Tuberculosis (TB) is an infectious disease caused by mycobacterium tuberculosis bacteria (tubercle bacilli) and spreads through the air from an infectious person to a susceptible person. This research aims to investigate the transmission dynamics of tuberculosis (TB) using a mathematical model with optimal control and to estimate the reproduction number in the Werder district of the Somali regional state. To achieve the objective of the study, we employed different mathematical tools and available cumulative case data from the Werder district of the Somali regional state health officials. The model is developed using a system of nonlinear ordinary differential equations and basic reproduction number is determine to perform analysis for both local and global stability of the equilibrium points of the model. In addition to this, sensitivity analysis is studied based on the reproduction number to reveal influential parameters of transmission dynamics of tuberculosis. Then the optimal control strategy was found by minimizing the number of exposed and infected individual populations with tuberculosis taking into account the cost of implementation. The existence of optimal control and characterization was established with the help of Pontryagin's Maximum Principle and we conducted different numerical simulation cases to see agreements with analytical results. Moreover, the model was calibrated with real data collected from the Werder district of the Somali regional state health office from 2016 – 2022. The simulation results also clearly shown that the tuberculosis can be minimized by applying TB prevention programs to susceptible individuals and preventing the failure of treatment in active tuberculosis individuals through continuous supervision and helping patients during treatment period with optimal implementation cost. Furthermore, the optimal control model also predicted that tuberculosis is under control in Werder district.
| Published in | Science Development (Volume 7, Issue 1) |
| DOI | 10.11648/j.scidev.20260701.11 |
| Page(s) | 1-22 |
| 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), 2026. Published by Science Publishing Group |
Tuberculosis (TB), Differential Equations, Basic Reproduction Number, Stability, Optimal Control
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APA Style
Taddese, T. A., Silase, T. M. G., Tessema, T. M., Adem, S. D. (2026). Mathematical Modelling and Optimal Control of Tuberculosis Transmission Dynamics in Werder District, Somali Regional State, Ethiopia. Science Development, 7(1), 1-22. https://doi.org/10.11648/j.scidev.20260701.11
ACS Style
Taddese, T. A.; Silase, T. M. G.; Tessema, T. M.; Adem, S. D. Mathematical Modelling and Optimal Control of Tuberculosis Transmission Dynamics in Werder District, Somali Regional State, Ethiopia. Sci. Dev. 2026, 7(1), 1-22. doi: 10.11648/j.scidev.20260701.11
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Download@article{10.11648/j.scidev.20260701.11,
author = {Tasew Ayele Taddese and Tsega Mhretab Gebre Silase and Tsegaw Muche Tessema and Sadam Dawud Adem},
title = {Mathematical Modelling and Optimal Control of Tuberculosis Transmission Dynamics in Werder District, Somali Regional State, Ethiopia},
journal = {Science Development},
volume = {7},
number = {1},
pages = {1-22},
doi = {10.11648/j.scidev.20260701.11},
url = {https://doi.org/10.11648/j.scidev.20260701.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.scidev.20260701.11},
abstract = {Tuberculosis (TB) is an infectious disease caused by mycobacterium tuberculosis bacteria (tubercle bacilli) and spreads through the air from an infectious person to a susceptible person. This research aims to investigate the transmission dynamics of tuberculosis (TB) using a mathematical model with optimal control and to estimate the reproduction number in the Werder district of the Somali regional state. To achieve the objective of the study, we employed different mathematical tools and available cumulative case data from the Werder district of the Somali regional state health officials. The model is developed using a system of nonlinear ordinary differential equations and basic reproduction number is determine to perform analysis for both local and global stability of the equilibrium points of the model. In addition to this, sensitivity analysis is studied based on the reproduction number to reveal influential parameters of transmission dynamics of tuberculosis. Then the optimal control strategy was found by minimizing the number of exposed and infected individual populations with tuberculosis taking into account the cost of implementation. The existence of optimal control and characterization was established with the help of Pontryagin's Maximum Principle and we conducted different numerical simulation cases to see agreements with analytical results. Moreover, the model was calibrated with real data collected from the Werder district of the Somali regional state health office from 2016 – 2022. The simulation results also clearly shown that the tuberculosis can be minimized by applying TB prevention programs to susceptible individuals and preventing the failure of treatment in active tuberculosis individuals through continuous supervision and helping patients during treatment period with optimal implementation cost. Furthermore, the optimal control model also predicted that tuberculosis is under control in Werder district.},
year = {2026}
}
TY - JOUR T1 - Mathematical Modelling and Optimal Control of Tuberculosis Transmission Dynamics in Werder District, Somali Regional State, Ethiopia AU - Tasew Ayele Taddese AU - Tsega Mhretab Gebre Silase AU - Tsegaw Muche Tessema AU - Sadam Dawud Adem Y1 - 2026/01/16 PY - 2026 N1 - https://doi.org/10.11648/j.scidev.20260701.11 DO - 10.11648/j.scidev.20260701.11 T2 - Science Development JF - Science Development JO - Science Development SP - 1 EP - 22 PB - Science Publishing Group SN - 2994-7154 UR - https://doi.org/10.11648/j.scidev.20260701.11 AB - Tuberculosis (TB) is an infectious disease caused by mycobacterium tuberculosis bacteria (tubercle bacilli) and spreads through the air from an infectious person to a susceptible person. This research aims to investigate the transmission dynamics of tuberculosis (TB) using a mathematical model with optimal control and to estimate the reproduction number in the Werder district of the Somali regional state. To achieve the objective of the study, we employed different mathematical tools and available cumulative case data from the Werder district of the Somali regional state health officials. The model is developed using a system of nonlinear ordinary differential equations and basic reproduction number is determine to perform analysis for both local and global stability of the equilibrium points of the model. In addition to this, sensitivity analysis is studied based on the reproduction number to reveal influential parameters of transmission dynamics of tuberculosis. Then the optimal control strategy was found by minimizing the number of exposed and infected individual populations with tuberculosis taking into account the cost of implementation. The existence of optimal control and characterization was established with the help of Pontryagin's Maximum Principle and we conducted different numerical simulation cases to see agreements with analytical results. Moreover, the model was calibrated with real data collected from the Werder district of the Somali regional state health office from 2016 – 2022. The simulation results also clearly shown that the tuberculosis can be minimized by applying TB prevention programs to susceptible individuals and preventing the failure of treatment in active tuberculosis individuals through continuous supervision and helping patients during treatment period with optimal implementation cost. Furthermore, the optimal control model also predicted that tuberculosis is under control in Werder district. VL - 7 IS - 1 ER -
Department of Mathematics, Kebri Dehar University, Kebri Dehar, Ethiopia
Department of Mathematics, Kebri Dehar University, Kebri Dehar, Ethiopia
Department of Mathematics, Kebri Dehar University, Kebri Dehar, Ethiopia
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