# American Institute of Mathematical Sciences

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June  2018, 15(3): 653-666. doi: 10.3934/mbe.2018029

## Dynamics of an ultra-discrete SIR epidemic model with time delay

 1 Tokyo Metropolitan Ogikubo High School, 5-7-20, Ogikubo, Suginami-ku, Tokyo 167-0051, Japan 2 Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan 3 Department of Mathematics, Shimane University, 1600 Nishikawatsu-cho, 690-8504, Matsue, Japan

* Corresponding author: Masaki Sekiguchi

Received  March 12, 2017 Published  December 2017

We propose an ultra-discretization for an SIR epidemic model with time delay. It is proven that the ultra-discrete model has a threshold property concerning global attractivity of equilibria as shown in differential and difference equation models. We also study an interesting convergence pattern of the solution, which is illustrated in a two-dimensional lattice.

Citation: Masaki Sekiguchi, Emiko Ishiwata, Yukihiko Nakata. Dynamics of an ultra-discrete SIR epidemic model with time delay. Mathematical Biosciences & Engineering, 2018, 15 (3) : 653-666. doi: 10.3934/mbe.2018029
##### References:

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##### References:
Numerical experiments $x_{n}$ and $y_{n}$ with $\omega =0$.
Numerical experiments $x_{n}$ and $y_{n}$ with $\omega =10$.
A solution $w^{j}_{m}$ is constructed by two solutions $u^{j}_{m}$ and $v^{j}_{m}$
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