > 21 (11 + 5 5)a1 , then there is at most one stability switch of
(G∗ , I ∗ ). Specifically, if (G∗ , I ∗ ) is stable when b5 = 0, then there exists τ0 > 0 such that
(G∗ , I ∗ ) is stable for b5 ∈ [0, τ0 ) and unstable for b5 ≥ τ0 ; if (G∗ , I ∗ ) is unstable when
b5 = 0, then (G∗ , I ∗ ) is unstable for all b5 > 0.
√
Case (ii). If γ ≤ 12 (11 + 5 5)a1 , then the stability of (G∗ , I ∗ ) does not change
for all b5 ≥ 0.
97
Proof By Theorem 3.7.1, we need only to check the sign of
(f ′ (G∗ ) + gx (G∗ , I ∗ ))p′ (I ∗ ) − gy (G∗ , I ∗ )q ′ (G∗ ).
(3.7.1)
Notice that
.
G∗ = 2b7 ((b1 − αb7 ) +
q
(b1 − αb7 )2 + 4b7 (αb1 + b4 b6 /b2 )),
and I ∗ =
b6 ∗
G , (3.7.2)
b2
and
f ′ (x) = b1 ,
gx (x, y) =
p′ (x) = b2 ,
b4 y
,
(αx + 1)2
q ′ (x) = b6 ,
gy (x, y) =
b4 x
.
αx + 1
Let
∆
β = b4 b6 /b2 (= b1 γ).
Then (3.7.1) becomes
b4 b6 G∗
βG∗
−
b2 b1 +
(αG∗ + 1)2
αG∗ + 1
b2
(b1 (αG∗ + 1)2 − βαG∗2 )
=
(αG∗ + 1)2
q
q
q √
q √
b2
∗
∗
∗
b
β
β αG∗ ).
(αG
+
1)
+
=
αG
)(
b
(αG
+
1)
−
(
1
1
(αG∗ + 1)2
¶
µ
Let
w(α) =
q
=
q
∆
∆
=
b1 (αG∗ + 1) −
q √
β αG∗
√
√ √
b1 (1 + ( α − γ) αG∗ )
q
b1 w1 (α).
Then sign (5.3) = sign w1 (α). We have
√
√ √
w1 (α) = 1 + ( α − γ) αG∗
√ √
√
2b7 α( α − γ)
q
= 1+
b1 − αb7 + (b1 − αb7 )2 + 4b1 b7 (α + γ)
=
q
√
(b1 − αb7 )2 + 4b1 b7 (α + γ) + (b1 + b7 α − 2b7 γα)
b1 − αb7 +
q
(b1 − αb7 )2 + 4b1 b7 (α + γ)
.
98
Let
v(α) =
√
(b1 − 2b7 )2 + 4b1 b7 (α + γ) + (b1 + b7 α − 2b7 γα)
q
then sign (5.3) = sign v(α).
√
(b1 + αb7 )2 + 4b1 b7 γ + (b1 + b7 α − 2b7 γα)
√
√
4b1 b7 γ + 4b1 b7 γα + 4b27 γα3/2 − 4b27 γα
q
=
√
(b1 + 2b7 )2 + 4b1 b7 γ − (b1 + b7 α − 2b7 γα)
√
√
√
√
4b27 γ(α3/2 − γα + a1 α + a1 γ)
.
= q
√
(b1 + αb7 )2 + 4b1 b7 γ − (b1 + b7 α − 2b7 γα)
v(α) =
Let µ =
√
q
α and
J(µ) = µ3 −
√
√
γµ2 + a1 µ + a1 γ,
µ ≥ 0.
Then
sign (5.3) = sign v(α) = sign J(µ).
Notice that
√
J ′ (µ) = 3µ2 − 2 γµ + a1
∆J = 4γ − 12a1 = 4(γ − 3a1 )
J ′ (µ) = 0 gives two extreme points if and only if ∆J > 0. So if ∆J ≤ 0, J(µ) > 0. If
∆J > 0, J(µ) has the possibility to assume negative value.
Assume now that ∆J > 0, i.e., γ − 3a1 > 0. We shall find the minimum value of
J(µ), µ ≥ 0. Solving J ′ (µ) = 0, we obtain
√
√
√
√
µ1,2 = 16 (2 γ ± 2 γ − 3a1 ) = 31 ( γ ± γ − 3a1 ).
√
√
Clearly µ0 = 13 ( γ + γ − 3a1 ) is the minimum point of J(µ), µ ≥ 0.
√
√
J(µ0 ) = a1 γ + µ0 (µ20 − γµ0 + a1 )
99
√
= a1 γ +
1 √
( γ
27
+
q
√
q
γ − 3a1 )(γ + 2 γ(γ − 3a1 ) + γ − 3a1
−3γ − 3 γ(γ − 3a1 ) + 9a1 )
√
= a1 γ +
1 √
( γ
27
√
= a1 γ +
+
√
√
1
(−2γ γ
27
=
√
1
(36a1 γ
27
=
2
((18a1
27
γ − 3a1 )(−γ −
q
γ(γ − 3a1 ) + 6a1 )
√
√
+ 9a1 γ − 2(γ − 3a1 ) γ − 3a1 )
√
√
− 2γ γ − 2(γ − 3a1 ) γ − 3a1 )
√
√
− γ) γ − (γ − 3a1 ) γ − 3a1 )
≤ 0 if γ ≥ 18a1 .
Assume 3a1 < γ < 18a1 , then
J(µ0 ) =
2 (18a1 − γ)2 γ − (γ − 3a1 )2 (γ − 3a1 )
√
.
