Stability And Bifurcations Analysis And OGY Chaos Control Of An Discrete SIS Epidemic Model With Standard Incidence

In this paper, we have researched an discrete SIS epidemic model with stan-dard incidence by analysis and numerical stimulation, giving the codimension-one bifurcations at disease-free equilibrium and endemic equilibrium, that isthe fold, fip and Neimark-Sacker bifurcations and chaos. Besides, we also takeOGY control to eliminate the unstable periodic orbits in the chaotic orbits.At the same time, we also show the codimension-two bifurcations around thedisease-free equilibrium and endemic equilibrium, that is1:2strong resonance,1:4strong resonance. Besides, our numerical stimulation gives codimension-oneand codimension-two diagrams, as well as the phase portraits and the maxi-mum Lyapunov exponents diagrams for two diferent varying parameters in a3-dimension space. The results obtained in this paper show that a discrete SISepidemic model can have very rich dynamical behaviors.There are six sections in all. The frst section is introduction. The back-ground and purposes of the model, mainly describing the present investigationof the model. Finally, we give the structure of our papers.In the second section, we give an discrete SIS epidemic model with stan-dard incidence, that is the existence of the disease-free equilibrium and endemicequilibrium, and local stability. The sufcient conditions of codimension-onebifurcations and codimension-two bifurcations.In the third section, we prove the codimension-one bifurcations by usingbifurcation theory and normal measures. That is the existence of the fold, fip andNeimark-Sacker bifurcations. In addition, in numerical stimulation, we will givebifurcation diagrams, and portraits as well as the largest Lyapunov diagrams. In the fourth section, we will give the codimension-two bifurcations at disease-free equilibrium and endemic equilibrium. That is1:2strong resonance and1:4strong resonance bifurcations.In the sixth sections, we will give a conclusion. Giving a comparison betweenan discrete SIS epidemic model with the continuous model as well as showingthat the dynamical behaviors of the continuous ones are quite simple, the discreteones has more rich dynamical behaviors than the continuous model.