Dynamics of a SIR epidemic model with variable recruitment and quadratic treatment

The dynamical behavior of a variable recruitment SIR model has been investigated with the nonlinear incidence rate and the quadratic treatment function for a horizontally transmitted infectious disease that sustains for a long period(more than one year).For a long duration,we have incorporated human fertility in variable recruitment.The societal effort,i.e.all types of medical infrastructures,have a vital role in controlling such a disease.For this reason,we have considered the quadratic treatment function,which divides the system into two subsystems.We have established the existence and stability of different equilibrium points that depend mainly on the societal effort parameter in both subsystems and also global stability.Different rich dynamics such as forward bifurcation,Hopf bifurcation,limit cycle,and Bogdanov-Takens bifurcation of co-dimension 2 have been established by using bifurcation theory and the biological significance of these dynamics has been explained.Different numerical examples have been considered to illustrate the theoretical results.Finally,we have discussed the advantage of our model with the model by Eckalbar and Eckalbar[Nonlinear Anal.:Real World Appl.12(2011)320332].