Fig. 4.

(a) Graph of the control reproduction number ${\mathcal{R}}_{\psi }$ when vaccination is the sole personal protection strategy ($\eta =0$). The threshold value for the vaccination rate that is required to achieve herd immunity is denoted as ${\psi }_{HI}$. (b) Graph of the control reproduction number ${\mathcal{R}}_{\eta }$ when social distancing is the sole personal protection strategy ($\psi =\delta =0$). The threshold value for the social distancing level that is required to achieve herd immunity is denoted as ${\eta }_{HI}$.