# Modeling Impulsive Intake of Glucose: The Significance of Diet to Glucose-Insulin System

Authors: Xiangyun Shi; Jiaoyan Yao; Xueyong Zhou
Journal: Complexity
 (2020)

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## Abstract

It is important to get good dietary advice such as a daily value of exogenous glucose intake to keep blood glucose within reasonable range for everyone, especially for those with diabetes mellitus or with borderline diabetes mellitus. In this paper, two novel glucose-insulin models with impulsive glucose intake are considered to investigate the significance of diet to the glucose-insulin system. For the first model, we prove the positiveness, boundedness, and permanence of the solutions and then the existence and stability of the periodic solutions are demonstrated. For the second model, we consider the insulin secretion delay based on the first model. The sufficient delay-dependent conditions which determine the stability of the periodic solution are also obtained. In addition, the theoretical results are verified by numerical simulation. The results can give people some dietary advice how to prevent diabetes, and also, the intake amount and frequency of the exogenous glucose from diet are recommended for diabetic patients to assist in the treatment of diabetes.


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## Body

## 1. Introduction

During recent decades, diabetes mellitus has become an epidemic disease in the sense of life style [1]. At present, diabetes is the third category of noncommunicable diseases which harms human health after malignant tumors and cardiovascular diseases, and has developed into a worldwide public health problem. In the glucose-insulin regulatory system, elevated glucose levels trigger pancreatic β-cells to secrete insulin, which helps glucose levels return to normal levels. When the concentration of glucose decreases, the secretion of insulin stops gradually [2]. Studying the dynamics between glucose and its control hormone insulin is conducive to the prevention and treatment of diabetes.Many mathematical models have been developed to better understand the mechanisms of the glucose-insulin regulatory system ([3–12]). The most well-known clinical model is the Minimal Model, which was proposed by Bergman et al. in 1979 [6] and 1980 [7]. The Minimal Model is widely used in the intravenous glucose tolerance test (IVGTT). It was found that there is a time delay for insulin to release in responding to the elevated glucose concentration. And professionals point out that the ultradian oscillations of insulin secretion may be caused by the time delay. Therefore, a number of delayed models have been proposed to study the effect of time delay on the dynamics of the system. In 2000, De Gaetano and Arino [8] proposed a new aggregated delay differential equation model with a distributed time delay, and pointed that this model was more easier to analyze than the previous model mathematically. Later, Li et al. [9] in 2001 generalized the model to a general form. In 2007, De Gaetano et al. in their application work focused on a discrete delay model and claimed that such a model is good enough to test the insulin sensitivity [3]. In 2017, Shi et al. [5] proposed a novel approach to model the insulin secretion time delay, which uses two parameters to simulate both discrete time delay and distributed time delay. All these above models incorporate the insulin secretion delay implicitly in the ordinary differential equation (ODE) model by using compartment-split technique or explicitly in delay differential equation (DDE) models. Accounting for the effect of white noise, Shi et al. [12] discussed a stochastic IVGTT glucose-insulin model with discrete time delay.In recent years, impulsive differential equation models have been established to provide effective therapies for diabetic patients. The treatment of diabetes is mainly insulin injection or insulin intake of hypoglycemic drugs. Continuous subcutaneous injection of insulin (CSII) therapy is achieved by the use of an insulin pump, a medical device for administering insulin or its analogues. There are many mathematical models modeling the insulin pump control of glucose level by impulsive insulin injections ([13–17]), and also, a number of impulsive differential equation models have been developed for the artificial pancreas ([18, 19]). In addition, because the constant glucose exogenous infusion is usually a discrete process, in order to investigate how to prevent hyperinsulinemia and hyperglycaemia, Yang et al. [20] developed a model involving periodic intakes of glucose with insulin injections applied only when the blood glucose level reached a given critical glucose threshold. Their results can provide clinical strategies for insulin-administration practices.To the best of our knowledge, both diabetes and borderline diabetics need to control their glucose intake through diet control, and it is necessary to discuss the significance of diet to the glucose-insulin system. In many mathematical models of the glucose-insulin system, glucose exogenous infusion is always simulated by a constant which does not reflect the cyclical reality of eating. Therefore, in this paper, we propose novel impulsive differential equation model (1) to simulate the glucose intake from the daily diet with periodic impulses. Furthermore, to consider the effect of insulin secretion delay on the glucose and insulin regulation system in the process of glucose exogenous infusion, we add the time delay to the impulsive glucose-insulin model to study the effect of time delay on the stability of system (7).The paper is organized as follows. In Section2, we formulate two impulsive differential equation models to simulate the impulsive glucose intake. In Section 3, the qualitative analysis of the model is presented. The persistence, existence, and globally asymptotic stability of periodic solution of the system are discussed. In Section 4, the model with impulsive glucose intake and insulin secretion time delay is studied. We mainly prove the globally asymptotic stability of the periodic solution of the system under delay-dependent conditions. In Section 5, the numerical simulations are carried out, which not only verify the theoretical results but also are complementary to those theoretical results with specific features. A brief discussion is presented in the last section.


## 2. Model Formulation

In order to obtain the value of exogenous glucose intake which can keep the blood glucose in a certain range and give some reasonable dietary advice to borderline diabetic patients or diabetic patients, we propose a novel mathematical model for the periodic impulse glucose intake. The model is given by(1)dGtdt=−σ2Gt−ac+kltm+ItGt+ρXt+bdItdt=−diIt+σ1G2tα12+G2tdXtdt=−μXt,t≠nτ,ΔXt=Gin,t=nτ,with the initial condition G0=G0>0, I0=I0>0, and X0=X0>0, where Gt and It represent the glucose concentration and insulin concentration in the human body at time t≥0, respectively. Xt represents the exogenous glucose concentration at the moment t after a meal. σ2Gt and ac+klt/m+ItGt are the insulin-independent and insulin-dependent glucose consumption, respectively. b is the hepatic glucose production rate. σ1G2t/α12+G2t represents the insulin secretion stimulated by elevated glucose concentration. diIt is the insulin degradation with di>0 as a constant degradation rate. μXt represents the ingested glucose consumption. ρXt represents the ingested glucose conversion. Here, σ1, σ2, α1, a, c, m, b, di, k, μ, and ρ are positive constants. Since the loss of glucose absorption, we assume that ρ<μ. ΔXt represents the amount of glucose per exogenous intake. GinmU>0 is the intake value, and τmin>0 is the intake period. ΔXt=Xnτ+−Xnτ, that is, intake GinmU of glucose through the diet when t=nτ, n∈Z+=1,2,3,…. The moment immediately after the nth intake is denoted as t=nτ+ here.First, we consider the third and the forth equations of system (1); that is,(2)dXtdt=−μXt,t≠nτ,ΔXt=Gin,t=nτ.From the theoretical knowledge to impulsive differential systems, the readers may refer to [20–22]. It can be easily obtained that system (2) has a globally asymptotically stable positive periodic solution:(3)X˜t=Gine−μt−nτ1−e−μτ,t∈nτ,n+1τ,n∈Z+,X˜0+=Gin1−e−μτ.The proof process is similar to the proof in Appendix A in reference [20], and we omit it here. And the solution of system (2) takes the form Xt=X0−X˜0+e−μt+X˜t, which satisfies(4)limt⟶∞Xt=X˜t.Now, we consider the following system:(5)dGtdt=−σ2Gt−ac+kltm+ItGt+ρX˜t+b,dItdt=−diIt+σ1G2tα12+G2t.System (5) is the limiting system of system (1); that is, if G˜t,I˜t is a solution of system (5) and Gt,It is a solution of system (1), then(6)limt⟶∞Gt=G˜t,limt⟶∞It=I˜t.It is well known that the secretion of insulin has an impact on the production of liver glucose and the utilization of insulin-dependent glucose. When the plasma glucose concentration level is elevated, theβ-cells secrete insulin after a complex series of cascading physiological processes, and such a process can delay the release of insulin. The total delay time is about 5–15 minutes [23]. In order to further understand the mechanism of the glucose-insulin system, we consider the effect of insulin secretion delay on system (1) and propose a model with the discrete insulin secretion delay. The model is given by(7)dGtdt=−σ2Gt−ac+kltm+ItGt+ρXt+bdItdt=−diIt+σ1G2t−τsα12+G2t−τsdXtdt=−μXt,t≠nτ,ΔXt=Gin,t=nτ,with initial condition G0=G0>0, I0=I0>0, X0=X0>0, and t∈−τs,0, and τs represents the insulin secretion delay. Each parameter in system (7) has the same meaning as system (1). Similar to system (5), the limit system of system (7) is as follows:(8)dGtdt=−σ2Gt−ac+kltm+ItGt+ρX˜t+b,dItdt=−diIt+σ1G2t−τsα12+G2t−τs.In the following, for the sake of analyzing the impulsive differential systems, we will investigate the dynamical behavior of system (1) and system (7) by studying their limit systems (5) and (8).