√
27 (18a1 − γ) γ + (γ − 3a1 ) γ − 3a1
∆
Let h(a1 , γ) = γ(18a1 − γ)2 − (γ − 3a1 )(γ − 3a1 )2 , then sign J(µ0 ) = sign h(a1 , γ). We
have
h(a1 , γ) = γ(15a1 − (γ − 3a1 ))2 − (γ − 3a1 )(γ − 3a1 )2
= −27a1 (γ 2 − 11a1 γ − a22 )
√
√ ¶
µ
¶µ
−11 + 5 5
11 + 5 5
= −27a1 γ +
a1 γ −
a1
2
2
√
11 + 5 5
a1 .
< 0 iff γ >
2
This completes the proof.
8. Delay Dependent Stability Conditions
The stability results in the previous section do not depend on the values of the
delay(b5 ). However, they do suggest that for some parameter values(delay is included),
the positive steady state may become unstable. It is thus interesting to locate such
100
parameter values. Such a task turns out to be very complex. Instead, we shall single
out parameter region where no stability switch can take place. Specifically, we would
like to obtain some upper bound on the delay length(while holding other parameters
steady) for the positive steady state to remain stable. Specifically, we shall seek the
relationship of such upper bound with the value of α. We shall consider both model
(3.3.2) and (3.3.3). Recall that the following notations in Section 6,
A = f ′ (G∗ ) + gx (G∗ , I ∗ ), B = gy (G∗ , I ∗ ), C = q ′ (G∗ ), D = p′ (I ∗ ).
and
a = A + D,
c = AD,
d = BC,
τ = b5 .
Define
H(α) =
f ′ (G∗ ) + gx (G∗ , I ∗ ) + p′ (I ∗ )
.
gy (G∗ , I ∗ )q ′ (G∗ )
8.1. The case of discrete delay. We shall consider here the characteristic
equation (3.6.6) below. Recall that
D(λ) = λ2 + aλ + c + de−τ λ = 0.
If there exists a τ0 > 0 such that the trivial solution of (3.6.3) is unstable for all τ ≥ τ0 ,
then there must be an ω > 0 such that D(ωi) = 0. Thus
−ω 2 + aϕi + c + d(cos τ ω − i sin τ ω) = 0,
and hence




 ω 2 = c + d cos τ ω,



 aω = d sin τ ω.
Since a > 0 and ω > 0, (3.8.1) implies that τ ≥ a/d. This leads to the following
(3.8.1)
101
Theorem 3.8.1 In linear equation (3.6.5), if L(Gt ) = G(t − τ ), t > 0, τ > 0, and
τ < a/d, then the trivial solution of (3.6.5) is globally asymptotically stable.
Applying theorem 3.8.1 to model (3.3.2), we have
Corollary 3.8.1 In model (3.3.2), if L(Gt ) = G(t − τ ), t > 0, τ > 0, and τ < H(α),
then the equilibrium point (G∗ , I ∗ ) is locally asymptotically stable.
8.2. The case of distributed delay. We now consider the characteristic equation (3.6.7) of linear equation (3.6.5) for the case of L(Gt ) =
D̃(λ) = λ2 + aλ + c +
1
τ
Rt
t−τ
G(θ)dθ, t > 0, τ > 0,
d Z 0 λθ
e dθ = 0.
τ −τ
(3.8.2)
If the trivial solution of (3.6.5) is unstable for some τ > 0, then there exist u ≥ 0 and
v > 0 such that λ = u + iv is a solution of (3.8.2), i.e.,
d Z 0 θu
e (cos vθ + i sin vθ)dθ = 0.
(u + iv) + a(u + iv) + c +
τ −τ
2
Thus
d Z 0 θu
e cos vθdθ = 0,
u − v + au + c +
τ −τ
2
2
(3.8.3)
and
2uv + av +
d Z 0 θu
e sin vθdθ = 0.
τ −τ
Since v > 0, we see that (3.8.4) implies
2u + a = −
d Z 0 uθ sin vθ
dθ.
e
τ −τ
v
Therefore
d ¯¯ Z 0 uθ sin vθ ¯¯
dθ¯
2u + a ≤
e ·
τ ¯ −τ
v
1
dZ 0
|θ|dθ = dτ.
≤
τ −τ
2
¯
Hence we have u ≤
1
2
µ
1
τd
2
¶
¯
− a . This leads to the following
(3.8.4)
102
Theorem 3.8.2 In linear equation (3.6.5), if L(Gt ) = (1/τ )
R0
−τ
G(t + θ)dθ, t > 0,
τ > 0 and τ < 2a/d, then the trivial solution of (3.6.5) is globally asymptotically stable.
From Theorem 3.8.2, we have
Corollary 3.8.2 In model (3.3.2), if L(Gt ) = (1/τ )
R0
−τ
G(t + θ)dθ, t > 0, τ > 0, and
τ < 2H(α), then the equilibrium point (G∗ , I ∗ ) is locally asymptotically stable.
8.3. Expression of H(α). In the following, we shall give the explicit expression
of H(α) for the Model (3.3.2) and (3.3.3). This is useful in applications and in planning
computer simulations. Recall that Model (3.3.1) reduces to the Model (3.3.2) ((3.3.3))
if
f (x) = b1 x,
g(x, y) =
b4 xy
,
αx + 1
p(x) = b2 x,
and q(L(ut )) = b6 L(ut ), ut ∈ C.
In this case, we have
f ′ (x) = b1 ,
gx (x, y) =
and q ′ (L(ut )) = b6
b4 y
,
(αx + 1)2
gy (x, y) =
b4 x
,
αx + 1
p′ (x) = b2 ,
x≥0
for ut ∈ C.
