Summary: An adjacent vertex strongly distinguishing total-coloring of a graph \(G\) is a proper total-coloring such that no two adjacent vertices meet the same color set, where the color set of a vertex consists of all colors assigned on the vertex and its incident edges and neighbors. The minimum number of the colors required is called adjacent vertex strongly distinguishing total chromatic number, denoted by \(\chi_{ast}(G)\). Let \(\text{mad} (G)\) and \(\Delta(G)\) denote the maximum average degree and the maximum degree of graph \(G\), respectively. In this paper, we prove the following results. (1) If \(G\) is a graph with \(\text{mad} (G)<\frac{7}{3}\) and \(\Delta(G)\geq5\), then \(\chi_{ast}(G) \leq\max\{\Delta(G)+2, 8\}\). (2) If \(G\) is a graph with \(\text{mad} (G)<\frac{9}{4}\) and \(4\leq\Delta(G)\leq5\), then \(\chi_{ast}(G)\leq\Delta(G)+2\). (3) If \(G\) is a graph with \(\text{mad} (G)<\frac{9}{4}\) and \(\Delta(G)=3\), then \(\chi_{ast}(G)\leq6\).