论文标题
对流行动力学的退化PDE模型的定性分析
Qualitative analysis of solutions for a degenerate PDE model of epidemic dynamics
论文作者
论文摘要
隔室模型被广泛用于数学流行病学，以描述感染疾病的动力学。最近由Chalub和Souza得出的新的SIS-PDE模型基于众所周知的离散 - 时间Markov链SIS-DTMC模型中概率密度的扩散式饮食近似。由于原点扩散项的堕落，这种新的SIS-PDE模型是保守的。我们分析了一类退化PDE模型，并获得了具有某些规律性特性的经典解决方案的足够条件。另外，我们表明，在任何有限时间，在原点上，关于系数和初始数据经典解决方案的其他假设都消失了。解决方案的起源消失与模型的保护特性一致。本文的主要结果是：存在较弱的解决方案的足够条件，对起源的渐近行为的分析以及存在弱溶液的证明，将其收集到DIRAC DIRAC DELTA功能。此外，我们研究解决方案的长期行为，并通过数值计算确认我们的分析。
Compartmental models are widely used in mathematical epidemiology to describe dynamics of infection disease. A new SIS-PDE model, recently derived by Chalub and Souza, is based on a diffusion-drift approximation of probability density in a well-known discrete - time Markov chain SIS-DTMC model. This new SIS-PDE model is conservative due to degeneracy of the diffusion term at the origin. We analyze a class of degenerate PDE models and obtain sufficient conditions for existence of classical solutions with certain regularity properties. Also, we show that under some additional assumptions about coefficients and initial data classical solutions vanish at the origin at any finite time. Vanishing at the origin of solutions is consistent with the conservation property of the model. The main results of this article are: sufficient conditions for existence of weak solutions, analysis of their asymptotic behavior at the origin, and the proof of existence of weak solutions convergent to Dirac delta function. Moreover, we study long-time behavior of solutions and confirm our analysis by numerical computations.
