It's not terribly likely that a disease outbreak would start from an escaped zoo animal.
First, the monkeys would have to get a disease. The most likely way for that to happen is contact with other members of their own species. That's not going to happen a lot with zoo monkeys, unless they were recently captured in the wild (or had cage-mates who were). Lion-tailed macaques, the species to which these monkeys belong, are endangered, so it's not likely that zoos are capturing them in the wild. It's far more likely that zoos are breeding them with the intention of returning some of them to the wild. In Outbreak, the monkey that brought the disease to the US was a recently captured wild monkey, not a monkey from a zoo.
Next, the monkeys would have to escape. A sick monkey is probably less likely than a healthy monkey to be able to escape and evade capture. You'd probably have a harder time evading someone who wanted to capture you if you had the flu, the same is likely to be true of a sick monkey. In Outbreak, the monkey didn't escape on its own, it was smuggled by a person.
Then you have to have a virus that is easily transmitted between monkeys and humans, and between humans and humans. That's not that typical. Lots of viruses, including rabies and Ebola, require contact with bodily fluids to spread. Viruses don't generally mutate in ways that change how they are transmitted (the virus in Outbreak did this). Scientists are not terribly worried about Ebola mutating and becoming airborne like measles or the flu but still being as nasty as it is now. That's just not likely to happen. Even if you did have wild monkeys infected by an airborne virus that could be really nasty in humans, how likely is it that zoos would be capturing and exhibiting monkeys that were at all likely to have been exposed to that virus? Without any kind of quarantine period?