The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time


In this paper, a class of discrete deterministic SIR epidemic model with vertical and horizontal transmission is studied. Based on the population assumed to be a constant size, we transform the discrete SIR epidemic model into a planar map. Then we find out its equilibrium points and eigenvalues. From discussing the influence of the coefficient parameters effected on the eigenvalues, we give the hyperbolicity of equilibrium points and determine which point is saddle, node or focus as well as their stability. Further, by deriving equations describing flows on the center manifolds, we discuss the transcritical bifurcation at the non-hyperbolic equilibrium point. Finally, we give some numerical simulation examples for illustrating the theoretical analysis and the biological explanation of our theorem.

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Liao, X. , Wang, H. , Huang, X. , Zeng, W. and Zhou, X. (2015) The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time. Applied Mathematics, 6, 1665-1675. doi: 10.4236/am.2015.610148.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Kermack, W.O. and McKendrick, A.G. (1927) Contributions to the Mathematical Theory of Epidemics. Proceedings of the Royal Society of London, Series A, 115, 700-721.
[2] Chen, L.S. (1988) Models for Mathematical Ecology and Research Method. Science Press, Beijing. (In Chinese)
[3] Ma, Z., Zhou, Y., Wang, W. and Jin, Z. (2004) Mathematical Modelling and Research of Epidemic Dynamical Systems. Science Press, Beijing. (In Chinese)
[4] Allen, L.J.S. (1994) Some Discrete-Time SI, SIR and SIS Epidemic Models. Mathematical Biosciences, 124, 83-105.
[5] Anderson, R.M. and May, R.M. (1991) Infections Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford.
[6] Diekmann, O. and Heersterbeek, J.A.P. (2000) Mathematical Epidemiology of Infectious Diseases: Model Building Analysis, and Interpretation. John Wiley & Sons Ltd., Chichester.
[7] Hethcote, M. (1989) Three Basic Epidemiological Models. In: Levin, S., et al., Eds., Applied Mathematical Ecology, Springer, New York, 119-144.
[8] Meng, X.Z. and Chen, L.S. (2008) The Dynamics of a New SIR Epidemic Model Concerning Pulse Vaccination Strategy. Applied Mathematics and Computation, 197, 582-597.
[9] Zhou, X.L., Li, X.P. and Wang, W.S. (2014) Bifurcations for a Deterministic SIR Epidemicmodel in Discrete Time. Advances in Difference Equations, 2014, 168.
[10] Robinson, R.C. (2004) An Introduction to Dynamical Systems: Continuous and Discrete. Pearson Prentice Hall, Upper Saddle River.
[11] Wiggins, S. (1990) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York.

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