DYNAMICS OF A STOCHASTIC GLUCOSE-INSULIN MODEL WITH IMPULSIVE INJECTION OF INSULIN

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, the dynamics of a stochastic glucose-insulin model with impulsive injection of insulin are investigated analytically and numerically. Firstly, we show that the system admits unique positive global solution starting from the positive initial value, which is a prerequisite for analyzing the long-term behavior of the stochastic model. Then, according to the theory of Khasminskii, we show that there exists at least one nontrivial positive periodic solution. Finally, numerical simulations are carried out to support our theoretical results. It is found that: (i) The presence of environmental noises is capable of supporting the irregular oscillation of blood glucose level, and the average level of the glucose always increases with the increase in noise intensity. (ii) The higher the volatility of the environmental noises, the more difficult the prediction of the peak size of blood glucose level.


INTRODUCTION
Diabetes mellitus, a metabolic disorder is one of the major problems in global public health.
It is caused by the fact that the pancreas aren't able to produce enough insulin which is the only 1 diabetes mellitus (T1DM) and type 2 diabetes mellitus (T2DM). T1DM is a metabolic disorder characterized by insufficient or insufficient insulin secretion, resulting in elevated plasma glucose levels and the inability of beta cells to respond appropriately to metabolic stimuli. It is an autoimmune disease that causes insulin deficiency by autoimmune destruction of islet beta cells. T2DM (also known as adult or non-insulin-dependent diabetes mellitus) is characterized by a patient's insensitivity to insulin secreted in the body resulting in elevated blood glucose levels. It can be controlled through regular exercise and healthy eating.
The blood glucose level is regulated by two negative feedback loops, where short-term hyperglycemia stimulates islet beta cells to secrete insulin and simultaneously inhibits the secretion of glucagon in islet A cells, thereby lowering blood sugar. In order to better understand the dynamics of insulin and blood glucose concentration, many scholars have established mathematical models to describe the principle of insulin and blood glucose, and given insulin injection strategies through mathematical theory and numerical simulation then to better control blood sugar levels. For instance, Li et al. [1] have reviewed multiple models of subcutaneous injection of regular insulin and insulin analogues, and found that these models provide key building blocks for some important endeavors into physiological questions of insulin secretion and action. Li and Kuang [2] proposed two systemic models to model the subcutaneous injection of rapid-acting insulin analogues and long-acting insulin analogues, respectively. Their work shows the two models will be good choices in practical applications. Particularly, Huang et al. [3] formulated a physiological and metabolic model by a semicontinuous dynamical system. They found that the glucose level of a diabetic can be controlled within a desired level by adjusting the values of two model parameters, injection period and injection dose. The model in DYNAMICS OF A STOCHASTIC GLUCOSE-INSULIN MODEL WITH IMPULSIVE INJECTION OF INSULIN 3 [3] is given by where G(t), I(t) are the concentration of blood glucose and blood insulin at time t, respectively.
G in is the estimated average constant rate of glucose input, αG(t) is the insulin-independent glucose uptake, a(c + mI(t) n+I(t) )G(t) stands for the insulin-dependent glucose utilization, b is the hepatic glucose production, and γI(t) indicates the insulin degradation with γ as the constant degradation rate, T and q represent the period of the impulsive injection of insulin and the insulin input amount every time, respectively. ∆ρ(t) = ρ(t + ) − ρ(t), n ∈ {1, 2, 3, ...}. All parameters above are positive. The other parameters can be seen in [3].
It is well known that meals and exercise, the age and weight of the patient also affect the insulin/glucose dynamics. Liu et al. [4] pointed out that glucose tolerance, insulin response to the glucose challenge, insulin sensitivity and β cell morphology can be affected by environmental noise. These daily and hourly fluctuations of patient parameters can create difficulties in continuous glucose control. Hence, it is necessary and important to study the impact of those uncertain factors on the insulin/glucose level in the body of the patients. In present paper, we intend to consider model (1.1) incorporating the influence of those uncertain factors, moreover, we take into account the effect of randomly factors into model (1.1) by assuming α → α + σ dB(t), then we can obtain the SDE model as follows where B(t) is a real-valued standard Brownian motion defined on the complete probability space (Ω, F , {F t } t≥0 , P) with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is right continuous and F 0 contains all P-null sets); σ 2 represents the intensities of the white noise. Our main purpose is to investigate the effect of random fluctuations on the glucose dynamics based on realistic parameters obtained from previous literatures. The rest of this article is organized as follows: In Section 2, we present some preliminaries which will be used in our following analysis. In Section 3, we present the detailed proof of the theoretical results. Section 4 is devoted to illustrate our analytical results by using some numerical examples. In Section 5, we provide a brief discussion and summary of main results.

