Technical Abstract: Rift Valley fever virus (RVFV) has been expanding its geographical distribution with important implications for both human and animal health. The emergence of Rift Valley fever (RVF) in the Middle East, and its continuing presence in many areas of Africa, has negatively impacted both medical and veterinary infrastructures and human morbidity, mortality, and economic endpoints. Furthermore, worldwide attention should be directed towards the broader infection dynamics of RVFV, because suitable host, vector and environmental conditions for additional epidemics likely exist on other continents; including Asia, Europe and the Americas. We propose a new compartmentalized model of RVF and the related ordinary differential equations to assess disease spread in both time and space; with the latter driven as a function of contact networks. Humans and livestock hosts and two species of vector mosquitoes are included in the model. The model is based on weighted contact networks, where nodes of the networks represent geographical regions and the weights represent the level of contact between regional pairings for each set of species. The inclusion of human, animal, and vector movements among regions is new to RVF modeling. The movement of the infected individuals is not only treated as a possibility, but also an actuality that can be incorporated into the model. We have tested, calibrated, and evaluated the model using data from the recent 2010 RVF outbreak in South Africa as case study; mapping the epidemic spread within and among three South African provinces. An extensive set of simulation results shows the potential of the proposed approach for accurately modeling the RVF spreading process in additional regions of the world. The benefits of the proposed model are twofold: not only can the model differentiate the maximum number of infected individuals among different provinces, but also it can reproduce the different starting times of the outbreak in multiple locations.