On the stability of a mathematical model for HIV(AIDS) - cancer dynamics

: pp. 783–796
Received: May 23, 2021
Accepted: June 07, 2021
Mathematical department, Salahaddin University-Erbil, Erbil Kurdistan region Iraq
Laboratoire de Mathematiques Jean Leray, Universite de Nantes, France

In this work, we study an impulsive mathematical model proposed by Chavez et al. [1] to describe the dynamics of cancer growth and HIV infection, when chemotherapy and HIV treatment are combined.  To better understand these complex biological phenomena, we study the stability of equilibrium points.  To do this, we construct an appropriate Lyapunov function for the first equilibrium point while the indirect Lyapunov method is used for the second one.  None of the equilibrium points obtained allow us to study the stability of the chemotherapeutic dynamics, we then propose a bifurcation of the model and make a study of the bifurcated system which contributes to a better understanding of the underlying biochemical processes which govern this highly active antiretroviral therapy.  This shows that this mathematical model is sufficiently realistic to formulate the impact of this treatment.

  1. Chavez J. P., Gurbuz B., Pinto C. M.  The effect of aggressive chemotherapy in a model for HIV/AIDS-cancer dynamics.  Communications in Nonlinear Science and Numerical Simulation. 75, 109–120 (2019).
  2. Lifson J. D., Engleman E. G.  Role of CD4 in Normal Immunity and HIV infection.  Immunological Reviews. 109 (1), 93–109 (1989).
  3. Walker B., McMichael A.  The T-cell response to HIV.  Cold Spring Harbor Perspectives in Medicine. 2 (11), a00754 (2012).
  4. World Health Organization. HIV Country Profiles.  http://www.who.int.
  5. Adams B., Banks H., Davidian M., Kwon H.-D., Tran H., Wynne S., Rosenberg E.  HIV dynamics : Modeling, data analysis, and optimal treatment protocols.  Journal of Computational and Applied Mathematics. 184 (1), 10–49 (2005).
  6. Joshi H. R.  Optimal control of an HIV immunology model.  Optim. Control Appl. Meth. 23, 199 (2002).
  7. Souza M., Zubelli J.  Global stability for a class of virus models with cyto-toxic T lymphocyte immune response and antigenic variation.  Bulletin of Math. Biol. 73, 609–625 (2011).
  8. Arsenashvili A., Nachaoui A., Tadumadze T.  On approximate solution of an inverse problem for linear delay differential equations.  Bulletin of the Georgian National Academy of Sciences. 2 (2), 24–28 (2008).
  9. Mardanov M. J., Nachaoui A., Velieva N. I., Gasimov Y. S., Niftili A. A.  Methods for the solution of the optimal stabilization problem for the descriptor systems.  Dokl. Nats. Akad. Nauk Azerb. 64 (4), 8–13 (2008).
  10. Nachaoui A., Shavadze T., Tadumadze T.  The local representation formula of solution for the perturbed controlled differential equation with delay and discontinuous initial condition.  Mathematics. 8 (10), 1845 (2020).
  11. Wodarz D., Nowak M.  Specific therapy regimes could lead to long-term immunological control of HIV.  Proceedings of the National Academy of Sciences. 96 (25), 14464–14469 (1999).
  12. Lyapunov A. M.  The General Problem of the Stability of Motion.  Taylor and Francis, London (1992).
  13. Salih H. W., Nachaoui A.  Computing general form of the focal value and Lyapunov function for the lopsiderd system in degree eight.  Advanced Mathematical Models & Applications. 5 (1), 121–130 (2020).
  14. Salih H. W., Nachaoui A.  Study of the stability for Drug Delivery Models.  Journal of Physics: Conference Series. 1743, 012019 (2021).
  15. Brown R. J.  A modern introduction to dynamical systems.  Oxford University Press, Oxford (2018).
  16. Wiggins S.  Introduction to applied nonlinear dynamical systems and chaos.  Texts in Applied Mathematics. Springer-Verlag, New York (2003).
  17. Kooi B. W., Poggiale J. C.  Modelling, singular perturbation and bifurcation analyses of bitrophic food chains.  Mathematical Biosciences. 301, 93–10 (2018).
Mathematical Modeling and Computing, Vol. 8, No. 4, pp. 783–796 (2021)

Uncaught exception thrown in session handler.

PDOException: SQLSTATE[23000]: Integrity constraint violation: 1062 Duplicate entry '0' for key 'uid': INSERT INTO {sessions} (sid, ssid, uid, cache, hostname, session, timestamp) VALUES (:db_insert_placeholder_0, :db_insert_placeholder_1, :db_insert_placeholder_2, :db_insert_placeholder_3, :db_insert_placeholder_4, :db_insert_placeholder_5, :db_insert_placeholder_6); Array ( [:db_insert_placeholder_0] => ftGzjUT2GpAOsa341nKGq-FYzvlw__X4iDu1oaUzvas [:db_insert_placeholder_1] => ftGzjUT2GpAOsa341nKGq-FYzvlw__X4iDu1oaUzvas [:db_insert_placeholder_2] => 0 [:db_insert_placeholder_3] => 0 [:db_insert_placeholder_4] => [:db_insert_placeholder_5] => nodeviewcount_views_limit|a:1:{i:25128;O:8:"DateTime":3:{s:4:"date";s:26:"2023-03-28 06:26:39.000000";s:13:"timezone_type";i:3;s:8:"timezone";s:15:"Europe/Helsinki";}} [:db_insert_placeholder_6] => 1679973999 ) in _drupal_session_write() (line 209 of /home/science/public_html/includes/session.inc).