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

Masaki Sekiguchi; Emiko Ishiwata; Yukihiko Nakata

doi:10.3934/mbe.2018029

Article Contents

2018, Volume 15, Issue 3: 653-666. Doi: 10.3934/mbe.2018029

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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
* Corresponding author: Masaki Sekiguchi

Received: March 12, 2017

Published: June 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 37N25; Secondary: 39B82.

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Figure 1. Numerical experiments $x_{n}$ and $y_{n}$ with $\omega =0$.

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Figure 2. Numerical experiments $x_{n}$ and $y_{n}$ with $\omega =10$.

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Figure 3. A solution $w^{j}_{m}$ is constructed by two solutions $u^{j}_{m}$ and $v^{j}_{m}$

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