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1 SIAM J. APPL. MATH. Vol. 72, No. 5, pp c 2012 Society for Industrial and Applied Mathematics MODELING IMPULSIVE INJECTIONS OF INSULIN: TOWARDS ARTIFICIAL PANCREAS MINGZHAN HUANG, JIAXU LI, XINYU SONG, AND HONGJIAN GUO Abstract. We propose two novel mathematical models with impulsive injection of insulin or its analogues for type 1 and type 2 diabetes mellitus. One model incorporates the periodic impulsive injection of insulin. We analytically show the existence and uniqueness of a positive globally asymptotically stable periodic solution for type 1 diabetes, which implies that the perturbation caused by insulin injection will not disturb homeostasis but amend the glucose from a pathological level toward a healthier level. We also show that the system is uniformly permanent for type 2 diabetes, that is, the glucose concentration level is uniformly bounded above and below. The other model determines the insulin injection by closely monitoring the glucose level when it reaches or passes a predefined but adjustable threshold value. We analytically prove the existence and stability of the order one periodic solution, which ensures that the perturbation by the injection in such an automated way can keep the blood glucose concentration under control. Our numerical analysis confirms and further enhances the usefulness and robustness of our models. The first model has clinical implications in that the glucose level of a diabetic can be controlled within a desired level by adjusting the values of two model parameters, injection period annjection dose. A method for choosing these two parameters is described. The second model is, to the best of our knowledge, the first attempt to conquer some critical issues in the design of an artificial pancreas with closed-loop approach. For the former case, our numerical analysis reveals that a regime with a smaller insulin dose and more frequent deliveries is more efficient and effective, but surprisingly, for the latter case, a larger dose is more effective. This can be significant in the design and development of the insulin pump and artificial pancreas. Key words. glucose level control, insulin impulsive delivery, insulin pump, artificial pancreas, semicontinuous dynamical system, orbital stability AMS subject classifications. 92C50, 34C60, 92D25 DOI / Introduction. Diabetes mellitus is a disease in which the plasma glucose concentration level mostly remains above normal range. Diabetes mellitus is typically classified as type 1 diabetes, type 2 diabetes, and gestational diabetes. Type 1 diabetes is mainly due to the fact that almost all β-cells in the pancreas are lost or dysfunctional, and thus no insulin can be synthesized and secreted from the pancreas. Type 2 diabetes is thought to be due to the dysfunction of the glucose-insulin regulatory system, for example, insulin resistance, so that insulin cannot be utilized sufficiently by cells to uptake glucose. To compensate for the insulin resistance, β-cells need to synthesize and secrete more insulin. Typical diagnostics in type 2 diabetes are Received by the editors December 27, 2011; accepted for publication (in revised form) July 13, 2012; published electronically October 4, College of Mathematics and Information Science, Xinyang Normal University, Xinyang , Henan, People s Republic of China (huangmingzhan@163.com, xbghj@163.com). Department of Mathematics, University of Louisville, Louisville, KY (jiaxu.li@louisville. edu). The work of this author was supporten part by NIH grants R01-DE and P30ES01443, and DOE grant DE-EM College of Mathematics and Information Science, Xinyang Normal University, Xinyang , Henan, People s Republic of China (xysong88@163.com). The work of this author was supporten part by the National Natural Science Foundation of China ( ), the Innovation Scientists and Technicians Troop Construction Projects of Henan Province of China ( ), and Program for Innovative Research Team in Science and Technology of Universities of Henan Province of China (2010IRTSTHN006). 1524

2 MODELING INSULIN IMPULSIVE INJECTION 1525 hyperglycemia and hyperinsulinemia. Researchers and medical doctors have worked for a long time to discover how the system becomes dysfunctional and how to provide effective and efficient therapies for diabetic patients. The most common regimens are subcutaneous injection of insulin analogues either daily or continuously. The continuous subcutaneous insulin infusion (CSII) therapy is achieved by using an insulin pump, which is a medical device for administration of insulin or its analogues. The use of an insulin pump not only has greatly increased for type 1 diabetes (see [21]), but also provides a feasible alternative for type 2 diabetes (see [7], [27], [29], [31]) to exogenous injection of insulin or its analogues. All insulin pumps used by diabetic treatment nowadays follow the so-called open-loop approach; that is, insulin is injected without knowledge of plasma glucose level. While these therapies provide important anmproved treatments for diabetic patients, such regimes change the life styles of patients; for example, a patient has to inject insulin manually before or after meal ingestion to avoid hyperglycemia, and the dose has to be carefully computed based on the carbohydrate to be ingested. A risk in the open-loop control is hypoglycemia. In recent years, to improve the life styles of patients or to return their life styles to normal as much as possible, researchers have been making great efforts in developing technology that will close the loop, that is, to develop an artificial pancreas (see [32], [33]). The artificial pancreas, which is still in development, is a controller that would replace the endocrine functionality of a real and healthy pancreas for diabetic patients and automatically keep their plasma glucose level under control (see [3]). Equipped with the endocrine functionality of a healthy pancreas, the patient would be relieved from the inconvenience in measuring foontake and manually injecting insulin. The major impediments in the development of the artificial pancreas include (a) the need for reliable predictive models, (b) lack of effective and efficient control algorithms, and (c) unreliable real time glucose monitoring system (see [33]). However, issues (a) and (b) have not yet been fully resolven open-loop control devices. In this paper we propose two models that simulate impulsive injection of insulin in open-loop control in the fashion of periodic impulses, ann closed-loop control in view of the feedback from glucose monitoring systems. To the best of our knowledge, this paper is the first attempt to conquer issues (a) and (b) in a mathematical model with impulsive administration of insulin for the development of an artificial pancreas. We shall show the existence of periodic solutions and the global stability, or permanence, of the system, which ensure the possibility and feasibility of such insulin administrations. The analytical results and numerical observations have great implications in the development of the artificial pancreas. The paper is organized as follows. In section 2, we formulate two impulsive differential equation models to simulate the impulsive insulin injection for diabetic patients. In section 3, the qualitative analysis of the model with periodic impulsive injection of insulin is given, and the theoretical and practical regime in controlling glucose levels within an ideal range is also discussed. In section 4, we mainly discuss the existence and stability of the order one periodic solution of the models with state-dependent impulsive injection of insulin for type 1 and type 2 diabetes mellitus by differential equation geometry theory and the method of successor functions. Then the numerical simulations are carried out in section 5, which not only confirm the theoretical results, but also are complementary to those theoretical results with specific features. We finish this paper by discussing the implications of these models in the development of an artificial pancreas and point out the future work that is needed to improve these models.

