A simple chaotic model for development of HIV virus

Document Type : Article


1 Department of Biomedical Engineering, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran 159163-4311, Iran

2 School of Engineering, Monash University, Selangor, Malaysia

3 Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA

4 - Health Technology Research Institute, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran 159163-4311, Iran - Department of Biomedical Engineering, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran 159163-4311, Iran



Studying the growth of HIV virus in the human body as one of the fastest infectious viruses is very important. Using mathematical modeling can make experimental tests easier to process and evaluate. It can also help to predict the disease progress and provide a better insight into the virus development. In this study, a new nonlinear differential equation model is introduced to investigate the interaction of the HIV virus with the body immune system. This is a physiological-based model capable of representing complex behaviors. The bifurcation analysis with a variation of activated healthy T cells is carried out. It is shown that the chaotic development of the virus is available for some ranges of activated healthy T cells. This may explain why the virus develops differently in different individuals or under different circumstances. The chaotic region contains some narrow periodic windows, in which the chaotic mode suddenly ends at some critical points, and the system starts a periodic behavior for a tiny range of active healthy T cells. This may indicate the possibility of controllable development of the HIV virus even when it is in the random-like phase of the disease. For more illustration, the system's state space is represented.


References   1. Ippolito, G., Levy, J.A., Sonnerborg, A., et al. AIDS   and HIV Infection after Thirty Years", AIDS Research   and Treatment, 2013, p. 731983 (2013).   F. Parastesh et al./Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1643{1652 1651   2. Kim, A., Kormyshev, V., Kwon, H., et al. HIVinfection   model stabilization", IFAC-Papers OnLine,   48(25), pp. 214{217 (2015).   3. Li, M.Y. and Wang, L. Backward bifurcation in a   mathematical model for HIV infection in vivo with   anti-retroviral treatment", Nonlinear Analysis: Real   World Applications, 17, pp. 147{160 (2014).   4. Chen, X., Huang, L., and Yu, P. Dynamic behaviors   of a class of HIV compartmental models", Communications   in Nonlinear Science and Numerical Simulation,   23(1{3), pp. 115{128 (2015).   5. Raue, A., Schilling, M., Bachmann, J., et al. Lessons   learned from quantitative dynamical modeling in systems   biology", PloS One, 8(9), p. e74335 (2013).   6. Hug, S., Schmidl, D., Li, W.B., et al. Bayesian model   selection methods and their application to biological   ODE systems", In: Geris L., Gomez-Cabrero D.   (Eds), Uncertainty in Biology, Studies in Mechanobiology,   Tissue Engineering and Biomaterials, 17,   Springer, Cham. (2016). https://doi.org/10.1007/978-   3-319-21296-8-10   7. Sprott, J.C., Elegant Chaos: Algebraically Simple   Chaotic Flows, World Scienti_c (2010).   8. Perc, M. and Marhl, M. Chaos in temporarily destabilized   regular systems with the slow passage e_ect",   Chaos, Solitons & Fractals, 27(2), pp. 395{403 (2006).   9. Panahi, S., Aram, Z., Jafari, S., et al. Modeling of   epilepsy based on chaotic arti_cial neural network",   Chaos, Solitons & Fractals, 105, pp. 150{156 (2017).   10. Fister Jr, I., Perc, M., Kamal, S.M., et al. A   review of chaos-based _rey algorithms: perspectives   and research challenges", Applied Mathematics and   Computation, 252, pp. 155{165 (2015).   11. Yao, Y. and Freeman, W.J. Model of biological   pattern recognition with spatially chaotic dynamics",   Neural Networks, 3(2), pp. 153{170 (1990).   12. Chiu, Y.-T., Lee, J.-C., Cheng, A., et al. Epstein-Barr   virus-associated smooth muscle tumor as the initial   presentation of HIV infection: A case report", Journal   of the Formosan Medical Association, 117(1), pp. 82{   84 (2018).   13. Lund, O., Mosekilde, E., and Hansen, J. Period   doubling route to chaos in a model of HIV infection   of the immune system: A comment on the Andersonmay   model", Simulation Practice and Theory, 1(2),   pp. 49{55 (1993).   