Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics

Bassey Echeng Bassey; Adagba Odey Henry

doi:doi:10.11648/j.pamj.20211001.12

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Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics

Bassey Echeng Bassey, Adagba Odey Henry

Published in Pure and Applied Mathematics Journal (Volume 10, Issue 1)

Received: 24 December 2020 Accepted: 20 February 2021 Published: 4 March 2021

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Abstract

Devastatingly, in spite of the long standing research works on HIV/AIDS infection and treatment dynamics, reviews of existing models clearly shown that the behavioral attitude to treatment consistency by those screened to become aware and those receiving treatment have not been given the desired attention. Moreso, the inconsistency following avoidable treatment truncation and later resumption of treatment by these classes of infectives, which could lead to colossal drug abuse is also not accorded the much expected consideration. Therefore, in this present study, we sought and formulated a nonlinear 6-Dimensional deterministic mathematical HIV/AIDS dynamic model that accounted for the global stability analysis of the role of antiretroviral therapy abuse for the treatment of HIV/AIDS epidemic. The model is structured upon dynamical interactions between 6-subpopulations and HI-virus under bilinear control functions with constant screening of the susceptible. It is assumed that the rate of resumption of ART upon truncation is less than initial ART truncation following the incorporation of HIV aware infectives not ready to receive ART treatment and HIV aware infectives with truncated treatment protocol The system mathematical well-posedness was investigated and model reproduction number determined for both off- treatment (with value ) and for onset-treatment (with value ). We considered the model for off-treatment and thereafter by incorporating LaSalle’s invariant principle into classical Lyapunov function method, we presented an approach for the global stability analysis of the role of ART abuse in HIV/AIDS treatment. Furthermore, the analysis and results of this paper presented a dynamic methodological application of bilinear control functions and an impeccable understanding of the fundamental mechanism in HIV/AIDS treatment in the presence of ART abuse. Using in-built Runge-Kutta of order of precision 4 in a Mathcad surface, numerical validity of model is conducted to investigate the study theoretical and analytical predictions. Results shows that application of onset-treatment functions with trend of ART abuse yield tremendous reduction in HIV/AIDS infection epidemic following the recovery rate of the susceptible population with value increasing from 0.5 cells/mm³ to 1.203 cells/mm³ within the first months and attained stability of 0.62 cells/mm³ through the time interval of 20- 30 months.

Published in	Pure and Applied Mathematics Journal (Volume 10, Issue 1)
DOI	10.11648/j.pamj.20211001.12
Page(s)	9-31
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Antiretroviral-Therapy-abuse, Global-Stability-Conditions, Lyapunov-Function, Bilinear-Control-Functions, Upper-Triangular-Matrix, Mass-Action, Theoretical Predictions, Analytical Results

