SEIRS model for TB transmission dynamics incorporating the environment and optimal control

Tuberculosis (TB) remains a significant global health challenge, claiming more than 2 million lives annually, predominantly among adults. Existing studies often neglect the environment, reinfection, relapse/reactivation, and model calibration, thus limiting their applicability. This study presents a novel data-driven model that incorporates these factors to analyze the dynamics of TB transmission. Using the next-generation matrix approach, a basic reproduction number ( $$ ) of 1.737266 was calculated, indicating that active TB disease will persist in the human population without robust public health interventions. The model equations were numerically solved using fourth- and fifth-order Runge-Kutta methods. The model was calibrated to the historical TB incidence data for Kenya, spanning 2000 to 2022, using least squares curve fitting. The fitting algorithm yielded a mean absolute error (MAE) of 0.01% when comparing the actual data points with the results of the simulated model. This finding indicates that the proposed mathematical model closely aligns with the recorded TB incidence data. The optimal values of the model parameters were estimated from the fitting algorithm, and future TB transmission dynamics was projected for the next two decades. Key findings indicate that a 10% decrease in transmission rate, while maintaining other parameters constant, would result in a 10% reduction in TB transmission in Kenya. In addition, the incidence of tuberculosis in Kenya is expected to decrease to an estimated 35 cases per 100,000 people by 2045 with sustained efforts in Bacillus Calmette-Guérin (BCG) vaccination programs and public awareness campaigns. BCG vaccination emerges as the most cost-effective strategy to combat TB transmission in Kenya. Policymakers should prioritize investing in BCG vaccination programs to achieve optimal public health outcomes and economic benefits, aligning with Kenya's Vision 2030.