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We introduce a new model for contagion spread using a network of interactingfinite memory two-color P\'{o}lya urns, which we refer to as the finite memoryinteracting P\'{o}lya contagion network. The urns interact in the sense thatthe probability of drawing a red ball (which represents an infection state) fora given urn, not only depends on the ratio of red balls in that urn but also onthe ratio of red balls in the other urns in the network, hence accounting forthe effect of spatial contagion. The resulting network-wide contagion processis a discrete-time finite-memory ($M$th order) Markov process, whose transitionprobability matrix is determined. The stochastic properties of the networkcontagion Markov process are analytically examined, and for homogeneous systemparameters, we characterize the limiting state of infection in each urn. Forthe non-homogeneous case, given the complexity of the stochastic process, andin the same spirit as the well-studied SIS models, we use a mean-field typeapproximation to obtain a discrete-time dynamical system for the finite memoryinteracting P\'{o}lya contagion network. Interestingly, for $M=1$, we obtain alinear dynamical system which exactly represents the corresponding Markovprocess. For $M>1$, we use mean-field approximation to obtain a nonlineardynamical system. Furthermore, noting that the latter dynamical system admits alinear variant (realized by retaining its leading linear terms), we study theasymptotic behavior of the linear systems for both memory modes andcharacterize their equilibrium. Finally, we present simulation studies toassess the quality of the approximation purveyed by the linear and non-lineardynamical systems.

A Finite Memory Interacting P\'{o}lya Contagion Network and its Approximating Dynamical Systems