Research Article
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An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications

Year 2023, Issue: 45, 105 - 119, 31.12.2023

İlkem Turhan Çetinkaya

https://doi.org/10.53570/jnt.1391403

Abstract

In this study, a mathematical model describing diabetes mellitus and its complications in a population is considered. Since standard numerical methods can lead to numerical instabilities, it aims to solve the problem using a nonstandard method. Among the nonstandard methods, nonstandard finite difference (NSFD) schemes that satisfy dynamical consistency are preferred to make the model discrete. Both continuous and discrete models are analyzed to show the stability of the model at the equilibrium points. The Schur-Cohn criterion is used to perform stability analysis at the equilibrium point of the discretized model. Thus, asymptotically stability of the model is presented. Moreover, the advantages of the NSFD method are emphasized by comparing the stability for different step sizes with classical methods, such as Euler and Runge-Kutta. It has been observed that the NSFD method is convergence for larger step sizes. In addition, the numerical results obtained by NSFD schemes are compared with the Runge–Kutta–Fehlberg (RKF45) method in graphical forms. The accuracy of the NSFD method is observed.

Keywords

Diabetes mellitus nonstandard finite difference scheme stability analysis

References

  • A. Boutayeb, E. H. Twizell, K. Achouayb, A. Chetouani, A Mathematical Model for the Burden of Diabetes and Its Complications, Biomedical Engineering Online 3 (2004) 1--8.
  • V. O. Akinsola, T. O. Oluyo, Analytic Solution of Mathematical Model of the Complications and Control of Diabetes Mellitus Using Fundamental Matrix Method, Journal of Interdisciplinary Mathematics 23 (4) (2020) 877--884.
  • V. O. Akinsola, T. O. Oluyo, A Note on the Divergence of the Numerical Solution of the Mathematical Model for the Burden of Diabetes and Its Complications Using Euler Method, International Journal of Mathematics and Computer Applications Research 5 (3) (2015) 93--100.
  • V. O. Akinsola, T. O. Oluyo, Mathematical Analysis with Numerical Solutions of the Mathematical Model for the Complications and Control of Diabetes Mellitus, Journal of Statistics and Management Systems 22 (5) (2019) 845--869.
  • M. AlShurbaji, L.A. Kader, H. Hannan, M. Mortula, G. A. Husseini, Comprehensive Study of a Diabetes Mellitus Mathematical Model Using Numerical Methods with Stability and Parametric Analysis, International Journal of Environmental Research and Public Health 20 (2) (2023) 939 23 pages.
  • R. Vanitha, R. Porchelvi, A Linear Population Model for Diabetes Mellitus, Bulletin of Pure and Applied Sciences-Mathematics and Statistics 36 (2) (2017) 311--315.
  • A. Boutayeb, A. Chetouani, A. Achouyab, E. H. Twizell, A Nonlinear Population Model of Diabetes Mellitus, Journal of Applied Mathematics and Computing 21 (2006) 127--139.
  • S. R. de Oliveira, S. Raha, D. Pal, Global Asymptotic Stability of a Nonlinear Population Model of Diabetes Mellitus, in: S. Pinelas, T. Caraballo, P. Kloeden, J. Graef (Eds.), Differential and Difference Equations with Applications: ICDDEA 2017, Springer Proceedings in Mathematics and Statistics, Vol 230. Springer, Cham, 2018, pp. 351--357.
  • W. Boutayeb, M. E. N. Lamlili, A. Boutayeb, M. Derouich, The Dynamics of a Population of Healthy People, Pre-diabetics and Diabetics with and without Complications with Optimal Control, in: A. El Oualkadi, F. Choubani, A. El Moussati (Eds.), Proceedings of the Mediterranean Conference on Information and Communication Technologies, Vol. 380 of Lecture Notes in Electrical Engineering, Springer, Cham, 2016, pp. 463--471.
  • A. H. Permatasari, R. H. Tjahjana, T. Udjiani. Global Stability for Linear System and Controllability for Nonlinear System in the Dynamics Model of Diabetics Population, Journal of Physics: Conference Series, Vol. 1025 (1), IOP Publishing, 2018.
  • P. Widyaningsih, R. C. Affan, D. R. S. Saputro, A Mathematical Model for the Epidemiology of Diabetes Mellitus with Lifestyle and Genetic Factors, Journal of Physics: Conference series, Vol. 1028 (1) IOP Publishing, 2018.
  • P. O. Aye, Stability Analysis of Mathematical Model for the Dynamics of Diabetes Mellitus and Its Complications in a Population, Data Analytics and Applied Mathematics 3 (1) (2022) 20--27.
  • P. O. Aye, K. A. Adeyemo, A. S. Oke, A. E. Omotoye, Analysis of Mathematical Model for the Dynamics of Diabetes Mellitus and Its Complications, Applied Mathematics and Computational Intelligence, 10 (1) (2021) 57--77.
  • P. O. Aye, Mathematical Analysis of the Effect of Control on the Dynamics of Diabetes Mellitus and Its Complications, Earthline Journal of Mathematical Sciences 6 (2) (2021) 375--395.
