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Although many mathematical models have been presented for glucose and insulin interaction, none of these models can describe diabetes disease completely. In this work, the dynamical behavior of a regulatory system of glucose-insulin incorporating time delay is studied and a new property of the presented model is revealed. This property can describe the diabetes disease better and therefore may help us in deeper understanding of diabetes, interactions between glucose and insulin, and possible cures for this widespread disease.

Diabetes, technically called diabetes mellitus, is referred to types of disorders in the metabolic processes of the human body in which the controlling mechanism of sugar level in blood is disrupted. In these cases, insulin, the mainstay controlling element, is either not secreted or its presence is ignored by body cells [

The function of insulin alters for each organ in the human body, so the effects of environmental factors like stress and nourishing habits may cause blood glucose shift. As observed in various countries, diabetes is discernibly widespread and there is an increasing number of people suffering from this. Hence, the potentially lethal symptoms of the illness necessitate more meticulous treatments and precautionary activities. The essence for such cure procedures is even more accentuated in contemporary hectic life in which people have an increasing penchant to be nourished by artificially cultivated foods and do less exercise. The number of people suffering from this disease was approximately 415 million in 2015 with equal shares of both genders, which accounted for 8.3% of the overall adult population of the world. And nearly 1.5 to 5 million people have died because of diabetes every year between years 2012 and 2015 worldwide [

Speaking of the reasons triggering this illness, many elements can cause this irregularity behavior in the body, such as genetic factors inherited through generations that fertilize the body for other factors of the disease to easily disrupt the metabolic system, obesity due to malnutrition and urbanization as consequent of modern lifestyle, side effects of taking specific drugs like glucocorticoids and thyroid hormone, progression of other illnesses, and many other factors which cannot be wholly included [

Besides the paramount and distinctive importance of experimental researches for developing effective treatment protocols, studying and developing mathematical models of glucose-insulin bilateral interplay have had an essential role in accelerating the research processes and making breakthroughs in this field by saving both money and time. Conventionally, it was believed that a linear relationship defines the mechanism of glucose-insulin negative feedback system. A linear model for diabetes assumes that the relationship between glucose and insulin concentration could be studied in isolation from other components [

As delineated in the preceding paragraph, various mathematical models have been proposed in attempts to simulate the relation between plasma glucose concentration and plasma insulin concentration more accurately, so that scientists will be able to have an elaborate perspective of this metabolic interaction [

The investigation of chaotic dynamics has attracted the foci of many scientists, and a great deal of effort is put in this field as it has provided a successful method for studying biological systems [

Because of the complexity of the system, the model that has been studied in this paper is a nonlinear model. The nonlinear model that we study reflects the relationship between injected insulin and blood glucose response. The studies about variation in the blood glucose indicate a chaotic component.

In the second section, the dynamical properties of the last presented model for glucose and insulin concentration are investigated. Eventually, conclusion remarks are given in Section

In 1964, Ackerman et al. [

In 1987, Bajaj et al. [

In the current research, we study a nonlinear mathematical model for glucose-insulin feedback control system by incorporating the enhanced delay differential equations embracing

Two time delays in glucose-insulin system [

According to the model presented by Molnar et al. [

Based on the study by Chuedoung et al. [

In current research, the new capability of the mentioned model is revealed. By increasing the insulin secretion delay by

The model bifurcation diagrams based on different values of parameter

With the increase in glucose response time caused by insulin secretion (

The model bifurcation diagrams based on different values of parameter

In the present study, we used the insulin-glucose model involving

A new sliding mode control scheme for a class of uncertain time-delay chaotic systems is proposed in [

In this section, we design the adaptive sliding mode controllers to suppress the chaotic oscillations in the model presented in (

We define the integral sliding mode surface as

The sliding surface dynamics can be derived as

The parameter estimation errors are defined as

The first derivatives of the estimation errors are

Consider the following Lyapunov function:

The first derivative of the Lyapunov candidate function is

Applying (

By introducing uncertainties without changing the definition in (

After some mathematical simplifications, let us define the adaptive sliding mode controllers as

The parameter estimate laws can be defined as

Using (

As

Time history of the states with control in action at

Time history of the states with control in action at

Time history of parameter estimates.

Implementation of chaotic and hyperchaotic systems using Field Programmable Gate Arrays (FPGA) has been widely investigated [

In this section, we implement a circuit for the model (

Overall RTL schematics of (

RTL schematics of the implemented delay (

Xilinx RTL schematics of the controllers with sliding surfaces and parameter estimate laws.

Time history of the delay (

Time history of the parameter estimates with controllers in action at

By studying presented mathematical models for glucose-insulin interaction, according to the value of parameters, a chaotic model for describing the glucose-insulin regulatory system was found. In the present study, it is expected to observe periodic behavior in the proposed system under normal metabolic conditions and chaotic behavior under abnormal metabolic conditions. It is noteworthy to say that the chaotic behavior of a system is a sign of a faulty condition in the biological systems [

The effect of two time delays on glucose-insulin regulatory system was investigated. Two main results of this study are listed as below.

If the time lag of insulin response to glucose increases, system exits from periodic region and enters to chaotic region, and if there is more delay on response of glucose to insulin secretion, the system displays chaotic manner, which was in line with previous studies [

If there is more delay on the response of glucose to insulin secretion, the behavior of system alters from periodic to chaotic

The proposed system can explain the interaction between glucose and insulin concentration in both normal and abnormal (diabetes disease) situations. Also, a control method was investigated with the hope of possible clinical applications.

The authors declare that they have no conflicts of interest.