## 3. Qualitative Analyses of System (5)

### 3.1. Permanence

In reality, the concentrations of glucoseGt and insulin It cannot be negative at any time t and are always maintained within a certain range. Now, we will show the positiveness and boundedness of system (5).Proposition 1.
Positiveness. Suppose that Gt,It is a solution of system (5) with the initial value G0=G0>0 and I0=I0>0, and then Gt>0 and It>0 for all t>0.Proof.
For system (5), except that X˜t is a periodic function, and all the other parameters are positive constants. Since X˜t is continuous, and it is easy to verify that the each function on the right side of system (5) satisfies the Lipschitz condition with respect to the variable G or I; thus, for any initial value G0=G0>0, I0=I0>0, there exists a unique solution for system (5). Let Gt,It be a solution of system (5) with G0>0, I0>0. Now, we show that it is nonnegative. If Gt is not nonnegative, then there exists t>0 such that Gt<0. Let t∗=inft:Gt≤0, then Gt∗=0 and dGt∗/dt∗≤0. From the first equation of system (5), we have dGt∗/dt∗=ρX˜t∗+b>0, which is a contradiction.
If there existst>0 such that It<0, let t∗=inft:It≤0, then It∗=0, and limt⟶t∗−dIt/dt<0. From the second equation of system (5), we have limt⟶t∗−dIt/dt>0, which is also a contradiction.
From above, we knowGt≥0 for all t≥0, and it is clear that dGt/dt>0 when Gt=0. Therefore, we have Gt>0 for all t>0 if G0=G0>0. Besides, we know It≥0 for all t≥0, and dIt/dt>0 when It=0, Gt>0. Therefore, we have It>0 for all t>0 if G0=G0>0 and I0=I0>0.
Next, we show that the solution of system (5) is bounded. Let us start with a useful lemma which will be used to prove Proposition 2 and Theorem 1.Lemma 1 (see [1]).
Ifdht/dt≤p−qht or dht/dt≥p−qhtallt≥0, where p,q≥0, then(9)ht≤pq+h0−pqe−qt,ht≥pq+h0−pqe−qt.Proposition 2.
Boundedness. For a solution Gt,It of system (5) with positive initial values, there exist positive constants M1 and t0 such that Gt≤M1 and It≤M1 for all t>t0.Proof.
From the first equation of system (5), we have(10)dGtdt≤supt∈0,τρX˜t−acGt+b,=ρGin+b1−e−μτ1−e−μτ−acGt.
From Lemma1, we can get(11)Gt≤ρGin+b1−e−μτ1−e−μτac+G0−ρGin+b1−e−μτ1−e−μτace−act.
Thus, for sufficient small constantε1>0, there exists N1>0 such that(12)Gt≤ρGin+b1−e−μτ1−e−μτac+ε1,t>N1.
Denote(13)MG=maxρGin+b1−e−μτ1−e−μτac+ε1,G0.
Then,(14)Gt≤MG,t>N1.
From the second equation of system (5), we have(15)dItdt≤σ1MG2α12+MG2−diIt.
From Lemma 1, we can get(16)It≤σ1MG2α12+MG2di+I0−σ1MG2α12+MG2die−dit.
Thus, for sufficient small constantε2>0, there exists N2>0 such that(17)It≤σ1MG2α12+MG2di+ε2,t>N2.
Denote(18)MI=maxσ1MG2α12+MG2di+ε2,I0.
Then,(19)It≤MI,t>N2.
Let(20)M1=maxMG,MI,t0=maxN1,N2.
We can getGt≤M1 and It≤M1 for all t>t0.Theorem 1.
System (5) is permanent; that is, for the solution Gt,It of system (5) with positive initial values, there exist positive constants M1, M20<M2<M1, and T0 such that M2≤Gt,It≤M1, and t>T0.Proof.
From Lemma1 and Proposition 2, we only need to prove that there exist positive constants M2 and T0, such that Gt≥M2 and It≥M2 for t>T0.
From the first equation of system (5), we have(21)dGtdt≥inft∈0,τρX˜t+b−σ2Gt−ac+kGt,=ρGine−μτ+b1−e−μτ1−e−μτ−σ2+ac+akGt,and then(22)Gt≥ρGine−μτ+b1−e−μτ1−e−μτσ2+ac+ak+G0−ρGine−μτ+b1−e−μτ1−e−μτσ2+ac+ake−σ2+ac+akt.
So, for sufficient small constantε3>0, there exists N3>0 such that(23)Gt≥ρGine−μτ+b1−e−μτ1−e−μτσ2+ac+ak−ε3,t>N3.
Denote(24)mG=minρGine−μτ+b1−e−μτ1−e−μτσ2+ac+ak−ε3,G0.
We have(25)Gt≥mG,t>N3.
From the second equation of system (5), we have(26)dItdt≥σ1mG2α12+mG2−diIt,and hence,(27)It≥σ1mG2α12+mG2di+I0−σ1mG2α12+mG2die−dit.
Thus, for sufficient small constantε4>0, there exists N4>0 such that(28)It≥σ1mG2α12+mG2di−ε4,t>N4.
Denote(29)mI=minσ1mG2α12+mG2di−ε4,I0,and then(30)It≥mI,t>N4.
Let(31)M2=minmG,mI,t1=maxN3,N4.
We can getGt≥M2 and It≥M2, t>t1. Let T0=maxt0,t1, then M2≤Gt,It≤M1, t>T0. Therefore, system (5) is permanent.


