Pang Guodong, Pardoux Étienne
Department of Computational Applied Mathematics and Operations Research, George R. Brown College of Engineering, Rice University, Houston, TX 77005 USA.
Aix-Marseille Univ, CNRS, I2M, Marseille, France.
Appl Math Optim. 2023;87(3):50. doi: 10.1007/s00245-022-09963-z. Epub 2023 Mar 13.
We study epidemic models where the infectivity of each individual is a random function of the infection age (the elapsed time since infection). To describe the epidemic evolution dynamics, we use a stochastic process that tracks the number of individuals at each time that have been infected for less than or equal to a certain amount of time, together with the aggregate infectivity process. We establish the functional law of large numbers (FLLN) for the stochastic processes that describe the epidemic dynamics. The limits are described by a set of deterministic Volterra-type integral equations, which has a further characterization using PDEs under some regularity conditions. The solutions are characterized with boundary conditions that are given by a system of Volterra equations. We also characterize the equilibrium points for the PDEs in the SIS model with infection-age dependent infectivity. To establish the FLLNs, we employ a useful criterion for weak convergence for the two-parameter processes together with useful representations for the relevant processes via Poisson random measures.
其中每个个体的感染力是感染年龄（自感染以来经过的时间）的随机函数。为了描述流行病的演变动态，我们使用一个随机过程，该过程跟踪在每个时间点感染时间小于或等于特定时长的个体数量，以及总感染力过程。我们为描述流行病动态的随机过程建立了泛函大数定律（FLLN）。极限由一组确定性的沃尔泰拉型积分方程描述，在某些正则性条件下，这些方程可以用偏微分方程进一步刻画。解由一个沃尔泰拉方程组给出的边界条件来刻画。我们还刻画了具有感染年龄依赖感染力的SIS模型中偏微分方程的平衡点。为了建立FLLN，我们采用了一个关于双参数过程弱收敛的有用准则，以及通过泊松随机测度对相关过程的有用表示。
