## Dynamics Analysis of a Stochastic Glucose-Insulin Model

Li Jiang, Lan Guijie, Zhang Shuwen, Wei Chunjin,

School of Science, Jimei University, Fujian Xiamen 361021

 基金资助: 福建省自然科学基金项目.  2018J01418

 Fund supported: the NSF of Fujian Province.  2018J01418

Abstract

In this paper, we investigate the global dynamics of a glucose-insulin model and its corresponding stochastic differential equation version. For the deterministic model, we show that there exists a unique equilibrium point, which is globally asymptotically stable for all parameter values. For the stochastic model, we show that the system admits unique positive global solution starting from the positive initial value and derive the stochastic permanence of the solutions of the stochastic system. In addition, by using Hasminskiis methods, we prove that there exists a unique stationary distribution and it has ergodicity. Finally, numerical simulations are carried out to support our theoretical results. It is found that: (ⅰ) the difficulty of the prediction of the peak size of the plasma glucose concentration always increases with the increase of the intensity of environmental fluctuations; (ⅱ) environmental fluctuations can result in the irregular oscillating of the plasma glucose concentration and plasma insulin concentration. Moreover, the volatility of the plasma glucose concentration and plasma insulin concentration always increase with the increase of the intensity of environmental fluctuations.

Keywords： Type 2 diabetes mellitus ; Global stability ; Environmental fluctuations ; Stationary distribution and ergodicity

Li Jiang, Lan Guijie, Zhang Shuwen, Wei Chunjin. Dynamics Analysis of a Stochastic Glucose-Insulin Model. Acta Mathematica Scientia[J], 2021, 41(6): 1937-1949 doi:

## 1 引言

$$$\left\{ \begin{array}{llc} { } {\rm d}G(t) = \Big(G_{in}-S_gG(t)-S_iG(t)\frac{I(t)}{n+I(t)}+\frac{S_k}{r^m+I^m(t)}\Big){\rm d}t, \\ { } {\rm d}I(t) = \Big(\frac{\sigma G^2(t)}{\alpha^2+G^2(t)}-d_iI(t)\Big){\rm d}t, \end{array} \right.$$$

$$$\left\{ \begin{array}{llc} { } {\rm d}G(t) = \Big(G_{in}-S_gG(t)-S_iG(t)\frac{I(t)}{n+I(t)}+\frac{S_k}{r^m+I^m(t)}\Big){\rm d}t+\alpha_1G(t){\rm d}B_1(t), \\ { } {\rm d}I(t) = \Big(\frac{\sigma G^2(t)}{\alpha^2+G^2(t)}-d_iI(t) \Big){\rm d}t+\alpha_2I(t){\rm d}B_2(t), \end{array}\right.$$$

## 2 确定性模型

$$$G_{in}-S_gG^*-S_iG^*\frac{I^*}{n+I^*} +\frac{S_k}{r^m+(I^*)^m} = 0, \frac{\sigma (G^*)^2}{\alpha^2+(G^*)^2}-d_iI^* = 0.$$$

$$$G_{in}-S_gG^*-S_iG^*\frac{\frac{\sigma (G^*)^2}{d_i(\alpha^2+(G^*)^2)}}{n+ \frac{\sigma (G^*)^2}{d_i(\alpha^2+(G^*)^2)}} +\frac{S_k}{r^m+(\frac{\sigma (G^*)^2}{d_i(\alpha^2+(G^*)^2)})^m} = 0.$$$

### 3.1 全局正解

$$$V[G(\tau_k\wedge T), I(\tau_k\wedge T)]\leq V[G(0), I(0)]+\int_0^{\tau_k\wedge T} M {\rm d}s+M_1+M_2,$$$

$$$E[V(G(\tau_k\wedge T), I(\tau_k\wedge T))]\leq V[G(0), I(0)]+M(\tau_k\wedge T).$$$