### 3.2. Existence and Stability of the Periodic Solution

In this section, we will prove that system (5) exists a positive periodic solution by Arzela–Ascoli theorem and Krasnoselskii’s fixed point theorem.Lemma 2 (Arzela–Ascoli theorem, see [24]).
Let sequencefn1∞ be a uniformly bounded and equicontinuous sequence of functions on the closed interval a,b. There exists a subsequence of fn1∞ which converges uniformly on a,b.Lemma 3 (Krasnoselskii’s fixed point theorem, see [25]).
LetB be a Banach space and K⊂B be a cone in B. Assume that Ω1 and Ω2 are open subsets of B with 0∈Ω1, Ω¯1∈Ω2. And let(32)T:K∩Ω¯2\Ω1⟶K,be a completely continuous operator, such that either(1)
Tx≤x, x∈K∩∂Ω1 and Tx≥x, x∈K∩∂Ω2 or(2)
Tx≥x, x∈K∩∂Ω1 and Tx≤x, x∈K∩∂Ω2 is true.
Then,T has a fixed point in K∩Ω¯2\Ω1.Theorem 2.
If(33)Gin≥σ1expdiτexpσ2τ−11−exp−μτρexpσ2τexpdiτ−1−bρ1−exp−μτ,then system (5) has a positive periodic solution G∗t,I∗t with period τ.Proof.
To prove the theorem, we consider the Banach space(34)B=ut,vtu∈CR,R,v∈CR,R,ut+τ=ut,vt+τ=vt,with(35)u,v=maxsupt∈0,τut,supt∈0,τvt.
Define a coneK in B by(36)K=u,v∈B:ut≥ABsupt∈0,τut,vt≥CDsupt∈0,τvt,where A, B, C, and D are defined by(37)A=1expσ2τ−1>0,B=expσ2τexpσ2τ−1>0,C=1expdiτ−1>0,D=expdiτexpdiτ−1>0.
We haveA<B and C<D. Let(38)r1=Aτinft∈0,τρX˜t−ac+kMG+b=AτρGinexp−μτ1−exp−μτ−ac+kMG+b,r2=Bτsupt∈0,τρX˜t+b=BτρGin1−exp−μτ+b.
Then, define two open setsΩr1 and Ωr2 by(39)Ωr1=u,v∈B:u,v<r1,Ωr2=u,v∈B:u,v<r2,and thus, ∂Ωri=u,v∈B:u,v=ri,i=1,2, and K∩Ω¯r2\Ωr1=u,v∈B:r1≤u,v≤r2. Define the map Tu,v=T1u,v,T2u,v:K∩Ω¯r2\Ωr1⟶K by(40)T1ut,vt=∫tt+τUut,sρX˜s−ac+kvsm+vsus+bds,T2ut,vt=∫tt+τUvt,sσ1u2sα12+u2sds,where(41)Uut,s=expσ2s−texpσ2τ−1,Uvt,s=expdis−texpdiτ−1.
Obviously, we haveA≤Uut,s≤B,t≤s≤t+τ, C≤Uvt,s≤D,t≤s≤t+τ.
In order to use Krasnoselskii’s fixed point theorem, we need to prove the mappingT is completely continuous. Firstly, we prove that the mapping T is continuous.
In view of the definition ofK, if u,v∈K, then(42)T1u,vt+τ=∫t+τt+2τUut+τ,sρX˜s−ac+kvsm+vsus+bds,=∫tt+τUut+τ,θ+τρX˜θ+τ−ac+kvθ+τm+vθ+τuθ+τ+bdθ=T1u,vt,T2u,vt+τ=∫t+τt+2τUvt+τ,sσ1u2sα12+u2sds=∫tt+τUvt+τ,θ+τσ1u2θ+τα12+u2θ+τdθ=∫tt+τUvt,sσ1u2sα12+u2sds=T2u,vt.
Besides, for everyu,v∈K∩Ω¯2\Ω1,(43)T1u,vt≥A∫tt+τρX˜s−ac+kvsm+vsus+bds=ABB∫0τρX˜s−ac+kvsm+vsus+bds≥ABsupt∈0,τT1u,vt,T2u,vt≥C∫tt+τσ1u2sα12+u2sds=CDD∫0τσ1u2sα12+u2sds≥CDsupt∈0,τT2u,vt.
Thus,T:K∩Ω¯2\Ω1⊂K.
It is easy to know thatT is continuous. Next, we show T is compact by proving that it is uniformly bounded and equicontinuous.
LetS⊆K∩Ω¯2\Ω1 be a bounded set. For any ut,vt∈S and t<nτ<t+τ, we know(44)Tut,vt=maxsupt∈0,τT1ut,vt,supt∈0,τT2ut,vt.
Besides,(45)T1u,vt≤B∫tt+τρX˜s−ac+kvsm+vsus+bds=B∫0τρX˜s−ac+kvsm+vsus+bds≤Bτsupt∈0,τρX˜s+b=BτρGin1−exp−μτ+Bτb,T2u,vt≤D∫tt+τσ1G2sα12+G2sds≤Dτσ1.
Therefore,(46)Tut,vt≤maxBτρGin1−exp−μτ+Bτb,Dτσ1,which means TS is uniformly bounded.
For everyut,vt∈S, it is easy to know that the derivative functions of Tiut,vt, i=1,2, are uniformly bounded, so all functions in TS are equicontinuous.
From Arzela–Ascoli theorem, we knowT is compact. Thus, the compact continuous operator T is a completely continuous operator.
Now, we prove thatu,v is the fixed point of T.
Ifu,v∈∂Ωr1, then u,v=r1, and(47)T1u,vt≥A∫tt+τρX˜s−ac+kvsm+vsus+bds≥Aτinft∈0,τρX˜s−ac+kMG+b=r1.
So,(48)Tu,v≥u,v.
Ifu,v∈∂Ωr2, then u,v=r2, and(49)T1u,vt≤B∫tt+τρX˜s−ac+kvsm+vsus+bds≤Bτsupt∈0,τρX˜s+b=r2,T2u,vt≤D∫tt+τσ1u2sα12+u2sds≤Dτσ1.
From condition (33), we know Dτσ1=expdiττσ1/expdiτ−1≤r2; thus, Tu,v≤u,v, where u,v∈∂Ω2. Therefore, T has a fixed point u,v in K∩Ω¯2\Ω1 by Krasnoselskii’s fixed point theorem. Next, we will prove u,v is the positive periodic solution of system (5).
Sinceu,v∈K∩Ω¯2\Ω1 and T1u,v=u, T2u,v=v,(50)dutdt=ddt∫tt+τUut,sρX˜s−ac+kvsm+vsus+bds=Uut,t+τρX˜t+τ−ac+kvt+τm+vt+τut+τ+b−Uut,tρX˜t−ac+kvtm+vtut+b+∫tt+τdUudtρX˜s−ac+kvsm+vsus+bds=Uut,t+τ−Uut,tρX˜t−ac+kvtm+vtut+b−σ2ut=ρX˜t−σ2ut−ac+kvtm+vtut+b,dvtdt=ddt∫tt+τUvt,sσ1u2sα12+u2sds=Uvt,t+τσ1u2t+τα12+u2t+τ−Uvt,tσ1u2tα12+u2t+∫tt+τdUvdtσ1u2sα12+u2sds=Uvt+τ−Uvt,tσ1u2tα12+u2t+∫tt+τ−diUvσ1u2sα12+u2sds=σ1u2tα12+u2t−divt.
Thus,u,v is a positive periodic solution of system (5), which means system (5) has a positive periodic solution G∗t,I∗t with period τ. This completes the proof of Theorem 2.
Then, we will prove the globally asymptotic stability of the positive periodic solution of system (5).Theorem 3.
If there existε1>0 and ε2>0 such that(51)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2≥0,di−mkaMG2m+mI21ε1−3σ116α11ε2≥0,then the positive periodic solution G∗t,I∗t of system (5) is globally asymptotically stable.Proof.
SupposeGt,It is the solution of system (5). We define(52)pt=Gt−G∗t,qt=It−I∗t.
Letf2G=aG, f3I=c+ki/m+I, f2′G=df2G/dG, and f3′I=df3I/dI, and we can get(53)dptdt=dGtdt−dG∗tdt=−σ2Gt−G∗t−ac+kItm+ItGt−ac+kI∗tm+I∗tG∗t=−σ2pt−f2Gf3I−f2G∗f3I∗=−σ2pt−Gt−G∗tf2′ξ2f3I−It−I∗tf3′ξ3f2G∗=−σ2pt−ptac+kItm+It+qtkmm+ξ32aG∗=−σ2+ac+akItm+Itpt−kmm+ξ32aG∗tqt,dqtdt=dItdt−dI∗tdt=−diIt−I∗t+σ1Gt2α12+Gt2−σ1G∗t2α12+G∗t2=−diqt+f1G−f1G∗=−diqt+Gt−G∗tf1′ξ1=pt2α12σ1ξ1α12+ξ122−diqt,where ξ1 and ξ2 are between Gt and G∗t, and ξ3 is between It and I∗t.
LetVt=1/2p2t+1/2q2t, and we use the inequality 2ab≤εa2+b2/ε in the following derivation; we have(54)dVtdt=ptdptdt+qtdqtdt=−σ2+ac+akItm+Itp2t−mkaG∗tm+ξ32ptqt+2α12σ1ξ1α12+ξ122ptqt−diq2t<−σ2+ac+akItm+Itp2t+mkaG∗tm+ξ32ptqt+2α12σ1ξ1α12+ξ122ptqt−diq2t<−σ2+ac+akmIm+mIp2t+mkaMGm+mI2ptqt+33σ18α1ptqt−diq2t<−σ2+ac+akmIm+mIp2+mkaMGm+mI2ε12p2t+12ε1q2t+33σ116α1ε2p2t+33σ116α11ε2q2t−diq2t=−σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2p2t−di−mkaMG2m+mI21ε1−3σ116α11ε2q2t.
If we chooseε1>0 and ε2>0 such that(55)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2≥0,di−mkaMG2m+mI21ε1−3σ116α11ε2≥0.
Then,dvt/dt≤0, and p,qdvt/dt=0 contains no other trajectories other than p,q=0,0; that is, the positive periodic solution of system (5) is globally asymptotically stable.Remark 1.
The existence and globally asymptotic stability of the periodic solution of system (5) indicate that the glucose-insulin system can be regulated in the form of continuous oscillations by controlling the amount and period of glucose intake.


