Transmission Dynamics of Tuberculosis Model with Control Strategies

M. L. Olaosebikan, M. K. Kolawole, K. A. Bashiru


Tuberculosis (TB) is a global health concern, with a significant proportion of the population at severe risk of infection. Mathematical models can provide valuable insights into the transmission dynamics of TB, especially with the use of vaccination and the mixed proportional incidence rate. In this study, we developed a compartmental model to analyze the impact of mixing proportional incidence rates with vaccination on TB transmission. We conducted a qualitative study on the mathematical model, which included showing that it is unique, positively invariant, and bounded, showing that it is epidemiologically sound to study the physical transmission of TB. We used the homotopy perturbation method to obtain numerical solutions to the model. Using python software, we simulated the obtained results, and our results show that increasing vaccination coverage is an effective measure for reducing TB incidence. Furthermore, our analysis suggests that the mixing proportional incidence rate can be used to predict the spatial spread of TB in a population. It was concluded that vaccination and proportional incidence rate mixing are critical factors to be considered when developing effective TB control strategies.


Mathematical Model; Tuberculosis; Basic Reproduction Number; Local Stability; Global Stability; Sensitivity Analysis; Python Software

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B. M. Murphy, B. H. Singer, and D. Kirschner, “On treatment of tuberculosis in heterogeneous populations,” Journal of Theoretical Biology, vol. 223, no. 4, pp. 391–404, 2003. DOI:10.1016/s0022-5193(03)00038-9

T. M. Daniel, J. H. Bates, and K. A. Downes, History of Tuberculosis. John Wiley & Sons, Ltd, 1994, ch. 2, pp. 13–24. ISBN 9781683672753. DOI:10.1128/9781555818357.ch2

W. H. Organization et al., “World health organization global tuberculosis report 2021,” URL: https://www. who. int/teams/global-tuberculosis-programme/tbreports/global-tuberculosis-report-2021, Accessed on 14 October 2021.

K. A. Bashiru, O. Fasoranbaku, T. Ojurongbe, M. Lawal, A. Abiona, and B. Oluwasanmi, “A Stochastic Model to Analyze and Predict Transmission Dynamics of Tuberculosis in Ede Kingdom of Osun State,” Fountain Journal of Natural and Applied Sciences, vol. 7, no. 1, pp. 12–19, 2018. DOI:10.53704/fujnas.v7i1.182

S. Khajanchi, D. K. Das, and T. K. Kar, “Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation,” Physica A: Statistical Mechanics and its Applications, vol. 497, pp. 52–71, 2018. DOI:10.1016/j.physa.2018.01.014

S. Bisuta, P. Kayembe, M. Kabedi, H. Situakibanza, J. Ditekemena, A. Bakebe, G. Lay, G. Mesia, J. Kayembe, and S. Fueza, “Trends of bacteriologically confirmed pulmonary tuberculosis and treatment outcomes in democratic republic of the congo: 2007–2017,” Ann. Afr. Med, vol. 11, no. 4, pp. 2974–2985, 2018.

M. Dauda, A. Magaji, P. Okolo, J. Bulus, and U. Shehu, “Analyzing the transmission dynamics of tuberculosis in kaduna metropolis, nigeria,” Science World Journal, vol. 15, no. 4, pp. 76–82, 2020.

F. Firmansyah and Y. M. Rangkuti, “Sensitivity Analysis and Optimal Control of Covid 19 Model,” Jambura Journal of Biomathematics (JJBM), vol. 4, no. 1, pp. 95–102, 2023. DOI:10.34312/jjbm.v4i1.19025

M. Manaqib, M. Mahmudi, and G. Prayoga, “Mathematical Model and Simulation of the Spread of COVID-19 with Vaccination, Implementation of Health Protocols, and Treatment,” Jambura Journal of Biomathematics (JJBM), vol. 4, no. 1, pp. 69–79, 2023. DOI:10.34312/jjbm.v4i1.19162

R. Resmawan, L. Yahya, R. S. Pakaya, H. S. Panigoro, and A. R. Nuha, “Analisis Dinamik Model Penyebaran COVID-19 dengan Vaksinasi,” Jambura Journal of Biomathematics (JJBM), vol. 3, no. 1, 2022. DOI:10.34312/jjbm.v3i1.13176

Fatmawati, M. A. Khan, C. Alfiniyah, and E. Alzahrani, “Analysis of dengue model with fractal-fractional Caputo–Fabrizio operator,” Advances in Difference Equations, vol. 2020, no. 1, pp. 1–23, 2020. DOI:10.1186/s13662-020-02881-w

T. A. Ayoola, M. K. Kolawole, and A. O. Popoola, “Mathematical model of covid-19 transmission dynamics with double dose vaccination,” Tanzania Journal of Science, vol. 48, no. 2, pp. 499–512, 2022. DOI:10.4314/tjs.v48i2.23

Y. Yang, J. Li, Z. Ma, and L. Liu, “Global stability of two models with incomplete treatment for tuberculosis,” Chaos, Solitons & Fractals, vol. 43, no. 1-12, pp. 79–85, 2010. DOI:10.1016/j.chaos.2010.09.002

J. Zhang, Y. Li, and X. Zhang, “Mathematical modeling of tuberculosis data of china,” Journal of theoretical biology, vol. 365, pp. 159–163, 2015. DOI:10.1016/j.jtbi.2014.10.019

A. Egonmwan and D. Okuonghae, “Mathematical analysis of a tuberculosis model with imperfect vaccine,” International Journal of Biomathematics, vol. 12, no. 07, p. 1950073, 2019. DOI:10.1142/S1793524519500736

I. Syahrini, Sriwahyuni, V. Halfiani, S. M. Yuni, T. Iskandar, Rasudin, and M. Ramli, “The epidemic of tuberculosis on vaccinated population,” in Journal of Physics: Conference Series, vol. 890, no. 1, pp. 1–6, 2017. DOI:10.1088/1742-596/890/1/012017

M. K. Kolawole, M. O. Olayiwola, A. I. Alaje, H. O. Adekunle, and K. A. Odeyemi, “Conceptual analysis of the combined effects of vaccination, therapeutic actions, and human subjection to physical constraint in reducing the prevalence of covid-19 using the homotopy perturbation method,” Beni-Suef University Journal of Basic and Applied Sciences, vol. 12, no. 1, pp. 1–20, 2023. DOI:10.1186/s43088-023-00343-2

A. A. Ayoade, O. J. Peter, A. I. Abioye, T. Adinum, and O. A. Uwaheren, “Application of homotopy perturbation method to an sir mumps model,” Advances in Mathematics: Scientific Journal, vol. 9, no. 3, pp. 1329–1340, 2020. DOI:10.37418/amsj.9.3.57

M. Ibrahim, O. Peter, O. Ogwumu, and O. Akinduko, “On the homotopy analysis method for solving pstir typhoid model,” Trans. Nigerian Assoc. Math. Phys, vol. 4, pp. 51–56, 2017.

O. Peter and A. Awoniran, “Homotopy perturbation method for solving sir infectious disease model by incorporating vaccination,” The Pacific Journal of Science and Technology, vol. 19, no. 1, pp. 133–140, 2018.

L. M. Erinle-Ibrahim, W. O. Lawal, O. Adebimpe, and G. R. Sontan, “A susceptible exposed infected recovered susceptible (seirs) model for the transmission of tuberculosis,” Tanzania Journal of Science, vol. 47, no. 3, pp. 917–927, 2021. DOI: 10.4314/tjs.v47i3.4


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