3 1526 M. HUANG, J. LI, X. SONG, AND H. GUO 2. Model formulation. The terminology artificial pancreas can be traced back to 1974 (see [2]). Although the PID (proportional-integrative-derivative) controller is considered to be the best controller when the underlying mechanisms are not completely known in applications [6], model-based control is preferred due to limitations existing in PID controllers [8], and optional control of the system or its stability is not guaranteed by PID algorithms [26]. Model-based algorithms require reliable models that determine the time and the dose for insulin injections. Several such models have been proposed (see, e.g., [13], [36], [37]). Such models reflect the physiology of insulin secretion stimulated by glucose, and of glucose metabolism assisted by insulin or exogenously deliverensulin or its analogues. The metabolic model proposen [13] used data from conducted closed-loop insulin delivery trials to describe intraday variation of model parameters and concluded that the model systems do not have to be composed of a large number of compartments and variables; that is, the differential equation system does not need to be high dimensional. Another related closely relevant type of models computes glucose fluxes after meal ingestion (see, for example, Hovorka et al. [12]), which creates a computational framework by combining maximum likelihood theory with the ordinary differential equation system. Two critical, and harmful, episodes in therapies of insulin administration are hypoglycemia, caused by overdosing, and hyperglycemia, caused by underdosing. The model-based algorithms used by insulin pumps and the future artificial pancreas should be designed to avoid such episodes. To carefully determine the correct dose of insulin and the right timing of injection, it is necessary to understand the dynamics of the regulations between glucose annsulin in physiology. Therefore, a model based on physiology is much needed, ant is extremely important to study the model analytically so that the episodes of hyperglycemia and hypoglycemia can be avoided. To this end, we extend the model proposed by Li, Kuang, and Mason [19] and Li and Kuang [17], which models the physiological oscillatory insulin secretion stimulated by elevated glucose, and we form a new model, taking into account impulsive insulin injection either periodically or by monitoring the plasma glucose concentration level. A similar effort has been attempted by Wang, Li, and Kuang [36], [37], in which periodic insulin administration was employed to mimic impulsive injection for the regime of type 1 or type 2 diabetes mellitus. The model in [19] or [17] is given by dg = G in f 2 (G(t)) f 3 (G(t))f 4 (I(t)) + f 5 (I(t τ 2 )), (2.1) di = f 1(G(t τ 1 )) I, where G in is the estimated average constant rate of glucose input, f 1 (G(t τ 1 )) is insulin secretion stimulated by elevated glucose concentration with a time delay of τ 1 caused by complex pathways including chemical-electrical processes, f 2 (G) is the insulin-independent glucose uptake, f 3 (G)f 4 (I) stands for the insulin-dependent glucose utilization, f 5 (I(t τ 2 )) is the hepatic glucose production (HGP) with time delay τ 2,and I indicates the insulin degradation with > 0 as the constant degradation rate. Refer to Figure 3 in [19] for the shapes and forms of the response functions f 1,f 2,f 3,f 4,andf 5, which were originally determined by Sturis et al. in [34]. We modify the model diagram given in Figure 2 in [19] and obtain the following model diagram in Figure 2.1 including exogenous insulin injection and the feedback of monitored glucose concentration level.

4 MODELING INSULIN IMPULSIVE INJECTION 1527 Exogenous intake Glucose monitor Glucose Insulin helps cells to consume glucose Glucose u liza ons Liver Glucagon α β Pancreas Exogenous injec on Insulin Degrada on Fig Model diagram. For type 1 and type 2 diabetes, β-cells either secrete no insulin or secrete insufficient insulin. Insulin is injected exogenously. Therefore the repression of insulin on hepatic glucose production and α-cells secreting glucagon is insignificant. Insulin injection can be adjusted based on the input from the glucose monitoring device. In normal subjects, the liver is the first organ to which the newly secretensulin arrives, and the HGP is controlled by insulin. For type 1 and type 2 diabetics using an artificial pancreas, little or no insulin is secreted from β-cells and the exogenous insulin is injected subcutaneously. Therefore the control power of HGP by insulin level is not significant (refer to Figure 2.1). Thus, we assume that the HGP f 5 (I(t τ 2 )) is at a constant rate b>0. Furthermore, according to [14], the shapes of the response functions, instead of the forms, are important. So for simplicity, we assume that f 1 (x) = σ 1x 2 α x2, f 2(x) =σ 2 x, f 3 (x) =ax, f 4 (x) =c + mx n + x, where σ 1,σ 2,α 1,a,c,m,andn are positive constant parameters that are chosen and adjusted from [10], [17], [19], [20], [23], [24], [34], and [35] (refer to Table 5.1). Thus we first formulate the following two-compartment model with periodic impulsive injection of exogenous insulin: (2.2) dg(t) di(t) ( = G in σ 2 G a c + mi ) G + b n + I = σ 1G 2 α G2 I(t) G(t + )=G(t) I(t + )=I(t)+σ, t = kτ,, t kτ, with initial condition G(0) = G 0 > 0,I(0) = I 0 > 0, where σ (mu) > 0 is the dose in each injection and τ (min) > 0 is the period of the impulsive injection. That is, σ is insulin injected as an impulse at discrete times t = kτ, k Z + = {1, 2, 3,...}. The moment immediately after the kth injection is denoted here as t = kτ +.

5 1528 M. HUANG, J. LI, X. SONG, AND H. GUO The main feature of model (2.2) is that insulin is injected subcutaneously periodically, which is in agreement with how the insulin pump works in an open-loop fashion. We shall perform analytical analysis in section 3 for the existence, uniqueness, and stability of the periodic solution with two adjustable parameters σ and τ in a reasonable region. The periodic solution reflects the dynamics of the glucose annsulin for the patients, and doctors can adjust these two parameters to determine the dose and periodic timing of injection so that the range of plasma glucose can be controlled within a desirable range. Although insulin pumps with open-loop technique have been available for clinical use, the ideal treatment is that in which the insulin administration can be automatically determined by the so-called closed-loop technique integrated with the glucose monitoring system. This would build an artificial pancreas. In the design of an artificial pancreas, it is critical to inject insulin or its analogues according to the observed glucose level from a monitoring system, and that the injections are prompt. Based on the model (2.1), we propose a novel model simulating the injection of insulin as impulse in observing the glucose level, and we investigate its dynamical behavior. The model is given by (2.3) dg(t) di(t) ( = G in σ 2 G a c + mi ) G + b n + I = σ 1G 2 α G2 I(t) G(t + )=G(t) I(t + )=I(t)+σ, G = L G and I I C,, G < L G or I>I C, with initial condition G(0) = G 0 L G,I(0) = I 0, where the constant I C = nk 0 /(m k 0 ),k 0 = a 1 L 1 G (G in + b σ 2 L G ) c, which is determined by the intersection of the null-cline G in σ 2 G a(c + mi n+i )G + b = 0 and the horizontal line G = L G in the (I,G)-plane. L G is an adjustable constant threshold value for glucose level when the glucose level reaches the threshold value, the impulsive injection of insulin with dose σ (mu) is performed. It is easy to see that the glucose level must decrease when the insulin level surpasses the point I C. In section 4 we show that a periodic solution exists with orbital stability. Remark. For type 1 diabetes, all or most β-cells are dysfunctional and thus secrete no insulin. Such a case corresponds to σ 1 = 0 in models (2.2) and (2.3). 3. Analysis of model (2.2) for open-loop control. In this section, we consider system (2.2) in two cases. We first consider the case σ 1 =0,whichmeans that the pancreas does not release insulin and the patient can rely only on exogenous insulin through subcutaneous injection. We show that the system has a globally asymptotically stable positive periodic solution. Second, we consider the case σ 1 > 0, which means that the pancreas may release some insulin; we show that system (2.2) is permanent, that is, the glucose concentration level is bounded above by a constant and below by another constant Preliminaries. For the sake of convenience, the following notation and definitions are assumed throughout the paper. Let R + =[0, ), R+ 2 = {(x, y) R2 + : x 0,y 0}, Ω=intR2 +. Denote by f =(g, h) T the mapping defined by the right-hand side of system (2.2).