## 3.1. Permanence

In reality, the concentrations of glucoseGt and insulin It cannot be negative at any time t and are always maintained within a certain range. Now, we will show the positiveness and boundedness of system (5).Proposition 1.
Positiveness. Suppose that Gt,It is a solution of system (5) with the initial value G0=G0>0 and I0=I0>0, and then Gt>0 and It>0 for all t>0.Proof.
For system (5), except that X˜t is a periodic function, and all the other parameters are positive constants. Since X˜t is continuous, and it is easy to verify that the each function on the right side of system (5) satisfies the Lipschitz condition with respect to the variable G or I; thus, for any initial value G0=G0>0, I0=I0>0, there exists a unique solution for system (5). Let Gt,It be a solution of system (5) with G0>0, I0>0. Now, we show that it is nonnegative. If Gt is not nonnegative, then there exists t>0 such that Gt<0. Let t∗=inft:Gt≤0, then Gt∗=0 and dGt∗/dt∗≤0. From the first equation of system (5), we have dGt∗/dt∗=ρX˜t∗+b>0, which is a contradiction.
If there existst>0 such that It<0, let t∗=inft:It≤0, then It∗=0, and limt⟶t∗−dIt/dt<0. From the second equation of system (5), we have limt⟶t∗−dIt/dt>0, which is also a contradiction.
From above, we knowGt≥0 for all t≥0, and it is clear that dGt/dt>0 when Gt=0. Therefore, we have Gt>0 for all t>0 if G0=G0>0. Besides, we know It≥0 for all t≥0, and dIt/dt>0 when It=0, Gt>0. Therefore, we have It>0 for all t>0 if G0=G0>0 and I0=I0>0.
Next, we show that the solution of system (5) is bounded. Let us start with a useful lemma which will be used to prove Proposition 2 and Theorem 1.Lemma 1 (see [1]).
Ifdht/dt≤p−qht or dht/dt≥p−qhtallt≥0, where p,q≥0, then(9)ht≤pq+h0−pqe−qt,ht≥pq+h0−pqe−qt.Proposition 2.
Boundedness. For a solution Gt,It of system (5) with positive initial values, there exist positive constants M1 and t0 such that Gt≤M1 and It≤M1 for all t>t0.Proof.
From the first equation of system (5), we have(10)dGtdt≤supt∈0,τρX˜t−acGt+b,=ρGin+b1−e−μτ1−e−μτ−acGt.
From Lemma1, we can get(11)Gt≤ρGin+b1−e−μτ1−e−μτac+G0−ρGin+b1−e−μτ1−e−μτace−act.
Thus, for sufficient small constantε1>0, there exists N1>0 such that(12)Gt≤ρGin+b1−e−μτ1−e−μτac+ε1,t>N1.
Denote(13)MG=maxρGin+b1−e−μτ1−e−μτac+ε1,G0.
Then,(14)Gt≤MG,t>N1.
From the second equation of system (5), we have(15)dItdt≤σ1MG2α12+MG2−diIt.
From Lemma 1, we can get(16)It≤σ1MG2α12+MG2di+I0−σ1MG2α12+MG2die−dit.
Thus, for sufficient small constantε2>0, there exists N2>0 such that(17)It≤σ1MG2α12+MG2di+ε2,t>N2.
Denote(18)MI=maxσ1MG2α12+MG2di+ε2,I0.
Then,(19)It≤MI,t>N2.
Let(20)M1=maxMG,MI,t0=maxN1,N2.
We can getGt≤M1 and It≤M1 for all t>t0.Theorem 1.
System (5) is permanent; that is, for the solution Gt,It of system (5) with positive initial values, there exist positive constants M1, M20<M2<M1, and T0 such that M2≤Gt,It≤M1, and t>T0.Proof.
From Lemma1 and Proposition 2, we only need to prove that there exist positive constants M2 and T0, such that Gt≥M2 and It≥M2 for t>T0.
From the first equation of system (5), we have(21)dGtdt≥inft∈0,τρX˜t+b−σ2Gt−ac+kGt,=ρGine−μτ+b1−e−μτ1−e−μτ−σ2+ac+akGt,and then(22)Gt≥ρGine−μτ+b1−e−μτ1−e−μτσ2+ac+ak+G0−ρGine−μτ+b1−e−μτ1−e−μτσ2+ac+ake−σ2+ac+akt.
So, for sufficient small constantε3>0, there exists N3>0 such that(23)Gt≥ρGine−μτ+b1−e−μτ1−e−μτσ2+ac+ak−ε3,t>N3.
Denote(24)mG=minρGine−μτ+b1−e−μτ1−e−μτσ2+ac+ak−ε3,G0.
We have(25)Gt≥mG,t>N3.
From the second equation of system (5), we have(26)dItdt≥σ1mG2α12+mG2−diIt,and hence,(27)It≥σ1mG2α12+mG2di+I0−σ1mG2α12+mG2die−dit.
Thus, for sufficient small constantε4>0, there exists N4>0 such that(28)It≥σ1mG2α12+mG2di−ε4,t>N4.
Denote(29)mI=minσ1mG2α12+mG2di−ε4,I0,and then(30)It≥mI,t>N4.
Let(31)M2=minmG,mI,t1=maxN3,N4.
We can getGt≥M2 and It≥M2, t>t1. Let T0=maxt0,t1, then M2≤Gt,It≤M1, t>T0. Therefore, system (5) is permanent.


