Analysis of mathematical transmission in the SEI<b> <i>L</i> </b>1<b> <i>L</i> </b>2R tuberculosis model

Tuberculosis (TB) remains a major worldwide health threat, with growing concern for its transmission pattern and persistence. In this paper, we develop a mathematical model of TB transmission as an SEIL1L2R compartment system that includes temporal subordinate boundaries in order to model the dynamics of disease spread more realistically. We split the population into six different compartments based on the phases of disease development and risk factors. The main objective of this work is to examine the stability of the model, calculate the basic reproduction number (R0), and investigate conditions for both local and global asymptotic stability at the infection-free equilibrium and endemic equilibrium states. Numerical computations show that the system is locally stable if R0 ≈ 0.846 (R0 &amp;lt; 1), which implies the non-persistence of TB, and globally stable if R0 &amp;gt; 1, indicating the possibility of endemic transmission of TB. In addition, bifurcation analysis through center manifold theory exhibits a forward bifurcation when R0 crosses the critical threshold, highlighting the transition from disease-free to endemic states. These results are highly informative regarding TB transmission dynamics and are critical to inform policy choices for TB control interventions, vaccination policy, and risk factor management in high-prevalence areas.