Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment

Ram Singh; Shoket Ali; Madhu Jain; Rakhee

doi:doi:10.11648/j.ajam.20160405.14

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Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment

Ram Singh, Shoket Ali, Madhu Jain, Rakhee

Published in American Journal of Applied Mathematics (Volume 4, Issue 5)

Received: 30 August 2016 Accepted: 23 September 2016 Published: 14 October 2016

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Abstract

The mathematical model for analyzing the transmission dynamics of HIV/AIDS epidemic with treatment is studied by considering the three latent compartments for slow, medium and fast progresses of developing the AIDS. By constructing the system of differential equations for the different population groups namely susceptible, three types of latent individuals, symptomatic stage group and full blown AIDS individuals, the mathematical analysis is carried out in order to understand the dynamics of disease spread. By determining the basic reproduction number (R₀), the model examines the two equilibrium points (i) the disease free equilibrium and (ii) the endemic equilibrium. It is established that if R₀ <1, the disease free equilibrium is locally and globally asymptotically stable. The stability of endemic equilibrium has also been discussed.

Published in	American Journal of Applied Mathematics (Volume 4, Issue 5)
DOI	10.11648/j.ajam.20160405.14
Page(s)	222-234
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), 2016. Published by Science Publishing Group

Keywords

Transmission Dynamic, HIV/AIDS, Latent Compartments, Reproduction Number, Stability

References

[1]	May R. M. and Anderson R. M., “Transmission dynamics of HIV infection”, Nature. 236, 137-142 (1987).
[2]	Anderson R. M., Medly G. F., May R. M. and Johnson A. M., “A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS”, IMA J. Math. Appl. Med. Biol. 3, 229–263 (1986).
[3]	Anderson R. M., “The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS”, J. AIDS. 1, 241–256 (1988).
[4]	Morgan D., Mahe C., Mayanja B., Okongo J. M., Lubega R. and Whitworth J. A., “HIV-1infection in rural Africa: is there a difference in median time to AIDS and survival compared with that in industrialized countries”, AIDS. 16, 597–632 (2002).
[5]	Bachar M. and Dorfmayr A., “HIV treatment models with time delay”, C. R. Biol. 327, 983–994 (2004).
[6]	Stoddart C. A. and Reyes R. A., “Models of HIV-1 disease: A review of current status, Drug Discov”, Today: Disease Models. 3, 113–119 (2006).
[7]	Alexander M. E. and Moghadas S. M., “Bifurcation analysis of an SIRS epidemic model with generalised incidence”, SIAM J. Appl. Math. 65, 1794-1816 (2005).
[8]	Cai L., Li X. and Ghosh M., “Global stability of a stage-structured epidemic model with a nonlinear incidence”, Appl. Math. Comput. 214, 73-82 (2009).
[9]	Cai L., Li X., Ghosh M. and Guo B., “Stability analysis of an HIV/AIDS epidemic model with treatment”, J. Comput. Appl. Math. 229, 313-323 (2009).
[10]	Huo H.F. and Feng L.X., “Global stability for an HIV/AIDS epidemic model with different latent stages and treatment”, Appl. Math. Model. 37, 1480-1489 (2013).
[11]	Okosun K. O., Makinde O. D. and Takaidza I., “Impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives”, Appl. Math. Model. 37, 3802-3820 (2013).
[12]	Defeng L. and Wang B., “A novel time delayed HIV/AIDS model with vaccination & antiretroviral therapy and its stability analysis”, Appl. Math. Model. 37, 4608-4625 (2013).
[13]	Bhunu C. P. and Mushayabasa S., “Modelling the transmission dynamics of HIV/AIDS and hepatitis C virus co-infection”, HIV/AIDS Rev. 12(2), 37-42 (2013).
[14]	Cai L., Guo S. and Wang S., “Analysis of an extended HIV/AIDS epidemic model with treatment”, Appl. Math. Comput. 336, 621-627 (2014).
[15]	Kaur N., Ghosh M. and Bhatia S. S., “Mathematical analysis of the transmission dynamics of HIV/AIDS: Role of female sex workers”, Appl. Math. Inf. Sci. 8 (5), 2491-2501 (2014).
[16]	Elaiw A. M. and Almuallem N. A., “Global properties of delayed-HIV dynamics models with differential drug efficacy in cocirculating target cells”, Appl. Math. Comput. 265, 1067-1089 (2015).
[17]	Wang J., Zhang R. and Kuniya T., “Global dynamics for a class of age-infection HIV models with nonlinear infection rate”, Math. Analysis and Appl. 432 (1), 289-313 (2015).
[18]	Shen M., Xiao Y. and Rong L., “Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics”, Math. Biosci. 263, 37-50 (2015).
[19]	Huo H. F., Chen R. and Wang X. Y., “Modelling and stability of HIV/AIDS epidemic model with treatment”, Appl. Math. Model. 40(13-14), 6550–6559 (2016).
[20]	Ven den Driessche P. and Watmough J., “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”, Math. Biosci. 180, 29–48 (2002).
[21]	Lakshmikantham V., Leela S. and Martynyuk A. A., “Stability Analysis of Nonlinear Systems”, Marcel Dekker Inc. New York (1989).
[22]	Mushayabasa S., Tchuenche J. M., Bhunu C. P. and Gwasira-Ngarakana E., “Modelling gonorrhoea and HIV co-interaction”, Bio. Systems. 103 (1), 27-37 (2011).
[23]	LaSalle J. P., “The stability of dynamical systems”, Regional Conference Series in Appl. Math. SIAM. Philadelphia (1976).

