Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach

Erick Mutwiri Kirimi; Jeconiah Okelo; Mark Kimathi; Kenneth Ngure

doi:doi:10.11648/j.acm.20251404.15

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Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach

Erick Mutwiri Kirimi^*

, Jeconiah Okelo

, Mark Kimathi

, Kenneth Ngure

Published in Applied and Computational Mathematics (Volume 14, Issue 4)

Received: 7 July 2025 Accepted: 22 July 2025 Published: 20 August 2025

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Abstract

This study presents a novel mathematical model that captures the co-dynamics of pulmonary tuberculosis in the presence of opportunistic pneumonia. The model integrates opportunistic pneumonia into the classical pulmonary tuberculosis transmission framework, based on clinical evidence of a lethal synergism between the two diseases. A Holling-type saturation function is utilized to reflect the impact of natural immunity on the progression from latent tuberculosis infection to active disease. Asymptomatic infectious individuals can transmit the infection unnoticed, thereby significantly contributing to the high rate of transmission. Thus, the screening of asymptomatic individuals is incorporated into the model as a strategy to mitigate the prevalence of co-infections. A comprehensive sensitivity analysis and numerical simulations were conducted to evaluate the effects of various interventions, including enhancing natural immunity, screening asymptomatic and latently infected individuals, improving vaccine efficacy, and treating patients with severe pulmonary tuberculosis. The sensitivity analysis reveals that improving vaccine efficacy is the most effective strategy for minimizing co-infections. Simulations also demonstrate that increased screening and treatment rates significantly reduce the burden of pulmonary tuberculosis-pneumonia co-infections. These results underscore the need to develop vaccines with higher efficacy to reduce co-infections of pulmonary tuberculosis and opportunistic pneumonia. Additionally, the study recommends expanded screening of asymptomatic tuberculosis cases and the strengthening of immunity among latently infected populations.

Published in	Applied and Computational Mathematics (Volume 14, Issue 4)
DOI	10.11648/j.acm.20251404.15
Page(s)	223-241
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

Pulmonary Tuberculosis, Asymptomatic Infectious, Opportunistic Pneumonia, Latently Infected, Natural Immunity, Numerical Simulation

References

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APA Style

Kirimi, E. M., Okelo, J., Kimathi, M., Ngure, K. (2025). Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach. Applied and Computational Mathematics, 14(4), 223-241. https://doi.org/10.11648/j.acm.20251404.15

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ACS Style

Kirimi, E. M.; Okelo, J.; Kimathi, M.; Ngure, K. Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach. Appl. Comput. Math. 2025, 14(4), 223-241. doi: 10.11648/j.acm.20251404.15

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AMA Style

Kirimi EM, Okelo J, Kimathi M, Ngure K. Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach. Appl Comput Math. 2025;14(4):223-241. doi: 10.11648/j.acm.20251404.15

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@article{10.11648/j.acm.20251404.15,
  author = {Erick Mutwiri Kirimi and Jeconiah Okelo and Mark Kimathi and Kenneth Ngure},
  title = {Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {4},
  pages = {223-241},
  doi = {10.11648/j.acm.20251404.15},
  url = {https://doi.org/10.11648/j.acm.20251404.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251404.15},
  abstract = {This study presents a novel mathematical model that captures the co-dynamics of pulmonary tuberculosis in the presence of opportunistic pneumonia. The model integrates opportunistic pneumonia into the classical pulmonary tuberculosis transmission framework, based on clinical evidence of a lethal synergism between the two diseases. A Holling-type saturation function is utilized to reflect the impact of natural immunity on the progression from latent tuberculosis infection to active disease. Asymptomatic infectious individuals can transmit the infection unnoticed, thereby significantly contributing to the high rate of transmission. Thus, the screening of asymptomatic individuals is incorporated into the model as a strategy to mitigate the prevalence of co-infections. A comprehensive sensitivity analysis and numerical simulations were conducted to evaluate the effects of various interventions, including enhancing natural immunity, screening asymptomatic and latently infected individuals, improving vaccine efficacy, and treating patients with severe pulmonary tuberculosis. The sensitivity analysis reveals that improving vaccine efficacy is the most effective strategy for minimizing co-infections. Simulations also demonstrate that increased screening and treatment rates significantly reduce the burden of pulmonary tuberculosis-pneumonia co-infections. These results underscore the need to develop vaccines with higher efficacy to reduce co-infections of pulmonary tuberculosis and opportunistic pneumonia. Additionally, the study recommends expanded screening of asymptomatic tuberculosis cases and the strengthening of immunity among latently infected populations.},
 year = {2025}
}

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TY  - JOUR
T1  - Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach

