Analysis of a stochastic IVGTT glucose-insulin model with time delay

Xiangyun Shi; Qi Zheng; Jiaoyan Yao; Jiaxu Li; Xueyong Zhou; Xiangyun Shi; Qi Zheng; Jiaoyan Yao; Jiaxu Li; Xueyong Zhou

doi:10.3934/mbe.2020123

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 3: 2310-2329. doi: 10.3934/mbe.2020123

Previous Article Next Article

Research article

Analysis of a stochastic IVGTT glucose-insulin model with time delay

1.
School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
2.
Department of Bioinformatics and Biostatistics, University of Louisville, Louisville, KY 40202, USA
3.
Department of Mathematics, University of Louisville, Louisville, KY 40292, USA

Received: 31 October 2019 Accepted: 03 January 2020 Published: 19 January 2020

Diabetes mellituse has been one of the major diseases in the world due to the high percentage of diabetics in the global population and the increasing growth rate of its onset. Identifying individual physiological characteristics, e.g., insulin sensitivity and glucose effectiveness and others, is extremely important in developing effective drugs and investigating genetic pathways causing the defects in these physiological responses. Intravenous glucose tolerance test (IVGTT) is such a protocol to determine an individual insulin sensitivity and glucose effectiveness indices. In this paper, we propose a stochastic delay differential equation model for the IVGTT protocol attempting to develop a method to increase the accuracy of parameter estimation. We first study the existence and uniqueness of the global positive solution and its asymptotic behavior of the stochastic path close to the steady state of the corresponding deterministic model. Then we develop a maximum likelihood estimation method to estimate the parameters involved in the proposed model. Our simulation studies numerically confirm our theoretical findings and demonstrate that the proposed model with estimated parameters can improve the fitness of clinical data.
- IVGTT model,
- stochastic perturbation,
- time delay,
- Itô’s formula,
- MLE method
Citation: Xiangyun Shi, Qi Zheng, Jiaoyan Yao, Jiaxu Li, Xueyong Zhou. Analysis of a stochastic IVGTT glucose-insulin model with time delay[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2310-2329. doi: 10.3934/mbe.2020123

Related Papers:

Abstract

Diabetes mellituse has been one of the major diseases in the world due to the high percentage of diabetics in the global population and the increasing growth rate of its onset. Identifying individual physiological characteristics, e.g., insulin sensitivity and glucose effectiveness and others, is extremely important in developing effective drugs and investigating genetic pathways causing the defects in these physiological responses. Intravenous glucose tolerance test (IVGTT) is such a protocol to determine an individual insulin sensitivity and glucose effectiveness indices. In this paper, we propose a stochastic delay differential equation model for the IVGTT protocol attempting to develop a method to increase the accuracy of parameter estimation. We first study the existence and uniqueness of the global positive solution and its asymptotic behavior of the stochastic path close to the steady state of the corresponding deterministic model. Then we develop a maximum likelihood estimation method to estimate the parameters involved in the proposed model. Our simulation studies numerically confirm our theoretical findings and demonstrate that the proposed model with estimated parameters can improve the fitness of clinical data.