Recall that Model (3.3.2) ((3.3.3)) assumes the existence of an unique steady
state (G∗ , I ∗ ) in R2+ , where G∗ and I ∗ are given by (5.4). Let S(α) = α +
S(α) = α +
=
1
,
G∗
q
1
(b1 − αb7 ) + (b1 − αb7 )2 + 4b7 (αb1 + b4 b6 /b2 )
2b7
·
q
i
1 h
(b1 + b7 α) + (b1 + b7 α)2 + 4b4 b6 b7 /b2 .
2b7
Therefore
f ′ (G∗ ) + gx (G∗ , I ∗ ) + p′ (I ∗ )
H(α) =
gy (G∗ , I ∗ )q ′ (G∗ )
b1 + b2 + (b4 I ∗ /(αG∗ + 1)2 )
=
(b6 b4 G∗ /αG∗ + 1)
¶¸
then
103
b1 + b2 αG∗ + 1
1
+
∗
b4 b6
G
b2 (αG∗ + 1)
b1 + b2
(G∗ )−1
=
S(α) +
.
b4 b6
b2 S(α)
=
Let ∆ = (b1 + b7 α)2 + 4b4 b6 b7 /b2 . Notice that
∗ −1
(G )
S(α)
q
(b1 − αb7 )2 + 4b7 (αb1 + (b4 b6 /b2 ))
√
(b1 + b7 α) + ∆
√
√
[(b1 − αb7 ) + ∆][(b1 + b7 α) − ∆]
=
4b4 b6 b7 /b2
√
√
b2
[∆ + (b1 − b7 α) ∆ − (b1 + b7 α) ∆ − (b21 − b27 α2 )]
=
4b4 b6 b7
√
b2
[b27 α2 − b7 α ∆ + b1 b7 α + 2b4 b6 b7 /b2 ].
=
2b4 b6 b7
=
(b1 − αb7 ) +
Thus,
√
b1 + b2
[(b1 + b7 α) + ∆]
2b4 b6 b7
√
1
+
[b27 α2 − b7 α ∆ + b1 b7 α + 2b4 b6 b7 /b2 ]
2b4 b6 b7
√
1
=
[b27 α2 + (b1 + b2 − b7 α) ∆
2b4 b6 b7
H(α) =
+(2b1 + b2 )b7 α + 2b4 b6 b7 /b2 ].
9. Numerical Simulations
In a clinic study reported in [23], ten healthy volunteers participated(5 males
and 5 females). All of them had negative family and personal histories for diabetes
mellitus and other endocrine diseases, were on no medications and had maintained
a constant weight for the six months preceding that study. For detailed experiment
description, see [23]. They were able to show that the dynamic model does produce
solutions that fit well with the data collected from that experiment. The parameter
values for these individuals are listed in their Table 1 ([23]). Using these parameters
and Corollary 3.5.4, we found that for all these persons except the sixth person (female)
104
Table 3.9.1.
P
U
6
7
Parameters for subjects 6 and 7 in IVGTT Models (b5 = 23min.)
Gb
Ib
mg pM
dl
88 68.6
87 37.9
b0
b1
b2
mg
1
1
min
min
dl
209 2E-4 0.0422
311 1E-4 0.2196
b3
dl·pM
mg
1.64
0.64
b4
1
min·pM
1.09E-4
3.73E-4
b6
dl·pM
mg·min
0.033
0.096
b7
mg
dl·min
0.68
1.24
and the seventh person (male), and all values of α and delay lengths, solutions always
tend to the positive steady state for both models (3.3.2) and (3.3.3). Since one of our
main objectives is to see if unstable positive steady state is possible for our models and
some of these ten persons, and our theoretical results failed to exclude such possibility,
we thus focused our simulations on the subjects 6 and 7 (sixth and seventh persons).
The relevant parameter values (obtained through the above mentioned clinic study) for
these two persons are summarized in table 3.9.1 (De Gaetano and Arino [23]), where P
stands for parameters, U stands for units and 6, 7 stand for subject 6, 7, respectively..
Even for subjects 6 and 7, according to our extensive simulation work, the steady
state is globally asymptotically stable in all practically meaningful values of time delay
τ = b5 and α. Nevertheless, we did find that the positive steady state can be unstable(if
a1 < γ and α ∈ (a1 , γ) provided the delay is large enough). Figures in figure 3.9.1
illustrate such instability with nontrivial periodic solutions for the discrete delay model
(3.3.2), using the data (except b5 and α) given in the table 1 for subjects 6 and 7. For
the discrete delay model (3.3.2), we need the time delay to be as close to 550 minutes
in order to observe sustainable oscillatory solutions for both subjects 6 (550) and 7
(600). The values of α are 0.01 and 0.05 for subject 6 and subject 7, respectively. Since
for both subjects, the actual delay length is only 23 minutes, we believe that it is very
unlikely to observe any sustainable oscillatory solutions in real life experiments for these
105
300
900
800
250
700
200
600
G, I
G, I
500
150
400
100
300
200
50
100
0
1.1
1.2
1.3
1.4
1.5
1.6
time t
Figure 3.9.1.
1.7
1.8
1.9
2
x 10
0
0.5
0.6
0.7
4
0.8
0.9
1
time t
1.1
1.2
1.3
1.4
1.5
x 10
4
Periodic solutions for the discrete delay model (3.3.2) for subject 6 and 7
Left: The large amplitude curve (solid line) is G, the other one (dash-dot line) is I. Here α = 0.01,
and b5 = 550. Right: The large amplitude curve (solid) is G, the other one (dash-dot line) is I. Here
α = 0.05, and b5 = 600.
subjects. In other words, the global asymptotical results reported in [23] is confirmed
at least practically by the more general model (3.3.1).
