Stochastic SEIRD model

The model is SEIRD model, where infected individuals can survive or die at different rates, with waning immunity. The full model specification is:

There are no birth of natural death processes in this model. Parameters are:

$\beta$ : rate of infection
$\delta$ : rate at which symptoms appear (i.e inverse of mean incubation period)
$\gamma_R$ : recovery rate
$\gamma_D$ : death rate
$\mu$ : case fatality ratio (proportion of cases who die)
$\epsilon$ : import rate of infected individuals (applies to $E$ and $I$ )
$\omega$ : rate waning immunity

The model will be written as:

$S_{t+1} = S_t - \beta \frac{S_t (I_{R,t} + I_{D,t})}{N_t} + \omega R_t$

$E_{t+1} = E_t + \beta \frac{S_t (I_{R,t} + I_{D,t})}{N_t} - \delta E_t + \epsilon$

$I_{R,t+1} = I_{R,t} + \delta (1 - \mu) E_t - \gamma_R I_{R,t} + \epsilon$

$I_{D,t+1} = I_{D,t} + \delta \mu E_t - \gamma_D I_{D,t} + \epsilon$

$R_{t+1} = R_t + \gamma_R I_{R,t} - \omega R_t$

$D_{t+1} = D_t + \gamma_D I_{D,t}$