  • R. E. Mickens, Difference Equations Theory and Applications, Chapman and Hall, Atlanta, 1990.
  • R. E. Mickens, Advances in the Applications of Nonstandard Finite Difference Schemes, Wiley-Interscience, Singapore, 2005.
  • R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific Publishing Company, New Jersey, 1993.
  • R. E. Mickens, Nonstandard Finite Difference Schemes for Differential Equations, Journal of Difference Equations and Applications 8 (9) (2002) 823--847.
  • R. E. Mickens, Calculation of Denominator Functions for Nonstandard Finite Difference Schemes for Differential Equations Satisfying a Positivity Condition, Numerical Methods for Partial Differential Equations 23 (3) (2006) 672--691.
  • R. E. Mickens, Exact Solutions to a Fnite-Difference Model of a Nonlinear Reaction-Advection Equation: Implications for Numerical Analysis, Numerical Methods for Partial Differential Equations 5 (4) (1989) 313--325.
  • K. C. Patidar, On the Use of Nonstandard Finite Difference Methods, Journal of Difference Equations and Applications 11 (8) (2005) 735--758.
  • K. C. Patidar, Nonstandard Finite Difference Methods: Recent Trends and Further Developments, Journal of Difference Equations and Applications 22 (6) (2016) 817--849.
  • O. Adekanye, T. Washington, Nonstandard Finite Difference Scheme for a Tacoma Narrows Bridge Model, Applied Mathematical Modelling 62 (2018) 223--236.
  • R. Anguelov, T. Berge, M. Chapwanya, J. K. Djoko, P. Kama, J. S. Lubuma, Y. Terefe, Nonstandard Finite Difference Method Revisited and Application to the Ebola Virus Disease Transmission Dynamics, Journal of Difference Equations and Applications 26 (6) (2020) 818--854.
  • A. J. Arenas, G. Gonzalez-Parra, B. M. Chen-Charpentier, Construction of Nonstandard Finite Difference Schemes for the SI and SIR Epidemic Models of Fractional Order, Mathematics and Computers in Simulation 121 (2016) 48--63.
  • D. Baleanu, S. Zibaei, M. Namjoo, A. Jajarmi, A Nonstandard Finite Difference Scheme for the Modeling and Nonidentical Synchronization of a Novel Fractional Chaotic System, Advances in Difference Equations 2021 (1) (2021) Article Number 308 19 pages.
  • Q. A. Dang, M. T. Hoang, Lyapunov Direct Method for Investigating Stability of Nonstandard Finite Difference Schemes for Metapopulation models, Journal of Difference Equations and Applications 24 (1) (2018) 15--47.
  • Q. A. Dang, M. T. Hoang, Numerical Dynamics of Nonstandard Finite Difference Schemes for a Computer Virus Propagation Model, International Journal of Dynamics and Control 8 (3) (2020) 772--778.
  • M. Kocabıyık, N. Özdoğan, M. Y. Ongun, Nonstandard Finite Difference Scheme for a Computer Virus Model, Journal of Innovative Science and Engineering 4 (2) (2020) 96--108.
  • N. Özdoğan, M. Y. Ongun, Dynamical Behaviours of a Discretized Model with Michaelis-Menten Harvesting Rate, Journal of Universal Mathematics 5 (2) (2022) 159--176.
  • M. Kocabıyık, M. Y. Ongun, Construction a Distributed Order Smoking Model and Its Nonstandard Finite Difference Discretization, AIMS Mathematics 7 (3) (2021) 4636–-4654.
  • M. Y. Ongun, D. Arslan, Explicit and Implicit Schemes for Fractional–order Hantavirus Model, Iranian Journal of Numerical Analysis and Optimization 8 (2) (2018) 75--94.
  • M. S. Shabbir, Q. Din, M. Safeer, M. A. Khan, K. Ahmad, A Dynamically Consistent Nonstandard Finite Difference Scheme for a Predator–prey Model, Advances in Difference Equations 2019 (1) (2019) 1--17.
  • S. Vaz, D. F. Torres, A Dynamically-consistent Nonstandard Finite Difference Scheme for the SICA Model, Mathematical Biosciences and Engineering 18 (4) (2021) 4552--4571.
  • O. Egbelowo, C. Harley, B. Jacobs, Nonstandard Finite Difference Method Applied to a Linear Pharmacokinetics Model, Bioengineering 4 (2) (2017) 40 21 pages.
  • I. U. Khan, S. Mustafa, A. Shokri, S. Li, A. Akgül, A. Bariq, The Stability Analysis of a Nonlinear Mathematical Model for Typhoid Fever Disease, Scientific Reports 13 (1) (2023) Article Number 15284 15 pages.
  • F. Özköse, M. Yavuz, Investigation of Interactions between COVID-19 and Diabetes with Hereditary Traits Using Real Data: A Case Study in Turkey, Computers in Biology and Medicine 141 (2022) 105044 22 pages.
  • A. Atangana, S. İğret Araz, Mathematical Model of COVID-19 Spread in Turkey and South Africa: Theory, Methods, and Applications, Advances in Difference Equations 2020 (1) (2020) 1--89.
  • Ö. A. Gümüs, Q. Cui, G. M. Selvam, A. Vianny, Global Stability and Bifurcation Analysis of a Discrete Time SIR Epidemic Model, Miskolc Mathematical Notes 23 (1) (2022) 193--210.
  • S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 2003.
  • S. Elaydi, An Introduction to Difference Equations, Springer, New York, 1999.

