Inaba Hisashi
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan.
J Math Biol. 2019 Jul;79(2):731-764. doi: 10.1007/s00285-019-01375-y. Epub 2019 May 13.
In the previous paper (Inaba in J Math Biol 65:309-348, 2012), we proposed a new (most biologically natural) definition of the basic reproduction number for structured population in general time-heterogeneous environments based on the generation evolution operator. Using the mathematical definition for cone spectral radius, we show that our is given by the spectral radius of the generation evolution operator in the time-state space. Then as far as we consider linear population dynamics, our is a threshold value for population extinction and persistence in time-heterogeneous environments. Next we prove that even for nonlinear systems, our plays a role of a threshold value for population extinction in time-heterogeneous environments. For periodic systems, we can show that supercritical condition implies existence of positive periodic solution. Finally using the idea of in time-heterogeneous environment, we examine existence and stability of periodic solution in the age-structured SIS epidemic model with time-periodic parameters.
在之前的论文中（稻叶，《数学生物学杂志》65:309 - 348，2012年），我们基于世代演化算子，针对一般时变环境中的结构化种群，提出了基本再生数的一种新的（最符合生物学自然规律的）定义。利用锥谱半径的数学定义，我们证明了我们所定义的基本再生数由时 - 态空间中世代演化算子的谱半径给出。那么，就我们所考虑的线性种群动力学而言，我们所定义的基本再生数是时变环境中种群灭绝和持续存在的一个阈值。接下来我们证明，即使对于非线性系统，我们所定义的基本再生数在时变环境中也起着种群灭绝阈值的作用。对于周期系统，我们可以证明超临界条件意味着正周期解的存在。最后，利用时变环境中的相关思想，我们研究了具有时间周期参数的年龄结构SIS传染病模型中周期解的存在性和稳定性。