6 MODELING INSULIN IMPULSIVE INJECTION 1529 Let V : R + R 2 + R +.ThenV is said to belong to class V 0 if (i) V is continuous on (kτ, (k+1)τ] R 2 +, and lim (t,y) (kτ +,x) V (t, y) =V (kτ +,x) exists ans finite; (ii) V is locally Lipschitzian in x. Definition 3.1. Let V V 0 ; then for V (t, x) (kτ, (k +1)τ] R 2 +,theupper right derivative of V (t, x) with respect to the impulsive differential system (2.2) is defined as D + V (t, x) = lim sup 1 [V (t + h, x + hf(t, x)) V (t, x)]. h 0 h Definition 3.2 (see Bainov and Simeonov [4]). Let r(t) =r(t, t 0,x 0 ) be a solution of system (2.2) on [t 0,t 0 + l). r(t) is called the maximal solution of system (2.2) if for any solution x(t, t 0,x 0 ) of system (2.2) existing on [t 0,t 0 + l), we have x(t) r(t), t [t 0,t 0 + l). The minimal solution ρ(t) can be defined similarly. Definition 3.3. System (2.2) is said to be uniformly persistent if there is a q>0 (independent of the initial conditions) such that every solution (G(t),I(t)) of system (2.2) satisfies lim inf G(t) q, lim t inf I(t) q. t Definition 3.4. System (2.2) is said to be permanent if every solution (G(t),I(t)) is bounded below by a positive constant and above by another positive constant. Lemma 3.5. Let m V 0, and assume that { D + m(t) g(t, m(t)), t t k, k =1, 2,..., m(t + k ) ψ k(m(t k )), t = t k, k =1, 2,..., where g C(R + R +,R), ψ k C(R, R), andψ k (u) is nondecreasing in u for each k =1, 2,... Letr(t) be the maximal solution of the scalar impulsive differential equation (3.1) u = g(t, u), t t k, k =1, 2,..., u(t + k )=ψ k(u(t k )), t = t k,t k >t 0 0, k =1, 2,..., u(t 0 )=u 0, which exists on [t 0, ). Then, m(t + 0 ) u 0 implies that m(t) r(t) for t t 0. Similar result can be obtained when all directions of the inequalities in the lemma are reversed and ψ k (u) is nonincreasing. Remark. In Lemma 3.5, if g is smooth enough to guarantee the existence and uniqueness of the solution for the initial value problem of system (3.1), then r(t) is indeed the unique solution of (3.1). Next we show the positivity and the boundedness of the solutions of system (2.2). Let x(t) =(G(t),I(t)) T be a solution of system (2.2). Notice that it is continuous on (kτ, (k +1)τ], k Z +, and x(kτ + ) = lim t kτ + x(t) exists. Thus the global existence and uniqueness of solutions of system (2.2) is ensured by the smoothness of f =(g, h) T (see [4], [5]).

7 1530 M. HUANG, J. LI, X. SONG, AND H. GUO Clearly, dg(t)/ > 0whenG(t) = 0, and di(t)/ 0whenI(t) = 0. Therefore we have the following. Proposition 3.6 (positivity). Suppose that x(t) is a solution of system (2.2) with x(0 + ) 0, and then x(t) 0 for all t 0. Further,ifx(0 + ) > 0, thenx(t) > 0 for all t>0. The following lemma summarizes some basic properties of linear system (3.2). We state it without proof. Interested readers can refer to [30]. Lemma 3.7. The linear system u(t) =a 1 b 1 u(t), t kτ, (3.2) u(t + )=u(t)+p, t = kτ, u(0 + )=u 0 0 has a unique positive periodic solution ũ(t) with period τ and for every solution u(t) of (3.2) such that u(t) ũ(t) 0 as t,where ũ(t) = a 1 + p exp( b 1(t kτ)), t (kτ, (k +1)τ], k Z +, b 1 1 exp( b 1 τ) ũ(0 + )= a 1 p + b 1 1 exp( b 1 τ), and ũ(t) is globally asymptotically stable. In addition, we have u(t) = (u(0 + ) ũ(0 + )) exp( b 1 t)+ũ(t), andlim t u(t) =ũ(t). Especially, if a 1 =0,system(3.2) has a unique positive periodic solution ũ(t) = p exp( b 1 (t kτ))/(1 exp( b 1 τ)) with initial value ũ(0 + )=p/(1 exp( b 1 τ)), andũ(t) is globally asymptotically stable. Now we can show the boundedness of the solutions of system (2.2). Proposition 3.8 (boundedness). For a solution (G(t),I(t)) of system (2.2) with positive initial values, there exists a positive constant M such that G(t) M and I(t) M for all t 0. Proof. From the first equation of system (2.2) we have dg(t) (G in + b) (σ 2 + ac)g, and then there exists a positive number M 1 > 0 such that G(t) M 1,t 0. From the second and fourth equations of system (2.2), we have di(t) σ 1 I(t), I(t + )=I(t)+σ, I(0) = I 0 > 0. Now we consider the impulsive differential equation (3.3) di 2 (t) t kτ, t = kτ, = σ 1 I 2 (t), t kτ, I 2 (t + )=I 2 (t)+σ, I 2 (0 + )=I 0 > 0. t = kτ,