Year 2023, Issue: 45, 105 - 119, 31.12.2023

İlkem Turhan Çetinkaya

https://doi.org/10.53570/jnt.1391403

Abstract

References

  • A. Boutayeb, E. H. Twizell, K. Achouayb, A. Chetouani, A Mathematical Model for the Burden of Diabetes and Its Complications, Biomedical Engineering Online 3 (2004) 1--8.
  • V. O. Akinsola, T. O. Oluyo, Analytic Solution of Mathematical Model of the Complications and Control of Diabetes Mellitus Using Fundamental Matrix Method, Journal of Interdisciplinary Mathematics 23 (4) (2020) 877--884.
  • V. O. Akinsola, T. O. Oluyo, A Note on the Divergence of the Numerical Solution of the Mathematical Model for the Burden of Diabetes and Its Complications Using Euler Method, International Journal of Mathematics and Computer Applications Research 5 (3) (2015) 93--100.
  • V. O. Akinsola, T. O. Oluyo, Mathematical Analysis with Numerical Solutions of the Mathematical Model for the Complications and Control of Diabetes Mellitus, Journal of Statistics and Management Systems 22 (5) (2019) 845--869.
  • M. AlShurbaji, L.A. Kader, H. Hannan, M. Mortula, G. A. Husseini, Comprehensive Study of a Diabetes Mellitus Mathematical Model Using Numerical Methods with Stability and Parametric Analysis, International Journal of Environmental Research and Public Health 20 (2) (2023) 939 23 pages.
  • R. Vanitha, R. Porchelvi, A Linear Population Model for Diabetes Mellitus, Bulletin of Pure and Applied Sciences-Mathematics and Statistics 36 (2) (2017) 311--315.
  • A. Boutayeb, A. Chetouani, A. Achouyab, E. H. Twizell, A Nonlinear Population Model of Diabetes Mellitus, Journal of Applied Mathematics and Computing 21 (2006) 127--139.
  • S. R. de Oliveira, S. Raha, D. Pal, Global Asymptotic Stability of a Nonlinear Population Model of Diabetes Mellitus, in: S. Pinelas, T. Caraballo, P. Kloeden, J. Graef (Eds.), Differential and Difference Equations with Applications: ICDDEA 2017, Springer Proceedings in Mathematics and Statistics, Vol 230. Springer, Cham, 2018, pp. 351--357.
  • W. Boutayeb, M. E. N. Lamlili, A. Boutayeb, M. Derouich, The Dynamics of a Population of Healthy People, Pre-diabetics and Diabetics with and without Complications with Optimal Control, in: A. El Oualkadi, F. Choubani, A. El Moussati (Eds.), Proceedings of the Mediterranean Conference on Information and Communication Technologies, Vol. 380 of Lecture Notes in Electrical Engineering, Springer, Cham, 2016, pp. 463--471.
  • A. H. Permatasari, R. H. Tjahjana, T. Udjiani. Global Stability for Linear System and Controllability for Nonlinear System in the Dynamics Model of Diabetics Population, Journal of Physics: Conference Series, Vol. 1025 (1), IOP Publishing, 2018.
  • P. Widyaningsih, R. C. Affan, D. R. S. Saputro, A Mathematical Model for the Epidemiology of Diabetes Mellitus with Lifestyle and Genetic Factors, Journal of Physics: Conference series, Vol. 1028 (1) IOP Publishing, 2018.
  • P. O. Aye, Stability Analysis of Mathematical Model for the Dynamics of Diabetes Mellitus and Its Complications in a Population, Data Analytics and Applied Mathematics 3 (1) (2022) 20--27.
  • P. O. Aye, K. A. Adeyemo, A. S. Oke, A. E. Omotoye, Analysis of Mathematical Model for the Dynamics of Diabetes Mellitus and Its Complications, Applied Mathematics and Computational Intelligence, 10 (1) (2021) 57--77.
  • P. O. Aye, Mathematical Analysis of the Effect of Control on the Dynamics of Diabetes Mellitus and Its Complications, Earthline Journal of Mathematical Sciences 6 (2) (2021) 375--395.
  • R. E. Mickens, Difference Equations Theory and Applications, Chapman and Hall, Atlanta, 1990.
  • R. E. Mickens, Advances in the Applications of Nonstandard Finite Difference Schemes, Wiley-Interscience, Singapore, 2005.
  • R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific Publishing Company, New Jersey, 1993.
  • R. E. Mickens, Nonstandard Finite Difference Schemes for Differential Equations, Journal of Difference Equations and Applications 8 (9) (2002) 823--847.
  • R. E. Mickens, Calculation of Denominator Functions for Nonstandard Finite Difference Schemes for Differential Equations Satisfying a Positivity Condition, Numerical Methods for Partial Differential Equations 23 (3) (2006) 672--691.
