
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

38580678
58132
10.1038/s41598-024-58132-5
Article
Analysis and dynamical structure of glucose insulin glucagon system with Mittage-Leffler kernel for type I diabetes mellitus
Batool Maryam 1
Farman Muhammad farmanlink@gmail.com

12
Ghaffari Abdul Sattar 1
Nisar Kottakkaran Sooppy 34
Munjam Shankar Rao 4
1 https://ror.org/0161dyt30 grid.510450.5 Institute of Mathematics, Khawaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, Pakistan
2 https://ror.org/00hqkan37 grid.411323.6 0000 0001 2324 5973 Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
3 https://ror.org/04jt46d36 grid.449553.a 0000 0004 0441 5588 Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj, 11942 Saudi Arabia
4 https://ror.org/01j4v3x97 grid.459612.d 0000 0004 1767 065X School of Technology, Woxsen University, Hyderabad, Telangana 502345 India
5 4 2024
5 4 2024
2024
14 805817 7 2023
26 3 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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.

Keywords

Mittage-Leffler Kernel
Boundedness
Uniqueness
Lyapunov Stability
GIG system
Subject terms

Applied mathematics
Physiology
Prince Sattam Bin Abdulaziz UniversityRSP2023R167 Nisar Kottakkaran Sooppy issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Specifically, blood sugar levels are regulated by two hormones that have antagonistic effects on the human body. In contrast to insulin, which promotes the uptake of blood sugar by muscles and adipose tissues and stores it as glycogen in the liver, glucagon is secreted to treat hypoglycemia following fasting or meals without carbohydrates, but insulin is released to prevent blood sugar levels from going above a specific threshold after meals. Because of this, it will be challenging to control glycemia whenever glucagon or insulin secretion is disrupted. Research on α-cells and glucagon is less important than that on β-cells and insulin, despite the significance glucagon plays in the regulation of blood sugar1. According to studies, type 2 diabetic participants do not display the same glucagon suppression in response to the glucose stimulus as healthy people. Plasma glucagon levels significantly decrease, plasma insulin levels significantly rise, and as a result, plasma glucose levels are normal in non-diabetic individuals. The opposite is true for those with type 2 diabetes, who also have post-prandial hyperglycemia and low plasma insulin levels that are at or above pre-prandial levels2. According to the bi-hormonal hypothesis proposed by Unger and Orci in 1975, hyperglycemia is caused by both an excess of glucagon and insulin insufficiency or resistance, which cause the liver to create more glucose than is needed for the utilization of the glucose, leading to diabetes3. In contrast to α-cells and glucagon, mathematical models of the dynamics of glucose, insulin, and β-cells have recently drawn a lot of attention4–7. In this study, we provided a mathematical model of the coupled dynamics of glucose, insulin, α and β cells, and glucagon8.

Atangana9 has proposed an entirely novel technique for fractional calculus called the fractal fractional derivative. The idea behind this subject is frequently quite helpful for solving some difficult issues. Two orders of the operator are the fractal dimension and the fractional order. The conventional method is outperformed by this new approach to fractal fractions10–12. This is so that one can simultaneously study fraction operators and fractal dimensions while dealing with fractal fractional derivatives. The huge advantage of this operator is that it enables you to construct models that more precisely describe systems with memory effects. In addition, there are other real-world concerns that call for knowledge of a system’s information capacity. utilised unique applications and a variety of kernels to report some breakthroughs in fractal-fractional differential equations13,14. Fractional calculus is parqueting the interest of scientists all around the world because of its many benefits and useful applications in physics and engineering. Hereditary traits, memory, and crossover behaviour can only be observed using a model with a fractional-order system15–17. The behaviour of a fractional-order diabetes model was examined in the research paper. The fractional order derivative of the glucose concentration was taken into account along with the interactions between glucose, insulin, and glucagon. The model’s equilibrium points were examined for stability, and the scientists also looked at how the fractional order derivative affected the dynamics of the system. The outcomes demonstrated that the model’s behaviour could be influenced by the fractional order derivative, producing dynamics that were more complex than in the traditional integer order scenario18. The literature on fractional order modelling and analysis of diabetes was thoroughly reviewed by them. The authors explored a range of modelling techniques for diabetes, such as conventional integer order models, fractional order models, and hybrid models. They emphasised the benefits of employing fractional order models, including their capacity to reproduce memory and non-locality effects observed in biological systems19. Researchers have been examining the intricate interactions between glucose-insulin dynamics, metabolic regulation, and patient behaviour in the study of behavioral dynamics in the fractional order diabetes model in recent years20. Researchers have been able to capture the non-integer order derivatives that result from patient behavior by introducing fractional calculus into diabetes modelling, providing a more accurate picture of the dynamics of the disease21. The existing fractional order models for diabetes and their uses in evaluating the course of the disease, the effectiveness of insulin treatment, and glucose control were also reviewed by some writers22. Nevertheless, conventional or classical fractional-order models find it challenging to effectively articulate the concept of piecewise fractional-order derivatives to get over this limitation due to the complexity of the epidemic, particularly during crossover periods. With this approach, we want to provide a more realistic picture of the intricate crossover behaviours present in the dynamics of the pandemic and some real life application through different kinds of fractional order derivative28–32.

Fractional calculus has a strong history and plays a major role in the simulation of physical phenomenon of real life. Recently, there has been a lot of interest in the study of fractional calculus. Fractional calculus and fractional processes have become one of the most useful approaches to deal with a variety of problems in applied sciences due to memory and hereditary properties. A number of studies for fractional order linear, non-linear and complex dynamical mathematical models have been presented with interesting results during recent years. Therefore, compared to the traditional integer order models, fractional order models seem to be more objective and flexible. There is a long history of mathematical modelling on the topic of glucose metabolism. There are various reasons for using models. Models have been used to infer physiologically significant factors from experimental data in an indirect manner, to provide a clear quantitative depiction of pathophysiological pathways, and to derive clinically valuable indexes from basic experimental procedures. As the impact of type 1 diabetes on society grows, models related to the disruption of the glucose homeostasis system are being developed and utilised more frequently. Currently, diabetes mellitus is one of the major issues all over the world. We construct a fractional-order glucose-insulin-glucagon system with a novel fractional technique. For this purpose, Section “Introduction” is an introduction, and Section “Basic concepts has some useful definitions”. A mathematical model with a description is presented in Section “Diabetes model with fractional derivative”. An analysis with different aspects and stability is given in Section “Analysis of proposed model”. The advanced numerical scheme fractal fractional is present in Section “Computational analysis with fractal fractional operator for reliable solutions”. A numerical simulation is given in Section “Numerical simulation and discussion” to see its physical interpretation and conclusion in Section “Conclusion”.