DYNAMICS OF A STOCHASTIC GLUCOSE-INSULIN MODEL WITH IMPULSIVE INJECTION OF INSULIN 5
Let U be a given open set in the d-dimensional Euclidean space is the family functions on E which are twice continuously differentiable with respect to x ∈ R d and continuously differentiable with respect to t ∈ [0, ∞).
Lemma 2.1. [5] Suppose that the coefficients of (2.1) are T -periodic in t and satisfy the conditions (2.2) in every cylinder I ×U, and assume further there exists a function V (t, x) ∈ C 2,1 , which is T -periodic in t and satisfies, Then system (2.1) has at least a T -periodic Markov process.
Proof. Since the coefficients of model ( For each integer k ≥ k 0 , define the stopping time Throughout this paper we set inf / 0 = ∞ (as usual / 0 denotes the empty set). Clearly, τ k is s.. If we can show that τ ∞ = ∞ a.s., then In other words, to complete the proof all we need to show is that τ ∞ = ∞ a.s.. If this statement is false, then there is a pair of constants T > 0 and ε ∈ (0, 1) such that Define a function V : The nonnegativity of this function can be seen from u − 1 − ln u ≥ 0 on u > 0. Then, by Itô's formula, one can see Substituting this inequality into Eq. (2.5), we see that which implies that Taking the expectations of the above inequality leads to Set Ω k = {τ k ≤ T } for k ≥ k 1 and from (2.4), we have P(Ω k ) ≥ ε. Note that for every ω ∈ Ω k , there is at least one ofḠ τ k (ω) andĪ τ k (ω) equaling either k or 1 k , hence It then follows from (2.7) that So we must have τ ∞ = ∞. The conclusion is confirmed.
Now, we give some basic properties of the following subsystem of model (1.2), which are very important for obtaining our main results.
Remark 2.1. Substituting I * (t) into the first equation of system (1.2) for I(t), we obtain the following system where G in = G in + b, α = α + ac, a = am. Next, we will consider the system (2.10).

MAIN RESULTS
First of all, we show that there is a unique positive solution, which is a prerequisite for analyzing the long-term behavior of the stochastic model (1.2).
Theorem 3.1. For any given initial value X(0) = G(0), I(0) ∈ R 2 + , there is a unique solution X(t) = G(t), I(t) of system (1.2) and the solution will remain in R 2 + with probability 1, that is X(t) ∈ R 2 + for all t ≥ 0 almost surely.
Proof. For t ∈ [0, T ] and for any initial condition X(0) = G(0), I(0) ∈ R 2 + and by Lemma 2.2, Eq. (2.3) has a unique global solutionX t; 0, X(0) ∈ R 2 + that is defined and continuous on interval [0, T ], hence Eq. (1.2) also has a unique global solution X t; 0, By the Lemma 2.2 and the same deduction, we get there is a unique global solution X t; T, X(T + ) = X t; T, X(T + ) that is defined on [T + , 2T ] and X(2T + ) = Ḡ (2T ),Ī(2T ) + q ∈ R 2 + . It is easy to see the above deduction can go on infinitely. The proof is completed.
Theorem 3.2. Suppose that α > σ 2 2 and G(t) is the solution of stochastic model (2.10) with initial value G(0) > 0, then the following statement is valid with probability 1: The proof is the application of the well-known comparison theorem for stochastic differential equation, Lemmas 2.4 and 2.5. Here it is omitted. Proof. By Lemma 2.1, one can see that in order to verify Theorem 3.3, it suffices to find a C 2,1function V (x,t) which is T -periodic in t and a closed set U ∈ R + such that conditions (Q 1 ) and (Q 2 ) of Lemma 2.1 hold.
Define a C 2,1 -function V : R + × R + → R + as follows where ω(t) is positive T -periodic continuous functions. Obviously, V (G,t) is T periodic in t and satisfies lim inf , which shows that (Q 1 ) in Lemma 2.1 holds. Making use of Itô's formula, we have Consider the bounded open subset where 0 < ε < 1 is a sufficiently small number. In the set D C ε = R + \D ε , let us choose sufficiently small ε such thatω For convenience, we divide D C ε into two domains, Case 2. On domain D 2 , one can see that that is, the condition (Q 2 ) holds. Hence in view of Lemma 2.1, we obtain that system (2.10) has a nontrivial positive T -periodic solution. In addition, one can see that for any initial value G(0) ∈ R + system (2.10) has a unique global positive solution and so system (2.10) has at least one nontrivial positive T -periodic solution. This completes the proof.

NUMERICAL SIMULATIONS
In this section, we provide numerical simulation results to substantiate the analytical findings for the stochastic model (1.2) reported in the previous sections by using the Milstein method mentioned in [8].

DISCUSSION
Over the decades subcutaneous injection of insulin analogues was considered as the most widely method in treating diabetes. In the point of view of treating diabetes, the ultimate purpose of the subcutaneous injection of insulin analogues is to increase the plasma insulin concentration and thus lower blood glucose to maintain normal glycemia. In the real ecological systems, meals and exercise, the age and weight of the patient also affect the insulin/glucose dynamics. These daily and hourly fluctuations of patient parameters can create difficulties in continuous glucose control. In order to better understand the dynamics of the insulin analogues  from subcutaneous injection to absorption, we have considered the basic features of insulinglucose stochastic model of subcutaneous injection of regular insulin. Our results show that there exists at least one nontrivial positive periodic solution for model (1.2), which means that the glucose and insulin will exhibit periodicity in the long run. In addiion, we found that the noise has great effects on the diabetic patient, such as, (i) the presence of environmental noises is capable of supporting the irregular oscillation of blood glucose level, and the average level of the glucose always increases with the increment in noise intensity; (ii) the higher the volatility of the environmental noises, the more difficult the prediction of the peak size of blood glucose level. Hence, diabetic patient should avoid the influence of uncertain factors on them, such as, mood and stress.

ACKNOWLEDGMENTS
This work was supported by Fujian provincial Natural science of China (2018J01418) and National Natural Science Foundation Breeding Program of Jimei University (ZP2020064).

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.