References

[1]	UNAIDS (2019). Fact Sheet – World Aids Day 2019, (On-line: https://www.firsnet.org/images/worlddays/World_AIDS_Day_Fact_Sheet.pdf)
[2]	Ouattara, D. A. (2005). Mathematical Analysis of the HIV-1 Infection: Parameter Estimation, Therapies Effectiveness and Therapeutical Failures. Proceedings of the 2005 IEEE, Engineering in Medicine and Biology, 27th Annual Conference Shanghai, China, September 1-4.
[3]	Seyed, M. M., Abba B. G., Robert G. M. and Richard G. (2003). Could condom stop the AIDS epidemic?, Theor. Med., 5 (3-4), 171-181.
[4]	Mukandavire, Z., Garira W., Tchuenche, J. M. (2009). Modelling effects of public-health educational campaigns on HIV/AIDS transmission dynamics, Applied Mathematical Modelling 33, 1-5.
[5]	Anderson, R. M. and May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, N.Y., 290-292.
[6]	Greenhalgh, D., Doyle, M. and Lewis, F. (2001). Mathematical treatment of AIDS and condom use, IMA. J. Math. Appl. Med. Biol., 18, 225-262.
[7]	Hyman, J. M. and Stanley E. N. (2003). Modeling the impact of random screening and contact tracing in reducing the spread of HIV, Math. Biosc., 181, 17-54.
[8]	Anderson, R. M., May, R. M., Medley, G. F. and Johnson, A. (1986). A preliminary study of the transmission of the human immunodeficiency virus (HIV), the causative agent of AIDS. IMA J. Math. Appl. Med. Biol., 3, 229-263.
[9]	Knox, E. R. (1986). A transmission model for AIDS. European J. Epidemiology, 2, 165-177.
[10]	May, R. M. and Anderson, R. M. (1987). Transmission dynamics of HIV infection. Nature, 3426, 137-142.
[11]	Hsieh, H. Y. and Valasco-Hernandez, J. X. (1991). Modeling the effect of treatment and behavioral change in HIV transmission dynamics. (On-line: https://ecommons.cornell.edu/bitstream/handle/1813/31721/BU-1143M.pdf%3Bjsessionid%3D0053CF3ED9DA07868A1BBB4D92321815?sequence%3D1
[12]	Moghadas, S. M., Gumel, A. B., Mcleod, R. G. and Gordon, R. (2003). Could condom stop the AIDS epidemic? Journal of Theoretical Medicine, 5 (3-4), 171-181.
[13]	Kimbir, A. R., Musa, S. and Bassey, E. B. (2006). On a two-sex Mathematical model for the prevention of HIV/AIDS in a varying population. J. Math. Assoc. Nig., 33 (201), 1-13.
[14]	Bassey, E. B. and Lebedev, K. A. (2015a). On mathematical model of the impact of non-compliance with preventive measures for the prevention of the spread of HIV/AIDS among heterosexual population. Scientific Journal of KubSAU, 108 (04), 1-8.
[15]	Kefale, B. and Kefale, Y. (2013). Knowledge, attitude, practice and determinant of condom use among people living with HIV/AIDS in Gondar University Hospital North West Ethiopia. J. Phys Pharm Adv, 3 (10), 247-260.
[16]	Bracher, M., Santon G., Watkins, S. C. (2004). Assessing the potential of condom use to prevent the spread of HIV: a micro-simulation study. Stud. Fam. Plan., 35 (1), 48-64.
[17]	Agha, S. (1998). Sexuality activity and condom use in Lusaka, Zambia. (On-line: https://www.guttmacher.org/pubs/journals2403298.html)
[18]	Bassey B. E. and Lebedev K. A. (2015b). On mathematical model of the impact of heterosexual use of condom and antiretroviral therapy for the prevention of HIV/AIDS epidemic, International Journal of Applied and Fundamental Research, 1 (2015), 1-17.
[19]	Kimbir, A. R. and Oluwole, H. K. (2008). A mathematical model of HIV transmission dynamics considering counseling and antiretroviral therapy (ART). J. Mod. Math. Stat., 2 (5), 166-169.
[20]	Bassey, B. E. and Lebedev, K. A. (2016). On numerical analysis of the impact of condom use and counseling for the prevention of heterosexual transmission of HIV/AIDS infection// Proceedings of the XXII-th international scientific conference "the Potential of modern science (Russian Federation, Lipetsk, March 28, 2016.) / Edited by M. J. Levin. – Lipetsk: "maximum information technology", 2 (19), 7-21. http://elibrary.ru/author_items.asp?authorid=80 1157
[21]	Tripathi, A., Naresh, R. and Sharma, D. (2007). Modelling the effect screening of unaware infectives on the spread of HIV infection. Applied Mathematics and Computation 184 (2007) 1053-1068.
[22]	Zahedi, M. S. and Kargar, N. S. (2017). The Volterra-Lyapunov matrix theory for global stability analysis of a model of the HIV/AIDS. Int. J. Biomath., 10 (1): 1-21.
[23]	Osman, S., Makinde, O. D. and Theuri, D. M. (2018). Stability analysis and modeling of listeriosis dynamics in human and animal populations. Global Journal of Pure and Applied Mathematics, 14 (1), 115-138.
[24]	Mafuta, P., Mushanyu, J. and Nhawu, G. (2014). Invariant Region, Endemic Equilibria and Stability Analysis. IOSR Journal of Mathematics, 10 (2), 118-120.
[25]	Akanni, J. O. and Akinpelu, F. O. (2016). Sensitivity analysis of HIV/AIDS model with vertical transmission, treatment and progression rate. International Journal of Scientific and Engineering Research, 7 (8), 497-514.
[26]	Edward, S. and Nyerere, N. (2015). A Mathematical Model for the Dynamics of Cholera with Control Measures. Applied and Computational Mathematics, 4 (2), 53-63.
[27]	Van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180 (2002), 29–48.
[28]	Simon, C. P. and Blume, L. (1994). Mathematics for Economists. New York, W. W. Norton.
[29]	Kahuru, J., Luboobi, L. and Nkansah-Gyekye, Y. (2017a). Stability analysis of the dynamics of tungiasis transmission in endemic areas. Asian Journal of Mathematics and Applications, 2017, 1-24.
[30]	Castillo-Chavez, C., Feng, Z. and Huang, W. (2002). On the computation of R₀ and its role on global stability in Mathematical Approaches for Emerging and Re-emerging Infectious Disease: An Introduction. IMA Volumes in Mathematical and its Approaches, 125, Springer, 229-250.
[31]	LaSalle, J. P. (1976). The stability of dynamical systems. CBMS-NSF regional conference series in applied mathematics 25. SIAM, Philadelphia.
[32]	Lyapunov, A. M. (1992). The General Problem of the Stability of motion, (A. T. Fuller trans.) Taylor and Francis, London.
[33]	Hattaf, K. and Yousfi, N. (2012a). Two optimal treatments of HIV infection model. World Journal of Modeling and Simulation, 8 (1), 27-35.
[34]	Culshaw, R. V., Ruan, S. and Spiteri, R. J. (2004). Optimal HIV treatment by maximizing immune response, Journal of Mathematical Biology, 48 (5), 545-562.
[35]	Bassey, E. B. (2018). Dynamic optimal control model for dual-pair treatment functions of dual-delayed HIV-pathogen infections. Journal of Mathematical Sciences: Advances and Applications, 51 (1), 1-50.
[36]	Zarei, H., Kamyad, A. V. and Effati, S. (2010). Maximizing of asymptomatic stage of fast progressive HIV infected patient using embedding methods. Intelligent Control and Automation, 1 (1), 48-58.
[37]	Hattaf, K. and Yousfi, N. (2012b). Optimal control of a delayed HIV infection model with immune response using an efficient numerical method. ISRN Biomathematics, dio: 10.5402/2012/215124, 1-7.
[38]	Landi, A., Mazzoldi, A. Andreoni, C., Bianchi, M., Cavallini, A., Laurino, M., Ricotti, L., Iuliano, R., Matteoli, B., Ceccherini-Nelli, L. (2008). Modeling and control of HIV dynamics. Computer Method and Programs in Biomedicine, 89 (2), 162-168.

