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Dynamical analysis of a multi-strain model of HIV in the presence of anti-retroviral drugs. (English) Zbl 1154.92033
Summary: One major drawback associated with the use of anti-retroviral drugs in curtailing HIV spread in a population is the emergence and transmission of HIV strains that are resistant to these drugs. This paper presents a deterministic HIV treatment model, which incorporates a wild (drug sensitive) and a drug-resistant strain, for gaining insights into the dynamical features of the two strains, and determining effective ways to control HIV spread under this situation. Rigorous qualitative analysis of the model reveals that it has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (\(\mathcal R^t_0\)) is less than unity and that the disease will persist in the population when this threshold exceeds unity. Further, for the case where \(\mathcal R^t_0 > 1\), it is shown that the model can have two co-existing endemic equilibria, and the competitive exclusion phenomenon occurs whenever the associated reproduction number of the resistant strain (\(\mathcal R^t_{\text r}\)) is greater than that of the wild strain (\(\mathcal R^t_{\text w}\)).
Unlike in the treatment model, it is shown that the model without treatment can have a family of infinitely many endemic equilibria when its associated epidemiological threshold \((\mathcal R_{0})\) exceeds unity. For the case when \(\mathcal R_{\text w^t} > \mathcal R^t_{\text r}\) and \(R^t_{\text w} > 1\), it is shown that the widespread use of treatment against the wild strain can lead to its elimination from the community if the associated reduction in infectiousness of infected individuals (treated for the wild strain) does not exceed a certain threshold value (in this case, the use of treatment is expected to make \(\mathcal R_{\text w^t} > \mathcal R^t_{\text r}\)).

MSC:
92C60 Medical epidemiology
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
37N25 Dynamical systems in biology
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[1] Blower S. M., Curr. Opin. Infect. Dis 15 pp 609– (2002) · doi:10.1097/00001432-200212000-00009
[2] DOI: 10.1038/nm0901-1016 · doi:10.1038/nm0901-1016
[3] Castillo-Chavez C., SIAM J. Appl. Math 5 pp 1790– (1991)
[4] DOI: 10.1136/bmj.324.7331.232 · doi:10.1136/bmj.324.7331.232
[5] DOI: 10.1007/s11538-005-9057-5 · Zbl 1334.91060 · doi:10.1007/s11538-005-9057-5
[6] Fleck F., BMJ 329 pp . 129– (2004)
[7] DOI: 10.1097/00126334-200406010-00007 · doi:10.1097/00126334-200406010-00007
[8] Group of the Office of AIDS Research Advisory Council, Guidelines for the use of antiretroviral agents in HIV-1 infected adults and adolescents (2006) Available athttp://aidsinfo.nih.gov/Guidelines/GuidelineDetail.aspx?MenuItem=Guidelines&Search=Off&GuidelineID=7&ClassID=1
[9] Hale J. K., Ordinary Differential Equations (1969) · Zbl 0186.40901
[10] DOI: 10.1137/S0036144500371907 · Zbl 0993.92033 · doi:10.1137/S0036144500371907
[11] Kgosimore M., Math. Biosci. Engr 3 pp 297– (2006)
[12] Lakshmikantham V., Stability Analysis of Nonlinear Systems (1989) · Zbl 0676.34003
[13] DOI: 10.1097/00007435-200009000-00012 · doi:10.1097/00007435-200009000-00012
[14] Lasalle, J. P. The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics. Philadelphia: SIAM. · Zbl 0153.40602
[15] DOI: 10.1097/00002030-200107060-00011 · doi:10.1097/00002030-200107060-00011
[16] DOI: 10.1016/j.matcom.2006.01.004 · Zbl 1087.92040 · doi:10.1016/j.matcom.2006.01.004
[17] Moore H., Math. Biosci. Eng 2 pp 363– (2005)
[18] Nicolosi A., J. Acquir. Immun. Defic. Syndr 7 pp 296– (1994)
[19] O’Brien T., J. Acquir. Immune. Defic. Syndr 7 pp 705– (1994)
[20] Shiri T., Math. Biosci. and Engr 2 pp 811– (2005)
[21] DOI: 10.1016/S1473-3099(04)01148-X · doi:10.1016/S1473-3099(04)01148-X
[22] DOI: 10.1016/j.bulm.2004.10.001 · Zbl 1334.92426 · doi:10.1016/j.bulm.2004.10.001
[23] DOI: 10.1080/08898480009525481 · Zbl 0961.92021 · doi:10.1080/08898480009525481
[24] United Nations Department of Economic and Social Affairs/Population Division, The Impact of AIDS, United Nations, New York, 2004
[25] DOI: 10.1016/S0025-5564(02)00108-6 · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[26] DOI: 10.1006/tpbi.1994.1017 · Zbl 0804.92024 · doi:10.1006/tpbi.1994.1017
[27] Confronting AIDS: Public Priorities in a Global Epidemic (1997)
[28] The world health report: Changing history (2004)
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