论文标题
长期存在具有非本地术语对免疫反应建模的反应扩散系统的解决方案 - 一种以解释为导向的证明
Longterm existence of solutions of a reaction diffusion system with non-local terms modeling an immune response -- an interpretation-orientated proof
论文作者
论文摘要
本文显示了反应扩散系统建模肝感染的全球存在和界限。存在证明是逐步提出的，重点在于中间的解释在建模的肝感染中。来自淋巴中免疫反应的病毒和免疫系统细胞之间动力学的非本地效应导致了整个派对微分方程。尽管抛物线偏微分方程的存在定理是现场的教科书示例，但附加的积分术语需要新的方法来证明解决方案的全球存在。这允许建立一个存在证明，重点是解释，从而在系统中和建模的角度进行更多的见解。 我们在$ l^1（ω）$和$ l^2（ω）$ - 规范中显示了解决方案的界限，并使用这些结果来证明解决方案的全局存在和界限。证明的一个核心要素是在反应项中处理反对作用机制，这在所有种群动力学模型中都发生，并且以相反的单调性行为导致反应术语。在对肝感染进行建模的背景下，$ l^\ infty（ω）$ - 标准的界限具有实际相关性：较大的免疫反应会导致肝组织的强烈炎症。强烈的炎症会对感染者的健康产生负面影响，并导致严重的继发性疾病。将获得的粗略估计与数值测试进行了比较。
This paper shows the global existence and boundedness of solutions of a reaction diffusion system modeling liver infections. The existence proof is presented step by step and the focus lies on the interpretation of intermediate results in the context of liver infections which is modeled. Non-local effects in the dynamics between the virus and the immune system cells coming from the immune response in the lymphs lead to an integro-partial differential equation. While existence theorems for parabolic partial differential equations are textbook examples in the field, the additional integral term requires new approaches to proving the global existence of a solution. This allows to set up an existence proof with a focus on interpretation leading to more insight in the system and in the modeling perspective at the same time. We show the boundedness of the solution in the $L^1(Ω)$- and the $L^2(Ω)$-norms, and use these results to prove the global existence and boundedness of the solution. A core element of the proof is the handling of oppositely acting mechanisms in the reaction term, which occur in all population dynamics models and which results in reaction terms with opposite monotonicity behavior. In the context of modeling liver infections, the boundedness in the $L^\infty(Ω)$-norm has practical relevance: Large immune responses lead to strong inflammations of the liver tissue. Strong inflammations negatively impact the health of an infected person and lead to grave secondary diseases. The gained rough estimates are compared with numerical tests.