## 3.2. Existence and Stability of the Periodic Solution

In this section, we will prove that system (5) exists a positive periodic solution by Arzela–Ascoli theorem and Krasnoselskii’s fixed point theorem.Lemma 2 (Arzela–Ascoli theorem, see [24]).
Let sequencefn1∞ be a uniformly bounded and equicontinuous sequence of functions on the closed interval a,b. There exists a subsequence of fn1∞ which converges uniformly on a,b.Lemma 3 (Krasnoselskii’s fixed point theorem, see [25]).
LetB be a Banach space and K⊂B be a cone in B. Assume that Ω1 and Ω2 are open subsets of B with 0∈Ω1, Ω¯1∈Ω2. And let(32)T:K∩Ω¯2\Ω1⟶K,be a completely continuous operator, such that either(1)
Tx≤x, x∈K∩∂Ω1 and Tx≥x, x∈K∩∂Ω2 or(2)
Tx≥x, x∈K∩∂Ω1 and Tx≤x, x∈K∩∂Ω2 is true.
Then,T has a fixed point in K∩Ω¯2\Ω1.Theorem 2.
If(33)Gin≥σ1expdiτexpσ2τ−11−exp−μτρexpσ2τexpdiτ−1−bρ1−exp−μτ,then system (5) has a positive periodic solution G∗t,I∗t with period τ.Proof.
To prove the theorem, we consider the Banach space(34)B=ut,vtu∈CR,R,v∈CR,R,ut+τ=ut,vt+τ=vt,with(35)u,v=maxsupt∈0,τut,supt∈0,τvt.
Define a coneK in B by(36)K=u,v∈B:ut≥ABsupt∈0,τut,vt≥CDsupt∈0,τvt,where A, B, C, and D are defined by(37)A=1expσ2τ−1>0,B=expσ2τexpσ2τ−1>0,C=1expdiτ−1>0,D=expdiτexpdiτ−1>0.
We haveA<B and C<D. Let(38)r1=Aτinft∈0,τρX˜t−ac+kMG+b=AτρGinexp−μτ1−exp−μτ−ac+kMG+b,r2=Bτsupt∈0,τρX˜t+b=BτρGin1−exp−μτ+b.
Then, define two open setsΩr1 and Ωr2 by(39)Ωr1=u,v∈B:u,v<r1,Ωr2=u,v∈B:u,v<r2,and thus, ∂Ωri=u,v∈B:u,v=ri,i=1,2, and K∩Ω¯r2\Ωr1=u,v∈B:r1≤u,v≤r2. Define the map Tu,v=T1u,v,T2u,v:K∩Ω¯r2\Ωr1⟶K by(40)T1ut,vt=∫tt+τUut,sρX˜s−ac+kvsm+vsus+bds,T2ut,vt=∫tt+τUvt,sσ1u2sα12+u2sds,where(41)Uut,s=expσ2s−texpσ2τ−1,Uvt,s=expdis−texpdiτ−1.
Obviously, we haveA≤Uut,s≤B,t≤s≤t+τ, C≤Uvt,s≤D,t≤s≤t+τ.
In order to use Krasnoselskii’s fixed point theorem, we need to prove the mappingT is completely continuous. Firstly, we prove that the mapping T is continuous.
In view of the definition ofK, if u,v∈K, then(42)T1u,vt+τ=∫t+τt+2τUut+τ,sρX˜s−ac+kvsm+vsus+bds,=∫tt+τUut+τ,θ+τρX˜θ+τ−ac+kvθ+τm+vθ+τuθ+τ+bdθ=T1u,vt,T2u,vt+τ=∫t+τt+2τUvt+τ,sσ1u2sα12+u2sds=∫tt+τUvt+τ,θ+τσ1u2θ+τα12+u2θ+τdθ=∫tt+τUvt,sσ1u2sα12+u2sds=T2u,vt.
Besides, for everyu,v∈K∩Ω¯2\Ω1,(43)T1u,vt≥A∫tt+τρX˜s−ac+kvsm+vsus+bds=ABB∫0τρX˜s−ac+kvsm+vsus+bds≥ABsupt∈0,τT1u,vt,T2u,vt≥C∫tt+τσ1u2sα12+u2sds=CDD∫0τσ1u2sα12+u2sds≥CDsupt∈0,τT2u,vt.
Thus,T:K∩Ω¯2\Ω1⊂K.
It is easy to know thatT is continuous. Next, we show T is compact by proving that it is uniformly bounded and equicontinuous.
LetS⊆K∩Ω¯2\Ω1 be a bounded set. For any ut,vt∈S and t<nτ<t+τ, we know(44)Tut,vt=maxsupt∈0,τT1ut,vt,supt∈0,τT2ut,vt.
Besides,(45)T1u,vt≤B∫tt+τρX˜s−ac+kvsm+vsus+bds=B∫0τρX˜s−ac+kvsm+vsus+bds≤Bτsupt∈0,τρX˜s+b=BτρGin1−exp−μτ+Bτb,T2u,vt≤D∫tt+τσ1G2sα12+G2sds≤Dτσ1.
Therefore,(46)Tut,vt≤maxBτρGin1−exp−μτ+Bτb,Dτσ1,which means TS is uniformly bounded.
For everyut,vt∈S, it is easy to know that the derivative functions of Tiut,vt, i=1,2, are uniformly bounded, so all functions in TS are equicontinuous.
From Arzela–Ascoli theorem, we knowT is compact. Thus, the compact continuous operator T is a completely continuous operator.
Now, we prove thatu,v is the fixed point of T.
Ifu,v∈∂Ωr1, then u,v=r1, and(47)T1u,vt≥A∫tt+τρX˜s−ac+kvsm+vsus+bds≥Aτinft∈0,τρX˜s−ac+kMG+b=r1.
So,(48)Tu,v≥u,v.
Ifu,v∈∂Ωr2, then u,v=r2, and(49)T1u,vt≤B∫tt+τρX˜s−ac+kvsm+vsus+bds≤Bτsupt∈0,τρX˜s+b=r2,T2u,vt≤D∫tt+τσ1u2sα12+u2sds≤Dτσ1.
From condition (33), we know Dτσ1=expdiττσ1/expdiτ−1≤r2; thus, Tu,v≤u,v, where u,v∈∂Ω2. Therefore, T has a fixed point u,v in K∩Ω¯2\Ω1 by Krasnoselskii’s fixed point theorem. Next, we will prove u,v is the positive periodic solution of system (5).
Sinceu,v∈K∩Ω¯2\Ω1 and T1u,v=u, T2u,v=v,(50)dutdt=ddt∫tt+τUut,sρX˜s−ac+kvsm+vsus+bds=Uut,t+τρX˜t+τ−ac+kvt+τm+vt+τut+τ+b−Uut,tρX˜t−ac+kvtm+vtut+b+∫tt+τdUudtρX˜s−ac+kvsm+vsus+bds=Uut,t+τ−Uut,tρX˜t−ac+kvtm+vtut+b−σ2ut=ρX˜t−σ2ut−ac+kvtm+vtut+b,dvtdt=ddt∫tt+τUvt,sσ1u2sα12+u2sds=Uvt,t+τσ1u2t+τα12+u2t+τ−Uvt,tσ1u2tα12+u2t+∫tt+τdUvdtσ1u2sα12+u2sds=Uvt+τ−Uvt,tσ1u2tα12+u2t+∫tt+τ−diUvσ1u2sα12+u2sds=σ1u2tα12+u2t−divt.
Thus,u,v is a positive periodic solution of system (5), which means system (5) has a positive periodic solution G∗t,I∗t with period τ. This completes the proof of Theorem 2.
Then, we will prove the globally asymptotic stability of the positive periodic solution of system (5).Theorem 3.
If there existε1>0 and ε2>0 such that(51)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2≥0,di−mkaMG2m+mI21ε1−3σ116α11ε2≥0,then the positive periodic solution G∗t,I∗t of system (5) is globally asymptotically stable.Proof.
SupposeGt,It is the solution of system (5). We define(52)pt=Gt−G∗t,qt=It−I∗t.
Letf2G=aG, f3I=c+ki/m+I, f2′G=df2G/dG, and f3′I=df3I/dI, and we can get(53)dptdt=dGtdt−dG∗tdt=−σ2Gt−G∗t−ac+kItm+ItGt−ac+kI∗tm+I∗tG∗t=−σ2pt−f2Gf3I−f2G∗f3I∗=−σ2pt−Gt−G∗tf2′ξ2f3I−It−I∗tf3′ξ3f2G∗=−σ2pt−ptac+kItm+It+qtkmm+ξ32aG∗=−σ2+ac+akItm+Itpt−kmm+ξ32aG∗tqt,dqtdt=dItdt−dI∗tdt=−diIt−I∗t+σ1Gt2α12+Gt2−σ1G∗t2α12+G∗t2=−diqt+f1G−f1G∗=−diqt+Gt−G∗tf1′ξ1=pt2α12σ1ξ1α12+ξ122−diqt,where ξ1 and ξ2 are between Gt and G∗t, and ξ3 is between It and I∗t.
LetVt=1/2p2t+1/2q2t, and we use the inequality 2ab≤εa2+b2/ε in the following derivation; we have(54)dVtdt=ptdptdt+qtdqtdt=−σ2+ac+akItm+Itp2t−mkaG∗tm+ξ32ptqt+2α12σ1ξ1α12+ξ122ptqt−diq2t<−σ2+ac+akItm+Itp2t+mkaG∗tm+ξ32ptqt+2α12σ1ξ1α12+ξ122ptqt−diq2t<−σ2+ac+akmIm+mIp2t+mkaMGm+mI2ptqt+33σ18α1ptqt−diq2t<−σ2+ac+akmIm+mIp2+mkaMGm+mI2ε12p2t+12ε1q2t+33σ116α1ε2p2t+33σ116α11ε2q2t−diq2t=−σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2p2t−di−mkaMG2m+mI21ε1−3σ116α11ε2q2t.
If we chooseε1>0 and ε2>0 such that(55)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2≥0,di−mkaMG2m+mI21ε1−3σ116α11ε2≥0.
Then,dvt/dt≤0, and p,qdvt/dt=0 contains no other trajectories other than p,q=0,0; that is, the positive periodic solution of system (5) is globally asymptotically stable.Remark 1.
The existence and globally asymptotic stability of the periodic solution of system (5) indicate that the glucose-insulin system can be regulated in the form of continuous oscillations by controlling the amount and period of glucose intake.


