knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
$g$ --- vaccination lag period.\ $e$ --- vaccine effectiveness.\ $v$ --- vaccination rate.\ $r$ --- infection rate.\
$$ \begin{aligned} \frac{dS^u}{dt} &= -vS^u-rS^u\ \frac{dS^g}{dt} &= +vS^u-rS^g-g^{-1}S^g\ \frac{dS^v}{dt} &= +g^{-1}S^g-r(1-e)S^v\ \frac{dI^u}{dt} &= +rS^u-d^{-1}I^u\ \frac{dI^v}{dt} &= +rS^g+r(1-e)S^v-d^{-1}I^v\ \frac{dR^u}{dt} &= +d^{-1}I^u-vR^u\ \frac{dR^v}{dt} &= +d^{-1}I^v+vR^u\ \end{aligned} $$
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