Table 3.4.

No interior equilibrium and the related global dynamics for the systems (2.7)–(2.8) and (2.9)–(2.10)

Cases	The system (2.7)–(2.8)	The system (2.9)–(2.10)
No interior equilibrium	Case (1): $R_0^h\le 1$; Case (2): $\frac{1-\rho }{\beta -d}<\frac{4}{{(1-\theta )}^2}$; Case (3): $\frac{4}{{(1-\theta )}^2}<\frac{1-\rho }{\beta -d}<R_0^{\upsilon }$.	Case (1): $R_0^h\le 1$; Case (2): $\frac{1-\rho }{\beta \theta -d}<\frac{4}{{(1-\theta -\frac{\beta }{1-\rho })}^2}$; Case (3): $\text{max}\{\frac{1-\rho }{d},R_0^h,\frac{1-\rho }{\beta +\rho -1}\}>1/\theta ,R_0^{\upsilon }>\frac{4}{{(1-\theta )}^2},{\theta }_2<N_i^{*}<K_2$, i = 1, 2; Case (4): $R_0^h<1/\theta ,R_0^{\upsilon }>\frac{4}{{(1-\theta )}^2}$, θ2 < N2 < K2
Disease-free dynamics	$R_0^h\le 1$	$R_0^h\le 1$
Susceptible-free dynamics	$\text{min}\{R_0^h,\frac{{(1-\theta )}^2}{4d}\}>\frac{1}{\rho }$.	$R_0^{\upsilon }>\text{max}\{\frac{1}{\beta K_2},\frac{4}{{(1-\theta )}^2}\}$ and Conditions of Case (2) or Case (3) or Case (4)
Disease-driven extinction	$\frac{4}{{(1-\theta )}^2}>\text{max}\{\frac{1-\rho }{\beta -d},R_0^{\upsilon }\}$	$R_0^{\upsilon }>\frac{4}{{(1-\theta )}^2}$ and $\frac{1-\rho }{\beta \theta -d}<\frac{4}{{(1-\theta -\frac{\beta }{1-\rho })}^2}$.
Permanence	N(0) ∈ (θ2, K2) and $R_0^h<\frac{1}{\rho }<\text{min}\{\frac{R_0^h}{\rho },\frac{{(1-\theta )}^2}{4d}\}$.	N(0) ∈ (θ2, K2) and $R_0^h<\frac{1}{\rho K_2}<\text{min}\{\frac{R_0^h}{\rho K_2},\frac{{(1-\theta )}^2}{4d{K}_2}\}$.