An SEITR Model for Tuberculosis Incorporating Migration, General Non-Linear Incidence, and Treatment Effectiveness
5 affiliations
- 1Quantitative Technics Department, Faculty of Economics and Management, University of Nouakchott, Nouakchott P.O. Box 880, Mauritania
- 2Analysis and Modeling for Environment and Health (UMR-AMES), Nouakchott P.O. Box 880, Mauritania
- 3Department of Industrial Engineering, Higher Institute of Industrial Engineering, Nouakchott P.O. Box 880, Mauritania
- 4Department of Mathematics, Faculty of Sciences and Technics, University of Nouakchott, Nouakchott P.O. Box 880, Mauritania
- 5Department of Statistics and Operation Research, College of Sciences, Qassim University, P.O. Box 6688, Buraydah 51452, Saudi Arabia
Abstract
Tuberculosis (TB) remains a major global public health challenge, particularly in populations affected by migration and limited access to healthcare. In this study, we formulate and analyze a compartmental SEITR model (Susceptible–Exposed–Infectious–Treated–Recovered) for tuberculosis transmission, incorporating a general nonlinear incidence function of the form f1(S)f2(I)+f3(T). The model accounts for migration-related population influx and distinguishes between scenarios with and without influx into disease-related compartments. For the model with migration, we establish the existence and global asymptotic stability of a unique endemic equilibrium under suitable conditions. For the reduced model without direct influx into disease-related compartments, we derive the basic reproduction number R0 using the next-generation matrix method and establish the global asymptotic stability of the disease-free equilibrium for R0<1 and the endemic equilibrium for R0>1, using suitable Lyapunov functions and LaSalle’s invariance principle. We further employ the center manifold approach of Castillo-Chavez and Song to demonstrate a forward transcritical bifurcation at the threshold R0=1. The model explicitly accounts for incomplete and fully effective treatment, allowing us to investigate how treatment outcomes influence disease persistence and control. A global sensitivity analysis based on Sobol indices assesses the relative influence of model parameters on R0. Notably, we identify a critical treatment-efficacy threshold pc : when pc>1, even perfect treatment (p=1) cannot drive R0 below unity, implying that treatment alone cannot eliminate TB in high-burden regimes. Finally, numerical simulations investigate the effects of migration-related influx, bilinear versus saturated incidence, and treatment effectiveness on the epidemic dynamics. These findings contribute to the mathematical understanding of tuberculosis transmission and highlight the importance of treatment outcomes and migration in assessing disease control strategies.
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BibTeX
@article{mohamed2026seitr,
title = {{An SEITR Model for Tuberculosis Incorporating Migration, General Non-Linear Incidence, and Treatment Effectiveness}},
author = {Yahya Mohamed and Aziza Ahmedou and Lativa Sid Ahmed Abdellahi and Sultana DIDI and Ahmed Jidou El Bechir},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213874},
url = {https://www.mdpi.com/2227-7390/14/21/3874},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - An SEITR Model for Tuberculosis Incorporating Migration, General Non-Linear Incidence, and Treatment Effectiveness
AU - Yahya Mohamed
AU - Aziza Ahmedou
AU - Lativa Sid Ahmed Abdellahi
AU - Sultana DIDI
AU - Ahmed Jidou El Bechir
PY - 2026
DA - 2026/10/23
JO - Mathematics
DO - 10.3390/math14213874
UR - https://www.mdpi.com/2227-7390/14/21/3874
AB - Tuberculosis (TB) remains a major global public health challenge, particularly in populations affected by migration and limited access to healthcare. In this study, we formulate and analyze a compartmental SEITR model (Susceptible–Exposed–Infectious–Treated–Recovered) for tuberculosis transmission, incorporating a general nonlinear incidence function of the form f1(S)f2(I)+f3(T). The model accounts for migration-related population influx and distinguishes between scenarios with and without influx into disease-related compartments. For the model with migration, we establish the existence and global asymptotic stability of a unique endemic equilibrium under suitable conditions. For the reduced model without direct influx into disease-related compartments, we derive the basic reproduction number R0 using the next-generation matrix method and establish the global asymptotic stability of the disease-free equilibrium for R0<1 and the endemic equilibrium for R0>1, using suitable Lyapunov functions and LaSalle’s invariance principle. We further employ the center manifold approach of Castillo-Chavez and Song to demonstrate a forward transcritical bifurcation at the threshold R0=1. The model explicitly accounts for incomplete and fully effective treatment, allowing us to investigate how treatment outcomes influence disease persistence and control. A global sensitivity analysis based on Sobol indices assesses the relative influence of model parameters on R0. Notably, we identify a critical treatment-efficacy threshold pc : when pc>1, even perfect treatment (p=1) cannot drive R0 below unity, implying that treatment alone cannot eliminate TB in high-burden regimes. Finally, numerical simulations investigate the effects of migration-related influx, bilinear versus saturated incidence, and treatment effectiveness on the epidemic dynamics. These findings contribute to the mathematical understanding of tuberculosis transmission and highlight the importance of treatment outcomes and migration in assessing disease control strategies.
SN - 2227-7390
ER -