一类随机葡萄糖-胰岛素模型动力学分析

摘要/Abstract

摘要：

该文研究了一类葡萄糖-胰岛素模型及其相应的随机模型的全局动力学.对于确定性模型，证明存在唯一的平衡点，且该平衡点是全局渐近稳定.对于随机模型，证明系统全局正解的存在唯一性以及系统解的随机持久性.利用Hasminskiis方法，证明系统存在唯一遍历的平稳分布.最后，通过数值模拟验证了理论结果并发现：（i）预测血糖浓度峰值大小的难度总是随着环境波动强度的增加而增加；（ii）环境波动会导致血糖浓度和胰岛素浓度的不规则振荡.此外，血糖浓度和胰岛素浓度的震荡幅度总是随着环境波动强度的增加而增加.

关键词: 2型糖尿病, 全局稳定性, 环境波动, 平稳分布和遍历性

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.

Key words: Type 2 diabetes mellitus, Global stability, Environmental fluctuations, Stationary distribution and ergodicity

中图分类号:

O211.6

李江,蓝桂杰,张树文,魏春金. 一类随机葡萄糖-胰岛素模型动力学分析[J]. 数学物理学报, 2021, 41(6): 1937-1949.

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

图/表 4

表 1

图 1

图 2

图 3

参考文献 35

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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]

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