Table 1 Description of model parameters and their associated values and ranges [[15]-[17]]

q _s ,q _v ,q _r

Fraction of individuals receiving vaccination

0 - 1

σ

Vaccine efficacy

0.6 – 0.95

μ

Fraction of population vaccinated

Variable

m ₁

Number of seasons that the protection acquired from natural infection lasts

2 - 8 seasons

m ₂

Number of seasons that the protection acquired from vaccination lasts

2 - 4 seasons

γ

Recovery rate from infection

0.2 day ^-1

ϵ _j

Amplitude of seasonal fluctuation

0.85

α

Age since last infection

Variable

${\beta }_s,{\beta }_v^i\left(t\right),{\beta }_r\left(a\right)$

i = 0, 1, ⋯, m ₂ - 1

Baseline transmission rates of infection for susceptible, vaccinated, and recovered individuals, respectively

β _s  = 150/N year ^-1 (N: total population size)

${\overline{\beta }}_v\left(t\right),{\overline{\beta }}_r$

Baseline transmission rates of infection for previously vaccinated and recovered individuals, respectively, who also received vaccine for the current season

Variable

κ _s (t) = β _s  · P _s (t)

Transmission rate of infection for susceptible class

Variable

${\kappa }_v^0\left(t\right)={\beta }_v^0\left(t\right)P_s\left(t\right)$

Transmission rate of infection for newly vaccinated from susceptible class

Variable

${\kappa }_v^i\left(t\right)={\beta }_v^i\left(t\right)P_v\left(t\right)$

Transmission rate of infection for previously vaccinated individuals who received their last vaccine i seasons ago

Variable

i = 1, 2, ⋯, m ₂ - 1

${\overline{\kappa }}_v\left(t\right)={\overline{\beta }}_v·P_v\left(t\right)$

Transmission rate of infection for previously vaccinated individuals who also received vaccine for the current season

Variable

κ _r (t, a) = β _r (a) · P _r (t)

Transmission rate of infection for recovered individuals with the recovery age α since last infection

Variable

${\overline{\kappa }}_r\left(t\right)={\overline{\beta }}_r·P_r\left(t\right)$

Transmission rate of infection for recovered individuals who received vaccine for the current season

Variable