Modeling glucose and insulin in diabetic human during physical activity

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1 331 Modeling glucose and insulin in diabetic human during physical activity Jean Marie Ntaganda Benjamin Mampassi ICM 2012, March, Al Ain Abstract This paper aims at designing a two compartmental mathematical model for determining the response of plasma glucose and insulin sensitivity in Type II diabetes case to heart rate and alveolar ventilation that are cardiovascular and respiratory control respectively A two nonlinear coupled ordinary differential equations is provided Stability conditions are established The inverse techniques are used for identifying model parameters The validation of the model is achieved by considering a woman in physical activity for testing its efficient Keywords: Glucose, insulin, parameters identification, modelling, numerical simulation 2000 Mathematics Subject classification: 92C30, 49J15 1 Introduction Human bodies need to maintain glucose concentration level in a narrow range mg/dl or mmol/l If one s glucose concentration level is significantly out of the normal range, this person is considered to have the plasma glucose problem: Hyperglycemia (greater than140 mg/dl or 78 mmol/l after an Oral Glucose Tolerance Test, or greater than mg/dl or 55 mmol/l after a Fasting Glucose Tolerance Test) or hypoglycemia (less than 40 mg/dl or 22 mmol/l) The major long-term effects of diabetes are caused by hyperglycemia Prolonged hyperglycemia can cause complications, which may lead to kidney disease, blindness, loss of limbs, and so on The hypoglycemia can lead to dizziness, coma, or even death The key organs that control blood glucose are the pancreas and the liver The key hormones are insulin and glucagon In the pancreas, there are clusters of endocrine cells scattered throughout the tissue These are the α-cells and the β-cells The α-cells produce glucagon and the β-cells produce insulin The pancreas secretes these antagonistic hormones into the extracellular fluid, which then enters the circulatory system and regulates the concentration of glucose in the blood For biologists, this is known as a simple endocrine pathway Diabetes Mellitus is an endocrine disorder caused by a deficiency of insulin (Type I Diabetes) or a decreased response to insulin in target tissues (Type II Diabetes) [10] Type I Diabetes was previously called insulin-dependent diabetes mellitus (IDDM) or juvenile-onset diabetes It is an autoimmune disorder in which the immune system destroys the β-cells of the pancreas As a result, the person s ability to produce insulin is greatly inhibited Diagnosis usually occurs in early childhood and is treated with insulin injections Type II 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 [28], and resistance to the action of insulin or a reduced responsiveness of insulin target cells due to some change in the insulin receptors Heredity can play a role, but research indicates that excess body weight and lack of exercise significantly increases risk It generally appears after age 40, but young people who are

2 332 Modeling glucose and insuline in diabetic human overweight and sedentary can develop the disease The relative contribution and interaction of these defects in the pathogenesis of this disease remains to be clarified [12] About 90% to 95% of people with diabetes have type II and many can manage their blood glucose level with regular exercise and a healthy diet; however, some require drug therapy Type II diabetes is associated with older age, obesity, family history of diabetes, prior history of gestational diabetes, impaired glucose tolerance, physical inactivity, and race or ethnicity Many researches are motivated by the large population of diabetes patients in the world and the big health expenses to study the glucose-insulin endocrine metabolic regulatory system [6], [33], [35], [36], [38], what cause the dysfunctions of the system [10], how to detect the onset of the either type of diabetes including the so called prediabetes [7], [8], [16], [29], and eventually provide more reasonable, more effective, more efficient and more economic treatments to diabetics Research clearly shows that achieving good control early on prevents Type II diabetic complications, including nerve, kidney, eye and heart disease, up to twenty years later The burden of serious complications can be considerable for the individual and for the healthcare system Many of these complications can be prevented or their progression halted with good early management of the condition, including effective management of blood glucose levels In addition, people with poorly controlled diabetes are at higher risk of cardiovascular disease events, which are a major cause of morbidity and mortality Large-scale clinical trials have demonstrated the benefits of tight control in Type II diabetes, minimizing disease complications and improving quality of life [39] Intensive insulin therapy involves three to four daily blood glucose measurements by finger pricking, followed by subcutaneous insulin injection; these usually correspond to meal times and bed time [4] Although this process is adequate to maintain the blood glucose level within an acceptable range, wide fluctuations persist throughout the day as a consequence of personal daily life activities (such as food intake and exercise) occurring between glucose measurements Since the 1960s, mathematical models have been used to describe glucose insulin dynamics [11] Bergman et al [9] proposed a three-compartment minimal model to analyze the glucose disappearance and insulin sensitivity during an intravenous glucose tolerance test Modifications have been made to the original minimal model to incorporate various physiological effects of glucose and insulin Cobelli and co-workers [13] developed a revised minimal model in order to separate the effects of glucose production from utilization The overestimation of glucose effectiveness and the underestimation of insulin sensitivity by the minimal model were addressed in yet another publication by Cobelli et al [14] where a second nonaccessible glucose compartment was added to the original model Hovorka and co-workers [23] extended the original minimal model by adding three glucose and insulin subcompartments in order to capture absorption, distribution, and disposal dynamics, respectively Anirban Roy et al presented a three compartimental model to capture the changes in glucose and insulin dynamics due to exercise [5] This model incorporates the effects of physiological exercise into the Bergman minimal model [9] in order to capture the plasma glucose and insulin dynamics during, as well as after, periods of mild-to-moderate exercise The purpose of this article is to highlight how physical activity plays a crucial role in decreasing plasma glucose and increasing insulin sensitivity in Type II diabetes This paper is organised as follows In section 2, we build the mathematical model to be considered as well as the inverse technique for computing unknown constants and functions of the model Model parameters are computed in section 3 In section 4 we present numerical results for a healthy subject The concluding remarks are presented in section 5

