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Mathematical analysis of epidemic models formulated by delay equations: loss of immunity and instability

Research Project

Project/Area Number 16K20976
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Mathematical analysis
Research InstitutionAoyama Gakuin University (2020)
Shimane University (2016-2019)

Principal Investigator

Nakata Yukihiko  青山学院大学, 理工学部, 准教授 (30741061)

Project Period (FY) 2016-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords時間遅れをもつ微分方程式 / 感染症の数理モデル / 周期解 / 構造化個体群モデル / 特性方程式 / 安定性 / 免疫 / 感染症数理モデル / 時間遅れ / 微分方程式 / 平衡解 / 楕円関数 / 免疫低下 / 数理モデル / 遅延微分方程式 / 力学系 / 感染症 / 爆発解 / 大域漸近安定性 / 可積分系 / 積分方程式 / 遅延方程式
Outline of Final Research Achievements

Analyzing delay models of epidemic models, we have obtained insight into disease transmission dynamics caused by loss of immunity and boosting of immunity of individuals. Using a simple mathematical model, we show that heterogeneity of susceptible individuals can cause an epidemic outbreak with time delays (delayed outbreak). We also formulate a mathematical model that incorporates decline and boosting of immunity using a Volterra type integral equation, and obtain a sufficient condition for the uniqueness of the equilibrium. Furthermore, for a logistic equation with time delay, which is derived from an epidemic model that explain periodicity of a childhood infectious disease, we show the existence of a period-2 solution expressed by Jacobi elliptic functions.

Academic Significance and Societal Importance of the Research Achievements

感染症問題への取り組みは現代社会における喫緊の課題である。感染症の流行現象は、多くの要素の相互作用による、マルチスケールで非線形な現象であり、その理解や制御において、数理モデルが果たす役割は大きい。感染症流行のメカニズムの説明においては、現象をミクロな個体レベルから構成的に人口動態を記述した構造化個体群モデルの活用が重要であり、このようなモデルの定式化から、時間遅れをもつ微分方程式など、現代においても解析が困難な方程式が現れる。これらの方程式に対する数学的な問題は、現象理解における課題でもあり、数理モデルの性質を明らかにすることは、現象理解に大きな役割を果たすと考えられる。

Report

(6 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (45 results)

All 2021 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (9 results) Journal Article (11 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 11 results,  Open Access: 5 results,  Acknowledgement Compliant: 1 results) Presentation (20 results) (of which Int'l Joint Research: 10 results,  Invited: 5 results) Remarks (5 results)

  • [Int'l Joint Research] セゲド大学(ハンガリー)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] 東北師範大学(中国)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] University of Szeged/University of Pannonia(ハンガリー)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] TU Dresden(ドイツ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of Helsinki(フィンランド)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] York University(カナダ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of Udine(イタリア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of Szeged(ハンガリー)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] University of Szeged, Bolyai Institute(ハンガリー)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Note on the uniqueness of an endemic equilibrium of an epidemic model with boosting of immunity2021

    • Author(s)
      Liu Yang, Yukihiko Nakata
    • Journal Title

      Journal of Biological Systems

      Volume: in press(特別号) Issue: 02 Pages: 1-12

    • DOI

      10.1142/s0218339021400076

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Existence of a period two solution of a delay differential equation2020

    • Author(s)
      Y. Nakata
    • Journal Title

      DCDS-S

      Volume: -

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Epidemic dynamics with time-varying susceptibility due to repeated infections2019

    • Author(s)
      Y. Nakata, R. Omori
    • Journal Title

      Journal of Biological Dynamics

      Volume: 13 Issue: 1 Pages: 567-585

    • DOI

      10.1080/17513758.2019.1643043

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The change of susceptibility following infection can induce failure to predict outbreak potential by R02019

    • Author(s)
      Y. Nakata, R. Omori
    • Journal Title

      Mathematical Biosciences and Engineering

      Volume: 16(2) Issue: 2 Pages: 813-830

    • DOI

      10.3934/mbe.2019038

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Unbounded and blow-up solutions for a delay logistic equation with positive feedback2018

    • Author(s)
      I. Gyori, Y. Nakata, G. Rost
    • Journal Title

      Communications on Pure and Applied Analysis

      Volume: 17(6) Issue: 6 Pages: 2848-2854

    • DOI

      10.3934/cpaa.2018134

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] An explicit periodic solution of a delay differential equation2018

    • Author(s)
      Y. Nakata
    • Journal Title

      Journal of Dynamics and Diffrential Equations

      Volume: 印刷中 Issue: 1 Pages: 163-179

    • DOI

      10.1007/s10884-018-9681-z

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods2018

    • Author(s)
      Ph. Getto, M. Gyllenberg, Y. Nakata, F. Scarabel
    • Journal Title

      Journal of Mathematical Biology

      Volume: 印刷中 Issue: 1 Pages: 281-328

    • DOI

      10.1007/s00285-019-01357-0

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Global stability of an SIS epidemic model with a finite infectious period2018

    • Author(s)
      Yukihiko Nakata, Gergely Rost
    • Journal Title

      Differential Integral Equations

      Volume: 31 Pages: 161-172

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Dynamics of an ultra-discrete SIR epidemic model with time delay2018

    • Author(s)
      Masaki Sekiguchi, Emiko Ishiwata, Yukihiko Nakata
    • Journal Title

      Mathematical Biosciences and Engineering

      Volume: 15 Pages: 653-666

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Convergence of a Logistic Type Ultradiscrete Model2017

    • 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

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Note on stability conditions for structured population dynamics models2016

    • Author(s)
      Yukihiko Nakata
    • Journal Title

      Electron. J. Qual. Theory Differ. Equ.

