Singular Perturbation Analysis for Identification of Dynamic Behaviour and Stability of a Nonlinear Model of Long Term Progression of Diabetes Mellitus CHONTITA RATANAKUL 1,2 , YONGWIMON LENBURY* 1,2 , JEERAWAN SUKSAMRAN 2,3 1 Department of Mathematics, Faculty of Science, Mahidol University Rama 6 rd., Bangkok 10400, THAILAND 2 Centre of Excellence in Mathematics, PERDO, THAILAND 3 Department of Mathematics, Faculty of Science King Mongkut’s University of Technology North Bangkok, Bangkok 10800, THAILAND *corresponding author: scylb@mahidol.ac.th http://www.sc.mahidol.ac.th/scma Abstract: - There have been numerous attempts to model the progression of Diabetes Mellitus, which is a disease suffered by those with eating disorders with prevalence in the aged population. Models in the past have not been very successful in discovering the future development of the symptoms in a long term prediction. This is due to the fact that the state variables under consideration change in drastically different time scales, and the models that do not take careful account of this are not able to provide sufficiently accurate forecast that can be of satisfactory assistance to physicians taking care of their patients. In this work, we use the singular perturbation method to analyse a model of insulin and glucose interaction, incorporating beta cell dynamics and the pancreatic reserve, proposed by De Gaetano et al. in 2008. Different dynamic behaviour will be identified and numerical simulations will be carried out in support of our theoretical predictions. Key-Words: - nonlinear model, stability, singular perturbation, Diabetes Mellitus, dynamic behavior Received: April 27, 2020. Revised: September 23, 2020. Accepted: October 30, 2020. Published: November 2, 2020. 1 Introduction Drug resistance has been a grave concern for many. According to [1], diabetes is on the rise across the globe. IDF’s statistics shows that at the present time it is estimated that every seven seconds someone dies from diabetes or its complications, 50% of which deaths occurring to those under the age of 60 years, amounting to a total of 4 million deaths per year. The global diabetes prevalence is 8.8% (95% confidence interval 7.2-11.3%) of the world population in 2017, standardized for the age group 20-79 years, according to [2]. In [3], the statistics indicates that, in 2017, Thailand has a population of 4.8 million who suffer from diabetes, only half of which has been diagnosed with the disease. In addition, it was reported in [4] that one in eleven people in the Thai population, at a mature age, is diagnosed with diabetes in 2016, while the World Health Organization reported that during 2009 - 2014, the number of diabetic patients has risen 4 folds and over 70 thousand died from illnesses related to diabetes each year [4]. Glycemia and insulinemia are regulated through a negative feedback loop in which  -cells are stimulated by plasma glucose to release insulin leading to insulin-mediated increased tissue glucose uptake and decreased liver gluconeogenesis and glycogenolysis [5]. In 2001, Lenbury et al. [6] proposed a nonlinear mathematical model of the glucose-insulin control mechanism, incorporating the function of beta-cells in maintaining and regulating plasma insulin level in human. A gastrointestinal absorption term is utilized to model glucose absorption by the intestine and the entry of glucose into the bloodstream, assuming that this process takes place at a given rate initially but declining exponentially with time. The model is analysed using the singular perturbation method by which the delineating conditions on the system parameters are derived to identify different dynamic behaviour, including the existence of limit cycles in the system model which mimic oscillatory patterns often observed in clinical data. A sinusoidal term is WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.57 Chontita Ratanakul, Yongwimon Lenbury, Jeerawan Suksamran E-ISSN: 2224-2880 523 Volume 19, 2020