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CD4+T Hücrelerindeki HIV Enfeksiyonunun Yayılım Modelinin Chebyshev Operasyonel Matris Metodu ile bir Nümerik Uygulaması

Year 2023, Volume: 13 Issue: 1&2, 59 - 73, 31.12.2023

Yalçın Öztürk , Ayşe Anapalı Şenel , Ceren Limoncu , Mustafa Gülsu

https://doi.org/10.37094/adyujsci.1351204

Abstract

Bu çalışma, CD4+T hücrelerinde HIV virüsünün matematiksel yayılım modeli için yaklaşık çözümler elde etmeyi amaçlamaktadır. Nümerik çözümler Chebyshev polinomları ile operasyonel matris metodunun CD4+T hücrelerinde HIV virüsünün matematiksel yayılım modeline uygulanması ile elde edilecektir. Önerilen method modele ait nümerik çözümlerin bir Chebyshev serisi formunda yazılarak, Chebyhev serisi içindeki bilinmeyen katsayıları içeren lineer olmayan bir denklem sistemi inşa atmeyi amaçlar. Yöntemin doğruluğunu kontrol etmek için nümerik sonuçlar var olan nümerik yöntemlerle karşılaştırılmıştır.

Keywords

HIV Enfenksiyon Modeli CD4+T hücreleri Operasyonel matris metot Chebyshev polinomları Lineer olmayan diferansiyel denklem sistemleri

