Yes, as in the OP, S curves can be challenging to work with, in particular, to use, say, early in the history of smart phones, to make long term projections fro

Yes, as in the OP, S curves can be challenging to work with, in particular, to use, say, early in the history of smart phones, to make long term projections from the number of smart phones sold each day for each of the last 30 days. But there is some good news: The data used can vary, and in some cases good projections can be easier to make. Can see, e.g., for COVID-19 the recent https://news.ycombinator.com/item?id=22898015 https://news.ycombinator.com/item?id=22898015 https://news.ycombinator.com/item?id=22897967 https://news.ycombinator.com/item?id=22897967 https://news.ycombinator.com/item?id=22900104 https://news.ycombinator.com/item?id=22900104 https://news.ycombinator.com/item?id=22902667 https://news.ycombinator.com/item?id=22902667 In the third one of those we have that the projection from a FedEx case is the solution to the first order ordinary differential equation initial value problem y'(t) = k y(t) (b - y(t)) There for data we used y(0) and b. Then we guessed at k. Had we used values of y for the past month, we could have picked a better, likely fairly good, value for k. Lesson: Fitting an S curve does not have to be terribly bad. The key here is the b: The S curve of the solution is the logistic curve, and it rises to be asymptotic to b from below. Knowing b helps a LOT! When have b, are no longer doing a projection or extrapolation but nearly just an interpolation -- much better. For FedEx, the b was the capacity of the fleet. For COVID-19 the b would be the population needed for herd immunity (from recovering from the virus, from therapeutics that confer immunity, and a vaccine that confers immunity). Knowing b makes the fitting much easier/better. To know b, likely need to look at the real situation, e.g., population of candidate smart phone users, candidate TV set owners, market potential of FedEx (as it was planned at the time), or population needed for herd immunity for the people in some relatively isolated geographic area. Then in TeX source code, the solution is y(t) = { y(0) b e^{bkt} \over y(0) \big ( e^{bkt} - 1 \big ) + b} Can also use a continuous time discrete state space Markov process subordinated to a Poisson process. Here's how that works: Have some states, right, they are discrete. For FedEx, that would be (i) the number of customers talking about the service and (ii) the number of target customers listening. Then the time to the next customer is much like the time to the next click of a Geiger counter, that is, has exponential distribution, that is, is the time of the next arrival in a Poisson arrival process (e.g., the time of the next arrival at the Google Web site). So at this arrival, the process moves to a new state where we have 1 more current customer and 1 less target customer. Then start again to get the next new customer. The Markov assumption is that the past and future of the process are conditionally independent given the present state; so that justifies our getting to the next state using only the current state -- given the current state, for predicting the future, everything before that is irrelevant. What is a Markov process, what satisfies that the Markov assumption, can depend on what we select for the state -- roughly the more we have in the state, the closer we are to Markov. In particular, if we take the whole past history of the process as the state, IIRC every process is Markov. But Markov helps in something like the FedEx application since that state is so simple. We get to use continuous time since the time to the next change of state is from a Poisson process whose arrival times are the continuum -- that is, we don't have to make time discrete although it is true that the history of the process (one sample path) has state changes only at discrete times. So, for state change, and for some positive integer n we have some n possible states, then for i, j = 1, 2, ..., n, we can have some p(i,j) which is the probability of jumping from state i to state j, that is, we have an n x n matrix of transition probabilities. [p(i,j) is the conditional probability of entering state j given that the last state was i.] For two jumps, square that matrix. Now there is a lot of pretty math -- get some limits and eigenvectors of states, etc. Actually fairly generally there is a closed form solution to the process. Alas, often in practice that closed form is useless because the n and the n x n are so large, maybe n^2 in the trillions. E.g., in a problem I solved for war at sea, there were Red weapons, Blue weapons, on each side some number types and some number of weapons of that type. The states were the combinatorial explosion. Then there were the one on one Red-Blue encounters where one died, the other died, both died, or neither died. The time to an encounter was the next arrival of Poisson processes, also Poisson. Well, that was an example where there was a closed form solution but n and n x n were wildly too large for the closed form solution but running off, say, 500 sample paths via Monte-Carlo was easy to program and fast for the computer. So, sure the software reported the average of the 500 sample paths. On a PC today, my software would be done before could get finger off the mouse button or the Enter key. This approach is fairly general. And since what I did included attack submarines, SSBN submarines, anti-submarine destroyer ships, long range airplanes, etc., there should be no difficulty building such a model for COVID-19 that included babies, grade school kids, ..., nursing home residents, people at home, people working nearly alone on farms, .... Back to S curves, IIRC dropping out of the math for the n x n matrix and its powers is an S curve. So, in a broad range of cases, always get an S curve although a different curve depending on, yes, the p(i,j) and the initial state. Uh, when no one is left sick, the Markov process handles that as an absorbing state -- once get there, don't leave. For the n in the billions, the n x n is really a biggie. So, for the submarine problem I did, J. Keilson, Green's Function Methods in Probability Theory. asked "How can you possibly fathom that enormous state space?". That is a good question, and my answer was: "After, say, 5 days, the number of SSBNs left is a random variable. It is bounded. So it has finite variance. So, both the strong and weak laws of large numbers apply. So, run off 500 sample paths, average them, and get the expectation within a gnat's ass nearly all the time. Intuitively, Monte Carlo puts the effort where the action is.". Keilson was offended by "gnat's ass" but liked the math and approved my work for the US Navy. That question and answer are good to keep in mind. There is more in, say, Erhan Çinlar, Introduction to Stochastic Processes, ISBN 0-13-498089-1, Prentice-Hall, Englewood Cliffs, NJ, 1975. For why the arrival times have exponential distribution and why we get a Poisson process, Çinlar has a nice simple, intuitive, useful axiomatic derivation. There is more via the renewal theorem in William Feller, An Introduction to Probability Theory and Its Applications, Second Edition, Volume II, ISBN 0-471-25709-5, John Wiley & Sons, New York, 1971.