8 MODELING INSULIN IMPULSIVE INJECTION 1531 By Lemma 3.7, we know that system (3.3) has a globally asymptotically stable positive periodic solution Ĩ 2 (t) = σ 1 + σ exp( (t kτ)), t (kτ, (k +1)τ], k Z +, 1 exp( τ) Ĩ 2 (0 + )= σ 1 σ + 1 exp( τ), and the solution of system (3.3) has the form which satisfies lim t I 2 (t) =Ĩ2(t). By Lemma 3.5, we have I 2 (t) =(I 2 (0 + ) Ĩ2(0 + )) exp( t)+ĩ2(t), I(t) I 2 (t) σ 1 σ + 1 exp( τ) + I 2(0 + ) Ĩ2(0 + ) for t 0, and then there exists a positive number M M 1 such that I(t) M, t Existence and stability of the periodic solution for type 1 diabetes: The case σ 1 =0. Little or no insulin is released from the pancreas in type 1 diabetes. This can be modelen (2.2) by assuming that the maximum insulin secretion rate σ 1 = 0. In this subsection we prove that a globally asymptotically stable periodic solution exists if exogenous insulin is injecten the fashion of periodic impulse. Theorem 3.9. If σ 1 =0,thensystem(2.2) has a unique positive periodic solution ( G(t), Ĩ(t)) with period τ. Proof. Note that the variable G does not appear in the second equation of system (2.2), and hence for the dynamics of insulin I(t) we need only consider the subsystem (3.4) di(t) = I(t), t kτ, I(t + )=I(t)+σ, t = kτ, I(0) = I 0 > 0. According to Lemma 3.7, system (3.4) has a unique periodic solution Ĩ(t) = σ exp( (t kτ)), t (kτ, (k +1)τ], k Z +, 1 exp( τ) with period τ, andĩ(t) is globally asymptotically stable. Substituting Ĩ(t) into the first equation of (2.2) for I(t), we have (3.5) dg(t) =(G in + b) (σ 2 + ac)g amgĩ(t) n + Ĩ(t), G(t + )=G(t), t kτ, t = kτ.

9 1532 M. HUANG, J. LI, X. SONG, AND H. GUO Integrating and solving (3.5) between pulses, we can get, for kτ < t (k +1)τ, (3.6) G(t) =G(kτ)exp +(G in + b) [ t kτ + t kτ + ( σ 2 + ac + amĩ(s) ) n + Ĩ(s) [ ( t exp = G(kτ)exp[ (σ 2 + ac)(t kτ)] exp u ] ds ( σ 2 + ac + amĩ(s) ) n + Ĩ(s) [ am t kτ + t { +(G in + b) exp[ (σ 2 + ac)(t u)] exp kτ + From system (3.4) we get, for kτ + b 1 b 2 (k +1)τ, (3.7) [ b2 exp am b 1 Ĩ(t) ] [ am b2 n + Ĩ(t) =exp b 1 [ am b2 =exp b 1 )] ds du Ĩ(s) ] n + Ĩ(s)ds [ t am u Ĩ(t) ] n + Ĩ(t) ( d ln(n + Ĩ(t)) am ( n + Ĩ(b 2 ) )] =exp[ ln n + Ĩ(b 1) ( n + Ĩ(b 2 ) ) am =. n + Ĩ(b 1) By (3.6) and (3.7), for kτ < t (k +1)τ, weget ( n + Ĩ(t) ) am G(t) =G(kτ)exp[ (σ 2 + ac)(t kτ)] n + Ĩ(0+ ) t { ( n + Ĩ(t) ) am +(G in + b) exp[ (σ 2 + ac)(t u)] kτ + n + Ĩ(u) ( n + Ĩ(t) ) am = G(kτ)exp[ (σ 2 + ac)(t kτ)] n + Ĩ(0+ ) am t exp[ (σ d 2 + ac)(t u)] +(G in + b)(n + Ĩ(t)) i am du, kτ + d (n + Ĩ(u)) i and (3.8) ( n + Ĩ(τ) G((k +1)τ) =G(kτ)exp[ (σ 2 + ac)τ] n + Ĩ(0+ ) am d +(G in + b)(n + Ĩ(τ)) i f ( G(kτ)), where f(g) =A G + B and τ ) am Ĩ(s) ]} n + Ĩ(s)ds du. ) ] } du exp[ (σ 2 + ac)(τ u)] am du 0 + d (n + Ĩ(u)) i ( n + Ĩ(τ) ) am d 0 <A=exp[ (σ 2 + ac)τ] i < 1, n + Ĩ(0+ )

10 MODELING INSULIN IMPULSIVE INJECTION 1533 and am τ exp[ (σ d 2 + ac)(τ u)] B =(G in + b)(n + Ĩ(τ)) i am du > d (n + Ĩ(u)) i B 1 A Equation (3.8) has a unique fixed point Ḡ = > 0. If 0 <G<Ḡ, then G<f(G) < Ḡ, andifg>ḡ, then Ḡ<f(G) <G,so we know that Ḡ is globally asymptotically stable, and the corresponding period solution G(t) of system (3.5) is also globally asymptotically stable, where G(t) = B ( n + 1 A exp[ (σ Ĩ(t) 2 + ac)(t kτ)] n + Ĩ(0+ ) am d +(G in + b)(n + Ĩ(t)) i t kτ + ) am exp[ (σ 2 + ac)(t u)] am du d (n + Ĩ(u)) i B 1 A. for kτ < t (k +1)τ and k Z +, with initial value G(0 + )= According to the above discussion, we get that system (2.2) has a unique positive periodic solution ( G(t), Ĩ(t)). This completes the proof. Theorem If σ 1 =0, then the positive periodic solution ( G(t), Ĩ(t)) of system (2.2) is globally asymptotically stable. Proof. We first prove the local stability of the τ-period solution and then prove that the stability is a global behavior. The local stability of the τ-period solution ( G(t), Ĩ(t)) may be determined by considering the behavior of small-amplitude perturbations (v 1 (t),v 2 (t)) T of the solution. Define G(t) = G(t)+v 1 (t), I(t) =Ĩ(t)+v 2(t), where v 1 (t), v 2 (t) are small perturbations, which can be written as ( ) ( ) v1 (t) v1 (0) =Φ(t), v 2 (t) v 2 (0) where Φ(t) satisfies dφ(t) ( (σ 2 + ac) amĩ(t) n+ĩ(t) am G(t) n+ĩ(t) = Φ(t), 0 with Φ(0) = I, the identity matrix. The resetting conditions of system (2.2) become ( ) ( )( ) v1 (kτ + ) 1 0 v1 (kτ) v 2 (kτ + =. ) 0 1 v 2 (kτ) Hence, according to the Floquet theory (see [4]), if the absolute values of both eigenvalues of ( ) 1 0 M = Φ(τ) =Φ(τ) 0 1 are less than one, then the τ-period solution is locally stable. Clearly, ( exp( ) τ Φ(τ) = 0 [ (σ 2 + ac) amĩ(t) ]) n+ĩ(t). 0 e diτ )

