Mathematical Analysis of Multi-Route Tuberculosis Transmission with Airborne Environmental Reservoirs and Treatment Failure

Despite the availability of effective treatment, tuberculosis (TB) remains a major public health problem worldwide. Related factors that promote continued transmission include treatment failure, residual infectiousness during treatment, environmental contamination, and progression from latent to active TB in the absence of tuberculosis preventive treatment (TPT). This study aims to investigate the interplay of these factors with regard to TB transmission and identify intervention measures that can shift the disease from persistence to elimination. To achieve this objective, we formulate and analyze an SLIQTPR compartmental model that incorporates latent infection, preventive treatment of latently infected individuals, treatment of infectious individuals, treatment failure, residual infectiousness among treated individuals, and an environmental pathogen reservoir. The model accounts for both direct transmission from infectious and treated individuals and indirect transmission through environmental contamination. Positivity and boundedness of the model solutions are established, and the disease-free and endemic equilibria are characterized. The basic reproduction number, R0, is derived using the next-generation matrix method and used to determine the threshold behavior of the model. Lyapunov function and LaSalle-invariance arguments are employed to establish the global asymptotic stability of the disease-free equilibrium when R0≤1 and of the endemic equilibrium when R0>1. Normalized local and global sensitivity analyses are performed to identify the parameters exerting the greatest influence on TB transmission. In addition, an explicit threshold relationship between treatment failure and the residual infectiousness of treated individuals is derived to quantify their joint effect on the transition between disease elimination and persistence. Numerical simulations support the analytical findings and illustrate the effects of active treatment, TPT, and environmental pathogen clearance. The results indicate that TPT expansion, treatment efficacy improvement, treatment failure and residual infectiousness minimization, and speeding up environmental pathogen clearance may substantially decrease TB transmission. These results provide a quantitative basis for the development of integrated TB-control strategies that simultaneously target latent infection, treatment outcomes, and environmental transmission.