10. Discussion
From a purely theoretical point of view, we can still make a few insightful statements. For both very large or small values of α, the positive steady state is believed to
be globally asymptotically stable for any conceivable person. For relatively small α and
large delay, oscillatory solutions become possible. This indicates that it is important to
have α in the model, since the value of α is most likely small, but not too small. Indeed,
from the simulation results (Figure 3.9.1), we see α is in the range of 0.01 to 0.05, which
translates into a range of 20 to 100 for half saturation value of G. This range is very
close or comparable to the experiment values of G depicted in figures 2 and 3 in [23],
where G starts at values around 250 and drop down quickly (in about 40 minutes) to
values close to 100. The added work for measuring α should not be overwhelming, since
106
more than enough measurements are recorded. The additional work is mathematical.
We would like to point out here that including α in the model may very well change
the values of other parameters, in particular that of b4 .
From Corollary 3.7.1, we know that the positive steady state may be unstable if
√
√
γ > 21 (11 + 5 5)a1 , which is equivalent to b4 b6 b7 > 12 (11 + 5 5)b21 b2 . In view of the fact
that the values of b1 (from 0.0001 to 0.0565) and b4 (from 3.51E-08 to 3.73E-04) change
significantly from one person to another, we see that the likely candidates for having
sustainable oscillatory glucose and insulin levels are subjects that have high values of
b4 (insulin-dependent glucose disappearance rate) and low values of b1 (the spontaneous
or insulin-independent glucose first order disappearance rate). That is, those subjects
who can not process their glucose quick enough when insulin level is low but respond
very well with added insulin.
Roughly speaking, when the delay is short (less than 60 minutes), we probably
will not see any sustainable oscillatory solutions. However, for large enough delays, this
may be possible. In such cases, the larger the half-saturation constant (i.e., 1/α), the
more likely that the steady state is to be unstable. Even when the delay is small and the
steady state is stable, the solutions may converge to the steady state in an oscillatory
way. This can give rise to subsequent peaks of glucose and insulin, as observed in the
experiment of De Gaetano and Arino [23].
If α = 0, then we know the positive steady state is always locally stable. If
α > 0, the positive steady state remains stable for small and medium delays. Only
for large values of delay and appropriate values of α, the positive steady state may
lose its stability. Therefore, delay or the saturating glucose uptake functional response
(G/(αG + 1)) alone will not destabilize the steady state. In other words, for model
107
(3.3.1), oscillatory solutions resulted from delayed suitable nonlinear glucose uptake
mechanism. Refer [79], [84] and Chapter 2 and Chapter 4. This is in striking contrast
to the well known predator-prey dynamics (see [54]), where either alone will be enough
to produce periodic solutions.
A potential application of our study here is to find better ways of delivering
insulin and timing of the intake of glucose. Previous studies are largely done on linear
compartment models ([80], [81]) or on the minimal type models [36]. Our theoretical and
simulation work shows that in clinic applications, we can safely assume that after about
40 minutes, solutions are close and stay close to steady state levels. Another possible
usage of our work is to design effective ways to estimate the involved parameters for
clinic applications. For minimal model, such effort is documented in [24]. The recent
development of PET technology provides a possibility of effectively monitor subjects’
glucose and insulin levels through noninvasive method [18]. This should be very helpful
in estimating parameters for individuals in order to design proper controls for their
glucose and insulin levels.
CHAPTER 4
The Effects of Active β-Cells: A Preliminary Study
1. Introduction
It is known that the β cells are the only hormone that secretes insulin ([89]).
Dysfunctional β cells can lead to diabetes ([9]) due to the plasma glucose can not
be utilized sufficiently and cause hyperglycemia. Hyperglycemia is known to induce
insulin resistance ([71]) and a deficit in the mass of β cells, reduced insulin secretion
[53]. This supports the hypothesis that a primary insulin secretory defect that causes
hyperglycemia could lead to insulin resistance and diabetes via increased glucose levels
([9]). It is believed that insulin resistance can cause β-cell defects, and hence diabetes
either by overworking the β cells or by toxic effects of hyperglycemia on the β cells ([9]).
The β cells are contained in the Langerhans islets through out the pancreas.
(Refer to Figure 3.1 for Langerhans islet.) The insulin is contained in the granules in
the β cells. There are about one million of the Langerhans islets and each islet contains
approximately three hundreds of β cells. Each β cell contains around one thousand of
granules. The distribution is very complex ([85]).
The formation and growth (via neogenesis or replication) and death of the β cells
are still not completely clear ([9], [35], [28], [85] and [12]). It is believed that the β cells of
each individual, shortly after his/her birth, do not replicate. When a β cell is damaged,
109
it will not be able to function as it did originally. If one’s β cells are damaged in large
amounts, he/she would have to suffer diabetes due to β cell dysfunction. However, this
hypothesis is challenged by S Bonner-Weir [12] in 2000. The new perspective ([12],
2000) assumes that the β-cells can be replicated and neogenesis, which is supported by
in − vivo and in − vitra experiments ([13], [14], [34], [82] and [49]).
The purpose of this chapter is to study the effects of the β-cell mass to the
glucose-insulin endocrine regulatory system. Due to it is still not clear if the β-cells can
replicated or neogenerated (for example, from stem cells) ([28], [12]), we consider only
the active β-cell mass in the dynamics. Next section will introduce the current research
status and we present a model taking active β-cell mass as a factor in the glucose-insulin
regulatory system in Section 3. In Section 4 we will exhibit our numerical simulation.
Then we will discuss our simulation results in Section 5.
2. Current Research Status
The first study on β-cell mass kinetics was given in 1995 by D. T. Finegood,
L. Scaglia and S. Bonner-Weir ([35], 1995). Based on experimental data, a simple
single variable (β-cell mass) system was developed as follows to study the β cell mass
growth and renewal (neogenesis or replication) vs. death.
β ′ (t) = REP (t) + N EO(t) − DEAT H(t)
where
• REP (t) - replication rate = 16.10e−0.065t + 2.31
• N EO(t) - islet neogenesis rate
110
• DEAT H(t) - death rate.