Project Number

22/139/05/1

References

  • Perelson, A.S., Modelling the interaction of the immune system with H.I.V., in: C. Castillo-Chavez (Ed.), Mathematical and Statistical Approaches to AIDS Epidemiology, Springer, Berlin, 1989.
  • Perelson, A.S., Kirschner, D.E., De Boer, R., Dynamics of H.I.V. infection of CD4+ T- cells, Mathematical Biosciences, 114: 81, 1983.
  • Zapata, H.D., Salazar C.A., Mayorga, E.M., Mathematical model describing HIV infection with time-delayed CD4 T-cell activation, Processes, 8(7), 782, 2020.
  • Arafa, A.A.M., Rida, S.Z., Khalil, M., Fractional modeling dynamics of H.I.V. and CD4+ T-cells during primary infection, Nonlinear Biomed. Phys., 6(1), 1–7, 2012.
  • Huo, H.F., Chen, R., Wang, X.Y., Modelling and stability of HIV/AIDS epidemic model with treatment, Appl. Mathe. Model. 40(13–14), 6550–6559, 2016.
  • Wang, L., Li, M.Y., Mathematical analysis of the global dynamics of a model for H.I.V. infection of CD4+ T cells, Math. Biosci., 200(1), 44–57, 2006.
  • lba-Pérez, J., Macías-Díaz, J.E., A finite-difference discretization preserving the structure of solutions of a diffusive model of type-1 human immunodeficiency virüs, Advances in Difference Equations, (1):1–19, 2021.
  • Crandall, K.A., The Evolution of H.I.V., Johns Hopkins University Press, Baltimore, Maryland, 1999.
  • Culshaw, R.V., Ruan, S., A delay-differential equation model of H.I.V. infection of CD4+ T-cells, Mathematical Biosciences, 165, 27–39, 2000.
  • Wang, X., Song, X., Global stability and periodic solution of a model for H.I.V. infection of CD4+ T-cells, Appl. Math. Model., 189, 1331–1340, 2007.
  • Ghoreishi, M., Ismail, A.I., Alomari, A.K., Application of the homotopy analysis method for solving a model for H.I.V. infection of CD4(+) T-cells, Math. Comp. Model., 54, 3007- 3015, 2011.
  • Ongun, M.Y., The Laplace adomian decomposition method for solving a model for H.I.V. infection of CD4+T cells, Math. Comput. Model., 53(6), 597–603, 2011.
  • Merdan, M., Gökdogan, A., Yıldırım, A., On the numerical solution of the model for H.I.V. infection of CD4+ T cells, Comput. Math. Appl. 62(1), 118-123, 2011.
  • Yüzbaşı, Ş., A numerical approach to solve the model for H.I.V. Infection of CD4+T cells, Appl. Math. Model., 36(12), 2576-5890, 2012.
  • Beler, A., Şimşek, G.Ö., Gümgüm, S., Numerical solutions of the HIV infection model of CD4+ cells by Laguerre wavelets, Math. Comp. Simulat., 209, 205-219, 2023.
  • Mehregan, F.S., Assari, P., Dehghan, M., On the approximate solution of dynamic systems derived from the HIV infection of CD4+ T-cells using the LRBF-collocation scheme, Engin. Anal. Bound. Elem., 153, 39-50, 2023.
  • Parand, K., Kalantari, Z., Delkhosh, D., Quasilinearization-Lagrangian method to solve the H.I.V. infection model of CD4+ T cells, SeMA J., 75(2), 271-283, 2007.
  • Guerrero–Sánchez, Y., Umar, M., Sabir, Z., Juan, L.G., Raja, M.A.Z., Solving a class of biological HIV infection model of latently infected cells using heuristic approach, Discrete and Continuous Dynamical Systems - S, 14(10), 3611-3628, 2021.
  • Yüzbaşı, Ş., Ismailov, N., A numerical method for the solutions of the H.I.V. infection model of CD4+ T-cells, Int. J. Biomath. 10(7), 1750098, 2017.
  • Khan, H., Shah, R., Arif, M., Bushnaq, S., The Chebyshev wavelet method for the numerical solution of fractional H.I.V. infection of CD4+T cells model, Int. J. Appl. Comput. Math., 6, 34, 2020.
  • Merdan, M., Homotopy perturbation method for solving a model for hiv infection of CD4+ T-cells, Istanbul Commerce University Journal of Science, 12, 39–52, 2007.
  • Ali, N., Ahmad, S., Aziz, S., Zaman, G., The Adomian decomposition method for solving HIV infection model of latently infected cells, Matrix Science Mathematic, 3, 5-8, 2019.
  • N. Doğan, Numerical treatment of the model for H.I.V. infection of CD4+T-cells by using multistep Laplace Adomian decomposition method, Discrete Dyna. Mature Soci., 976352, 2012.
  • Khalid, M., Sultana, M., Zaidi, F., Khan, F.S., A numerical solution of a model for H.I.V. infection CD4+ T-cell, Inter. J. Inno. Scie. Rese., 16(1), 79-85, 2015.
  • Gandomani, M.R., Kajani, M.T., Numerical solution of a fractional order model of H.I.V. infection of CD4+T cells using Müntz-Legendre polynomials, Int. J. Bioa., 20(2), 193-204 2016.
  • Ullah, R., Ellahi, R., Said M.S., Din, S.T., On the fractional-order model of HIV infection of CD4+ T*cells under the influence of antiviral drug treatment, Journal of Taibah University for Science, 1, 50-59, 2020.
  • Deniz, S., Semi-analytical approach for solving a model for HIV infection of CD4+ T-cells, TWMS J. Appl. And Eng. Math. 1,273-281, 2021.
  • Kumbinarasaiah, S., Manohara, G., Modified Bernoulli wavelets functional matrix approach for the HIV infection of CD4+ cells model, Results in Control Optim., 10, 100197, 2023.
  • Attaullah, A., Muhammad, S., Mathematical modeling and numerical simulation of HIV infection model, Results in Control Optim., 7, 100118, 2022.
  • Mason, J.C., Handscomb, D.C., Chebyshev polynomials, Chapman and Hall/C.R.C., New York, 2003.
  • Body, J.P., Chebyshev and fourier spectral methods, University of Michigan, New York, 2000.
  • Epperson, J.F., An Introduction to numerical methods and analysis, Second Edition, Jhon Wiley & Sons, New Jersey, 2013.
  • Öztürk, Y., Gülsu, M., An operational matrix method to solve linear Fredholm- Volterra integro-differential equations with piecewise intervals, Math. Scie. 15, 189-197, 2021.
  • Öztürk, Y., Gülsu, M., An operational matrix method for solving Lane-Emden equations arising in astrophysics, Mathematical Methods in Applied Sciences 37, 2227-2235, 2014.
  • Öztürk, Y., An efficient numerical algorithm for solving system of Lane-Emden type equations arising in engineering, Nonlinear Engineering 8, 429-439, 2019.
  • Baleanu, D., Shiri, B., Srivasta H.M., Qurashi, A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel, Advances in Difference Equations, 353, 2017.
  • Doha, E. H., Bhrawy, A. H., Ezz-Eldien, S. S., A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Computers & Mathematics with Applications, 62(5), 2364–2373, 2011.
  • Raslan, K.R., Ali, K.K., Mohammed, E:M:, Younis, J.A., Elsalam, M.A., An operational matrix technique based on Chebyshev polynomias for solving mixed Volterra- Fredholm delay integro differential equations of variable-order, Advan. Frac. Func. Anal., 6203440, 2023.

