HIV drug resistance: insights from mathematical modelling. (English) Zbl 1481.92066
Summary: In 2013 the World Health Organization recommended the initiation of antiretroviral therapy (ART) to any person who tests HIV positive irrespective of his/her CD\(4^+\) count. However, implementation of the new guidelines poses a lot of challenges especially in Sub-Sahara Africa such as: drug side effects, drug resistance-mutations and significant financial burdens. Most importantly, it has been established that HIV resistance and subsequent virologic failure occur in a substantial proportion of HIV-infected patients receiving HAART. This study therefore, seeks to investigate the emergence of drug resistant HIV virus during treatment with the aim of determining the proper use of HIV therapy that would lessen drug resistance. To carry out the analysis a ten dimensional in-vivo mathematical model is proposed for HIV dynamics. The model is formulated in such away that it takes into account two virus strain, that is, the wild type as well as the naive type HIV virus. The in-vivo model is shown to be both biologically meaningful and mathematically well posed. The existence of unique infection-free equilibrium point is determined and both its local and global stability investigated. In addition, the basic reproduction number for each viral strain is computed using the next generation matrix method. An optimal control model is proposed and analysed by applying Pontryagin maximum principle, to obtain the optimal drug combination for HIV treatment. Here two drugs, that is, Reverse Transcriptase inhibitor and Protease inhibitor are used as the controls in the model. We provide an objective function for the minimisation of the number of wild type HIV virus and the drug resistant virus as well as the costs associated with the use of Reverse Transcriptase inhibitor and protease inhibitor. The forward backward sweep method is applied to numerically solve the optimality system. From the numerical simulations, it is evident that protease inhibitor is the most effective drug in controlling HIV infection. The results suggest that prolonged use of HAART leads to development of drug resistant and that people with drug-resistant infection could play a core role in the epidemic of HIV.
MSC:
| 92C50 | Medical applications (general) |
| 49N90 | Applications of optimal control and differential games |
Keywords:
wild type virus; drug resistant virus; virion free equilibrium; reverse transcriptase inhibitor; protease inhibitor; optimal controlReferences:
| [1] | UNAIDSUNICEF and World Health Organization, Global HIV/AIDS Response: Epidemic Update and Health Sector Progress Towards Universal Access: Progress Report (2011) |
| [2] | Lehman, D. A.; Baeten, J. M.; McCoy, C. O.; Weis, J. F.; Peterson, D.; Mbara, G.; Donnell, D.; Thomas, K. K.; Hendrix, C. W.; Marzinke, M. A., Risk of drug resistance among persons acquiring HIV within a randomized clinical trial of single-or dual-agent preexposure prophylaxis, J. Infect. Dis., 211, 8, 1211-1218 (2015) |
| [3] | Drake, J. W.; Holland, J. J., Mutation rates among RNA viruses, Proc. Natl. Acad. Sci., 96, 24, 13910-13913 (1999) |
| [4] | Rong, L.; Feng, Z.; Perelson, A. S., Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol., 69, 6, 2027-2060 (2007) · Zbl 1298.92053 |
| [5] | Lima, V. D.; Gill, V. S.; Yip, B.; Hogg, R. S.; Montaner, J. S.; Harrigan, P. R., Increased resilience to the development of drug resistance with modern boosted protease inhibitor-based highly active antiretroviral therapy, J. Infect. Dis., 198, 1, 51-58 (2008) |
| [6] | Tarfulea, N.; Read, P., A mathematical model for the emergence of HIV drug resistance during periodic bang-bang type antiretroviral treatment, Involve: J. Math., 8, 3, 401-420 (2015) · Zbl 1354.92093 |
| [7] | Tarfulea, N.; Blink, A.; Nelson, E.; Turpin Jr., D., A CTL-inclusive mathematical model for antiretroviral treatment of HIV infection, Int. J. Biomath., 4, 01, 1-22 (2011) · Zbl 1403.92122 |
