Bifurcation of an SIS model with nonlinear contact rate

The bifurcation and dynamics of SIS models with nonlinear effective contact rate can be very complicated. We study an SIS model with nonlinear contact rate describing behavior change effect of susceptible individual when infectious population increases. By the qualitative and bifurcation analyses, we show that the maximal multiplicity of weak focus is 2, i.e. at most 2 limit cycles can arise from this weak focus. In the meanwhile, we also prove that the model can undergo a Bogdanov–Takens bifurcation of codimension 2, while the responding parameter bifurcation diagrams are presented. These results illustrate that the behavior change of the susceptible individuals may affect the final spread level.