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APA Style

Ram Singh, Shoket Ali, Madhu Jain, Rakhee. (2016). Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment. American Journal of Applied Mathematics, 4(5), 222-234. https://doi.org/10.11648/j.ajam.20160405.14

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ACS Style

Ram Singh; Shoket Ali; Madhu Jain; Rakhee. Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment. Am. J. Appl. Math. 2016, 4(5), 222-234. doi: 10.11648/j.ajam.20160405.14

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AMA Style

Ram Singh, Shoket Ali, Madhu Jain, Rakhee. Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment. Am J Appl Math. 2016;4(5):222-234. doi: 10.11648/j.ajam.20160405.14

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@article{10.11648/j.ajam.20160405.14,
  author = {Ram Singh and Shoket Ali and Madhu Jain and Rakhee},
  title = {Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment},
  journal = {American Journal of Applied Mathematics},
  volume = {4},
  number = {5},
  pages = {222-234},
  doi = {10.11648/j.ajam.20160405.14},
  url = {https://doi.org/10.11648/j.ajam.20160405.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160405.14},
  abstract = {The mathematical model for analyzing the transmission dynamics of HIV/AIDS epidemic with treatment is studied by considering the three latent compartments for slow, medium and fast progresses of developing the AIDS. By constructing the system of differential equations for the different population groups namely susceptible, three types of latent individuals, symptomatic stage group and full blown AIDS individuals, the mathematical analysis is carried out in order to understand the dynamics of disease spread. By determining the basic reproduction number (R0), the model examines the two equilibrium points (i) the disease free equilibrium and (ii) the endemic equilibrium. It is established that if R0 <1, the disease free equilibrium is locally and globally asymptotically stable. The stability of endemic equilibrium has also been discussed.},
 year = {2016}
}