| [8] | Ngina, P.; Mbogo, R. W.; Luboobi, L. S., Modelling optimal control of in-host HIV dynamics using different control strategies, Comput. Math. Methods Med., 2018, 1-18 (2018) · Zbl 1411.92177 |
| [9] | Ngina, P.; Mbogo, R. W.; Luboobi, L. S., The in vivo dynamics of HIV infection with the influence of cytotoxic T lymphocyte cells, Int. Sch. Res. Not., 2017, 1-10 (2017) |
| [10] | Li, Z.; Teng, Z.; Miao, H., Modeling and control for HIV/AIDS transmission in China based on data from 2004 to 2016, Comput. Math. Methods Med., 2017, 1-13 (2017) · Zbl 1397.92648 |
| [11] | Cesari, L., Optimizationtheory and Applications: Problems with Ordinary Differential Equations, 17 (2012), Springer Science & Business Media |
| [12] | Buonomo, B.; Manfredi, P.; dOnofrio, A., Optimal time-profiles of public health intervention to shape voluntary vaccination for childhood diseases, J. Math. Biol., 73, 1-25 (2018) |
| [13] | Ogunlaran, O.; Oukouomi, S., Mathematical model for an effective management of HIV infection, BioMed Res. Int., 2016, 1-6 (2016) |
| [14] | Ali, N.; Zaman, G.; Alshomrani, A. S., Optimal control strategy of HIV-1 epidemic model for recombinant virus, Cogent Math., 4, 1, 1293468-1293478 (2017) · Zbl 1426.92069 |
| [15] | Ngina, P. M.; Mbogo, R. W.; Luboobi, L. S., Mathematical modelling of in-vivo dynamics of hiv subject to the influence of the cd8+ t-cells, Appl. Math., 8, 08, 1153 (2017) |
| [16] | Van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 1, 29-48 (2002) · Zbl 1015.92036 |
| [17] | Brauer, F.; Castillo-Chavez, C.; Castillo-Chavez, C., Mathematical Models in Population Biology and Epidemiology, 40 (2001), Springer · Zbl 0967.92015 |
| [18] | Heesterbeek, J.; Dietz, K., The concept of Ro in epidemic theory, Stat. Neerl., 50, 1, 89-110 (1996) · Zbl 0854.92018 |
| [19] | Castillo-Chavez, C.; Feng, Z.; Huang, W., On the computation of RO and its role on, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, 1, 229-254 (2002), Springer Science & Business Media |
| [20] | Neilan, R. L.M.; Schaefer, E.; Gaff, H.; Fister, K. R.; Lenhart, S., Modeling optimal intervention strategies for cholera, Bull. Math. Biol., 72, 8, 2004-2018 (2010) · Zbl 1201.92045 |
| [21] | Pontryagin, L. S., Mathematical Theory of Optimal Processes (1987), CRC Press |
| [22] | Fleming, W. H.; Rishel, R. W., Deterministic and Stochastic Optimal Control, 1 (2012), Springer Science & Business Media |
| [23] | Rahmoun, A.; Ainseba, B.; Benmerzouk, D., Bang bang control applied on an HIV 1 within host model, Mediterr. J. Model. Simul., 5, 1, 59-75 (2016) |
| [24] | Li, M. Y.; Wang, L., Backward bifurcation in a mathematical model for HIV infection in-vivo with anti-retroviral treatment, Nonlinear Anal.: Real World Appl., 17, 147-160 (2014) · Zbl 1325.92054 |
| [25] | Mbogo, R. W.; Luboobi, L. S.; Odhiambo, J., Mathematical model for HIV and CD4 cells dynamics in vivo, Int. Electron. J. Pure Appl. Math., 6, 2, 83-103 (2013) · Zbl 1399.92022 |
| [26] | Wodarz, D.; Nowak, M. A., Immune responses and viral phenotype: do replication rate and cytopathogenicity influence virus load?, Comput. Math. Methods Med., 2, 2, 113-127 (2000) · Zbl 0943.92024 |
| [27] | Kirschner, D. E.; Webb, G.; Cloyd, M., Model of HIV-1 disease progression based on virus-induced lymph node homing and homing-induced apoptosis of CD4+ lymphocytes, J. Acquir. Immune Defic. Syndr, 24, 4, 352-362 (2000) |
| [28] | Zarei, H.; Kamyad, A. V.; Effati, S., Multiobjective optimal control of HIV dynamics, Math. Prob. Eng., 2010, 1-29 (2010) · Zbl 1213.49048 |
| [29] | McLean, A. R., Infectious disease modeling, Encyclopedia of Sustainability Science and Technology, 5347-5357 (2012) |
| [30] | Arruda, E. F.; Dias, C. M.; de Magalhães, C. V.; Pastore, D. H.; Thomé, R. C.; Yang, H. M., An optimal control approach to HIV immunology, Appl. Math., 6, 06, 1115 (2015) |
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