The Mars Global Surveyor mission has imaged slope streaks, some of which have formed in periods as short as 109 days. These features are one of the most currently active surface processes on Mars. Some slope streaks have flow-like morphologic characteristics, which include anastomosing patterns influenced by small topographic barriers. In order to understand what processes gave rise to these specific features, we applied viscoplastic flow numerical modeling techniques to simulated Martian surfaces. We simulate Martian surfaces with observed slope streaks by using Mars Orbiter Laser Altimeter measurements to obtain large-scale slope measurements and a photoclinometry technique on Mars Orbiter Camera images to obtain meter-scale topographic information. Our numerical simulations of slow-moving plastic flows show that a fluid rheology and a short formation period are necessary to explain these features. We estimate that the typical values of bulk viscosity and bulk yield strength are less than 10 Pa s and less than 10 Pa, respectively. The fluid rheology can be explained by a water-related flow with a solid content less than about 20%. An alternative explanation is a dry grain flow with extremely low cohesion and friction angle supported by dispersive pressure or a lubricant, such as electric conditions of particles. The continuous nature of anastomosing slope streaks that originate from point sources is best explained by continuous discharges of material or lubricant. In this case, the estimated flow rate is less than several cubic meters per second, and the flow duration is estimated to be less than a day.