Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment

Eshetu Dadi Gurmu; Boka Kumsa Bole; Purnachandra Rao Koya

doi:doi:10.11648/j.ajam.20200801.16

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Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment

Eshetu Dadi Gurmu, Boka Kumsa Bole, Purnachandra Rao Koya

Published in American Journal of Applied Mathematics (Volume 8, Issue 1)

Received: 7 November 2019 Accepted: 2 January 2020 Published: 13 February 2020

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Abstract

This paper examines a mathematical modelling of HIV/AIDS transmission dynamics with drug resistance compartment. A nonlinear deterministic mathematical model for the problem is proposed using a system of ordinary differential equations. The aim of this study is to investigate the role of passive immunity and drug therapy in reducing the viral replication and transmission of the disease. The well possedness of the formulated model equations was proved and the equilibrium points of the model have been identified. In addition, the basic reproductive number that governs the disease transmission is obtained from the largest eigenvalue of the next-generation matrix. Both local and global stability of the disease free equilibrium and endemic of the model was established using basic reproduction number. The results show that the disease free equilibrium is locally asymptotically stable if the basic reproduction number is less than unity and unstable if the basic reproduction number is greater than unity. It is observed that if the basic reproduction is less than one then the solution converges to the disease free steady state i.e., disease will wipe out and thus the drug therapy 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 drug therapy is said to be unsuccessful. Sensitivity analysis of the model is performed on the key parameters to determine their relative importance and potential impact on the transmission dynamics of HIV/AIDS. Numerical results of the model show that a combination of passive immunity and drug therapy is the best strategy to reduce the disease from the community.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 1)
DOI	10.11648/j.ajam.20200801.16
Page(s)	34-45
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), 2020. Published by Science Publishing Group

Keywords

HIV, Reproductive Number, Stability Analysis, Drug Therapy

References

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[2]	Nowak, M. A., and May, R. M. (2000). Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford, U. K.
[3]	Nowak, M. A., and Bangham, C. R. M. (1996). Population dynamics of immune responses to persistent viruses. Science 272, 5258, 74.
[4]	N. Rom, S. B. Markowitz (2007). Environmental and Occupational Medicine, Lippincott Williams & Wilkins.
[5]	UNAIDSDATA (2019).
[6]	Robert J. Smith, Jing Li, Jun Mao and Beni Sahai (2013). Using within-host Mathematical Modelling to predict the long-term outcome of Human Papillomavirus Vaccines. volume 38 Canadian Applied Mathematics Quarterly, 21 (2).
[7]	K. O. Okosun, O. D. Makinde, I. Takaidza (2013). Impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives. Applied Mathematical Modelling 37, 3802–3820.
[8]	Karrakchou, M. Rachik, S. Gourari (2006). Optimal control and infectiology: application to an HIV/ AIDS model, Appl. Math. Comput. 177, 807–818.
[9]	B. M. Adams, H. T. Banks, Kwon Hee-Dae, T. T. Hien T (2004). Dynamic multidrug therapies for HIV: Optimal and STI control approaches. Mathematical Biosciences and Engineering. 1 and 2, 223–241.
[10]	Silva, CJ, Torres, DF (2017). Modeling and optimal control of HIV/AIDS prevention through PrEP. arXiv: 1703. 06446.
[11]	Mukandavire, Z, Mitchell, KM (2016). Comparing the impact of increasing condom use or HIV pre-exposure prophylaxis (PrEP) use among female sex workers. Epidemics 14, 62-70.
[12]	Grant, H, Mukandavire, Z, Eakle, R, Prudden, H, Gomez, GB, Rees, H, Watts, C (2017). When declines in condom use while using are PrEP a concern Modelling insights from a Hillbrow. South Africa case study. J. Int. AIDS Soc. 20 (1), 21744.
[13]	Eshetu Dadi Gurmu, Purnachandra Rao Koya (2019). Sensitivity Analysis and Modeling the Impact of Screening on the Transmission Dynamics of Human Papilloma Virus (HPV). American Journal of Applied Mathematics. 7 (3), pp. 70-79.
[14]	Eshetu Dadi Gurmu and Purnachandra Rao Koya (2019). Impact of Chemotherapy treatment of SITR Compartmentalization and Modeling of Human Papilloma Virus (HPV). IOSR Journal of Mathematics (IOSR–JM). 15 (3), Pp. 17–29.
[15]	Chitnis, N., Hyman, J. M., and Cusching, 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.

