RUI: Interacting Particle Systems in Biology

相互作用粒子系统是可数集合（通常是晶格）上的空间随机模型。研究者使用这些系统来模拟几种生物现象。许多这些模型已经在数学生物学和物理学文献中被考虑过，但很少有严谨的结果得到证明。到目前为止，大多数关于相互作用粒子系统的数学理论都涉及单调（或吸引）系统。当人们处理流行病时，单调性就不成立了。证明非单调系统的结果是至关重要的，因为我们世界中的许多现象在任何（数学上）有用的方式上都不是单调的。相互作用粒子系统在研究流行病方面非常有用。例如，研究者为流行病的传播提出了以下模型。有一个站点网格，每个站点要么是空的，健康的，要么是受感染的。健康和受感染的个体在空旷的最近的邻近地点产下健康的个体。健康的人会被最近的感染者感染。健康个体和受感染个体的死亡率不同。这似乎对艾滋病这样的流行病是足够的，因为受感染的人可以生下健康的人。研究者证明的一个结果是，如果感染率足够大，当且仅当受感染个体的出生率足够高时，这种流行病才会持续存在。虽然所考虑的模型是对现实世界的粗略简化，但它确实提供了一个有趣的想法来进行测试。也就是说，艾滋病流行最具破坏性的国家是否就是受感染者生育率最高的国家？

9625126 Schinazi ABSTRACT Interacting particle systems are spatial stochastic models on a countable set (usually the lattice). The investigator uses these systems to model several biological phenomena. Many of these models have already been considered in the mathematical biology and physics literature but very few rigorous results have been proved. So far most of the mathematical theory about interacting particle systems deals with monotone (or attractive) systems. When one deals with epidemics, monotonicity does not hold. To prove results for non-monotone systems is crucial since so many phenomena in our world are not monotone in any (mathematically ) useful way. Interacting particle systems are very useful in studying epidemics. For instance, The investigator proposes the following model for the spread of an epidemic. There is a grid of sites and each site is either empty, healthy or infected. Healthy and infected individuals give birth to healthy individuals on empty nearest neighbor sites. Healthy individuals get infected by infected nearest neighbors. Healthy and infected individuals die at different rates. This seems adequate for epidemics like the AIDS epidemic for which infected individuals may give birth to healthy individuals. One of the results proved by The investigator is that, given an infection rate which is large enough, the epidemic will persist if and only if the rate at which infected individuals give birth is high enough. While the model considered is a gross simplification of the real world it does give an interesting idea to test. Namely, is it true that the countries where the AIDS epidemic is the most devastating are the countries where the rate at which infected individuals give birth is the highest?