Yes, the logistic curves are
lazy S curves and approach a limit asymptotically from below.
Here's the calculus: At FedEx, the BoD wanted some revenue projections. We knew (1) the current revenue and (2) the revenue for serving all the target customers. So, for time t in days, let y(t) be the revenue per day at time t, t = 0 corresponds to the present, y(0) is the current revenue. Let b be the revenue when serving all the target customers. Assume that the rate of growth in revenue, in dollars per day, that is, the calculus
y'(t) = d/dt y(t)
is directly proportional to (a) the current number of customers talking about the service and (b) the number of target customers not yet customers hearing about the service, that is proportional to y(t) and (b - y(t)).
Then for some constant of proportionality k we have the first order ordinary differential equation initial value problem
y'(t) = k y(t) (b - y(t))
Then in TeX source code, the solution is
y(t) =
{ y(0) b e^{bkt} \over y(0) \big ( e^{bkt} - 1 \big ) + b}
which is the famous logistic curve. So, this curve starts as an exponential and convex, rises, becomes concave, and approaches b asymptotically from below.
[On a Friday, I did the calculus, and a Senior VP of Planning and I picked a reasonable value of k, and I submitted the graph.
The next day at 8 AM was a BoD meeting. The graph was presented at the start, and two guys from BoD Member General Dynamics asked how the graph was constructed. At noon in my office I got a call from another SVP asking if I knew about the graph and could come to the meeting. When I arrived, everyone was unhappy; the two General Dynamics were standing in the hall with their bags packed and plane tickets back to Texas. FedEx was about to die. I calculated a few points on the graph; the General Dynamics guys stayed; and the rest is history.
The problem was, from the beginning of the meeting at 8 AM until noon, no one at the meeting could explain how the graph was calculated. The General Dynamics guys lost patience with FedEx, went to their rooms, packed their bags, and as a last effort returned to the meeting.]
Well the little calculus exercise is, say, an axiomatic derivation of viral growth, say, as for TV sets in US homes a few years around 1950. Well, a glance at the derivation shows that the solution should also be good for viral growth of viruses.
Can get the same thing from a continuous time, discrete state space Markov process subordinated to a Poisson process as in the book on stochastic processes by E. Çinlar, long at Princeton in operations research, etc.