Table 1.

Parameters definition of system in Eq. (1).

Parameter	Description
$\mu$	Natural death rate
${\beta }_S\left({\beta }_A\right)$	Symptomatic (Asymptomatic) transmission contact rate
${\psi }_V$	Vaccination rate
${\omega }_V$	Waning rate of vaccine. $1∕{\omega }_V$ is the average time to lose vaccine-induced immunity
${\epsilon }_V$	Vaccine efficacy
${\sigma }_E$	Latency rate. $1∕{\sigma }_E$ is the average latency period
$p$	Exposed individuals’ fraction who become symptomatic infectious
${\alpha }_S$	Transition rate from symptomatic to recover or death. $1∕{\alpha }_S$ is the average output time of symptomatic individuals class
$\theta$	Proportion of symptomatic individuals who die due to the disease
${\alpha }_A$	Recovery rate of asymptomatic individuals. $1∕{\alpha }_A$ is the average time which asymptomatic individuals leave being infectious
${\sigma }_R$	Rate of loss of natural immunity. $1∕{\sigma }_R$ is the natural immunity period