This study investigates the flow of a magnetohydrodynamic (MHD) Maxwell fluid over a stretching sheet using a Darcy-Forchheimer (DF) model. We employ numerical analysis with a copper (Cu) nanofluid suspended in water, considering Cattaneo–Christov heat flow, viscous dissipation, and joule heating. Nonlinear ordinary differential equations (ODEs) are solved using the bvp4c method in Matlab and we examine the normalized shear stress, temperature profile, and heat flux rate. Our findings reveal insights for practical applications, showing how parameters such as the relaxation Prandtl number, magnetic parameter, Eckert number parameter, and radiation parameter impact system behaviour.

Dengue fever continues to be a major global health concern, particularly in tropical and subtropical regions. This research develops an updated fractional-order dengue transmission model that incorporates vector control methods with vertical transmission in mosquitoes. Memory effects are considered in the model formulation, which are examined using the Caputo fractional derivative. Through vertical transmission, Infected mosquitoes transmit the virus to their offspring, which keeps the infection active when humans are not available as hosts. Spraying insecticides together with biological control measures is part of vector control efforts that help assess how well the disease prevalence decreases. The Banach fixed-point theorem provides a basis to prove existence and uniqueness solutions. The mathematical model uses the basic reproduction number ($ \mathscr{R}_0$) to determine disease persistence, while stability conditions depend on equilibrium points. The investigation of local stability uses fractional-order stability theory together with a suitable Lyapunov function to establish global stability. The disease dynamics display changes through numerical results where fractional order variations combine with vector control measures. The simulation data demonstrates that vertical disease transmission functions as a crucial factor for maintaining dengue outbreaks, but strong vector control programs can significantly reduce spreading rates. The presented model functions as a foundation to establish optimal preventive strategies against dengue fever. The subsequent part of this work investigates optimal control models.