International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064

Index Copernicus Value (2015): 78.96 | Impact Factor (2015): 6.391

Volume 6 Issue 2, February 2017 www.ijsr.net

Licensed Under Creative Commons Attribution CC BY

Mathematical Model for Diabetes Use Glucose Tolerance Test (GTT)

Usha Rani M.1, Komahan G.2

1Research Scholar, A.V.V.M. Sri Pushpam College (Autonomous), Poondi, 613 503 Thanjavur (India)

2Associate Professor and Head, Department of Mathematics, A.V.V.M. Sri Pushpam College (Autonomous), Poondi, 613 503 Thanjavur (India)

Abstract: Diabetes is disease of metabolism whose essential feature is excessive sugar in the blood and urine. Second order differentialequation has been formulated to describe the performance of BGRS during GTT and its solution has been shown over here. Interstingpoints in that the sociological factors play an important role in the Blood Glucose Rogulatory system.

Keywords: Mathematical Modeing, Diabetes, Second order differential equation, and Glucose Tolerance Test

1. Introduction

Diabetes

Most of the food we eat is turned into glucose (sugar) for our bodies to use for energy. The pancreas, an organ near the stomach, makeser hormone called insulin to help glucose get into our body cells. When you have diabetes, your body either doesn’t make enough insulin or can’t use its own insulin very well. This problem causes glucose to build up in your blood.

Diabetes means that a person’s blood sugar is too high. Your blood always has some sugar in it because the body needs sugar for energy to keep you going. But too much sugar in the blood can cause serious damage to the eyes, Kidneys, nerves and heart.

Signs and symptoms of Diabetes

You may recall having some of these signs before you found out you had diabetes: Being very thirsty. Urinating a lot – often at night Having blurry vision from time to time. Feeling very tired much of the time Losing weight without trying Having very dry skin Having sores that are slow to heal.

Types of Diabetes

There are two main types of diabetes Type 1 Diabetes Type 2 Diabetes Another type of diabetes appears during pregnancy in some women. It’s called gestational diabetes.

Type 1 Diabetes One out of 10 people with diabetes has type 1 diabetes. These people usually find out they have diabetes when they are children or young adults. People with type 1 diabetes

must inject insulin every day to live. The pancreas of a person with type 1 make little (or) no insulin.

Type 2 Diabetes Most people with diabetes 9 out of 10 have type 2 diabetes. The pancreas of people with type 2 diabetes keeps making insuling for some time, but the body can’t use it very well. Most people with type 2 find out about their diabetes after age 30 or 40.

Certain risk factors make people more likely to develop type 2 diabetes. Some of these are. A family history of diabetes. Lack of exercise. Weighing too much Being of African American. Being of African American, American India, Asian /

pacific Island heritage Gestational diabetes history.

Glucose Tolerance Lest (GTT) To diagnose diabetes is by the Glucose Tolerance test (GTT) in which the patient is called the hospital after overnight fasting. On his arrival he is given a large dose of glucose (the form of sugar in which it occurs in the blood streem) and then several measurements of the concentration of glucose in the patient’s blood is taken during next 3 to 5 hours. Based on his measurements, the physician makes the diagnosis of diabetes which obviously depends on his interpretation of the results.

Formulation of Mathematical Model: We formulate an appropriate model in two steps.

Step 1 Background Information on the model We assume that the following two concentrations adequately describe the performance of blood glucose regulatory system (BGRS).(i) Concentration of glucose in the Blood (g). (ii) Net Hormonal concentration (h).

Paper ID: 31011701 430

Introduction

Most of the food we eat is turned into glucose (sugar) for our bodies to use for energy. The pancreas, an organ near the stomach, makeser hormone called insulin to help glucose get into our body cells. When you have diabetes, your body either doesn’t make enough insulin or can’t use its own

insulin very well. This problem causes glucose to build up

Diabetes means that a person’s blood sugar is too high.

Your blood always has some sugar in it because the body needs sugar for energy to keep you going. But too much sugar in the blood can cause serious damage to the eyes, Kidneys, nerves and heart.

