## Journal of Applied Mathematics

### Optimal Treatment Strategies for HIV with Antibody Response

#### Abstract

Numerical analysis and optimization tools are used to suggest improved therapies to try and cure HIV infection. An HIV model of ordinary differential equation, which includes immune response, neutralizing antibodies, and multidrug effects, is improved. For a fixed time, single-drug and two-drug treatment strategies are explored based on Pontryagin’s maximum principle. Using different combinations of weight factor pairs combining with special upper-bound pairs for controls, nine types of treatment policies are determined and different therapy effects are numerically simulated with a gradient projection method. Some strategies are effective, but some strategies are not particularly helpful for the therapy of HIV/AIDS. Comparing the effective treatment strategies, we find a more appropriate strategy with maximizing the number of uninfected CD4+T-cells and minimizing the number of active virus.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 685289, 13 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305897

Digital Object Identifier
doi:10.1155/2014/685289

Mathematical Reviews number (MathSciNet)
MR3232926

#### Citation

Zhou, Yinggao; Yang, Kuan; Zhou, Kai; Wang, Chunling. Optimal Treatment Strategies for HIV with Antibody Response. J. Appl. Math. 2014 (2014), Article ID 685289, 13 pages. doi:10.1155/2014/685289. https://projecteuclid.org/euclid.jam/1425305897

#### References

• B. M. Adams, H. T. Banks, H. Kwon, and H. T. Tran, “Dyna-mic multidrug therapies for HIV: optimal and STI control approaches,” Mathematical Biosciences and Engineering, vol. 1, no. 2, pp. 223–241, 2004.
• R. Zurakowski and A. R. Teel, “A model predictive control based scheduling method for HIV therapy,” Journal of Theoretical Biology, vol. 238, no. 2, pp. 368–382, 2006.
• J. Karrakchou, M. Rachik, and S. Gourari, “Optimal control and infectiology: application to an HIV/AIDS model,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 807–818, 2006.
• W. Garira, S. D. Musekwa, and T. Shiri, “Optimal control of combined therapy in a single strain HIV-1 model,” Electronic Journal of Differential Equations, vol. 2005, no. 52, pp. 1–22, 2005.
• D. Hull, Optimal Control Theory for Applications, Springer, New York, NY, USA, 2003.
• M. I. Kamien and N. L. Schwartz, Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, Advanced Textbooks in Economics, North-Holland Press, Amsterdam, Netherlands, 1991.
• L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, Boca Raton, Fla, USA, 1987.
• Y. Lin and P. Zhong, “Impact, mechanissm and laboratory detection of neu tralizing antibody in HIV-1,” Chinese Journal of AIDS & STD, vol. 14, no. 6, pp. 634–636, 2008 (Chinese).
• J. S. McLellan, M. Pancera, C. Carrico et al., “Structure of HIV-1 gp120 V1/V2 domain with broadly neutralizing antibody PG9,” Nature, vol. 480, no. 7377, pp. 336–343, 2011.
• C. Ren, Y. Li, and H. Ling, “Neutralizing anti bodies responses during natural HIV-1 infection,” International Journal of Immunology, vol. 3, no. 35, pp. 176–179, 2012 (Chinese).
• Y. Zhou, Y. Liang, and J. Wu, “An optimal strategy for HIV multitherapy,” Journal of Computational and Applied Mathematics, vol. 263, pp. 326–337, 2014.
• J. M. Orellana, “Optimal control for HIV multitherapy enhancement,” Comptes Rendus Mathematique, vol. 348, pp. 1179–1183, 2011.
• H. Wu, H. Zhu, H. Miao, and A. S. Perelson, “Parameter identi-fiability and estimation of HIV/AIDS dynamic models,” Bulletin of Mathematical Biology, vol. 70, no. 3, pp. 785–799, 2008.
• M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho, and A. S. Perelson, “Modeling plasma virus concentration during primary HIV infection,” Journal of Theoretical Biology, vol. 203, no. 3, pp. 285–301, 2000.
• W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, NY, USA, 1975.
• D. L. Lukes, “Differential equations: classical to controlled,” in Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1982.
• A. S. Perelson, D. E. Kirschner, and R. D. Boer, “Dynamics of HIV infection of CD4$^{+}$ T cells,” Mathematical Biosciences, vol. 114, no. 1, pp. 81–125, 1993. \endinput