A DISCRETE MODEL OF GLUCOSE-INSULIN INTERACTION AND STABILITY ANALYSIS A. & B.

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1 A DISCRETE MODEL OF GLUCOSE-INSULIN INTERACTION AND STABILITY ANALYSIS A. George Maria Selvam* & B. Bavya** Sacre Heart College, Tirupattur, Vellore, Tamilnau Abstract: The stability of a iscrete-time Glucose Insulin interaction system is consiere in this paper. The system is moele with ifference equations. Local stability conitions about the equilibrium points are obtaine. The phase portraits are obtaine for ifferent sets of parameter values. Also bifurcation iagrams are provie for selecte range of parameter. Numerical simulations are carrie out an graphs are also generate to inicate the role of insulin in the regulation process of glucose in the human boy. Introuction: Diabetes is a metabolic isease characterize by high levels of bloo glucose (bloo sugar), which leas to severe amage to heart, bloo vessels, eyes, kineys, an nerves. It is a isease that occurs either because the pancreas oes not prouce enough insulin or because the boy's cells cannot effectively use the insulin it prouces. Type iabetes is characterize by eficiency in insulin prouction an requires aily aministration of insulin. Type 2 iabetes results from the boy's inability to use insulin effectively. Diabetes is on the rise. The number of people with iabetes has risen from 08 million in 980 to 422 million in 204. Diabetes cause.5 million eaths in 202. Diabetes is an epiemic in Inia with more than 69. million iabetic iniviuals currently iagnose with the isease. A boy's homeostatic mechanism, when operating normally, restores the bloo sugar level to a narrow range of about 4.4 to 6. mmol/l (79.2 to 0 mg/l). For the majority of healthy iniviuals, normal bloo sugar levels are as follows: between 4.0 to 6.0 mmol/l (72 to 08mg/L) when fasting, up to 7.8 mmol/l (40 mg/l) 2 hours after eating. Mathematical moels have been use to escribe an unerstan iabetes ynamics. There are various moels base on glucose an insulin istributions an those moels have been use to explain glucose-insulin interaction. The most wiely use moel in the stuy of iabetes is the minimal moel which is use in the interpretation of the intravenous glucose tolerance test (IVGTT) [2]. A glucose tolerance test measures how well a human boy is able to break own glucose. Moel of Glucose an Insulin Interaction: Sometimes, it is esirable to use a set of ifference equations for which time is a iscrete variable. Also, iscrete time moels governe by ifference equations are more appropriate than continuous ones. Discrete time systems can also provie efficient computational moels of continuous ones of numerical simulations. Several authors [,3,5,6] have propose more popular, general an realistic moels with results consistent with physiology. The moeling of the glucose-insulin system has become an interesting topic an several moels have been propose an stuie with the purpose of unerstaning the system better, investigating possible pathways to iabetes as well as proviing better insulin aministration practices. In the glucose regulatory system, insulin an glucagon play vital role in balancing metabolism. Insulin an glucagon act together to balance metabolism. Insulin helps control bloo glucose levels by signaling the liver an muscle an fat cells to take in glucose from the bloo. Glucose is store in the boy as glycogen. The liver is an important storage site for glycogen. In this paper, 67

2 we consier the following system of ifference equations as the iscrete version of the moel stuie in [7]. x( n ) ( a) x( n) bx( n) y( n) c () y( n ) ( ) y( n) ex( n) where x0, y 0. Variables Description x Glucose Concentration y Insulin Concentration a Insulin Inepenent Glucose Disappearance b Insulin Depenent Glucose Disappearance c Glucose Infusion Rate Insulin Degraation e Insulin Prouction ue to Glucose Stimulation Equilibrium Points of the System: In orer to stuy the qualitative behavior of the solutions of the system of nonlinear Difference equations (), we efine the equilibrium points of the ynamic system as a non-negative fixe point of the map, i.e. the solutions of the following nonlinear algebraic system. x ( a) x bxy c : y ( ) y ex The system () has two equilibrium points E0 0,0 an a a 4 bce 4 E, a a bce. 2be 2b In this iscussion, we are intereste in the interior positive equilibrium point E. Stability Analysis: In this section, we investigate the local behavior of the moel aroun each fixe point. The local stability analysis of the moel can be stuie by computing the variation matrix corresponing to each fixe point [4]. The Jacobian matrix J for the system () is a by J (x,y) = bx (2) e Trace J( x, y) 2 ( a ) by an Det J( x, y) ( a by)( ) bex. For the system (), we have the following analysis. From (2), using the equilibrium pointe 0, Jacobian matrix for E 0 is given by J (E 0 ) = a 0 e Trace J( E0) 2 ( a ) an Det J( E0) ( a)( ). The Eigen values of the matrix J(E 0 ) are a an 2. Jacobian matrix for E is given by Where A= a 4 bce, J E J (E ) = a A e a A 2e a A Trace ( ) 2 an a Det J ( E) ( ) A 2 2a Stability of Equilibrium Points: The characteristic roots λ an λ 2 of p(λ) = 0 are calle eigen values of the fixe point (x*, y*). Then the fixe point (x*,y*) is a sink if λ,2 <. Hence the sink is locally asymptotically stable. The fixe point (x*, y*) is a source if λ,2 >. The source is locally unstable. The fixe point (x*, y*) is a sale if λ > an λ < (or λ < an 68

