An Isolation Model for Tuberculosis Dynamics with Optimal Control Application

Abstrak

Tuberculosis (TB) remains a persistent global health challenge, worsened by asymptomatic carriers who contribute to undetected transmission. An SIQR mathematical model that classifies infected individuals into symptomatic and asymptomatic classes, with isolation as the primary intervention, is formulated in this study. We establish the positivity and invariant region to ensure epidemiological relevance and derive the basic reproduction number, R0, as a threshold for disease persistence. The model analysis reveals that the diseasefree equilibrium is both locally and globally asymptotically stable if R0 < 1, while an endemic equilibrium also exists if R0 > 1. The key parameters influencing transmission dynamics are identified through sensitivity analysis. Furthermore, an optimal control framework is formulated using the Pontryagin’s maximum principle to assess the efficacy of isolation in reducing disease burden while minimizing associated costs. Numerical simulations demonstrate that well-implemented isolation significantly curtails TB spread, highlighting its potential as a targeted intervention.