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Abstract

Various infectious diseases, such as Tuberculosis and Hepatitis B virus (HBV), caused by Mycobacterium bacteria and hepatitis B virus, respectively, pose serious threats worldwide. This paper examines the spread of these diseases using a mathematical model that incorporates control measures, including vaccination and awareness programs. We use a system of differential equations with control variables and apply Pontryagin's principle to determine optimal control strategies. The stability of the Disease Free Equilibrium (DFE) has been analysed and demonstrated to be both locally and globally asymptotically stable when the basic reproduction number remains below unity, thereby ensuring disease elimination. Furthermore, the Endemic Equilibrium (EE) has been established to be locally and globally stable when the reproduction number exceeds unity, indicating persistent disease dynamics under insufficient control measures. The goal of this study is to reduce both the number of infections and the cost of applying these controls. Numerical simulations have shown that the coordinated implementation of immunization and behavioral prevention strategies yields more effective infection control than singular intervention. The findings highlight that integrated preventive controls can be effectively utilized to enhance disease containment and long-term epidemiological stability.

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