I've used a variation of this epidemiology model scaled to dimensionless independent variables by dividing through by population: SIR Model, A standard SIR model for 3 independent variables.
When I set $\beta$ and $\gamma$ to any value giving an $R_0$>1.7, the horizontal asymptote for the recovery curve is greater than 50%. I compare those curves to the country data here, e.g. Italy Covid Cases. It looks like the green recovery curve will top at at some value under 250,000. The population of Italy is about 60 million. So it looks like the recovery curve will level off at under 0.5%, a tiny fraction of predicted values.
What can be off with the model to make it more accurate?
Some guesses:
Separate $R$ compartment into Recovered and Dead. Pros, there are probably more accurate records of death by covid than confirmed infections, so treating the death curve separately would probably improve fit. Con, the quantity of deaths is probably insufficient to account for the large discrepancy.
Add terms $\alpha R$ to move people from the $R$ compartment to the $S$ compartment to represent formerly cured who are once again susceptible, i.e. model temporary immunity. Analysis: Still working on this version of the model. Not sure if it will compensate for the discrepancy. Is this added feature likely to bear fruit?
Add some term to factor in testing.
Pro, will make explicit inaccuracies in data collection and might allow for greater latitude of solutions.
Con, unless there are ~10 times as many cases as recorded this probably won't cut it either.
Are any of these changes worth pursuing? Are there some more reasonable changes I'm neglecting?