$$S_{t+1}=S_t-b\frac{\left(p_{t-{\tau }_p}{I}_t+r_{t-{\tau }_r}{E}_t\right)}{N}{S}_t,$$ (1a)

$$E_{t+1}=(1-\sigma -{\sigma }^{\prime })E_t+b\frac{\left(p_{t-{\tau }_p}{I}_t+r_{t-{\tau }_r}{E}_t\right)}{N}{S}_t,$$ (1b)

$$I_{t+1}=(1-\gamma -\mu )I_t+\sigma E_t,$$ (1c)

$$R_{t+1}=R_t+\gamma I_t+{\sigma }^{\prime }{E}_t,$$ (1d)

$$D_{t+1}=D_t+\mu I_t,$$ (1e)