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

Parameter Description Value (range)
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