Skip to main content

Advertisement

SpringerLink
Log in

An Optimization Method Based on the Generalized Polynomials for a Model of HIV Infection of \(\hbox {CD4}^{+}\) T Cells

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

In this paper, an optimization method based on the generalized polynomials (GP) including the unknown free coefficients and control parameters has been proposed to approximate the solution of a model of HIV infection of \(\hbox {CD4}^{+}\) T cells (HIV-I-CD4T). First, the operational matrices (OM) of derivatives are derived. Then, based on these OM and the Lagrange multipliers method, an optimization method is presented to approximate solution of a model of HIV-I-CD4T. An illustrative example is given to demonstrate the efficiency and accuracy of the proposed method and confirms that results are in good accuracy in comparisons with other numerical approaches.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Abbreviations

T(t):

The concentration of healthy CD4T at time t

I(t):

The concentration of I-CD4T at time t

V(t):

The concentration of free HIV at time t

s :

The source of CD4T from the precursors

\(\alpha \) :

The natural death rate of CD4T

r :

Growth rate of CD4T population

\(T_{\mathrm{max}}\) :

The maximal population level of CD4T

\(k^{*}\) :

The rate of I-CD4T with free virus present in the environment and hence is a plus term for I-CD4T

\(\beta \) :

The overall death rate for I-CD4T

N :

Virus particles are released by each dying (infected) CD4T

\(\gamma \) :

The death rate of viruses

References

  • Abdel-Halim Hassan IH (2008) Application to differential transformation method for solving systems of differential equations. Appl Math Model 32:2552–2559

    MathSciNet  MATH  Google Scholar 

  • Ablinger J, Blümlein J, Marquard P, Rana N, Schneider C (2019) Automated solution of first order factorizable systems of differential equations in one variable. Nucl Phys B 939:253–291

    MathSciNet  MATH  Google Scholar 

  • Alliera CHD, Amster P (2018) Systems of delay differential equations: analysis of a model with feedback. Commun Nonlinear Sci 65:299–308

    MathSciNet  MATH  Google Scholar 

  • Arafa AAM, Rida SZ, Khalil M (2012) Fractional modeling dynamics of HIV and \(\text{ CD4 }^+\) T-cells during primary infection. Nonlinear Biomed Phys 6(1):1–7

    Google Scholar 

  • Arafa AAM, Rida SZ, Khalil M (2014) A fractional-order model of HIV infection: numerical solution and comparisons with data of patients. Int J Biomath 7(4):1450036

    MathSciNet  MATH  Google Scholar 

  • Avazzadeh Z, Hassani H (2019) Transcendental Bernstein series for solving reaction-diffusion equations with nonlocal boundary conditions through the optimization technique. Numer Methods. Partial Differ Equ 35(6):2258–2274

    MathSciNet  MATH  Google Scholar 

  • Ball JM (1983) Systems of nonlinear partial differential equations. Springer, Berlin

    MATH  Google Scholar 

  • Dahaghin MS, Hassani H (2017a) A new optimization method for a class of time fractional convection-diffusion-wave equations with variable coefficients. Eur Phys J Plus 132:130

    Google Scholar 

  • Dahaghin MS, Hassani H (2017b) An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation. Nonlinear Dyn 88(3):1587–1598

    MathSciNet  MATH  Google Scholar 

  • Feng TF, Chang CH, Chen JB, Zhang HB (2019) The system of partial differential equations for the \(c_0\) function. Nucl Phys B 940:130–189

    MATH  Google Scholar 

  • Gimena L, Gonzaga P, Gimena FN (2014) Boundary equations in the finite transfer method for solving differential equation systems. Appl Math Model 38:2648–2660

    MathSciNet  MATH  Google Scholar 

  • Hassani H, Avazzadeh Z (2019) Transcendental Bernstein series for solving nonlinear variable order fractional optimal control problems. Appl Math Comput. https://doi.org/10.1016/j.amc.2019.124563

    Article  MathSciNet  MATH  Google Scholar 

  • Hassani H, Tenreiro Machado JA, Naraghirad E (2019a) Generalized shifted Chebyshev polynomials for fractional optimal control problems. Commun Nonlinear Sci 75:50–61

    MathSciNet  Google Scholar 

  • Hassani H, Avazzadeh Z, Tenreiro Machado JA (2019b) Solving two-dimensional variable-order fractional optimal control problems with transcendental Bernstein series. J Comput Nonlinear Dyn 14(6):061001

