Local stability analysis for tuberculosis epidemic with SI1I2R model

Abstract

TB is an infectious disease that attacks parts of the body, one of the lungs. This disease is caused by the bacterium Mycobacterium Tuberculosis through the droplet nucleus which spreads when the patient coughs or sneezes. This TB epidemic model is classified into four subpopulations, namely the susceptible subpopulation, the symptomatic infection subpopulation, the asymptomatic infection subpopulation, and the recovered subpopulation, by introducing treatment at each stage of infection and analysing dynamic models. The analysis was carried out to determine the local stability balance point which can be determined through the basic reproduction number (ℛ0) obtained through the Next Generation Matrix (NGM). When the value of ℛ0 <1, the disease-free equilibrium point will be asymptotically stable using the Routh-Hurwitz criteria, and when ℛ0> 1, the endemic equilibrium point will be asymptotically stable locally using the Manifold Center. The Routh-Hurwitz criterion is used to investigate the stability of the disease-free equilibrium point if all parts of the real eigenvalues of the jacobian matrix are negative. The manifold center method is used to investigate the stability of the endemic equilibrium point at ℛ0> 1 which is determined by the number of roots of the characteristic equation with zero root real parts. Numerical simulations are used to make it easier to explain the dynamic behavior of a system and describe its analytical results.