Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment

Kumama Regassa; Purnachandra Rao Koya

doi:doi:10.11648/j.ajam.20190704.14

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Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment

Kumama Regassa, Purnachandra Rao Koya

Published in American Journal of Applied Mathematics (Volume 7, Issue 4)

Received: 28 June 2019 Accepted: 3 August 2019 Published: 20 September 2019

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Abstract

In this paper, a mathematical model has been formulated to describe the population dynamics of human cells pertaining to the HIV/AIDS disease with ART as treatment and is analyzed. The human cells have been divided into four compartments Susceptible – Asymptomatic – Symptomatic – AIDS (SAIV). The well posedness of the four dimensional dynamical system is proved and the steady states of the model are identified. Additionally, parametric expression for the basic reproduction number is constructed following next generation matrix method and analyzed its stability using Routh Hurwitz criterion. From the analytical and numerical simulation studies it is observed that if the basic reproduction is less than one unit then the solution converges to the disease free steady state i.e., disease will wipe out and thus the treatment is said to be successful. On the other hand, if the basic reproduction number is greater than one then the solution converges to endemic equilibrium point and thus the infectious cells continue to replicate i.e., disease will persist and thus the treatment is said to be unsuccessful. Sensitivity analysis of the model parameters is conducted and their impact on the reproduction number is analyzed. Finally, the model of the present study simulated using MATLAB. The results and observations have been included in the text of this paper lucidly.

Published in	American Journal of Applied Mathematics (Volume 7, Issue 4)
DOI	10.11648/j.ajam.20190704.14
Page(s)	127-136
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), 2019. Published by Science Publishing Group

Keywords

HIV, ART, Basic Reproduction Number, Stability Analysis, Routh Hurwitz Criterion

References

[1]	Brauer F., P. van den Driessche and W. Jianhong. Mathematical Epidemiology, volume 1945. Springer-Verlag Berlin Heidelberg, Canada, 2008.
[2]	W. S. Ronald and H. James. Mathematical biology: An Introduction with Maple and Matlab. Springer Dordrecht Heidelberg, Boston, (1996).
[3]	J. Robertson. World Health Organization. AIDS epidemic update: November in 2009 UNAIDS/09.36E / JC1700E. ISBN 978 92 9173 832 8.
[4]	A. A. Ejigu. Mathematical Modeling of HIV/AIDS transmission under treatment structured by age of infection, Stellenbosch University (2010).
[5]	Alemu Geleta Wedajo, Boka Kumsa Bole, Purnachandra Rao Koya. The Impact of Susceptible Human Immigrants on the Spread and Dynamics of Malaria Transmission. American Journal of Applied Mathematics. Vol. 6, No. 3, 2018, Pp. 117-127. doi: 10.11648/j.ajam.20180603.13.
[6]	Alemu Geleta Wedajo, Boka Kumsa Bole, Purnachandra Rao Koya. Analysis of SIR Mathematical Model for Malaria disease with the inclusion of Infected Immigrants. IOSR Journal of Mathematics (IOSR-JM). Volume 14, Issue 5 Ver. I (Sep - Oct 2018), PP 10-21 DOI: 10.9790/5728-1405011021.
[7]	Geremew Kenassa Edessa, Boka Kumsa, Purnachandra Rao Koya. Dynamical behavior of Susceptible prey – Infected prey – Predator Populations. IOSR Journal of Mathematics (IOSR-JM). Volume 14, Issue 4 Ver. III (Jul - Aug 2018), PP 31-41. DOI: 10.9790/5728-1404033141.
[8]	Solomon Tolcha, Boka Kumsa, Purnachandra Rao Koya. Modeling and Simulation Study of Mutuality Interactions with Type II functional Response and Harvesting. American Journal of Applied Mathematics. Vol. 6, No. 3, 2018, pp. 109-116. doi: 10.11648/j.ajam.20180603.
[9]	Chitnis N., Hyman J. M., and Cushing J. M. (2008). Determining important Parameters in the spread of malaria through the sensitivity analysis of a mathematical Model. Bulletin of Mathematical Biology 70 (5): 1272–12.
[10]	P. van den Driesch and James Warmouth. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences 180 (2002) 29–48.
[11]	Jones E., P., Roemer. Department of Mathematics United States Naval Academy, 572C Holloway Road, Chauvenet Hall, Annapolis, MD 21402 (peteusna@gmail.com, raghupat@usna.edu).