References

[1]	R. A. DeFronzo, J. D. Tobin, R. Andres, Glucose clamp technique: A method for quantifying insulin secretion and resistance, Am. J. Physiol. Endocrinol. Metab., 237 (1979), 214-223.
[2]	R. N. Bergman, Y. Z. Ider, C. R. Bowden, Quantitative estimation of insulin sensitivity, Am. J. Physiol. Endocrinol. Metab., 236 (1979), E667-677.
[3]	G. Toffolo, R. N. Bergman, D. T. Finegood, C. R.Bowden, C. Cobelli, Quantitative estimation of beta cell sensitivity to glucose in the intact organism: A minimal model of insulin kinetics in the dog, Diabetes, 29 (1980), 979-990.
[4]	A. De Gaetano, O. Arino, Mathematical modelling of the intravenous glucose tolerance test, J. Math. Biol., 40 (2000), 136-168.
[5]	J. Li, Y. Kuang, B. Li, Analysis of IVGTT glucose-insulin interaction models with time delay, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 103-124.
[6]	G. Toffolo, C. Cobelli, The hot IVGTT two-compartment minimal model: An improved version, Am. J. Physiol. Endocrinol. Metab., 284 (2003), 317-321.
[7]	A. Mukhopadhyay, A. De Gaetano, O. Arino, Modeling the intra-venous glucose tolerance test: A global study for a single-distributed-delay model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 407-418.
[8]	S. Panunzi, P. Palumbo, A. De Gaetano, A discrete single delay model for the intra-venous glucose tolerance test, Theor. Biol. Med. Modell., 4 (2007), 35.
[9]	S. Panunzi, A. De Gaetano, G. Mingrone, Advantages of the single delay model for the assessment of insulin sensitivity from the intravenous glucose tolerance test, Theor. Biol. Med. Modell., 7 (2010), 9.
[10]	J. Li, M. H. Wang, A. De Gaetano, P. Palumbo, S. Panunzi, The range of time delay and the global stability of the equilibrium for an IVGTT model, Math. Biosci., 235 (2012), 128-137.
[11]	M. Pitchaimani, P. Krishnapriya, C. Monica, Mathematical modeling of intra-venous glucose tolerance test model with two discrete delays, J. Biol. Syst., 23 (2015), 1-30.
[12]	X. Y. Shi, Y. Kuang, A. Makroglou, S. Mokshagundam, J. Li, Oscillatory dynamics of an intravenous glucose tolerance test model with delay interval, Chaos: Interdiscip. J. Nonlinear Sci., 27 (2017), 114324.
[13]	Y. Zhang, T. A. Holt, N. Khovanova, A data driven nonlinear stochastic model for blood glucose dynamics, Comput. Methods Programs Biomed., 125 (2016), 18-25.
[14]	A. K. Duun-Henriksen, S. Schmidt, R. R. Meldgaard, J. B. Moller, K. Norgaard, J. B. Jorgensen, et al., Model identification using stochastic differential equation grey-box models in diabetes, J. Diabetes Sci. Technol., 7 (2013), 431-440.
[15]	B. Benyó, B. Paláncz, Á. Szlávecz, K. Stewart, J. Homlok, C. G. Pretty, et al., Analysis of stochastic noise of blood-glucose dynamics, IFAC Pap. OnLine, 50 (2017), 15157-15162.
[16]	M. De la Sen, On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays, Appl. Math. Comput., 190 (2007), 382-401.
[17]	M. De la Sen, Absolute stability of feedback systems independent of internal point delays, IEE Proc. Control Theory Appl., 152 (2005), 567-574.
[18]	A. L. Murillo, J. Li, C. Castillo-Chavez, Modeling the dynamics of glucose, insulin, and free fatty acids with time delay: The impact of bariatric surgery on type 2 diabetes mellitus, Math. Biosci. Eng., 16 (2019), 5765.
[19]	P. Palumbo, P. Pepe, S. Panunzi, A. De Gaetano, Time-delay model-based control of the glucoseCinsulin system, by means of a state observer, Eur. J. Control, 18 (2012), 591-606.
[20]	J. Sturis, K. S. Polonsky, E. Mosekilde, E. Van Cauter, Computer model for mechanisms underlying ultradian oscillations of insulin and glucose, Am. J. Physiol. Endocrinol. Metab., 260 (1991), 801-809.
[21]	J. Li, Y. Kuang, C. C. Mason, Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays, J. Theor. Biol., 242 (2006), 722-735.
[22]	J. Li, Y. Kuang, Analysis of a model of the glucose-insulin regulatory system with two delays, SIAM J. Appl. Math., 67 (2007), 757-776.
[23]	X. R. Mao, Stochastic Differential Equations and Applications, 2nd edition, Woodhead Publishing, Chichester, 2008.
[24]	J. L. Lv, K. Wang, Almost sure permanence of stochastic single species models, Osaka J. Math., 422 (2015), 675-683.
[25]	R. Liptser, A. N. Shiryayev, Theory of Martingales, Springer, 1989.
[26]	K. Bahlali, Backward stochastic differential equations with locally Lipschitz coefficient, C. R. Acad. Sci., Ser. I: Math., 333 (2001), 481-486.
[27]	N. Ikeda, S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.
[28]	E. L. Lehmann, G. Casella, Theory of Point Estimation, Springer, 2006.
[29]	P. E. Greenwood, L. F. Gordillo, Stochastic epidemic modeling, in Mathematical and Statistical Estimation Approaches in Epidemiology (eds. G. Chowell, J. M. Hyman, L. M. A. Bettencourt, C. Castillo-Chavez), Springer, (2009), 31-52.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)