Research Article | | Peer-Reviewed

A Mathematical Model for the Co-infection Dynamics of Pneumocystis Pneumonia and HIV/AIDS with Treatment

Received: 19 June 2024     Accepted: 16 July 2024     Published: 27 July 2024
Views:       Downloads:
Abstract

The control of opportunistic infections among HIV infected individuals should be one of the major public health concerns in reducing mortality rate of individuals living with HIV/AIDS. In this study a deterministic co-infection mathematical model is developed to provide a quantification of treatment at each contagious stage against Pneumocystis Pneumonia (PCP) among HIV infected individuals on ART. The goal is to minimize the co-infection burden by putting the curable PCP under control. The disease-free equilibria for the HIV/AIDS sub-model, PCP sub-model and the co-infection model are shown to be locally asymptotically stable when their associated disease threshold parameter is less than a unity. By use of suitable Lyapunov functions, the endemic equilibria corresponding to HIV/AIDS and PCP sub-models are globally asymptotically stable whenever the HIV/AIDS related basic reproduction number R0H and the PCP related reproduction number R0P are respectively greater than a unity. The sensitivity analysis results implicate that the effective contact rates are the main mechanisms fueling the proliferation of the two diseases and on the other hand treatment efforts play an important role in reducing the incidence. The model numerical results reveal that PCP carriers have a considerable contribution in the transmission dynamics of PCP. Furthermore, treatment of PCP at all contagious phases significantly reduces the burden with HIV/AIDS and PCP co-infection.

Published in Science Journal of Applied Mathematics and Statistics (Volume 12, Issue 4)
DOI 10.11648/j.sjams.20241204.11
Page(s) 48-63
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, PCP Carriers, Basic Reproduction Number(s), Co-infection, Stability, Sensitivity Analysis

