Study of the properties of some difference equations and application of computation

• Project/Area Number: 26400212
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Tokyo University of Science
• Principal Investigator: Ishiwata Emiko 東京理科大学, 理学部第一部応用数学科, 教授 (30287958)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 時間遅れ / 差分方程式 / 数理生物モデル / 超離散化 / 安定性 / 数値計算 / 超離散方程式 / 感染症モデル / 数理モデル / 安定性解析 / 病理モデル

Outline of Final Research Achievements

Due to the reality of mathematical models of infections, the effects of time delays are often studied recently. Proper discretization to keep the solution positive is important to understand the behavior of such a model.
We newly tried to apply these ideas to the ultra-discrete models. As a result, clear correspondence in the qualitative property was confirmed among the continuous model, the discretized model, and the ultra-discrete model including time delay.
Moreover, through the help of graduate students, we derived results such as convergence behavior for a kind of the integrable discrete hungry systems and formulation of calculation procedure of eigenpairs. Related papers have been published.

Academic Significance and Societal Importance of the Research Achievements

昨今のウィルスの感染伝播の脅威に対し，数理モデルによる解明は重要視されている。これらの時間発展による具体的な挙動の把握に差分方程式の解明は役立つと考える。

Research Products

• [Journal Article] Global stability of an SIS epidemic model with delays (2018)
  • Author(s): Kei Fushimi, Yoichi Enatsu, Emiko Ishiwata
  • Journal Title: MATHEMATICAL METHODS IN THE APPLIED SCIENCES Volume: 2018 Issue: 13 Pages: 5345-5354
  • DOI: 10.1002/mma.5084
  • Peer Reviewed
• [Journal Article] Dynamics of an ultra-discrete SIR epidemic model with time delay (2018)
  • Author(s): Masaki Sekiguchi, Emiko Ishiwata, Yukihiko Nakata
  • Journal Title: Mathematical Biosciences and Engineering Volume: 2018 Pages: 653-666
  • Peer Reviewed / Open Access
• [Journal Article] Convergence of a Logistic Type Ultradiscrete Model (2017)
  • Author(s): Sekiguchi Masaki, Ishiwata Emiko, Nakata Yukihiko
  • Journal Title: Discrete Dynamics in Nature and Society Volume: 2017 Pages: 1-6
  • DOI: 10.1155/2017/7893049
  • Peer Reviewed / Open Access
• [Journal Article] Computation of Eigenvectors for a Specially Structured Banded Matrix (2017)
  • Author(s): Takeuchi Hiroshi, Aihara Kensuke, Fukuda Akiko, Ishiwata Emiko
  • Journal Title: Lecture Notes in Computational Science and Engineering Volume: 117 Pages: 143-155
  • DOI: 10.1007/978-3-319-62426-6_10
  • ISBN: 9783319624242, 9783319624266
  • Peer Reviewed / Open Access
• [Journal Article] Computation of eigenvectors for a specially structured banded matrix (2017)
  • Author(s): Hiroshi Takeuchi, Kensuke Aihara, Akiko Fukuda, Emiko Ishiwata
  • Journal Title: Lect. Notes Comput. Sci. Eng. Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Conserved quantities of the integrable discrete hungry systems (2015)
  • Author(s): Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura
  • Journal Title: Discrete and Continuous Dynamical Systems - Series S (DCDS-S) Volume: 8 Pages: 889-899
  • NAID: 120005674655
  • Peer Reviewed / Open Access
• [Journal Article] Eigenvalue computation of totally nonnegative upper Hessenberg matrices based on a variant of the discrete hungry Toda equation (2015)
  • Author(s): Ryo Sumikura, Akiko Fukuda, Emiko Ishiwata, Yusaku Yamamoto, Masashi Iwasaki, Yoshimasa Nakamura
  • Journal Title: AIP Conference Proceedings (ICNAAM-2014) Volume: 1648 Pages: 690006-690006
  • Peer Reviewed
• [Journal Article] 病理モデルにおける連続型と超離散型との対応関係 (2015)
  • Author(s): 関口 真基、石渡 恵美子
  • Journal Title: 九州大学応用力学研究所 研究集会報告 Volume: 26AO-S2 Pages: 127-132
  • Peer Reviewed
• [Presentation] Global stability of an SEIS epidemic model with delays (2017)
  • Author(s): Kei Fushimi, Yoichi Enatsu, Emiko Ishiwata
