Type 1 diabetes (also referred to as juvenile-onset or insulin-dependent diabetes) is due to autoimmune attack on the insulin secreting β-cells [1] . Type 1 diabetes affects people under the age of 40, and represents 10% to 15% of the diabetic population [4] . Patients with type 1 diabetes are recommended to take insulin injections [1] .

Type 2 diabetes (also referred to as adult-onset or non-insulin-dependent diabetes) is associated with a deficit in the mass of β-cells due to extreme elevated blood glucose level which reduced insulin secretion and resistance to the action of insulin [1] . Type 2 diabetes represents 85% - 90% of the diabetic population [4] .

The pre-disposing historical high-risk factors for T2DM include increasing age (old age), central obesity, dietary polyphagia of animal fat, carbonate drink, lack of physical exercise, familiar genetic factor, history of gestational diabetes, polycystic ovary syndrome, and severe mental illness, presence of hypertension, hyperlipidaemia, ethnicity and cardio-metabolic factors. Recent emerging risk factors include sleeping deprivation, drug-induced metabolic changes, environmental pollutants, low birth weight and fatal malnutrition [7] .

Gestational diabetes can occur temporally during pregnancy which is due to hormonal changes and usually begins in the fifth or sixth month of pregnancy (between the 24^th and 28^th weeks). Gestational diabetes usually resolves once the baby is born. However, 25% - 50% of women with gestational diabetes will eventually develop diabetes later in their life, especially in those who require insulin during pregnancy and those who are overweight after their delivery [1] .

Diabetes mellitus, commonly known as diabetes, is a syndrome of disordered metabolism, usually due to a combination of hereditary and environmental causes, resulting in abnormally high blood sugar levels known as hyperglycemia [8] .

Blood glucose levels are regulated by two negative feedback loops. In the short term, hyperglycemia stimulates a rapid increase in insulin release from the pancreatic β-cells. The associated increase in blood insulin levels causes increased glucose uptake and decrease glucose production by the liver leading to a reduction in blood glucose [1] . Recent evidence suggests that chronic hyperglycemia may contribute to a second negative feedback loop by increasing the mass of insulin secreting β-cells, through changes in the rates of β-cell replication and death [1] . An increased β-cell mass represents an increase capacity for insulin secretion which, in turn, would lead to a decrease in blood glucose. Type 2 diabetes has been associated with defects in components of both the short-term and chronic negative feedback loops. Although type 2 diabetes is associated with insulin resistance, insulin secretory defects, and insufficient β-cell mass, each of these defects can also be found in people without diabetes.

Insulin-stimulated glucose disposal is reduced by 50% - 100% in patients with type 2 diabetes as compared to non-diabetic controls. However, insulin resistance of a similar magnitude also has been documented in many no diabetic individuals including obese subjects, or during pregnancy, puberty, and aging [1] . Thus, normoglycemia can be maintained in subjects with insulin resistance via increase in blood insulin levels. Defects of insulin secretion have been demonstrated in some people with type 2 diabetes [1] . Even more severe defects in insulin secretion are present in patients with type 1 diabetes following islet transplantation, when normoglycemia is maintained in the absence of exogenous insulin treatment [1] . This suggests that glucose homeostasis can be maintained despite significant loss of β-cell function when an individual has normal insulin sensitivity. [1] observed that β-cell mass is reduced by 40% - 50% in patients with type 2 diabetes when compared with weight matched non-diabetic subjects. In comparison, approximately 80% - 90% of the β-cell mass is lost before the onset of hyperglycemia in individuals who develop type 1 diabetes, suggesting that a greater β-cell mass is required in the presence of insulin resistance [1] . This is also consistent with the observation of a 43% higher β-cell mass in normoglycemia for subjects with insulin resistance due to obesity [1] . Although these data suggest that multiple defects are required for the onset of type 2 diabetes, it is unclear if these defects have a single causal origin or if they occur independently. Experimental induction of insulin resistance using either high fat feeding, glucocorticoid administration, or genetically induced obesity has been shown to cause type 2 diabetes under certain circumstances. This supports the hypothesis that insulin resistance can cause β-cell defects, and hence diabetes, either by overworking the β-cell (β-cell exhaustion) or by toxic effects of hyperglycemia on the β-cell (glucose toxicity). However, the existence of normoglycemia in humans and animals highly resistant to insulin suggests independent defects in insulin sensitivity and β-cell function are required for type 2 diabetes. Finally, hyperglycemia is known to induce insulin resistance. This supports the hypothesis that a primary insulin secretory defect that causes hyperglycemia could lead to insulin resistance and diabetes via increased glucose levels [1] .