Basic concepts

Definition 1

23,24 For a power law kernel in the Riemann–Liouville sense is given as;1 0FFPDtα,ηx(t)=1Γ(1-α)ddtη∫0t(t-ς)-αx(ς)dς

with 0≤α,η≤1. Wheredf(t)dtη=limt→t1f(t)-f(t1)t2-η-t12-η(2-η)

The corresponding power law kernel fractal-fractional integral of order (α,η) is given as2 0FFPItα,ηx(t)=1Γ(α)∫0t(t-ς)α-1ς1-ηx(ς)dς

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;3 0FFEDtα,ηx(t)=H(α)Γ(1-α)ddtη∫0texp-α1-α(t-ς)-αx(ς)dς

where α>0,η≤1, and H(0)=1=H(1). The corresponding exponential decay kernel fractal-fractional integral of order (α,η) is given as4 0FFEIα,ηx(t)=η(1-α)tη-1x(t)H(α)+αηH(α)∫0tςα-1x(ς)dς

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;5 0FFMDtα,ηx(t)=AB(α)(1-α)ddtη∫0tEα-α1-α(t-ς)αx(ς)dς

where 0<α,η≤1, Eα is the Mittag-Leffler function and AB(α)=1-α+αΓ(α) is a normalization function.

The corresponding Mittag-Leffler kernel fractal-fractional integral of order (α,η) is given as6 0FFMIα,ηx(t)=η(1-α)AB(α)t1-ηx(t)+ηαAB(α)Γ(α)∫0t(t-ς)α-1ς1-ηx(ς)dς

Diabetes model with fractional derivative

For our motivation, we consider the glucose-insulin-glucagon system given in8, 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 equations7 0FFMDtψ,ηG=ω-bG-δIGαG+1+δiJ0FFMDtψ,ηI=γβG2e+G2-μI0FFMDtψ,ηβ=ρβ1-βv0FFMDtψ,ηα=ρiα1-αvi0FFMDtψ,ηJ=-γiα(G-Gt)-μiJ

with initial condition G(0)=G0≥0,I(0)=I0≥0,β(0)=β0≥0,α(0)=α0≥0,J(0)=J0≥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 δiJ refers to glucagon’s effect on liver gluconeogenesis, which produces glucose. The blood glucose level increases at a rate ω (the rate of glucose generation 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 ρi represent the growth rates of the β and the α cell masses, respectively. v and vi represent the carrying 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<Gl).

Analysis of proposed model

Positivity and boundedness of solutions

Here, we demonstrate the suggested model’s positivity and boundedness.

Theorem 1

Assume the initial condition be8 {G(0),I(0),β(0),α(0),J(0)}⊂ϖ,

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 real-world issues with positive values using the methodology described in25,26. This section looks at the conditions necessary for the proposed model to provide positive results. We’ll describe the norm9 ‖h‖∞=supt∈Dh|h(t)|

where the domain of h is Dh. Let’s begin with the G(t) class10 0FFMDtψ,ηG=ω-bG-δIGαG+1+δiJ,∀t≥0≥-b+δ|I|αG+1G,∀t≥0≥-b+δsupt∈DI|I|αG+1G,∀t≥0≥-b+δ‖I‖∞αG+1G,∀t≥0

This yield11 G(t)≥G(0)Eψ-r1-ηψb+δ‖I‖∞αG+1tψAB(ψ)-(1-ψ)b+δ‖I‖∞αG+1,∀t≥0

where the time component is r. This illustrates that for any t≥0, G(t) is positive. For the function I(t)12 0FFMDtψ,ηI=γβG2e+G2-μI,∀t≥0≥-(μ)I,∀t≥0

This yield13 I(t)≥I(0)Eψ-r1-ηψμtψAB(ψ)-(1-ψ)μ,∀t≥0

where the time component is r. This illustrates that for any t≥0, I(t) is positive. For the function β(t)14 0FFMDtψ,ηβ=ρβ1-βv,∀t≥0≥-ρ|β|v-1β,∀t≥0≥-ρsupt∈Dβ|β|v-1β,∀t≥0≥-ρ‖β‖∞v-1β,∀t≥0

This yield15 β(t)≥β(0)Eψ-r1-ηψρ‖β‖∞v-1tψAB(ψ)-(1-ψ)ρ‖β‖∞v-1,∀t≥0

where the time component is r. This illustrates that for any t≥0, β(t) is positive. For the function α(t)16 0FFMDtψ,ηα=ρiα1-αvi,∀t≥0≥-ρi|α|vi-1α,∀t≥0≥-ρisupt∈Dα|α|vi-1α,∀t≥0≥-ρi‖α‖∞vi-1α,∀t≥0

This yield17 α(t)≥α(0)Eψ-r1-ηψρi‖α‖∞vi-1tψAB(ψ)-(1-ψ)ρi‖α‖∞vi-1,∀t≥0

where the time component is r. This illustrates that for any t≥0, α(t) is positive. For the function J(t)18 0FFMDtψ,ηJ=-γiα(G-Gt)-μiJ,∀t≥0≥-(μi)J,∀t≥0

This yield19 I(t)≥I(0)Eψ-r1-ηψμtψAB(ψ)-(1-ψ)μ,∀t≥0

where the time component is r. This illustrates that for any t≥0, J(t) is positive. □

Positive invariant regions

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:20 0FFMDtψ,ηG(t)|G=0=ω+δiJ≥0,

21 0FFMDtψ,ηI(t)|I=0=γβG2e+G2≥0,

22 0FFMDtψ,ηβ(t)|β=0=0,

23 0FFMDtψ,ηα(t)|α=0=0,

24 0FFMDtψ,ηJ(t)|J=0=γiα(Gt-G)≥0.