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Eshetu Dadi Gurmu, Boka Kumsa Bole, Purnachandra Rao Koya. (2020). Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment. American Journal of Applied Mathematics, 8(1), 34-45. https://doi.org/10.11648/j.ajam.20200801.16

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ACS Style

Eshetu Dadi Gurmu; Boka Kumsa Bole; Purnachandra Rao Koya. Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment. Am. J. Appl. Math. 2020, 8(1), 34-45. doi: 10.11648/j.ajam.20200801.16

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AMA Style

Eshetu Dadi Gurmu, Boka Kumsa Bole, Purnachandra Rao Koya. Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment. Am J Appl Math. 2020;8(1):34-45. doi: 10.11648/j.ajam.20200801.16

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@article{10.11648/j.ajam.20200801.16,
  author = {Eshetu Dadi Gurmu and Boka Kumsa Bole and Purnachandra Rao Koya},
  title = {Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {1},
  pages = {34-45},
  doi = {10.11648/j.ajam.20200801.16},
  url = {https://doi.org/10.11648/j.ajam.20200801.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200801.16},
  abstract = {This paper examines a mathematical modelling of HIV/AIDS transmission dynamics with drug resistance compartment. A nonlinear deterministic mathematical model for the problem is proposed using a system of ordinary differential equations. The aim of this study is to investigate the role of passive immunity and drug therapy in reducing the viral replication and transmission of the disease. The well possedness of the formulated model equations was proved and the equilibrium points of the model have been identified. In addition, the basic reproductive number that governs the disease transmission is obtained from the largest eigenvalue of the next-generation matrix. Both local and global stability of the disease free equilibrium and endemic of the model was established using basic reproduction number. The results show that the disease free equilibrium is locally asymptotically stable if the basic reproduction number is less than unity and unstable if the basic reproduction number is greater than unity. It is observed that if the basic reproduction is less than one then the solution converges to the disease free steady state i.e., disease will wipe out and thus the drug therapy 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 drug therapy is said to be unsuccessful. Sensitivity analysis of the model is performed on the key parameters to determine their relative importance and potential impact on the transmission dynamics of HIV/AIDS. Numerical results of the model show that a combination of passive immunity and drug therapy is the best strategy to reduce the disease from the community.},
 year = {2020}
}

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TY  - JOUR
T1  - Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment
AU  - Eshetu Dadi Gurmu
AU  - Boka Kumsa Bole
AU  - Purnachandra Rao Koya
Y1  - 2020/02/13
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200801.16
DO  - 10.11648/j.ajam.20200801.16
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 34
EP  - 45
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200801.16
AB  - This paper examines a mathematical modelling of HIV/AIDS transmission dynamics with drug resistance compartment. A nonlinear deterministic mathematical model for the problem is proposed using a system of ordinary differential equations. The aim of this study is to investigate the role of passive immunity and drug therapy in reducing the viral replication and transmission of the disease. The well possedness of the formulated model equations was proved and the equilibrium points of the model have been identified. In addition, the basic reproductive number that governs the disease transmission is obtained from the largest eigenvalue of the next-generation matrix. Both local and global stability of the disease free equilibrium and endemic of the model was established using basic reproduction number. The results show that the disease free equilibrium is locally asymptotically stable if the basic reproduction number is less than unity and unstable if the basic reproduction number is greater than unity. It is observed that if the basic reproduction is less than one then the solution converges to the disease free steady state i.e., disease will wipe out and thus the drug therapy 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 drug therapy is said to be unsuccessful. Sensitivity analysis of the model is performed on the key parameters to determine their relative importance and potential impact on the transmission dynamics of HIV/AIDS. Numerical results of the model show that a combination of passive immunity and drug therapy is the best strategy to reduce the disease from the community.
VL  - 8
IS  - 1
ER  -