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TY  - JOUR
T1  - Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment
AU  - Ram Singh
AU  - Shoket Ali
AU  - Madhu Jain
AU  - Rakhee
Y1  - 2016/10/14
PY  - 2016
N1  - https://doi.org/10.11648/j.ajam.20160405.14
DO  - 10.11648/j.ajam.20160405.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 222
EP  - 234
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20160405.14
AB  - The mathematical model for analyzing the transmission dynamics of HIV/AIDS epidemic with treatment is studied by considering the three latent compartments for slow, medium and fast progresses of developing the AIDS. By constructing the system of differential equations for the different population groups namely susceptible, three types of latent individuals, symptomatic stage group and full blown AIDS individuals, the mathematical analysis is carried out in order to understand the dynamics of disease spread. By determining the basic reproduction number (R0), the model examines the two equilibrium points (i) the disease free equilibrium and (ii) the endemic equilibrium. It is established that if R0 <1, the disease free equilibrium is locally and globally asymptotically stable. The stability of endemic equilibrium has also been discussed.
VL  - 4
IS  - 5
ER  -

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Author Information

Ram Singh

Department of Mathematical Sciences, Baba Ghulam Shah Badshah, University, Rajouri, Jammu and Kashmir, India
Shoket Ali

Department of Mathematical Sciences, Baba Ghulam Shah Badshah, University, Rajouri, Jammu and Kashmir, India
Madhu Jain

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttrakhand, India
Rakhee

Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan, India

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APA Style

Ram Singh, Shoket Ali, Madhu Jain, Rakhee. (2016). Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment. American Journal of Applied Mathematics, 4(5), 222-234. https://doi.org/10.11648/j.ajam.20160405.14

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ACS Style

Ram Singh; Shoket Ali; Madhu Jain; Rakhee. Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment. Am. J. Appl. Math. 2016, 4(5), 222-234. doi: 10.11648/j.ajam.20160405.14

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AMA Style

Ram Singh, Shoket Ali, Madhu Jain, Rakhee. Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment. Am J Appl Math. 2016;4(5):222-234. doi: 10.11648/j.ajam.20160405.14

Copy | Download

@article{10.11648/j.ajam.20160405.14,
  author = {Ram Singh and Shoket Ali and Madhu Jain and Rakhee},
  title = {Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment},
  journal = {American Journal of Applied Mathematics},
  volume = {4},
  number = {5},
  pages = {222-234},
  doi = {10.11648/j.ajam.20160405.14},
  url = {https://doi.org/10.11648/j.ajam.20160405.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160405.14},
  abstract = {The mathematical model for analyzing the transmission dynamics of HIV/AIDS epidemic with treatment is studied by considering the three latent compartments for slow, medium and fast progresses of developing the AIDS. By constructing the system of differential equations for the different population groups namely susceptible, three types of latent individuals, symptomatic stage group and full blown AIDS individuals, the mathematical analysis is carried out in order to understand the dynamics of disease spread. By determining the basic reproduction number (R0), the model examines the two equilibrium points (i) the disease free equilibrium and (ii) the endemic equilibrium. It is established that if R0 <1, the disease free equilibrium is locally and globally asymptotically stable. The stability of endemic equilibrium has also been discussed.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment
AU  - Ram Singh
AU  - Shoket Ali
AU  - Madhu Jain
AU  - Rakhee
Y1  - 2016/10/14
PY  - 2016
N1  - https://doi.org/10.11648/j.ajam.20160405.14
DO  - 10.11648/j.ajam.20160405.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 222
EP  - 234
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20160405.14
AB  - The mathematical model for analyzing the transmission dynamics of HIV/AIDS epidemic with treatment is studied by considering the three latent compartments for slow, medium and fast progresses of developing the AIDS. By constructing the system of differential equations for the different population groups namely susceptible, three types of latent individuals, symptomatic stage group and full blown AIDS individuals, the mathematical analysis is carried out in order to understand the dynamics of disease spread. By determining the basic reproduction number (R0), the model examines the two equilibrium points (i) the disease free equilibrium and (ii) the endemic equilibrium. It is established that if R0 <1, the disease free equilibrium is locally and globally asymptotically stable. The stability of endemic equilibrium has also been discussed.
VL  - 4
IS  - 5
ER  -

Copy | Download