A fractional-order control model for diabetes with restraining and time-delay

Even though diabetes is a silent killer and one of the world's leading public health issues, people can take preventative measures by becoming aware of its causes. This study aims to identify the importance of treatment function and then control the complications of various individuals. We present a mathematical model of diabetes (type-2 diabetes) based on insulin therapy as a controlling factor. With fractional-order delay differential equations, four parts of the population control the dynamic system. The well-posedness (positivity, boundedness) of the model is examined to show that it is biologically and mathematically relevant. According to the characteristics equations for the model, certain sufficient conditions must be met for diabetic-free, endemic equilibrium points to be stable locally. To assess the imbalanced glucose level and treatment over a finite time period, we construct an optimal control problem based on treatment control and awareness program control as time-dependent control parameters. A necessary and sufficient condition for optimality is examined. In order to determine the most cost-effective treatment strategy with limited resources, we assessed the effectiveness and costs of treatments. The theoretical findings are verified by numerical simulations.