Table 2.

Parameters and their meanings.

Parameters	Meaning
$b$	Natural birth rate per unit time
$d$	The rate at which vaccinated individuals but they lose their immunity per unit time
${\mu }_1$	Natural death rate per unit time
${\mu }_2$	Disease induced death rate per unit time
${\beta }_1$	Disease transmission rate from unvaccinated–infected to susceptible per unit time
${\beta }_2$	Disease transmission rate from vaccinated–infected to susceptible per unit time
$\sigma$	Vaccination rate per unit time
$c$	The rate at which recovered individuals lose their immunity per unit time
$p$	The protection rate of vaccine
$w$	Recovery rate per unit time
$m$	The incubation period