Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics

Sammy Maingi; Benjamin Kikwai; Mark Kimathi

doi:doi:10.11648/j.ajmcm.20251002.14

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Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics

Sammy Maingi^*

, Benjamin Kikwai

, Mark Kimathi

Published in American Journal of Mathematical and Computer Modelling (Volume 10, Issue 2)

Received: 22 May 2025 Accepted: 13 June 2025 Published: 30 June 2025

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Abstract

The persistence of the HIV/AIDS epidemic is significantly challenged by difficulties in maintaining long-term adherence to antiretroviral therapy (ART), leading to treatment defaults hence hindering disease control. Defaulter tracing is a crucial intervention aimed at returning patients back to care. This paper develops and analyzes a deterministic mathematical model for HIV/AIDS transmission dynamics, explicitly incorporating defaulter tracing. The model utilizes a system of ordinary differential equations to describe the transitions between susceptible (S_p), infected (I_T), infected on ART (I_ARV), infected not on ART (I_NARV), and individuals under defaulter tracing (D_TR) compartments. Mathematical analysis includes establishing the positivity and boundedness of solutions, determination of the Disease-Free Equilibrium (DFE) and the existence of an Endemic Equilibrium (EE). The basic reproduction number (R₀) is also derived using the next-generation matrix method. Local stability analysis of the DFE shows it is asymptotically stable if R₀ < 1 and unstable otherwise. Sensitivity analysis identified parameters related to transmission (λ, c, π) as having a positive impact on R₀, while parameters associated with treatment, defaulter tracing, and mortality were found to negatively influence the R₀. This study shows that improving treatment uptake and retention to care, through defaulter tracing efforts, can contribute to reduction of R₀ and in turn controll the epidemic.

Published in	American Journal of Mathematical and Computer Modelling (Volume 10, Issue 2)
DOI	10.11648/j.ajmcm.20251002.14
Page(s)	74-83
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), 2025. Published by Science Publishing Group

Keywords

HIV/AIDS, Mathematical Model, Defaulter Tracing, Basic Reproduction Number, Stability Analysis

References

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APA Style

Maingi, S., Kikwai, B., Kimathi, M. (2025). Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics. American Journal of Mathematical and Computer Modelling, 10(2), 74-83. https://doi.org/10.11648/j.ajmcm.20251002.14

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ACS Style

Maingi, S.; Kikwai, B.; Kimathi, M. Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics. Am. J. Math. Comput. Model. 2025, 10(2), 74-83. doi: 10.11648/j.ajmcm.20251002.14

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AMA Style

Maingi S, Kikwai B, Kimathi M. Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics. Am J Math Comput Model. 2025;10(2):74-83. doi: 10.11648/j.ajmcm.20251002.14

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@article{10.11648/j.ajmcm.20251002.14,
  author = {Sammy Maingi and Benjamin Kikwai and Mark Kimathi},
  title = {Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics
},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {10},
  number = {2},
  pages = {74-83},
  doi = {10.11648/j.ajmcm.20251002.14},
  url = {https://doi.org/10.11648/j.ajmcm.20251002.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20251002.14},
  abstract = {The persistence of the HIV/AIDS epidemic is significantly challenged by difficulties in maintaining long-term adherence to antiretroviral therapy (ART), leading to treatment defaults hence hindering disease control. Defaulter tracing is a crucial intervention aimed at returning patients back to care. This paper develops and analyzes a deterministic mathematical model for HIV/AIDS transmission dynamics, explicitly incorporating defaulter tracing. The model utilizes a system of ordinary differential equations to describe the transitions between susceptible (Sp), infected (IT), infected on ART (IARV), infected not on ART (INARV), and individuals under defaulter tracing (DTR) compartments. Mathematical analysis includes establishing the positivity and boundedness of solutions, determination of the Disease-Free Equilibrium (DFE) and the existence of an Endemic Equilibrium (EE). The basic reproduction number (R0) is also derived using the next-generation matrix method. Local stability analysis of the DFE shows it is asymptotically stable if R0 λ, c, π) as having a positive impact on R0, while parameters associated with treatment, defaulter tracing, and mortality were found to negatively influence the R0. This study shows that improving treatment uptake and retention to care, through defaulter tracing efforts, can contribute to reduction of R0 and in turn controll the epidemic.
},
 year = {2025}
}

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TY  - JOUR
T1  - Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics

AU  - Sammy Maingi
AU  - Benjamin Kikwai
AU  - Mark Kimathi
Y1  - 2025/06/30
PY  - 2025
N1  - https://doi.org/10.11648/j.ajmcm.20251002.14
DO  - 10.11648/j.ajmcm.20251002.14
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 74
EP  - 83
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20251002.14
AB  - The persistence of the HIV/AIDS epidemic is significantly challenged by difficulties in maintaining long-term adherence to antiretroviral therapy (ART), leading to treatment defaults hence hindering disease control. Defaulter tracing is a crucial intervention aimed at returning patients back to care. This paper develops and analyzes a deterministic mathematical model for HIV/AIDS transmission dynamics, explicitly incorporating defaulter tracing. The model utilizes a system of ordinary differential equations to describe the transitions between susceptible (Sp), infected (IT), infected on ART (IARV), infected not on ART (INARV), and individuals under defaulter tracing (DTR) compartments. Mathematical analysis includes establishing the positivity and boundedness of solutions, determination of the Disease-Free Equilibrium (DFE) and the existence of an Endemic Equilibrium (EE). The basic reproduction number (R0) is also derived using the next-generation matrix method. Local stability analysis of the DFE shows it is asymptotically stable if R0 λ, c, π) as having a positive impact on R0, while parameters associated with treatment, defaulter tracing, and mortality were found to negatively influence the R0. This study shows that improving treatment uptake and retention to care, through defaulter tracing efforts, can contribute to reduction of R0 and in turn controll the epidemic.

