零程过程中的传染病相变
Epidemic Phase Transitions in the Zero-Range Process
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中文总结 AI 辅助
研究\(\mathbb{Z}^d\)上相互作用粒子系统内感染传播模型的灭绝 - 存活相图，通过分析粒子感染与治愈情况，证明不同密度和治愈速率下感染的存亡情况，确立了非平凡临界密度。
中文摘要 AI 辅助
我们考虑一个在\(\mathbb{Z}^d\)上相互作用粒子系统内感染传播的模型，将Kesten和Sidoravicius引入的框架推广到密度\(\rho>0\)的平衡零程过程。在我们的模型中，个体在任何时候要么健康要么被感染。当被感染和健康粒子占据同一位置时感染瞬间传播，而被感染粒子以速率\(\delta>0\)独立治愈。我们从仅原点处粒子被感染的构型开始研究此过程的灭绝 - 存活相图。对于每个固定的正治愈速率，我们证明如果密度\(\rho\)足够小，感染几乎肯定灭绝；如果\(\rho\)足够大，感染以正概率存活，确立了非平凡临界密度的存在。在足够高的密度下，即使瞬间治愈感染也会存活。我们还表明，对于每个正密度，当治愈速率足够小时感染会存活。
英文摘要
We consider a model for the spread of an infection within an interacting particle system on $\mathbb{Z}^d$, generalizing a framework introduced by Kesten and Sidoravicius to a zero-range process in equilibrium with density $ρ> 0$. In our model, at any time, individuals are either healthy or infected. The infection spreads instantaneously whenever infected and healthy particles occupy the same site, while infected particles heal independently at rate $δ> 0$. We investigate the extinction-survival phase diagram of this process starting from a configuration where only the particles at the origin are infected. For every fixed positive healing rate, we prove that the infection becomes extinct almost surely if the density $ρ$ is sufficiently small and survives with positive probability if $ρ$ is sufficiently large, establishing the existence of a non-trivial critical density. At sufficiently high densities, survival occurs even under instantaneous healing. We also show that, for every positive density, the infection survives when the healing rate is sufficiently small.
