tailieunhanh - Đề tài " A shape theorem for the spread of an infection "

In [KSb] we studied the following model for the spread of a rumor or infection: There is a “gas” of so-called A-particles, each of which performs a continuous time simple random walk on Zd , with jump rate DA . We assume that “just before the start” the number of A-particles at x, NA (x, 0−), has a mean μA Poisson distribution and that the NA (x, 0−), x ∈ Zd , are independent. In addition, there are B-particles which perform continuous time simple random walks with jump rate DB . We start with a finite number of B-particles. | Annals of Mathematics A shape theorem for the spread of an infection By Harry Kesten and Vladas Sidoravicius Annals of Mathematics 167 2008 701 766 A shape theorem for the spread of an infection By HARRy Kesten and Vladas SiDORAVicius Abstract In KSb we studied the following model for the spread of a rumor or infection There is a gas of so-called A-particles each of which performs a continuous time simple random walk on Zd with jump rate Da. We assume that just before the start the number of A-particles at x Na x 0 has a mean HA Poisson distribution and that the Na x 0 x E Zd are independent. In addition there are B-particles which perform continuous time simple random walks with jump rate Db. We start with a finite number of B-particles in the system at time 0. The positions of these initial B-particles are arbitrary but they are nonrandom. The B-particles move independently of each other. The only interaction occurs when a B-particle and an A-particle coincide the latter instantaneously turns into a B-particle. KSb gave some basic estimates for the growth of the set B t x E Zd a B-particle visits x during 0 t . In this article we show that if Da Db then B t B t 2 1 d grows linearly in time with an asymptotic shape . there exists a nonrandom set B0 such that 1 t B t B0 in a sense which will be made precise. 1. Introduction We study the model described in the abstract. One interpretation of this model is that the B-particles represent individuals who are infected and the A-particles represent susceptible individuals see KSb for another interpretation. B t represents the collection of sites which have been visited by a B-particle during 0 t and B t is a slightly fattened up version of B t obtained by adding a unit cube around each point of B t . This fattened up version is introduced merely to simplify the statement of our main result. It is simpler to speak ofthe shape of the set 1 t B t as a subset of Rd than of the discrete set 1 t B t . The aim of this paper