According to report by [9] , the symptoms of diabetes are as follows:

1) Frequent urination and excessive thirst;

2) Disproportionate thirst;

3) Intense hunger;

4) Weight gain;

5) Unusual weight loss;

6) Increase fatigue;

7) Irritability;

8) Blurred vision;

9) Sexual dysfunction among men;

10) Numberless or tingling, especially in a feet or hands, etc.

Diabetes if not properly controlled it will lead to the following complications:

1) Kidney failure;

2) Blindness;

3) Amputation;

4) Cardiovascular diseases [4] .

[1] Obtained the model equations describing the β-cell mass, insulin and glucose dynamics which we described in the figures bellow as in the absence of epinephrine as follows:

d G d t = R 0 − ( E G O + S I I ) G , d I d t = β σ G 2 α + G 2 − K I , d β d t = ( − d + r 1 G − r 2 G 2 ) β (3.1)

1) Only trauma, excitement and stress are factors that trigger epinephrine secretion from kidney;

2) Epinephrine induced glucose production from kidney is negligible and ignored;

3) Epinephrine induced glucose production from the liver is considered;

4) Blood insulin level will also be decreased by Epinephrine suppression on insulin secretion.

We present the model diagram for the dynamics of β -Cell mass, insulin, glucose and epinephrine as in Figure 1.

Insulin is secreted by pancreatic β-cell, cleared by the liver, kidneys, and insulin receptors, and distributed into several compartments (e.g. Portal vein, peripheral blood, and interstitial fluid) of the body. Our main concern is the long-time evolution of fasting insulin levels in peripheral blood. Since the dynamics of fasting insulin levels on this time-scale are slow, we use a single-compartment equation given by

d I d t = Secretion − Clearance (3.7)

where Secretion and Clearance are rates normalized by insulin’s volume of distribution [1] .

Insulin secretion

The rate of insulin secretion from pancreatic β-cells is a sigmoidal function of glucose production [1] . Hence, we have

d I d t = β σ G 2 α + G 2 (3.8)

where β is the mass of pancreatic β-cells, all β-cells are assumed to secrete insulin at the same maximal rate σ , α is a constant and G 2 α + G 2 is a Hill function that describes the sigmoid nature of insulin secretion [1] .

When epinephrine is secreted into the blood stream, the insulin secretion by pancreatic β-cells will be suppressed by 41% [5] . This means insulin sensitivity to trigger glucose utilization by cells will be reduced. Let ρ be amount of insulin suppression by epinephrine, then we have a negative effect on insulin secretion thus:

AmountofInsulinsecretionsuppressed = − ρ I (3.9)

Since ρ is the epinephrine effectiveness in suppressing insulin secretion. We incorporate ( − ρ I ) to insulin secretion in Equation (3.9) to obtain the total insulin secretion when epinephrine is secreted. Therefore, the rate of insulin secretion in the presence of epinephrine secretion is given by

d I d t = β σ G 2 α + G 2 − ρ I (3.10)

The rate of insulin clearance is proportional to blood insulin levels when the system is near steady state [1] . Clearance ∝ I

Thus,

Clearance = K I (3.11)

where K is the insulin clearance constant by liver, kidney, muscle, and insulin receptors [1] .

Substituting Equations (3.10) and (3.11) into Equation (3.7), we obtain the equation governing insulin secretion and clearance in the presence of epinephrine, as

d I d t = β σ G 2 α + G 2 − ρ I − K I d I d t = β σ G 2 α + G 2 − ( ρ + K ) I (3.12)

where β is the mass of pancreatic β-cells. σ is a rate of maximum secretion by the pancreatic β-cells and G 2 α + G 2 Hill function that describes the sigmoid

ranging from 0 to 1, ρ is the parameter incorporated to insulin secretion which is epinephrine effectiveness in suppressing insulin secretion, and K is insulin clearance constant by liver, kidney, muscle, and insulin receptors, and I is insulin.

Despite a complex distribution of pancreatic β-cells throughout the pancreas, β-cell mass dynamics have been successfully quantified with a single compartment model [1] .

d β d t = Formation − Loss (3.13)

where Formation and Loss represent the rates at which β-cell mass is added to or removed from the population, respectively [1] .

