Global analysis of a network-based SIR epidemic model with a saturated treatment function

In this paper, we study a network-based SIR epidemic model with a saturated treatment function in which a parameter $\alpha$ is introduced to measure the extent of the effect of the infected being delayed for treatment. Our aim is to present a global analysis and to investigate how the parameter $\alpha$ affects the spreading of diseases. Our main results are as follows: (1) In the case of the threshold value $R_0<1$, there exist two values of $\alpha$: ${\alpha }_c$ and ${\alpha }_0$, such that the disease-free equilibrium is globally asymptotically stable when $\alpha \le {\alpha }_c$ and multiple endemic equilibria exist when $\alpha \ge {\alpha }_0$. This means that the parameter $\alpha$ has an essential influence on the spreading of the disease. (2) In the case of the threshold value $R_0>1$, if the model has only one endemic equilibrium, then the unique endemic equilibrium is globally attractive. In this case, it is also proved that if $\alpha \le {\alpha }_c$, then the endemic equilibrium has only one, so is globally attractive. In addition, numerical simulation is performed to illustrate our theoretical results.