## 4. Qualitative Analyses of System (8)

### 4.1. Existence and Stability of the Periodic Solution

System (8) satisfies positiveness, boundedness, and permanence, which can be proved in the same way as in system (5). And system (8) also has a positive periodic solution with period τ.Theorem 4.
If(56)Gin≥σ1expdiτexpσ2τ−11−exp−μτρexpσ2τexpdiτ−1−bρ1−exp−μτ,then system (8) has a positive periodic solution G∗t,I∗t with period τ.The proof of Theorem4 is similar to Theorem 2; here, we omit it.Remark 2.
Let(57)G^τ=σ1expdiτexpσ2τ−11−exp−μτρexpσ2τexpdiτ−1−bρ1−exp−μτ.
Theorems1 and 2 show that if Gin>G^τ, then system (5) and system (8) exist periodic solutions when the intake period is constant.
Next, we prove that the positive periodic solutionG∗t,I∗t of system (8) is globally asymptotically stable.Theorem 5.
If there existε1>0, ε2>0, and ε3>0 such that(58)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2−33σ1τs16α1ε3σ2+ac+akMIm+MI≥0,(59)di−mkaMG2m+mI21ε1−33σ116α11ε2−33σ1τs16α1σ2+ac+akMIm+MIε3−33σ1τs8α1kmaMGm+mI2≥0,then the positive periodic solution G∗t,I∗t of system (8) is globally asymptotically stable.Proof.
SupposeGt,It is a solution of system (8) with Gt=G0>0 and It=I0>0 for all t∈−τ,0. We define(60)p¯t=Gt−G∗t,q¯t=It−I∗t.
Letf2G=aG, f3I=c+kI/m+I, f2′G=df2G/dG, and f3′I=df3I/dI, and we get(61)dp¯tdt=dGtdt−dG∗tdt=−σ2Gt−ac+kItm+ItGt+ρX˜t+b−−σ2G∗t−ac+kI∗tm+I∗tG∗t+ρX˜t+b=−σ2Gt−G∗t−ac+kItm+ItGt−ac+kI∗tm+I∗tG∗t=−σ2p¯t−f2Gf3I−f2G∗f3I∗=−σ2p¯t−Gt−G∗tf2′ξ2f3It−It−I∗tf3′ξ3f2G∗t=−σ2p¯t−p¯tac+kItm+It+q¯tkmm+ξ32aG∗t=−σ2+ac+akItm+Itp¯t−kmm+ξ32aG∗tq¯t,dq¯tdt=dItdt−dI∗tdt=−diIt+σ1Gt−τs2α12+Gt−τs2−−diI∗t+σ1G∗t−τs2α12+G∗t−τs2=−diIt−I∗t+σ1Gt−τs2α12+Gt−τs2−σ1G∗t−τs2α12+G∗t−τs2=−diq¯t+f1Gt−τs−f1G∗t−τs=−diq¯t+Gt−τs−G∗t−τsf1′ξ1=−diq¯t+Gt−τs−G∗t−τs2α12σ1ξ1σ12+ξ122=p¯t−τs2α12σ1ξ1α12+ξ122−diq¯t,where ξ1 and ξ2 are between Gt and G∗t, and ξ3 is between It and I∗t.
LetV1t=1/2p¯2t+1/2q¯2t, and we make use of the inequality 2ab≤εa2+b2/ε in the following derivation; then, we can have(62)dV1tdt=p¯tdp¯tdt+q¯tdq¯tdt=−σ2+ac+akItm+Itp¯2t−mkaG∗tm+ξ32p¯tq¯t−diq¯2t+2α12σ1ξ1α12+ξ122p¯t−τsq¯t<−σ2+ac+akItm+Itp¯2t+mkaG∗tm+ξ32p¯tq¯t−diq¯2t+2α12σ1ξ1α12+ξ122p¯t−τsq¯t<−σ2+ac+akmIm+mIp¯2t+mkaMGm+mI2p¯tq¯t−diq¯2t+33σ18α1p¯t−τsq¯t=−σ2+ac+akmIm+mIp¯2t+mkaMGm+mI2p¯tq¯t−diq¯2t+33σ18α1q¯tp¯t−∫t−τstdp¯sdsds<−σ2+ac+akmIm+mIp¯2t+mkaMGm+mI2ε12p¯2t+12ε1q¯2t+33σ1ε216α1p¯2t+33σ116α1ε2q¯2t−diq¯2t−33σ18α1q¯t∫t−τstdp¯sdsds=−σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2p¯2t−di−mkaMG2m+mI21ε1−33σ116α11ε2q¯2t−33σ18α1q¯t∫t−τstdp¯sdsds,−33σ18α1q¯t∫t−τstdp¯sdsds=−33σ18α1q¯t∫t−τst−σ2+ac+akItm+Itp¯s−kmm+ξ32aG∗q¯sds=33σ18α1q¯t∫t−τstσ2+ac+akItm+Itp¯s+kmm+ξ32aG∗q¯sds≤3σ18α1q¯t∫t−τstσ2+ac+akMIm+MIp¯s+kmaMGm+mI2q¯sds=33σ18α1σ2+ac+akMIm+MI∫t−τstq¯tp¯sds+33σ18α1kmaMGm+mI2∫t−τstq¯tq¯sds≤33σ18α1σ2+ac+akMIm+MI∫t−τstε32q¯t2+12ε3p¯s2ds+33σ18α1kmaMGm+mI2∫t−τst12q¯t2+12q¯s2ds=33σ1τs16α1σ2+ac+akMIm+MIε3+kmaMGm+mI2q¯t2+33σ116α1ε3σ2+ac+akMIm+MI∫t−τstp¯s2ds+33σ116α1kmaMGm+mI2∫t−τstq¯s2ds.
Then,(63)dV1tdt≤−σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2p¯2t−di−mkaMG2m+mI21ε1−33σ116α11ε2−33σ1τs16α1σ2+ac+akMIm+MIε3+kmaMGm+mI2q¯2t+33σ116α1ε3σ2+ac+akMIm+MI∫t−τstp¯s2ds+33σ116α1kmaMGm+mI2∫t−τstq¯s2ds.
Let(64)V2t=33σ116α1ε3σ2+ac+akMIm+MI∫t−τst∫γtp¯s2dsdγ,V3t=33σ116α1kmaMGm+mI2∫t−τst∫γtq¯s2dsdγ.
We can get(65)dV2tdt=−33σ116α1ε3σ2+ac+akMIm+MI∫t−τstp¯s2ds+33σ1τs16α1ε3σ2+ac+akMIm+MIp¯t2,dV3tdt=−33σ116α1kmaMGm+mI2∫t−τstq¯s2ds+33σ1τs16α1kmaMGm+mI2q¯t2.
We define(66)Vt=V1t+V2t+V3t.
Then,(67)dVtdt≤−σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2−33σ1τs16α1ε3σ2+ac+akMIm+MIp¯2t−di−mkaMG2m+mI21ε1−33σ116α11ε2−33σ1τs16α1σ2+ac+akMIm+MIε3−33σ1τs8α1kmaMGm+mI2q¯2t.
If we chooseε1>0, ε2>0, and ε3>0 such that(68)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2−33σ1τs16α1ε3σ2+ac+akMIm+MI≥0,di−mkaMG2m+mI21ε1−33σ116α11ε2−33σ1τs16α1σ2+ac+akMIm+MIε3−33σ1τs8α1kmaMGm+mI2≥0.
Thus,dVt/dt≤0, and p,qdvt/dt=0 contains no other trajectories other than p,q=0,0; that is, the positive periodic solution G∗t,I∗t of system (8) is globally asymptotically stable. This completes the proof.
From (58), we know(69)τs≤163α19σ1m+MIm+mIσ2+ac+m+MIakmIm+MIm+mIσ2+ac+m+mIakMIε3−83α19σ1mkaMGm+MIm+mI2m+MIσ2+ac+akMIε1ε3−m+MIm+MIσ2+ac+akMIε2ε3≜a˜.
Lets˜=σ2+ac+akMI/m+MIε3+2kmaMG/m+mI2, and from (59), we get(70)τs≤163α1di9σ1s˜−83α1mkaMG9σ1m+mI2s˜1ε1−1s˜1ε2≜b˜.Remark 3.
From Theorem 5, we know that the globally asymptotic stability of the periodic solution of system (8) is related to the time delay. And the periodic solution is globally asymptotically stable if the insulin secretion delay τs<mina˜,b˜.