Signs and symptoms of Diabetes

You may recall having some of these signs before you found

often at night Having blurry vision from time to time. Feeling very tired much of the time

must inject insulin every day to live. The pancreas of a person with type 1 make little (or) no insulin.

Type 2 Diabetes Most people with diabetes 9 out of 10 have type 2 diabetes. The pancreas of people with type 2 diabetes keeps making insuling for some time, but the body can’t use it very well.

Most people with type 2 find out about their diabetes after age 30 or 40.

Certain risk factors make people more likely to develop type 2 diabetes. Some of these are. A family history of diabetes. Lack of exercise. Weighing too much Being of African American. Being of African American, American India, Asian /

pacific Island heritage Gestational diabetes history.

Glucose Tolerance Lest (GTT) To diagnose diabetes is by the Glucose Tolerance test (GTT) in which the patient is called the hospital after overnight fasting. On his arrival he is given a large dose of glucose

International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064

Index Copernicus Value (2015): 78.96 | Impact Factor (2015): 6.391

Volume 6 Issue 2, February 2017 www.ijsr.net

Licensed Under Creative Commons Attribution CC BY

Since, the variable g and h change with time, we consider g and has dependent variables while t(time) as the independent variable. From the elementary concentration of the biological facts, stated above, the logistic law govering the performance of BGRs may be written as

1( , ) ( )dg E g h F tdt

(1)

2 ( , )dh E g hdt

(2)

Where E1 and E2 are the same functions of g and h, while F(t) is the external rate at which the Blood Glucose concentration (BGC) is being increased.

2. Mathematical Model use Glucose Tolerence Test

Formulate a second order differential equations model to describe the performance of BGRS during a GTT. Let g0 and h0 be the optimal value of g and h respectively. We set g = G – G0 and h = H – H0

Substituting these values of g and h in equations (1) and (2) and using Taylor’s Expansion, we get.

1 0 10 0

1 1( , )odg E EE G H G H ddt g g

+ F(t)

(3)

2 0 0 20 0

2 2( , )dh E EE G H G h ddt g g

(4)

Where 1

0

Eg

, and 2Eg

denotes g = G0 and h = H0

and d1, d2 contains terms of second and higher powers in g and h. (i) Assume that if E1(G0, H0) = 0 and E2(G0, H0) =0, because

it is assumed that g and h have their optimal values G0and H0 respectively by the time the fasting patient arrives at the hospital and (ii) E1 and E2 being small quantities may be neglected and g and h are very small.

Substituting these two conditions in equation (3) and (4) we get

1 1

00

E Edg G Hdt g h

+ F(t) (5)

2 2

00

E Edh G Hdt g h

(6)

To find the value of

1

0

Eg

, 1

0

Eh

, 2

0

Eg

and 2

0

Eh

.

Case (i)

We consider g >0, h =0 (excessive glucose) ie 0.dgdt

The equation (5) implies that 1

0

Eh

may be negative.

Case (ii)

If h >0, g =0 (excessive insulin) ie., 0dhdt

.

Then the equation (5) implies that 1

0

Eh

may be

negative.

Similarly, 1

0

Eh

and 2Eh

also be negative.

Therefore equation (5) and (6) can be written as dgdt

= - A1g – A2h + F(t) (7)

dhdt

= A3g – A4h (8)

Where are all positive constants. A1, A2, A3 and A4.

Since it is the BGS that can be measured easily therefore we attempt to eliminate h if possible, between equations (7) and (8).

Differenciate equation (7), (8) with respect to t. 2

1 22

d g dg dh dFA Adt dt dt dt

(9)

2

3 42

d h dg dhA Adt dt dt

(10)

Substitute - equation (7) the value of dhdt

in equation (9) we

get 2

1 2 3 42 ( )d g dg dFA A A g A hdt dt dt

2

1 2 3 2 42

d g dg dFA A A g A A h dtdt dt (11)

Substitute – the value of A2h in (7) in (11) we get 2

1 2 3 1 42 ( ( ) )d g dg dg dFA A A g A g F t A dtdt dt dt

2

1 4 2 3 1 4 42 ( ) ( ) ( )d g dg dFA A g A A A A F t A dtdt dt

2

202 2

d g wdt

= m(t) (12)

