  Electric current has been experimentally demonstrated to be able to drive the
insulator-to-metal transition (IMT) in VO$_2$. The main mechanisms involved are
believed to be the Joule heating effect and the strong electron-correlation
effect. These effects are often entangled with each other in experiments, which
complicates the understanding of the essential nature of the observations. We
formulate a phase-field model to investigate theoretically in mesoscale the
pure correlation effect brought by the current on the IMT in VO$_2$, i.e., the
isothermal process under the current. We find that a current with a large
density ($\sim 10^1$ nA/nm$^2$) induces a few-nanosecond ultrafast switch in
VO$_2$, in agreement with the experiment. The temperature-current phase diagram
is further calculated, which reveals that the current may induce the M2 phase
at low temperatures. The current is also shown capable of driving domain walls
to move. Our work may assist related experiments and provide guidance to the
engineering of VO$_2$-based electric switching devices.
