AM  Vol.6 No.10 , September 2015
The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time
ABSTRACT
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.

Cite this paper
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.
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