Our objective in this study is to investigate the behavior of a nonlinear terminal fractional system under $w$-Hilfer fractional derivative in different weighted Banach spaces. We examine the system's dynamics and understand the effects of different weighted Banach spaces on the properties of solutions, including existence, uniqueness, stability, and symmetry. We derive the equivalent integral equations and employ the Schauder and Banach fixed point theorems. Additionally, we discuss three symmetric cases of the system to show how the choice of the weighted function $w(\iota )$ impacts the solutions and their symmetry properties. We study the stability of the solutions in the Ulam sense to assess the robustness and reliability of these solutions under various conditions. Finally, to understand the system's behavior, we present an illustrative example with graphs of the symmetric cases.

Dengue is a viral illness transmitted by Aedes mosquitoes and is a significant global threat. In this study, we developed a model of the dengue epidemic that incorporates larvicide and adulticide, as well as the harmonic mean incidence rate under fractal-fractional derivatives. We examined various theoretical aspects of the model, including nonnegativity, boundedness, existence, uniqueness, and stability. We computed the basic reproduction number ${\mathrm{\Re }}_0$ using the next-generation matrix. The model has two disease-free equilibriums, a trivial equilibrium, and a biologically realistic, along with one endemic equilibrium point. These findings enhanced our understanding of dengue transmission, providing valuable insights for awareness campaigns, control strategies, intervention approaches, decision support, guiding public health planning, and resource allocation to manage dengue effectively.