Correction to: Optimal control analysis of COVID-19 vaccine epidemic model: a case study Arshad Alam Khan, Saif Ullah and Rohul Amin Eur. Phys. J. Plus, 137 2 (2022) 198 Published online: 04 February 2022 DOI: 10.1140/epjp/s13360-022-02420-4