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 theorem27. Consider the system (7) as25 0FFMDtψ,ηG=ω-bG-δIGαG+1+δiJ=G¯(t,G(t)),0FFMDtψ,ηI=γβG2e+G2-μI=I¯(t,I(t)),0FFMDtψ,ηβ=ρβ1-βv=β¯(t,β(t)),0FFMDtψ,ηα=ρiα1-αvi=α¯(t,α(t)),0FFMDtψ,ηJ=-γiα(G-Gt)-μiJ=J¯(t,J(t)).

The following is a reformulation of (25) in the form of a Fractal-Fractional integral for the Mittag-Leffler kernel as expressed in (6).26 G(t)=G(0)+η(1-ψ)t1-ηAB(ψ)G¯(t,G(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηG¯(ς,G(ς))dς=A1+A2,I(t)=I(0)+η(1-ψ)t1-ηAB(ψ)I¯(t,I(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηI¯(ς,I(ς))dς=B1+B2,β(t)=β(0)+η(1-ψ)t1-ηAB(ψ)β¯(t,β(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηβ¯(ς,β(ς))dς=C1+C2,α(t)=α(0)+η(1-ψ)t1-ηAB(ψ)α¯(t,α(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηα¯(ς,α(ς))dς=D1+D2,J(t)=J(0)+η(1-ψ)t1-ηAB(ψ)J¯(t,J(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηJ¯(ς,J(ς))dς=E1+E2,

where27 A1=G(0)+η(1-ψ)t1-ηAB(ψ)G¯(t,G(t)),A2=ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηG¯(ς,G(ς))dςB1=I(0)+η(1-ψ)t1-ηAB(ψ)I¯(t,I(t)),B2=ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηI¯(ς,I(ς))dςC1=β(0)+η(1-ψ)t1-ηAB(ψ)β¯(t,β(t)),C2=ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηβ¯(ς,β(ς))dςD1=α(0)+η(1-ψ)t1-ηAB(ψ)α¯(t,α(t)),D2=ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηα¯(ς,α(ς))dςE1=J(0)+η(1-ψ)t1-ηAB(ψ)J¯(t,J(t)),E2=ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηJ¯(ς,J(ς))dς

We prove the primary component of governing Eq. (26), M(A1,B1,C1,D1,E1) as contraction maps and N(A2,B2,C2,D2,E2) as continuous compact integral parts using Krasnoselski’s fixed point theorem.

Theorem 3

The non-linear map M(A1,B1,C1,D1,E1):[0,T]×R×R→R5 given in (27) ensures Lipschitz contractive condition for constants PA,PB,PC,PD,PE>0.

Proof

Consider the operator M(A1,B1,C1,D1,E1):[0,T]×R×R→R5 defined on a fully normed space. Where the norm is28 ‖(G,I,β,α,J)‖=maxt∈[0,T]‖G(t)+I(t)+β(t)+α(t)+J(t)‖,G,I,β,α,J∈[0,T]

(i) Firstly, we will show that M(A1,B1,C1,D1,E1) is a contraction map. For G(t) and G^(t), we have 29 ‖A(G,I,β,α,J)(t)-A(G^,I,β,α,J)(t)‖=‖{ω-bG-δIGαG+1+δiJ}-{ω-bG^-δIG^αG+1+δiJ}‖=‖-b(G-G^)-δIαG+1(G-G^)‖≤‖b+δIαG+1‖‖(G-G^)‖≤PA‖(G-G^)‖

where PA=‖b+δIαG+1‖. Using this approach, we have ‖B(G,I,β,α,J)(t)-B(G,I^,β,α,J)(t)‖≤PB‖(I-I^)‖‖C(G,I,β,α,J)(t)-C(G,I,β^,α,J)(t)‖≤PC‖(β-β^)‖‖D(G,I,β,α,J)(t)-D(G,I,β,α^,J)(t)‖≤PD‖(α-α^)‖‖E(G,I,β,α,J)(t)-E(G,I,β,α,J^)(t)‖≤PE‖(J-J^)‖

where PB=‖μ‖,PC=‖ρ1-βv‖,PD=‖ρi1-αvi‖,PE=‖μi‖ This implies that, for the operator M(G,I,β,α,J), we have 30 ‖M(G,I,β,α,J)-M(G^,I^,β^,α^,J^)‖=η(1-ψ)tη-1AB(ψ)maxt∈[0,T]|(G,I,β,α,J)(t)-(G^,I^,β^,α^,J^)(t)|≤η(1-ψ)tη-1AB(ψ)‖(G,I,β,α,J)(t)-(G^,I^,β^,α^,J^)(t)‖≤η(1-ψ)tη-1AB(ψ)P

where P=max[PA,PB,PC,PD,PE]<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(A2,B2,C2,D2,E2) is continuously compact. The absolute modulus of all positively bounded continuous operators A, B, C, D, E specified in (27) given by the non-zero positive constants ℘A,℘B,℘C,℘D,℘E,ℵA,ℵB,ℵC,ℵD,ℵE meeting the following bounded-ness inequalities, illustrates the compactness of the operator N(A2,B2,C2,D2,E2). 31 |A(t,G)|≤℘A‖G‖+ℵA|B(t,I)|≤℘B‖I‖+ℵB|C(t,β)|≤℘C‖β‖+ℵC|D(t,α)|≤℘D‖α‖+ℵD|E(t,J)|≤℘E‖J‖+ℵE

