MATHEMATICAL MODEL FOR THE TRANSMISSION DYNAMICS OF TUBERCULOSIS WITH PASSIVE IMMUNITY, DRUG SENSITIVE TUBERCULOSIS AND DRUG-RESISTANT TUBERCULOSIS

Özet

This study presents a mathematical model for the transmission dynamics of tuberculosis in the drug sensitive TB and drug-resistant TB. Eight (8) compartments were considered, namely: passively immune infants, susceptible, latently effected with DS-TB, latently infected with DR-TB, infectious with DS-TB, infectious with DR-TB, recovered with DS-TB and recovered with DR-TB and we mathematically modelled the natural growth, the interactions between these populations and the effects of treatments. The disease-free equilibrium (DFE) and endemic equilibrium (EE) were obtained. We obtained the basic reproduction number, cwhich can be used to control the transmission dynamics of the disease and thus, established the conditions for local and global stability of the disease free- equilibrium using Routh- Hurwitz criterion using Routh-Hurwitz theorem and Lasalle-Lyapunov function respectively. The result of the analysis of the stability of the disease-free equilibrium state that tuberculosis can totally be eradicated if effort is made to ensure that the rate of recovery infected individuals with DS-TB and DR-TB and the rate of natural death must have a lower bound. Numerical analysis for the model was carried out and it demonstrated that in the case of patients with both DS-TB and DR-TB, tuberculosis will be eradicated if effort is intensified in bringing down both the transmission rate of DS-TB and DR-TB through the using of BCG vaccine and continuous treatment.