Including a graph of your predictions versus the test set might be a good idea.
However, I suspect that your analysis might not yield the desired results for a few reasons:
It is not definitive as to whether COVID-19 follows a seasonal pattern - i.e. spikes at certain parts of the year. There is no way to know for sure with less than two years of data. Therefore, a seasonal factor cannot be known definitively.
Leading on from point 1 - this might be why you are getting a flat line for the three months of predictions. There is a lot of volatility in the data and ARIMA cannot predict such volatility.
Additionally, ARIMA relies on past data to predict the future. Using such a model is not a good idea if there are interventions that can affect future trends. For instance, a significant portion of people globally are now vaccinated against COVID-19, but this was not the case during the period across your training set. This means that past data will not incorporate such information that will be reflected in future data.
As was mentioned in the comments, you should refrain from adding predictions to your training set. This influences the predictions that will be made in subsequent time periods. If the prediction added to the training set was inaccurate, then subsequent predictions will also be inaccurate. The training set and test set also need to be kept entirely separate to avoid data leakage - where the model gives the illusion that it is predicting well but in fact it is because data from the test set is influencing the model.
I don't know if your end goal is simply to predict COVID-19 cases or if this is part of a wider analysis, but you might wish to reconsider your approach given the points above.
As a disclaimer, I am not a health specialist and none of the above is meant to be of any material advice regarding COVID-19 specifically. This question is simply answered from the point of view of explaining time series principles.