AU  - Erick Mutwiri Kirimi
AU  - Jeconiah Okelo
AU  - Mark Kimathi
AU  - Kenneth Ngure
Y1  - 2025/08/20
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251404.15
DO  - 10.11648/j.acm.20251404.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 223
EP  - 241
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251404.15
AB  - This study presents a novel mathematical model that captures the co-dynamics of pulmonary tuberculosis in the presence of opportunistic pneumonia. The model integrates opportunistic pneumonia into the classical pulmonary tuberculosis transmission framework, based on clinical evidence of a lethal synergism between the two diseases. A Holling-type saturation function is utilized to reflect the impact of natural immunity on the progression from latent tuberculosis infection to active disease. Asymptomatic infectious individuals can transmit the infection unnoticed, thereby significantly contributing to the high rate of transmission. Thus, the screening of asymptomatic individuals is incorporated into the model as a strategy to mitigate the prevalence of co-infections. A comprehensive sensitivity analysis and numerical simulations were conducted to evaluate the effects of various interventions, including enhancing natural immunity, screening asymptomatic and latently infected individuals, improving vaccine efficacy, and treating patients with severe pulmonary tuberculosis. The sensitivity analysis reveals that improving vaccine efficacy is the most effective strategy for minimizing co-infections. Simulations also demonstrate that increased screening and treatment rates significantly reduce the burden of pulmonary tuberculosis-pneumonia co-infections. These results underscore the need to develop vaccines with higher efficacy to reduce co-infections of pulmonary tuberculosis and opportunistic pneumonia. Additionally, the study recommends expanded screening of asymptomatic tuberculosis cases and the strengthening of immunity among latently infected populations.
VL  - 14
IS  - 4
ER  -

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Author Information

Erick Mutwiri Kirimi

Department of Mathematics, Pan African University Institute for Basic Sciences, Technology, and Innovation, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0009-0496-9592
Jeconiah Okelo

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0002-9439-4110
Mark Kimathi

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

Contact Email

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

Department of Public and Community Health, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

Contact Email

http://orcid.org/0000-0002-8062-0933

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APA Style

Kirimi, E. M., Okelo, J., Kimathi, M., Ngure, K. (2025). Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach. Applied and Computational Mathematics, 14(4), 223-241. https://doi.org/10.11648/j.acm.20251404.15

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ACS Style

Kirimi, E. M.; Okelo, J.; Kimathi, M.; Ngure, K. Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach. Appl. Comput. Math. 2025, 14(4), 223-241. doi: 10.11648/j.acm.20251404.15

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AMA Style

Kirimi EM, Okelo J, Kimathi M, Ngure K. Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach. Appl Comput Math. 2025;14(4):223-241. doi: 10.11648/j.acm.20251404.15

Copy | Download

@article{10.11648/j.acm.20251404.15,
  author = {Erick Mutwiri Kirimi and Jeconiah Okelo and Mark Kimathi and Kenneth Ngure},
  title = {Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {4},
  pages = {223-241},
  doi = {10.11648/j.acm.20251404.15},
  url = {https://doi.org/10.11648/j.acm.20251404.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251404.15},
  abstract = {This study presents a novel mathematical model that captures the co-dynamics of pulmonary tuberculosis in the presence of opportunistic pneumonia. The model integrates opportunistic pneumonia into the classical pulmonary tuberculosis transmission framework, based on clinical evidence of a lethal synergism between the two diseases. A Holling-type saturation function is utilized to reflect the impact of natural immunity on the progression from latent tuberculosis infection to active disease. Asymptomatic infectious individuals can transmit the infection unnoticed, thereby significantly contributing to the high rate of transmission. Thus, the screening of asymptomatic individuals is incorporated into the model as a strategy to mitigate the prevalence of co-infections. A comprehensive sensitivity analysis and numerical simulations were conducted to evaluate the effects of various interventions, including enhancing natural immunity, screening asymptomatic and latently infected individuals, improving vaccine efficacy, and treating patients with severe pulmonary tuberculosis. The sensitivity analysis reveals that improving vaccine efficacy is the most effective strategy for minimizing co-infections. Simulations also demonstrate that increased screening and treatment rates significantly reduce the burden of pulmonary tuberculosis-pneumonia co-infections. These results underscore the need to develop vaccines with higher efficacy to reduce co-infections of pulmonary tuberculosis and opportunistic pneumonia. Additionally, the study recommends expanded screening of asymptomatic tuberculosis cases and the strengthening of immunity among latently infected populations.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Pulmonary Tuberculosis and Pneumonia Co-Dynamics: A Mathematical Modelling Approach

AU  - Erick Mutwiri Kirimi
AU  - Jeconiah Okelo
AU  - Mark Kimathi
AU  - Kenneth Ngure
Y1  - 2025/08/20
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251404.15
DO  - 10.11648/j.acm.20251404.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 223
EP  - 241
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251404.15
AB  - This study presents a novel mathematical model that captures the co-dynamics of pulmonary tuberculosis in the presence of opportunistic pneumonia. The model integrates opportunistic pneumonia into the classical pulmonary tuberculosis transmission framework, based on clinical evidence of a lethal synergism between the two diseases. A Holling-type saturation function is utilized to reflect the impact of natural immunity on the progression from latent tuberculosis infection to active disease. Asymptomatic infectious individuals can transmit the infection unnoticed, thereby significantly contributing to the high rate of transmission. Thus, the screening of asymptomatic individuals is incorporated into the model as a strategy to mitigate the prevalence of co-infections. A comprehensive sensitivity analysis and numerical simulations were conducted to evaluate the effects of various interventions, including enhancing natural immunity, screening asymptomatic and latently infected individuals, improving vaccine efficacy, and treating patients with severe pulmonary tuberculosis. The sensitivity analysis reveals that improving vaccine efficacy is the most effective strategy for minimizing co-infections. Simulations also demonstrate that increased screening and treatment rates significantly reduce the burden of pulmonary tuberculosis-pneumonia co-infections. These results underscore the need to develop vaccines with higher efficacy to reduce co-infections of pulmonary tuberculosis and opportunistic pneumonia. Additionally, the study recommends expanded screening of asymptomatic tuberculosis cases and the strengthening of immunity among latently infected populations.
VL  - 14
IS  - 4
ER  -

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