Complex dynamics in an SIS epidemic model induced by nonlinear incidence

Ruixia Yuan; Zhidong Teng; Jinhui Li

In this paper, by combing qualitative and bifurcation analyses, we study an SIS epidemic model with the incidence rate aI 2 c+I 2 + bI c+I 2 , which can describe the inhibition effect from the behavioral change and interpret the "psychological" effect. In Section 2, we give a full-scale analysis for the types and stability of equilibria E i (i = 0, 1, 2, 3, 4, 5). We prove that system (2) can occur backward bifurcation and the backward bifurcation will disappear if a = 0. At equilibrium E i (i = 3, 5), degenerate Hopf bifurcation arises under certain conditions. When the critical condition Ψ i (i = 3, 5) satisfied, we calculate the Liapunov value of the weak focus and obtain the value 2 for the maximal multiplicity of weak focus, indicating that there exist at most two limit cycles around E i (i = 3, 5). In Fig. 6, and Fig. 8 , we give the phase portrait corresponding to equilibrium E 3 and E 5 about exhibiting a unique limit cycle and adding a new limit cycle after small perturbation of parameter Λ and c. In Subsection 3.3, we proved that the model exhibits Bogdanov-Takens bifurcation of codimension 2 and codimension 3, under certain conditions. If parameter a = 0, the model can just have Bogdanov-Takens bifurcation of codimension 2, shown in [20] .