The transcritical Bogdanov Takens bifurcation with boundary due to the risk perception on a recruitment epidemiological model

We analyze an epidemiological model with treatment and recruitment considering the risk perception. In this model, we consider an exponential function as a recruitment rate. We have found that this model undergoes the transcritical Bogdanov-Takens bifurcation with boundary, where the system experiences the transcritical bifurcation between the disease-free equilibrium point and the endemic equilibrium point. The Hopf bifurcation also arises at the endemic equilibrium point, this is, the appearance or disappearance of a limit cycle, and finally, the Homoclinic bifurcation which transforms the limit cycle into a homoclinic cycle, starting and ending at the disease-free equilibrium point