Where 2 = (A1 + A4), 20w = (A2A3+A1A4) and m(t) = F(t)

A4+ dF dt . 2

202 2 0

d g dg w gdt dt

= m(t) (13)

Where m(t) is identically zero. Equation (13) is known as second order differential equation with constant coefficient

Paper ID: 31011701 431

describe the performance of BGRS during a GTT. be the optimal value of g and h respectively.

and h = H – H0

Substituting these values of g and h in equations (1) and (2) and using Taylor’s Expansion, we get.

dg E E dg E E 1 1 1 1 1 1 1 1dg E E dg E E dg E E dg E E1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1dg E E dg E E dg E E dg E E dg E E dg E E dg E E dg E E1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 0 1 1 0 1E G H G H d E G H G H d1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1

dg E E

dg E Edg E EE G H G H ddg E E dg E EE G H G H ddg E E

dg E E dg E E

dg E E dg E E

1 1 1 1

1 1 1 11 1dg E E1 1 1 1dg E E1 1

1 1dg E E1 1 1 1dg E E1 1E G H G H d

E G H G H d E G H G H d

E G H G H d1 1E G H G H d1 1 1 1E G H G H d1 1

1 1E G H G H d1 1 1 1E G H G H d1 11 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1

1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1dg E E dg E E

dg E E dg E Edg E EE G H G H ddg E E

dg E EE G H G H ddg E E dg E EE G H G H ddg E E

dg E EE G H G H ddg E E 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 11 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1

1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1dg E E dg E E dg E E dg E E

dg E E dg E E dg E E dg E Edg E E dg E E dg E E dg E E

dg E E dg E E dg E E dg E E1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1

1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E

dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1

1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1dg E E

dg E E dg E E dg E E dg E E dg E E dg E E dg E E

dg E E dg E E dg E E dg E E dg E E dg E E dg E E dg E E1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1

1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 1 1 1dg E E1 11 0 1 1 0 11 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1E G H G H d E G H G H d E G H G H d E G H G H d1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1E G H G H d E G H G H d E G H G H d E G H G H d1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1

dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E EE G H G H d E G H G H d E G H G H d E G H G H ddg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E EE G H G H d

E G H G H d E G H G H d

E G H G H d E G H G H d

E G H G H d E G H G H d

E G H G H d1 1E G H G H d1 1

1 1E G H G H d1 1 1 1E G H G H d1 1

1 1E G H G H d1 1 1 1E G H G H d1 1

1 1E G H G H d1 1 1 1E G H G H d1 1

1 1E G H G H d1 1dg E EE G H G H ddg E E dg E EE G H G H ddg E E

dg E EE G H G H ddg E E dg E EE G H G H ddg E E

dg E EE G H G H ddg E E dg E EE G H G H ddg E E

dg E EE G H G H ddg E E dg E EE G H G H ddg E E1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1

1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1

1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1

1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E

dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E

dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E

dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1

1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1

1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1

1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 1 1 1dg E E1 1E G H G H d1 1dg E E1 11 0 1 1 0 1 1 0 1 1 0 11 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1E G H G H d E G H G H d E G H G H d E G H G H d E G H G H d E G H G H d E G H G H d E G H G H dE G H G H d E G H G H d E G H G H d E G H G H d E G H G H d E G H G H d E G H G H d E G H G H dE G H G H d E G H G H d E G H G H d E G H G H d E G H G H d E G H G H d E G H G H d E G H G H d1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1 1 0 1E G H G H d1 0 1

dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E dg E EE G H G H ddg E E

1 0 1 1 0 1

1 0 1 1 0 11 0 1 1 0 1

1 0 1 1 0 11 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 11 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 11 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 11 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 11 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 11 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 0 0 0 0 1 0 1

1 0 1

1 0 1

1 0 1

0 0 0 0 0 0 0 0 1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1 dt g g dt g g dt g g dt g g