3 JM Ntaganda, B Mampassi Model design 21 Outline of the model In this section we would like to design a mathematical model for determining blood partial pressures with respect to heart rate and alveolar ventilation Let us describe briefly the physiology of exercise on dynamic of glucose and insulin Physiological exercise induces several fundamental metabolic changes in the body [42] Elevated physical activity promotes a drop in plasma insulin concentration from its basal level [19] The researches show that an insulin clamp at its basal level disrupted the human plasma glucose homeostasis [1], [45] An increase physical activity amplifies glucose uptake by the working tissues [43] With increasing work intensity, the plasma glucose is maintained homeostasis by the increasing of hepatic glucose release also increases [41] During a mild-to-moderate work load, the major source of increased splanchnic glucose release is contributed by elevated hepatic glycogenolysis As the duration of exercise increases, the rate of hepatic glycogenolysis diminishes because of the limited supply of liver glycogen stores [2] Simultaneously, hepatic gluconeogenesis is stimulated [3] However, the rate of glucose produced via liver gluconeogenesis does not fully compensate for the decrease in glucose release by liver glycogenolysis (the former is a slower process), thereby resulting in a net decrease in hepatic glucose release during prolonged exercise [22] Because of this imbalance between glucose uptake and hepatic glucose release, the plasma glucose concentration declines and hypoglycemia occurs [2], [3] It is also known that in glucose regulation liver glycogen content declines more rapidly with increasing exercise intensity [2], [31] During the recovery period after short-term exercise, both the elevated glucose uptake rate by working muscles and the rate of hepatic glucose release decline gradually to their respective basal levels However, glucose fluxes after prolonged exercise are quite different Because of the substantial depletion of liver glycogen stores during prolonged exercise, the rate of glycogenolysis is suppressed significantly, leading to a net decrease in the hepatic glucose release rate During the recovery period, the elevated muscle glucose uptake rate gradually declines to the basal level; however, the already suppressed net hepatic glucose release rate is elevated significantly as a consequence of an increase in hepatic gluconeogenesis [22]21 In vivo studies have revealed a significant increment in hepatic lactate consumption immediately after prolonged exercise, and this lactate serves as a substrate for enhanced post-exercise gluconeogenesis [44] Physical exercise is an important factor in diabetes management Physical exercise of diverse forms gives diverse results of glycemia Although physical activity may be sometimes a risky method to loose weight while being ill with Type II diabetes, regular exercises are the basis in diabetes treatment Appropriately selected physical activity increases the sensibility of cells to insulin, enhances the homeostasis of glucose and helps to reduce medicine doses [25] During exercising the uptake of glucose to muscles increases proportionally to the intensity and duration of physical load Exercises both stimulate glucose uptake independently from insulin stimulation and neutralise cell immunity to insulin [24] While evaluating the type, duration and intensity of physical exercises it is necessary to consider the level of physical state of patients Diabetics perform aerobical and anaerobical exercises Anaerobical exercises last less than 2 minutes, for example short distance running, swimming or weight lifting In this case namely cells receive ATP energy from sebum and carbohydrates from glycogen that accumulates in muscles Aerobical exercises last over 2 minutes, for example, long distance running and other sporting fields The consequence of frequent intervals between intensive activity, that is interval between rapid warmup exercises and heavy training, is a prominent glycogen consumption in muscles that highly increases insulin sensibility after active activity During the period of long lasting active activity glucose in blood drops down significantly and the resources of glycogen in muscles exhaust rather considerably In case the activity is not coordinated with insulin doses the volume of glycogen in