  • R. E. Mickens, Exact Solutions to a Fnite-Difference Model of a Nonlinear Reaction-Advection Equation: Implications for Numerical Analysis, Numerical Methods for Partial Differential Equations 5 (4) (1989) 313--325.
  • K. C. Patidar, On the Use of Nonstandard Finite Difference Methods, Journal of Difference Equations and Applications 11 (8) (2005) 735--758.
  • K. C. Patidar, Nonstandard Finite Difference Methods: Recent Trends and Further Developments, Journal of Difference Equations and Applications 22 (6) (2016) 817--849.
  • O. Adekanye, T. Washington, Nonstandard Finite Difference Scheme for a Tacoma Narrows Bridge Model, Applied Mathematical Modelling 62 (2018) 223--236.
  • R. Anguelov, T. Berge, M. Chapwanya, J. K. Djoko, P. Kama, J. S. Lubuma, Y. Terefe, Nonstandard Finite Difference Method Revisited and Application to the Ebola Virus Disease Transmission Dynamics, Journal of Difference Equations and Applications 26 (6) (2020) 818--854.
  • A. J. Arenas, G. Gonzalez-Parra, B. M. Chen-Charpentier, Construction of Nonstandard Finite Difference Schemes for the SI and SIR Epidemic Models of Fractional Order, Mathematics and Computers in Simulation 121 (2016) 48--63.
  • D. Baleanu, S. Zibaei, M. Namjoo, A. Jajarmi, A Nonstandard Finite Difference Scheme for the Modeling and Nonidentical Synchronization of a Novel Fractional Chaotic System, Advances in Difference Equations 2021 (1) (2021) Article Number 308 19 pages.
  • Q. A. Dang, M. T. Hoang, Lyapunov Direct Method for Investigating Stability of Nonstandard Finite Difference Schemes for Metapopulation models, Journal of Difference Equations and Applications 24 (1) (2018) 15--47.
  • Q. A. Dang, M. T. Hoang, Numerical Dynamics of Nonstandard Finite Difference Schemes for a Computer Virus Propagation Model, International Journal of Dynamics and Control 8 (3) (2020) 772--778.
  • M. Kocabıyık, N. Özdoğan, M. Y. Ongun, Nonstandard Finite Difference Scheme for a Computer Virus Model, Journal of Innovative Science and Engineering 4 (2) (2020) 96--108.
  • N. Özdoğan, M. Y. Ongun, Dynamical Behaviours of a Discretized Model with Michaelis-Menten Harvesting Rate, Journal of Universal Mathematics 5 (2) (2022) 159--176.
  • M. Kocabıyık, M. Y. Ongun, Construction a Distributed Order Smoking Model and Its Nonstandard Finite Difference Discretization, AIMS Mathematics 7 (3) (2021) 4636–-4654.
  • M. Y. Ongun, D. Arslan, Explicit and Implicit Schemes for Fractional–order Hantavirus Model, Iranian Journal of Numerical Analysis and Optimization 8 (2) (2018) 75--94.
  • M. S. Shabbir, Q. Din, M. Safeer, M. A. Khan, K. Ahmad, A Dynamically Consistent Nonstandard Finite Difference Scheme for a Predator–prey Model, Advances in Difference Equations 2019 (1) (2019) 1--17.
  • S. Vaz, D. F. Torres, A Dynamically-consistent Nonstandard Finite Difference Scheme for the SICA Model, Mathematical Biosciences and Engineering 18 (4) (2021) 4552--4571.
  • O. Egbelowo, C. Harley, B. Jacobs, Nonstandard Finite Difference Method Applied to a Linear Pharmacokinetics Model, Bioengineering 4 (2) (2017) 40 21 pages.
  • I. U. Khan, S. Mustafa, A. Shokri, S. Li, A. Akgül, A. Bariq, The Stability Analysis of a Nonlinear Mathematical Model for Typhoid Fever Disease, Scientific Reports 13 (1) (2023) Article Number 15284 15 pages.
  • F. Özköse, M. Yavuz, Investigation of Interactions between COVID-19 and Diabetes with Hereditary Traits Using Real Data: A Case Study in Turkey, Computers in Biology and Medicine 141 (2022) 105044 22 pages.
  • A. Atangana, S. İğret Araz, Mathematical Model of COVID-19 Spread in Turkey and South Africa: Theory, Methods, and Applications, Advances in Difference Equations 2020 (1) (2020) 1--89.
  • Ö. A. Gümüs, Q. Cui, G. M. Selvam, A. Vianny, Global Stability and Bifurcation Analysis of a Discrete Time SIR Epidemic Model, Miskolc Mathematical Notes 23 (1) (2022) 193--210.
  • S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 2003.
  • S. Elaydi, An Introduction to Difference Equations, Springer, New York, 1999.
There are 41 citations in total.

