Effect of vaccination rate in multi-wave compartmental model V. R. Saiprasad, V. Vikram, R. Gopal, D. V. Senthilkumar and V. K. Chandrasekar Eur. Phys. J. Plus, 138 11 (2023) 995 Published online: 08 November 2023 DOI: 10.1140/epjp/s13360-023-04634-6