A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-cells

Year 2023, Volume: 13 Issue: 1&2, 59 - 73, 31.12.2023

Yalçın Öztürk , Ayşe Anapalı Şenel , Ceren Limoncu , Mustafa Gülsu

https://doi.org/10.37094/adyujsci.1351204

Abstract

In this research study, we aim to approximate a solution for the mathematical model of the Human Immunodeficiency Virus (HIV) infection of CD4+T-cells. An operational matrix method based on Chebyshev orthogonal polynomials has been adapted to obtain numerical solutions for the model of HIV infection of CD4+T-cells. The proposed numerical scheme is built on a system of a nonlinear algebraic equation, including coefficients of a finite Chebyshev series that represent the approximate solutions of the model. Results are compared to existing methods to verify the accuracy of the numerical scheme.

Keywords

Model of the HIV infection CD4+T cells Operational matrix method Chebyshev polynomials Nonlinear system of differential equations

Supporting Institution

Muğla Sıtkı Kocman University Research Projects Coordination Office

Project Number

22/139/05/1

Thanks

Muğla Sıtkı Kocman University Research Projects Coordination Office has granted this paper. Project Grant Number 22/139/05/1 and title “Extending the Operational Matrix Method to Nonlinear System Models.”

References

  • Perelson, A.S., Modelling the interaction of the immune system with H.I.V., in: C. Castillo-Chavez (Ed.), Mathematical and Statistical Approaches to AIDS Epidemiology, Springer, Berlin, 1989.
  • Perelson, A.S., Kirschner, D.E., De Boer, R., Dynamics of H.I.V. infection of CD4+ T- cells, Mathematical Biosciences, 114: 81, 1983.
  • Zapata, H.D., Salazar C.A., Mayorga, E.M., Mathematical model describing HIV infection with time-delayed CD4 T-cell activation, Processes, 8(7), 782, 2020.
  • Arafa, A.A.M., Rida, S.Z., Khalil, M., Fractional modeling dynamics of H.I.V. and CD4+ T-cells during primary infection, Nonlinear Biomed. Phys., 6(1), 1–7, 2012.
  • Huo, H.F., Chen, R., Wang, X.Y., Modelling and stability of HIV/AIDS epidemic model with treatment, Appl. Mathe. Model. 40(13–14), 6550–6559, 2016.
  • Wang, L., Li, M.Y., Mathematical analysis of the global dynamics of a model for H.I.V. infection of CD4+ T cells, Math. Biosci., 200(1), 44–57, 2006.
  • lba-Pérez, J., Macías-Díaz, J.E., A finite-difference discretization preserving the structure of solutions of a diffusive model of type-1 human immunodeficiency virüs, Advances in Difference Equations, (1):1–19, 2021.
  • Crandall, K.A., The Evolution of H.I.V., Johns Hopkins University Press, Baltimore, Maryland, 1999.
  • Culshaw, R.V., Ruan, S., A delay-differential equation model of H.I.V. infection of CD4+ T-cells, Mathematical Biosciences, 165, 27–39, 2000.
  • Wang, X., Song, X., Global stability and periodic solution of a model for H.I.V. infection of CD4+ T-cells, Appl. Math. Model., 189, 1331–1340, 2007.
  • Ghoreishi, M., Ismail, A.I., Alomari, A.K., Application of the homotopy analysis method for solving a model for H.I.V. infection of CD4(+) T-cells, Math. Comp. Model., 54, 3007- 3015, 2011.
  • Ongun, M.Y., The Laplace adomian decomposition method for solving a model for H.I.V. infection of CD4+T cells, Math. Comput. Model., 53(6), 597–603, 2011.
  • Merdan, M., Gökdogan, A., Yıldırım, A., On the numerical solution of the model for H.I.V. infection of CD4+ T cells, Comput. Math. Appl. 62(1), 118-123, 2011.
  • Yüzbaşı, Ş., A numerical approach to solve the model for H.I.V. Infection of CD4+T cells, Appl. Math. Model., 36(12), 2576-5890, 2012.
  • Beler, A., Şimşek, G.Ö., Gümgüm, S., Numerical solutions of the HIV infection model of CD4+ cells by Laguerre wavelets, Math. Comp. Simulat., 209, 205-219, 2023.
  • Mehregan, F.S., Assari, P., Dehghan, M., On the approximate solution of dynamic systems derived from the HIV infection of CD4+ T-cells using the LRBF-collocation scheme, Engin. Anal. Bound. Elem., 153, 39-50, 2023.
