Before revision:  Our algorithm is based on the dynamic primal-dual framework, and it carefully combines the ideas from the dynamic vertex cover algorithm of Bhattacharya and Kulkarni [SODA ` 19] and the dynamic set cover algorithm of Bhattacharya et al.
After revision:  Our algorithm is based on the dynamic primal-dual framework, and it carefully combines the ideas from the dynamic vertex cover algorithm of Bhattacharya and Kulkarni [SODA ' 19] and the dynamic set cover algorithm of Bhattacharya et al.