1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 11 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1dt g g dt g g dt g g

dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g1 0 1dt g g1 0 1 1 0 1dt g g1 0 1

1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1dt g g dt g g dt g g

dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g1 0 1dt g g1 0 1 1 0 1dt g g1 0 1

1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 11 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 11 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0dt g g

dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1

1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 11 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1 1 0 1dt g g1 0 1

+ F(t)

(3) dh E E dh E E 2 2 2 2 2 2 2 2dh E E dh E E dh E E dh E E2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2dh E E dh E E dh E E dh E E dh E E dh E E dh E E dh E E2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 0 0 2 2 0 0 2E G H G h d E G H G h d2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2

dh E E

dh E EE G H G h d E G H G h ddh E EE G H G h ddh E E dh E EE G H G h ddh E E

dh E E dh E E

dh E E dh E E

2 2 2 2

2 2 2 22 2dh E E2 2 2 2dh E E2 2

2 2dh E E2 2 2 2dh E E2 2E G H G h d

E G H G h d E G H G h d

E G H G h d2 2E G H G h d2 2 2 2E G H G h d2 2

2 2E G H G h d2 2 2 2E G H G h d2 22 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2

2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2dh E E dh E E

dh E E dh E Edh E EE G H G h ddh E E

dh E EE G H G h ddh E E dh E EE G H G h ddh E E

dh E EE G H G h ddh E E 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 22 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2

2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2dh E E dh E E dh E E dh E E

dh E E dh E E dh E E dh E Edh E E dh E E dh E E dh E E

dh E E dh E E dh E E dh E E2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2

2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E

dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2

2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2dh E E

dh E E dh E E dh E E dh E E dh E E dh E E dh E E

dh E E dh E E dh E E dh E E dh E E dh E E dh E E dh E E2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2

2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 2 2 2dh E E2 22 0 0 2 2 0 0 22 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2E G H G h d E G H G h d E G H G h d E G H G h d2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2E G H G h d E G H G h d E G H G h d E G H G h d2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2E G H G h d E G H G h d E G H G h d E G H G h dE G H G h d

E G H G h d E G H G h d

E G H G h d E G H G h d

E G H G h d E G H G h d

E G H G h d2 2E G H G h d2 2

2 2E G H G h d2 2 2 2E G H G h d2 2

2 2E G H G h d2 2 2 2E G H G h d2 2

2 2E G H G h d2 2 2 2E G H G h d2 2

2 2E G H G h d2 2dh E EE G H G h ddh E E dh E EE G H G h ddh E E

dh E EE G H G h ddh E E dh E EE G H G h ddh E E

dh E EE G H G h ddh E E dh E EE G H G h ddh E E

dh E EE G H G h ddh E E dh E EE G H G h ddh E E2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2

2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2

2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2

2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2E G H G h d

E G H G h d E G H G h d

E G H G h d E G H G h d

E G H G h d E G H G h d

E G H G h d E G H G h d

E G H G h d E G H G h d

E G H G h d E G H G h d

E G H G h d E G H G h d

E G H G h ddh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E

dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E

dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E

dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E dh E EE G H G h ddh E E2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2

2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2

2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2

2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 2 2 2dh E E2 2E G H G h d2 2dh E E2 22 0 0 2 2 0 0 2 2 0 0 2 2 0 0 22 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2E G H G h d E G H G h d E G H G h d E G H G h d E G H G h d E G H G h d E G H G h d E G H G h dE G H G h d E G H G h d E G H G h d E G H G h d E G H G h d E G H G h d E G H G h d E G H G h dE G H G h d E G H G h d E G H G h d E G H G h d E G H G h d E G H G h d E G H G h d E G H G h d2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2 2 0 0 2E G H G h d2 0 0 2

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 22 0 0 2 2 0 0 2

2 0 0 2 2 0 0 22 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 22 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 22 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 22 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 22 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 22 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 0 0 0 0 2 0 0 2

2 0 0 2

2 0 0 2

2 0 0 2

0 0 0 0 0 0 0 0 2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2 dt g g dt g g dt g g dt g g