## 4.1. Existence and Stability of the Periodic Solution

System (8) satisfies positiveness, boundedness, and permanence, which can be proved in the same way as in system (5). And system (8) also has a positive periodic solution with period τ.Theorem 4.
If(56)Gin≥σ1expdiτexpσ2τ−11−exp−μτρexpσ2τexpdiτ−1−bρ1−exp−μτ,then system (8) has a positive periodic solution G∗t,I∗t with period τ.The proof of Theorem4 is similar to Theorem 2; here, we omit it.Remark 2.
Let(57)G^τ=σ1expdiτexpσ2τ−11−exp−μτρexpσ2τexpdiτ−1−bρ1−exp−μτ.
Theorems1 and 2 show that if Gin>G^τ, then system (5) and system (8) exist periodic solutions when the intake period is constant.
Next, we prove that the positive periodic solutionG∗t,I∗t of system (8) is globally asymptotically stable.Theorem 5.
If there existε1>0, ε2>0, and ε3>0 such that(58)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2−33σ1τs16α1ε3σ2+ac+akMIm+MI≥0,(59)di−mkaMG2m+mI21ε1−33σ116α11ε2−33σ1τs16α1σ2+ac+akMIm+MIε3−33σ1τs8α1kmaMGm+mI2≥0,then the positive periodic solution G∗t,I∗t of system (8) is globally asymptotically stable.Proof.
SupposeGt,It is a solution of system (8) with Gt=G0>0 and It=I0>0 for all t∈−τ,0. We define(60)p¯t=Gt−G∗t,q¯t=It−I∗t.
Letf2G=aG, f3I=c+kI/m+I, f2′G=df2G/dG, and f3′I=df3I/dI, and we get(61)dp¯tdt=dGtdt−dG∗tdt=−σ2Gt−ac+kItm+ItGt+ρX˜t+b−−σ2G∗t−ac+kI∗tm+I∗tG∗t+ρX˜t+b=−σ2Gt−G∗t−ac+kItm+ItGt−ac+kI∗tm+I∗tG∗t=−σ2p¯t−f2Gf3I−f2G∗f3I∗=−σ2p¯t−Gt−G∗tf2′ξ2f3It−It−I∗tf3′ξ3f2G∗t=−σ2p¯t−p¯tac+kItm+It+q¯tkmm+ξ32aG∗t=−σ2+ac+akItm+Itp¯t−kmm+ξ32aG∗tq¯t,dq¯tdt=dItdt−dI∗tdt=−diIt+σ1Gt−τs2α12+Gt−τs2−−diI∗t+σ1G∗t−τs2α12+G∗t−τs2=−diIt−I∗t+σ1Gt−τs2α12+Gt−τs2−σ1G∗t−τs2α12+G∗t−τs2=−diq¯t+f1Gt−τs−f1G∗t−τs=−diq¯t+Gt−τs−G∗t−τsf1′ξ1=−diq¯t+Gt−τs−G∗t−τs2α12σ1ξ1σ12+ξ122=p¯t−τs2α12σ1ξ1α12+ξ122−diq¯t,where ξ1 and ξ2 are between Gt and G∗t, and ξ3 is between It and I∗t.
LetV1t=1/2p¯2t+1/2q¯2t, and we make use of the inequality 2ab≤εa2+b2/ε in the following derivation; then, we can have(62)dV1tdt=p¯tdp¯tdt+q¯tdq¯tdt=−σ2+ac+akItm+Itp¯2t−mkaG∗tm+ξ32p¯tq¯t−diq¯2t+2α12σ1ξ1α12+ξ122p¯t−τsq¯t<−σ2+ac+akItm+Itp¯2t+mkaG∗tm+ξ32p¯tq¯t−diq¯2t+2α12σ1ξ1α12+ξ122p¯t−τsq¯t<−σ2+ac+akmIm+mIp¯2t+mkaMGm+mI2p¯tq¯t−diq¯2t+33σ18α1p¯t−τsq¯t=−σ2+ac+akmIm+mIp¯2t+mkaMGm+mI2p¯tq¯t−diq¯2t+33σ18α1q¯tp¯t−∫t−τstdp¯sdsds<−σ2+ac+akmIm+mIp¯2t+mkaMGm+mI2ε12p¯2t+12ε1q¯2t+33σ1ε216α1p¯2t+33σ116α1ε2q¯2t−diq¯2t−33σ18α1q¯t∫t−τstdp¯sdsds=−σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2p¯2t−di−mkaMG2m+mI21ε1−33σ116α11ε2q¯2t−33σ18α1q¯t∫t−τstdp¯sdsds,−33σ18α1q¯t∫t−τstdp¯sdsds=−33σ18α1q¯t∫t−τst−σ2+ac+akItm+Itp¯s−kmm+ξ32aG∗q¯sds=33σ18α1q¯t∫t−τstσ2+ac+akItm+Itp¯s+kmm+ξ32aG∗q¯sds≤3σ18α1q¯t∫t−τstσ2+ac+akMIm+MIp¯s+kmaMGm+mI2q¯sds=33σ18α1σ2+ac+akMIm+MI∫t−τstq¯tp¯sds+33σ18α1kmaMGm+mI2∫t−τstq¯tq¯sds≤33σ18α1σ2+ac+akMIm+MI∫t−τstε32q¯t2+12ε3p¯s2ds+33σ18α1kmaMGm+mI2∫t−τst12q¯t2+12q¯s2ds=33σ1τs16α1σ2+ac+akMIm+MIε3+kmaMGm+mI2q¯t2+33σ116α1ε3σ2+ac+akMIm+MI∫t−τstp¯s2ds+33σ116α1kmaMGm+mI2∫t−τstq¯s2ds.
Then,(63)dV1tdt≤−σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2p¯2t−di−mkaMG2m+mI21ε1−33σ116α11ε2−33σ1τs16α1σ2+ac+akMIm+MIε3+kmaMGm+mI2q¯2t+33σ116α1ε3σ2+ac+akMIm+MI∫t−τstp¯s2ds+33σ116α1kmaMGm+mI2∫t−τstq¯s2ds.
Let(64)V2t=33σ116α1ε3σ2+ac+akMIm+MI∫t−τst∫γtp¯s2dsdγ,V3t=33σ116α1kmaMGm+mI2∫t−τst∫γtq¯s2dsdγ.
We can get(65)dV2tdt=−33σ116α1ε3σ2+ac+akMIm+MI∫t−τstp¯s2ds+33σ1τs16α1ε3σ2+ac+akMIm+MIp¯t2,dV3tdt=−33σ116α1kmaMGm+mI2∫t−τstq¯s2ds+33σ1τs16α1kmaMGm+mI2q¯t2.
We define(66)Vt=V1t+V2t+V3t.
Then,(67)dVtdt≤−σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2−33σ1τs16α1ε3σ2+ac+akMIm+MIp¯2t−di−mkaMG2m+mI21ε1−33σ116α11ε2−33σ1τs16α1σ2+ac+akMIm+MIε3−33σ1τs8α1kmaMGm+mI2q¯2t.
If we chooseε1>0, ε2>0, and ε3>0 such that(68)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2−33σ1τs16α1ε3σ2+ac+akMIm+MI≥0,di−mkaMG2m+mI21ε1−33σ116α11ε2−33σ1τs16α1σ2+ac+akMIm+MIε3−33σ1τs8α1kmaMGm+mI2≥0.
Thus,dVt/dt≤0, and p,qdvt/dt=0 contains no other trajectories other than p,q=0,0; that is, the positive periodic solution G∗t,I∗t of system (8) is globally asymptotically stable. This completes the proof.
From (58), we know(69)τs≤163α19σ1m+MIm+mIσ2+ac+m+MIakmIm+MIm+mIσ2+ac+m+mIakMIε3−83α19σ1mkaMGm+MIm+mI2m+MIσ2+ac+akMIε1ε3−m+MIm+MIσ2+ac+akMIε2ε3≜a˜.
Lets˜=σ2+ac+akMI/m+MIε3+2kmaMG/m+mI2, and from (59), we get(70)τs≤163α1di9σ1s˜−83α1mkaMG9σ1m+mI2s˜1ε1−1s˜1ε2≜b˜.Remark 3.
From Theorem 5, we know that the globally asymptotic stability of the periodic solution of system (8) is related to the time delay. And the periodic solution is globally asymptotically stable if the insulin secretion delay τs<mina˜,b˜.