Suppose that χ is a closed subset of Z as 32 χ={(A,B,C,D,E)∈Z/‖A,B,C,D,E|≤Λ,Λ>0}

For (A,B,C,D,E)∈χ, we find 33 ‖A2(t,G)‖=maxt∈[0,T]|ψηAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηA(ς,G(ς))dς|≤τψ,ηAB(ψ)Γ(ψ)∫0τ(τ-ς)ψ-1ς1-ηA(ς,G(ς))dς|≤τψ,ηAB(ψ)Γ(ψ)℘AΛ+ℵA

Similarly, we find ‖B2(t,I)‖≤τψ,ηAB(ψ)Γ(ψ)℘BΛ+ℵB‖C2(t,β)‖≤τψ,ηAB(ψ)Γ(ψ)℘CΛ+ℵC‖D2(t,α)‖≤τψ,ηAB(ψ)Γ(ψ)℘DΛ+ℵD‖E2(t,J)‖≤τψ,ηAB(ψ)Γ(ψ)℘EΛ+ℵE

proceeding this process, we find the maximum norm of ‖Ξ(A2,B2,C2,D2,E2)‖ as, 34 ‖Ξ(A2,B2,C2,D2,E2)‖≤{[℘A+℘B+℘C+℘D+℘E]Λ+ℵA+ℵB+ℵC+ℵD+ℵE}=ξ

where ξ is a positive constant. Therefore, 35 ‖Ξ(A2,B2,C2,D2,E2)‖≤ξ⇒Ξ

is a uniformly bounded operator. Now we will prove that Ξ is equi-continuous for tx<ty∈[0,T]. For this purpose, we have for tx<ty∈[0,T]36 |A2(t2,G)-A2(t1,G)|=ψηAB(ψ)Γ(ψ)|∫0y(t-ς)ψ-1ς1-ηA(ς,G(ς))dς-∫0x(t-ς)ψ-1ς1-ηA(ς,G(ς))dς|≤ψηAB(ψ)Γ(ψ)∫0y(t-ς)ψ-1ς1-ηdς-∫0x(t-ς)ψ-1ς1-ηdς(℘AΛ+ℵA)≤℘AΛ+ℵAAB(ψ)Γ(ψ)t2ψ,η-t1ψ,η

Similarly, |B2(t2,I)-B2(t1,I)|≤℘BΛ+ℵBAB(ψ)Γ(ψ)t2ψ,η-t1ψ,η,|C2(t2,β)-C2(t1,β)|≤℘CΛ+ℵCAB(ψ)Γ(ψ)t2ψ,η-t1ψ,η,|D2(t2,α)-D2(t1,α)|≤℘DΛ+ℵDAB(ψ)Γ(ψ)t2ψ,η-t1ψ,η,|E2(t2,J)-E2(t1,J)|≤℘EΛ+ℵEAB(ψ)Γ(ψ)t2ψ,η-t1ψ,η.

Since t2→t1 is independent of (G,I,β,α,J). This implies that 37 ‖Ξ(A2,B2,C2,D2,E2)(t2)-Ξ(A2,B2,C2,D2,E2)(t1)‖→0

⇒Ξ(A2,B2,C2,D2,E2) is a completely continuous, equi-continuous operator. ⇒Ξ(A2,B2,C2,D2,E2) 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.

□

Theorem 4

The model (7) has a unique solution if38 τψ,ηAB(ψ)Γ(ψ)P≤1

where P=max{PA,PB,PC,PD,PE}.

Proof

Establish an operator H=(H1,H2,H3,H4,H5):Z→Z utilizing (31) as:39 H1(G,I,β,α,J)(t)=G(0)+η(1-ψ)t1-ηAB(ψ)A(t,G(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηA(ς,G(ς))dς,H2(G,I,β,α,J)(t)=I(0)+η(1-ψ)t1-ηAB(ψ)B(t,I(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηB(ς,I(ς))dς,H3(G,I,β,α,J)(t)=β(0)+η(1-ψ)t1-ηAB(ψ)C(t,β(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηC(ς,β(ς))dς,H4(G,I,β,α,J)(t)=α(0)+η(1-ψ)t1-ηAB(ψ)D(t,α(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηD(ς,α(ς))dς,H5(G,I,β,α,J)(t)=J(0)+η(1-ψ)t1-ηAB(ψ)E(t,J(t))+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-ηE(ς,J(ς))dς.

For (G,I,β,α,J),(G¯,I¯,β¯,α¯,J¯)∈Z, and utilizing (39) we have,40 ‖H1(G,I,β,α,J)(t)-H1(G¯,I¯,β¯,α¯,J¯)(t)‖=η(1-ψ)t1-ηAB(ψ)‖A(t,G(t))-A(t,G¯(t))‖+ηψAB(ψ)Γ(ψ)∫0t(t-ς)ψ-1ς1-η‖A(ς,G(ς))-A(ς,G¯(ς))‖dς,≤η(1-ψ)t1-ηAB(ψ)PA‖G-G¯‖+ςη,ψAB(ψ)Γ(ψ)PA‖G-G¯‖≤η(1-ψ)t1-ηAB(ψ)+ςη,ψAB(ψ)Γ(ψ)PA‖G-G¯‖

‖G-G¯‖→0 when G→G¯. Hence41 ‖H1(G,I,β,α,J)(t)-H1(G¯,I¯,β¯,α¯,J¯)(t)‖≤η(1-ψ)t1-ηAB(ψ)+ςη,ψAB(ψ)Γ(ψ)PA≤1

with42 ‖H1(G,I,β,α,J)(t)-H1(G¯,I¯,β¯,α¯,J¯)(t)‖1-η(1-ψ)t1-ηAB(ψ)+ςη,ψAB(ψ)Γ(ψ)PA≤0

