传染病传播的形状定理
A shape theorem for the spread of an infection
摘要：
我们考虑以下相互作用粒子系统：有一个“气体”的粒子，其中每个执行一个连续的时间简单的随机游动的d维晶格。这些粒子被称为A粒子，它们相互独立地运动。我们假设系统开始时每个格点x上有一个泊松数的粒子，不同x的i.i.d.上的粒子数。此外，还有有限个B粒子，它们可以像A粒子一样进行连续时间的简单随机游动。A粒子和B粒子分别被解释为健康或受感染的个体。B粒子彼此独立运动。唯一的相互作用是，当B粒子和A粒子重合时，后者瞬间变成B粒子。设B（t）是B粒子在[0，t]期间访问的站点的集合。我们证明了B（t）随时间线性增长并具有渐近形状;更确切地说，存在一个非随机凸紧集B_0，使得几乎必然地，对所有0 < a <1，（1-a）tB_0包含在B（t）中，并且B（t）最终包含在（1+a）tB_0中。
英文摘要：
We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous time simple random walk on the d-dimensional lattice. These particles are called A-particles and move independently of each other. We assume that we start the system with a Poisson number of particles at each lattice site x, with the number of particles at different x's i.i.d. In addition, there are a finite number of B-particles which perform the same continuous time simple random walks as the A-particles. A- and B-particles are interpreted as individuals who are healthy or infected, respectively. The B-particles move independently of each other. The only interaction is that when a B-particle and an A-particle coincide, the latter instantaneously turns into a B-particle. Let B(t) be the set of sites visited by a B-particle during [0,t]. We show that B(t) grows linearly in time and has an asymptotic shape; more precisely, there exists a non-random convex, compact set B_0 such that almost surely, for all 0 < a <1, (1-a)tB_0 is contained in B(t) and B(t) is contained in (1+a)tB_0 eventually.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