  • Parand, K., Kalantari, Z., Delkhosh, D., Quasilinearization-Lagrangian method to solve the H.I.V. infection model of CD4+ T cells, SeMA J., 75(2), 271-283, 2007.
  • Guerrero–Sánchez, Y., Umar, M., Sabir, Z., Juan, L.G., Raja, M.A.Z., Solving a class of biological HIV infection model of latently infected cells using heuristic approach, Discrete and Continuous Dynamical Systems - S, 14(10), 3611-3628, 2021.
  • Yüzbaşı, Ş., Ismailov, N., A numerical method for the solutions of the H.I.V. infection model of CD4+ T-cells, Int. J. Biomath. 10(7), 1750098, 2017.
  • Khan, H., Shah, R., Arif, M., Bushnaq, S., The Chebyshev wavelet method for the numerical solution of fractional H.I.V. infection of CD4+T cells model, Int. J. Appl. Comput. Math., 6, 34, 2020.
  • Merdan, M., Homotopy perturbation method for solving a model for hiv infection of CD4+ T-cells, Istanbul Commerce University Journal of Science, 12, 39–52, 2007.
  • Ali, N., Ahmad, S., Aziz, S., Zaman, G., The Adomian decomposition method for solving HIV infection model of latently infected cells, Matrix Science Mathematic, 3, 5-8, 2019.
  • N. Doğan, Numerical treatment of the model for H.I.V. infection of CD4+T-cells by using multistep Laplace Adomian decomposition method, Discrete Dyna. Mature Soci., 976352, 2012.
  • Khalid, M., Sultana, M., Zaidi, F., Khan, F.S., A numerical solution of a model for H.I.V. infection CD4+ T-cell, Inter. J. Inno. Scie. Rese., 16(1), 79-85, 2015.
  • Gandomani, M.R., Kajani, M.T., Numerical solution of a fractional order model of H.I.V. infection of CD4+T cells using Müntz-Legendre polynomials, Int. J. Bioa., 20(2), 193-204 2016.
  • Ullah, R., Ellahi, R., Said M.S., Din, S.T., On the fractional-order model of HIV infection of CD4+ T*cells under the influence of antiviral drug treatment, Journal of Taibah University for Science, 1, 50-59, 2020.
  • Deniz, S., Semi-analytical approach for solving a model for HIV infection of CD4+ T-cells, TWMS J. Appl. And Eng. Math. 1,273-281, 2021.
  • Kumbinarasaiah, S., Manohara, G., Modified Bernoulli wavelets functional matrix approach for the HIV infection of CD4+ cells model, Results in Control Optim., 10, 100197, 2023.
  • Attaullah, A., Muhammad, S., Mathematical modeling and numerical simulation of HIV infection model, Results in Control Optim., 7, 100118, 2022.
  • Mason, J.C., Handscomb, D.C., Chebyshev polynomials, Chapman and Hall/C.R.C., New York, 2003.
  • Body, J.P., Chebyshev and fourier spectral methods, University of Michigan, New York, 2000.
  • Epperson, J.F., An Introduction to numerical methods and analysis, Second Edition, Jhon Wiley & Sons, New Jersey, 2013.
  • Öztürk, Y., Gülsu, M., An operational matrix method to solve linear Fredholm- Volterra integro-differential equations with piecewise intervals, Math. Scie. 15, 189-197, 2021.
  • Öztürk, Y., Gülsu, M., An operational matrix method for solving Lane-Emden equations arising in astrophysics, Mathematical Methods in Applied Sciences 37, 2227-2235, 2014.
  • Öztürk, Y., An efficient numerical algorithm for solving system of Lane-Emden type equations arising in engineering, Nonlinear Engineering 8, 429-439, 2019.
  • Baleanu, D., Shiri, B., Srivasta H.M., Qurashi, A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel, Advances in Difference Equations, 353, 2017.
  • Doha, E. H., Bhrawy, A. H., Ezz-Eldien, S. S., A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Computers & Mathematics with Applications, 62(5), 2364–2373, 2011.
  • Raslan, K.R., Ali, K.K., Mohammed, E:M:, Younis, J.A., Elsalam, M.A., An operational matrix technique based on Chebyshev polynomias for solving mixed Volterra- Fredholm delay integro differential equations of variable-order, Advan. Frac. Func. Anal., 6203440, 2023.
There are 38 citations in total.