4 334 Modeling glucose and insuline in diabetic human muscles during active activity is consumed faster than its resources are accumulated, therefore hypoglycemia may develop When active activity lasts for a short period of time and is very intensive than the replenishing of human body with carbohydrates is the only efficient way to maintain the normal level of glycaemia [15] It is known that heart rate and alveolar ventilation are two controls of cardiovascular respiratory system The knowledge of the control mechanism of the cardiovascular and respiratory system is very helpful for improving diagnostics and treatments of diseases related to this system During a moderate physical activity we are interested in determining these controls for controlling plasma glucose and insulin sensitivity which was quantified by Quicki, as inverse of the logarithm of the product of plasma glucose and plasma insulin at baseline [26] Both in vivo [17] and in vitro [34] studies have demonstrated the ability of exercise to increase insulin sensitivity Based on physiology properties and the role of the human cardiovascular and respiratory system during physical activity we propose a two compartmental model composed of the liver compartment (LC) and the pancreas compartment (PC) as shown in the figure 1 Cardiovascular-Respiratory system Lungs V A Q r Q l Heart H P V P A Gl Liver Compartment f(h,v A ) g(h,v A ) Tissues I Pancreas Compartment Figure 1: A schematic diagram of two compartments for modeling human glucose-insulin Q l and Q r are left and right cardiac flow respectively H is heart rate and V A denotes alveolar ventilation P A and P V represent arterial and venous pressure respectively The model has two compartments: liver compartment (LC) and pancreas compartment (PC) Blood flows between lungs and heart due to left (Q l ) and right (Q r ) cardiac output The arterial pressure (P A ) leads the tissues to receive the blood from cardiovascular respiratory system whereas the blood comes to cardiovascular respiratory system from tissues due to arterial pres-

5 JM Ntaganda, B Mampassi 335 sure (P V ) The respiratory control system varies the ventilation rate in response to the levels of dioxide CO 2 and oxygen O 2 gases Consequently, it arises the ventilation rate and cardiac output influence mutually It is then obvious that exchanges between LC and PC are controlled by heart rate (H) and alveolar ventilation ( V A ) functions The mechanism of this control is not direct and can be represented by outflow functions between systemic arterial and venous compartments that depend on heart rate alveolar ventilation (figure 1) Therefore a nonlinear compartment analysis leads on the following new global model d dt G l(t) = G l (t) + (P vs ) δ f(h(s), V A (s)) (1) d dt I(t) = I(t) + (G l(t)) σ g(h(s), V A (s) (2) where the functions G l (t) and I(t) denote respectively glucose and insulin at time t, δ and σ are model constants and f, g model functions to be identified Equation (1) and (2) arise from straightforward development of mass balance between glucose and insulin compartments 22 Stability analysis Let H e, VAe, Gl e and I e be the equilibrium states According to equations (1) and (2) we have { Gle + (I e ) δ f e = 0 I e + (Gl e ) σ (3) g e = 0 where f e = f(h e, V Ae ) and g e = g(h e, V Ae ) Since it is known that glucose and insulin take the values strictly that is Gl(t) > 0 and I(t) > 0, t it follows that the equilibrium state is determined by σ 1 I e = f e g e 1 δ Gl e = f g Proposition 1 Let us assume that then the equilibrium state defined by (3) is stable e e (δσ 1) (4) 0 < δσ < 1 (5) Proof According to the dynamic system theory, the stability of an equilibrium state of two ordinary differential equations is determined by analysing the behavour of the Hessian matrix When it is definite negative this equilibrium is stable Since the corresponding Hessian matrix defined by the the equilibrium state (4) is given as follows 1 σ δ 1 1 δf e g e H = σ 1 1 δ σf g 1 e e