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APA Style

Bassey Echeng Bassey, Adagba Odey Henry. (2021). Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics. Pure and Applied Mathematics Journal, 10(1), 9-31. https://doi.org/10.11648/j.pamj.20211001.12

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ACS Style

Bassey Echeng Bassey; Adagba Odey Henry. Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics. Pure Appl. Math. J. 2021, 10(1), 9-31. doi: 10.11648/j.pamj.20211001.12

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AMA Style

Bassey Echeng Bassey, Adagba Odey Henry. Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics. Pure Appl Math J. 2021;10(1):9-31. doi: 10.11648/j.pamj.20211001.12

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@article{10.11648/j.pamj.20211001.12,
  author = {Bassey Echeng Bassey and Adagba Odey Henry},
  title = {Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics},
  journal = {Pure and Applied Mathematics Journal},
  volume = {10},
  number = {1},
  pages = {9-31},
  doi = {10.11648/j.pamj.20211001.12},
  url = {https://doi.org/10.11648/j.pamj.20211001.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20211001.12},
  abstract = {Devastatingly, in spite of the long standing research works on HIV/AIDS infection and treatment dynamics, reviews of existing models clearly shown that the behavioral attitude to treatment consistency by those screened to become aware and those receiving treatment have not been given the desired attention. Moreso, the inconsistency following avoidable treatment truncation and later resumption of treatment by these classes of infectives, which could lead to colossal drug abuse is also not accorded the much expected consideration. Therefore, in this present study, we sought and formulated a nonlinear 6-Dimensional deterministic mathematical HIV/AIDS dynamic model that accounted for the global stability analysis of the role of antiretroviral therapy abuse for the treatment of HIV/AIDS epidemic. The model is structured upon dynamical interactions between 6-subpopulations and HI-virus under bilinear control functions with constant screening of the susceptible. It is assumed that the rate of resumption of ART upon truncation is less than initial ART truncation following the incorporation of HIV aware infectives not ready to receive ART treatment and HIV aware infectives with truncated treatment protocol The system mathematical well-posedness was investigated and model reproduction number determined for both off- treatment (with value ) and for onset-treatment (with value ). We considered the model for off-treatment and thereafter by incorporating LaSalle’s invariant principle into classical Lyapunov function method, we presented an approach for the global stability analysis of the role of ART abuse in HIV/AIDS treatment. Furthermore, the analysis and results of this paper presented a dynamic methodological application of bilinear control functions and an impeccable understanding of the fundamental mechanism in HIV/AIDS treatment in the presence of ART abuse. Using in-built Runge-Kutta of order of precision 4 in a Mathcad surface, numerical validity of model is conducted to investigate the study theoretical and analytical predictions. Results shows that application of onset-treatment functions with trend of ART abuse yield tremendous reduction in HIV/AIDS infection epidemic following the recovery rate of the susceptible population with value increasing from 0.5 cells/mm3 to 1.203 cells/mm3 within the first  months and attained stability of 0.62 cells/mm3 through the time interval of 20- 30 months.},
 year = {2021}
}