Details

Primary Language English
Subjects Mathematical Methods and Special Functions
Journal Section Mathematics
Authors

Yalçın Öztürk 0000-0002-4142-5633

Ayşe Anapalı Şenel 0000-0002-0198-1862

Ceren Limoncu 0000-0002-8396-5945

Mustafa Gülsu 0000-0001-6139-0266

Project Number 22/139/05/1
Publication Date December 31, 2023
Submission Date August 28, 2023
Acceptance Date October 20, 2023
Published in Issue Year 2023 Volume: 13 Issue: 1&2

Cite

APA Öztürk, Y., Anapalı Şenel, A., Limoncu, C., Gülsu, M. (2023). A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-cells. Adıyaman University Journal of Science, 13(1&2), 59-73. https://doi.org/10.37094/adyujsci.1351204
AMA Öztürk Y, Anapalı Şenel A, Limoncu C, Gülsu M. A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-cells. ADYU J SCI. December 2023;13(1&2):59-73. doi:10.37094/adyujsci.1351204
Chicago Öztürk, Yalçın, Ayşe Anapalı Şenel, Ceren Limoncu, and Mustafa Gülsu. “A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-Cells”. Adıyaman University Journal of Science 13, no. 1&2 (December 2023): 59-73. https://doi.org/10.37094/adyujsci.1351204.
EndNote Öztürk Y, Anapalı Şenel A, Limoncu C, Gülsu M (December 1, 2023) A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-cells. Adıyaman University Journal of Science 13 1&2 59–73.
IEEE Y. Öztürk, A. Anapalı Şenel, C. Limoncu, and M. Gülsu, “A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-cells”, ADYU J SCI, vol. 13, no. 1 & 2, pp. 59–73, 2023, doi: 10.37094/adyujsci.1351204.
ISNAD Öztürk, Yalçın et al. “A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-Cells”. Adıyaman University Journal of Science 13/1 & 2 (December 2023), 59-73. https://doi.org/10.37094/adyujsci.1351204.
JAMA Öztürk Y, Anapalı Şenel A, Limoncu C, Gülsu M. A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-cells. ADYU J SCI. 2023;13:59–73.
MLA Öztürk, Yalçın et al. “A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-Cells”. Adıyaman University Journal of Science, vol. 13, no. 1&2, 2023, pp. 59-73, doi:10.37094/adyujsci.1351204.
Vancouver Öztürk Y, Anapalı Şenel A, Limoncu C, Gülsu M. A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-cells. ADYU J SCI. 2023;13(1&2):59-73.
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