Pugliese Andrea, Milner Fabio
Dipartimento di Matematica, Università degli Studi di Trento, Trento, Italy.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, USA.
J Math Biol. 2018 Dec;77(6-7):2079-2102. doi: 10.1007/s00285-018-1246-6. Epub 2018 May 9.
A structured population model is described and analyzed, in which individual dynamics is stochastic. The model consists of a PDE of advection-diffusion type in the structure variable. The population may represent, for example, the density of infected individuals structured by pathogen density x, [Formula: see text]. The individuals with density [Formula: see text] are not infected, but rather susceptible or recovered. Their dynamics is described by an ODE with a source term that is the exact flux from the diffusion and advection as [Formula: see text]. Infection/reinfection is then modeled moving a fraction of these individuals into the infected class by distributing them in the structure variable through a probability density function. Existence of a global-in-time solution is proven, as well as a classical bifurcation result about equilibrium solutions: a net reproduction number [Formula: see text] is defined that separates the case of only the trivial equilibrium existing when [Formula: see text] from the existence of another-nontrivial-equilibrium when [Formula: see text]. Numerical simulation results are provided to show the stabilization towards the positive equilibrium when [Formula: see text] and towards the trivial one when [Formula: see text], result that is not proven analytically. Simulations are also provided to show the Allee effect that helps boost population sizes at low densities.
描述并分析了一个结构化种群模型，其中个体动态是随机的。该模型由结构变量中的对流扩散型偏微分方程组成。例如，种群可以表示由病原体密度(x)构建的感染个体的密度，[公式：见正文]。密度为[公式：见正文]的个体未被感染，而是易感或已康复。它们的动态由一个带有源项的常微分方程描述，该源项是扩散和对流的精确通量，即[公式：见正文]。然后通过概率密度函数在结构变量中分布这些个体的一部分，将其转移到感染类别中来对感染/再感染进行建模。证明了全局时间解的存在性，以及关于平衡解的一个经典分岔结果：定义了一个净繁殖数[公式：见正文]，它将([公式：见正文])时仅存在平凡平衡的情况与([公式：见正文])时存在另一个非平凡平衡的情况区分开来。提供了数值模拟结果，以显示当([公式：见正文])时向正平衡的稳定以及当([公式：见正文])时向平凡平衡的稳定，该结果未经过解析证明。还提供了模拟结果以显示有助于在低密度时增加种群规模的阿利效应。
