Modelling the Dynamics of Tuberculosis A Multi-Stage Compartmental Approach
DOI:
https://doi.org/10.5614/j.math.fund.sci.2025.57.2.5Keywords:
diagnosed TB, immunity loss, latent infection, multi-stage model, TB relapse, tuberculosis (TB), undiagnosed TBAbstract
We propose a robust multi-stage compartmental model to study
tuberculosis (TB) transmission dynamics in Namibia, incorporating multiple stages
of latent infection, along with undiagnosed and diagnosed active cases. The model
distinguishes between early and late latent stages, allowing for a more accurate
representation of TB progression. By capturing the complex interactions between
disease progression, diagnosis, treatment, and relapse, the model provides valuable
insights into TB dynamics. Mathematical analyses and key simulation parameters
are discussed, offering a framework that can inform public health strategies for
effective TB control and intervention in Namibia.
References
Castillo-Chavez, C. & Song, B., Dynamical Models of Tuberculosis and Their
Applications, Mathematical Biosciences and Engineering, 1(2), pp.361?404,
Roeger, L.-I., Feng, Z., & Castillo-Chavez, C., Modeling TB and HIV
Co-infections, Mathematical Biosciences and Engineering, 6(4), pp. 815?837,
Jabbari, A., Levin, S.A., Castillo-Chavez, C., Nazari, F., Song, B. &
Kheiri, H., A Two-Strain TB Model with Multiple Latent Stages, Mathematical
Biosciences and Engineering, 13(4), pp. 741?785, 2016.
Adetunde, I.A., The Mathematical Models of the Dynamical Behaviour of
Tuberculosis Disease in the Upper Region of the Northern Part of Ghana: A Case
Study of Bawku, Current Research in Tuberculosis, 1(1), pp. 1?6, 2008.
Kasereka, S.K., Multi-scale Environment, arXiv preprint, arXiv:2411.04297,
Sy, K.T.L., Leavitt, S.V., de Vos, M., Dolby, T., Bor, J., Horsburgh Jr, C.R.,
Warren, R.M., Streicher, E.M., Jenkins, H.E. & Jacobson, K.R., Spatial
Heterogeneity of Extensively Drug-Resistant Tuberculosis in Western Cape Province,
South Africa, Scientific Reports, 12(1), p.10844, 2022.
Herrera, M., Bosch, P., Najera, M., & Aguilera, X., Modeling the Spread
of Tuberculosis in Semiclosed Communities, Computational and Mathematical
Methods in Medicine, 648291, pp. 1?19, 2013.
Ochieng, F.O., Mathematical Modeling of Tuberculosis Transmission Dynamics
With Reinfection and Optimal Control, Engineering Reports, 7:e13068, 2025.
Pienaar, E., Fluitt, A.M., Whitney, S.E., Freifeld, A.G. & Viljoen, H.J.,
A Model of Tuberculosis Transmission and Intervention Strategies in an Urban
Residential Area, Computational Biology and Chemistry, 34(2), pp. 86?96,
Pedrazzoli, D., Boccia, D., Dodd, P.J., Lonnroth, K., Dowdy, D.W., Siroka,
A., Kimerling, M.E., White, R.G. & Houben, R.M.G.J., Modelling the
Social and Structural Determinants of Tuberculosis: Opportunities and Challenges,
International Jurnal of Tuberculosis and Lung Disease 21(9), pp. 957?964,
Eegunjobi, A.S. & Makinde, O.D., Impact of Corruption in a Society with
Exposed Honest Individuals: A Mathematical Model, Asia Pacific Journal of
Mathematics, 10(18), pp. 1?22, 2023.
Eegunjobi, A.S & Makinde, O.D., Mathematical Analysis of Two Strains
Covid-19 Disease Using SEIR Model, Journal of Mathematical and Fundamental
Sciences, 54(2), pp. 211?232, 2022.
Anyanwu, M.C., Neossi-Nguetchue, S.N., Moremedi, G.M., Eegunjobi, A.S.,
Mathematical Model and Optimal Control of Covid-19 in Nigeria, Earthline Journal
of Mathematical Science, 14(3), pp. 421?441, 2024.
Moreno, V., Espinoza, B., Barley, K., Paredes, M., Bichara, D., Mubayi,
A. & Castillo-Chavez, C., The Role of Mobility and Health Disparities on
the Transmission Dynamics of Tuberculosis, Theoretical Biology and Medical
Modelling, 14(3), 2017.
Kasereka, S.K., Goufo,E.F.D. & Tuong, V.H., Analysis and Simulation of a
Mathematical Model of Tuberculosis Transmission in Democratic Republic of the
Congo, Advances in Difference Equations, p. 642, 2020.
Diekmann, O., Heesterbeek, J.A.P. & Metz, J.A.J., On the Definition and the
Computation of the Basic Reproduction Ratio R0 in Models for Infectious Diseases in
Heterogeneous Populations, Journal of Mathematical Biology, 28, pp. 365?382,
Van den Driessche, P. &Watmough, J., Reproduction Numbers and Sub-threshold
Endemic Equilibria for Compartmental Models of Disease Transmission,
Mathematical Biosciences, 180, pp. 29?48, 2002.
Korobeinikov, A., Global Stability of SIR and SIRS Epidemiological Models
with Nonlinear Incidence Rates, Mathematical Medicine and Biology, 19, pp.
?134, 2002.
Li, M.Y. & Thieme, H.R., Global Dynamical Behavior of an Epidemic Model
in a Patchy Environment, SIAM Journal on Mathematical Analysis, 32, pp.
?1102, 2001.
World Health Organization, WHO Regional Office for Africa: Tuberculosis Report,
World Health Organization, https://www.afro.who.int/, (14 November 2025).
Namibia Statistics Agency, Life Expectancy Report for Namibia, Demographic
and Health Surveys - Namibia Statistics Agency, https://www.nsa.org.na/,
(14 November 2025).
World Health Organization, Global Tuberculosis Report, WHO Publications,
https://www.who.int/publications/, (14 November 2025).
Centers for Disease Control and Prevention, Tuberculosis Data and Statistics,
CDC TB Reports, https://www.cdc.gov/tb/statistics/, (14 November 2025).
Ministry of Health and Social Services, Namibia, National Tuberculosis Control
Program Annual Report, MoHSS Publications, https://www.mhss.gov.na/, (14
November 2025).
Mathematical Modeling Assumptions, Parameters for TB Transmission Models,
Theoretical Epidemiology Journal, https://www.modeling-tb.com/, (14
November 2025).
Chitnis, N., Hyman, J.M. & Cushing, J.M., Determining Important Parameters
in the Spread of Malaria through the Sensitivity Analysis of a Mathematical Model,
Bulletin of Mathematical Biology, 70, pp. 1272-1296, 2008.
Berhe, H.W., Makinde, O.D. & Theuri,D.M., Parameter Estimation and
Sensitivity Analysis of Dysentery Diarrhea Epidemic Model, Journal of Applied
Mathematics, 8465747, 2019.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Journal of Mathematical and Fundamental Sciences
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