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Author Information

Eshetu Dadi Gurmu

Department of Mathematics, Wollega University, Nekemte, Ethiopia
Boka Kumsa Bole

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

Department of Mathematics, Wollega University, Nekemte, Ethiopia

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APA Style

Eshetu Dadi Gurmu, Boka Kumsa Bole, Purnachandra Rao Koya. (2020). Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment. American Journal of Applied Mathematics, 8(1), 34-45. https://doi.org/10.11648/j.ajam.20200801.16

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ACS Style

Eshetu Dadi Gurmu; Boka Kumsa Bole; Purnachandra Rao Koya. Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment. Am. J. Appl. Math. 2020, 8(1), 34-45. doi: 10.11648/j.ajam.20200801.16

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AMA Style

Eshetu Dadi Gurmu, Boka Kumsa Bole, Purnachandra Rao Koya. Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment. Am J Appl Math. 2020;8(1):34-45. doi: 10.11648/j.ajam.20200801.16

Copy | Download

@article{10.11648/j.ajam.20200801.16,
  author = {Eshetu Dadi Gurmu and Boka Kumsa Bole and Purnachandra Rao Koya},
  title = {Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {1},
  pages = {34-45},
  doi = {10.11648/j.ajam.20200801.16},
  url = {https://doi.org/10.11648/j.ajam.20200801.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200801.16},
  abstract = {This paper examines a mathematical modelling of HIV/AIDS transmission dynamics with drug resistance compartment. A nonlinear deterministic mathematical model for the problem is proposed using a system of ordinary differential equations. The aim of this study is to investigate the role of passive immunity and drug therapy in reducing the viral replication and transmission of the disease. The well possedness of the formulated model equations was proved and the equilibrium points of the model have been identified. In addition, the basic reproductive number that governs the disease transmission is obtained from the largest eigenvalue of the next-generation matrix. Both local and global stability of the disease free equilibrium and endemic of the model was established using basic reproduction number. The results show that the disease free equilibrium is locally asymptotically stable if the basic reproduction number is less than unity and unstable if the basic reproduction number is greater than unity. It is observed that if the basic reproduction is less than one then the solution converges to the disease free steady state i.e., disease will wipe out and thus the drug therapy 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 drug therapy is said to be unsuccessful. Sensitivity analysis of the model is performed on the key parameters to determine their relative importance and potential impact on the transmission dynamics of HIV/AIDS. Numerical results of the model show that a combination of passive immunity and drug therapy is the best strategy to reduce the disease from the community.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment
AU  - Eshetu Dadi Gurmu
AU  - Boka Kumsa Bole
AU  - Purnachandra Rao Koya
Y1  - 2020/02/13
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200801.16
DO  - 10.11648/j.ajam.20200801.16
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 34
EP  - 45
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200801.16
AB  - This paper examines a mathematical modelling of HIV/AIDS transmission dynamics with drug resistance compartment. A nonlinear deterministic mathematical model for the problem is proposed using a system of ordinary differential equations. The aim of this study is to investigate the role of passive immunity and drug therapy in reducing the viral replication and transmission of the disease. The well possedness of the formulated model equations was proved and the equilibrium points of the model have been identified. In addition, the basic reproductive number that governs the disease transmission is obtained from the largest eigenvalue of the next-generation matrix. Both local and global stability of the disease free equilibrium and endemic of the model was established using basic reproduction number. The results show that the disease free equilibrium is locally asymptotically stable if the basic reproduction number is less than unity and unstable if the basic reproduction number is greater than unity. It is observed that if the basic reproduction is less than one then the solution converges to the disease free steady state i.e., disease will wipe out and thus the drug therapy 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 drug therapy is said to be unsuccessful. Sensitivity analysis of the model is performed on the key parameters to determine their relative importance and potential impact on the transmission dynamics of HIV/AIDS. Numerical results of the model show that a combination of passive immunity and drug therapy is the best strategy to reduce the disease from the community.
VL  - 8
IS  - 1
ER  -

Copy | Download