In this paper, we propose a new dynamical system model pertaining to Dengue transmission, and investigate its consequent morphology. We present and study various ramifications of our mathematical model for Dengue spread, encapsulated in a spatio-temporal differential system made of reaction-diffusion equations. Diffusion terms are incorporated into the said model by using specific derivations for infected mosquitoes, and infected humans, as well. Moreover, mechanisms for the nearest neighbor(s) infections are integrated into the model. Furthermore, using adaptive multigrid finite difference with decoupling and quasi-linearization techniques, we investigate two main factors for Dengue spatial propagation. We determine the effects of temperature variations, and the mobility of infectious agents, be they mosquitoes or humans. Finally, the proposed model-based analytico-numerical results are obtained, and rendered in graphical profiles, which show the major role the climate temperature and the mobility of infected humans have on the spread and speed of the disease. The consequent proposed model outcomes and health-based ramifications are then raised, discussed, and then validated.