AN ORDINARY DIFFERENTIAL EQUATION MODEL OF DIABETIC POPULATION IN NEW DELHI

Transcription

1 Indian Journal of Mathematics and Mathematical Sciences Vol. 7, No. 1, (June 2011) : AN ORDINARY DIFFERENTIAL EQUATION MODEL OF DIABETIC POPULATION IN NEW DELHI Sandhya, Deepak Kumar & Prerna Pandit Abstract This is a population based mathematical model of Diabetes Mellitus patients. The Mathematical model explained in Ordinary Differential Equations which contain percentages of uncontrolled diabetic patients and controlled diabetic patients in Delhi. By suitable definition of a parameter the solution discussed the critical nature of percentages of uncontrolled diabetic patients and controlled diabetic patients are analyzed for stability. 1. INTRODUCTION It was noted in BBC WORLD web page, that the incidence of diabetes is increasing because people are living longer, getting fatter and leading inactive lifestyle. Diabetes is caused by increased level of glucose in blood. When carbohydrate is taken by body it get converted in to glucose which is transmitted to blood cells through blood with the help of insulin. Insulin has an important role for the control of intermediatery metabolism. In case of normal human being, insulin is released from beta cells in the pancreas which usually absorb the excess glucose present in the blood and convert it into energy or fuel. Insulin also stimulates liver to absorb & store any glucose left over. Diabetes mellitus is a metabolic disorder characterized by the decreased ability or complete inability of the tissues to utilize carbohydrate. The disorder is due to a deficiency or diminished effectiveness of the hormone insulin. Diabetes mellitus can be classified as: In 1980 expert committee of WHO (World Health Organization) proposed 2 classification of Diabetes Melitus and named it as IDDM (Insulin Dependent Diabetes Mellitus) or Type1 and NIDDM (Non Insulin Dependent Diabetes Mellitus) or Type2. But in 1985 Type 1 and Type 2 names were omitted and only IDDM & NIDDM were known as the types of Diabetes Mellitus [1]. IDDM or Type 1 is usually occurs before the age of 40 years. The disease usually has an abrupt onset and the abnormality of carbohydrate metabolism is severe due to lack of endogenous Insulin to control blood glucose levels. In this case insulin is provided

2 46 Sandhya, Deepak Kumar & Prerna Pandit by outside sources like injections. These types of patients usually are obese and may even be underweight. NIDDM or Type 2 usually occurs in the middle life or beyond the age of 40 years although it may also occur in younger age group. The patients are most oftnely obese. In this case, the body is able to produce insulin but that insulin is not affective because of destruction of beta-cells. Majority of patients of type 2 improve with weight loss and are maintained on diet therapy, which may sometimes be assisted by an oral hypoglycemic drug. The majority of people with diabetes mellitus are NIDDM (Non Insulin Dependent Diabetes Mellitus). Sometimes it becomes difficult to distinguish between IDDM and NIDDM. IDDM is usually diagnosed before the age of 40 and is treated by insulin injection and Diet but NIDDM is usually diagnosed after the age of 40 and may be treated by diet alone or by diet and tablets both [2]. Thus the people with NIDDM who are treated with insulin may be classified wrongly as having IDDM and the overall ratio of NIDDM and IDDM will be distorted. Also sometimes the symptoms of NIDDM are not evident until the complications develop. A. Boytabyeb et al., (2006). 1.1 Diabetes Mellitus in India Diabetes mellitus is the disease of millions and if the projections are right, a quarter billion people across the globe will be diabetic by the year The prevalence of DM for all age groups worldwide was estimated to be 2.8% in 2000 and an anticipated 4.4% in The total number of people in the world with diabetes is projected to rise from 171 million in 2000 to 366 million in 2030 [3]. India ranked number one as the country with the highest number of diabetes patients in 1995, at 31.7 million in 2000, with a projected 57.2 million in 2025, and 79.4 million in 2030, retaining its top position [4]. 1.2 Diabetes Mellitus in Delhi There is a genuine secular increase in incidence of DM in Delhi. It was noted in BBC WORLD web page, that the incidence of diabetes is increasing because people are living longer, getting fatter and leading inactive lifestyle. According to the study of SITE (Screening India s Twin Epidemic) 62% of the diabetic patients in Delhi having complications and rest are having no complications this report is generated by Aventis Pharma Limited (Sanofi-Aventis Group (2010)). The concept of steady state has relevance in many fields, in particular thermodynamics and economics. Steady state is a more general situation than dynamic equilibrium. If a system is in steady state, then the recently observed behavior of the system will continue into the future. In stochastic systems, the probabilities that various different states will be repeated will remain constant. A