6 336 Modeling glucose and insuline in diabetic human the characteristic equation becomes λ 2 + 2λ + = 0 (6) where λ is eigenvalue It is easy to verify that the equation (6) has two strictly negative real roots if and only if 0 < δσ < 1 so that the proposition 1 yields The consequence of the proposition 1 is that the solutions of proposal model (1) and (2) converge toward the equilibrium state Therefore we have the following result Proposition 2 Assume that f and g are positive functions and differentiable with respect to their argument, ( then for given positive constants Gl 0 and I 0, there exist a couple of control functions H(s), V ) A (s) with s [0, S] such that the system (1)-(2) admits a unique positive solution (Gl(t), I(t)) (C 1 (0, T )) 2 that satisfies Gl(0) = Gl 0 and I(0) = I 0 Furthermore this solution is asymptotically stable We can refer to [27] and [40] for the proof of the proposition 2 It is should be mentioned that the bifurcation analysis technique may predict the existence of the Hopf bifurcation at parameter values where the equilibrium loses its stability and periodical stable solutions exist when the value of parameter increases In this work we mainly focus our attention on the identification of the model parameters that leads to asymptotically stable solutions 3 Computing model parameters Let us be interested in identifying the constant δ and σ and the functions f and g We take T max and S max as a positive time parameters and N and M as integer parameters such that M < N We consider = (G µ l (t 1),, G µ l (t N)) T G µ l I µ = (I µ (t 1 ),, I µ (t N )) T where G µ l (t k) and I µ (t k ) are measured data at the time t k = kt max representing ideal values N G l (t k ) and; µ is the perturbation parameter due to some imprecisions on measured data Mathematically the identification problem can be formulated as follows Find u= ( δ, σ, f, ḡ) solution of the output least squares problem where and J(u) = min J(u) (7) u=(δ,σ,f,g) J(u) = G µ l G µ 2 l + I µ I µ 2 (8) f = (f(h(s k ), V A (s k )) T, g = (g(h(s k ), V A (s k )) T (9) and where G l and I are R N vector solutions at time grid points of the system (1)-(2) depending of the parameter vector u and H(s k ) and V A (s k ) are the values of cardiovascular respiratory

7 JM Ntaganda, B Mampassi 337 system control H and V A at the time s k = ks max M respectively Since M < N we take s k = t k to simplify We should mention that (7) is a nonlinear inverse problem that is generally ill-posed in the sense that a couple (G l, I) does not depend continuously on u That is, a little perturbation on data produces a solution that is very different of the original ones For getting a well posed problem the regularization techniques are used [18, 20] Therefore based upon Tikhonov regularization [20], we consider the problem of finding ū η solution of J(ū η ) = min J η (u) (10) u=(δ,σ,f,g) where we have set J η (u) = G l G µ l 2 + I µ I µ 2 + η Lu 2 (11) for a given η such that u η converges toward the solution u as η 0 Here L is an operator used for stabilization (ie, L is the identity, a differentiation operator, etc) Our numerical simulations aims at identification of coefficients and functions parameters of the proposal model It is for this purpose we consider the control observed data corresponding to physical activity for a 30 - years old women that are generated via a global cardiovascular and respiratory exercise model from [37] The behavour of these control vis-a-vis P V O max 2 leads to get the observed data of glucose and insulin by using a minimal exercise model for plasma and insulin levels presented in [32] where P V O max 2 is percentage of maximum rate of oxygen consumption (V O 2 ) for an individual during exercise Thereafter numerical solutions are carry out using a collection of MaTlaB routines [21] for solving the optimization problem (10) and the ordinary differential system (1)-(2) Observed data of the heart rate and the alveolar ventilation are plotted in figures 2 The solution of glucose obtained from the model and its observed data are given in figures 3 Computed values obtained with MaTlaB routines are δ = and σ = (12) and the corresponding identified functions f and g are represented in figures (a) 74 (b) H(t) (beats/mn) V A (L/mn) Figure 2: Observed data for heart rate (a) and alveolar ventilation (b) to jogging exercise for a 30 - years old woman

