I have a large list of equations at zunzun.com/AllEquations/2/Standard - this list might give you some ideas.

I would use R-squared to mean "What fraction of the variance in Y is explained by the model?" so that if the same x is used in two different models of the same data "y = f1(x)" and "y = f2(x)", my understanding is that I could use R-squared to tell me if model f1 or model f2 explains more of the variance in y when using the same data set.

Would you please post the data, or a link to the data?

In this case, the parameter "a" should be the initial number of known infections when data collection starts, so that parameter is required.

RMSE is similar to "average size of regression error", so if you fit a straight line to the data in each scatterplot the RMSE using index (time) would be very much smaller than the RMSE using sensor readings.

Upvoted for the clarity and excellence of the answer.

Upvoted for awesomely cool graphic.

If you could predict whether or not demand was zero at certain times, you could use that to filter the data and make two separate models. For example, if you could accurately determine 50 percent of the zero demand, you could then use a different model for the remaining data. My guess is that you also need different models for workdays and non-workdays.

I used the "function finder" on my zunzun.com open source curve fitting web site, looking for equations with three or less parameters. This equation seemed like a good candidate.

I have experience with exposing statistical web applications to the wider internet - see my zunzun.com web site - and if possible I recommend restricting access.

Instead of smoothing the data, I would try using only the adjacent peaks, or the maximum values found within some narrow sliding window.

Scaling the income data to units of thousands, a 2nd order (quadratic) polynomial fit as well as a 3rd or 4th order polynomial.

Upvoted for speed of correct reply and clarity.

Upvoted for combination of excellent question and follow-up discussion.