  We prove the quasimodularity of generating functions for counting torus
covers, with and without Siegel-Veech weight. Our proof is based on analyzing
decompositions of flat surfaces into horizontal cylinders. The quasimodularity
arise as contour integral of quasi-elliptic functions. It provides an
alternative proof of the quasimodularity results of Bloch-Okounkov,
Eskin-Okounkov and Chen-Moeller-Zagier, and generalizes the results of
Boehm-Bringmann-Buchholz-Markwig for simple ramification covers.
