Abstract
The dynamical behaviors of a discrete-time SIS epidemic model are investigated in this paper. The result indicates that the model undergoes a flip bifurcation and a Hopf bifurcation, as found by using the center manifold theorem and bifurcation theory. Numerical simulations not only illustrate our results, but they also exhibit the complex dynamical behaviors, such as the period-doubling bifurcation in period-2, -4, -8, quasi-periodic orbits and the chaotic sets. Specifically, when the parameters A, , , r, λ are fixed at some values and the bifurcation parameter h changes with different values, there exist local stability, Hopf bifurcation, 3-periodic orbits, 7-periodic orbits, period-doubling bifurcation and chaotic sets. These results reveal far richer dynamical behaviors of the discrete epidemic model compared with the continuous epidemic models although the discrete epidemic model is simple. Finally, the feedback control method is used to stabilize chaotic orbits at an unstable endemic equilibrium.
Highlights
1 Introduction In the theoretical studies of epidemic dynamical models, there are two kinds of mathematical models: the continuous-time models described by differential equations, and the discrete-time models described by difference equations
They claimed that when the time step h is small (h < h∗) the dynamical behaviors are similar with the continuous-time model, and when the time step h is increasing (h > h∗) in the discrete epidemic model appears a flip bifurcation, a Hopf bifurcation, chaos, and more complex dynamical behaviors by the numerical simulations
We present the numerical simulations, which illustrate our results with the theoretical analysis, but we exhibit the complex dynamical behaviors such as the cascade of period-doubling bifurcation in period, quasi-periodic orbits, -periodic orbits, -periodic orbits and chaotic sets
Summary
In the theoretical studies of epidemic dynamical models, there are two kinds of mathematical models: the continuous-time models described by differential equations, and the discrete-time models described by difference equations. The authors in [ – ] discussed the stabilities of the disease-free equilibrium and the endemic equilibrium for some SI, SIS, SIR, and SIRS type discrete-time epidemic models. For the discrete population models [ – ] approached by the forward Euler scheme, there existed a flip bifurcation, a Hopf bifurcation and chaos dynamical behaviors which are different from the dynamical behaviors in the corresponding continuous-time models. In [ ] the authors used the forward Euler scheme to obtain a class of discrete SIRS epidemic models They claimed that when the time step h is small (h < h∗) the dynamical behaviors are similar with the continuous-time model, and when the time step h is increasing (h > h∗) in the discrete epidemic model appears a flip bifurcation, a Hopf bifurcation, chaos, and more complex dynamical behaviors by the numerical simulations.
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