References
[1] Bhunu, C. P., Garira, W., & Mukandavire, Z. (2009). Modeling HIV/AIDS and tuberculosis coinfection. Bulletin of mathematical biology, 71(7), 1745.
[2] Bhunu, C. P., Mushayabasa, S., Kojouharov, H., & Tchuenche, J. M. (2011). Mathematical analysis of an HIV/AIDS model: impact of educational programs and abstinence in sub-Saharan Africa. Journal of Mathematical Modelling and Algorithms, 10, 31-55.
[3] Bozorgomid, A., Hamzavi, Y., Khayat, S. H., Mahdavian, B., & Bashiri, H. (2019). Pneumocystis jirovecii pneumonia and human immunodeficiency virus co- infection in Western Iran. Iranian Journal of Public Health, 48(11), 2065.
[4] Carreto-Binaghi, L. E., Morales-Villarreal, F. R., GarcÃa-de la Torre, G., Vite-GarÃn, T., Ramirez, J. A., Aliouat, E. M.,... & Taylor, M. L. (2019). Histoplasma capsulatum and Pneumocystis jirovecii coinfection in hospitalized HIV and non- HIV patients from a tertiary care hospital in Mexico. International Journal of Infectious Diseases, 86, 65-72.
[5] Castillo-Chavez, C., Feng, Z., Haung, W. (2002). On the computation of R0 and its role on global stability. Mathematical approaches for emerging and reemerging infectious diseases: an introduction, 1, 229.
[6] Chitnis, N., Hyman, J. M., & 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, 1272-1296.
[7] Cilloniz, C., Dominedo, C., Alvarez-Martinez, M. J., Moreno, A., GarcÃa, F., Torres, A., & Miro, J. M. (2019). Pneumocystis pneumonia in the twenty-first century: HIV infected versus HIV-uninfected patients. Expert review of anti-infective therapy, 17(10), 787-801.
[8] DeJesus, E. X., & Kaufman, C. (1987). Routh- Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations. Physical Review A, 35(12), 5288.
[9] Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. (1990). Onthe definitionandthe computationofthe basic reproduction ratio R0in models for infectious diseases in heterogeneous populations. Journal of mathematical biology, 28, 365-382.
[10] Ega, T. T., & Ngeleja, R. C. (2022). Mathematical Model Formulation and Analysis for COVID-19 Transmission with Virus Transfer Media and Quarantine on Arrival. Computational and Mathematical Methods, 2022.
[11] Endashaw, E. E., & Mekonnen, T. T. (2022). Modeling the effect of vaccination and treatment on the transmission dynamics of hepatitis B virus and HIV/AIDS coinfection. Journal of Applied Mathematics, 2022, 1-27.
[12] Fauci, A. S., & Lane, H. C. (2020). Four decades of HIV/AIDS much accomplished, much to do. New England Journal of Medicine, 383(1), 1-4.
[13] Fishman, J. A. (2020, February). Pneumocystis jiroveci. In Seminars in Respiratory and Critical Care Medicine (Vol. 41, No. 01, pp. 141-157). Thieme Medical Publishers.
[14] Ho, D. D. (1995). Time to hit HIV, early and hard. New England Journal of Medicine, 333(7), 450-451.
[15] Huang, Y. S., Yang, J. J., Lee, N. Y., Chen, G. J., Ko, W. C., Sun, H. Y., & Hung, C. C. (2017). Treatment of Pneumocystis jirovecii pneumonia in HIV-infected patients: a review. Expert review of anti-infective therapy, 15(9), 873-892.
[16] Kamara, T., Byamukama, M., & Karuhanga, M. (2022). Modelling the Role of Treatment, Public Health Education, and Chemical Control Strategies on Transmission Dynamics of Schistosomiasis. Journal of Mathematics, 2022.
[17] Kato, H., Samukawa, S., Takahashi, H., & Nakajima, H. (2019). Diagnosis and treatment of Pneumocystis jirovecii pneumonia in HIV-infected or non-HIV infected patients-difficulties in diagnosis and adverse effects of trimethoprim sulfamethoxazole. Journal of Infection and Chemotherapy, 25(11), 920-924.
[18] LaSalle, J. P. (1976). Stability theory and invariance principles. In Dynamical systems (pp. 211-222). Academic Press.
[19] Lowe, D. M., Rangaka, M. X., Gordon, F., James, C. D., & Miller, R. F. (2013). Pneumocystis jirovecii pneumonia in tropical and low and middle income countries: a systematic review and meta-regression. PloS one, 8(8), e69969.
[20] Mbabazi, F. K., Mugisha, J. Y. T., & Kimathi, M. (2020). Global stability of pneumococcal pneumonia with awareness and saturated treatment. Journal of Applied Mathematics, 2020, 1-12.
[21] Medrano, F. J., Montes-Cano, M., Conde, M., De La Horra, C., Respaldiza, N., Gasch, A.,... & Calderon, E. J. (2005). Pneumocystis jirovecii in general population. Emerging infectious diseases, 11(2), 245.
[22] Nah, K., Nishiura, H., Tsuchiya, N., Sun, X., Asai, Y., & Imamura, A. (2017). Test-and-treat approach to HIV/AIDS: a primer for mathematical modeling. Theoretical Biology and Medical Modelling, 14(1), 1-11.
[23] Nannyonga, B., Mugisha, J. Y. T., & Luboobi, L. S. (2011). The role of HIV positive immigrants and dual protection in a co-infection of malaria and HIV/AIDS. Applied Mathematical Sciences, 5(59), 2919-2942.
[24] Nthiiri, J. K., Lavi, G. O., & Manyonge, A. (2015). Mathematical model of pneumonia and HIV/AIDS coinfection in the presence of protection.
[25] Omondi, E. O., Mbogo, R. W., & Luboobi, L. S. (2018). Mathematical modelling of the impact of testing, treatment and control of HIV transmission in Kenya. Cogent Mathematics & Statistics,5(1), 1475590.
[26] Oshinubi, K., Peter, O. J., Addai, E., Mwizerwa, E., Babasola, O., Nwabufo, I. V.,... & Agbaje, J. O. (2023). Mathematical modelling of tuberculosis outbreak in an East African country incorporating vaccination and treatment. Computation, 11(7), 143.
[27] Rodriguez, Y. D. A., Wissmann, G., Muller, A. L., Pederiva, M. A., Brum, M. C., Brackmann, R. L.,...& Calderon, E. J. (2011). Pneumocystis jirovecii pneumonia in developing countries. Parasite: journal de la Societe Francaise de Parasitologie, 18(3), 219.
[28] Salzer, H. J., SchÃfer, G., Hoenigl, M., GÃnther, G., Hoffmann, C., Kalsdorf, B.,... & Lange, C. (2018). Clinical, diagnostic, and treatment disparities between HIV-infected and non-HIV-infected immunocompromised patients with Pneumocystis jirovecii pneumonia. Respiration, 96(1), 52-65.
[29] Tasaka, S. (2020). Recent advances in the diagnosis and management of Pneumocystis pneumonia. Tuberculosis and Respiratory Diseases, 83(2), 132.
[30] Teklu, S. W., & Mekonnen, T. T. (2021). HIV/AIDS- pneumonia coinfection model with treatment at each infection stage: mathematical analysis and numerical simulation. Journal of Applied Mathematics, 2021, 1-21.
[31] Teklu, S. W., & Rao, K. P. (2022). HIV/AIDS-pneumonia codynamics model analysis with vaccination and treatment. Computational and Mathematical Methods in Medicine, 2022.
[32] Tilahun, G. T., Makinde, O. D., & Malonza, D. (2017). Modelling and optimal control of pneumonia disease with cost-effective strategies. Journal of Biological Dynamics, 11(sup2), 400-426.
[33] UNAIDS, Global HIV statistics 2022 fact sheet, updated July 2022.
[34] Van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2), 29-48.
[35] Wills, N. K., Lawrence, D. S., Botsile, E., Tenforde, M. W., & Jarvis, J. N. (2021). The prevalence of laboratory-confirmed Pneumocystis jirovecii in HIV- infected adults in Africa: A systematic review and meta-analysis. Medical mycology, 59(8), 802-812.
[36] Yoshimura, K. (2017). Current status of HIV/AIDS in the ART era. Journal of Infection and Chemotherapy, 23(1), 12-16.
Cite This Article
  • APA Style

