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C型肝炎ウイルス感染動態のマルチスケールモデルの数理解析

北川, 耕咲 KITAGAWA, Kosaku キタガワ, コウサク 九州大学

2021.03.24

概要

生命現象は往々にして多階層的な構造を持つ。生命は階層内および階層間の動的な相互作用によって形成されるマルチスケールな高次元システムとして成り立っている。非常に大きなブラックボックスである生命現象に対して、古くから数理モデルは観察可能なデータから背景を推察することで一歩一歩生命現象の解明に貢献してきた。特に対象の変化を表現する微分方程式モデルはシステマチックな生命現象との相性が非常に良く、多くの結果を残してきた。また、近年になって計算機の性能が向上し、これによってより生命に近づいた詳細な数理モデルを膨大なデータと共に扱う大規模な計算が現実的なものとなってきている。

本論文では、特に C 型肝炎ウイルス(Hepatitis C Virus, HCV)の感染現象を表現するマルチスケールモデルを対象として、近年の課題である大規模なデータの取り扱いを視野に入れ、モデルの数理的な解析を行う。第一章では、偏微分方程式(Partial Differential Equation, PDE)によって記述される HCV の感染現象を表現するマルチスケールモデルに対し、数学的にパラメータの情報を失わない変換を行うことによって同値な現象を表現する常微分方程式(Ordinary DifferentialEquation, ODE)によるモデルを導出した。これにより、PDE モデルを直接計算することに比べてODE モデルの計算において計算時間が短縮されたことから、大規模なデータ解析に用いた際に非常に大きなコストの削減がなされる可能性が示唆された。また、ODE の計算に関しては PDE 計算と比較して種々の計算ソフトのパッケージが充実しており、さらに数理解析や理論も容易である。このように、モデルの表現する情報を保持する簡単なモデルを導出することを示した。第二章では、第一章で導出した ODE モデルに対して平衡点の大域漸近安定性を証明した。これによって第一章の結果が数理解析において有用であることの支持とした。また平衡点の大域漸近安定性はウイルス感染を治療の見地から考えた時、ウイルスの排除のため条件を与えることができる。ここで示した大域漸近安定性はモデルパラメータの式で表される基本再生産数の値による条件付きで証明される。

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