American Journal of Applied Mathematics

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Modeling the Combined Effect of Vertical Transmission and Variable Inflow of Infective Immigrants on the Dynamics of HIV/AIDS

Received: 11 December 2015    Accepted: 23 December 2015    Published: 11 January 2016
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Abstract

In this paper, a Non linear Mathematical model is proposed and studied the combined effect of vertical transmission (MTCT) and variable inflow of infective immigrants on the dynamics of HIV/AIDS. Vertical transmission means propagation of the disease from mother to children. ‘Variable inflow of infective immigrants’ includes both the aware and unaware infected immigrants. The equilibrium points of the model are found and the stability analysis of the model around these equilibrium points is conducted. The stability analysis on the model shows that the disease free equilibrium point E0 is locally asymptotically stable when R0<1. The positive endemic equilibrium point E* is shown to be locally asymptotically stable when R0>1. Further it is shown that R0>0, this shows that the basic reproduction number of the present model is greater than the one which is obtained from the model modeled without vertical transmission. Through vertical transmission the disease flows from infected mother to children. That is, Vertical transmission contributes positively to the spread of the disease. Numerical simulation of the model is carried out to assess the effect of unaware HIV infective immigrants and vertical transmission (MTCT) in the spread of HIV/AIDS disease. The result showed that HIV infective immigrants and vertical transmission (MTCT) significantly affects the spread of the disease. Screening of the disease reduces the spread of HIV and also prevents mother to child transmission. It is well accepted that both vertical transmission and immigration contribute positively to the spread of the disease and these two parameters cannot be avoided in practice. Hence, the purpose of this study is to investigate the combined effect of vertical transmission, unaware and aware infected immigrants on the spread of HIV/AIDS and offers possible intervention strategies.

DOI 10.11648/j.ajam.20160401.12
Published in American Journal of Applied Mathematics (Volume 4, Issue 1, February 2016)
Page(s) 11-19
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), 2024. Published by Science Publishing Group

Keywords

HIV/AIDS, Unaware and Aware Infective Immigrant, Vertical Transmission (MTCT), Screening, Local Stability, Reproduction Number

References
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[2] Dabbo M. I. and Seidu, Modelling the combined effects of careless Susceptible and infective immigrants on the transmission dynamics of HIV/AIDS. Journal of public health and epidemiology, Vol. 3 (1), 362-369, 2013.
[3] Dabbo M. I. and Seidu, Modelling the effect of irresponsible infective immigrants on the dynamics of HIV/AIDS. Advances in Applied Math. Biosci, 3 (1): 31-40, 2012.
[4] E. Chinwendu, A. Martins. A Mathematical Modelling to study the impact of irresponsible infective immigrants and vertical transmission on the dynamics of HIV/AIDS. J. Math. comp. Sci. vol. (1): 72-90, 2015.
[5] Issa Shabani and S. Massawe, Modeling the effect of screening on the spread of HIV infection in a population with variable inflow of infective immigrants. Scientific research and essays, Vol. 20, 4397-4405, 2011.
[6] Ochejele S. Overview of PMTCT of HIV infection. Update course of the national postgraduate medical college of Nigeria, Makurdi. March, 18, 2011.
[7] Anderson R. M. and May R. M., (1991). Infectious Diseases of Human, Dynamics and Control. Oxford: Oxford university press.
[8] Onoja Matthew Akpa and B. A. Oyejola, Modeling the transmission dynamics of HIV/AIDS epidemics, an introduction and review, J. Infect DEv Ctries V (10), 597-608.
[9] Dejen Ketema Mamo, Purnachandra Rao Koya. Mathematical Modeling and Simulation Study of SEIR disease and Data Fitting of Ebola Epidemic in West Africa. Journal of Multidisciplinary Engineering Science and Technology (JMEST). Vol. 2, Issue 1, January 2015, pp 106 – 14. ISSN: 3159 – 0040. http://www.jmest.org/wp-content/uploads/JMESTN42350340.pdf.
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[11] Abdulsamad Engida Sado, Purnachandra Rao Koya. Application of Brody Growth Function to Describe Dynamics of Breast Cancer Cells. American Journal of Applied Mathematics (AJAM). Vol. 3, No. 3, 2015, pp. 138-145. Doi:10.11648/j.ajam.20150303.20.
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[14] Dancho Desaleng, Purnachandra Rao Koya. The Role of Polluted Air and Population Density in the Spread of Mycobacterium Tuberculosis Disease. Journal of Multidisciplinary Engineering Science and Technology (JMEST). Vol. 2, Issue 5, May – 2015, Pp 1212 – 20. ISSN: 3159 – 0040. http://www.jmest.org/wp-content/uploads/JMESTN42350782.pdf.
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[16] S. Al-sheikh, F. Musali, M. Alsomali, Properties of Stability and Hope Bifurcation for an HIV/AIDS epidemic model with screening and time delay, Vol-11, 87-99, 2012.
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Author Information
  • School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

