The SIR model is too simple and the data is tainted. Meaning that fitting the model to available real-world data will almost certainly lead to surprisingly unrealistic parameters.
Viral respiratory infection has 2 phases. In a first phase viral particles reproduce rapidly in infected cells and in the second phase the immune reaction sets in and infected cells get destroyed and cleaned up. The "I" in the model is mostly concerned with the first, infectious phase, which in most cases is asymptomatic or with weak cold symptoms. However, only the second, much less infectious phase leads to "serious symptoms" and secondary bacterial infections, only patients in the second phase will be tested and thus registered in the data collection.
Note that testing and the bureaucratic registration of the case takes time, so that the daily case numbers are almost meaningless. Even if you use the cleaned up, back-dated data, it will be back-dated to the onset of the heavy symptoms, that is, to a time when the body has started to fight and push back the virus.
You have to also be aware that the RT-PCR test does not really prove viral infection, it just shows that some short pieces of RNA from a bat sample are similar to RNA fragments that float around in the human test sample. In the end the test might only prove that more extra-cellular RNA is in the sample, which would be a symptom of illness. 80%-95% of the population may be incompatible to the bat RNA in any case. This would have a heavy influence on the total number of the initial "S"usceptable population relative to the data collection.
Evaluating the test is highly subjective, depending on state regulations, procedures recommended by the producer of the test, the lab, the technician performing it. The same sample can lead to wildly different results in different labs, or the same lab at a different time, or using a different test kit.
Also check if cases are counted without performing any test or without waiting for test results. So there are both sources for over- and under-counting present. You might get more consistent results if you make a model for the data of all respiratory diseases, with a more complex model that more closely captures the course of a viral infection.
Best Answer
SEIR models are not typically "solved". Finding a nice solution might not be possible. You could make some assumptions which reduce the number of variables (like close the system so $S+E+I+R=1$), but still the nonlinear parts $S(t)I(t)$ will remain a pain to deal with. Here is my advice: