Surveys in Mathematics and its Applications


ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 12 (2017), 165 -- 178

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

MATHEMATICAL MODELING APPLIED TO UNDERSTAND THE HOST-PATHOGEN INTERACTION OF HIV INFECTION IN BANGLADESH

S. K. Sahani, A. Islam and M. H. A. Biswas

Abstract. The most urgent public health problem today is to devise effective strategies to minimize the destruction caused by the AIDS epidemic. The understanding of HIV infection through mathematical modeling have made a significant contribution. The interaction of host to pathogen have been determined by fitting mathematical models to experimental data. In Bangladesh, the increasing rate of HIV infection comparing to the other countries of the world is not so high. Among the most at risk population of Bangladesh the HIV prevalent is still considered to be low with prevalence <1%. In this paper, the current situation of HIV infection in Bangladesh have been shown and a mathematical representation of HIV has been discussed. We have determined the basic reproduction number R0 and shown the local and global stability at disease free and chronic infected equilibrium points. Also we have shown that if the basic reproduction number R0≤1, then HIV infection is cleared from T cell population and it converges to disease free equilibrium point. Whereas if R0>1 then HIV infection persists.

2010 Mathematics Subject Classification: 97M60; 92B05; 34G20; 34A60.
Keywords: CD4+ T cells; dynamical systems; basic reproduction number; equilibrium points; stability analysis.

Full text

References

  1. B. M. Adams, H. T. Banks, M. Davidiana, H. D Kwon, H. T. Tran, S. N. Wynne and E. S. Rosenberg, HIV dynamics modeling, data analysis and optimal treatment protocols, Journal of Computational and Applied Mathematics, 184(2004), 10-49. MR2160056.Zbl 1075.92030

  2. H. T. Banks and D. M. Bortz, A parameter sensitivity methodology in the context of HIV delay equation models, J. Math. Biol. 50 (2005), 6, 607--625. MR2211638.Zbl 1083.92025

  3. M. H. A. Biswas, Necessary Conditions for Optimal Control Problems with State Constraints: Theory and Applications, PhD Thesis, Department of Electrical and Computer Engineering, Faculty of Engineering, University of Porto, Portugal, (2013).

  4. M. H. A. Biswas, AIDS epidemic worldwide and the millennium development strategies: A light for lives, HIV and AIDS Review, 11(4)(2012), 87-94.

  5. M. H. A. Biswas, L. T. Paiva and M. d. R. de Pinho, A SEIR Model for Control of Infectious Diseases with Constraints, Mathematical Biosciences and Engineering, 11(4)(2014), 761-784. MR3181992. Zbl 1327.92055.

  6. M. H. A. Biswas, On the Evolution of AIDS/HIV Treatment: An Optimal Control Approach, Current HIV Research, 12(1)(2014), 1-12. Zbl 1327.92055.

  7. M. H. A. Biswas, On the Immunotherapy of HIV Infections via Optimal Control with Constraint, Proceedings of the 18th International Mathematics Conference, (2013), 51-54.

  8. M. H. A. Biswas, Optimal Chemotherapeutic Strategy for HIV Infections-State Constrained Case, Proceedings of the 1st PhD Students Conference in Electrical and Computer Engineering, Department of Electrical and Computer Engineering, Faculty of Engineering, University of Porto, Portugal on 28-29 June, (2012).

  9. M. H. A. Biswas, Model and Control Strategy of the Deadly Nipah Virus (NiV) Infections in Bangladesh, Research & Reviews in Biosciences, 6(12)(2012), 370-377.

  10. L. Cai, X. Li, M. Ghosh and B. Guo. Stability analysis of an HIV/AIDS epidemic model with treatment, Journal of Computational and Applied Mathematics, 229(2009), 313-323. Zbl 1257.92034.

  11. D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64(2001), 29-64. Zbl 1334.92227.

  12. W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Health, Boston, (1995).

  13. R. P. Duffinin and R. H. Tullis, Mathematical Models of the Complete Course of HIV Infection and AIDS, Journal of Theoretical Medicine, 4(4)(2002), 215-221. Zbl 1059.92030.

  14. K. P. Gupta, Topology, Sixteenth edition, Pragati Prakashan, India, (2007).

  15. Z. Mukandavire, P. Das, C. Chiyaka and F. Nyabadza, Global analysis of an HIV/AIDS epidemic model, World Journal of Modeling and Simulation, 6(3)(2010), 1-10.

  16. A. S. Perelson, D. E. Kirschner and R. D. Boer, Dynamics of HIV infection of CD4+ T cells, Mathematical Biosciences, 114(1993), 81-125.

  17. A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41(1999), 3-44. MR1669741. Zbl 1078.92502.

  18. S. L. Ross, \textquotedblleft Differential Equation\textquotedblright, Third edition, Jhon Wiley & Sons Inc., U.K., (2004).

  19. L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Mathematical Biosciences, 200(2006), 44-57. MR2211927.

  20. National AIDS/STD Program, http://bdnasp.org/, 20/12/2014.

  21. World Health Organization, http://www.who.int/mediacentre/news/releases/2014/ world-health-statistics-2014/en/, 05/01/2015.

  22. Wikipedia, https://en.wikipedia.org/wiki/Host-pathogen-interaction.




S. K. Sahani
Mathematics Discipline, Khulna University, Khulna-9208, Bangladesh.
e-mail: mhabiswas@yahoo.com

A. Islam
Mathematics Discipline, Khulna University, Khulna-9208, Bangladesh.

M. H. A. Biswas (Corresponding author)
Mathematics Discipline, Khulna University, Khulna-9208, Bangladesh.


http://www.utgjiu.ro/math/sma