Dynamical analysis of a network-based SIR model with saturated incidence rate and nonlinear recovery rate: an edge-compartmental approach

A new network-based SIR epidemic model with saturated incidence rate and nonlinear recovery rate is proposed. We adopt an edge-compartmental approach to rewrite the system as a degree-edge-mixed model. The explicit formula of the basic reproduction number $ \mathit{\boldsymbol{R_{0}}} $ is obtained by renewal equation and Laplace transformation. We find that $ \mathit{\boldsymbol{R_{0}}} < 1 $ is not enough to ensure global asymptotic stability of the disease-free equilibrium, and when $ \mathit{\boldsymbol{R_{0}}} > 1 $, the system can exist multiple endemic equilibria. When the number of hospital beds is small enough, the system will undergo backward bifurcation at $ \mathit{\boldsymbol{R_{0}}} = 1 $. Moreover, it is proved that the stability of feasible endemic equilibrium is determined by signs of tangent slopes of the epidemic curve. Finally, the theoretical results are verified by numerical simulations. This study suggests that maintaining sufficient hospital beds is crucial for the control of infectious diseases.