VL  - 10
IS  - 2
ER  -

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Author Information

Sammy Maingi

Deparment of Mathematics and Statistics, Machakos University, Machakos, Kenya

Contact Email

http://orcid.org/0009-0005-1960-4896
Benjamin Kikwai

Deparment of Mathematics and Statistics, Machakos University, Machakos, Kenya

http://orcid.org/0000-0002-6741-5011
Mark Kimathi

Deparment of Mathematics and Statistics, Machakos University, Machakos, Kenya

http://orcid.org/0000-0002-5822-8687

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APA Style

Maingi, S., Kikwai, B., Kimathi, M. (2025). Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics. American Journal of Mathematical and Computer Modelling, 10(2), 74-83. https://doi.org/10.11648/j.ajmcm.20251002.14

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ACS Style

Maingi, S.; Kikwai, B.; Kimathi, M. Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics. Am. J. Math. Comput. Model. 2025, 10(2), 74-83. doi: 10.11648/j.ajmcm.20251002.14

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AMA Style

Maingi S, Kikwai B, Kimathi M. Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics. Am J Math Comput Model. 2025;10(2):74-83. doi: 10.11648/j.ajmcm.20251002.14

Copy | Download

@article{10.11648/j.ajmcm.20251002.14,
  author = {Sammy Maingi and Benjamin Kikwai and Mark Kimathi},
  title = {Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics
},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {10},
  number = {2},
  pages = {74-83},
  doi = {10.11648/j.ajmcm.20251002.14},
  url = {https://doi.org/10.11648/j.ajmcm.20251002.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20251002.14},
  abstract = {The persistence of the HIV/AIDS epidemic is significantly challenged by difficulties in maintaining long-term adherence to antiretroviral therapy (ART), leading to treatment defaults hence hindering disease control. Defaulter tracing is a crucial intervention aimed at returning patients back to care. This paper develops and analyzes a deterministic mathematical model for HIV/AIDS transmission dynamics, explicitly incorporating defaulter tracing. The model utilizes a system of ordinary differential equations to describe the transitions between susceptible (Sp), infected (IT), infected on ART (IARV), infected not on ART (INARV), and individuals under defaulter tracing (DTR) compartments. Mathematical analysis includes establishing the positivity and boundedness of solutions, determination of the Disease-Free Equilibrium (DFE) and the existence of an Endemic Equilibrium (EE). The basic reproduction number (R0) is also derived using the next-generation matrix method. Local stability analysis of the DFE shows it is asymptotically stable if R0 λ, c, π) as having a positive impact on R0, while parameters associated with treatment, defaulter tracing, and mortality were found to negatively influence the R0. This study shows that improving treatment uptake and retention to care, through defaulter tracing efforts, can contribute to reduction of R0 and in turn controll the epidemic.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Analysis of an HIV/AIDS Epidemic Model with Defaulter Tracing Dynamics

AU  - Sammy Maingi
AU  - Benjamin Kikwai
AU  - Mark Kimathi
Y1  - 2025/06/30
PY  - 2025
N1  - https://doi.org/10.11648/j.ajmcm.20251002.14
DO  - 10.11648/j.ajmcm.20251002.14
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 74
EP  - 83
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20251002.14
AB  - The persistence of the HIV/AIDS epidemic is significantly challenged by difficulties in maintaining long-term adherence to antiretroviral therapy (ART), leading to treatment defaults hence hindering disease control. Defaulter tracing is a crucial intervention aimed at returning patients back to care. This paper develops and analyzes a deterministic mathematical model for HIV/AIDS transmission dynamics, explicitly incorporating defaulter tracing. The model utilizes a system of ordinary differential equations to describe the transitions between susceptible (Sp), infected (IT), infected on ART (IARV), infected not on ART (INARV), and individuals under defaulter tracing (DTR) compartments. Mathematical analysis includes establishing the positivity and boundedness of solutions, determination of the Disease-Free Equilibrium (DFE) and the existence of an Endemic Equilibrium (EE). The basic reproduction number (R0) is also derived using the next-generation matrix method. Local stability analysis of the DFE shows it is asymptotically stable if R0 λ, c, π) as having a positive impact on R0, while parameters associated with treatment, defaulter tracing, and mortality were found to negatively influence the R0. This study shows that improving treatment uptake and retention to care, through defaulter tracing efforts, can contribute to reduction of R0 and in turn controll the epidemic.

VL  - 10
IS  - 2
ER  -

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