$$ \begin{aligned} \frac{dS_I}{dt} &= \psi_I - \mu_I S_I - \Bigg[\beta_{BI}\left(\frac{F I_B + I_B}{N_B}\right) f_t f_h + \beta_I \left(\frac{I_I}{N_I}\right)\Bigg] S_I, \\ \frac{dE_I}{dt} &= \Bigg[\beta_{BI}\left(\frac{F I_B + I_B}{N_B}\right) f_t f_h + \beta_I \left(\frac{I_I}{N_I}\right)\Bigg] S_I - \mu_I E_I - \rho_I E_I - \sigma_I E_I, \\ \frac{dI_I}{dt} &= \rho_I E_I - (\gamma_I + \omega) I_I - (\delta_I + \mu_I) I_I, \\ \frac{dR_I}{dt} &= (\gamma_I + \omega) I_I + \sigma_I E_I - \mu_I R_I. \end{aligned} $$

$$ \begin{aligned} \frac{dS_H}{dt} &= \psi_H - \mu_H S_H - \Bigg[\beta_{BH}\left(\frac{F I_B + I_B}{N_B}\right) f_t f_h + \beta_I \left(\frac{I_I}{N_I}\right) + \beta_H \left(\frac{k I_H + (1-k) I_H}{N_H}\right)\Bigg](1-\theta \pi \varepsilon)\, S_H, \\ \frac{dE_H}{dt} &= \Bigg[\beta_{BH}\left(\frac{F I_B + I_B}{N_B}\right) f_t f_h + \beta_I \left(\frac{I_I}{N_I}\right) + \beta_H \left(\frac{k I_H + (1-k) I_H}{N_H}\right)\Bigg](1-\theta \pi \varepsilon)\, S_H - \rho_H E_H - \mu_H E_H - \sigma_H E_H, \\ \frac{dI_H}{dt} &= \rho_H E_H - (\gamma_H + \alpha) I_H - (\delta_H + \mu_H) I_H, \\ \frac{dR_H}{dt} &= (\gamma_H + \alpha) I_H + \sigma_H E_H - \mu_H R_H. \end{aligned} $$