$\Omega_k = \{\tau_k\leq T\}$($k\geq k_1$), 则由式(3.1), $P(\Omega_k)\geq\varepsilon$. 注意到, 对任意的$\omega\in \Omega_k$, 至少有一个$G(\tau_k, \omega) $$I(\tau_k, \omega) 等于 k$$ \frac{1}{k}$, 因此$V(G, I)$不小于$k-\ln k-1 $$\frac{1}{k}+\ln k-1 , 即 由式(3.4) 可知 其中 1_{\Omega_k}$$ \Omega_k$的示性函数. 令$k\rightarrow \infty$可得$\infty>V\big(G(0), I(0)\big)+MT = \infty.$因此有$\tau_\infty = \infty$, 系统(1.2) 存在唯一的全局正解.

$$$\begin{array} {ll} { } \liminf\limits_{t\rightarrow \infty} P\{G(t)\geq\delta\}\geq1-\epsilon, \quad \quad \liminf\limits_{t\rightarrow \infty} P\{I(t)\geq\delta\}\geq1-\epsilon, \\ { } \liminf\limits_{t\rightarrow \infty} P\{G(t)\leq\bar{\delta}\}\geq1-\epsilon, \quad \quad \liminf\limits_{t\rightarrow \infty} P\{I(t)\leq\bar{\delta}\}\geq1-\epsilon, \end{array}$$$

设$(G(t), I(t))$是系统(1.2) 具有任意正初始值$(G(0), I(0)) \in {\Bbb R}^2_+$的解, 我们将定理的证明分为两步.

$$$\liminf\limits_{t\rightarrow \infty} P\{G\geq\delta_1\}\geq1-\epsilon_1.$$$

$$$\liminf\limits_{t\rightarrow \infty} P\{I(t)\leq\bar{\delta}_2\}\geq 1-\epsilon_2.$$$

$\begin{eqnarray} \liminf\limits_{t\rightarrow \infty} P\{G\geq\delta_2\}\geq 1-\epsilon_2. \end{eqnarray}$

### 3.3 平稳分布

$X(t) $${\Bbb R}^d 中的自治 Markov 过程, 可表示为如下随机微分方程 其扩散矩阵为 引理3.1[26] 对Markov过程 X(t) , 若存在具有正则边界的有界区域 U \in {\Bbb R}^d 具有如下性质: \rm (P.1) 对任意的 x \in U , 扩散矩阵 A(X) 为严格正定的; \rm (P.2) 存在非负 C^2 -函数 V , 使得在 {\Bbb R}^d \backslash U$$ LV$为负数;

为了证明定理3.3, 我们需要验证引理3.1中的条件$\rm (P.1) $$\rm (P.2) . 系统(1.2) 的扩散矩阵为 显然满足条件 \rm (P.1) . 下面我们验证条件 \rm (P.2) . 由定理3.1, 对任意初始值 \big(G(0), I(0)\big)\in {\Bbb R}^2_+ , 系统(1.2) 存在唯一的全局正解 \big(G(t), I(t)\big) . 定义函数 V(G, I) \in C^2({\Bbb R}^2_+, {\Bbb R}_+) : 由It \hat{\rm o} 公式可得 其中 M_1' = {G_{in}+\frac{S_k}{r^m}+S_g+S_i+\sigma+d_i+\frac{1}{2}(\alpha_1^2+\alpha_2^2)} . 考虑如下开集 U_{\epsilon} = \{(G, I)\in {\Bbb R}^2_+|\epsilon<G<\frac{1}{\epsilon}, \epsilon^3<I<\frac{1}{\epsilon^3}\} , 其中 0<\epsilon<1 为一个充分小的常数且满足 U^{C}_{\epsilon} 分为如下四个区域: 于是有 情形1 在区域 U_1 上, 有 情形2 在区域 U_2 上, 有 \rm 情形 3 在区域 U_3 上, 有 \rm 情形 4. 在区域 U_4 上, 有 综上所述, 对任意的 (G, I)\in U^{C}_{\epsilon} , 有 LV(G, I)<0 , 即引理3.1中条件 (P_2) 成立, 因此系统(1.2) 存在唯一遍历平稳分布 \mu(\cdot) . ## 4 数值模拟 为了验证上述理论结果, 我们将给出对确定性模型(1.1) 以及随机模型(1.2) 数值模拟的结果. 对于随机系统(1.2), 我们采用Milstein高阶方法[27], 其对应的离散方程为 $$\left\{ \begin{array}{rl} G(j+1) = &{ } G(j)+\Big(G_{in}-S_gG(j)-\frac{S_iG(j)I(j)}{n+I(j)} +\frac{S_k}{r^m+I^m(j)}\Big)\Delta t+\alpha_1G(j)\sqrt{\Delta t}\xi_j\\ &{ } +\frac{\alpha_1^2}{2}G(j)(\xi_j^2-1)\Delta t, \\ I(j+1) = &{ } I(j)+\Big(\frac{\sigma G^2(j)}{\alpha^2+G^2(j)}+d_iI(j) \Big)\Delta t +\alpha_2I(j)\sqrt{\Delta t}\zeta_j+\frac{\alpha_2^2}{2}I(j)(\zeta_j^2-1)\Delta t, \end{array} \right.$$ 其中 \xi_j$$ \zeta_j$, $j = 1, 2, \cdots , n$, 为相互独立的高斯随机变量. 本节的主要目的是通过探究确定性(无噪声) 和对应的随机(有噪声) 葡萄糖-胰岛素系统的全局动力学, 进一步研究环境波动如何影响胰岛素对血糖的产生和利用.

