A Mathematical Analysis of Tuberculosis Transmission Using a Two-Age Group Compartmental Model

Abstract

In this study, a two-age-group compartmental model for tuberculosis transmission is developed and an alyzed, distinguishing between individuals below 10 years and those above this age. The model includes key epidemiological features such as reinfection within both age groups and the transition of treated individuals back to full susceptibility, representing realistic post-treatment outcomes. Essential analytical properties like non-negativity, existence, and uniqueness of solutions are established to confirm the model’s biological validity. Equilibrium analysis identifies both disease-free and endemic steady states, and the basic reproduc tion number, R_0, is derived using the next-generation matrix method. Local stability analysis indicates that the disease-free equilibrium is asymptotically stable when R_0 < 1, suggesting effective disease control, while instability occurs when R_0 > 1, resulting in disease persistence. Additionally, the global stability of both equilibrium states is rigorously proven-employing the Metzler matrix method for the disease-free case and a Lyapunov function for the endemic state-demonstrating the model’s strong dynamical behavior. Sensitivity analysis of R_0 identifies parameters that significantly impact tuberculosis transmission dynamics, offering in sights for targeted intervention strategies. Scenario analyses, supported by three-dimensional plots, illustrate how variations in parameters influence R_0 and the spread of infection. Numerical simulations conducted in Python validate the analytical findings, indicating an increase in immunity as individuals age from under 10 to above 10 years, while higher contact rates among children considerably enhance transmission potential. This study provides a deeper understanding of age-dependent tuberculosis dynamics and offers relevant implications for disease control policies across different age groups.