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Mean-field deterministic epidemic models have been successful in uncovering several important dynamic properties of stochastic epidemic spreading processes over complex networks. In particular, individual-based epidemic models isolate the impact of the network topology on spreading dynamics. In this paper, the existing models are generalized to develop a class of models that includes the spreading process in multilayer complex networks. We provide a detailed description of the stochastic process at the agent level where the agents interact through different layers, each represented by a graph. The set of differential equations that describes the time evolution of the state occupancy probabilities has an exponentially growing state-space size in terms of the number of the agents. Based on a mean-field type approximation, we developed a set of nonlinear differential equations that has linearly growing state-space size. We find that the latter system, referred to as the generalized epidemic mean-field (GEMF) model, has a simple structure characterized by the elements of the adjacency matrices of the network layers and the Laplacian matrices of the transition rate graphs. Finally, we present several examples of epidemic models, including spreading of virus and information in computer networks and spreading of multiple pathogens in a host population .