Table 2.

Parameters and parameter definition.

Parameters	Parameters Definition
P	proportion of the population that have been vaccinated before and after the outbreak.
$1-p$	proportion of the population that are susceptible.
ω	Recruitment rate.
q	proportion susceptible decrease to become symptomatic carriers.
1-q	proportion of susceptible decrease to become asymptomatic carriers.
φ	rate at which susceptible population decrease to symptomatic or asymptomatic carriers.
β	the rate at which symptomatic carriers decrease to become infected.
α	the rate at which asymptomatic carriers decrease to become infected.
σ	the rate at which symptomatic carriers decrease to recovered stage after treatment.
Parameters	Parameters Definition
τ	the rate at which asymptomatic carriers decrease to recovery stage after treatment.
θ	the rate at which symptomatic carriers become asymptomatic carries.
$\text{Ω}$	the rate at which asymptomatic carriers become symptomatic carries.
μ	the natural mortality rate.
ϕ	the rate of loss of temporary immunity to become susceptible.
k	the proportion of vaccinated population decrease to become symptomatic carriers.
1-k	the proportion of vaccinated population decrease to become asymptomatic carriers.
η	rate at which vaccinated population decrease to symptomatic and asymptomatic.
δ	the rate at which vaccinated population decrease to become susceptible.
γ	the rate at which infected individuals turn to recovered state.
ρ	the rate at which vaccinated population decrease to become infected.
$\text{Π}$	the rate at which the symptomatic carriers decrease to become susceptible.
ξ	the rate at which the asymptomatic carriers decrease to become susceptible.
a	the rate at which infected decrease to asymptomatic carrier.
$\ell$	the rate at which infected decrease to symptomatic.
χ	the rate at which susceptible become infected.
ν	the rate at which susceptible become vaccinated.
ς	the disease induced death rate.