ANALYSIS OF A MATHEMATICAL MODEL ON THE SPREAD AND CONTROL OF TUBERCULOSIS IN THE CASE OF BISHOFTU TOWN

Abstract

TB is an air born communicable infectious disease caused by MBT. In Ethiopia TB is the major cause of morbidity and mortality. In this thesis, we have considered a nonlinear dynamical system. We divide the total population of Bishoftu town into five compartments namely Susceptible class (𝑆), Latent class (𝐿) ,Infectious class, treatment class(T) and Recovered class (R), to study the dynamics of tuberculosis in the case of the community of Bishoftu town. We calculated that two equilibrium points namely, the disease free equilibrium point and the endemic equilibrium point. We also calculate the basic reproduction number of the dynamical system, 𝑅0 = 𝛬𝛽(𝜇+𝛿+𝜓+𝛼2)[𝜇+𝜀− 𝑝𝜇] 𝜇[(𝜇+𝜀)(𝜇+𝛾+𝛼1)(𝜇+𝛿+𝜓+𝛼2)−𝛿𝛾(𝑘𝜇+𝜀)] , which depends on eleven parameters. The numerical value of the reproduction number based on real data collected from Bishoftu town is 𝑅0 = 1.39373995 > 1. This shows that tuberculosis disease spread in Bishoftu town community. We also proved that the disease free equilibrium point is unstable and the endemic equilibrium point is stable. The basic control parameters that can help us to control the spread of the tuberculosis disease are the transmission coefficient 𝛽 = 0.20372671. For values 𝛽 > 0.20372671 the reproduction number is greater than one. So, to control the spread of tuberculosis, the transmission coefficient should not exceed more than 0.20372671. The effects of the remaining parameters on the basic reproduction number are discussed in detail in their subsections.