Qualitative and bifurcation analysis using an SIR model with a saturated treatment function

In this paper, we introduce a saturated treatment function into the SIR epidemic model with a bilinear incidence rate and density-dependent demographics, where the treatment function is limited for increasing number of infected individuals. By carrying out global qualitative and bifurcation analysis, it is shown that the system exhibits some new and complicated behaviors: if the basic reproduction number is larger than unity, the number of infected individuals will show persistent behavior, either converging to some positive constant or oscillating; and if the basic reproduction number is below unity, the model may exhibit complicated behaviors including: (i) backward bifurcation; (ii) almost sure disease eradication where the number of infective individuals tends to zero for all initial positions except the interior equilibria; (iii) “oscillating” backward bifurcation where either the number of infective individuals oscillates persistently, if the initial position lies in a region covering the stable endemic equilibrium, or disease eradication, if the initial position lies outside this region; (iv) disease eradication for all initial positions if the basic reproduction number is less than a turning point value. Numerical simulations are presented to illustrate the conclusions.