11 1534 M. HUANG, J. LI, X. SONG, AND H. GUO There is no need to calculate the exact form of ( ) as it is not requiren the following analysis. Then the eigenvalues of M denoted by λ 1,λ 2 are { τ [ λ 1 =exp (σ 2 + ac) amĩ(t) ] } 0 n + Ĩ(t) < 1 and λ 2 =exp( τ) < 1. So the periodic solution ( G(t), Ĩ(t)) is locally asymptotically stable. In the following, we show that the periodic solution is a global attractor. According to Proposition 3.8 we have lim t I(t) =Ĩ(t). If ε 1 and ε 2 > 0 are small enough, then there exists a t 1 > 0 such that (1 ε 1 )Ĩ(t) <I(t) < (1 + ε2)ĩ(t) for all t>t 1. From the first equation of system (2.2), we have dg (G am(1 + ε2)gĩ(t) in + b) (σ 2 + ac)g n +(1+ε 2 )Ĩ(t) and dg (G am(1 ε1)gĩ(t) in + b) (σ 2 + ac)g n +(1 ε 1 )Ĩ(t). Consider the impulsive equations dg 1 =(G (3.9) in + b) (σ 2 + ac)g 1 am(1 + ε 2)G 1 Ĩ(t) n +(1+ε 2 )Ĩ(t), t kτ, G 1 (t + )=G 1 (t), t = kτ, and (3.10) dg 2 =(G in + b) (σ 2 + ac)g 2 am(1 ε 1)G 2 Ĩ(t) n +(1 ε 1 )Ĩ(t), G 2 (t + )=G 2 (t), t kτ, t = kτ. According to the proof of Theorem 3.9, both systems (3.9) and (3.10) have a unique globally asymptotically stable positive periodic solution G 1 (t) = B ( 1 n +(1+ε2 exp[ (σ 2 + ac)(t kτ)] )Ĩ(t) 1 A 1 n +(1+ε 2 )Ĩ(0+ ) am d +(G in + b)(n +(1+ε 2 )Ĩ(t)) i exp[ (σ 2 + ac)(t u)] am du kτ + d (n +(1+ε 2 )Ĩ(u)) i for kτ < t (k +1)τ, where ( n +(1+ε2 0 <A 1 =exp[ (σ 2 + ac)τ] )Ĩ(τ) ) am < 1, n +(1+ε 2 )Ĩ(0+ ) and t ) am am τ d B 1 =(G in + b)(n +(1+ε 2 )Ĩ(τ)) i exp[ (σ 2 + ac)(τ u)] am du > 0, 0 + d (n +(1+ε 2 )Ĩ(u)) i G 2 (t) = B ( 2 n +(1 ε1 exp[ (σ 2 + ac)(t kτ)] )Ĩ(t) 1 A 2 n +(1 ε 1 )Ĩ(0+ ) am d +(G in + b)(n +(1 ε 1 )Ĩ(t)) i t kτ + ) am exp[ (σ 2 + ac)(t u)] am du d (n +(1 ε 1 )Ĩ(u)) i

12 for kτ < t (k +1)τ, where MODELING INSULIN IMPULSIVE INJECTION 1535 and ( n +(1 ε1 0 <A 2 =exp[ (σ 2 + ac)τ] )Ĩ(τ) ) am < 1 n +(1 ε 1 )Ĩ(0+ ) am τ d B 2 =(G in + b)(n +(1 ε 1 )Ĩ(τ)) i exp[ (σ 2 + ac)(τ u)] am du > d (n +(1 ε 1 )Ĩ(u)) i By Lemma 3.5, we get, for ε>0 small enough, that there exists a t 2 >t 1 such that G 1 (t) ε<g 1 (t) G(t) G 2 (t) < G 2 (t)+ε, t > t 2. If ε, ε 1,ε 2 0, we get G 1 (t) G(t) and G 2 (t) G(t), and then G(t) G(t) as t. That completes the proof Permanence for type 2 diabetes: The case σ 1 > 0. Diagnostics of type 2 diabetes and prediabetes are hyperglycemia and hyperinsulinemia, which are most likely caused by insulin resistance. In this case, pancreatic β-cells still secrete insulin, and might possibly secrete extra insulin to compensate for the insulin resistance (see [1], [28]), although the compensation is not sufficient for glucose uptake. Therefore we have the maximum insulin secretion rate σ 1 > 0 of model (2.2). We study the range of variation for plasma glucose concentration G(t) annsulin concentration I(t) under impulsive injection of insulin for sufficiently large t>0. Such a qualitative result has implications in design regimes of exogenous insulin injection, in which both hyperglycemia and hypoglycemia can be avoided. We show the following. Theorem If σ 1 > 0, system(2.2) is permanent; that is, the solutions are bounded below and above by some constants. Proof. From the second and fourth equations of system (2.2) we have d i I(t) di(t) σ 1 I(t), t kτ, I(t + )=I(t)+σ, t = kτ, I(0) = I 0 > 0. Now we consider the impulsive differential equation di 1 (t) = I 1 (t), t kτ, (3.11) I 1 (t + )=I 1 (t)+σ, t = kτ. By Lemma 3.7 and Proposition 3.8, both systems (3.3) and (3.11) have unique globally asymptotically stable positive periodic solutions Ĩ 1 (t) = σ exp( (t kτ)), t (kτ, (k +1)τ], k Z +, 1 exp( τ) Ĩ 2 (t) = σ 1 + σ exp( (t kτ)), t (kτ, (k +1)τ], k Z +. 1 exp( τ)

13 1536 M. HUANG, J. LI, X. SONG, AND H. GUO According to Lemmas 3.5 and 3.7, for sufficiently small ε>0, there exists a t 0 > 0 such that and then we have (3.12) Ĩ 1 (τ) = Ĩ 1 (t) ε<i 1 (t) I(t) I 2 (t) < Ĩ2(t)+ε, t t 0, σ exp( τ) 1 exp( τ) = lim inf Ĩ1(t) lim inf I(t) lim sup I(t) t t t lim t sup Ĩ2(t) = σ 1 + σ 1 exp( τ) = Ĩ2(0 + ). By (3.12) and the first equation of system (2.2), we know for t large enough that (3.13) Let (3.14) (G in + b) (σ 2 + ac)g amgĩ2(0 + ) n + Ĩ2(0 + ) dg(t) G m (σ, τ) By (3.13) we get = G in + b σ 2 + ac + amĩ2(0+ ) n+ĩ2(0+ ) ( (G in + b) n + σ1 ( (σ 2 + ac) n + σ1 σ + 1 exp( τ ) (G in + b) (σ 2 + ac)g amgĩ1(τ) n + Ĩ1(τ). σ + 1 exp( τ ) G in + b G M (σ, τ) σ 2 + ac + amĩ1(τ ) n+ĩ1(τ ) ( ) (G in + b) n + σ exp( diτ ) 1 exp( τ ) = ( ) (σ 2 + ac) n + σ exp( diτ ) 1 exp( τ ) + am ) ) + am( σ1 + σ 1 exp( τ ) ( σ exp( diτ ) 1 exp( τ ) (3.15) G m (σ, τ) lim t inf G(t) lim t sup G(t) G M (σ, τ). According to (3.12) and (3.15), we know that system (2.2) is permanent. This completes the proof. Remark. According to (3.15), G m (σ, τ)andg M (σ, τ) can be considered as a lower bound and an upper bound of glucose homeostasis, respectively A method to select injection period τ and dose σ. In this subsection, we sketch a method for selecting insulin injection dose σ annjection period τ to control glucose levels within an ideal range based on (3.14) and (3.15) to avoid both hyperglycemia and hypoglycemia. ). ),