We studied the equilibrium solution of the system in Equation (3.17) by splitting the glucose regulatory system into subsystems; slow (β-cell mass) subsystems and fast (glucose/insulin) subsystems.

From Equation (3.17) we have

d G d t = R 0 + G e − ( E G O + S I I ) G d I d t = β σ G 2 ( α + G 2 ) − ( ρ − K ) I d β d t = ( − d 0 + r 1 G − r 2 G 2 ) β

Putting

d β d t = 0

We have the following:

( − d 0 + r 1 G − r 2 G 2 ) β = 0

Either β = 0 or ( − d 0 + r 1 G − r 2 G 2 ) = 0 (4.1)

We have

r 2 G 2 − r 1 G + d 0 = 0

Solving for G

G 1 , 2 = r 1 ± r 2 2 − 4 × r 1 × d 0 2 r 2 (4.2)

Either

G 1 = r 1 + r 2 2 − 4 × r 1 × d 0 2 r 2

or

G 2 = r 1 − r 2 2 − 4 × r 1 × d 0 2 r 2

Equating d G d t = d I d t = 0

We have

β σ G 2 ( α + G 2 ) − ( ρ − K ) I = 0 (4.3)

From 4.1, β = 0 hence 4.2 gives

I = 0 (4.4)

For glucose dynamics

At equilibrium point, we have

R 0 + G e − ( E G O + S I I ) G = 0 (4.5)

Using 4.3 in 4.4, we have

R 0 + G e − E G O G = 0 E G O G = R 0 + G e G = R 0 + G e E G O

Therefore, the equilibrium point of the fast (glucose/insulin) subsystems is

given by

G * , I * = ( R 0 + G e E G O , 0 )

From Equation (3.17) we should find the stability of the fast (glucose/insulin) subsystems

d G d t = R 0 + G e − ( E G O + S I I ) G

d I d t = β σ G 2 α + G 2 − ( ρ + K ) I

Using Jacobian Matrix to find the stability

J = [ ∂ F 1 ∂ G ∂ F 1 ∂ I ∂ F 2 ∂ G ∂ F 2 ∂ I ] (4.6)

Let F 1 = R 0 + G e − ( E G O + S I I ) G and

F 2 = β σ G 2 α + G 2 − ( ρ + K ) I

Therefore, the Jacobian matrix in (4.5) becomes

J = [ − ( E G O + S I I ) − S I G − 2 α β σ G ( α + G 2 ) 2 − ( ρ + K ) ]

At equilibrium of the fast subsystems

J 0 = [ − E G O − S I ( R 0 + G e ) E G O 0 − ( ρ + K ) ]

Using characteristics equation.

We have

| J ( E o ) − λ I | = 0

Putting the Jacobian into characteristics equation

| ( − E G O − S I ( R 0 + G e ) E G O 0 − ( ρ + K ) ) − λ ( 1 0 0 1 ) | = 0

| − E G O − λ − S I ( R 0 + G e ) E G O 0 − ( ρ + K ) − λ | = 0

( − E G O − λ ) ( − ( ρ + K ) − λ ) = 0

− E G O − λ = 0     or     − ( ρ + K ) − λ = 0

λ 1 = − E G O     and     λ 2 = − ( ρ + K )

Therefore, the fast (Glucose and Insulin) subsystem is locally and asymptotically stable since all the values of λ i < 0 for i = 1, 2 were negative.

For slow (β-cell mass) subsystem, we have the following:

either β = 0 or − d 0 + r 1 G − r 2 G 2 = 0

λ = − d 0 + r 1 G − r 2 G 2

The slow (β-cell mass) subsystems is locally and asymptotically stable since the Eigenvalue was negative, at r 2 G 2 + d 0 > r 1 G .

Since the values of h r 2 = 0.0000024 are in Table 2. We substituted the values

in Table 2 into (4.1) we have

− 0.06 + 0.00084 G − 0.0000024 G 2 = 0

24 G 2 − 8400 G + 600000 = 0

Putting the above quadratic equation in (4.2), we have

G 1 , 2 = 8400 ± 8400 2 − 4 × 24 × 600000 2 × 24

Either

G 1 = 12000 48 = 250

or

G 2 = 4800 48 = 100

The one dimensional subsystem had three steady state solutions which were β = 0 , G = 100 and G = 250 . These three steady states were referred to as the pathological, physiological and unstable steady state [1] .

