Roundup consists of glyphosate which is not as effective in penetrating the leaves' water-repellant cuticle. You can use Triclopyr, but it will kill any shrubs or trees it comes in contact with, and you need to be sure to follow the directions and wear appropriate protection for your eyes, skin and lungs. The time to spray it is now.
Manual removal of the plants and roots combined with close mowing, is the best way to eventually get rid of the plants. Attached is a pamphlet on removing invasive species, including periwinkle.