The net growth or death rate is given by
N EO(t) − DEAT H(t) = β ′ (t) − 16.10e−0.065t + 2.31
The increased β-cell death rate is consistent with experiment reports of increased islet
cell apoptosis ([9] and its references). However, this model consists of only β cell mass as
its variable. The β-cell mass is related to the glucose concentration level ([82], [49] and
[85]) and the glucose concentration level is regulated by the insulin secreted from the
β celss. Obviously it is necessary to consider glucose, insulin and β-cell mass together
as a whole system when study the pathways to the Type 2 diabetes which is caused by
β-cell dysfunction and insulin resistance ([9], [85] and their references).
In 2000, B. Topp, K. Promislow, G. De Vries, R. M. Miura and D. T. Finegood
([85]) introduced a novel model, according to Mari ([60]), taking β cell mass as a
dynamically growing variable together with the glucose and insulin concentrations. The
model takes following form.
where




G′ (t) = R0 − EG0 G(t) − SI I(t)G(t),















σβ(t)G2 (t)
′
I
(t)
=
− kI(t),


α + G2 (t)















 β ′ (t) = (−d0 + r1 G(t) − r2 G2 (t))β(t)
• SI (mlµU −1 d−1 ) is the total insulin sensitivity.
• EG0 (d−1 ) is the total glucose effectiveness at zero insulin.
(4.2.1)
111
Table 4.2.1.
Parameters of the Model 4.2.1
Parameters Values
Units Parameters
Values
Units
−1 −1
SI
0.72 mlµU d
k
432
d−1
EG0
1.44
d−1
d0
0.06
d−1
R0
864
mgdl−1
r2 0.24x10−5 mg−2 dl2 d−1
σ
43.2 µUml−1 d−1
r1 0.84x10−3 mg−2 dl2 d−1
2 −2
α
2000
mg dl
• R0 (mgdl−1 ) is the net rate of production at zero glucose.
• The term
σβ(t)G2 (t)
α+G2 (t)
indicates the insulin secretion. All β cells are assumed to
secrete insulin at the same maximal rate σ(µU ml−1 d−1 ). The term kI stands for
the insulin clearance by the liver, kidneys and insulin receptors at the rate k(d−1 )
• The term (−d0 + r1 G(t) − r2 G2 (t)) is the β cell growth/death rate which is dependent on glucose concentration G(t). d0 (d−1 ) is the death rate at zero glucose
concentration.
All the parameter values of the model (4.2.1) are listed in Table 4.2.1.
B. Topp, K. Promislow, G. De Vries, R. M. Miura and D. T. Finegood [85] studied
the above model assuming β-cell reaction to the insulin-glucose slowly, namely, β-cell
dynamics moves slowly and try to understand the complex dynamics regarding β-cell
mass, insulin and glucose. This model predicts, by the numerical study results, that
rapidly falling insulin sensitivity leads to the onset of hyperglycaemia and then β-cell
dysfunction ([85], [9]). Another result induced from this model is that the increases in
insulin sensitivity mediated by agents such as rosiglitazone prevents the plasma glucose
levels to go higher and thus preserves β cell functions from overworking ([85], [9]).
112
3. Active β-Cell Model
While the model (4.2.1) exhibits some of the physiological meanings, the model
leaves few places to improve. The authors assume the β-cell mass growth-death rate
take a quadratic function due to that the growth rate was shown to be non-linear
([82], [49], [85]. The authors stated that extremely high hyperglycemia may reduce
the replication of the β cells ([85]). But the lower glucose concentration level does not
trigger the β cells to release insulin. This appears not considered by the βGI mode
(4.2.1. We present a new model to consider the effect of β-cell mass dynamics in the
glucose-insulin regulatory system. Instead of considering the whole β-cell mass, we only
let the healthy and active β cells be involved in the our model. This is due to the fact
that if a β cell has been damaged, it does not secret insulin even when the glucose
concentration level is high.





G′ = Gin − f2 (G(t)) − f3 (G(t))f4 (I(t)) + f5 (I(t − τ2 )),














I ′ = σf1 (G(t − τ1 ))β(t) − p(I(t)),

















 β ′ = g(G(t − τ1 )) − kβ(t),
(4.3.1)
where the initial condition I(0) = I0 > 0, G(0) = G0 > 0, G(t) ≡ G0 for all t ∈ [−τ1 , 0]
and I(t) ≡ I0 for t ∈ [−τ2 , 0] with τ1 , τ2 > 0. In model (4.3.1), G (mg/dl/min) and
I (µU/ml/min) stands for the glucose and insulin concentration, respectively. The β
(mg) stands for the mass of active β cells which will release insulin into the bloodstream
when stimulated by the glucose concentration level. Additionally, in the third equation,
the term g(G(t − τ1 )) mimics that the transient and mild glucose can increase active β
113
Liver
Delay
Glucose Infusion:
meal ingenstion,
oral intake,
enteral nutrition,
constant infusion
Liver converts
glucagon and
glycogen to
glucose
Insulin Controls
Hepatic
glucose production
Glucagon
secrete
Pancreas
Active
ß-cell mass
Insulin secretion
Glucose
utilization
Glucose
Insulin
Insulin helps cells consume glucose
Figure 4.3.1.