  • Organizer: 第27回日本数理生物学会年会 (JSMB2017)
• [Presentation] Mosquito model for Wolbachia infection with hatching delays (2017)
  • Author(s): Yoichi Enatsu, Masatoshi Kanamori, Emiko Ishiwata
  • Organizer: 第27回日本数理生物学会年会 (JSMB2017)
• [Presentation] 箱と玉の両方に番号が付いた箱玉系について (2016)
  • Author(s): 福田亜希子、山本有作、岩崎雅史、石渡恵美子、中村佳正
  • Organizer: 平成28年度 九州大学応用力学研究所 共同利用研究集会「非線形波動研究の深化と展開」
  • Place of Presentation: 九州大学筑紫地区
  • Year and Date: 2016-11-04
• [Presentation] Effect of time delay for an ultra-discretized logistic equation (2016)
  • Author(s): Masaki Sekiguchi, Emiko Ishiwata, Yukihiko Nakata
  • Organizer: The 26th annual meeting of the Japanese Society for Mathematical Biology (JSMB2016)
  • Place of Presentation: Kyushu University
  • Year and Date: 2016-09-07
• [Presentation] Delay-induced blow-up in some limit-cycle oscillation model: mathematical and numerical analyses (2016)
  • Author(s): 石渡恵美子、石渡哲哉、中田行彦、三木勝博
  • Organizer: 2016年度 応用数学合同研究集会
  • Place of Presentation: 龍谷大学瀬田キャンパス
• [Presentation] 時間遅れをもつ超離散型SIS病理モデル (2015)
  • Author(s): 関口真基, 石渡恵美子
  • Organizer: 明治大学MIMS共同研究集会「可積分系が拓く現象数理モデル」（ポスターセッション）
  • Place of Presentation: 明治大学中野キャンパス （東京都中野区）
  • Year and Date: 2015-11-06
• [Presentation] 超離散型ロジスティック方程式における時間遅れの影響 (2015)
  • Author(s): 関口真基, 石渡恵美子, 中田行彦
  • Organizer: 明治大学MIMS共同研究集会「可積分系が拓く現象数理モデル」（ポスターセッション）
  • Place of Presentation: 明治大学中野キャンパス （東京都中野区）
  • Year and Date: 2015-11-06
• [Presentation] On eigenvectors of structured nonsymmetric matrices related to the integrable discrete systems of hungry type (2015)
  • Author(s): Hiroshi Takeuchi, Kensuke Aihara, Akiko Fukuda, Emiko Ishiwata
  • Organizer: International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2015)
  • Place of Presentation: Tsukuba International Congress Center (Tsukuba, Japan)
  • Year and Date: 2015-09-08
  • Int'l Joint Research
• [Presentation] Effects of time delays in ultra-discrete epidemic models (2015)
  • Author(s): Masaki Sekiguchi, Emiko Ishiwata
  • Organizer: 2015 Joint Meeting of the 5th China-Japan-Korea Colloquium on Mathematical Biology and the Japanese Society for Mathematical Biology (CJK-JSMB 2015)
  • Place of Presentation: Doshisha University (Kyoto, Japan)
  • Year and Date: 2015-08-26
  • Int'l Joint Research
• [Presentation] ある種の構造を持つ非対称帯行列に対する固有ベクトル計算 (2015)
  • Author(s): 竹内弘史, 相原研輔, 福田亜希子, 石渡恵美子
  • Organizer: 日本応用数理学会2014年度研究部会連合発表会「行列・固有値の解法とその応用」研究部会
  • Place of Presentation: 明治大学中野キャンパス
  • Year and Date: 2015-03-06 – 2015-03-07
• [Presentation] 離散戸田方程式のある拡張に基づくTotally Nonnegative 行列の固有値計算アルゴリズム (2014)
  • Author(s): 隅蔵亮, 福田亜希子, 石渡恵美子, 山本有作, 岩崎雅史, 中村佳正
  • Organizer: 九州大学応用力学研究所共同利用研究集会「非線形波動研究の現状―課題と展望を探る―」
  • Place of Presentation: 九州大学筑紫地区 筑紫ホール
  • Year and Date: 2014-10-30 – 2014-11-01
• [Presentation] 病理モデルにおける連続型と超離散型との対応関係 (2014)
  • Author(s): 関口真基, 石渡恵美子
  • Organizer: 九州大学応用力学研究所共同利用研究集会「非線形波動研究の現状―課題と展望を探る―」
  • Place of Presentation: 九州大学筑紫地区 筑紫ホール
  • Year and Date: 2014-10-30 – 2014-11-01
• [Presentation] 離散ハングリーロトカ・ボルテラ系に基づく非対称帯行列の全固有対の計算 (2014)
  • Author(s): 竹内弘史, 相原研輔, 福田亜希子, 石渡恵美子
  • Organizer: 九州大学応用力学研究所共同利用研究集会「非線形波動研究の現状―課題と展望を探る―」
  • Place of Presentation: 九州大学筑紫地区 筑紫ホール
  • Year and Date: 2014-10-30 – 2014-11-01
• [Presentation] 複数遅れを持つ離散ロジスティック方程式の安定領域の可視化 (2014)
  • Author(s): 谷津田直之, 中田行彦, 石渡恵美子
  • Organizer: 京都大学数理解析研究所共同利用研究集会 「第11回生物数学の理論とその応用」 (研究代表者: 今隆助)
  • Place of Presentation: 京都大学数理解析研究所420号室
  • Year and Date: 2014-09-16 – 2014-09-19
• [Presentation] Dynamics of ultra discrete epidemic models (2014)
  • Author(s): Masaki Sekiguchi, Emiko Ishiwata
  • Organizer: The Joint Annual Meeting of the Japanese Society for Mathematical Biology and the Society for Mathematical Biology, (JSMB/SMB Osaka 2014)
  • Place of Presentation: Osaka International Convention Center
  • Year and Date: 2014-07-28 – 2014-08-01