      Volume: 78 Issue: 78 Pages: 1-14

    • DOI

      10.14232/ejqtde.2016.1.78

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Presentation] 感染症の数理モデルから現れる時間遅れをもつ微分方程式と解のダイナミクス2021

    • Author(s)
      中田行彦
    • Organizer
      日本数学会(応用数学分科会)
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] 階段形の非線形関数をもつ遅延微分方程式について2020

    • Author(s)
      中田行彦, Gabriella Vas
    • Organizer
      日本応用数理学会年会
    • Related Report
      2020 Annual Research Report
  • [Presentation] 分布型の時間遅れをもつ微分方程式の対称的な周期解について2020

    • Author(s)
      中田行彦
    • Organizer
      北陸応用数理研究会2020
    • Related Report
      2019 Research-status Report
  • [Presentation] Period two solutions of distributed delay differential equations2019

    • Author(s)
      Y. Nakata
    • Organizer
      11th Colloquium on the Qualitative Theory of Differential Equations
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stability analysis of an epidemic model with boosting of immunity2019

    • Author(s)
      Y. Nakata, R. Omori, L. Yang
    • Organizer
      The fifth conference on computational and mathematical population dynamics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] 分布型の時間遅れをもつ微分方程式の周期解について2019

    • Author(s)
      中田行彦
    • Organizer
      日本数学会 2019年度秋季総合分科会
    • Related Report
      2019 Research-status Report
  • [Presentation] Period two solutions of distributed delay differential equations2019

    • Author(s)
      Y. Nakata
    • Organizer
      Equadiff 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] 免疫減衰をもつ感染症数理モデル における周期流行2019

    • Author(s)
      中田行彦, 大森亮介, Yang Liu
    • Organizer
      日本応用数理学会2019年度年会
    • Related Report
      2019 Research-status Report
  • [Presentation] Periodic solutions of a delay differential equation.2018

    • Author(s)
      Y. Nakata
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Taipei
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] ある遅延微分方程式の陽的な周期解について2018

    • Author(s)
      中田行彦
    • Organizer
      日本応用数理学会
    • Related Report
      2018 Research-status Report
  • [Presentation] ある遅延微分方程式の陽的な周期解について2018

    • Author(s)
      中田行彦
    • Organizer
      日本数学会
    • Related Report
      2018 Research-status Report
  • [Presentation] Period two solutions of a class of delay differential equations.2018

    • Author(s)
      Y. Nakata
    • Organizer
      常微分方程式の定性的理論および数理モデル研究への応用, 京都大学数理解析研究所
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Delay Equations for Epidemic Models with Waning Immunity2017

    • Author(s)
      Yukihiko Nakata
    • Organizer
      SIAM Conference on Applications of Dynamical Systems
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Delay Equations for Reinfection Dynamic2017

    • Author(s)
      中田行彦
    • Organizer
      日本数学会2017年度秋季総合分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] Infection and reinfection dynamics in a heterogeneous susceptible population2017

    • Author(s)
      中田行彦
    • Organizer
      第14回「生物数学の理論とその応用」- 構造化個体群モデルとその応用-
    • Related Report
      2017 Research-status Report
  • [Presentation] Epidemic models for delay equations2016

    • Author(s)
      Yukihiko Nakata
    • Organizer
      RIMS研究集会 常微分方程式の定性的理論とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-11-16
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Epidemic models with waning immunity2016

    • Author(s)
      Yukihiko Nakata
    • Organizer
      International Conference for the 70th Anniversary of Korean Mathematical Society
    • Place of Presentation
      ソウル大学、韓国
    • Year and Date
      2016-10-20
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Epidemic Models with Waning Immunity2016

    • Author(s)
      Yukihiko Nakata
    • Organizer
      China-Japan Joint Workshop on Mathematics & Statistics
    • Place of Presentation
      School of Mathematics and Statistics, Northeast Normal University, China
    • Year and Date
      2016-10-09
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Epidemic models for delay equations2016

    • Author(s)
      Yukihiko Nakata
    • Organizer
      JSMB2016
    • Place of Presentation
      九州大学
    • Year and Date
      2016-09-07
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stability of a logistic equation with multiple delays2016

    • Author(s)
      Yukihiko Nakata
    • Organizer
      The 22nd International Conference on Difference Equations and Applications
    • Place of Presentation
      I-siteなんば(大阪府立大学)
    • Year and Date
      2016-07-16
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Remarks] 青山学院大学理工学部数理サイエンス学科中田行彦

    • URL

      http://www.math.aoyama.ac.jp/users/ynakata/

    • Related Report
      2020 Annual Research Report
  • [Remarks]

    • URL

      http://www.math.shimane-u.ac.jp/~ynakata/

    • Related Report
      2019 Research-status Report
  • [Remarks]

    • URL

      http://www.math.shimane-u.ac.jp/~ynakata/

    • Related Report
      2018 Research-status Report
  • [Remarks]

    • URL

      http://www.math.shimane-u.ac.jp/~ynakata/

    • Related Report
      2017 Research-status Report
  • [Remarks]

    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2022-01-27  

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