Analysing the system as a whole, consider (3.17) above

F 1 = R 0 + G e − ( E G O + S I I ) G F 2 = β σ G 2 ( α + G 2 ) − ( ρ − K ) I F 3 = ( − d 0 + r 1 G − r 2 G 2 ) β

Thus, the Jacobian Matrix J of the above system is given by

J = ( ∂ F 1 ∂ G ∂ F 1 ∂ I ∂ F 1 ∂ β ∂ F 2 ∂ G ∂ F 2 ∂ I ∂ F 2 ∂ β ∂ F 3 ∂ G ∂ F 3 ∂ I ∂ F 3 ∂ β ) (4.4)

J = ( − ( E G O + S I I ) − S I G 0 − 2 α β σ G ( α + G 2 ) 2 − ( ρ + K ) σ G 2 α + G 2 ( r 1 − 2 r 2 G ) β 0 ( − d 0 + r 1 G − r 2 G 2 ) )

At equilibrium points, the whole system becomes

J ( E 0 ) = ( − E G O − S I ( R 0 + G e ) E G O 0 0 − ( ρ + K ) σ ( R 0 + G e ) 2 α E G O − ( R 0 + G e ) 2 0 0 − d 0 + r 1 ( R 0 + G e ) E G O − r 2 ( R 0 + G e ) E G O 2 )

The characteristics equation is given by

| J ( E o ) − λ I | = 0

| − E G O − λ − S I ( R 0 + G e ) E G O 0 0 − ( ρ + K ) − λ σ ( R 0 + G e ) 2 α E G O − ( R 0 + G e ) 2 0 0 − d 0 + r 1 ( R 0 + G e ) E G O − r 2 ( R 0 + G e ) E G O 2 − λ | = 0

− E G O − λ | − ( ρ + K ) − λ σ ( R 0 + G e ) 2 α E G O + ( R 0 + G e ) 2 0 − d 0 + r 1 ( R 0 + G e ) E G O − r 2 ( R 0 + G e ) E G O − λ |

Therefore,

λ 1 = − E G O , λ 2 = − ( ρ + K ) and λ 3 = − ( d 0 + r 2 ( R 0 + G e ) E G O − r 1 ( R 0 + G e ) E G O )

Since all the values of λ i < 0 for i = 1, 2, 3 the whole system is locally and asymptotically stable.

The results of the performance of the existing and modified models are shown in Figures 2-15 and MATLAB 2015a was being used to generate the simulations.

In this work, we incorporated a parameter ρ by defining it as effectiveness of epinephrine in suppressing insulin secretion and a parameter G_e by defining it as epinephrine induced glucose increase as the factors that affect glucose and insulin homeostasis. The model which consists of a system of three nonlinear ordinary differential equations was used to investigate the effect of epinephrine on glucose, insulin and beta cell mass dynamics. The result of the study showed that; in the presence of epinephrine there is increase in blood glucose level and increase in Beta cell mass which increase blood insulin level but not enough to bring down blood glucose level into homeostasis. As a result of that, there is an extreme hyperglycemia in all three distinct states. Furthermore, the result of the numerical experiment carried out indicated that frequent epinephrine secretion into the blood induced prolong and extreme hyperglycemia. An extreme hyperglycemia induced both beta cells death and insulin resistance which would lead to diabetes. In conclusion, frequent state of trauma, excitement and/or stress induced diabetes. In view of the findings of this study, we recommended that there should be a massive and continuous health education especially for communities living in the areas where the stated agents (trauma, excitement and stress) of epinephrine secretion are common, so that the people will understand the effect of these agents on their healthy living and this will minimize the occurrence of diabetic among the people. The government, non-governmental organization and public health workers should focus on public enlightenment programs aimed at enabling individuals in the society to understand how these three factors increase the risk of having diabetes in their lives.

We modified [10] mathematical model for beta-cell mass, insulin and glucose kinetics by incorporating the parameter “G_e” and “ ρ ” as the rate of glucose production due to epinephrine secretion and epinephrine effectiveness in suppressing insulin secretion to examine the effect of two parameters on glucose regulatory system. Analytical studies were carried out using Jacobian matrix. The equilibrium points were obtained and our results showed that the equilibrium point of the system is locally asymptotically stable since all the Eigen values were negative; this showed that the disease will be controlled to avoid further spread. The result of the numerical experiment carried out indicated that frequent epinephrine secretion into the blood induced prolong and extreme hyperglycemia. An extreme hyperglycemia induced both beta-cells death and insulin resistance which can lead to both type 1 and type 2 diabetes.