    Byamukama, M., Kajunguri, D., Karuhanga, M. (2024). A Mathematical Model for the Co-infection Dynamics of Pneumocystis Pneumonia and HIV/AIDS with Treatment. Science Journal of Applied Mathematics and Statistics, 12(4), 48-63. https://doi.org/10.11648/j.sjams.20241204.11

    Copy | Download

    ACS Style

    Byamukama, M.; Kajunguri, D.; Karuhanga, M. A Mathematical Model for the Co-infection Dynamics of Pneumocystis Pneumonia and HIV/AIDS with Treatment. Sci. J. Appl. Math. Stat. 2024, 12(4), 48-63. doi: 10.11648/j.sjams.20241204.11

    Copy | Download

    AMA Style

    Byamukama M, Kajunguri D, Karuhanga M. A Mathematical Model for the Co-infection Dynamics of Pneumocystis Pneumonia and HIV/AIDS with Treatment. Sci J Appl Math Stat. 2024;12(4):48-63. doi: 10.11648/j.sjams.20241204.11

    Copy | Download

  • @article{10.11648/j.sjams.20241204.11,
      author = {Michael Byamukama and Damian Kajunguri and Martin Karuhanga},
      title = {A Mathematical Model for the Co-infection Dynamics of Pneumocystis Pneumonia and HIV/AIDS with Treatment},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {12},
      number = {4},
      pages = {48-63},
      doi = {10.11648/j.sjams.20241204.11},
      url = {https://doi.org/10.11648/j.sjams.20241204.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20241204.11},
      abstract = {The control of opportunistic infections among HIV infected individuals should be one of the major public health concerns in reducing mortality rate of individuals living with HIV/AIDS. In this study a deterministic co-infection mathematical model is developed to provide a quantification of treatment at each contagious stage against Pneumocystis Pneumonia (PCP) among HIV infected individuals on ART. The goal is to minimize the co-infection burden by putting the curable PCP under control. The disease-free equilibria for the HIV/AIDS sub-model, PCP sub-model and the co-infection model are shown to be locally asymptotically stable when their associated disease threshold parameter is less than a unity. By use of suitable Lyapunov functions, the endemic equilibria corresponding to HIV/AIDS and PCP sub-models are globally asymptotically stable whenever the HIV/AIDS related basic reproduction number R0H and the PCP related reproduction number R0P are respectively greater than a unity. The sensitivity analysis results implicate that the effective contact rates are the main mechanisms fueling the proliferation of the two diseases and on the other hand treatment efforts play an important role in reducing the incidence. The model numerical results reveal that PCP carriers have a considerable contribution in the transmission dynamics of PCP. Furthermore, treatment of PCP at all contagious phases significantly reduces the burden with HIV/AIDS and PCP co-infection.},
     year = {2024}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - A Mathematical Model for the Co-infection Dynamics of Pneumocystis Pneumonia and HIV/AIDS with Treatment
    AU  - Michael Byamukama
    AU  - Damian Kajunguri
    AU  - Martin Karuhanga
    Y1  - 2024/07/27
    PY  - 2024
    N1  - https://doi.org/10.11648/j.sjams.20241204.11
    DO  - 10.11648/j.sjams.20241204.11
    T2  - Science Journal of Applied Mathematics and Statistics
    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
    SP  - 48
    EP  - 63
    PB  - Science Publishing Group
    SN  - 2376-9513
    UR  - https://doi.org/10.11648/j.sjams.20241204.11
    AB  - The control of opportunistic infections among HIV infected individuals should be one of the major public health concerns in reducing mortality rate of individuals living with HIV/AIDS. In this study a deterministic co-infection mathematical model is developed to provide a quantification of treatment at each contagious stage against Pneumocystis Pneumonia (PCP) among HIV infected individuals on ART. The goal is to minimize the co-infection burden by putting the curable PCP under control. The disease-free equilibria for the HIV/AIDS sub-model, PCP sub-model and the co-infection model are shown to be locally asymptotically stable when their associated disease threshold parameter is less than a unity. By use of suitable Lyapunov functions, the endemic equilibria corresponding to HIV/AIDS and PCP sub-models are globally asymptotically stable whenever the HIV/AIDS related basic reproduction number R0H and the PCP related reproduction number R0P are respectively greater than a unity. The sensitivity analysis results implicate that the effective contact rates are the main mechanisms fueling the proliferation of the two diseases and on the other hand treatment efforts play an important role in reducing the incidence. The model numerical results reveal that PCP carriers have a considerable contribution in the transmission dynamics of PCP. Furthermore, treatment of PCP at all contagious phases significantly reduces the burden with HIV/AIDS and PCP co-infection.
    VL  - 12
    IS  - 4
    ER  - 

    Copy | Download

Author Information
  • Sections