 参数 数值 单位 参考 $G_{in}$ $2.16$ $mg/dl/\min$ [9, 21] $S_k$ $2$ $\min^{-1}$ — $S_g$ $5\times 10^{-6}$ $\min^{-1}$ [9, 21] $S_i$ $0.1$ $\min^{-1}$ [9, 21] $n$ $80$ $mg$ [9, 21] $m$ $4$ — — $r$ $80$ $mg$ [9, 21] $\sigma$ $6.27$ $mU/\min$ [9, 21] $\alpha$ $105$ $mg$ [9, 21] $d_i$ $0.08$ $\min^{-1}$ [9, 21]

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

National institute of diabetes and digestive and kidney diseases

Wang L , Huang Y M .

National institute of diabetes and digestive and kidney diseases

Chinese Journal of Internal Medicine, 2010, (8): 695- 695

Tfayli H , Arslanian S .

Pathophysiology of type 2 in youth: the evolving chameleon

Arq Bras Endocrinol, 2009, 53 (2): 165- 174

Bolie V W .

Coefficients of normal blood glucose regulation

J Appl Physiol, 1961, 16 (5): 783- 790

Ackerman E , Rosevear J W , Mcguckin W F .

A mathematical model of the glucose-tolerance test

Phys Med Biol, 1964, 9 (2): 203- 213

Ackerman E , Gatewood L C , Rosevear J W , et al.

Model studies of blood-glucose regulation

Bull Math Biophys, 1965, 27 (1): 21- 37

Sulston K W , Ireland W P , Praught J C .

Hormonal effects on glucose regulation

Atl Electron J Math, 2006, 1 (1): 31- 46

Divanovi H, Muli D, Padalo A, et al. Effects of electrical stimulation as a new method of treating diabetes on animal models: Review// Badnjevic A, et al. CMBEBIH. Singapore: Springer, 2017: 253-258

Wang H , Li J , Yang K .

Mathematical modeling and qualitative analysis of insulin therapies

Math Biosci, 2007, 210 (1): 17- 33

Li J , Kuang Y .

Analysis of a model of the glucose-insulin regulatory system with two delays

Siam J Appl Math, 2007, 67 (3): 757- 776

Topp B , Promislow K , Devries G , et al.

A Model of β-cell mass, insulin, and glucose kinetics: pathways to diabetes

J Theor Bio, 2000, 206 (4): 605- 619

Hernandez R D, Lyles D J, Rubin D B, et al. A model of β-cell mass, insulin, glucose and receptor dynamics with applications to diabetes[R]. Technical Report, Biometric Department, MTBI Cornell University, 2001

Gallenberger M , Castell WZ .

