Title: An SIR-like kinetic model tracking individuals' viral load Show Chinese title

Title: 一种跟踪个体病毒载量的类似SIR的动力学模型

Abstract: Mathematical models are formal and simplified representations of the knowledge related to a phenomenon. In classical epidemic models, a neglected aspect is the heterogeneity of disease transmission and progression linked to the viral load of each infectious individual. Here, we attempt to investigate the interplay between the evolution of individuals' viral load and the epidemic dynamics from a theoretical point of view. In the framework of multi-agent systems, we propose a particle stochastic model describing the infection transmission through interactions among agents and the individual physiological course of the disease. Agents have a double microscopic state: a discrete label, that denotes the epidemiological compartment to which they belong and switches in consequence of a Markovian process, and a microscopic trait, representing a normalized measure of their viral load, that changes in consequence of binary interactions or interactions with a background. Specifically, we consider Susceptible--Infected--Removed--like dynamics where infectious individuals may be isolated from the general population and the isolation rate may depend on the viral load sensitivity and frequency of tests. We derive kinetic evolution equations for the distribution functions of the viral load of the individuals in each compartment, whence, via suitable upscaling procedures, we obtain a macroscopic model for the densities and viral load momentum. We perform then a qualitative analysis of the ensuing macroscopic model, and we present numerical tests in the case of both constant and viral load-dependent isolation control. Also, the matching between the aggregate trends obtained from the macroscopic descriptions and the original particle dynamics simulated by a Monte Carlo approach is investigated. Show Chinese abstract

Abstract: 数学模型是对与现象相关的知识的正式且简化的表示。 在经典流行病模型中，一个被忽视的方面是疾病传播和进展的异质性，这与每个感染个体的病毒载量有关。 在这里，我们试图从理论角度研究个体病毒载量的演变与流行病动力学之间的相互作用。 在多智能体系统的框架下，我们提出了一种粒子随机模型，描述了通过智能体之间的相互作用以及个体生理病程的感染传播。 智能体具有双微观状态：一个离散标签，表示它们所属的流行病学类别，并由于马尔可夫过程而切换；一个微观特征，表示其病毒载量的归一化度量，由于二元相互作用或与背景的相互作用而发生变化。 具体而言，我们考虑易感-感染-移除类似的动力学，其中感染个体可能从一般人群中隔离，隔离率可能取决于病毒载量的敏感性和检测频率。 我们推导了每个类别中个体病毒载量分布函数的动力学演化方程，从而通过适当的放大程序，得到密度和病毒载量动量的宏观模型。 然后，我们对由此产生的宏观模型进行定性分析，并在恒定和依赖于病毒载量的隔离控制情况下进行数值测试。 此外，还研究了从宏观描述中获得的总体趋势与通过蒙特卡罗方法模拟的原始粒子动力学之间的匹配情况。