Curiously enough, after running some simulations in a supposed realistically connected population model (but much smaller n), I found linear growth occurring soon after the exponential phase.

I suspect this might be a topological condition (i.e. the due to the distribution of connections) that limits the growth, in the same way a rope will burn linearly if you set fire to one end of it. Of course, a society is more connected than a rope (in the sense we usually have more contacts that just 'up the rope' and 'down the rope'), but continuing exponential growth requires 'fresh' connections at each generation (so no cycles in the graph) - maybe this limits the acceleration of the epidemic.