Details

Primary Language English
Subjects Numerical Solution of Differential and Integral Equations, Numerical Analysis, Dynamical Systems in Applications
Journal Section Research Article
Authors

İlkem Turhan Çetinkaya 0000-0002-5520-310X

Early Pub Date December 30, 2023
Publication Date December 31, 2023
Submission Date November 15, 2023
Acceptance Date December 27, 2023
Published in Issue Year 2023 Issue: 45

Cite

APA Turhan Çetinkaya, İ. (2023). An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications. Journal of New Theory(45), 105-119. https://doi.org/10.53570/jnt.1391403
AMA Turhan Çetinkaya İ. An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications. JNT. December 2023;(45):105-119. doi:10.53570/jnt.1391403
Chicago Turhan Çetinkaya, İlkem. “An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications”. Journal of New Theory, no. 45 (December 2023): 105-19. https://doi.org/10.53570/jnt.1391403.
EndNote Turhan Çetinkaya İ (December 1, 2023) An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications. Journal of New Theory 45 105–119.
IEEE İ. Turhan Çetinkaya, “An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications”, JNT, no. 45, pp. 105–119, December 2023, doi: 10.53570/jnt.1391403.
ISNAD Turhan Çetinkaya, İlkem. “An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications”. Journal of New Theory 45 (December 2023), 105-119. https://doi.org/10.53570/jnt.1391403.
JAMA Turhan Çetinkaya İ. An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications. JNT. 2023;:105–119.
MLA Turhan Çetinkaya, İlkem. “An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications”. Journal of New Theory, no. 45, 2023, pp. 105-19, doi:10.53570/jnt.1391403.
Vancouver Turhan Çetinkaya İ. An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications. JNT. 2023(45):105-19.
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