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APA Style

Kumama Regassa, Purnachandra Rao Koya. (2019). Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment. American Journal of Applied Mathematics, 7(4), 127-136. https://doi.org/10.11648/j.ajam.20190704.14

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ACS Style

Kumama Regassa; Purnachandra Rao Koya. Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment. Am. J. Appl. Math. 2019, 7(4), 127-136. doi: 10.11648/j.ajam.20190704.14

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AMA Style

Kumama Regassa, Purnachandra Rao Koya. Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment. Am J Appl Math. 2019;7(4):127-136. doi: 10.11648/j.ajam.20190704.14

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@article{10.11648/j.ajam.20190704.14,
  author = {Kumama Regassa and Purnachandra Rao Koya},
  title = {Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment},
  journal = {American Journal of Applied Mathematics},
  volume = {7},
  number = {4},
  pages = {127-136},
  doi = {10.11648/j.ajam.20190704.14},
  url = {https://doi.org/10.11648/j.ajam.20190704.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190704.14},
  abstract = {In this paper, a mathematical model has been formulated to describe the population dynamics of human cells pertaining to the HIV/AIDS disease with ART as treatment and is analyzed. The human cells have been divided into four compartments Susceptible – Asymptomatic – Symptomatic – AIDS (SAIV). The well posedness of the four dimensional dynamical system is proved and the steady states of the model are identified. Additionally, parametric expression for the basic reproduction number is constructed following next generation matrix method and analyzed its stability using Routh Hurwitz criterion. From the analytical and numerical simulation studies it is observed that if the basic reproduction is less than one unit then the solution converges to the disease free steady state i.e., disease will wipe out and thus the treatment is said to be successful. On the other hand, if the basic reproduction number is greater than one then the solution converges to endemic equilibrium point and thus the infectious cells continue to replicate i.e., disease will persist and thus the treatment is said to be unsuccessful. Sensitivity analysis of the model parameters is conducted and their impact on the reproduction number is analyzed. Finally, the model of the present study simulated using MATLAB. The results and observations have been included in the text of this paper lucidly.},
 year = {2019}
}

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TY  - JOUR
T1  - Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment
AU  - Kumama Regassa
AU  - Purnachandra Rao Koya
Y1  - 2019/09/20
PY  - 2019
N1  - https://doi.org/10.11648/j.ajam.20190704.14
DO  - 10.11648/j.ajam.20190704.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 127
EP  - 136
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20190704.14
AB  - In this paper, a mathematical model has been formulated to describe the population dynamics of human cells pertaining to the HIV/AIDS disease with ART as treatment and is analyzed. The human cells have been divided into four compartments Susceptible – Asymptomatic – Symptomatic – AIDS (SAIV). The well posedness of the four dimensional dynamical system is proved and the steady states of the model are identified. Additionally, parametric expression for the basic reproduction number is constructed following next generation matrix method and analyzed its stability using Routh Hurwitz criterion. From the analytical and numerical simulation studies it is observed that if the basic reproduction is less than one unit then the solution converges to the disease free steady state i.e., disease will wipe out and thus the treatment is said to be successful. On the other hand, if the basic reproduction number is greater than one then the solution converges to endemic equilibrium point and thus the infectious cells continue to replicate i.e., disease will persist and thus the treatment is said to be unsuccessful. Sensitivity analysis of the model parameters is conducted and their impact on the reproduction number is analyzed. Finally, the model of the present study simulated using MATLAB. The results and observations have been included in the text of this paper lucidly.
VL  - 7
IS  - 4
ER  -

