A mathematical model on HIV/AIDS with fusion effect: Analysis and homotopy solution

  • Praveen Kumar GuptaEmail author
  • Ajoy Dutta
Regular Article


In this study, first we analyzed the dynamical behaviour of a deterministic dynamic model for HIV-infected CD4+ T cells in a population of three classes of cell compartments: uninfected, infected and virus, with fusion effect. The qualitative analysis of the model, i.e., positivity and boundedness of the solutions have been discussed. For the proposed model, the non-infected and endemic equilibrium points are recognized and their local stability examined by the Jacobian matrix. The Lyapunov functional and geometric approaches are discussed in detail in order to show the global stability of the non-infected and endemic equilibrium states, respectively. Additional to this qualitative analysis, the approximate analytical solution was obtained for the proposed model with the help of the homotopy analysis method (HAM), and we demonstrated the convergence region by the ℏ curve. The residual error is also calculated for the HAM solution. We have drawn the numerical solutions to confirm all analytical and HAM solution, which reflects the reliability of all solutions.


  1. 1.
    H. Nishikawa, S. Oishi, M. Fujita, K. Watanabe, R. Tokiwa, H. Ohno, E. Kodama, K. Izumi, K. Kaziwara, T. Naitoh, M. Matsuoka, A. Otaka, N. Fujii, Bioorg. Med. Chem. 16, 9184 (2018)CrossRefGoogle Scholar
  2. 2.
    R.V. Patel, S.W. Park, Bioorg. Med. Chem. 23, 5247 (2015)CrossRefGoogle Scholar
  3. 3.
    A.S. Perelson, D.E. Kirschner, R. de Boer, Math. Biosci. 114, 81 (1993)CrossRefGoogle Scholar
  4. 4.
    G. Haas, A. Hosmalin, F. Hadida, J. Duntze, P. Debré, B. Autran, Immun. Lett. 57, 63 (1997)CrossRefGoogle Scholar
  5. 5.
    D. Wodarz, D.H. Hamer, Math. Biosci. 209, 14 (2017)CrossRefGoogle Scholar
  6. 6.
    Y. Emvudu, D. Bongor, R. Koina, Appl. Math. Model. 40, 9131 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    O.M. Otunuga, Math. Biosci. 299, 138 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Adak, N. Bairagi, Appl. Math. Model. 54, 517 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Burg, L. Rong, A.U. Neumann, H. Dahari, J. Theor. Bio. 259, 751 (2009)CrossRefGoogle Scholar
  10. 10.
    H.-F. Huo, R. Chen, X.-Y. Wang, Appl. Math. Model. 40, 6550 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Dutta, P.K. Gupta, Chin. J. Phys. 56, 1045 (2018)CrossRefGoogle Scholar
  12. 12.
    P.K. Srivastava, P. Chandra, Nonlin. Anal. Real World Appl. 11, 612 (2010)CrossRefGoogle Scholar
  13. 13.
    P. Essunger, A.S. Perelson, J. Theor. Bio. 170, 367 (1994)CrossRefGoogle Scholar
  14. 14.
    K. Hattaf, N. Yousfi, A. Tridane, Appl. Math. Comp. 221, 514 (2013)CrossRefGoogle Scholar
  15. 15.
    M.Y. Li, J.S. Muldowney, SIAM J. Math. Anal. 27, 1070 (1996)MathSciNetCrossRefGoogle Scholar
  16. 16.
    B. Buonomo, D. Lacitignola, J. Math. Anal. Appl. 348, 255 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD Thesis, Shanghai Jiao Tong University, Shanghai, (1992)Google Scholar
  18. 18.
    S. Das, P.K. Gupta, R. Kumar, Int. J. Chem. Reac. Eng. 9, A15 (2011)Google Scholar
  19. 19.
    P.K. Gupta, S. Verma, Z. Naturforsch. A 67a, 621 (2012)CrossRefGoogle Scholar
  20. 20.
    Z. Yang, S.J. Liao, Comm. Nonlin. Sci. Numer. Simulat. 53, 249 (2017)CrossRefGoogle Scholar
  21. 21.
    N.H.A. Wahab, A. Salah, Eur. Phys. J. Plus 130, 92 (2015)CrossRefGoogle Scholar
  22. 22.
    E.K. Ghiasi, R. Saleh, Eur. Phys. J. Plus 134, 32 (2019)CrossRefGoogle Scholar

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© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologySilcharIndia

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