Global dynamics of target-mediated drug disposition models and their solutions by nonstandard finite difference method
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Global dynamics of target-mediated drug disposition models and their solutions by nonstandard finite difference method Oluwaseun Francis Egbelowo1
· Manh Tuan Hoang2
Received: 26 April 2020 / Revised: 14 October 2020 / Accepted: 21 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract The aim of this work is to study global dynamics of target-mediated drug disposition (TMDD) models and their solutions by nonstandard finite difference (NSFD) schemes. Firstly, we use comparison principles and the Lyapunov stability theory for ODEs to establish positivity, boundedness, local and global asymptotic stability of the TMDD models. Secondly, positivity-preserving NSFD schemes are proposed and their dynamical properties are analysed rigorously. Lastly, we perform a set of numerical simulations to support and illustrate the theoretical results and to show advantages of the NSFD schemes over standard ones. The results show that there is a good agreement between the numerical results and theoretical ones. In addition, the numerical simulations indicate that the constructed NSFD schemes are dynamically stable and efficient in replicating the complex dynamical properties of the continuous models. Keywords Target-mediated drug disposition · Nonstandard finite difference method · Global asymptotic stability · Lyapunov stability theorem · Dynamic consistency
1 Introduction The concept of target mediated drug disposition (TMDD) was first introduced by Levy [32] to describe his observation of drug bounding to high affinity. The TMDD can be considered as a good explaination for the non-linear behaviour displayed by
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Oluwaseun Francis Egbelowo [email protected] Manh Tuan Hoang [email protected]; [email protected]
1
Division of Clinical Pharmacology, Department of Medicine, University of Cape Town, Cape Town, South Africa
2
Department of Mathematical Methods in IT, Institute of Information Technology, Vietnam Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
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O. F. Egbelowo, M. T. Hoang
some drugs and occurs when the binding of the drug to it target affects the distribution and elimination of the drug [33]. The general model exhibiting TMDD was developed by Mager et al. [33], while Aston et al. [6] and Dua [17] provide compartmental representations of the physiological-based approach described by Levy [32]. These compartmental representations are described by nonlinear ODEs, and therefore, mathematical analyses and efficient numerical methods are required for the models. This fact motivate us to perform this research. The success of TMDD modeling requires accurate predictions of concentration in each compartment. However, for the case of TMDD models, obtaining the predictions is very challenging because the systems are nonlinear, and most standard numerical methods, typically the explicit forward Euler and Runge-Kutta schemes are unable to capture dynamics of the continuous models. Also, these standard methods are known to induce
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