The studies on discrete epidemic models are relatively few compared with those on continuous epidemic models.
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.
In view of the numerical simulations, we note that the discrete epidemic models have more complex dynamic behaviors than the corresponding continuous epidemic models.
In order to investigate the dynamics of infectious diseases, many authors analyzed the local and global stability of equilibria of a class of discrete and continuous epidemic models.
Lastly, the examples and numerical simulations carried out to illustrate the feasibility of the main results and revealed the far richer dynamical behaviors of the discrete epidemic model compared with the corresponding continuous epidemic models.
Remark 1 For the discrete model (2), the 3-period orbits, 7-period orbits and complex dynamical behaviors are obtained in this paper which reveal far richer dynamical behaviors than the continuous epidemic model (1).
For the discrete epidemic model with immigration of infectives, by adopting the means of the nonstandard discretization method from continuous epidemic, Jang and Eiaydi in [12] studied the globally asymptotic stability of the disease-free equilibrium, the locally asymptotic stability of the endemic equilibrium and the strong persistence of the susceptible class.
In [29, 30], the authors proposed a class of discrete SEIS epidemic models with bilinear incidence, which is established from the corresponding continuous SEIS epidemic model by applying the well-known backward difference scheme.
In this paper, we have proposed a discrete-time analogue of the continuous SIR epidemic model of childhood diseases with constant vaccination strategy which is derived by the NSFD scheme of Michens.
Applying the discrete Lyapunov functional technique (see [25, 26]) for both cases R 0 ≤ 0 and R 0 > 1, it shown that the global dynamics of this discrete-time analogue of the continuous SIR epidemic model is fully determined only by the basic reproduction number R 0. This shows dynamical consistency between the discrete SIR epidemic model and its corresponding continuous model.
