Abstract
The dynamics of discreteSIepidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interiorR+2, the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation have been derived. Numerical simulation not only presents our theoretical analysis but also exhibits rich and complex dynamical behavior existing in the case of the windows of period-1, period-3, period-5, period-6, period-7, period-9, period-11, period-15, period-19, period-23, period-34, period-42, and period-53 orbits. Meanwhile, there appears the cascade of period-doubling 2, 4, 8 bifurcation and chaos sets from the fixed point. These results show the discrete model has more richer dynamics compared with the continuous model. The computations of the largest Lyapunov exponents more than 0 confirm the chaotic behaviors of the systemx→x+δ[rN(1-N/K)-βxy/N-(μ+m)x],y→y+δ[βxy/N-(μ+d)y]. Specifically, the chaotic orbits at an unstable fixed point are stabilized by using the feedback control method.
Highlights
Within the field of epidemic theory, both the continuous-time model and the discrete-time model were described by the different equations
The dynamics of discrete SI epidemic model, which has been obtained by the forward Euler scheme, is investigated in detail
By using the center manifold theorem and bifurcation theorem in the closed first quadrant R+2, the specific conditions for the existence of flip bifurcation and NeimarkSacker bifurcation have been derived
Summary
Within the field of epidemic theory, both the continuous-time model and the discrete-time model were described by the different equations. There appear to be a number of articles about the flip bifurcation, fold bifurcation, and pitchfork bifurcation and in the sense of Marottos chaos of the discrete system which were presented in [17,18,19,20,21,22,23,24,25,26,27,28] Both a discrete-time predator-prey system without Allee effect and a discrete-time predator-prey system with Allee effect were studied in [22, 23]. In this paper by using the center manifold theorem and bifurcation theorem, the local stability and bifurcation behaviors of a simple SI epidemic model with surprising dynamic is investigated in detail.
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