    Google Scholar 

  • Heydari MH (2019) A direct method based on the Chebyshev polynomials for a new class of nonlinear variable-order fractional 2D optimal control problems. J Frankl Inst 365(15):8216–8236

    MathSciNet  MATH  Google Scholar 

  • Heydari MH, Hooshmandasl MR, Maalek Ghaini FM (2014) An efficient computational method for solving fractional biharmonic equation. Comput Math Appl 68(3):269–287

    MathSciNet  MATH  Google Scholar 

  • Heydari MH, Hooshmandasl MR, Shakiba A, Cattani C (2016) Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations. Nonlinear Dyn 85(2):1185–1202

    MathSciNet  MATH  Google Scholar 

  • Heydari MH, Avazzadeh Z, Yang Y (2019) A computational method for solving variable-order fractional nonlinear diffusion-wave equation. Appl Math Comput 352:235–248

    MathSciNet  MATH  Google Scholar 

  • Hooshmandasl MR, Heydari MH, Cattani C (2016) Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions. Eur Phys J Plus 131:268

    Google Scholar 

  • Hosseininia M, Heydari MH, Maalek Ghaini FM, Avazzadeh Z (2019) A wavelet method to solve nonlinear variable-order time fractional 2D Klein–Gordon equation. Comput Math Appl 78(12):3713–3730

    MathSciNet  Google Scholar 

  • Jaros J, Kusano T (2014) On strongly monotone solutions of a class of cyclic systems of nonlinear differential equations. J Math Anal Appl 417:996–1017

    MathSciNet  MATH  Google Scholar 

  • Khalid M, Sultana M, Zaidi F, Fareeha SK (2015) A numerical solution of a model for HIV infection \(\text{ CD4 }^+\) T-Cell”. Int J Innov Sci Res 16(1):79–85

    Google Scholar 

  • Kreyszig E (1978) Introductory functional analysis with applications. Wiley, New York

    MATH  Google Scholar 

  • Leon Pritchard F (2003) On implicit systems of differential equations. J Differ Equ 194:328–363

    MathSciNet  MATH  Google Scholar 

  • Liu Z, Lu P (2014) Stability analysis for HIV infection of \(\text{ CD4 }^+\) T-cells by a fractional differential time-delay model with cure rate. Adv Differ Equ 2014(298):1–20

    MathSciNet  Google Scholar 

  • Luo J, Wang W, Chen H, Fu R (2016) Bifurcations of a mathematical model for HIV dynamics. J Math Anal Appl 434(1):837–857

    MathSciNet  MATH  Google Scholar 

  • Merdan M (2007) Homotopy perturbation method for solving a model for HIV infection of \(\text{ CD4 }^+\) T-cells. Istanb Commer Univ J Sci 12:39–52

    Google Scholar 

  • Merdan M, Gökdogan A, Yildirim A (2011) On the numerical solution of the model for HIV infection of \(\text{ CD4 }^+\) T-cells. Comput Math Appl 62(1):118–123

    MathSciNet  MATH  Google Scholar 

  • Mirzaee F, Samadyar N (2019) On the numerical method for solving a system of nonlinear fractional ordinary differential equations arising in HIV infection of \(\text{ CD4 }^+\) T cells. Iran J Sci Technol Trans Sci 43(3):1127–1138

    MathSciNet  Google Scholar 

  • Mohammadi F, Cattani C (2018) A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations. J Comput Appl Math 339:306–316

    MathSciNet  MATH  Google Scholar 

  • Mohammadi F, Hassani H (2019) Numerical solution of two-dimensional variable-order fractional optimal control problem by generalized polynomial basis. J Optim Theory Appl 180(2):536–555

    MathSciNet  MATH  Google Scholar 

  • Mojaver A, Kheiri H (2015) Mathematical analysis of a class of HIV infection models of \(\text{ CD4 }^+\) T-cells with combined antiretroviral therapy. Appl Math Comput 259:258–270

    MathSciNet  MATH  Google Scholar 

  • Ongun MY (2011) The Laplace adomian decomposition method for solving a model for HIV infection of \(\text{ CD4 }^+\) T-cells. Math Comput Model 53:597–603

    MathSciNet  MATH  Google Scholar 

  • Perelson AS (1989) Modeling the interaction of the immune system with HIV. In: Mathematical and Statistical Approaches to AIDS Epidemiology. Springer, Berlin pp 350–370

  • Pfeiffer BM, Marquardt W (1996) Symbolic semi-discretization of partial differential equation systems. Math Comput Simul 42:617–628