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Author Information

Kumama Regassa

Department of Mathematics, Wollega University, Nekemte, Ethiopia
Purnachandra Rao Koya

Department of Mathematics, Wollega University, Nekemte, Ethiopia

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APA Style

Kumama Regassa, Purnachandra Rao Koya. (2019). Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment. American Journal of Applied Mathematics, 7(4), 127-136. https://doi.org/10.11648/j.ajam.20190704.14

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ACS Style

Kumama Regassa; Purnachandra Rao Koya. Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment. Am. J. Appl. Math. 2019, 7(4), 127-136. doi: 10.11648/j.ajam.20190704.14

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AMA Style

Kumama Regassa, Purnachandra Rao Koya. Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment. Am J Appl Math. 2019;7(4):127-136. doi: 10.11648/j.ajam.20190704.14

Copy | Download

@article{10.11648/j.ajam.20190704.14,
  author = {Kumama Regassa and Purnachandra Rao Koya},
  title = {Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment},
  journal = {American Journal of Applied Mathematics},
  volume = {7},
  number = {4},
  pages = {127-136},
  doi = {10.11648/j.ajam.20190704.14},
  url = {https://doi.org/10.11648/j.ajam.20190704.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190704.14},
  abstract = {In this paper, a mathematical model has been formulated to describe the population dynamics of human cells pertaining to the HIV/AIDS disease with ART as treatment and is analyzed. The human cells have been divided into four compartments Susceptible – Asymptomatic – Symptomatic – AIDS (SAIV). The well posedness of the four dimensional dynamical system is proved and the steady states of the model are identified. Additionally, parametric expression for the basic reproduction number is constructed following next generation matrix method and analyzed its stability using Routh Hurwitz criterion. From the analytical and numerical simulation studies it is observed that if the basic reproduction is less than one unit then the solution converges to the disease free steady state i.e., disease will wipe out and thus the treatment is said to be successful. On the other hand, if the basic reproduction number is greater than one then the solution converges to endemic equilibrium point and thus the infectious cells continue to replicate i.e., disease will persist and thus the treatment is said to be unsuccessful. Sensitivity analysis of the model parameters is conducted and their impact on the reproduction number is analyzed. Finally, the model of the present study simulated using MATLAB. The results and observations have been included in the text of this paper lucidly.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment
AU  - Kumama Regassa
AU  - Purnachandra Rao Koya
Y1  - 2019/09/20
PY  - 2019
N1  - https://doi.org/10.11648/j.ajam.20190704.14
DO  - 10.11648/j.ajam.20190704.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 127
EP  - 136
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20190704.14
AB  - In this paper, a mathematical model has been formulated to describe the population dynamics of human cells pertaining to the HIV/AIDS disease with ART as treatment and is analyzed. The human cells have been divided into four compartments Susceptible – Asymptomatic – Symptomatic – AIDS (SAIV). The well posedness of the four dimensional dynamical system is proved and the steady states of the model are identified. Additionally, parametric expression for the basic reproduction number is constructed following next generation matrix method and analyzed its stability using Routh Hurwitz criterion. From the analytical and numerical simulation studies it is observed that if the basic reproduction is less than one unit then the solution converges to the disease free steady state i.e., disease will wipe out and thus the treatment is said to be successful. On the other hand, if the basic reproduction number is greater than one then the solution converges to endemic equilibrium point and thus the infectious cells continue to replicate i.e., disease will persist and thus the treatment is said to be unsuccessful. Sensitivity analysis of the model parameters is conducted and their impact on the reproduction number is analyzed. Finally, the model of the present study simulated using MATLAB. The results and observations have been included in the text of this paper lucidly.
VL  - 7
IS  - 4
ER  -

Copy | Download