Mathematical analysis with numerical solutions of the mathematical model for the complications and control of diabetes mellitus

Abstract

IntroductionA model is an abstraction that reduces a problem to its essential characteristics. Mathematical models are useful because they exemplify the mathematical core of a situation without extraneous information. Models help to explain a system and to study the effects of different components, and to make predictions about behaviour. Analysis of model via computational and applied mathematical methods are ways to deduce the consequences of the interactions. It is the analysis of mathematical models that allows us to formalize the cause and effect process and tie it to the biological observations. Furthermore, model analysis yields insights into why a system behaves the way it does, thus providing links between network structure and behaviour.MethodologyStability natures of the critical points of the models at various values of the model parameters were investigated to determine the behaviour of the model solution. Eigenvalue sensitivity and Eigenvalue elasticity analyses were carried out to identify key parameters of the model which drive the solutions and to figure out the effect of proportional changes in parameter values on population growth of both diabetics with complications and diabetics with and without complications. Mathematical algorithms were coded in MATLAB computational environment to achieve these. The numerical solutions of the model at various values of the parameters were performed using Euler method and Runge-Kutta method of order four and compared with the analytic solution. The algorithms were coded with Maple software package.ResultThe stability analysis showed that the models were asymptotically stable at specified parameter values hence suitable for their intended purposes. The Eigenvalue sensitivity and Eigenvalue elasticity analyses showed the rate at which complications were controlled is the most important parameter of the model hence the policy lever for effective control of the size of diabetics with complications. The solutions were represented graphically at various values of prominent parameters.ConclusionThe population of diabetics will continue to increase for the time being, but the size of diabetics with complications can reduce drastically with comprehensive and concurrent treatment of Diabetes mellitus and its complications. Also with high rate of controlling complications of Diabetes mellitus and low probability of developing complications of the disease through interventions such as continuous education, reorientation, increase physical activities, balance nutrition, government and non-governmental support, the incidence of the disease reduce drastically.