We show that a certain model for the spread of an infection has a phase transition in the recuperation rate. The model is as follows: There are particles or individuals of type $A$ and type $B$, interpreted as healthy and infected, respectively. All particles perform independent, continuous time, simple random walks on $mathbb{Z}^d$ with the same jump rate $D$. The only interaction between the particles is that at the moment when a $B$-particle jumps to a site which contains an $A$-particle, or vice versa, the $A$-particle turns into a $B$-particle. All $B$-particles recuperate (that is, turn back into $A$-particles) independently of each other at a rate $la$. We assume that we start the system with $N_A(x,0-)$ $A$-particles at $x$, and that the $N_A(x,0-), , x in mathbb{Z}^d$, are i.i.d., mean $mu_A$ Poisson random variables. In addition we start with one additional $B$-particle at the origin. We show that there is a critical recuperation rate $la_c > 0$ such that the $B$-particles survive (globally) with positive probability if $la la_c$.
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机译：我们表明，某种感染传播的模型在恢复率上具有相变。模型如下：存在$ A $类型和$ B $类型的粒子或个体，分别被解释为健康和感染。所有粒子在$ mathbb {Z} ^ d $上以相同的跳跃率$ D $执行独立，连续的时间，简单的随机游动。粒子之间的唯一相互作用是，当$ B $粒子跳到包含$ A $粒子的位置时，反之亦然，而$ A $粒子变为$ B $粒子。所有$ B $粒子以$ la $的速率彼此独立地恢复（即变回$ A $粒子）。我们假设我们以$ x $处的$ N_A（x，0-）$ $ A $粒子启动系统，并假设$ N_A（x，0-），，x 在 mathbb {Z} ^中d $是iid，均值$ mu_A $泊松随机变量。另外，我们从原点开始再增加一个$ B $粒子。我们表明，存在临界的再生速率$ la_c> 0 $，使得如果$ la la_c $，$ B $粒子能够以正概率生存（全局）。
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