  • School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

  • Department of Mathematics, Debere Birhan University, Debere Birhan, Ethiopia

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    Tadele Tesfa Tegegne, Purnachandra Rao Koya, Temesgen Tibebu Mekonnen. (2016). Modeling the Combined Effect of Vertical Transmission and Variable Inflow of Infective Immigrants on the Dynamics of HIV/AIDS. American Journal of Applied Mathematics, 4(1), 11-19. https://doi.org/10.11648/j.ajam.20160401.12

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    Tadele Tesfa Tegegne; Purnachandra Rao Koya; Temesgen Tibebu Mekonnen. Modeling the Combined Effect of Vertical Transmission and Variable Inflow of Infective Immigrants on the Dynamics of HIV/AIDS. Am. J. Appl. Math. 2016, 4(1), 11-19. doi: 10.11648/j.ajam.20160401.12

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    AMA Style

    Tadele Tesfa Tegegne, Purnachandra Rao Koya, Temesgen Tibebu Mekonnen. Modeling the Combined Effect of Vertical Transmission and Variable Inflow of Infective Immigrants on the Dynamics of HIV/AIDS. Am J Appl Math. 2016;4(1):11-19. doi: 10.11648/j.ajam.20160401.12

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  • @article{10.11648/j.ajam.20160401.12,
      author = {Tadele Tesfa Tegegne and Purnachandra Rao Koya and Temesgen Tibebu Mekonnen},
      title = {Modeling the Combined Effect of Vertical Transmission and Variable Inflow of Infective Immigrants on the Dynamics of HIV/AIDS},
      journal = {American Journal of Applied Mathematics},
      volume = {4},
      number = {1},
      pages = {11-19},
      doi = {10.11648/j.ajam.20160401.12},
      url = {https://doi.org/10.11648/j.ajam.20160401.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajam.20160401.12},
      abstract = {In this paper, a Non linear Mathematical model is proposed and studied the combined effect of vertical transmission (MTCT) and variable inflow of infective immigrants on the dynamics of HIV/AIDS. Vertical transmission means propagation of the disease from mother to children. ‘Variable inflow of infective immigrants’ includes both the aware and unaware infected immigrants. The equilibrium points of the model are found and the stability analysis of the model around these equilibrium points is conducted. The stability analysis on the model shows that the disease free equilibrium point E0 is locally asymptotically stable when R0E* is shown to be locally asymptotically stable when R0>1. Further it is shown that R0>R´0, this shows that the basic reproduction number of the present model is greater than the one which is obtained from the model modeled without vertical transmission. Through vertical transmission the disease flows from infected mother to children. That is, Vertical transmission contributes positively to the spread of the disease. Numerical simulation of the model is carried out to assess the effect of unaware HIV infective immigrants and vertical transmission (MTCT) in the spread of HIV/AIDS disease. The result showed that HIV infective immigrants and vertical transmission (MTCT) significantly affects the spread of the disease. Screening of the disease reduces the spread of HIV and also prevents mother to child transmission. It is well accepted that both vertical transmission and immigration contribute positively to the spread of the disease and these two parameters cannot be avoided in practice. Hence, the purpose of this study is to investigate the combined effect of vertical transmission, unaware and aware infected immigrants on the spread of HIV/AIDS and offers possible intervention strategies.},
     year = {2016}
    }
    

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  • TY  - JOUR
    T1  - Modeling the Combined Effect of Vertical Transmission and Variable Inflow of Infective Immigrants on the Dynamics of HIV/AIDS
    AU  - Tadele Tesfa Tegegne
    AU  - Purnachandra Rao Koya
    AU  - Temesgen Tibebu Mekonnen
    Y1  - 2016/01/11
    PY  - 2016
    N1  - https://doi.org/10.11648/j.ajam.20160401.12
    DO  - 10.11648/j.ajam.20160401.12
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 11
    EP  - 19
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20160401.12
    AB  - In this paper, a Non linear Mathematical model is proposed and studied the combined effect of vertical transmission (MTCT) and variable inflow of infective immigrants on the dynamics of HIV/AIDS. Vertical transmission means propagation of the disease from mother to children. ‘Variable inflow of infective immigrants’ includes both the aware and unaware infected immigrants. The equilibrium points of the model are found and the stability analysis of the model around these equilibrium points is conducted. The stability analysis on the model shows that the disease free equilibrium point E0 is locally asymptotically stable when R0E* is shown to be locally asymptotically stable when R0>1. Further it is shown that R0>R´0, this shows that the basic reproduction number of the present model is greater than the one which is obtained from the model modeled without vertical transmission. Through vertical transmission the disease flows from infected mother to children. That is, Vertical transmission contributes positively to the spread of the disease. Numerical simulation of the model is carried out to assess the effect of unaware HIV infective immigrants and vertical transmission (MTCT) in the spread of HIV/AIDS disease. The result showed that HIV infective immigrants and vertical transmission (MTCT) significantly affects the spread of the disease. Screening of the disease reduces the spread of HIV and also prevents mother to child transmission. It is well accepted that both vertical transmission and immigration contribute positively to the spread of the disease and these two parameters cannot be avoided in practice. Hence, the purpose of this study is to investigate the combined effect of vertical transmission, unaware and aware infected immigrants on the spread of HIV/AIDS and offers possible intervention strategies.
    VL  - 4
    IS  - 1
    ER  - 

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