8 338 Modeling glucose and insuline in diabetic human Glucose (mg/dl) Figure 3: The glucose where dashed line denotes the observed data while solid line is the output of the model solution We see that the two curves are very closed (a) (b) f(h,v A ) g(h,v A ) V A H V A H Figure 4: The identified functions f and g The expressions of the functions f and g must be fitted to achieve the identification of our model Based on data illustrated in figure 4, we use numerical iterative techniques for the minimization of a merit function that gives information about the goodness of the fit The following are solutions of fitting curves of functions f and g 1 Walking case (a) Model 1 (b) Model 2 f(h, V A ) g(h, V A ) V A V A H + H V A H V A H H f(h, V A ) V A H V A V A H g(h, V A ) V A H V A H 08581

9 JM Ntaganda, B Mampassi Jogging case (a) Model 1 (b) Model 2 3 Running Fast case V f(h, V A ) A V A H + H g(h, V A ) V A H V A H H f(h, V A ) V A H V A V A H g(h, V A ) V A H V A H (a) Model 1 (b) Model 2 V f(h, V A ) A V A H + H g(h, V A ) V A H V A H H f(h, V A ) V A H V A V A H g(h, V A ) V A H V A H Test results To test our models we consider the acute cardiovascular respiratory response to graded dynamic exercise in a 30 - year old trained women whose mean values are given in table 1 [30] Exercise intensity Rest Walking Jogging Running Fast Ventilation (L/min) Heart rate (Beats /min) Table 1: Cardiovascular and respiratory responses to physical activity for a women with 30-yearold The case of aerobic exercise is considered These mean values are for healthy subjects The autoregulation process states that the cardiovascular and respiratory systems evolves in the optimal way toward these values This suggests us to solving the following optimal control problem: min Gl Gl e 2 + I I e 2 + H H e 2 + V A V 2 Ae subject to the system (1)-(2) in hyperglycemia state We consider initial values (Gl, I) = (150mg/dl, 25µU/dl) Here Gl e is taken as equal to 150mg/dl which corresponds to the value of healthy subject H e and V Ae are means values given in table 1 We consider a woman with 30 years old in moderate physical activity during the same period five days per week Therefore we take S equals to 30 minutes and a recovery period of 150 minutes Test results for our model in the case of walking, jogging and running fast exercises are plotted respectively in figures 5, 6, and 7 where the dotted lines correspond to the first model while the dashed lines are related to the second model The solid lines represent desired mean values for the system In these figures we have depicted the curves of optimal solutions for each model described above

10 340 Modeling glucose and insuline in diabetic human 90 (a) 9 (b) H(t) (beats/mn) V A (L/mn) mg/dl) (c) Time (Minutes) Insulin sensitivity (d) Time (Minutes) Figure 5: Dynamics optimal for 30-year-old a woman in the walking case with hyperglycemia 160 (a) 16 (b) H(t) (beats/mn) V A (L/mn) mg/dl) (c) Time (Minutes) Insulin sensitivity (d) Time (Minutes) Figure 6: Dynamics optimal for 30-year-old a woman in the jogging case with hyperglycemia 200 (a) 30 (b) 25 H(t) (beats/mn) 150 V A (L/mn) mg/dl) (c) Time (Minutes) Insulin sensitivity (d) Time (Minutes) Figure 7: Dynamics optimal for 30-year-old a woman in the running case with hyperglycemia