3 An Ordinary Differential Equation Model of Diabetic Population stable condition that does not change over time or in which change in one direction is continually balanced by change in another. 2. MATHEMATICAL MODEL This paper a mathematical model of diabetes mellitus will be analysed which contains diabetes mellitus with complication and without complication. The diabetic patient s condition is worsened by the appearance of complications of diabetes like Diabetic retinopathy, Diabetic neuropathy, Diabetic nephropathy, Diabetic foot ulcers. Whether these are the unavoidable consequences of the diabetic state over time or whether they may be influenced by controlling the diabetes through aggressive monitoring, treatment and life-style management, including diet and supplements, remains a central topic. This model will finally give the size of the people with complication and without complication. In this paper we use the Runge-Kutta Method to solve this mathematical model. C : D : I : C (t) : percentage of uncontrolled diabetic patients D (t) : percentage of controlled diabetic patients. Incidence of Diabetes Mellitus (constant){no. of cases diagnosed in time t & assumed that they have controlled diabetes }. µ : Natural Mortality Rate. λ : γ : ν : δ : Probability of Diabetic person developing uncontrolled diabetes. Rate at which uncontrolled diabetes are cured. The rate at which diabetics patients with uncontrolled diabetes becomes severely disabled. The mortality Rate Due to uncontrolled diabetes. Rate of Change is defined by ODE dd D (t) = I () D C dt = λ + µ + γ (1) dc C (t) = D () C dt = λ λ + µ + ν + δ (2) Let α = λ + µ and β = λ + γ + δ + µ. Then Equation (1) and (2) becomes D (t) = dd I D C dt = α + γ (3)

4 48 Sandhya, Deepak Kumar & Prerna Pandit Let C (t) = dc D C dt = λ β. (4) D = F (D, C) C = G (D, C). We assume the initial conditions as D (0) = D 0, C (0) = C 0. Now by R-K Method the solution of (3) and (4) is given by D n + 1 = D n (k 1 + 2k 2 + 2k 3 + k 4 ) (5) where, C n + 1 = C n (L 1 + 2L 2 + 2L 3 + L 4 ) (6) k 1 = hf (D 0 ); L 1 = hg (D 0 ) k 2 = k1 L1 hf D0 +, C ; L = k1 L1 hg 2 D 0 +, C k k 3 = 2 L hf 2 D0 +, C ; L = k2 L hg 2 3 D 0 +, C k 4 = hf(d 0 + k 3 + L 3 ); L 4 = hg(d 0 + k 3 + L 3 ) 3. NUMERICAL SOLUTIONS This figure shows the percentage of controlled diabetic patients for 100 years. Matlab 7.0 is used to solve the Eq. (2). The behavior of graph reached the steady state after 50 years. Figure 1: Graph of the Percentage of Controlled Diabetics Patients for 100 Years

5 An Ordinary Differential Equation Model of Diabetic Population This figure shows the percentage of uncontrolled diabetic patients for 100 years. Matlab 7.0 is used to solve the Eq. (1). The behavior of graph reached the steady state after 50 years. Figure 2: Graph of the Percentage of Uncontrolled Diabetics Patients for 100 Years 4. CONCLUSION AND DISCUSSION In this paper we analysed the non linear model of diabetic patients in New Delhi. To monitor the percentage of controlled and uncontrolled diabetic patients numerical methods were introduced. From the graph it is concluded that stability is obtained in the percentage of both types of controlled and uncontrolled diabetic patients. The analysis is done for the next 100 year. APPENDIX Detailed Solution of Mathematical Model: ν = 0.05, δ = 0.05, µ = 0.02, γ = 0.08, λ = βc N 0 0 = 62 = 62, D 0 = 38, N 0 = C 0 + D 0, α = λ + µ = = 0.64, β = γ + υ + µ + δ = = 0.2, I = 33 REFERENCE [1] Sarah Wild, Gojka Roglic, Anders Green, Richard Sicree, and Hilary King, (2004), Global Prevalence of Diabetes, Estimates for the Year 2000 and Projections for Diabetes Care, 27, [2] A. Boutayeb, A.Chetouani, A. Achouyab, and E. H. Twizell, (2006), A Non-Linear Population Model of Diabetes Mellitus, J. Appl. Math. & Computing, 21(1),

6 50 Sandhya, Deepak Kumar & Prerna Pandit [3] Jitender Nagpal, and Abhishek Bhartia, (2006), Quality of Diabetes Care in the Middleand High-Income Group Populace, Diabetes Care, 27, [4] Deepak Kumar, Sandhya, Anuj Kumar, and Prerna Pandit, (2009), A Computational Model for Diabetes Mellitus, Optimization-Journal of Research and Management, [5] Aparna THOMAS, (2010), Sanofi-Aventis India SITE Study Shows 62% of Diabetic Population in New Delhi has Uncontrolled Diabetes, Screening India s Twin Epidemic. [6] Anirudh B. Acharya, Aparna Satyanarayan, and Srinath L. Thakur, (2010), Status of Association Studies Linking Diabetes Mellitus and Periodontal Disease in India, International Journal of Diabetes in Developing Countries, 30, Sandhya &Deepak Kumar Department of Mathematics, Faculty of Engineering & Technology, Mriu Faridabad, Haryana, India. Prerna Pandit College of Pharmacy, IILM Academy of Higher Learning, Greater Noida, (U.P.), India.