Shewafera Wondimagegnhu Teklu ^* Abushet Hayalu Workie

• Debre Berhan University, Debre Berhan, Ethiopia

Studies investigated that the co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B virus (HBV) poses a major threat to the public health due to their combined negative impacts on health and increased risk of complications. Even though, some scholars formulated and analyzed the HBV and HIV co-infection model they did not consider the compartment that contains protected individuals against both HBV and HIV infections and incorporated the optimal control theory and cost-effectiveness analysis simultaneously. Having this gap in mind, we are motivated to formulate and analyze the HBV and HIV co-infection model by considering protected group and incorporating optimal control theory and cost-effectiveness. In this study, theoretically we have computed all the models disease-free equilibrium points, all the models effective reproduction numbers and unique endemic equilibrium points, the two sub-models disease-free equilibrium points are locally as well as globally asymptotically stable whenever their associated effective reproduction numbers are less than one. We re-formulated the optimal control problem by considering five time dependent control measures and carried out its theoretical analysis by applying the Pontryagin’s maximum principle. Using fourth order Runge-Kutta numerical method and MATLAB ODE45, we executed the numerical simulations with various combinations of control efforts to verify the theoretical results and investigate the impacts of the suggested protection and treatment control strategies for both the HBV and HIV diseases. Also, we carried out the cost-effectiveness analysis of the proposed control strategies. Eventually, we compared our model results with other researcher similar model results whenever cost-effectiveness analysis is not carried out the findings of this particular study suggest that implementing each of the proposed control strategies simultaneously has a high potential to reduce and control the HBV and HIV co-infection spreading in the community. Whereas the cost-effectiveness analysis suggest that implementing the HBV treatment and the HIV and HBV co-infection treatment measures has high potential effect to reduce and control the HBV and HIV co-infection transmission problem in the community.