Discretization and Stability Analysis for a Generalized Type Nonlinear Pharmacokinetic Models

Abstract

Estimating the effects of drugs at different stages is directly proportional to the duration of recovery and the duration of pulling through with the disease. It is very important to estimate the effects of drugs at different stages. For this reason, solving Pharmacokinetic models which investigate these effects are very important. In this study, numerical solutions of one, two, and three-compartment nonlinear Pharmacokinetic models have been studied. Distributed order differential equations have been used for the solution. Numerical solutions have been found with the density function contained in distributed order differential equations and different values of this function. A nonstandard finite difference scheme has been used for numerical solutions. Finally, stability analyses of equilibrium points of the obtained discretized system have also been researched with the help of the Matignon criterion.

Keywords

• Pharmacokinetic models
• Distributed order differential equations
• Stability Analysis
• Schur-Cohn criteria
• Nonstandard finite difference scheme

References

1. [1] Shargel, L., Yu, A.B.C., “Applied Biopharmaceutics and Pharmacokinetics”, 7th ed., McGraw-Hill, (2017).
2. [2] Michealis, L., Menten, M.L., “Die Kinetik der Invertinwirking”, Biochemische Zeitschrift, 49: 333–369, (1913).
3. [3] Widmark, E., Tandberg, J., “Uber die bedingungen f’tir die Akkumulation Indifferenter Narkoliken Theoretische Bereckerunger”, Biochemische Zeitschrift, 147: 358–369, (1924).
4. [4] Holford, N.H.G., Sheiner, L.B., “Kinetics of pharmacologic response”, Pharmacology and Therapeutics, 16: 143–166, (1982).
5. [5] Beringer, P., Nguyen, M., Hoem, N., Louie, S., Gill, M., Gurevitch, M., Wong-Beringer, A., “Absolute bioavailability and pharmacokinetics of linezolid in hospitalized patients given enteral feedings”. Antimicrobial Agents and Chemotherapy, 49(9): 3676-3681, (2005).
6. [6] Atlas, G., Dhar, S., “Development of a Recursive Finite Difference Pharmacokinetic Model from an Exponential Model: Application to a Propofol Infusion”, IAENG International Journal of Applied Mathematics, 40(1): 13-25, (2010).
7. [7] Egbelowo, O., Harley, C., Jacobs, B., “Nonstandard Finite Difference Method Applied to a Linear Pharmacokinetics Model”, Bioengineering, 4(40), (2017).
8. [8] Egbelowo, O., “Nonlinear Elimination of Drug in One-Compartment Pharmacokinetic Models: Nonstandard Finite Difference Approach for Various Routes of Administration”, Mathematical and Computational Applications, 23(27), (2018).