3 λ > ). Finally (x* y*) is calle non hyperbolic if either λ = or λ 2 =. For the system (), we have the following results [8, 9]. Proposition : The fixe point E 0 is a Sink if a > 0 an > 0. Source if a < 0 an < 0. Sale if a > 0 an < 0. Proposition 2: The fixe point E is a Sink if ( a)[ a ( a)] 3 [( ) 4(2 )]. a a c 4be 4be 2 Source if ( a)[ a ( a)] c 3 [( a ) 4(2 a)] an c. 4be 4be 2 Sale if ( a)[ a ( a)] c 3 [( a) 4(2 a)] an c. 4be 4be 2 Numerical Simulations: In this section, we provie the numerical simulations to illustrate the results of the previous sections. Mainly, we present the time plots of the solutions x an y with phase plane iagrams (aroun the equilibrium points) for the Glucose-Insulin systems. Dynamic nature of the system () about the equilibrium points uner ifferent sets of parameter values are presente in this section. Also the bifurcation iagram inicates the existence of chaos in both glucose an insulin interaction. Example : For the values a 0.99; b 0.00; c ; 0.99; e 0.007, it is the trivial fixe point. The Eigen values are,2 0.0so that, Hence the trivial fixe point is stable. The time plot an the phase iagram are presente, see Figure & 2. Example 2: For the values a.9; b.9; c.9;.25; e 0.65, we obtain E (0.43,0.22) which is an interior fixe point. The eigen values are, i so that, Hence the fixe point is stable. The time plot an the phase iagram are provie in Figure 3 & 4. Figure : Time Plot fore 0 for system () 69

4 Figure2: Phase Portrait E 0 for system() Figure 3: Time Plot for E for system () Figure 4: Phase Portrait E for system () Stuies in population ynamics aims at ientifying qualitative changes in the long-term ynamics preicte by the moel. Bifurcation theory eals with classifying, orering an stuying the regularity in the ynamical changes. Bifurcation iagrams 70

5 provie information about abrupt changes in the ynamics an the values of parameters at which such changes occur. Also they provie information about the epenence of the ynamics on a certain parameter. Qualitative changes are tie with bifurcation see Figure 5 & 6. Figure 5: Bifurcations for Glucose system Figure 6: Bifurcation for Insulin system References:. Ackerman E, Gatewoo L C, Rosevaer J W & Molnar G D, Moel stuies of bloogluocse regulation, Bull Math Biophys, 27, Suppl: 2 Suppl:37, Bolie V W, Coefficients of normal bloo glucose regulation, J Appl. Physiol, 6, , Chen C, Tsai H & Wong S, Moelling the physiological glucose-insulin ynamic system on iabetes, J Theor. Biol, 265, , Derouich M & Boutayeb A, The effect of physical exercise on the ynamics of glucose an insulin, J Biomech, 35, 9 97, Dubey B & Hussain J, Moels for the effect of environmental pollution on forestry resources with time elay, Nonlinear Analysis: Real worl Applications, 5, , Giang D V, Lenbury Y, De Gaetano A & Palumbo P, Delay moel of glucoseinsulin systems: global stability an oscillate solutions conitional on elays, J Math Anal Appl, 343, , Jamal Hussain an Denghmingliani Zaeng, A mathematical moel of glucoseinsulin interaction, Sci Vis, Vol 4, Issue No 2, April-June

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