    MathSciNet  MATH  Google Scholar 

  • Rahimkhani P, Ordokhani Y, Lima PM (2019) An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets. Appl Numer Math 145:1–27

    MathSciNet  MATH  Google Scholar 

  • Rong L, Perelson AS (2009) Modeling HIV persistence, the latent reservoir, and viral blips. J Theor Biol 260:308–3331

    MathSciNet  MATH  Google Scholar 

  • Roohi R, Heydari MH, Bavi O, Emdad H (2019) Chebyshev polynomials for generalized Couette flow of fractional Jeffrey nanofluid subjected to several thermochemical effects. Eng Comput. https://doi.org/10.1007/s00366-019-00843-9

    Article  Google Scholar 

  • Singla K, Gupta RK (2016) On invariant analysis of some time fractional nonlinear systems of partial differential equations. Int J Math Phys 57(10):101504. https://doi.org/10.1063/1.4964937

    Article  MathSciNet  MATH  Google Scholar 

  • Sun ZZ, Xu X (2006) A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math 56:193–209

    MathSciNet  MATH  Google Scholar 

  • Suryawansh GW, Hoffmann A (2015) A multi-scale mathematical modeling framework to investigate anti-viral therapeutic opportunities in targeting HIV-1 accessory proteins. J Theor Biol 386:89–104

    MathSciNet  MATH  Google Scholar 

  • Venkatesh SG, Raja Balachandar S, Ayyaswamy SK, Balasubramanian K (2016) A new approach for solving a model for HIV infection of \(\text{ CD4 }^+\) T-cells arising in mathematical chemistry using wavelets. J Math Chem 54(5):1072–1082

    MathSciNet  MATH  Google Scholar 

  • Wang X, Song X (2007) Global stability and periodic solution of a model for HIV infection of \(\text{ CD4 }^+\) T-cells. Appl Math Comput 189(2):1331–1340

    MathSciNet  MATH  Google Scholar 

  • Yan Y, Kou C (2012) Stability analysis for a fractional differential model of HIV infection of \(\text{ CD4 }^+\) T-cells with time delay. Math Comput Simul 82:1572–1585

    MathSciNet  MATH  Google Scholar 

  • Yüzbaşi S (2012) A numerical approach to solve the method for HIV infection of \(\text{ CD4 }^+\) T-cells. Appl Math Model 36:5876–5890

    MathSciNet  MATH  Google Scholar 

  • Yüzbaşi S (2016) An exponential collocation method for the solutions of the HIV infection model of \(\text{ CD4 }^+\) T-cells. Int J Biomath 9(3):1650036

    MathSciNet  MATH  Google Scholar 

  • Yüzbaşi S, Ismailov N (2017) A numerical method for the solutions of the HIV infection model of \(\text{ CD4 }^+\) T-cells. Int J Biomath 10(7):1750098

    MathSciNet  MATH  Google Scholar 

  • Yüzbaşi S, Karaçayir M (2017) An exponential Galerkin method for solutions of HIV infection model of \(\text{ CD4 }^{+}\) T-cells. Comput Biol Chem 67:205–212

    MathSciNet  MATH  Google Scholar 

  • Yüzbaşi S, Karaçayır M (2018) A Galerkin-type method for solving a delayed model on HIV infection of \(\text{ CD4 }^+\) T-cells. Iran J Sci Technol Trans Sci 42(3):1087–1095

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    H. Hassani

  2. Department of Internal Medicine, Shiraz University of Medical Sciences, Shiraz, Iran

    S. Mehrabi

  3. Department of Mathematics, Yasouj University, Yasouj, Iran

    E. Naraghirad

  4. Department of Nutrition, Faculty of Microbiology, Yasouj University of Medical Sciences, Yasouj, Iran

    M. Naghmachi

  5. Department of Mathematics, Faculty of Science, Akdeniz University, Akdeniz, Turkey

    S. Yüzbaşi

Authors
  1. H. Hassani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Mehrabi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Naraghirad
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Naghmachi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. S. Yüzbaşi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Hassani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hassani, H., Mehrabi, S., Naraghirad, E. et al. An Optimization Method Based on the Generalized Polynomials for a Model of HIV Infection of \(\hbox {CD4}^{+}\) T Cells. Iran J Sci Technol Trans Sci 44, 407–416 (2020). https://doi.org/10.1007/s40995-020-00833-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-020-00833-3

Keywords

Mathematics Subject Classification

Search

Navigation