I wouldn't bank on disease not playing a part in reducing the population. An out break of something like Spanish Flu could easily wipe out a billion now-a-days simply because it could spread so fast. With the best will in the world we are still really bad at tackling viruses and it takes weeks, often months, to come up with a flu vaccine that isn't 100% effective.
Having said that I don't think we are as close as most people think to person saturation. We're pretty crowded in the UK at 255/km^2 and if you look at just England that rises to nearly 400/km^2. That puts towards the top of the table which is mostly populated by tiny islands and tax havens. If you look down the table a bit you find huge places like Russia that only have 8.3/km^2. The world average is 57/km^2. I think it's easy to think the world is crowded because our bit of it is crowded.