14. Sun, H.-Y., Kung, H.-C., Ho, Y.-C., et al. Seroprevalence   of hepatitis a virus infection in persons with   HIV infection in Taiwan: implications for hepatitis   a vaccination", International Journal of Infectious   Diseases, 13(5), pp. e199{e205 (2009).   15. Dos Santos, R.M.Z., and Coutinho, S. Dynamics   of HIV infection: A cellular automata approach",   Physical Review Letters, 87(16), p. 168102 (2001).   16. Marinho, E., Bacelar, F., and Andrade, R. A model   of partial di_erential equations for HIV propagation in   lymph nodes", Physica A: Statistical Mechanics and   its Applications, 391(1{2), pp. 132{141(2012).   17. Khan, M.A. and Odinsyah, H.P. Fractional model   of HIV transmission with awareness e_ect", Chaos,   Solitons & Fractals, 138, p. 109967 (2020).   18. El-Dessoky, M. and Khan, M. Application of fractional   calculus to combined modi_ed function projective   synchronization of di_erent systems", Chaos,   29(1), p. 013107 (2019).   19. Atangana, A. and Khan, M.A. Validity of fractal   derivative to capturing chaotic attractors", Chaos,   Solitons & Fractals, 126, pp. 50{59 (2019).   20. Khan, M.A. The dynamics of a new chaotic system   through the Caputo-Fabrizio and Atanagan-Baleanu   fractional operators", Advances in Mechanical Engineering,   11(7), p. 1687814019866540 (2019).   21. Yavuz, M. and Sene, N. Stability analysis and numerical   computation of the fractional predator-prey model   with the harvesting rate", Fractal and Fractional, 4(3),   p. 35 (2020).   22. Yavuz, M. and Bonyah, E. New approaches to the   fractional dynamics of schistosomiasis disease model",   Physica A: Statistical Mechanics and its Applications,   525, pp. 373{393 (2019).   23. Naik, P.A., Zu, J., and Owolabi, K.M. Global dynamics   of a fractional order model for the transmission of   HIV epidemic with optimal control", Chaos, Solitons   & Fractals, 138, p. 109826 (2020).   24. Naik, P.A., Zu, J., and Owolabi, K.M. Modeling   the mechanics of viral kinetics under immune control   during primary infection of HIV-1 with treatment in   fractional order", Physica A, 545, p. 123816 (2020).   25. Naik, P.A., Yavuz, M., and Zu, J. The role of   prostitution on HIV transmission with memory: a   modeling approach", Alexandria Engineering Journal,   59(4), pp. 2513{2531 (2020).   26. Naik, P.A., Owolabi, K.M., Yavuz, M., et al. Chaotic   dynamics of a fractional order HIV-1 model involving   AIDS-related cancer cells", Chaos, Solitons & Fractals,   140, p. 110272 (2020).   27. Bonhoe_er, S., May, R.M., Shaw, et al. Virus dynamics   and drug therapy", Proceedings of the National   Academy of Sciences, 94(13), pp. 6971{6976 (1997).   28. Ho, C.Y.-F. and Ling, B.W.-K. Initiation of HIV   therapy", International Journal of Bifurcation and   Chaos, 20(04), pp. 1279{1292 (2010).   29. Blazek, D., Teque, F., Mackewicz, C., et al. The   CD8+ cell non-cytotoxic antiviral response a_ects   RNA polymerase II-mediated human immunode_-   ciency virus transcription in infected CD4+ cells",   Journal of General Virology, 97(1), pp. 220{224   (2016).   1652 F. Parastesh et al./Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1643{1652   30. Wang, L. Global dynamical analysis of HIV models   with treatments", International Journal of Bifurcation   and Chaos, 22(09), p. 1250227 (2012).   31. Hernandez-Vargas, E.A., Alanis, A.Y., and Sanchez,   E.N. Discrete-time neural observer for HIV infection   dynamics", Proc. World Automation Congress (WAC),   IEEE, pp. 1{6 (2012).   32. Revilla, T. and Garc__a-Ramos, G. Fighting a virus   with a virus: a dynamic model for HIV-1 therapy",   Mathematical Biosciences, 185(2), pp. 191{203 (2003).   33. Yu, P. and Zou, X. Bifurcation analysis on an HIV-1   model with constant injection of recombinant", International   Journal of Bifurcation and Chaos, 22(03), p.   1250062 (2012).   34. Greer, C. and Garc__a-Ramos, G. A hunter virus   that targets both infected cells and HIV free virions:   Implications for therapy", Theoretical Biology and   Medical Modelling, 9(1), p. 52 (2012).   35. Rahmani Doust, M. and Gholizade, S. The lotka-   Volterra predator-prey equations", Caspian Journal of   Mathematical Sciences (CJMS) , 3(2), pp. 221{225   (2014).