Similarly, we find43 ‖H2(G,I,β,α,J)(t)-H2(G¯,I¯,β¯,α¯,J¯)(t)‖1-η(1-ψ)t1-ηAB(ψ)+ςη,ψAB(ψ)Γ(ψ)PB≤0‖H3(G,I,β,α,J)(t)-H3(G¯,I¯,β¯,α¯,J¯)(t)‖1-η(1-ψ)t1-ηAB(ψ)+ςη,ψAB(ψ)Γ(ψ)PC≤0‖H4(G,I,β,α,J)(t)-H4(G¯,I¯,β¯,α¯,J¯)(t)‖1-η(1-ψ)t1-ηAB(ψ)+ςη,ψAB(ψ)Γ(ψ)PD≤0‖H5(G,I,β,α,J)(t)-H5(G¯,I¯,β¯,α¯,J¯)(t)‖1-η(1-ψ)t1-ηAB(ψ)+ςη,ψAB(ψ)Γ(ψ)PE≤0

Therefore,‖H(G,I,β,α,J)(t)-H(G¯,I¯,β¯,α¯,J¯)(t)‖≤η(1-ψ)t1-ηAB(ψ)+ςηψAB(ψ)Γ(ψ)P‖(G,I,β,α,J)-(G¯,I¯,β¯,α¯,J¯)‖

The contraction map H inherits the features of Schauder’s and Krasnoselski’s theorems and confirms our suggested model’s unique fixed point solution. □

Remark 1

The derived unique solution is attractiveif the zero solution (G,I,β,α,J)(t)=0 such that 44 ‖(G,I,β,α,J)‖≤ε,impliesthatlimt→∞(G,I,β,α,J)(t)=0

if the trivial solution φ(t)=0 such that ‖Z0‖≤ε⇒limt→∞z0=0.

Equilibrium points analysis

The system given in (7) is solved for equilibria. we have45 E1=ωb,0,0,0,0

46 E2=μiω+bviγiGtbμi+δiviγi,0,0,vi,-(ωviγi+δiviγiGt)bμi+δiviγi

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≥t0,47 0FFMDtψ,ηh(t)-h∗-h∗logh(t)h∗≤1-h∗h(t)0FFMDtψ,ηh(t)

h∗∈R+, ∀α∈(0,1).

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 R0>1.

Proof

Suppose that the Volterra-type Lyapunov function as:48 M=C1(G-G∗-G∗logG∗G)+C2(I-I∗-I∗logI∗I)+C3(β-β∗-β∗logβ∗β)+C4(α-α∗-α∗logα∗α)+C5(J-J∗-J∗logJ∗J)

Where Ci,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).49 0FFMDtψ,ηM≤C1G-G∗G0FFMDtψ,ηG+C2I-I∗I0FFMDtψ,ηI+C3β-β∗β0FFMDtψ,ηβ+C4α-α∗α0FFMDtψ,ηα+C5J-J∗J0FFMDtψ,ηJ

After the substituting the values of the derivative derivatives, we have.50 0FFMDtψ,ηM≤C1G-G∗G(ω-bG-δIGαG+1+δiJ)+C2I-I∗I(γβG2e+G2-μI)+C3β-β∗β×ρβ1-βv+C4α-α∗α(ρiα1-αvi)+C5J-J∗J(-γiα(G-Gt)-μiJ)

Replacing G=G-G∗,I=I-I∗,β=β-β∗,α=α-α∗,J=J-J∗, we can have the following51 0FFMDtψ,ηM≤C1G-G∗G(ω-b(G-G∗)-δ(I-I∗)(G-G∗)(α-α∗)(G-G∗)+1+δi(J-J∗))+C2I-I∗I×γ(β-β∗)(G-G∗)2e+(G-G∗)2-μ(I-I∗)+C3β-β∗βρ(β-β∗)1-(β-β∗)v+C4α-α∗α(ρi(α-α∗)1-(α-α∗)vi)+C5J-J∗J(-γiα((G-G∗)-Gt)-μi(J-J∗))

Now letC1=C2=C3=C4=C5=1. We can organize the above as follows52 0FFMDtψ,ηM≤ω-ωG∗G-b(G-G∗)2G-δ(I-I∗)(G-G∗)2G((α-α∗)(G-G∗)+1)+δiJ-δiJ∗-δiG∗GJ+δiG∗GJ∗+γ(β-β∗)(G-G∗)2e+(G-G∗)2-[γ(β-β∗)(G-G∗)2]I∗[e+(G-G∗)2]I-μ(I-I∗)2I+ρ(β-β∗)2β-ρ(β-β∗)3vβ+ρi(α-α∗)2α-ρi(α-α∗)3viα-γiαG+γiαG∗+γiαGt+γiα∗G-γiα∗G∗-γiα∗Gt+γiαGJ∗J-γiαG∗J∗J-γiαGtJ∗J-γiα∗GJ∗J+γiα∗G∗J∗J+γiα∗GtJ∗J-μi(J-J∗)2J

after simplification, we get53 0FFMDtψ,ηM≤Ω-Σ

where54 Ω=ω+δiJ+δiG∗GJ∗+γ(β-β∗)(G-G∗)2e+(G-G∗)2+ρ(β-β∗)2β+ρi(α-α∗)2α+γiαG∗+γiαGt+γiα∗G+γiαGJ∗J+γiα∗G∗J∗J+γiα∗GtJ∗J

and55 Σ=ωG∗G+b(G-G∗)2G+δ(I-I∗)(G-G∗)2G((α-α∗)(G-G∗)+1)+δiJ∗+δiG∗GJ+[γ(β-β∗)(G-G∗)2]I∗[e+(G-G∗)2]I+μ(I-I∗)2I+ρ(β-β∗)3vβ+ρi(α-α∗)3viα+γiαG+γiα∗G∗+γiα∗Gt+γiαG∗J∗J+γiαGtJ∗J+γiα∗GJ∗J+μi(J-J∗)2J

it is concluded that if Ω<Σ this yields 0FFMDtψ,ηM<0 however when G=G∗,I=I∗,β=β∗,α=α∗,J=J∗ so Ω-Σ=0 , 0FFMDtψ,ηM=0 □

