Mathematical Modeling of Tuberculosis Transmission Dynamics With Reinfection and Optimal Control

Abstract

Tuberculosis (TB) remains a significant global health challenge, claiming over 2 million lives annually, predominantly among adults. Existing TB models often neglect seasonal variations, optimal control, and reinfection, limiting their accuracy in predicting disease dynamics. This study presents a novel data-driven SVEITRS mathematical model incorporating these factors to analyze TB transmission dynamics. Employing the next-generation matrix approach, a basic reproduction number $\left(R_0\right)$ of 1.005341 was calculated, suggesting that without robust public health interventions, TB disease may persist in Kenya. The model equations were solved numerically using fourth- and fifth-order Runge–Kutta methods, with the forward–backward sweep technique applied to the optimal control problem. The model was fitted to historical TB incidence data for Kenya from 2000 to 2022 using lsqcurvefit algorithm in MATLAB software. The fitting algorithm yielded a mean absolute error (MAE) of 0.0069, demonstrating a close alignment between simulated and observed data. The optimized parameter values were used to project future TB dynamics. Key findings indicate that a 20% decrease in transmission rate coupled with a 5% increase in vaccine efficacy, while maintaining other parameters constant, would result in a 32.60% reduction in TB transmission in Kenya. Moreover, the incidence of TB in Kenya is expected to decrease to an estimated 17 cases per 100,000 people by 2045 with sustained efforts in vaccine development and public awareness campaigns. The development of highly efficacious vaccines emerges as the most cost-effective strategy in combating TB transmission in Kenya. Policymakers should prioritize investing in the development and deployment of highly efficacious vaccines to achieve optimal public health outcomes and economic benefits, aligning with Kenya's Vision 2030.