Can I synchronize Plague Inc. mobile version with the PC version?

This does not seem possible.

From what I understand, there are two versions of the game: Plague Inc. and Plague Inc: Evolved. Plague Inc. is for mobile devices, while Plague Inc: Evolved is for PC/Mac.

From the FAQS for Plague Inc: Evolved, question 4 asks:

Is Plague Inc. Evolved different from Plague Inc. on mobile?

The answer:

Yes – it is very different as we have been able to add a lot of things which would not work on mobile. Plague Inc: Evolved is a new game for PC, Mac and Linux which has been entirely redeveloped from the mobile game (Plague Inc.). It combines the original critically acclaimed gameplay of Plague Inc. with significant, all-new features...

Because of all the differences, there is no way to sync the two games. Without having support for cloud saves either makes it more challenging for the developer to allow for syncing the two games.