Insulin production
Glucose Controls
glucagon secretion
Glucose Controls
insulin secretion
Insulin independent:
brain cells, and
others
Delay
Insulin dependent:
fat cells, and
others
Insulin clearance
Glucose
production
α-cells
Insulin degradation:
receptor, enzyme, and
others
Glucose-Insulin with Active β-cell Interaction Diagram
The divide lines (dash-dot-dot) indicate insulin controlled hepatic glucose production with time delay;
the dash-dot lines indicate the insulin secretion from the active β-cells stimulated by elevated glucose
concentration level with time delay; the dashed lines indicate low glucose concentration level triggers αcells in pancreas to release glucagon; and the dot line indicates the insulin accelerates glucose utilization
in cells.
114
cell mass, but hyperglycemia (216 mg [85]) will cause β cells to overwork and damage
the cells. The term kβ(t) indicates the active β cell mass decreases at the rate k. The
term σf1 (G(t − τ1 ))β(t) in the second equation stands for insulin secretion from the
pancreas and is proportional to the active β-cell mass. σ > 0 (mg−1 ) is a parameter
indicates how efficient the active β-cells release the insulin in to blood stream. Insulin is
stored in β-cell granules. Glucose is the primary stimuli of insulin secretion from the β
cells. The delay is due to the complex electric processes inside of islet (refer to Chapter
2). The function f1 (G(t)), in sigmoidal shape, assumes the same condition as that in
the model (2.3.1) in Chapter 2. The term p(I(t)) indicates the insulin degradation and
p(0) = 0, p(x) > 0 and p′ (x) > 0 for x > 0. All terms in the first equation assume the
same physiological meanings and conditions as that in the model (2.3.1) in Section 3 of
Chapter 2. We do not repeat here.
The function g(G) takes the following form and the shape is shown in Figure
4.3.2.
g(G) =
γG5 (1 − G/e)
bG + 1
(4.3.2)
where γ = 1E − 6, b = 10E7 and e = 2.16E4.
4. Numerical Simulations
In this section, we show some simulation results of the GIβ model with a group
of particular functions (2.2.2) – (2.2.6) and parameters given in Table 2.2.1, 2.2.2 and
2.7.1.
Through out this section, when we let parameters be fixed, they always assume following values. τ1 = 5(min), τ2 = 36(min), Gin = 0.54(mg/(dl·min)), di =
115
8000
6000
4000
2000
0
0
5000
10000
15000
20000
G
-2000
Figure 4.3.2.
Function g(G) in GIβ-Model
where γ1 = 1E − 6, b = 10E7, and e = 2.16E4.
0.03846(µU/(ml·min), k = 0.1(mg/min), and σ = 0.005(mg−1 ).
4.1. Insulin Response Delay and Hepatic Glucose Production Are Critical for Sustain Insulin Secretion Oscillations. Insulin response time delay and
hepatic glucose production are critical for insulin secretion oscillation to sustain. Our
numerical simulation (Figure 4.4.1) indicate that the insulin secretion oscillations do
not take place if (a) the insulin response time delay is assumed to be zero (τ1 = 0) or
(b) there is no hepatic production (f5 ≡ 0).
4.2. Insulin Response Time Delay τ1 as a Bifurcation Parameter. In
last paragraph our simulation shows the insulin response time delay τ1 is critical to the
insulin secretion oscillations to sustain. In this paragraph, we take τ1 as a bifurcation
parameter and let τ1 change from 0 to 20 with step size 0.01 (min.) and try to find out
the bifurcation point at which a periodic solution can be bifurcated out. We let τ2 = 36,
Gin = 0.54, di = 0.03846, k = 0.1, and σ = 0.005 be fix and let τ1 = 5 change from 0.01
116
18
20
16
14
15
12
I
I
10
10
8
6
5
4
2
0
400
0
400
100
300
80
200
60
100
300
40
100
80
200
60
40
100
20
beta
0
0
20
G
Figure 4.4.1.
beta
0
0
G
Orbits of (G, I, β) of GIβ model
Left: the case τ1 = 0; right: the case f5 ≡ 0. These indicate that both hepatic production and insulin
response looks very necessary to the sustained oscillation.
to 20 (min). We observe from our numerical simulation (shown in Finger 4.4.2) that
when the delay τ1 changes from 0 to 20, when τ1 is sufficiently small (τ1 < 1.5), the
steady state of the model (4.3.1) is stable. When τ1 increases large enough (τ1 > 2), the
steady state becomes unstable and the model (4.3.1) has a stable periodic solution – the
insulin oscillation takes place. That means, there is one bifurcation point τ10 ∈ (1.5, 2.0)
for insulin response time delay parameter τ1 such that the steady state is always unstable
for τ1 > τ10 and thus the insulin oscillation sustains. The left of Figure 4.4.2 shows the
bifurcation diagram when τ1 changes from 0 to 20 and the right figure is the periods
of the periodic solutions when τ1 changes. Approximately speaking, the longer delay of
the insulin response to the glucose, the larger the period of the secretion oscillation.
4.3. Glucose Infusion Rate Gin as a Bifurcation Parameter. Let the
glucose infusion rate Gin vary from 0 to 3.0 (mg/dl·min) and all other parameters
be fixed. The numerical simulation indicates that there exists a bifurcation point
G0in ∈ (1.65, 1.75) such that the steady state is unstable when Gin ∈ [0, G0in ] and stable
117
150
250
100
50
200
0
2
4
6
8
10
12
14
16
18
20
100
50
0
0
2
4
6
8
10
12
14
16
18
20
Period of periodic solutions
0
150
100
1000
50
500
0
0
2
4
6
8
10
tau1
12
14
Figure 4.4.2.