11 JM Ntaganda, B Mampassi 341 The figures 5, 6 and 7 illustrate the dynamic optimal for a 30-year-old woman with hyperglycemia For different exercises (walking, jogging and running fast), we see that the heart rate and alveolar ventilation are increasing for oscillating during a period between the onset of physical exercise and ten minutes around the desired value where they stabilize themselves This is in accordance with the fact that during physical activity, heart rate and alveolar ventilation raise up and reach a level depending of the exercise intensity and before they stabilize themselves In physical activity the functions H(t) and V A (t) play a crucial role for controlling cardiovascular respiratory system parameters such as blood pressures [30] Consequently they have a great influence in controlling the diseases related to this system such as Type II diabetes In particular, it is required that these parameters must be stabilized to ensure good healthy conditions of patients Plasma glucose response to heart rate and alveolar ventilation functions are plotted in figures 5, 6 and 7 (c) They show that the plasma glucose decreases to reach a value less than the desired value in jogging (see the figure 6)and running (the figure 7) cases in the first fifty minutes since the onset of physical activity After this period it increases to be stabilized to desired value but the oscillations around this value occur for a period between 30 and 120 minutes in the jogging case The figure 5 (c) illustrates the decreasing of plasma glucose to be stabilized to desired value at 120 minutes of onset of physical activity We notice that in all cases of physical activity the plasma glucose is stabilized in recovery time that is after 30 minutes of onset onset of physical activity The response of heart rate and alveolar ventilation to plasma insulin is shown in the figures 5, 6 and 7 (d) where the insulin sensitivity increases when plasma glucose is decreasing This insulin sensitivity and plasma insulin are stabilized at the same time during the recovery period 5 Concluding remarks In this work we have investigated a mathematical model that describes plasma glucose and the plasma insulin sensitivity responses to cardiac and respiratory parameters (heart and ventilation rate) The cardiovascular and respiratory system is comprised of a multitude of elements The increasing necessity to interpret the meaning of measurable variables such as heart rate, ventilation capacity, and plasma glucose and plasma insulin under both physiological and pathological conditions has imposed the need for relatively simple models that should be able to describe as accurately as possible the mechanical behavior of the system The modelling technique used in present work provides interesting answers to the question of determining the best frequency and the breathing capacity during efforts Numerical simulations give rise to interesting conclusions Notably the model would helpful for the control of some sportsmen performances References [1] G Ahlborg, P Felig, L Hagenfeldt, R Hendler, J Wahren, Substrate turnover during prolonged exercise in man, J Clin Invest 1974 Apr;53(4): [2] G Ahlborg, P Felig, Lactate and glucose exchange across the forearm, legs, and splanchnic bed during and after prolonged leg exercise, J Clin Invest 1982 Jan;69(1):45-54 [3] G Ahlborg, J Wahren, P Felig, Splanchnic and peripheral glucose and lactate metabolism during and after prolonged arm exercise, J Clin Invest 1986 Mar;77(3):690-9

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14 344 Modeling glucose and insuline in diabetic human [37] S Timischl, A global Model for the Cardiovascular and Respiratory System, PhD thesis, Karl-Franzens-Universit of Graz, August 1998 [38] B Topp, K Promislow, G De Vries, R M Miura and D T Finegood, A Model of β- cell mass, insulin, and glucose kinetics: pathways to diabetes, J Theor Biol 206 (2000), [39] R C Turner, CA Cull, V Frighi, RR Holman, Glycemic control with diet, sulfonylurea, metformin, or insulin in patients with type 2 diabetes mellitus: progressive requirement for multiple therapies (UKPDS 49), UK Prospective Diabetes Study (UKPDS) Group, JAMA, 1999; 281: [40] M Vidyasagar, Nonlinear systems analysis, SIAM, Second Edition, 2002 [41] Wahren J, Felig R, Ahlborg G, Jorfeldt L, Glucose metabolism during leg exercise in man, J Clin Invest 1971 Dec;50(12): [42] Wasserman DH, Geer RJ, Rice DE, D Bracy, Flakoll PJ, Brown LL, Hill JO, N Abumrad, Interaction of exercise and insulin action in humans, Am J Physiol 1991 Jan;260(1 Pt 1):E37-45 [43] Wasserman DH, Cherrington AD, Hepatic fuel metabolism during muscular work: role and regulation, Am J Physiol 1991 Jun;260(6 Pt 1):E [44] Wasserman DH, Lacy DB, Green DR, Williams PE, Cherrington AD, Dynamics of hepatic lactate and glucose balances during prolonged exercise and recovery in the dog, J Appl Physiol 1987 Dec;63(6): [45] Wolfe RR, Nadel ER, Shaw JH, Stephenson LA, Wolfe MH, Role of changes in insulin and glucagon in glucose homeostasis in exercise, J Clin Invest 1986 Mar;77(3):900-7 Jean Marie Ntaganda Department of Applied Mathematics National University of Rwanda BP117, Butare, Rwanda jmnta@yahoofr Benjamin Mampassi Depatment of Mathematics and Computer Science Cheikh Anta Diop University Senegal mampassi@yahoofr