Sensitivity and Stability Analysis of Tuberculosis Disease with Infectious Latent

Abstract

Tuberculosis (TB) is a dangerous contagious disease which can even lead to death if no control measure is applied. The disease is caused by mycobacterium which generally affects lungs and other related organs such as lymph gland, intestine, kidneys, uterus, bone and brain. The spread of TB occurs via the bacteria contaminated air which is inhaled into the lungs. Cough, chest pain, shortness of breath, appetite loss, weight loss, fever, cold and fatigue are some of the symptoms of TB. However, we proposed a mathematical model to investigate the transmission dynamics of tuberculosis and it is investigated analytically that the endemic equilibrium point is stable with the help of Routh-Hurwitz criteria. The sensitivity analysis shows that there would be an epidemic if and only if β≈β^1, where β^1≤0.1. Finally, using Matlab, it is shown that the disease free equilibrium is unstable which the endemic equilibrium becomes stable beyond 60 days. In addition, the recovered population increased rapidly while the exposed population decreased steeply in the disease-free equilibrium. It is an indication that there will be no outbreak of the tuberculosis infection. Besides, an increased in the effective contact rate increases both the infected population and recovered population. It is equally inferred that the recovered population do not show a trend pattern as ∝ increases while the susceptible and infected populations increased and decreased respectively as ∝ is increased. The recovered population showed no response pattern for ∝ since recovered individuals do not obtain permanent immunity.