16
18
20
0
0
2
4
6
8
10
tau1
12
14
16
18
20
Bifurcation diagram of τ1 ∈ [0, 20]
Left: bifurcation diagrams (top - glucose; middle – insulin and bottom – active β-cell mass) of τ1 ∈
[0, 20] (min); right: periods of periodic solutions
when Gin > G0in . This shows that if the glucose infusion rate is not too high, then
the insulin oscillations sustain. When the infusion rate is large enough, the oscillations
disappear. This confirms again that the insulin secretion ultradian oscillations do not
take place in the intravenous glucose tolerance test (IVGTT) as a big bolus (0.33 mg
per kg of body weight [22] of glucose infusion. See the left figure in Figure 4.4.3.
Another agreement to the physiologic glucose-insulin regulatory system is exhibited in the right hand side figure in Figure 4.4.3. The top line is the time differences
(11 to 13 minutes) between the peaks of the glucose concentration and the peaks of the
active β-cell mass. The middle line is the time differences (5 to 7 minutes) between the
active β-cell mass and the glucose concentration. The bottom line is the time differences
(18 to 20 minutes) between the active β-cell mass and the insulin concentration. This
can be restated as follows. When the glucose infusion rate Gin varies in the range that
the insulin secretion oscillation sustain, in one cycle, the active β cells mass reaches its
peak about 11 to 13 minutes after the glucose concentration reaches its peak. Then it
takes about 5 to 7 minutes that the insulin concentration reaches its peak. Therefore
118
150
20
100
18
50
Bifurcation diagram
0
16
0
50
100
150
200
250
300
14
60
12
40
10
20
0
8
0
50
100
150
200
250
300
6
80
60
4
40
2
20
0
0
50
100
150
Gin
200
Figure 4.4.3.
250
300
0
0
50
100
150
200
250
300
Bifurcation diagram of Gin ∈ [0, 3.0]
Left: bifurcation diagram when Gin changes from 0 to 3.0(mg/dl·min). If Gin < 1.65, the oscillations
are sustained. Right: in one cycle, the active β cells mass reaches its peak about 11 to 13 minutes
after the glucose concentration reaches its peak (the middle line). Then it takes about 5 to 7 minutes
that the insulin concentration reaches its peak (the bottom line). Therefore it takes around 18 to 20
minutes from increased glucose concentration to the insulin is released (the top line).
it takes around 18 to 20 minutes from increased glucose concentration to the insulin is
released.
Several periodic solutions are shown in the left figure in Figure 4.4.4 when Gin
changes from 0 to 3.00 (mg/dl·min) while other parameters are fixed. The right hand
side figure shows the periods of periodic solutions decrease from around 138 minutes to
110 minutes when the Gin increases from 0 to G0in .
4.4. Peaks of Oscillations in One Cycle. Our simulation results exhibit that
in one oscillation cycle, the glucose concentration peaks before the active β-cell mass
peaks; the active β-cell mass peaks before the insulin concentration peaks. On the other
hand, the glucose concentration bottoms before the active β-cell mass does; the active
β-cell mass bottoms before the insulin concentration does. Figure 4.4.5 exhibits the
shifts among these three variables.
For example, the figure in the right hand side of Figure 4.4.3 demonstrates,
119
140
120
beta
500
0
0
30
5
40
10
50
15
Period of periodic solutions
100
1000
80
60
40
60
20
70
25
80
30
100
40
110
45
I
20
90
35
50
120
Figure 4.4.4.
G
0
0
50
100
150
Gin
200
250
300
Periodic solutions and periods when Gin ∈ [0, 3.0]
Left: periodic solutions; right: periods of periodic solutions. The periods decreases from 138 to 110
(min) along with the increase of Gin from 0 to G0in ∈ (1.65, 1.75) (mg/(dl·min)).
when the glucose infusion rate Gin changes from 0 to 3.00 (mg/(dl·min)), the glucose
concentration peaks around 11-13 minutes before the active β-cell mass peaks and the
active β-cell mass peaks around 5-7 minutes before the insulin concentration peaks.
This is in agreement with the physiological stimulation chain, that is, increased
glucose concentration level triggers active β cells to release insulin, and then the insulin
helps the cells to utilize the plasma glucose. These shifts among the peaks also reflect
the time delay of the insulin response to increased glucose concentration.
4.5. β-cell Deactivation Rate k ∈ [0.01, 2] as a Bifurcation Parameter.
The parameter k is the rate at which the β-cell deactivated. After the β cells release
the insulin contained in the granules, the cells have to reproduce insulin before next
secretion. We call this as β-cell deactivation. From the Figure 4.4.6, we can observe
that when k ∈ [0.01, 2] and τ1 = 5, τ2 = 36, Gin = 0.54, di = 0.03846 and σ = 0.005,
the insulin oscillation always sustain but the amplitudes and the periods of the periodic
solutions vary along with the changes of the β-cell deactivation rate parameter k from
120
150
mg/dl
G
500
100
450
50
400
0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
350
1500
beta
300
mg
1000
250
500
200
0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
150
150
mU/ml
I
100
100
50
50
0
0
500
1000
1500
2000
2500
3000
time (min)
3500
Figure 4.4.5.
4000
4500
5000
0
2050
2100
2150
2200
time (min)
Peaks of Oscillations in One Cycle
Left: a periodic solution of Model (4.3.1) when τ1 = 5(min), τ2 = 36(min), Gin = 0.54(mg/(dl· min)),
di = 0.03846(µU/(ml· min)), k = 0.1(mg/min) and σ = 0.005. G - top line; I - bottom line; and β middle line. Right: In one cycle, the glucose peaks before active β-cell mass peaks (middle line) and
the active β-cell mass peaks before Insulin concentration peaks (bottom line). (G and I are recalled
for comparison.)
0.01 to 2. It appears that the best range for k is around 0.05 so that the glucose
concentration level of subjects is within the normal range.