Second derivative of Lyapunov

56 0FFMDtψ,η[0FFMDtψ,η]M≤0FFMDtψ,ηGG2G∗+0FFMDtψ,ηII2I∗+0FFMDtψ,ηββ2β∗+0FFMDtψ,ηαα2α∗+0FFMDtψ,ηJJ2J∗+1+G∗G0FFMDtψ,η[0FFMDtψ,ηG]+1+I∗I0FFMDtψ,η[0FFMDtψ,ηI]+1+β∗β0FFMDtψ,η[0FFMDtψ,ηβ]+1+α∗α0FFMDtψ,η[0FFMDtψ,ηα]+1+J∗J0FFMDtψ,η[0FFMDtψ,ηJ]

where57 0FFMDtψ,η[0FFMDtψ,ηG]=-b(0FFMDtψ,ηG)+δi(0FFMDtψ,ηJ)-Φ(αG+1)20FFMDtψ,η[0FFMDtψ,ηI]=(e)(γ(0FFMDtψ,ηβ)G2+2γβG(0FFMDtψ,ηG))+γ(0FFMDtψ,ηβ)G4(e+G2)2-μ(0FFMDtψ,ηI)0FFMDtψ,η[0FFMDtψ,ηβ]=ρ(0FFMDtψ,ηβ)-2ρβ(0FFMDtψ,ηβ)v0FFMDtψ,η[0FFMDtψ,ηα]=ρi(0FFMDtψ,ηα)-2ρiα(0FFMDtψ,ηα)vi0FFMDtψ,η[0FFMDtψ,ηJ]=-γi(0FFMDtψ,ηα)G-γiα(0FFMDtψ,ηG)+γi(0FFMDtψ,ηα)Gt-μi(0FFMDtψ,ηJ)

where58 Φ=(αG+1)(δI(0FFMDtψ,ηG)+δ(0FFMDtψ,ηI)G)-δIG((0FFMDtψ,ηα)G+α(0FFMDtψ,ηG))

then we have59 0FFMDtψ,η[0FFMDtψ,η]M≤Π(G,I,β,α,J)+1+G∗G{-b(0FFMDtψ,ηG)+δi(0FFMDtψ,ηJ)-Φ(αG+1)2}+1+I∗I{(e)(γ(0FFMDtψ,ηβ)G2+2γβG(0FFMDtψ,ηG))+γ(0FFMDtψ,ηβ)G4(e+G2)2-μ(0FFMDtψ,ηI)}+1+β∗βρ(0FFMDtψ,ηβ)-2ρβ(0FFMDtψ,ηβ)v+1+α∗αρi(0FFMDtψ,ηα)-2ρiα(0FFMDtψ,ηα)vi+1+J∗J×{-γi(0FFMDtψ,ηα)G-γiα(0FFMDtψ,ηG)+γi(0FFMDtψ,ηα)Gt-μi(0FFMDtψ,ηJ)}

where60 Π(G,I,β,α,J)=0FFMDtψ,ηGG2G∗+0FFMDtψ,ηII2I∗+0FFMDtψ,ηββ2β∗+0FFMDtψ,ηαα2α∗+0FFMDtψ,ηJJ2J∗

now replacing 0FFMDtψ,ηG,0FFMDtψ,ηI,0FFMDtψ,ηβ,0FFMDtψ,ηα,0FFMDtψ,ηJ with their respective formula from the proposed model (7), we can get61 0FFMDtψ,η[0FFMDtψ,η]M≤Ψ1+Ψ2

Ψ1: represents all positive terms

Ψ2: represents all positive terms So thatIf Ψ1>Ψ2 then 0FFMDtψ,η[FFMDtψ,η]M>0

If Ψ1<Ψ2 then 0FFMDtψ,η[FFMDtψ,η]M<0

If Ψ1=Ψ2 then 0FFMDtψ,η[FFMDtψ,η]M=0

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.62 0FFMDtψ,ηG=G1(t,G,I,β,α,J)0FFMDtψ,ηI=I1(t,G,I,β,α,J)0FFMDtψ,ηβ=β1(t,G,I,β,α,J)0FFMDtψ,ηα=α1(t,G,I,β,α,J)0FFMDtψ,ηJ=J1(t,G,I,β,α,J)

where63 G1(t,G,I,β,α,J)=ω-bG-δIGαG+1+δiJI1(t,G,I,β,α,J)=γβG2e+G2-μIβ1(t,G,I,β,α,J)=ρβ1-βvα1(t,G,I,β,α,J)=ρiα1-αviJ1(t,G,I,β,α,J)=-γiα(G-Gt)-μiJ

With Mittag-Leffler kernel applying fractal-fractional integral , we get64 G(tφ+1)=G(0)+1-ψAB(ψ)tφ1-ηG1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φ∫tqq+1G1(t,G,I,β,α,J)ζ1-η(tφ+1-ζ)ψ-1dζ

65 I(tφ+1)=I(0)+1-ψAB(ψ)tφ1-ηI1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φ∫tqq+1I1(t,G,I,β,α,J)ζ1-η(tφ+1-ζ)ψ-1dζ

66 β(tφ+1)=β(0)+1-ψAB(ψ)tφ1-ηβ1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φ∫tqq+1β1(t,G,I,β,α,J)ζ1-η(tφ+1-ζ)ψ-1dζ

67 α(tφ+1)=α(0)+1-ψAB(ψ)tφ1-ηα1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φ∫tqq+1α1(t,G,I,β,α,J)ζ1-η(tφ+1-ζ)ψ-1dζ

68 J(tφ+1)=J(0)+1-ψAB(ψ)tφ1-ηJ1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φ∫tqq+1J1(t,G,I,β,α,J)ζ1-η(tφ+1-ζ)ψ-1dζ