14 MODELING INSULIN IMPULSIVE INJECTION 1537 G M (mg/ml) G m (mg/ml) Injection period (min) Dose σ (mu) Injection Period τ (min) G M (τ, σ) = 190 (mg/dl) The region W G m (τ, σ) = 115 (mg/dl) for σ 1 = 0 (mu) 5 G m (τ, σ) = 115 (mg/dl) for σ 1 = 6.27 (mu) Injection Dose σ (mu) Fig Left panel: The lower bound G m and upper bound G M definen (3.14) in (σ, τ )- plane. Right panel: Region W enclosed by G m(σ, τ ) = 115 and G M (σ, τ ) = 190 using the parameters given in Table 5.1. When the parameter values of insulin dose σ annjection period τ fall in the region W, the glucose homeostasis level will be controlled within 115 and 190 (mg/dl) according to (3.15). According to (3.15), the level of glucose at homeostasis is bounded by G m (σ, τ) and G M (σ, τ). The left panel of Figure 3.1 displays the surfaces of G m (σ, τ)and G M (σ, τ), respectively. Assume that an ideal range for glucose to vary is between B l and B h,whereb h >B l > 0. It is easy to see by the implicit function theorem that G m (σ, τ) = B l and G M (σ, τ) = B h will implicitly determine two curves in the (σ, τ)-plane. These two curves will form a region, denoted by W, so that the glucose at homeostasis is bounded by B l and B h if (σ, τ) W. The right panel of Figure 3.1 shows the region W formed by G m (σ, τ) = B l = 115 (mg/dl) and G M (σ, τ)=b h = 190 (mg/dl) with the parameters given in Table 5.1. Respectively, the lower curve of G m (σ, τ)=b l = 115 is for type 2 diabetes (σ 1 =6.27 mu), while the upper curve of G m (σ, τ)=b l = 115 is for type 1 diabetes (σ 1 =0mU).For example, if σ = 800 mu = 0.8 U,τ =30min,then(σ, τ) W, ann this case G 1 = and G 2 = for type 1 diabetes (σ 1 = 0) (refer to Figure 5.1). 4. Analysis of model (2.3) for closed-loop control. In this section, we mainly discuss the existence and stability of the order one periodic solution of model (2.3) by the geometric theory of differential equations and the method of successor functions. First, we consider the qualitative characteristics of system (2.3) without impulsive effect. In such a case, system (2.3) can be written as (4.1) dg(t) di(t) =(G in + b) (σ 2 + ac)g amgi n + I = P 1(G, I), = σ 1G 2 α G2 I(t) =Q 1 (G, I). Clearly, for σ 1 = 0, system (4.1) has a unique equilibrium E 0 (G 0, 0), where G 0 = (G in + b)/(σ 2 + ac) ande 0 is a global asymptotically stable node with two separatrixes I =0andG = ki, wherek = amg 0 /(n( σ 2 ac)). W

15 1538 M. HUANG, J. LI, X. SONG, AND H. GUO For σ 1 > 0, system (4.1) has a unique equilibrium E (G,I ) with G > 0and I > 0. The linearized system of (4.1) at the equilibrium E (G,I )isgivenby dg(t) = a 11 G + a 12 I, di(t) = a 21 G + a 22 I, where a 11 = (σ 2 + ac) ami n+i, a 12 = amng (n+i ), a 2 21 = 2α2 1 σ1g, a (α 2 1 +(G ) 2 ) 2 22 =. Let =(a 11 a 22 ) 2 +4a 12 a 21. When 0, E is a stable node, and when < 0, E is a stable focus. In other words, E is locally stable. Notice that P 1 G + Q 1 I = (σ 2 + ac) ami n + I < 0. Thus there exists no closed orbit for system (4.1) according to the Bendixson theorem (see [25]), and therefore E is global asymptotically stable. The isocline dg = P 1 (G, I) = 0 has an asymptotic line G = Gin+b σ G 2+ac+am s. Thus, in system (2.3), if L G G s, the horizontal lines G = L G would not intersect with the isocline dg = P 1(G, I) = 0, and we have I C =+, which implies that the trajectory from the initial point below the line G = L G will undergo infinite impulsive effect and remain in the line G = L G. So we assume that G s <L G <G 0 for the case σ 1 =0,andG s <L G <G for the case σ 1 > 0 throughout this section. Clinically, if L G <G s, the glucose level is probably not in a hyperglycemic state; if L G is above G 0 or G, some other medical treatment is required to bring the glucose level down in practice Definitions and notation of the geometric theory of the semicontinuous dynamical systems. We first introduce some notation and definitions of the geometric theory of semicontinuous dynamical systems, which will be useful for the discussion of system (2.3). Definitions and Lemma 4.6 are from Chen [9]. Definition 4.1. Consider the state-dependent impulsive differential equations (4.2) dx = P (x, y) dy = Q(x, y) x = α(x, y) y = β(x, y), (x, y) / M{x, y}, }, (x, y) M{x, y}. We define the dynamical system consisting of the solution mapping of system (4.2) as a semicontinuous dynamical system, denoted as (Ω,f,ϕ,M). We require that the initial point P of the system (4.2) not be in the set M{x, y}, that is, P Ω=R+ 2 \ M{x, y}, and the function ϕ is a continuous mapping that satisfies ϕ(m) =N. Here ϕ is called the impulse mapping, where M{x, y} and N{x, y} represent the straight lines or curves in the plane R+, 2 M{x, y} is called the impulse set, and N{x, y} is called the phase set. Remark. For system (2.3), Ω = {(I,G): G L G and 0 I< }, M = {(I,G): G = L G and I I C },andforany(i,g) M, wehaveϕ(i,g)=(i + kσ, G), where k is an integer such that I + kσ > I C, I +(k 1)σ I C.