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TY  - JOUR
T1  - Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics
AU  - Bassey Echeng Bassey
AU  - Adagba Odey Henry
Y1  - 2021/03/04
PY  - 2021
N1  - https://doi.org/10.11648/j.pamj.20211001.12
DO  - 10.11648/j.pamj.20211001.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 9
EP  - 31
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20211001.12
AB  - Devastatingly, in spite of the long standing research works on HIV/AIDS infection and treatment dynamics, reviews of existing models clearly shown that the behavioral attitude to treatment consistency by those screened to become aware and those receiving treatment have not been given the desired attention. Moreso, the inconsistency following avoidable treatment truncation and later resumption of treatment by these classes of infectives, which could lead to colossal drug abuse is also not accorded the much expected consideration. Therefore, in this present study, we sought and formulated a nonlinear 6-Dimensional deterministic mathematical HIV/AIDS dynamic model that accounted for the global stability analysis of the role of antiretroviral therapy abuse for the treatment of HIV/AIDS epidemic. The model is structured upon dynamical interactions between 6-subpopulations and HI-virus under bilinear control functions with constant screening of the susceptible. It is assumed that the rate of resumption of ART upon truncation is less than initial ART truncation following the incorporation of HIV aware infectives not ready to receive ART treatment and HIV aware infectives with truncated treatment protocol The system mathematical well-posedness was investigated and model reproduction number determined for both off- treatment (with value ) and for onset-treatment (with value ). We considered the model for off-treatment and thereafter by incorporating LaSalle’s invariant principle into classical Lyapunov function method, we presented an approach for the global stability analysis of the role of ART abuse in HIV/AIDS treatment. Furthermore, the analysis and results of this paper presented a dynamic methodological application of bilinear control functions and an impeccable understanding of the fundamental mechanism in HIV/AIDS treatment in the presence of ART abuse. Using in-built Runge-Kutta of order of precision 4 in a Mathcad surface, numerical validity of model is conducted to investigate the study theoretical and analytical predictions. Results shows that application of onset-treatment functions with trend of ART abuse yield tremendous reduction in HIV/AIDS infection epidemic following the recovery rate of the susceptible population with value increasing from 0.5 cells/mm3 to 1.203 cells/mm3 within the first  months and attained stability of 0.62 cells/mm3 through the time interval of 20- 30 months.
VL  - 10
IS  - 1
ER  -

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Author Information

Bassey Echeng Bassey

Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Abakaliki, Nigeria
Adagba Odey Henry

Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Abakaliki, Nigeria

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Bassey Echeng Bassey, Adagba Odey Henry. (2021). Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics. Pure and Applied Mathematics Journal, 10(1), 9-31. https://doi.org/10.11648/j.pamj.20211001.12

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ACS Style

Bassey Echeng Bassey; Adagba Odey Henry. Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics. Pure Appl. Math. J. 2021, 10(1), 9-31. doi: 10.11648/j.pamj.20211001.12

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AMA Style

Bassey Echeng Bassey, Adagba Odey Henry. Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics. Pure Appl Math J. 2021;10(1):9-31. doi: 10.11648/j.pamj.20211001.12