2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 22 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2dt g g dt g g dt g g

dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2

2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2dt g g dt g g dt g g

dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2

2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 22 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2

2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 22 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0 0 0dt g g0 0dt g g

dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g dt g g2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2

2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 22 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2 2 0 0 2dt g g2 0 0 2

(4)

2 2E E2E2 2E2 E EE E

2

2 2 2

2 2E E

E E2E2 2E2

2E2 2E2 E EE E

E EE E g g gg gg

denotes g = G0 and h = H0

contains terms of second and higher powers in g

, H0) = 0 and E2(G0, H0) =0, because it is assumed that g and h have their optimal values G0

respectively by the time the fasting patient arrives at the hospital and (ii) E1 and E2 being small quantities may be neglected and g and h are very small.

Substituting these two conditions in equation (3) and (4) we

dhdt

= A3g

Where are all positive constants. A

Since it is the BGS that can be measured easily therefore we attempt to eliminate h if possible, between equations (7) and (8).

Differenciate equation (7), (8) with respect to 2

1 22

d g dg dh dF2d g dg dh dF2d g dg dh dFA Ad g dg dh dFdt dt dt dt1 2dt dt dt dt1 22dt dt dt dt2

d g dg dh dF

d g dg dh dFA A A AA A A A1 2A A1 2 1 2A A1 2d g dg dh dFA Ad g dg dh dF d g dg dh dFA Ad g dg dh dF

2

3 42

d h dg dh2d h dg dh2 A A3 4A A3 4d h dg dhA Ad h dg dhdt dt dt3 4dt dt dt3 42dt dt dt2

A A A AA A A A

Substitute - equation (7) the value of

get 2

2

d g dg dF2d g dg dF2d g dg dFA A A g A hd g dg dF( )d g dg dF( )A A A g A h( )d g dg dF( )A A A g A h1 2 3 4A A A g A h1 2 3 41 2 3 4( )1 2 3 4A A A g A h1 2 3 4( )1 2 3 4dt dt dt1 2 3 4dt dt dt1 2 3 42dt dt dt2A A A g A h

dt dt dtA A A g A h1 2 3 4A A A g A h1 2 3 4dt dt dt1 2 3 4A A A g A h1 2 3 41 2 3 4( )1 2 3 4A A A g A h1 2 3 4( )1 2 3 4dt dt dt1 2 3 4

( )1 2 3 4A A A g A h1 2 3 4( )1 2 3 4 d g dg dF

d g dg dFA A A g A h A A A g A hA A A g A h A A A g A h1 2 3 4A A A g A h1 2 3 4 1 2 3 4A A A g A h1 2 3 4( )A A A g A h( ) ( )A A A g A h( )1 2 3 4( )1 2 3 4A A A g A h1 2 3 4( )1 2 3 4 1 2 3 4( )1 2 3 4A A A g A h1 2 3 4( )1 2 3 4d g dg dFA A A g A hd g dg dF d g dg dFA A A g A hd g dg dF( )d g dg dF( )A A A g A h( )d g dg dF( ) ( )d g dg dF( )A A A g A h( )d g dg dF( )

2

1 2 3 2 42

d g dg2d g dg2 A A A g A A h1 2 3 2 4A A A g A A h1 2 3 2 4d g dgA A A g A A hd g dgdt dt1 2 3 2 4dt dt1 2 3 2 42dt dt2

A A A g A A hdt dt

A A A g A A h1 2 3 2 4A A A g A A h1 2 3 2 4dt dt1 2 3 2 4A A A g A A h1 2 3 2 4

d g dg

d g dgA A A g A A h A A A g A A hA A A g A A h A A A g A A h1 2 3 2 4A A A g A A h1 2 3 2 4 1 2 3 2 4A A A g A A h1 2 3 2 4d g dgA A A g A A hd g dg d g dgA A A g A A hd g dg

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which governs, the BGRS after a heavy load of glucose is ingested.

3. Analysis of the Model

The mathematical model have been analyzed the following step.

The auxiliary equation of (13) is written as m2 + 2m + 20w = 0

Whose roots are given by

m = + 20w

Three cases can be considered.