Recall the Newton Polynomial:69 Q(t,G,I,β,α,J)≃Q(tφ-2,Gφ-2,Iφ-2,βφ-2,αφ-2,Jφ-2)+1Δt[Q(tφ-1,Gφ-1,Iφ-1,βφ-1,αφ-1,Jφ-1)-Q(tφ-2,Gφ-2,Iφ-2,βφ-2,αφ-2,Jφ-2)](ζ-tφ-2)+12Δt2[Q(tφ,Gφ,Iφ,βφ,αφ,Jφ)-2Q(tφ-1,Gφ-1,Iφ-1,βφ-1,αφ-1,Jφ-1)+Q(tφ-2,Gφ-2,Iφ-2,βφ-2,αφ-2,Jφ-2)]×(ζ-tφ-2)(ζ-tφ-1)

Replacing the Newton polynomial (69) into Eqs. (64)–(68), we have70 G(tφ+1)=G(0)+1-ψAB(ψ)tφ1-ηG1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φG1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Iq-2)tq-21-η∫tqq+1(tφ+1-ζ)ψ-1dζ+ψAB(ψ)Γ(ψ)∑q=2φ1Δt{tq-11-ηG1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)-tq-21-η×G1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(tφ+1-ζ)ψ-1dζ+ψAB(ψ)Γ(ψ)×∑q=2φ12Δt2{tq1-ηG1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-ηG1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηG1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(ζ-tq-1)(tφ+1-ζ)ψ-1dζ

71 I(tφ+1)=I(0)+1-ψAB(ψ)tφ1-ηI1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φI1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Iq-2)tq-21-η∫tqq+1(tφ+1-ζ)ψ-1dζ+ψAB(ψ)Γ(ψ)∑q=2φ1Δt{tq-11-ηI1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)-tq-21-η×I1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(tφ+1-ζ)ψ-1dζψAB(ψ)Γ(ψ)×∑q=2φ12Δt2{tq1-ηI1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-ηI1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηI1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(ζ-tq-1)(tφ+1-ζ)ψ-1dζ

72 β(tφ+1)=β(0)+1-ψAB(ψ)tφ1-ηβ1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φβ1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Iq-2)tq-21-η∫tqq+1(tφ+1-ζ)ψ-1dζ+ψAB(ψ)Γ(ψ)∑q=2φ1Δt{tq-11-ηβ1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)tq-21-η×β1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(tφ+1-ζ)ψ-1dζψAB(ψ)Γ(ψ)×∑q=2φ12Δt2{tq1-ηβ1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-ηβ1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηβ1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(ζ-tq-1)(tφ+1-ζ)ψ-1dζ

73 α(tφ+1)=α(0)+1-ψAB(ψ)tφ1-ηα1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φα1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Iq-2)tq-21-η∫tqq+1(tφ+1-ζ)ψ-1dζ+ψAB(ψ)Γ(ψ)∑q=2φ1Δt{tq-11-ηα1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)tq-21-η×α1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(tφ+1-ζ)ψ-1dζψAB(ψ)Γ(ψ)×∑q=2φ12Δt2{tq1-ηα1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-ηα1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηα1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(ζ-tq-1)(tφ+1-ζ)ψ-1dζ

74 J(tφ+1)=J(0)+1-ψAB(ψ)tφ1-ηJ1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψAB(ψ)Γ(ψ)∑q=2φJ1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Iq-2)tq-21-η∫tqq+1(tφ+1-ζ)ψ-1dζ+ψAB(ψ)Γ(ψ)∑q=2φ1Δt{tq-11-ηJ1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)tq-21-η×J1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(tφ+1-ζ)ψ-1dζψAB(ψ)Γ(ψ)×∑q=2φ12Δt2{tq1-ηJ1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-ηJ1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηJ1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}∫tqq+1(ζ-tq-2)(ζ-tq-1)(tφ+1-ζ)ψ-1dζ

Calculations for the integral in the Eqs. (70)–(74) are:75 ∫tqq+1(tφ+1-ζ)ψ-1dζ=(Δt)ψψ(φ-q+1)ψ-(φ-q)ψ

76 ∫tqq+1(ζ-tq-2)(tφ+1-ζ)ψ-1dζ=(Δt)ψ+1ψ(ψ+1){(φ-q+1)ψ(φ-q+3+2ψ)-(φ-q)ψ(φ-q+3+3ψ)}

77 ∫tqq+1(ζ-tq-2)(ζ-tq-1)(tφ+1-ζ)ψ-1dζ=(Δt)ψ+2ψ(ψ+1)(ψ+2){(φ-q+1)ψ(2(φ-q)2+(3ψ+10)(φ-q)+2ψ2+9ψ+12)-(φ-q)ψ×(2(φ-q)2+(5ψ+10)(φ-q)+6ψ2+18ψ+12)}

Hence, we get finally78 G(tφ+1)=G(0)+1-ψAB(ψ)tφ1-ηG1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψ(Δt)ψAB(ψ)Γ(ψ+1)∑q=2φG1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)tq-21-η(φ-q+1)ψ-(φ-q)ψ+ψ(Δt)ψAB(ψ)Γ(ψ+2)∑q=2φ{tq-11-ηG1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)-tq-21-η×G1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}{(φ-q+1)ψ(φ-q+3+2ψ)-(φ-q)ψ×(φ-q+3+3ψ)}+ψ(Δt)ψ2AB(ψ)Γ(ψ+3)∑q=2φ{tq1-ηG1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-η×G1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηG1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}×{(φ-q+1)ψ(2(φ-q)2(3ψ+10)(φ-q)+2ψ2+9ψ+12)-(φ-q)ψ(2(φ-q)2+(5ψ+10)(φ-q)+6ψ2+18ψ+12)}

79 I(tφ+1)=I(0)+1-ψAB(ψ)tφ1-ηI1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψ(Δt)ψAB(ψ)Γ(ψ+1)∑q=2φI1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)tq-21-η(φ-q+1)ψ-(φ-q)ψ+ψ(Δt)ψAB(ψ)Γ(ψ+2)∑q=2φ{tq-11-ηI1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)-tq-21-η×I1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}{(φ-q+1)ψ(φ-q+3+2ψ)-(φ-q)ψ×(φ-q+3+3ψ)}+ψ(Δt)ψ2AB(ψ)Γ(ψ+3)∑q=2φ{tq1-ηI1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-η×I1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηI1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}×{(φ-q+1)ψ(2(φ-q)2(3ψ+10)(φ-q)+2ψ2+9ψ+12)-(φ-q)ψ(2(φ-q)2+(5ψ+10)(φ-q)+6ψ2+18ψ+12)}

