Arrigoni F, Pugliese A
Università di Trento, Dipartimento di Matematica Via Sommarive 14, 38050 Povo di Trento, Italy.
J Math Biol. 2002 Nov;45(5):419-40. doi: 10.1007/s002850200155.
We start from a stochastic SIS model for the spread of epidemics among a population partitioned into M sites, each containing N individuals; epidemic spread occurs through within-site ('local') contacts and global contacts. We analyse the limit behaviour of the system as M and N increase to infinity. Two limit procedures are considered, according to the order in which M and N go to infinity; independently of the order, the limiting distribution of infected individuals across sites is a probability measure, whose evolution in time is governed by the weak form of a PDE. Existence and uniqueness of the solutions to this problem is shown. Finally, it is shown that the infected distribution converges, as time goes to infinity, to a Dirac measure at the value x(*), the equilibrium of a single-patch SIS model with contact rate equal to the sum of local and global contacts.
我们从一个随机SIS模型开始，该模型用于描述在一个被划分为M个场所的人群中传染病的传播，每个场所包含N个人；传染病的传播通过场所内（“局部”）接触和全局接触发生。我们分析当M和N增加到无穷大时系统的极限行为。根据M和N趋于无穷大的顺序考虑两种极限过程；无论顺序如何，跨场所感染个体的极限分布是一种概率测度，其随时间的演化由一个偏微分方程的弱形式支配。证明了该问题解的存在性和唯一性。最后，证明了随着时间趋于无穷大，感染分布收敛到值x()处的狄拉克测度，x()是具有等于局部和全局接触总和的接触率的单斑块SIS模型的平衡点。
