We introduce a new model for contagion spread using a network of interacting finite memory two-color P\'{o}lya urns, which we refer to as the finite memory interacting P\'{o}lya contagion network. The urns interact in the sense that the probability of drawing a red ball (which represents an infection state) for a given urn, not only depends on the ratio of red balls in that urn but also on the ratio of red balls in the other urns in the network, hence accounting for the effect of spatial contagion. The resulting network-wide contagion process is a discrete-time finite-memory ($M$th order) Markov process, whose transition probability matrix is determined. The stochastic properties of the network contagion Markov process are analytically examined, and for homogeneous system parameters, we characterize the limiting state of infection in each urn. For the non-homogeneous case, given the complexity of the stochastic process, and in the same spirit as the well-studied SIS models, we use a mean-field type approximation to obtain a discrete-time dynamical system for the finite memory interacting P\'{o}lya contagion network. Interestingly, for $M=1$, we obtain a linear dynamical system which exactly represents the corresponding Markov process. For $M>1$, we use mean-field approximation to obtain a nonlinear dynamical system. Furthermore, noting that the latter dynamical system admits a linear variant (realized by retaining its leading linear terms), we study the asymptotic behavior of the linear systems for both memory modes and characterize their equilibrium. Finally, we present simulation studies to assess the quality of the approximation purveyed by the linear and non-linear dynamical systems.
翻译:我们引入了一个新的传染传播模式, 使用一个互动的内存范围网, 两个颜色 P\ { { o} lya URns, 我们称之为“ 有限内存 互动 P\ { { o} lya 传染网络 ” 。 URns 互动的功能是, 为给定的 URn 绘制红球的概率( 代表感染状态) 不仅取决于 URn 红球的比例, 也取决于 网络中其他 URns 中红球的比例, 从而计算空间传染的影响。 由此形成的整个网络的传染过程是一个离子时间的有限内存( M$th 顺序 ) Markov 过程, 其过渡概率矩阵矩阵已经确定 。 网络传染 Markov 过程的随机特性是通过分析来绘制红球 红球, 红球 在网络中 红球, 红球 的 比例, 并用我们所研究的直线性 IS 模式, 我们用平均值 的直线性 直径 直线 直线 直线 直线 直线 直径 直径 直径 直径 直径 直径 直径 直径 直径 系统, 我们用 直 直 直 直 直 直 直 直, 直 直 直 直 直 直 直 的 流 流 系统 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流 流