16 MODELING INSULIN IMPULSIVE INJECTION 1539 Definition 4.2. For the semicontinuous dynamical system defined by the statedependent impulsive differential equations (4.2), the solution mapping f(p, t) :Ω Ω consists of two parts: (i) Let π(p, t) denote the Poincaré map with the initial point P of the system dx = P (x, y), dy = Q(x, y). If f(p, t) M{x, y} =, thenf(p, t) =π(p, t). (ii) If there exists a time point T 1 such that f(p, T 1 )=H M{x, y}, ϕ(h) = ϕ(f(p, T 1 )) = P 1 N{x, y} (the point H is called the impulse point of P, and the point P 1 is called the phase point of H), and f(p 1,t) M{x, y} =, then f(p, t) =π(p, T 1 )+f(p 1,t). Remark. For (ii) in Definition 4.2, if f(p 1,t) M{x, y}, anhereexists atimepointt 2 such that f(p 1,T 2 )=H 1 M{x, y} and ϕ(h 1 )=ϕ(f(p 1,T 2 )) = P 2 N{x, y}, thenf(p, t) =π(p, T 1 )+f(p 1,t)=π(P, T 1 )+π(p 1,T 2 )+f(p 2,t). Iteratively, if f(p i 1,t) M{x, y} for i =2, 3, 4,...,k for some k>1, then f(p, t) = k π(p i 1,T i )+f(p k,t), where P 0 = P. i=1 Definition 4.3. If there exists a point P N{x, y} and a time point T 1 such that f(p, T 1 )=H M{x, y} and ϕ(h) =ϕ(f(p, T 1 )) = P N{x, y}, thenf(p, t) is called an order one periodic solution of system (4.2) whose perios T 1. Definition 4.4. Suppose Γ=f(P, t) is an order one periodic solution of system (4.2). If for any ε>0, theremustexistδ>0and t 0 0 such that for any point P 1 U(P, δ) N{x, y}, we have ρ(f(p 1,t), Γ) <εfor t>t 0, then we call the order one periodic solution Γ orbitally asymptotically stable. Remark. Orbitally asymptotic stability is different from Lyapunov asymptotic stability. An order one periodic solution Γ = f(p, t) of system (4.2) is Lyapunov asymptotically stable if for any ε>0, there must exist δ>0an 0 0 such that, for the solution g(p 1,t) starting from any point P 1 U(P, δ), g(p 1,t) f(p, t) <ε for all t>t 0. Notice that when the impulse set M and the phase set N of system (4.2) are straight lines, a coordinate system can be well definen the phase set N. LetA N be a point, and let its coordinate be a. Assume that the trajectory from the point A intersects the impulse set M at a point A M, and, after impulsive effect, the point A is mapped to the point A 1 N with the coordinate a 1. Then we define the successor point and the successor function as follows. Definition 4.5. The point A 1 is called the successor point of A, anhefunction F 1 (A) =a 1 a is called the successor function of point A. Throughout this paper, in system (2.3), we define the coordinate of any point A N as the distance between A and the G-axis, denoted by I A. Therefore we have the following. Lemma 4.6. The successor function F (A) is continuous. A direct application of Lemma 4.6 leads to the following result that plays an important role in the study of model (2.3) in the next subsection. Lemma 4.7. For systems (2.3), if there exist two points A, B N such that F (A)F (B) < 0, then there must exist a point C N between the points A and B such that F (C) =0, and thus the system must have an order one periodic solution.

17 1540 M. HUANG, J. LI, X. SONG, AND H. GUO Proof. The continuousness of the successor function by Lemma 4.6 implies the existence of a point C N between the points A and B and F (C) =0. DenotebyC the impulse point of C. SinceF (C) =0,wehaveϕ(C )=C. According to Definition 4.3, Γ = f(c, t) is an order one periodic solution Existence, uniqueness, and stability of order one periodic solution. We study model (2.3) for the case of type 1 diabetes (σ 1 =0)anhecaseoftype2 diabetes (σ 1 > 0), respectively. Theorem 4.8. For the case σ 1 =0,ifG s <L G <G 0 and I C <σ,thensystem (2.3) has a unique order one periodic solution. Proof. Suppose that the horizontal line G = L G intersects dg = 0 at point C(L G,I C ). Select the point D(L G,I D ), where I D = I C + σ. Then the trajectory of system (2.3) through point D must intersect the line G = L G again at a point Q(L G,I Q ), where 0 <I Q <I C. The point Q is mapped to the point Q (L G,I Q ) after impulsive effect, where I D >I Q = I Q + σ>i C (because I C <σ,i D = I C + σ). Again, the trajectory of system (2.3) passing through point Q must intersect the line G = L G at a point Q 1 (L G,I Q1 ), and the point Q 1 is mapped to the point Q 1 (L G,I Q ) after impulsive effect, where I 1 Q = I Q1 + σ. Since distinct trajectories 1 do not intersect, we can easily have 0 <I Q <I Q1 <I C <I Q <I Q <I D. Since the 1 point D is in the phase set, point Q is the impulse point of point D, and point Q is the successor point of point D, and thus we can get the successor function of point D is F (D) =I Q I D < 0. Besides, for the point Q in the phase set, point Q 1 is the impulse point of point Q, and point Q 1 is the successor point of point Q,soweget that the successor function of point Q is F (Q )=I Q I 1 Q > 0. By Lemmas 4.6 and 4.7, there must exist a point M between the points Q and D such that F (M) =0, and thus system (2.3) has an order one periodic solution that has the point M as its phase point (refer to the left panel of Figure 4.1). Fig Existence and uniqueness of order one periodic solution of (2.3). In the following, we prove the uniqueness of the order one periodic solution. Arbitrarily choose two points A 1 and A 2 in the phase set, where I C <I A1 <I A2 <I D. Then the trajectories of system (2.3) through points A 1 and A 2 must intersect the line G = L G at some points A + 1 and A+ 2, respectively, which are in the impulse set and satisfy I Q <I A + <I 2 A + <I C. The points A + 1 and A+ 2 must be mapped to 1

18 MODELING INSULIN IMPULSIVE INJECTION 1541 two points in the phase set after impulsive effect which we denote as A 1 and A 2, respectively, where I A = I 1 A + + σ and I 1 A = I 2 A + + σ. Obviously, the point A + i is the 2 impulse point of A i,anhepointa i is the successor point of A i, i =1, 2. Then the successor functions of A 1 and A 2 satisfy F (A 2 ) F (A 1 )=(I A I A2 ) (I 2 A I A1 )= 1 (I A I 2 A )+(I A1 I A2 ) < 0, which means the successor function F (A) is monotone 1 decreasing in the segment CD, and thus there exists only one point M such that F (M) = 0 (refer to the right panel in Figure 4.1). That completes the proof. We show that the order one periodic solution is orbitally asymptotically stable. That implies that some perturbation will not drastically alter the effect of treatment. Theorem 4.9. For the case σ 1 =0,ifG s <L G <G 0 and I C <σ, then the order one periodic solution of system (2.3) is orbitally asymptotically stable. Proof. In the following discussion, for any point A in the impulse set, we denote its phase point as A, and we have I A = I A + σ. According to Theorem 4.8, system (2.3) has a unique order one periodic solution that has the point M(L G,I M ) as its phase point, where I Q <I M <I D. Consider the successor point Q of point D (which is definen Theorem 4.8; see Figure 4.2), we know I C <I Q <I M. The trajectory passing through the point Q must intersect the impulse set again at point Q 1 which is the impulse point of Q, and the point Q 1 must be mapped to point Q 1 after impulsive effect which is the successor point of Q. Besides, we denote the impulse point of the order one periodic solution as M. I Q <I M <I Q1 <I C and I M <I Q 1 Because distinct trajectories do not intersect, we can easily get <I D. Fig Illustration of the orbitally asymptotically stability of the order one periodic solution of system (2.3) when G s <L G <G 0 and I C <σ. Similarly, the trajectory passing through the point Q 1 must intersect the impulse set again at point Q 2 which is the impulse point of Q 1,anhepointQ 2 must be mapped to point Q 2 after impulsive effect which is the successor point of Q 1. We have I Q <I Q2 <I M and I Q <I Q <I M. 2 Repeating the above steps, we see that the trajectory from point D will come across the impulsive effect infinitely times. Denote the phase point corresponding to the ith impulsive effect as Q i 1, i =1, 2,...,whereQ 0 = Q.Wehave