80 β(tφ+1)=β(0)+1-ψAB(ψ)tφ1-ηβ1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψ(Δt)ψAB(ψ)Γ(ψ+1)∑q=2φβ1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)tq-21-η(φ-q+1)ψ-(φ-q)ψ+ψ(Δt)ψAB(ψ)Γ(ψ+2)∑q=2φ{tq-11-ηβ1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)-tq-21-η×β1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}{(φ-q+1)ψ(φ-q+3+2ψ)-(φ-q)ψ×(φ-q+3+3ψ)}+ψ(Δt)ψ2AB(ψ)Γ(ψ+3)∑q=2φ{tq1-ηβ1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-η×β1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηβ1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}×{(φ-q+1)ψ(2(φ-q)2(3ψ+10)(φ-q)+2ψ2+9ψ+12)-(φ-q)ψ(2(φ-q)2+(5ψ+10)(φ-q)+6ψ2+18ψ+12)}

81 α(tφ+1)=α(0)+1-ψAB(ψ)tφ1-ηα1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψ(Δt)ψAB(ψ)Γ(ψ+1)∑q=2φα1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)tq-21-η(φ-q+1)ψ-(φ-q)ψ+ψ(Δt)ψAB(ψ)Γ(ψ+2)∑q=2φ{tq-11-ηα1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)-tq-21-η×α1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}{(φ-q+1)ψ(φ-q+3+2ψ)-(φ-q)ψ×(φ-q+3+3ψ)}+ψ(Δt)ψ2AB(ψ)Γ(ψ+3)∑q=2φ{tq1-ηα1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-η×α1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηα1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}×{(φ-q+1)ψ(2(φ-q)2(3ψ+10)(φ-q)+2ψ2+9ψ+12)-(φ-q)ψ(2(φ-q)2+(5ψ+10)(φ-q)+6ψ2+18ψ+12)}

82 J(tφ+1)=J(0)+1-ψAB(ψ)tφ1-ηJ1(tφ,G(tφ),I(tφ),β(tφ),α(tφ),J(tφ))+ψ(Δt)ψAB(ψ)Γ(ψ+1)∑q=2φJ1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)tq-21-η(φ-q+1)ψ-(φ-q)ψ+ψ(Δt)ψAB(ψ)Γ(ψ+2)∑q=2φ{tq-11-ηJ1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)-tq-21-η×J1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}{(φ-q+1)ψ(φ-q+3+2ψ)-(φ-q)ψ×(φ-q+3+3ψ)}+ψ(Δt)ψ2AB(ψ)Γ(ψ+3)∑q=2φ{tq1-ηJ1(tq,Gq,Iq,βq,αq,Jq)-2tq-11-η×J1(tq-1,Gq-1,Iq-1,βq-1,αq-1,Jq-1)+tq-21-ηJ1(tq-2,Gq-2,Iq-2,βq-2,αq-2,Jq-2)}×{(φ-q+1)ψ(2(φ-q)2(3ψ+10)(φ-q)+2ψ2+9ψ+12)-(φ-q)ψ(2(φ-q)2+(5ψ+10)(φ-q)+6ψ2+18ψ+12)}

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 conditions8 are listed in table 1 that is given belowTable 1 Parameter values and biological interpretation.

Parameters	Biological interpretation	Values	Units	
ω	Production rate of glucose for G=0	864	mg/(dl · d)	
b	Glucose clearance rate independent of insulin	1.44	d-1	
δ	Glucose uptake rate induced by insulin	0.85	ml/(mU · d)	
δi	Glucose production rate induced by glucagon	1350	d-1	
γ	The maximum insulin secretory rate by β-cell	43.2	mU/(ml · d · mg)	
γi	The maximum glucagon secretory rate by α-cell	0.05	1/(mg · d)	
e	Inflection point for sigmoid	20000	mg2/dl2	
μ	Insulin clearance rate for complete body	432	d-1	
μi	Glucagon clearance rate for complete body	0.3	d-1	
v	Environmental capacity of β-cell mass	900	mg	
vi	α-cell mass environmental capacity	300	mg	
α	Half saturation inverse as a constant	0.01	mg	
Gt	Glycaemia minimum level	80	mg/dl	

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 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 Fig. 1, glucose level, insulin level, and α- cell mass with low glucose 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. 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 diabetes mellitus model simulation tests carried out in this work. 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.Figure 1 Simulation of the diabetes model compartments at initial condition (120, 07, 10, 30, 0.16).

Figure 2 Simulation of the diabetes model compartments at initial condition (120, 10, 05, 03, 0.16).

Figure 3 Simulation of the diabetes model compartments at initial condition (220, 15, 05, 03, 0.16).

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 a glucose-insulin pump plays an important role in overcoming the risk of hypoglycemia and hyperglycemia in humans. Research in this area will advance as a result of this method’s improved understanding of the dynamics and behaviour of diabetes mellitus. In the future, we will analyse the prediction model for treating and controlling diabetes in society with novel and modified fractional operators. Using non-local and non-singular kernel operators, such as Caputo–Fabrizio and ABC differential (integral) operators, can better capture empirical events than traditional mathematical operators, leading to deeper insights into the diabetes mellitus model. These fractional operators can be used to represent the diabetes mellitus disease, and their relative advantages and disadvantages can be evaluated. It would therefore be beneficial for aspiring young researchers to compare their findings with the findings of this study.

Acknowledgements

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

Author contributions

M.B.: Concept, original draft, M.F.: Software, original draft, analysis, A.G.: Writing and review, software, computations, K.S.N.: Writing and review, software, computations, S.M: Editing, software, Review.

Data availibility

All data generated or analysed during this study are included in this Manuscript.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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