Analysis and dynamical structure of glucose insulin glucagon system with Mittage-Leffler kernel for type I diabetes mellitus

In this paper, we propose a fractional-order mathematical model to explain the role of glucagon in maintaining the glucose level in the human body by using a generalised form of a fractal fractional operator. The existence, boundedness, and positivity of the results are constructed by fixed point theory and the Lipschitz condition for the biological feasibility of the system. Also, global stability analysis with Lyapunov’s first derivative functions is treated. Numerical simulations for fractional-order systems are derived with the help of Lagrange interpolation under the Mittage-Leffler kernel. Results are derived for normal and type 1 diabetes at different initial conditions, which support the theoretical observations. These results play an important role in the glucose-insulin-glucagon system in the sense of a closed-loop design, which is helpful for the development of artificial pancreas to control diabetes in society.


Basic concepts
Definition 1 23,24 For a power law kernel in the Riemann-Liouville sense is given as; with 0 ≤ α, η ≤ 1 .Where The corresponding power law kernel fractal-fractional integral of order (α, η) is given as Definition 2 23,24 Assume that x(t) is a function that is not constantly differentiable.For a exponential decay kernel in the Riemann-Liouville sense is given as; (1) where α > 0, η ≤ 1 , and H(0) = 1 = H(1) .The corresponding exponential decay kernel fractal-fractional inte- gral of order (α, η) is given as Definition 3 23,24 Assume that x(t) is a function that is not constantly differentiable.For a Mittag-Leffler kernel in the Riemann-Liouville sense is given as; where 0 < α, η ≤ 1 , E α is the Mittag-Leffler function and AB(α) = 1 − α + α Ŵ(α) is a normalization function.The corresponding Mittag-Leffler kernel fractal-fractional integral of order (α, η) is given as

Diabetes model with fractional derivative
For our motivation, we consider the glucose-insulin-glucagon system given in 8 , which explains the relation and role of insulin and glucagon in maintaining the glucose level in the human body to overcome the risk of death.We construct the fractional order model in followings equations with initial condition G(0) = G 0 ≥ 0, I(0) = I 0 ≥ 0, β(0) = β 0 ≥ 0, α(0) = α 0 ≥ 0, J(0) = J 0 ≥ 0 Assume that glucagon is produced by the α-cells at low glucose concentrations in order to increase hepatic glucose synthesis, which raises blood glucose levels.An excessive rise in blood glucose levels is prevented by the production of insulin by the β-cells.G(t) represents the dynamics of glucose.The term δ i J refers to glucagon's effect on liver gluconeogenesis, which produces glucose.The blood glucose level increases at a rate ω (the rate of glucose gen- eration by the liver and kidneys) and falls at a rate bG (independent of insulin) and δ (the rate of glucose uptake as a result of insulin sensitivity).where γ is the maximum rate of insulin secreted by β-cells and µI is the rate of kidney clearance of insulin.The dynamics of insulin are represented by I(t).We assumed a logistic equation, where ρ and ρ i represent the growth rates of the β and the α cell masses, respectively.v and v i represent the car- rying capacities of the β and the α cell masses, respectively.The glucagon J(t) is released if the glucose level falls below a specific threshold (G < G l ).

Positivity and boundedness of solutions
Here, we demonstrate the suggested model's positivity and boundedness.
Theorem 1 Assume the initial condition be then if the solutions {G, I, β, α, J} exist, they are all positive for all t ≥ 0.
Proof Start with the basic analysis to demonstrate that responses are superior because they demonstrate realworld issues with positive values using the methodology described in 25,26 .This section looks at the conditions necessary for the proposed model to provide positive results.We'll describe the norm where the domain of h is D h .Let's begin with the G(t) class (4) FFM 0 where the time component is r.This illustrates that for any t ≥ 0 , G(t) is positive.For the function I(t)

This yield
where the time component is r.This illustrates that for any t ≥ 0 , I(t) is positive.For the function β(t)

This yield
where the time component is r.This illustrates that for any t ≥ 0 , β(t) is positive.For the function α(t)

This yield
where the time component is r.This illustrates that for any t ≥ 0 , α(t) is positive.For the function J(t) , ∀t ≥ 0

Theorem 2
The diabetes fractional order model have distinct solution and constrained in R 5 + .
Proof System given in ( 7) is investigated with positive solution given as follows: If (G(0), I(0), β(0), α(0), J(0)) ∈ R 5 + , so that the solution must be from hyperplane.The domain R 5 + is a positive invariant with non-negative orthant because the vector field is enclosed with each hyperplane.

Existence and uniqueness analysis
The most crucial application of non-linear functional analysis is the use of fixed point theorems to demonstrate the existence of any non-linear system.Using fixed point contractions, non-linear functional analysis shows the point at which every given non-linear system exists.Fixed point mappings that are defined in Banach space ensure thorough investigation of the existence of unique solutions.The examined model (7) has at least one solution in [0, T] according to a fixed point mapping theorem 27 .Consider the system (7) as The following is a reformulation of (25) in the form of a Fractal-Fractional integral for the Mittag-Leffler kernel as expressed in (6).