19 1542 M. HUANG, J. LI, X. SONG, AND H. GUO I C <I Q <I 0 Q <I 2 Q < <I 4 Q <I 2k Q < <I M 2(k+1) and I D >I Q >I 1 Q >I 3 Q > >I 5 Q >I 2k+1 Q > >I M. 2(k+1)+1 Thus {I Q },k =0, 1, 2,..., is a monotonically increasing sequence, and {I 2k Q },k = 2k+1 0, 1, 2,..., is a monotonically decreasing sequence (see Figure 4.2), and furthermore, I Q I M as k ; I 2k Q I M as k. 2k+1 Choose an arbitrary point P 0 Q D, which is different from the point M. Without loss of generality, we assume that I Q <I P0 <I M (otherwise, I M <I P0 <I D,and the argument is similar). There must exist an integer k such that I Q <I P0 < 2k I Q. The trajectory from point P 0 will also undergo impulsive effect infinitely 2(k+1) many times. We denote the phase point corresponding to the lth impulsive effect as P l, l =0, 1, 2,...; then for any l, I Q 2(k+l) <I P2l <I Q 2(k+l+1) and I Q (2k+l+1)+1 < I P2l+1 <I Q 2(k+l)+1, and thus {I P2l },l =0, 1, 2,..., is also monotonically increasing, and {I P2l+1 },l=0, 1, 2,..., is also monotonically decreasing, and I P2l I M as l ; I P2l+1 I M as l. Therefore, in either case, the successor points of the phase points corresponding to the successive impulsive effect are attracted to the point M, which implies that the order one periodic solution of system (2.3) is orbitally asymptotically stable. By similar arguments, although slightly more complicated, we have the following results for type 2 diabetes. We omit the proof. Theorem When σ 1 > 0, ifg s <L G <G and I C <σ,thensystem(2.3) has a unique order one periodic solution, ant is orbitally asymptotically stable. Figure 4.3 displays one orbit produced by system (2.3), with initial condition below the threshold value, ants stability. Figure 5.3 in the next section shows that Glucose concentration (mg/dl) Upper bound for plasma glucose concentration Starting point Phase Portrait Trajectory approaches the stable periodic solution quickly Insulin Concentration (μu/ml) Fig Orbit in phase plane by model (2.3). It is shown that the solution is asymptotically orbital stable. It is clear that the glucose concentration is controlled by a preset threshold value.

20 MODELING INSULIN IMPULSIVE INJECTION 1543 the plasma glucose level is controlled under the predefined threshold value 190 mg/dl with various initial values. 5. Numerical simulations. The use of an insulin pump, also called continuous subcutaneous insulin infusion (CSII) therapy, has greatly increased for type 1 diabetes [21]. It can also provide a feasible alternative to insulin injections for type 2 diabetes [7], [27], [29], [31]. Model (2.2) is based on the physiology of the regulation of glucose annsulin while mimicking the subcutaneous injection of insulin with periodic impulse. Thus model (2.2) is close to the real-life clinical situation for the so-called open-loop administration. The loop can be closen clinical therapies by incorporating the feedback for blood sugar from an accurate glucose monitoring system. In such a case, model (2.3) provides a robust model with the most important and critical feature; that is, the timing of insulin injection is determined by the blood sugar level read from an accurate glucose monitor. A widely accepted agreement by researchers of the artificial pancreas is that for each model they should obtain a better understanding of the strengths and weaknesses of validating different control algorithms [33] and should develop clinical applicable controls. In this section, we apply models (2.2) and (2.3) under a few typical clinical situations and study the simulation results. Insulin analogues sold off-shelf are usually stored at temperatures between 36 F 46 F(2 C 8 C) in different concentrations, e.g., 40U or 100U in a 10ml vial. The units for the concentration of insulin and glucose are usually in μu/ml and mg/dl, respectively. So we need to make necessary conversions to obtain the dose as the model input of insulin injection. Two common methods, the weight method and an individual plan, are often employed to determine dose through insulin-to-carbohydrate ratios. Weight method assumes that insulin resistance increases along with weight, so the dose should be larger if a patient has gained weight. The individual plan method is an empirical process which deploys a series of daily doses and then adjusts accordingly to find the best plan. (Refer to [11].) In our simulation, we assume that the volume of the plasma compartment in humans is 10 liter (l). According to the dose calculator suggested by [22], we calculate that for dose as follows. If we inject xuforaykg subject every z minutes, then, each time, the injectensulin T = xu y 10 6 μu. For every 30 minutes, the total injected mass for 48 times in 24 hours is at m = T/48μU, so the concentration I = m( )μu/ml. For example, if we inject 0.5U for a 85kg subject every 30 minutes, then total injectensulin T =0.5U 85 = 42.5U, that is, T =42.5U 10 6 μu. For every 30 minutes, the injected mass is at m = T/48 = μU 0.89U, so the concentration I = m/( )= μU/ml. The parameter values in our simulations are determined either by comparing the response functions f 1,f 2,f 3,f 4,andf 5 in [34] and [35] or from the models for the intravenous glucose tolerance test (IVGTT) in [10], [20], [23], and [24] (see Table 5.1), which were estimated by experiments. Notice that the units of the parameters in Table 5.1 are for mass according to [34]. Therefore, unit conversion is made in simulation for display, which is the same way as that in [17] and [19]. For an insulin pump using the open-loop approach for type 1 diabetes, according to Theorems 3.9 and 3.10, a unique positive and globally stable periodic solution exists (refer to Figure 5.1). Figure 5.1 also reveals that, with different initial glucose levels, four delivery cycles are needen order for plasma glucose to be in oscillatory homeostasis. For type 2 diabetes (σ 1 > 0), Theorem 3.11 ensures that the solutions are bounded above and below (refer to Figure 5.2). We compare the profiles by setting the delivery