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@article{10.11648/j.pamj.20211001.12,
  author = {Bassey Echeng Bassey and Adagba Odey Henry},
  title = {Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics},
  journal = {Pure and Applied Mathematics Journal},
  volume = {10},
  number = {1},
  pages = {9-31},
  doi = {10.11648/j.pamj.20211001.12},
  url = {https://doi.org/10.11648/j.pamj.20211001.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20211001.12},
  abstract = {Devastatingly, in spite of the long standing research works on HIV/AIDS infection and treatment dynamics, reviews of existing models clearly shown that the behavioral attitude to treatment consistency by those screened to become aware and those receiving treatment have not been given the desired attention. Moreso, the inconsistency following avoidable treatment truncation and later resumption of treatment by these classes of infectives, which could lead to colossal drug abuse is also not accorded the much expected consideration. Therefore, in this present study, we sought and formulated a nonlinear 6-Dimensional deterministic mathematical HIV/AIDS dynamic model that accounted for the global stability analysis of the role of antiretroviral therapy abuse for the treatment of HIV/AIDS epidemic. The model is structured upon dynamical interactions between 6-subpopulations and HI-virus under bilinear control functions with constant screening of the susceptible. It is assumed that the rate of resumption of ART upon truncation is less than initial ART truncation following the incorporation of HIV aware infectives not ready to receive ART treatment and HIV aware infectives with truncated treatment protocol The system mathematical well-posedness was investigated and model reproduction number determined for both off- treatment (with value ) and for onset-treatment (with value ). We considered the model for off-treatment and thereafter by incorporating LaSalle’s invariant principle into classical Lyapunov function method, we presented an approach for the global stability analysis of the role of ART abuse in HIV/AIDS treatment. Furthermore, the analysis and results of this paper presented a dynamic methodological application of bilinear control functions and an impeccable understanding of the fundamental mechanism in HIV/AIDS treatment in the presence of ART abuse. Using in-built Runge-Kutta of order of precision 4 in a Mathcad surface, numerical validity of model is conducted to investigate the study theoretical and analytical predictions. Results shows that application of onset-treatment functions with trend of ART abuse yield tremendous reduction in HIV/AIDS infection epidemic following the recovery rate of the susceptible population with value increasing from 0.5 cells/mm3 to 1.203 cells/mm3 within the first  months and attained stability of 0.62 cells/mm3 through the time interval of 20- 30 months.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics
AU  - Bassey Echeng Bassey
AU  - Adagba Odey Henry
Y1  - 2021/03/04
PY  - 2021
N1  - https://doi.org/10.11648/j.pamj.20211001.12
DO  - 10.11648/j.pamj.20211001.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 9
EP  - 31
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20211001.12
AB  - Devastatingly, in spite of the long standing research works on HIV/AIDS infection and treatment dynamics, reviews of existing models clearly shown that the behavioral attitude to treatment consistency by those screened to become aware and those receiving treatment have not been given the desired attention. Moreso, the inconsistency following avoidable treatment truncation and later resumption of treatment by these classes of infectives, which could lead to colossal drug abuse is also not accorded the much expected consideration. Therefore, in this present study, we sought and formulated a nonlinear 6-Dimensional deterministic mathematical HIV/AIDS dynamic model that accounted for the global stability analysis of the role of antiretroviral therapy abuse for the treatment of HIV/AIDS epidemic. The model is structured upon dynamical interactions between 6-subpopulations and HI-virus under bilinear control functions with constant screening of the susceptible. It is assumed that the rate of resumption of ART upon truncation is less than initial ART truncation following the incorporation of HIV aware infectives not ready to receive ART treatment and HIV aware infectives with truncated treatment protocol The system mathematical well-posedness was investigated and model reproduction number determined for both off- treatment (with value ) and for onset-treatment (with value ). We considered the model for off-treatment and thereafter by incorporating LaSalle’s invariant principle into classical Lyapunov function method, we presented an approach for the global stability analysis of the role of ART abuse in HIV/AIDS treatment. Furthermore, the analysis and results of this paper presented a dynamic methodological application of bilinear control functions and an impeccable understanding of the fundamental mechanism in HIV/AIDS treatment in the presence of ART abuse. Using in-built Runge-Kutta of order of precision 4 in a Mathcad surface, numerical validity of model is conducted to investigate the study theoretical and analytical predictions. Results shows that application of onset-treatment functions with trend of ART abuse yield tremendous reduction in HIV/AIDS infection epidemic following the recovery rate of the susceptible population with value increasing from 0.5 cells/mm3 to 1.203 cells/mm3 within the first  months and attained stability of 0.62 cells/mm3 through the time interval of 20- 30 months.
VL  - 10
IS  - 1
ER  -

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