Confused about parameters of a discrete SIR infectious disease model In a discrete SIR infectious disease model:
$n$ = time in days (as would happen in the case of covid-19).
$S_n$ = number of susceptible in day n
$I_n$= number of infectives in day n
$R_n$ = number of recovered (or removed) in day n
And,
$S_{n+1}$=$S_n-\frac{\beta }{N}S_nI_n$ 
$I_{n+1}=I_n+\frac{\beta }{N}S_nI_n-\gamma I_n$] 
$R_{n+1}=R_n+\gamma I_n$
Where;
$\beta $ = infection rate (the number per day that susceptible people become infected)
$\gamma $ = recovery rate (the probability that an infected individual recovers). And therefore, $\frac{1}{\gamma }$ is the average length of the infectious period of the disease.
After simulating the SIR model and fitting it to available data using Excel's Solver (LSSE), the best parameter values were: $\beta $ = 3.993 and $\gamma $ = 3.517. I followed the process explained here:
https://jmahaffy.sdsu.edu/courses/f09/math636/lectures/SIR/sir.html
This implies that, if this data were to model Covid-19 cases in some city, the number of susceptible people who become infected per day is 3.993 and that the average length of the infectious period of the disease is $\frac{1}{\mathrm{3.517}}\approx 0.2843$ days. This does not make sense to me. I applied the same approach to covid-19 cases in some city and the results were similar. Am I interpreting these parameters correctly? Are my definitions of the parameters correct? - sorry two questions, but asking about the same thing.
Thank you in advance for your help.
 A: 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.
