“…Epidemic models with multiple vaccination options considering cost and efficacy were mentioned in ( Rajib Arefin et al., 2019 ). The vaccination behavior of humans and the waning immunity effect in epidemic models were also analyzed in ( Buonomo, 2020 ; Kabir, 2021 ; Khan et al., 2022a , 2023 ; Wu, Zhang, Song, & Xia, 2024 ; Zuo et al., 2022a , 2022b ). Some analytic and numerical simulations concerning the multistrain epidemic model were presented in ( Khan et al., 2022a , 2023 ; Rajib Arefin et al., 2019 ).…”