Theorem 3 The non-linear map
on a fully normed space.Where the norm is (i) Firstly, we will show that M(A 1 , B 1 , C 1 , D 1 , E 1 ) is a contraction map.For G(t) and Ĝ(t) , we have where P A = �b + δI αG+1 � .Using this approach, we have where This implies that, for the operator M(G, I, β, α, J) , we have where P = max[P A , P B , P C , P D , P E ] < 1 is a Lipschitz constant.This implies M(A, B, C, D, E) is a non- expansive operator.
(ii) Now we will show that N(A 2 , B 2 , C 2 , D 2 , E 2 ) is continuously compact.The absolute modulus of all posi- tively bounded continuous operators A, B, C, D, E specified in (27) given by the non-zero positive constants , ℵ E meeting the following bounded-ness inequalities, illustrates the compactness of the operator Similarly, we find proceeding this process, we find the maximum norm of where ξ is a positive constant.Therefore, is a uniformly bounded operator.Now we will prove that is equi-continuous for t x < t y ∈ [0, T] .For this purpose, we have for Similarly, Since t 2 → t 1 is independent of (G, I, β, α, J) .This implies that is relatively compact by Arzela's theorem.As a result, the Krasnoselski theorem follows, which states that the contraction and continuity of the operators M and N ensure the existence of a single unique solution.

Equilibrium points analysis
The system given in ( 7) is solved for equilibria.we have

Stability analysis
Global stability is analyzed for the proposed system as follows.
Lemma 1 Let h ∈ R + represents the continuous function for which any t ≥ t 0 ,

First derivative of Lyapunov
Lyapunov function for the endemic, {G, I, β, α, J} , L < 0 is the endemic equilibrium points E * .

Theorem 5
The endemic equilibria E * for the model are globally asymptotically stable, If the reproductive number R 0 > 1.
Proof Suppose that the Volterra-type Lyapunov function as: Where C i , i = 1, 2, 3, 4, 5 are positive constants will be considered later.Then putting Eq. (48) into main system and using Lemma (3.1).
After the substituting the values of the derivative derivatives, we have.
t J with their respective formula from the proposed model (7), we can get 1 : represents all positive terms 2 : represents all positive terms So that

Computational analysis with fractal fractional operator
In this section, By using Mittag-Leffler Kernel for diabetes model given in (7), we get the simplist form as follows.

Numerical simulation and discussion
In this section, the numerical simulation of the proposed method using the mittage-leffler kernel for the diabetes model is discussed.The system's parameter values and initial conditions 8 are listed in table 1 that is given below The effectiveness of the obtained theoretical outcomes is established by using advanced techniques.The mathematical analysis of diabetes with hormonal effects is analysed through simulation in Figs. 1, 2, 3 at different (81) initial conditions.In Figs. 1, 2, 3, solutions for all compartments are shown with different fractional order values at a fixed fractal dimension of the system.Matlab coding is employed to find the numerical simulation for the fractional-order diabetes model.A range of values for ψ(ψ = 0.85, 0.90, 0.95, 1) are shown to illustrate the dynamic effects.Interestingly, glucose, β mass, and α mass decrease while insulin and glucagon increase when taking into account a fractional order.Still, there is a marked drop in gulcagon and insulin as ψ gets closer to 1.The findings highlight how ψ affects the system's dynamics.The normal glucose concentration level in human blood is in a narrow range (80-180 mg/dl).In Similarly, by changing the initial condition again as a third case to be considered to see its behavior within the bounded domain.It is observed that glucose levels decrease due to the rise in insulin by β− cells, which will be helpful for diabetes patients.Diabetes patients will approach a stable position due to a rise in insulin level and maintain it after a certain period.It also demonstrates that when there is hypoglycemia, the α-cells secrete glucagon to keep the blood sugar levels within the normal range.This finding highlights the value of fractional calculus in explaining the intricate and persistent behaviour of the model and reveals the system's innate stability and resilience.As such, the fractional dimension ψ becomes crucial in the It predicts what should happen in the future through this research and how we will be able to reduce the spread of diabetes in society.The fractal-fractional method provides reliable findings for all compartments according to steady state at non-integer order derivatives as compared to classical derivatives.

Conclusion
In this work, the fractional order diabetes model is studied to impact inulin and glucagons for administrations of gulose in the human body.In this regard, qualitative and quantitative properties of the analysis, such as global stability, uniqueness of the solution, and positivity with fixed point theory, result.Results through figures are derived with the help of a fractal fractional operator utilising the Mittag-Leffler kernel, which provides us with continuous monitoring of the glucose-insulin relationship in the human body at different fractional order values.It is observed that maintaining the glucose level within the usual range is the major responsibility of the β and α cells in the pancreas to produce the hormones insulin and glucagon, respectively.However, diabetes can result from β-cell and α-cell malfunction.The research on how glucose, insulin, β-cells, α-cells, and glucagon interact has been avoided.These results play a key role in the study of the glucose-insulin-glucagon relationship, which is helpful for close-loop design (artificial pancreas) to control type 1 diabetes.A closed-loop design for

Fig. 1 ,
glucose level, insulin level, and α− cell mass with low glu- cose concentrations in people with diabetes decrease by decreasing fractional values, while the β− cell which produces insulin and glucagon, starts rising by decreasing fractional values.Similar behavior can be seen in Fig 2 with minor change in the initial condition.

Table 1 .
Parameter values and biological interpretation.