Abstract
In this article, two classes of sewage treatment models involving Caputo fractional derivative with impulses at variable times are established. Firstly, some new developed concepts such as critical points and noncritical points, () nonlinear functions sets, and ()-mild solutions are introduced. Secondly, local and global existence of solutions are obtained and other interesting properties such as local stability and periodic oscillatory phenomena of solutions are presented. Finally, a useful wastewater treatment model is given to demonstrate our theory results.