Qualitative And Bifurcation Analysis Of A SIRS Epidemic Model With A Nonmonotone Incidence Rate And Treatments

In this paper,we consider a SIRS epidemic model with a nonmonotone inci-dence rate and treatment.For the linear treatment,it is shown that the disease-free equilibrium is globally asymptotically stable in the first quadrant if the basic repro-duction number R0?1;the unique endemic equilibrium is globally asymptotically stable in the interior of first quadrant if R0>1,at this time the number of infec-tious individuals will decrease when the intensity of treatment increases.For the constant treatment,it is shown that the model can undergo saddle-node bifurcation,Bogdanov-Takens bifurcation and subcritical Hopf bifurcation as the parameters vary,and the model exhibits rich dynamics such as multiple coexistent steady states,co-existent periodic orbit,homoclinic orbit,etc.Numerical simulations are presented to illustrate the theoretical results.