Global stability of SEIM tuberculosis model with two infection phases and medication effects

Abstract

Tuberculosis (TB), caused by mycobacterium tuberculosis (MTB), remains a significant global health issue, leading to high morbidity and mortality rates despite being a preventable and curable disease. The dynamics of TB transmission and the effects of treatment are critical to improving disease management. This study aims to analyze the global stability of a susceptible, exposed, infected, medicated (SEIM) model for TB transmission, incorporating the effects of medication and infection phases on disease progression. A deterministic SEIM model is proposed, dividing the population into four compartments: susceptible, exposed, infected, and medicated. The model accounts for treatment effects, including non-permanent immunity and the potential dormancy of MTB. Stability analysis was conducted using Lyapunov functions to evaluate equilibrium points, and the basic reproduction number (ℜ0) was derived to determine disease dynamics. The analysis reveals that when ℜ0 < 1, the system is globally asymptotically stable at the non-endemic equilibrium, indicating disease eradication. Conversely, when ℜ0 >1, the system converges to the endemic equilibrium, signifying sustained transmission within the population. These findings highlight the critical role of treatment and infection dynamics in controlling TB spread. The SEIM model provides a comprehensive framework for understanding TB transmission dynamics and emphasizes the importance of reducing (ℜ0) through effective public health interventions. Further research is recommended to validate the model with empirical data and explore its applicability in different epidemiological settings.