Strengths:
A well-presented analysis on a topic of general interest.
Suggestions:
If you want to difference the data, 7-day differencing would remove the weekly periodicity. One could sum up cases over each week as another way to avoid dealign with day-of-week effects. This is simpler than the idea proposed in the conclusion of explicitly modeling day of week as a covariate.
Shapiro-Wilk test does not add much to the QQ plot here. The QQ plot tells you the nature of the non-normality (long tails both ends) which Shapiro-Wilk does not.
You reject the null hypothesis that the postulated model is reasonable, and then say “therefore, the model can be represented by …” which is not a clear conclusion.
It could be worth estimating \(E(t_0)\) and/or \(I(t_0)\) rather than fixing them. The model seems to struggle at the start of the wave.