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

In this paper, we present a study on a network-based susceptible-infected-recovered (SIR) epidemic model with a saturated treatment function. It is well known that treatment can have a specific effect on the spread of epidemics, and due to the limited resources of treatment, the number of patients during severe disease outbreaks who need to be treated may exceed the treatment capacity. Consequently, the number of patients who receive treatment will reach a saturation level. Thus, we incorporated a saturated treatment function into the model to characterize such a phenomenon. The dynamics of the present model is discussed in this paper. We first obtained a threshold value R₀, which determines the stability of a disease-free equilibrium. Furthermore, we investigated the bifurcation behavior at R₀=1. More specifically, we derived a condition that determines the direction of bifurcation at R₀=1. If the direction is backward, then a stable disease-free equilibrium concurrently exists with a stable endemic equilibrium even though R₀<1. Therefore, in this case, R₀<1 is not sufficient to eradicate the disease from the population. However, if the direction is forward, we find that for a range of parameters, multiple equilibria could exist to the left and right of R₀=1. In this case, the initial infectious invasion must be controlled to a lower level so that the disease dies out or approaches a lower endemic steady state.