9. [9] Saadah, A.M., Widodo, I., “Drug elimination in two-compartment pharmacokinetic models with nonstandard finite difference approach.”, IAENG International Journal of Applied Mathematics, 50(2): 1-7, (2020).
10. [10] Caputo, M., “Elasticita e dissipazione”, Zanichelli, (1969).
11. [11] Caputo, M., “Mean fractional-order-derivatives differential equations and filters”, Annali dell’Universita di Ferrara, 41(1): 73-84, (1995).
12. [12] Caputo, M., “Distributed order differential equations modelling dielectric induction and diffusion”, Fractional Calculus and Applied Analysis, 4(4): 421-442, (2001).
13. [13] Caputo, M., “Diffusion with space memory modelled with distributed order space fractional differential equations”, Annals of Geophysics, (2003).
14. [14] Bagley, R.L., Torvik, P.J., “On the existence of the order domain and the solution of distributed order equations-Part I, International Journal of Applied Mathematics”, 2(7): 865-882, (2000).
15. [15] Bagley, R.L., Torvik, P.J., “On the existence of the order domain and the solution of distributed order equations-Part II, International Journal of Applied Mathematics”, 2(8): 965-988, (2000).
16. [16] Diethelm, K., Ford, N.J., “Numerical analysis for distributed-order differential equations. Journal of Computational and Applied Mathematics”, 225(1): 96-104, (2009).
17. [17] Katsikadelis, J.T., “Numerical solution of distributed order fractional differential equations”, Journal of Computational Physics, 259: 11-22, (2014).
18. [18] Li, X.Y., Wu, B.Y., “A numerical method for solving distributed order diffusion equations”, Applied Mathematics Letters, 53: 92-99, (2016).
19. [19] Najafi, H.S., Sheikhani, A.R., Ansari, A., “Stability analysis of distributed order fractional differential equations”, In Abstract and Applied Analysis, Hindawi, (2011).
20. [20] Aminikhah, H., Refahi, S., Rezazadeh, H., “Stability analysis of distributed order fractional Chen system”, The Scientific World Journal, (2013).
21. [21] Hartley, T.T., Lorenzo, C.F., “Fractional-order system identification based on continuous order-distributions”, Signal Processing, 83(11): 2287-2300, (2003).
22. [22] Luchko, Y., “Boundary value problems for the generalized time-fractional diffusion equation of distributed order”, Fractional Calculus and Applied Analysis, 4: 409-422, (2009).
23. [23] Ford, N., Morgado, M., “Distributed order equations as boundary value problems”, Computers and Mathematics with Applications, 64(10): 2973-2981, (2012).
24. [24] Kocabıyık, M., Ongun, M.Y., Çetinkaya, İ.T., “Numerical analysis of distributed order SVIR model by nonstandard finite difference method”, Journal of Balıkesir University Institute of Science and Technology, 23(2): 577-591, (2021).
25. [25] Meerschaert, M.M., Tadjeran, C., “Finite difference approximations for fractional advection–dispersion flow equations”, Journal of Computational and Applied Mathematics, 172(1): 65-77, (2004).
26. [26] Dorciak, L., “Numerical models for simulation the fractional-order control systems”, UEF-04-94, The Academy of Sciences, Institute of Experimental Physic, Kosice, Slovak Republic, (1994).
27. [27] Mickens, R.E., “Exact solutions to a finite‐difference model of a nonlinear reaction‐advection equation: Implications for numerical analysis”, Numerical Methods for Partial Differential Equations, 5(4): 313-325, (1989).
28. [28] Mickens, R.E., “Nonstandard finite difference models of differential equations”, World scientific, (1994).
29. [29] Mickens, R.E., “Applications of nonstandard finite difference schemes”, World Scientific, (2000).
30. [30] Mickens, R.E., “Nonstandard finite difference schemes for differential equations”, Journal of Difference Equations and Applications, 8(9): 823-847, (2002).
31. [31] Mickens, R.E., “Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition”, Numerical Methods for Partial Differential Equations: An International Journal, 23(3): 672-691, (2007).
32. [32] Ongun, M.Y., Turhan, I., “A numerical comparison for a discrete HIV infection of CD4+ T-Cell model derived from nonstandard numerical scheme”, Journal of Applied Mathematics, 2013: 4, (2012).
33. [33] Khalsaraei, M. M., Jahandizi, R. S., “Efficient explicit nonstandard finite difference scheme with positivity-preserving property”, Gazi University Journal of Science, 30(1): 259-268, (2017).
34. [34] Ongun, M.Y., Arslan, D., “Explicit and Implicit Schemes for Fractional orders Hantavirus Model”, Iranian Journal of Numerical Analysis and Optimization, 8(2): 75–93, (2018).
35. [35] Kocabıyık, M., Özdoğan, N., Ongun, M.Y., “Nonstandard Finite Difference Scheme for a Computer Virus Model”, Journal of Innovative Science and Engineering (JISE), 4(2): 96-108, (2020).
36. [36] Zhang, Q., Ran, M., Xu, D., “Analysis of the compact difference scheme for the semi linear fractional partial differential equation with time delay”, Applicable Analysis, 96(11): 1867-1884, (2017).
37. [37] Hammouch, Z., Yavuz, M., Özdemir, N., “Numerical solutions and synchronization of a variable- order fractional chaotic system”, Mathematical Modelling and Numerical Simulation with Applications, 1(1): 11-23, (2021).
38. [38] Haq, I. U., Ali, N., Nisar, K. S., “An optimal control strategy and Grünwald-Letnikov finite- difference numerical scheme for the fractional-order COVID-19 model”, Mathematical Modelling and Numerical Simulation with Applications, 2(2): 108-116, (2022).
39. [39] Sene, N., “Numerical methods applied to a class of SEIR epidemic models described by the Caputo derivative", Methods of Mathematical Modelling, Academic Press, 23-40, (2022).
40. [40] Sene, N., "A Novel Fractional-Order System Described by the Caputo Derivative, Its Numerical Discretization, and Qualitative Properties", Handbook of Fractional Calculus for Engineering and Science, Chapman and Hall/CRC, 205-240, (2022).
41. [41] Sene, N., “Introduction to the fractional-order chaotic system under fractional operator in Caputo sense”, Alexandria Engineering Journal, 60(4): 3997-4014, (2021).
42. [42] Matignon, D., “Stability results for fractional differential equations with applications to control processing”, Computational Engineering in Systems Applications, 2(1), (1996).
43. [43] Naim, M., Sabbar, Y., Zeb, A., “Stability characterization of a fractional-order viral system with the non-cytolytic immune assumption”, Mathematical Modelling and Numerical Simulation with Applications, 2(3): 164-176, (2022).
44. [44] Joshi, H., Jha, B. K., Yavuz, M., “Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data”, Mathematical Biosciences and Engineering, 20(1): 213-240, (2023).
45. [45] Yavuz, M., Sene, N., “Stability analysis and numerical computation of the fractional predator–prey model with the harvesting rate”, Fractal and Fractional, 4(3): 35, (2020).
46. [46] Dimitrov, D.T., Kojouharov, H.V., “Nonstandard numerical methods for a class of predator-prey models with predator interference”, Electronic Journal of Differential Equations, 67-75, (2007).
47. [47] Dimitrov, D.T., Kojouharov, H.V., “Nonstandard finite-difference methods for predator–prey models with general functional response”, Mathematics and Computers in Simulation, 78(1): 1-11, (2008).
48. [48] Petráš, I., Magin, R.L., “Simulation of drug uptake in a two compartmental fractional model for a biological system”, Communications in Nonlinear Science and Numerical Simulation, 16(12): 4588-4595, (2011).
49. [49] Popović, J.K., Atanackovic, M.T., Pilipović, A.S., Rapaić, M. R., Pilipović, S., Atanacković, T. M., “A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac”, Journal of Pharmacokinetics and Pharmacodynamics, 37(2): 119-134, (2010).
50. [50] Bascı, Y., Ogrekci, S., Mısır, A. “Hyers-Ulam-Rassias Stability for Abel-Riccati Type First-Order Differential Equations”, Gazi University Journal of Science, 32(4): 1238-1252, (2019).