Stability analysis of the disease-free equilibrium of tuberculosis transmission dynamics incorporating drug resistance and vaccination

Abstract

Tuberculosis (TB) remains a major public health challenge, particularly in high-burden countries such as Nigeria. In this study, a deterministic compartmental model was developed and analyzed to investigate the transmission dynamics of TB incorporating vaccination, first-line and second-line treatment, and drug-resistant TB. The human population is stratified into eight compartments namely: susceptible, vaccinated, latent, infectious, first-line treatment, drug-resistant, second-line treatment, and recovered. The model captures key biological processes, including progression from latent to active TB, treatment failure leading to resistance, waning immunity, relapse, and TB-induced mortality. Analysis is performed to establish the positivity and boundedness of solutions, ensuring biological feasibility. The disease-free equilibrium (DFE) is derived, and the basic reproduction number ($R_0$) is computed using the next-generation matrix approach. Local stability of the DFE is analyzed using the Jacobian matrix and eigenvalue analysis, while global asymptotic stability is established using a Lyapunov function, showing that TB can be eliminated whenever $R_0 < 1$. Numerical simulations, calibrated with secondary data from the WHO and Nigeria’s National TB Programme (2020–2023), are used to show the model dynamics under different intervention scenarios. Results show that scaling up first-line and second-line treatment substantially reduces TB prevalence, infectious burden, drug-resistant TB, and cumulative mortality. Vaccination provides preventive intervention. Sensitivity analysis using partial rank correlation coefficients reveals that treatment-related parameters, particularly second-line treatment, exert the strongest influence on $R_0$. Overall, the findings highlight the necessity of integrated TB control strategies that combine effective treatment and vaccination to achieve sustained TB reduction and eventual elimination in Nigeria.