4.6. Parameter σ as a Bifurcation Parameter. We take the parameter
σ(mg−1 ) as the stimulation efficiency coefficient as its value controls how much insulin
is released from β cell mass. According to our numerical simulation result (shown
in the left figure in Figure 4.4.7), The larger the parameter σ, the higher the insulin
concentration, the less need of the β-cell mass and the lower the glucose concentration.
The other figure in Figure 4.4.7 indicates when the parameter σ changes from 0.001
to 0.1, the periods of periodic solutions increase from 119 minutes to 128 minutes
approximately. In particular, the increase of the periods near σ = 0.0001 is dramatic.
The Figure 4.4.8 simply shows, from two different directions, several limit cycles in this
parameter range.
121
150
140
100
120
50
100
0
0.5
1
1.5
2
30
20
10
0
0
0.5
1
1.5
80
2.5
2
2.5
Period of periodic solutions
0
800
0
0.5
1
1.5
2
2.5
0
0.5
1
1.5
2
2.5
0
0.5
1
1.5
2
2.5
140
120
100
80
140
600
120
400
100
200
0
0
0.5
1
1.5
2
80
2.5
Figure 4.4.6.
Bifurcation diagram of k ∈ [0.01, 2]
Left: bifurcation diagram; right: periods of periodic solutions
200
128
150
127
100
50
126
0
0.002
0.004
0.006
0.008
0.01
0.012
40
30
20
10
0
0
0.002
0.004
0.006
0.008
0.01
0.012
Period of periodic solutions
Bifurcation diagram
0
125
124
123
122
200
121
150
100
120
50
0
0
0.002
0.004
0.006
sigma
0.008
Figure 4.4.7.
0.01
0.012
119
0
0.002
0.004
0.006
sigma
0.008
Bifurcation diagram of σ ∈ [0.0001, 0.1]
Left: bifurcation diagram; right: periods of periodic solutions
0.01
0.012
122
1200
1000
1000
800
beta
1200
beta
800
600
400
600
200
400
0
20
200
40
0
0
140
60
10
120
80
20
100
80
100
30
60
40
20
120
40
I
140
G
G
Figure 4.4.8.
15
10
35
I
Limit Cycles when σ ∈ [0.0001, 0.1]
150
600
100
500
50
400
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
30
beta
0
0.02
5
0
30
25
20
300
200
20
100
10
0
0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0
5
600
10
400
15
200
0
0.02
20
25
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
I
110
100
Figure 4.4.9.
90
80
70
60
50
40
30
G
di
There is no bifurcation when di ∈ [0.005, 0.01]
Left: the amplitude of the periodic solutions; right: the limit cycles
4.7. The Changes of Insulin Degradation Rate di ∈ [0.025, 0.1] Do Not
Affect the Oscillations. Similar to the parameter σ ∈ [0.0001, 0.1], the changes of
insulin degradation rate di ∈ [0.025, 0.1] has no impact on stability of the stead state.
The insulin oscillations sustain. (Refer to Figure 4.4.9.)
123
5. Discussion
In this chapter, we tried to model the glucose-insulin endocrine regulatory system
involving the active β-cell mass. we performed numerical simulations for the model
(4.3.1) and have following observations.
• We confirmed that both the insulin response delay and the hepatic glucose production are critical to the oscillatory insulin secretion from the active β cells.
• We observed that the insulin response time delay needs to be big enough (e.g.,
τ1 ≥ 2 for the insulin secretion oscillations to be sustained. Besides, the longer
delay of the insulin response to the glucose, the larger the period of the oscillation.
• The glucose infusion rate has to be moderate to observe the insulin secretion
oscillations. When the Glucose infusion rate is extremely high, for example Gin =
2.16 mg/(dl·min), the insulin secretion becomes damped. This confirms again
that in IVGTT the hepatic glucose production is insignificant comparing to the
large glucose infusion.
• We observed that the glucose concentration peaks around 11-13 minutes before
the active β-cell mass peaks and the active β-cell mass peaks around 5-7 minutes
before the insulin concentration peaks when Gin changes. This is in agreement
with the physiological stimulation chains. Last, according to our numerical simulations, in the physiologically meaningful ranges, the insulin degradation rate
di ∈ [0.025, 0.1], the stimulation efficiency coefficient σ ∈ [0.0001, 0.1] and the βcell deactivation rate k ∈ [0.01, 2] do not affect the insulin secretion oscillations.
According to simulations we performed, no insulin secretion pacemaker is observed for the ultradian oscillations. More investigation is need to find out if the β-
124
100
74.9611
G
G
mg/dl
mg/dl
80
60
40
74.9611
74.9611
0
500
1000
1500
2000
2500
3000
3500
4000
4500
74.9611
3600
5000
300
3800
4000
4200
4400
4600
4800
5000
206.18
beta
beta
200
mg
mg
206.175
100
0
206.17
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
206.165
3600
15
3800
4000
4200
4400
4600
4800
I
mU/ml
mU/ml
I
10
5
5000
8.2859
0
500
1000
1500
2000
2500
3000
time (min)
3500
Figure 4.5.1.
4000
4500
5000
8.2859
8.2859
8.2859
3600
3800
4000
4200
4400
time (min)
4600
4800
5000
Possible β-cell pulsatile oscillation?
Left: solutions of Model (2.3.1 when τ1 = 5, τ2 = 36, Gin = 0.54, di = 0.03846, k = 0.1, and
σ = 0.005); right: Zoomed in figure which shows the active β-cell mass oscillates in a period around
50 minutes while the glucose concentration G and the insulin concentration I do not oscillate.
cells-selves can be the pacemaker only or some other factors have to be considered.
Nevertheless, an interesting numerical observation of the Model (4.3.1) (displayed in
Figure 4.5.1 is that the active β-cell mass oscillates in an almost invisible range while
both glucose and insulin concentrations do not oscillate. The period is around 50 minutes.
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