MATHEMATICAL MODELING OF TUBERCULOSIS WITH DRUG RESISTANCE IN THE PRESENCE OF OPTIMAL CONTROL: A CASE STUDY IN ETHIOPIA

Tuberculosis (TB) is a transmittable bacterial infection, and it is one of the main health problems worldwide. This infection is preventable and can be cured with an appropriate treatment. The mathematical modeling technique could be applied successfully to investigate the transmission dynamics and provide the proper control measures of transferable diseases inclusive TB. In this paper, we developed a mathematical model of TB with drug resistance TB (DR-TB) in the presence of optimal control. We considered the two diseases - drug sensitive TB (DS-TB) and drug resistance TB. They affect the country Ethiopia. The DS-TB can be cured by first-line anti-TB drugs. However, once the tubercle bacilli begins it resistance to one or more anti-TB drugs, the DR-TB appears. This type of TB is difficult for the physicians to detect the strains. It is also expensive to treat. We analyzed the model and discussed the basic elements such as equilibrium points, basic reproduction number, stabilities of equilibrium points and possibility of bifurcation. The analytical result showed that if the threshold quantity (R-0) < 1, the disease-free equilibrium is stable, whereas if R-0 > 1, the endemic equilibrium (EE) is stable. When R0 = 1, a backward bifurcation appears. We extended the model by proposing strategies such as preventive effort, case finding control and case holding control. In this study, four different strategies are introduced based on different combination measures. Moreover, the optimal control problem is examined both analytically and numerically. The finding suggested that optimal combination strategies are used to reduce both the disease burden and the cost of intervention.