Modern graphs are both large and dynamic, presenting significant challenges for fundamental queries, such as the Single-Source Shortest Path (SSSP) problem. Naively recomputing the SSSP tree after each topology change is prohibitively expensive, causing on-demand computation to suffer from high latency. Existing dynamic SSSP algorithms often cannot simultaneously handle both edge additions and deletions, operate in distributed memory, and provide low-latency query results. To address these challenges, this paper presents SSSP-Del, a new vertex-centric, asynchronous, and fully distributed algorithm for dynamic SSSP. Operating in a shared-nothing architecture, our algorithm processes streams of both edge insertions and deletions. We conduct a comprehensive evaluation on large real-world and synthetic graphs with millions of vertices, and provide a thorough analysis by evaluating result latency, solution stability, and throughput.