A Discrete Model of Glucose-insulin Interaction and Stability Analysis

The stability of a discrete-time Glucose Insulin interaction system is considered in this paper. The system is modeled with difference equations. Local stability conditions about the equilibrium points are obtained. The phase portraits are obtained for different sets of parameter values. Also bifurcation diagrams are provided for selected range of parameter. Numerical simulations are carried out and graphs are also generated to indicate the role of insulin in the regulation process of glucose in the human body. Introduction: Diabetes is a metabolic disease characterized by high levels of blood glucose (blood sugar), which leads to severe damage to heart, blood vessels, eyes, kidneys, and nerves. It is a disease that occurs either because the pancreas does not produce enough insulin or because the body's cells cannot effectively use the insulin it produces. Type 1 diabetes is characterized by deficiency in insulin production and requires daily administration of insulin. Type 2 diabetes results from the body's inability to use insulin effectively. Diabetes is on the rise. The number of people with diabetes has risen from 108 million in 1980 to 422 million in 2014. Diabetes caused 1.5 million deaths in 2012. Diabetes is an epidemic in India with more than 69.1 million diabetic individuals currently diagnosed with the disease. A body's homeostatic mechanism, when operating normally, restores the blood sugar level to a narrow range of about 4.4 to 6.1 mmol/L (79.2 to 110 mg/dL). For the majority of healthy individuals, normal blood sugar levels are as follows: between 4.0 to 6.0 mmol/L (72 to 108mg/dL) when fasting, up to 7.8 mmol/L (140 mg/dL) 2 hours after eating. Mathematical models have been used to describe and understand diabetes dynamics. There are various models based on glucose and insulin distributions and those models have been used to explain glucose-insulin interaction. The most widely used model in the study of diabetes is the minimal model which is used in the interpretation of the intravenous glucose tolerance test (IVGTT) [2]. A glucose tolerance test measures how well a human body is able to break down glucose. Model of Glucose and Insulin Interaction: Sometimes, it is desirable to use a set of difference equations for which time is a discrete variable. Also, discrete time models governed by difference equations are more appropriate than continuous ones. Discrete time systems can also provide efficient computational models of continuous ones of numerical simulations. Several authors [1,3,5,6] have proposed more popular, general and realistic models with results consistent with physiology. The modeling of the glucose-insulin system has become an interesting topic and several models have been proposed and studied with the purpose of understanding the system better, investigating possible pathways to diabetes as well as providing better insulin administration practices. In the glucose regulatory system, insulin and glucagon play vital role in balancing metabolism. Insulin and glucagon act together to balance metabolism. Insulin helps control blood glucose levels by signaling the liver and muscle and fat cells to take in glucose from the blood. Glucose is stored in the body as glycogen. The liver is an important storage site for glycogen. In this paper, International Journal of Multidisciplinary Research and Modern Education (IJMRME) ISSN (Online): 2454 6119 (www.rdmodernresearch.com) Volume II, Issue II, 2016 68 we consider the following system of difference equations as the discrete version of the model studied in [7]. ( 1) (1 ) ( ) ( ) ( ) ( 1) (1 ) ( ) ( ) x n a x n bx n y n c y n d y n ex n     