Dynamics of glucose and insulin concentration connected to the β-cell cycle: model development and analysis

Theor Biol Med Model, 2012, 9 (1): 46- 46

Bellazzi R , Nucci G , Cobelli C .

The subcutaneous route to insulin-dependent diabetes therapy

IEEE Eng Med Biol, 2001, 20 (1): 54- 64

Boutayeb A , Chetouani A .

A critical review of mathematical models and data used in diabetology

BioMed Eng OnLine, 2006, 5 (1): 43

Kansal A R .

Modeling approaches to type 2 diabetes

Diabetes Technol The, 2004, 6 (1): 39- 47

Landersdorfer C , Jusko W .

Pharmacokinetic/pharmacodynamic modelling in diabetes mellitus

Clin Pharmacokinet, 2008, 47 (7): 417- 448

Makroglou A , Li J , Yang K .

Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview

Appl Numer Math, 2006, 56 (3/4): 559- 573

Parker R S , Doyle F , Peppas N A .

The intravenous route to blood glucose control

IEEE Eng Med Biol, 2001, 20 (1): 65- 73

Pattaranit R , Berg H .

Mathematical models of energy homeostasis

J R Soc Interface, 2008, 5 (27): 1119- 1153

Li J , Yang K , Mason C C .

Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays

J Theor Biol, 2006, 242 (3): 722- 735

Huang M , Li J , Song X , et al.

Modeling impulsive injections of insulin: towards artificial pancreas

SIAM J Appl Math, 2012, 72 (5): 1524- 1548

Liu L , Wang F , Lu H , et al.

Effects of noise exposure on systemic and tissue-level markers of glucose homeostasis and insulin resistance in male mice

Environ Health Perspect, 2016, 124, 1390- 1398

Yu X , Yuan S , Zhang T .

The effects of toxin-producing phytoplankton and environmental fluctuations on the planktonic blooms

Nonlinear Dyn, 2018, 91, 1653- 1668

Nguyen D , Yin G , Zhu C .

Long-term analysis of a stochastic SIRS model with general incidence rates

Siam J Appl Math, 2020, 80 (2): 814- 838

Liu M , Fan M .

Permanence of stochastic lotka-volterra systems

J Nonl Sci, 2017, 27 (2): 425- 452

Khas'Miniskii R. Stochastic Stability of Differential Equations. Alphen aan den Rijn: Sijthoff and Noordhoff, 1980

Higham Desmond J .

An algorithmic introduction to numerical simulations of stochastic differentila equations

Siam Rev, 2001, 43, 525- 546

Liu M , Bai C .

Optimal harvesting of a stochastic mutualism model with regime-switching

Appl Math Comput, 2020, 373, 125040

Ji W , Wang Z , Hu G .

Stationary distribution of a stochastic hybrid phytoplankton model with allelopathy

Adv Differ Equ, 2020, 2020, 632

Wang Z , Deng M , Liu M .

Stationary distribution of a stochastic ratio-dependent predator-prey system with regime-switching

Chaos Solitons Fract, 2020, 2020, 110462

Yu X , Yuan S , Zhang T .

Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching

Commun Nonlinear Sci, 2018, 59, 359- 374

Zhao Y , Yuan S , Zhang T .

The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching

Commun Nonlinear Sci, 2016, 37, 131- 142

Xu C , Yuan S , Zhang T .

Average break-even concentration in a simple chemostat model with telegraph noise

Nonlinear Anal Hybr, 2018, 29, 373- 382

Lan G , Lin Z , Wei C , et al.

A stochastic SIRS epidemic model with non-monotone incidence rate under regime-switching

J Franklin Inst, 2019, 356 (16): 9844- 9866

Zhao D , Liu H .

Coexistence in a two species chemostat model with Markov switchings

Appl Math Lett, 2019, 94, 266- 271

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