The AI factions also play by the same set of victory conditions as you, so if you play a short campaign you'll periodically receive notifications that they're making progress towards winning. From what I seem to recall especially Egypt tends to do really well early on, so you often get a message about them moving closer to their goal.
If I recall correctly you get the message when they're 3 regions away from possessing the minimum amount of required regions, e.g. if Egypt needs 15 regions you'll get a message when Egypt owns 12 regions. What it doesn't take into account, however, is that all the factions also need to have eliminated or outlasted one or two factions
on top of also having at least 15 regions. This is usually the harder of the two objectives since you'll usually get the amount of required regions when you are in the process of taking out the rival factions.
If you do get a message telling you one of the AI factions is getting closer to winning, just check the factions rankings overview to check if their rival factions still are alive. If they are, then there's generally no cause for concern.