The role of the vaccine inhibiting the spread of COVID-19 virus under the frame of the SIR model

The subject is Applied Mathematics with knowledge in Programming and Data Visualization
Need to analyze the SIR model mathematically considering how the vaccination will affect the situation of COVID in the US
Please use the previous example of papers and the instruction as the guideline and focus mainly on the mathematical modeling.
(There is a similar example with topic of SIR model and three examples of different topics, please refer them with cautious)
Also need to use the format given by the instruction and provide detailed thinking process.
(The visualization of your concluded result is required either by python, R, matlab or any programming software you prefer)

As for the final conclusion, we want to prove that vaccine is vital for controlling the spread of covid using SIR model and also a insight of how much vaccination is needed to eventually eradicate the epidemic.

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