Asian Journal of Pure and Applied Mathematics

Tuberculosis remains a major public health challenge, and mathematical modelling provides a structured means of examining its transmission and control. This study formulates a deterministic compartmental model that incorporates susceptible, vaccinated, latently infected, treated latent, infectious, treated infectious, recovered, and relapsed population classes. The model is expressed as a system of nonlinear ordinary differential equations and includes vaccination, waning vaccine-induced immunity, progression from latent to active infection, treatment, recovery, relapse, natural mortality, and disease-induced mortality. Positivity and boundedness of the model solutions are considered to establish epidemiological feasibility. The disease-free and endemic equilibrium states are derived, while the effective reproduction number is obtained using the next-generation matrix method. Local stability is examined using the Routh–Hurwitz criterion and the trace-and-determinant approach, and global stability is assessed through an appropriate Lyapunov function. Sensitivity analysis identifies transmission-related parameters as important drivers of disease persistence, whereas vaccination and recovery-related parameters contribute to reducing tuberculosis prevalence. Numerical simulations further indicate that lower transmission, increased vaccination, and faster recovery reduce latent and infectious populations over time, while faster waning of vaccine-induced immunity may increase the latent infection burden. The findings support the combined use of vaccination, effective treatment, and measures that reduce infectious contact.