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The aggressive re-emergence of dengue fever during the last decade made it a principal topic of several research fields, especially biomathematics. In this vein, mathematical modeling is commonly viewed as a great option to get an idea of this disease's prevalence behavior. This work presents and analyzes a generalized stochastic dengue model that incorporates slight and huge environmental perturbations. In rigorous terms, our proposed model takes the form of a stochastic differential equations system that includes white noises and Lévy jumps. We first demonstrate its well-posedness then, and based on many novel analytical techniques, we provide some ameliorated necessary conditions for the extinction and the persistence in the mean. The theoretical findings show that the dynamics of our disturbed dengue model are mainly determined by the parameters related to the white noises and Lévy jumps intensities. In the end, certain numerical illustrative examples are depicted to confirm our findings and to highlight the effect of the adopted mathematical techniques on the results. • A stochastic model with Lévy jumps is proposed to describe Dengue fever propagation. • A mathematical analysis is performed to get an overview of Dengue spread behavior. • By using new and ameliorated techniques, interesting asymptotic properties are shown. • Some numerical simulation examples are introduced to support the findings. [ABSTRACT FROM AUTHOR]