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The farming of animals is one of the largest industries, with animal food products, milk, and dairy being crucial components of the global economy. However, zoonotic bacterial diseases, including brucellosis, pose significant risks to human health. The goal of this research is to develop a mathematical model to understand the spread of brucellosis in cattle populations, utilizing the Caputo-Fabrizio operator to control the disease's incidence rate. The existence and uniqueness of the model's solution are ensured through the Lipschitz conditions, the contraction mapping theorem, and the application of the kernel properties of the Caputo-Fabrizio operator. Sensitivity analysis is conducted to assess the impact of various factors on the disease's progression. This study performs a realistic stability analysis of both global and local stability at the disease-free and the endemic equilibrium point which give a more accurate understanding of the dynamism and behavior of the system. Stability analysis is performed using Picard stability in Banach spaces, and Lagrange's interpolation formula is employed to obtain initial approximations for successive fractional orders. The findings of this study demonstrate that fractional orders, along with memory effects, play a crucial role in describing the transmission dynamics of brucellosis. Sensitivity analysis helps identify the parameters most critical to the infection rate, providing essential data for potential control measures. The results highlight the applicability of the Caputo-Fabrizio operator in modeling the transmission of infectious diseases like brucellosis and offer a strong foundation for controlling disease spread within communities. [ABSTRACT FROM AUTHOR]