2- 20w < 0, 2 - 20w > 0.

And it is fact that equation (13) approaches to 0 as t and so our model confirme to reality in predicting that the BGC tends to return ultimately to its optimal concentration. So it passes the test of consistency. For the case 2- 20w < 0

g(t) = Ae -t Cos (wt - ) (14) where w = 20w -

20

Particular integral

P.I = 2 20

12D D w

e –at

= 2 20

12a w a

e –at

g(t) = g0 + Ae -t cos(wt - ) + 2 20

12a w a

e –at (15)

Equation (15) have unknown g0, , w0, and a.

g0, being Blood Glucose Concentration before the glucose load is ingested can be determined by measuring the patients Blood glucose concentration immediately upon this arrival at the hospital. gi(t) = g0 + Ae - ti cos (wti - )

+ 2 20

12a w a

e –ati,

i = 1, 2, 3, 4, 5, 6, ….n

By taking n measurements of g1, g2, ….gn of the patients BGC at time t1, t2, …tn respectively.

If we take n = 7 or 8 measurements of g1, g2, g3, g4, g5, g6, g7we find the optimal values of g0, , w0, a and . Such that the least square error given by

e = 201

cos( )n

ti i

jg g Ae wt

(16)

4. Conclusion

The aim of study can be concluded that the optimal value of g0, , w0, a, can be calculated in Blood Glucose regulatory system to a Glucose Tolerance Test. If w0 may be regarded

has been defined by T0 =0

2w

, w0 is the natural frequency

of the system and here it is considered “T0” as a suitable parameter for diagnosis of diabetes. If has been concluded that a value of less than four hours for T0 indicated normally suppose T0 is more than four hours implied mild diabetes.

References

[1] Sturis, Jeppe, Kenneth polonsky, Erik Mosekilde, and Eve van cauter, “Compute Model for Mechanisms underlying Ultradian osicillation of Insulin and glucose”American Physiological society (1991): E801 -809, Print.

[2] Keener, James P., and James Syneyd Mathematical Physiology, New York: Springer 1998, 594 -607.

[3] E.Ackerman, L.Gatewood, J.Rosevear and G.Molnar, Blood Glucose Regulation and diabetes in concepts models of Bio-Mathematics, F.Heinmets, Ed.Marcel Dekker, 1969, p.131-156.

[4] A.P. Verma and M.N.Mehta, Mathematics with application, Part II, Shivam Book centre, February 1997, p.69-111.

[5] Deo S.G., Raghavendra V. Ordinary Differential Equation and Stability theory, 7th Edition, 1993.

Paper ID: 31011701 432

And it is fact that equation (13) approaches to 0 as t and so our model confirme to reality in predicting that the BGC tends to return ultimately to its optimal concentration. So it passes the test of consistency.

g(t) = Ae -t Cos (wt - t Cos (wt - t ) (14)

2 20D D w

2 2D D w2 20D D w0D D w D D w2 2D D w2 2 2 2D D w2 2

e –at

a w aa w aa w ae –at

cos(wt - ) + 2 20

12a w a0a w a0 2a w a2a w aa w a

2 2 2 2a w a a w a0a w a0 0a w a0e –at (15) at (15) at

Equation (15) have unknown g0, , w0, and a.

, being Blood Glucose Concentration before the glucose load is ingested can be determined by measuring the patients Blood glucose concentration immediately upon this arrival

cos (wt )

American Physiological society (1991): E801 -809, Print.

[2] Keener, James P., and James Syneyd Mathematical Physiology, New York: Springer

[3] E.Ackerman, L.Gatewood, J.Rosevear and G.Molnar, Blood Glucose Regulation and diabetes models of Bio-Mathematics, F.Heinmets, Ed.Marcel of Bio-Mathematics, F.Heinmets, Ed.Marcel ofDekker, 1969, p.131-156.

[4] A.P. Verma and M.N.Mehta, Mathematics with application, Part II, Shivam Book centre, February 1997, p.69-111.

[5] Deo S.G., Raghavendra V.Equation and Stability theory, 7