## 5. Numerical Analysis

In this section, the theoretical results of Section3 and Section 4 will be verified by numerical simulations, and the effects of glucose intake period and glucose intake amount on the glucose-insulin system are discussed based on the following assumptions:(1)
By applying model (5) and model (8), once the range of glucose level is set, we can make appropriate dietary recommendations. In our simulations, we assume that the normal fasting glucose concentration is set at 70−110mg/dl and the postprandial glucose concentration is no more than 140mg/dl.(2)
We assume that the volume of the plasma compartment in humans is 10 liter. The units forXt and Gin are usually in mg/dl and mg, respectively. So, if we ingest Ymg glucose every 180 minutes, then the concentration dX=Y/10×10mg/dl.The parameter valuesσ2, α1, di, a, c, k, m, and b are referred from the references [18, 26]. The initial values are set at G0=100mg/dl, I0=70μU/ml, and X0=150mg/dl. The parameters μ and ρ represent the amount of glucose consumption and glucose conversion after the intake of glucose, respectively. Because of the individual differences, here we only consider the case of μ=0.06 and ρ=0.035. The parameters are as shown in Table 1.Table 1
Model parameter values.
ParametersValuesUnitsσ25×10−6min−1a2×10−5mg−1c40mg/mink900mg/minm80mgb100mg/minI070μU/mlσ190mU/minα1125mgdi0.08min−1μ0.06min−1ρ0.035min−1G0100mg/dlX0150mg/dlThe exogenous glucose concentrationXt after meals is first simulated numerically. Let Gin=20000mg, τ=180min, namely, glucose intake for 20000mg every three hours. Through the numerical simulation, it is found that Xt oscillates with the periodic τ and is globally asymptotically stable (refer to Figure 1).Figure 1
Change of glucose concentrationX (t) after glucose intake.Next, the existence and globally asymptotic stability of the positive periodic solutions of system (5) and system (8) are verified (see Figures 2–5). Let Gin=20000mg, τ=180min, and τs=15min, and other parameters for details are as shown in Table 1. According to the simulations, the dynamic relationship between glucose and insulin of system (5) and system (8) can be seen; that is, when the glucose concentration increases, the secretion of insulin will be stimulated, and when the insulin concentration increases, the glucose concentration will be reduced, which is consistent with the regulation system of glucose and insulin in the human body.Figure 2
Phase diagram of system (5).Figure 3
Phase diagram of system (8).Figure 4
Changes of glucose and insulin levels of system (5).Figure 5
Changes of glucose and insulin levels of system (8).In addition, substituting the above parameter values into the conditions of Theorem2, obviously, the conditions are satisfied. For Theorem 3, if we select ε1=0.2 and ε2=0.5, the calculation is shown as follows:(71)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2≈0.0029>0,di−mkaMG2m+mI21ε1−3σ116α11ε2≈0.0365>0.The conditions of Theorem3 are also satisfied. In order to verify Theorem 5, we choose ε1=0.2, ε2=0.5, and ε3=1 and substitute the parameter values into the conditions of Theorem 5, and we get(72)σ2+ac+akmIm+mI−mkaMG2m+mI2ε1−33σ116α1ε2−33σ1τs16α1ε3σ2+ac+akMIm+MI≈0.0025>0,di−mkaMG2m+mI21ε1−33σ116α11ε2−33σ1τs16α1σ2+ac+akMIm+MIε3−33σ1τs8α1kmaMGm+mI2≈0.0342>0.The conditions of the theorem are met. Therefore, based above discussion, the theoretical results of both system (5) and (8) are verified.Furthermore, the range of glucose intake amount to maintain normal glucose levels at different intake periods (refer to Figures6–8) is discussed. The glucose intake periods are set at 120, 180, and 240 min, respectively. It is clear that the glucose concentration is successfully controlled under the threshold value 140 mg/dl and above the fasting glucose concentration range, which avoid hyperglycemia and hypoglycemia.Figure 6
Range of glucose intake amount for maintaining normal glucose level whenτ=180min: (a) range of glucose intake amount of system (5) and (b) range of glucose intake amount of system (8).
(a)(b)Figure 7
Range of glucose intake amount for maintaining normal glucose level whenτ=120min: (a) range of glucose intake amount of system (5) and (b) range of glucose intake amount of system (8).
(a)(b)Figure 8
Range of glucose intake amount for maintaining normal glucose level whenτ=240min: (a) range of glucose intake amount of system (5) and (b) range of glucose intake amount of system (8).
(a)(b)We assume that the time quantum of glucose intake is from 8 : 00 to 20 : 00. By calculating, whenτ=120min, one can ingest seven times and the maximum total glucose intake is 7×17600=123200 mg. If τ=180 min, one can ingest five times and the maximum total glucose intake is 5×19500=98000 mg. If τ=240 min, one can ingest 4 times and the maximum total glucose intake is 4×20200=80800 mg. We notice that the smaller the intake period, the greater the amount of glucose available. Therefore, if one needs a sufficient amount of glucose, he had better eat multiple times.Lastly, we consider the effects of different insulin secretion delays on the glucose-insulin system (see Figure9). The delay of insulin secretion is set at 5, 10, and 15 minutes, respectively. The blue solid lines in the figures represent the solution curves of system (5), in which the delay of insulin secretion is set at τ=0 min. Letting Gin=15000 mg, τ=180 min, other parameter values are shown in Table 1. Figure 9(a) shows that insulin secretion delay has little influence on the glucose level. Figure 9(b) shows that there is a certain time lag from the rise of glucose level to the secretion of insulin.Figure 9
Effects of insulin secretion delay on system (5): (a) effects of insulin secretion delay on glucose level and (b) effects of insulin secretion delay on insulin level.
(a)(b)


## 6. Discussion

In this paper, two glucose-insulin models with impulsive intake of glucose are proposed and studied. The impulsive models that we presented can intuitively show the intake amount and intake period of exogenous glucose, which is convenient to implement numerical simulation and discuss. Different from the study in the literature [20] on pulse glucose injection in the course of diabetes physiotherapy in patients with type 1 and type 2 diabetes, we model daily dietary glucose intake on periodic pulses for general population, or those with diabetes mellitus or borderline diabetes mellitus. When the existence and stability of periodic solutions are analyzed theoretically, we do not use the usual theory of impulsive differential system but transform the impulsive systems into nonautonomous limit systems with periodic coefficients. Thus, we obtain the theoretical results which are not easy to obtain directly with impulse differential system theory.Firstly, qualitative analysis of the first impulsive differential equation of model (5) is given. It is proved that system (5) is persistence and has globally asymptotically stable positive periodic solution, which indicates that the glucose-insulin system (5) in the human body reaches a certain stable state after meals. And then, based on model (1), the second model with insulin secretion delay is analyzed. The delay system is more consistent with the biological background. Studies have shown that the system with delay has globally asymptotically stable positive periodic solution. From the conditions of Theorem 5, we find that the stability of periodic solution of delayed glucose-insulin system (7) is affected not only by the amount and period of glucose intake, but also by the time delay.Our numerical simulations verify these theorems of system (5) and (8) and discuss the effect of glucose intake amount and intake period on system (5) and system (8) when μ=0.06 and ρ=0.035. The recommended ranges of glucose intake when τ=120,180,240 min are given, respectively, which keep the glucose and insulin values of system (5) and system (8) at normal ranges. And we recommend that in order to avoid hypoglycemia, adequate glucose intake can be divided into several times.Daily diet is essential for everyone, so it is of great practical significance to use the impulsive differential system to simulate glucose intake through diet. The results obtained from the models can provide people some dietary advice how to prevent diabetes. In addition, it can also control the amount and frequency of diet of diabetic patients to assist in the treatment of diabetes.



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